tag:blogger.com,1999:blog-62785404216866299192018-11-17T20:17:48.939-05:00The Math SpotElementary math materials including ideas, resources and lessons for K-5 teachers. The Math Spot promotes hands-on learning of elementary math concepts. The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.comBlogger62125tag:blogger.com,1999:blog-6278540421686629919.post-16977402086856002162018-11-17T19:26:00.002-05:002018-11-17T20:17:48.863-05:00Multiplying Fractions by a Whole NumberFraction concepts are taken up a level in 5th grade and many of the topics are very abstract and difficult for students to understand. <b>Multiplying fractions by a whole number</b> doesn't need to be one of these difficult concepts! By using the CRA (Concrete-Representative-Abstract) framework your students will link what they already know about <u>multiplication</u> to the <u>multiplication of fractions</u> and will be successful in no time at all!<br /><br /><h3 style="text-align: center;">Concrete Multiplying Fractions by a Whole Number</h3><div>Before you begin with fractions, anchor your students back into what it means to multiply. Give them a pile of blocks and ask them to model something super simple such as "4x3" your 5th graders should quickly and easily be able to model this problem. Ask your students about what they built and how they know it represents the equation. </div><div><br /></div><div><div style="text-align: right;"><a href="https://2.bp.blogspot.com/-rTK6gBtx1dI/W_BJMALRL8I/AAAAAAAALWE/EQ86OXbcvhsVjUdF-L03ywAligT5AhmUwCLcBGAs/s1600/sixtimesrep.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="281" data-original-width="234" height="200" src="https://2.bp.blogspot.com/-rTK6gBtx1dI/W_BJMALRL8I/AAAAAAAALWE/EQ86OXbcvhsVjUdF-L03ywAligT5AhmUwCLcBGAs/s200/sixtimesrep.jpg" width="165" /></a></div>Go ahead and get out your pattern blocks. They are the PERFECT fraction representation. Let the hexagon represent a whole and allow your students to figure out which blocks represent 1/3 (rhombus), 1/2 (trapezoid) and 1/6 (triangles). Once your students are all set with their blocks, they are ready to begin! Don't have fraction blocks handy? Fraction circles would work just as well! </div><div><br /></div><div>Ask your students "If 4 groups of 3 blocks represented 4x3, how could we represent 4 x 1/3?" Allow students to use the rhombus pattern blocks to represent 4 groups of 1/3. The beauty of using blocks is that students can put these blocks back together to see both the improper fraction and mixed number that is created when fractions are multiplied together.<br /><br />**<span style="font-size: 12.8px; text-align: center;">This PICTURE is not a concrete </span><span style="font-size: 12.8px; text-align: center;">model, but if you are creating </span><span style="font-size: 12.8px; text-align: center;">this model out of fraction tiles, </span><span style="font-size: 12.8px; text-align: center;">your students are working at the </span><span style="font-size: 12.8px; text-align: center;">concrete level! **</span></div><div><br /></div><h3 style="text-align: center;">Representative Multiplying Fractions by a Whole Number</h3><div><a href="https://2.bp.blogspot.com/-gaQXVYMUonY/W_BJM6XIvaI/AAAAAAAALWI/Q9Zzgp_4Og4lB_ElRX66mHMXJHK01pP9QCEwYBhgL/s1600/sixtimesrepeated.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="284" data-original-width="224" height="200" src="https://2.bp.blogspot.com/-gaQXVYMUonY/W_BJM6XIvaI/AAAAAAAALWI/Q9Zzgp_4Og4lB_ElRX66mHMXJHK01pP9QCEwYBhgL/s200/sixtimesrepeated.jpg" width="157" /></a>Once your students are able to model 4 x 1/3 you want them to link this understanding to a representative model such as repeated addition. You may begin by asking your students to write a repeated addition equation to represent 4 x 3. This is easy for your students! Now, ask them to use what they know about multiplication to write a repeated addition equation that represents 4 x 1/3. </div><div><br /></div><h3 style="text-align: center;">Abstract Multiplying Fractions by a Whole Number </h3><div><a href="https://1.bp.blogspot.com/-FBv39Lw_vvU/W_BJMM0UK2I/AAAAAAAALWA/WEluanz0n3M6Qry240zlYgbFWUPasJXjgCEwYBhgL/s1600/sixtimesanswer.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="289" data-original-width="240" height="200" src="https://1.bp.blogspot.com/-FBv39Lw_vvU/W_BJMM0UK2I/AAAAAAAALWA/WEluanz0n3M6Qry240zlYgbFWUPasJXjgCEwYBhgL/s200/sixtimesanswer.jpg" width="166" /></a>The concrete and representative steps of this activity allow your students to clearly understand what is going on when multiplying a fraction by a whole number. After your students have had a good deal of exposure at the concrete and representative level, give them a new equation such as 4 x 2/8 and ask your students what they *think* the product will be. You are looking for your students to make generalizations about their multiplication and fraction understandings and to be able to explain their thinking. After a student shares their thinking, ask all students to model with concrete materials or a repeated addition equation to confirm the product!<br /><br />I have created a set of playing cards that includes multiplication equations, visual models, repeated addition and the resulting products. Your students can play so many traditional card games with this deck of cards - and I have included the instructions for 5 games to get you started! This resource is PERFECT for exploring the link between repeated addition and fraction multiplication. Click <a href="https://www.teacherspayteachers.com/Product/Multiplying-Fractions-Game-4203562?ref=feed/sellers_updates_product&aref=3m326jjt" target="_blank">HERE </a>to check it out! </div><hr /><div><b>Related Resources: </b><br /><div class="separator" style="clear: both; text-align: left;"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://3.bp.blogspot.com/-Hjv2iaLjJ7U/W_CxrYEiPnI/AAAAAAAALXU/gYDar_AbGj4bXP5j2AqZ9czpumkNhgnIgCLcBGAs/s200/Cover%2BUpdate%2B2.jpg" width="200" /><a href="https://www.teacherspayteachers.com/Product/Multiplying-Fractions-Game-4203562?ref=feed/sellers_updates_product&aref=3m326jjt" target="_blank">https://www.teacherspayteachers.com/Product/Multiplying-Fractions-Game-4203562?ref=feed/sellers_updates_product&aref=3m326jjt</a></div><br /></div><div><br /></div><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;">Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-58972352809520375322018-11-10T19:49:00.000-05:002018-11-10T19:49:09.640-05:00Subtraction With Regrouping for ALL LearnersIt's about that time of year when second graders begin to delve into subtraction with regrouping. Right off the bat, I want to mention that the standard algorithm is <b><i><u>NOT</u></i></b> a part of the second grade standards! That being said, so many of the programs that we use expect that students use the algorithm and even "exposure" sticks with our students. Luckily there is a way to teach subtraction with regrouping that honors the developmental needs of second grade students.<br /><br />The C-R-A (Concrete, Representative, Abstract) approach is the way to go here! Start students with hands on materials and, at first, don't expect any notation at all in terms of the standard algorithm, and allow students to explore the crossroads of operation and place value.<br /><br /><h3 style="text-align: center;">Concrete Materials </h3><div>When choosing concrete materials for your students to use when adding or subtracting multi-digit numbers consider the difference between groupable and pre-grouped models. Base ten blocks are a common tool used to represent subtraction for early learners. There is nothing wrong with using this tool. Be aware, however that base ten blocks are a <i>pre-grouped model </i>and are not appropriate for all learners. </div><div><br /></div><div>Some of your students may have difficulty understanding why you would "trade a ten for ten ones" when using base ten blocks. Consider starting your instruction using a groupable model such as linking cubes.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vX4v8ranmTA/W-d4m9YTt7I/AAAAAAAALOM/0TMDoalSNSMb3N-d6RDG0LWDedEQwl4_QCLcBGAs/s1600/groupable%2Bmodel.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="915" data-original-width="1600" height="366" src="https://1.bp.blogspot.com/-vX4v8ranmTA/W-d4m9YTt7I/AAAAAAAALOM/0TMDoalSNSMb3N-d6RDG0LWDedEQwl4_QCLcBGAs/s640/groupable%2Bmodel.png" width="640" /></a></div></div>You can see in the example above the strength of a groupable model. When subtracting 47 from 74, sutdents were able to break a ten into ones rather than "trading" one item for another. This is more true to what actually happens when subtracting!<br /><br /><h3 style="text-align: center;">Representative Models</h3><div>Once students understand what is happening when regrouping or unbundling in subtraction, they may be able to transition to subtracting without using a physical concrete material. One representative model that many students find success with is the place value chart. </div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-J4WTB5uFpJs/W-d5g3FD-5I/AAAAAAAALOU/sTBMBinYY1s9rh36H1hOaN32wO-hWKAtACLcBGAs/s1600/RepresentativeSubtraction.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="727" data-original-width="1125" height="412" src="https://1.bp.blogspot.com/-J4WTB5uFpJs/W-d5g3FD-5I/AAAAAAAALOU/sTBMBinYY1s9rh36H1hOaN32wO-hWKAtACLcBGAs/s640/RepresentativeSubtraction.png" width="640" /></a></div><div>An especially effective way of transitioning to a representative model is to ask students to use a concrete material and to "draw what is happening to your blocks". In the 3-digit example above, students would start by building 675 with a concrete material such as base ten blocks. Students would then draw the 6 hundreds, 7 tens and 5 ones that they have built. As they manipulate their blocks, they would also reflect these changes in their drawing as well. </div><div><br /></div><div>As with concrete materials, there are nuances in representations as well. Your students may start by showing hundreds, tens and ones as a place value drawing because they are a more direct representation of place value. As your students become more confident, they may be able to simply draw dots in each column of the place value chart to represent the number in each place value. </div><div><br /></div><h3 style="text-align: center;">Abstract</h3><div>I'll say, once again, that the standard algorithm is NOT expected in 2nd grade! That being said, programs, parents, and older brothers and sisters are all going to provide exposure to the algorithm. If you would like to introduce the algorithm in a more conceptual way, consider asking your students to "record what is happening with numbers". This can be done at either the concrete or representative stage. As students are regrouping or unbundling with blocks, they can show the changes in each place value numerically. Again, as students are using place value drawings to show unbundling, they can record these changes as they would be written in the traditional algorthim. </div><div><br /></div><hr /><div><b>Related Resources: </b><br /><div class="separator" style="clear: both; text-align: left;"><a href="https://www.teacherspayteachers.com/Product/24-Hour-Sale-Subtraction-With-Regrouping-Activities-4187210" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://1.bp.blogspot.com/-Mj6kRGoej3Y/W-d7RGsZp0I/AAAAAAAALOk/6jvWYY1Mti8DogI3sSKvR5rBmjjhfNlhgCLcBGAs/s200/Cover%2BUpdate%2B2.jpg" width="200" /></a></div><b><br /></b></div><div style="text-align: center;"><br /><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-65326202697880810952018-11-02T15:50:00.005-04:002018-11-04T13:01:30.305-05:00Teaching Fractions on a Number Line <div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-XNvuyV-6CHM/W98z8ZyJWLI/AAAAAAAALHE/Xi7DYdii0PsNuA7oWRSRbBgB4KxIyvEswCLcBGAs/s1600/Teaching%2BFractions%2Bon%2Ba%2BNumber%2BLine.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="800" height="320" src="https://4.bp.blogspot.com/-XNvuyV-6CHM/W98z8ZyJWLI/AAAAAAAALHE/Xi7DYdii0PsNuA7oWRSRbBgB4KxIyvEswCLcBGAs/s320/Teaching%2BFractions%2Bon%2Ba%2BNumber%2BLine.png" width="213" /></a></div>For primary students, it is important for students to not only know how to count from 1-10 but to also <i>understand </i>the count sequence. <a href="http://www.k5mathspot.com/2016/03/understanding-count-sequence.html" target="_blank">I have written on this in the past</a> and suggested number path activities for our youngest students.<br /><br />Once our students enter 3rd grade and begin to explore fractions on a number line, I recommend similar activities! For some students, transferring their understanding of whole number number lines to fractions will be quite simple but, for other students, there are common misconceptions that arise.<br /><br /><h4 style="text-align: center;">Misconception #1: Counting lines rather than understanding spaces</h4><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-i4Z3XHKuhhw/W9ymqKu27uI/AAAAAAAALE0/whyDV9PzazsPcRGjWMOCqq34fWjntQ0IQCLcBGAs/s1600/twothirds.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="131" data-original-width="477" height="87" src="https://4.bp.blogspot.com/-i4Z3XHKuhhw/W9ymqKu27uI/AAAAAAAALE0/whyDV9PzazsPcRGjWMOCqq34fWjntQ0IQCLcBGAs/s320/twothirds.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><div>Students who have number line misconceptions will often label this dot 3/4 instead of 2/3. We can support students in overcoming this misconception at the concrete, representative, and abstract level. </div><div><ul><li><b>Hands-On (Concrete)</b>: Give your students a strip of paper the SAME LENGTH as the number line they are working with. As them to fold the paper so that the segments match the number line. You may have to give your students multiple strips of paper so they can experiment and play around- give them the chance to persevere in figuring it out! They can then label the unit fractions on the paper strip and lay it against the number line. </li><li><b>Representative</b>: Ask your students to color each segment of the number line in a different color. How many colors did it take to represent the different sections of the number line? Ask your students to label the unit fractions within the number line. </li><li><b>Abstract</b>: Once your students have had a bit of time to explore the number line through concrete and/or representative means, ask them why this line could NOT represent 3/4. Allow your students to use math language think about and explicitly explain their misconception away. </li></ul><div><h4 style="text-align: center;">Misconception #2: We NEED to see the "lines" to label fractions on a number line </h4></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-Xv40iJVD8GM/W9yog2-h2rI/AAAAAAAALFE/Vj_r48qtL6YWlkuWuR3_ALzS_r4xAWmfwCLcBGAs/s1600/lacking%2Bmc.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="57" data-original-width="477" height="38" src="https://2.bp.blogspot.com/-Xv40iJVD8GM/W9yog2-h2rI/AAAAAAAALFE/Vj_r48qtL6YWlkuWuR3_ALzS_r4xAWmfwCLcBGAs/s320/lacking%2Bmc.png" width="320" /></a></div><div class="separator" style="clear: both; text-align: left;">It may be difficult for students to label this dot without any lines or delineations across the number line. That being said, there are things we know about the placement of this dot and we want our students to be critical thinkers when it comes to math! There are a number of questions you can ask your students to help them realize all that they can reason about fractions. Ask your students questions such as: </div><div class="separator" style="clear: both; text-align: left;"></div><ul><li>What do we know about where this dot is on the number line?</li><ul><li>The dot is just past the half way point. </li><li>The dot is not very close to either end of the number line.</li></ul><li>What fractions could NOT be represented by this dot? </li><ul><li>The dot can't be a fraction that is equal to or less than 1/2 </li><li>It couldn't, for example, be 9/10... the dot is too far away from 1 whole. </li></ul><li>Giving your students multiple choice options also allows them to think critically about fractions on a number line. </li></ul><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-n7mySPSlKm8/W9yofuSlajI/AAAAAAAALFI/ozCSrzMOJ-cDsSKZy0E-z-p3dRpx_8rEQCEwYBhgL/s1600/Multiple%2Bchoice.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="266" data-original-width="477" height="178" src="https://2.bp.blogspot.com/-n7mySPSlKm8/W9yofuSlajI/AAAAAAAALFI/ozCSrzMOJ-cDsSKZy0E-z-p3dRpx_8rEQCEwYBhgL/s320/Multiple%2Bchoice.png" width="320" /></a></div><div><br /></div><br /><div>Try using number lines in your classroom to explore fractions and clear up fraction misconceptions! </div><hr /><div><b>Related Resources: </b><br /><div class="separator" style="clear: both; text-align: left;"><a href="https://www.teacherspayteachers.com/Product/Fractions-on-a-Number-line-Worksheets-4164241" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://3.bp.blogspot.com/-j0MGkGcZs24/W9yqZbY4A0I/AAAAAAAALFQ/Pg3BM7pRA0YoxMhTZLOt9eS7FjTV-NjDQCLcBGAs/s200/Cover%2BUpdate%2B2.jpg" width="200" /></a></div><b><br /></b></div><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-56691536050767582392018-10-27T15:47:00.001-04:002018-10-27T15:47:39.153-04:00Teaching Fact FluencyA few years back I read the book<a href="https://amzn.to/2CK1J7x" target="_blank"> "Math Running Records in Action" by Dr. Nicki Newton</a>. What I appreciated most about this text wasn't necessarily the math running records process (although I'll tell you, it's a great way to look at assessing fluency!) What I really loved about the text was her description of how facts are acquired and how students become fluent within their facts.<br /><br />You have heard me say, time and time again, that CRA (concrete, representative, abstract) is the way to go when it comes to teaching math and fact fluency is no exception! When learning about math facts students first need to understand the operation at the concrete level, they can then link that understanding to representations and, eventually, they will be able to perform at the abstract level solving problems through mental processes.<br /><br />The cool thing about fact acquisition is that you can apply the CRA process over and over to different types of facts.<br /><br /><b>Teaching +1 Facts?</b><br /><br /><ul><li>Why not start on a <a href="https://amzn.to/2qfBbU8" target="_blank">Rekenrek</a>!</li><li>As your students become fluent with the beads, this will become a strategy students can do mentally- and quickly! </li></ul><b>Teaching Doubles? </b><br /><br /><ul><li>Think about using dice and dot patterns. </li><li>Subitizing activities paired with an equation will help move these facts to the "fluent" column in no time. </li></ul><b>What About Make a Ten? </b><br /><br /><ul><li>Start with linking cubes in columns of ten. </li><li>Think about how to show this on a ten frame</li><li>Eventually, your students will become fluent with the strategy and associated facts. </li></ul><span style="font-family: inherit;">No matter which strategies you choose to teach fact fluency, know that moving from concrete to abstract is going to take A LOT of practice! This is why Greg Tang created games like <a href="https://gregtangmath.com/kakooma" target="_blank">Kakooma</a>. He wanted students to practice a high volume of facts so that once they are at the abstract stage these facts can become internalized and students will become fluent. </span><br /><span style="font-family: inherit;"><br />If you are a first or second grade teacher looking for practice for your students, I have created this set of <a href="https://www.teacherspayteachers.com/Product/Addition-Facts-Games-for-Sums-to-20-4157697" target="_blank">20 addition games</a> that I know you will LOVE. All you need to do is print them out and they will be ready to go! Within the resource you will find <span style="background-color: white; color: #222222; font-size: inherit;">5 math games to be played with a paperclip (together with your pencil, it forms a spinner!), </span><span style="background-color: white; color: #222222; font-size: inherit;">5 math games to be played with dice, </span><span style="background-color: white; color: #222222; font-size: inherit;">5 math games to be played with playing cards or numeral card and </span><span style="background-color: white; color: #222222; font-size: inherit;">5 math games to be played with dominoes. </span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #222222; font-size: inherit;"><br /></span></span><h4><span style="font-family: inherit;"><span style="background-color: white; color: #222222; font-size: inherit;">Want to try before you buy? Click <a href="https://www.teacherspayteachers.com/Product/Addition-Fact-Fluency-Games-for-Sums-to-20-4157731" target="_blank">HERE</a> to grab the 5 spinner games for free! I know your students will love them and you will too! </span></span></h4><br /><br /><span style="font-family: inherit;">_</span>________________________________________________________________________________<br /><b>Related Resources: </b><br /><div class="separator" style="clear: both; text-align: left;"><a href="https://www.teacherspayteachers.com/Product/Addition-Facts-Games-for-Sums-to-20-4157697" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="320" src="https://3.bp.blogspot.com/--Qx9cqw-MMk/W9TAkaxTfjI/AAAAAAAAK0Y/5MY_YnW9q6Mr9z1L0Sh9vzPndZR9tDIigCLcBGAs/s320/Cover%2BUpdate%2B2.jpg" width="320" /></a></div><b><br /></b><br /><div style="text-align: center;"><i style="background-color: white; color: #3b3b3b; font-family: "Source Sans Pro", sans-serif; font-size: 16px; letter-spacing: 0.5px;"><span style="font-size: xx-small;"><span style="color: #333333; font-family: "sorts mill goudy", sans-serif; line-height: 33px;">A portion of the links and recommendations above are affiliate links. This means that at </span><span style="box-sizing: border-box; color: #333333; font-family: "sorts mill goudy", sans-serif; line-height: 33px;">no additional cost to you</span><span style="color: #333333; font-family: "sorts mill goudy", sans-serif; line-height: 33px;">, I will earn a commission if you choose to make a purchase. I have personally read each of these books or used these tools and would not recommend them if I did not appreciate the quality of the resource that I received. </span></span></i></div><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-40748805733994408622018-10-21T14:31:00.002-04:002018-10-23T16:46:00.775-04:00Understanding Comparision Word Problems <div class="separator" style="clear: both; text-align: center;"></div><a href="https://4.bp.blogspot.com/-JWbbS9E2ERU/W8-IdOk_DQI/AAAAAAAAKzk/DHLi3QHJC_4mI_DF65fPlIw0f9n4VqLYgCLcBGAs/s1600/Comparison%2BBlog%2B%25281%2529.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="800" height="320" src="https://4.bp.blogspot.com/-JWbbS9E2ERU/W8-IdOk_DQI/AAAAAAAAKzk/DHLi3QHJC_4mI_DF65fPlIw0f9n4VqLYgCLcBGAs/s320/Comparison%2BBlog%2B%25281%2529.png" width="213" /></a><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">Why is it that comparison word problems are particularly difficult for elementary school kiddos? There are a number of challenges present in these specific types of word problems that we don't have to contend with when we are doing more simple problem types such as put together, take apart, add to, or take from word problems! </span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><i><br /></i></span></span><br /><div style="text-align: center;"><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><i>**Hint: Read through to the bottom of this post- there's a HUGE freebie waiting for you!** </i></span></span></div><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span><br /><h3 style="text-align: center;"><span style="background-color: white;"><span style="font-family: inherit;">Challenge #1:</span></span></h3><div style="text-align: center;"><span style="background-color: white;"><span style="font-family: inherit;"><b>There are some specific vocabulary challenges around comparison word problems. A bit of vocabulary work with your students pointing out these challenges will help them go far when solving comparison word problems.</b></span></span></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-oSInbsdH3b8/W8oXbelj_GI/AAAAAAAAKts/TxveQFt4sNgk4VWopkq3QTVWS3ud3gXRACLcBGAs/s1600/Pencils%2Blabeled%2Bwith%2Bmeasurement.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="534" data-original-width="959" height="222" src="https://3.bp.blogspot.com/-oSInbsdH3b8/W8oXbelj_GI/AAAAAAAAKts/TxveQFt4sNgk4VWopkq3QTVWS3ud3gXRACLcBGAs/s400/Pencils%2Blabeled%2Bwith%2Bmeasurement.png" width="400" /></a></div><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">Show a picture of two different pencils, each with a measurement written alongside. Ask your students "How much longer is pencil B than pencil A?" I will bet that a large percentage of your class will tell you the <i>length</i> of pencil B rather than <i>how much longer</i> pencil B is! </span></span><br /><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><br /></span></span><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">Now try to ask this question in a different way. First, ask the students </span></span><br /><span style="background-color: white; color: #222222;"><i><span style="font-family: inherit;">How long is pencil A?</span></i></span><br /><span style="background-color: white; color: #222222;"><i><span style="font-family: inherit;">How long is pencil B?</span></i></span><br /><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">...I'll bet that all of your students get these answers correct! </span></span><br /><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">Now, ask your students which pencil is longer. Once they identify that pencil B is longer, ask </span><i style="color: #222222;">How much longer</i><span style="background-color: white; color: #222222;">? </span></span><br /><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">More of your students will get this question correct than the original question because you have walked them through the difference between long and longer.</span><span style="background-color: white; color: #222222;"> </span></span><br /><span style="font-family: inherit;"><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;"><i>Now, vocabulary challenges aside, this math concept is difficult and I am still sure that there are a group of students who we still need to help.</i></span></span><br /><div style="text-align: center;"><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><br /></span></span></div><div style="text-align: center;"></div><h3 style="text-align: center;"><span style="background-color: white;"><span style="font-family: inherit;">Challenge #2:</span></span></h3><span style="background-color: white; color: #222222;"></span> <br /><div style="text-align: center;"><span style="background-color: white; color: #222222;"><b style="font-family: inherit;">The second main difficulty with this type of problem is that a comparison is an abstract amount that doesn't necessarily exist! The number of centimeters fewer is not something our students can see or touch, it's an abstract concept. </b></span><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-VN4XC60oUB8/W8oYU1Mqp9I/AAAAAAAAKt4/owUo3LL1BbI2z7X06IMInL64UTJjR9fyACLcBGAs/s1600/Pencils%2Bwith%2Bcubes.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="734" data-original-width="1079" height="271" src="https://2.bp.blogspot.com/-VN4XC60oUB8/W8oYU1Mqp9I/AAAAAAAAKt4/owUo3LL1BbI2z7X06IMInL64UTJjR9fyACLcBGAs/s400/Pencils%2Bwith%2Bcubes.png" width="400" /></a></div></div><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">Lay centimeter cubes along the length of pencil A and pencil B. Ask your students again, </span></span><br /><span style="background-color: white; color: #222222;"><i><span style="font-family: inherit;">How long is pencil A?</span></i></span><br /><span style="background-color: white; color: #222222;"><i><span style="font-family: inherit;">How long is pencil B? </span></i></span><br /><span style="background-color: white; color: #222222;"><i><span style="font-family: inherit;">Which pencil is longer? </span></i></span><br /><span style="background-color: white; color: #222222;"><i><span style="font-family: inherit;">How much longer? </span></i></span><br /><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><br /></span></span><span style="background-color: white; color: #222222;"><span style="font-family: inherit;">This time, with the concrete modeling, many more of your students will be able to see how much longer pencil B is then pencil A!</span></span><br /><span style="background-color: white; color: #222222;"><span style="font-family: inherit;"><br /></span></span><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">If you need to make this even more clear, push the two pencils together so that the centimeter cubes are right alongside one another. Your students can see pencil A and pencil B are the same up to 5 cm but pencil B has one, two, three more centimeters than pencil A. Pencil B is 3 centimeters longer!</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">At this point many or most of your students will be very successful with the comparison language, however... You don't want them to be at the concrete stage forever so, what can we do? We can use a part part whole diagram to represent the same thinking. You can see below that when we draw a pencil A as a box and pencil B as a box we can use a dotted line to show the difference between the two. At this point, the diagram makes it very clear that pencil A and the difference put together are the same as the length as pencil B. Your students can write a missing addend equation to solve for the part that shows the difference in the length of the pencils.</span><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;"> </span><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;">This will still be a difficult concept for your students! Don't expect them to learn the skill overnight. Most of your students will need extensive hands-on practice linked to the part-part-whole diagram many, <i>many</i> times before this will be an internalized understanding.</span><br style="background-color: white; color: #222222;" /><br style="background-color: white; color: #222222;" /><span style="background-color: white; color: #222222;"> At this point, your students have done vocabulary work and have expanded their math understanding and they will be much more successful with comparison word problems! </span></span><br /><span style="font-family: inherit;"><span style="background-color: white; color: #222222;"><br /></span></span><br /><h4><a href="https://3.bp.blogspot.com/-qLeYllmOBIw/W8tpanrBPkI/AAAAAAAAKxo/FaW-6zE3lzQfTSMtRv1V0EK_5PE7LHGgQCLcBGAs/s1600/Comparison%2BFreebie%2BCover.jpg" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="720" data-original-width="960" height="240" src="https://3.bp.blogspot.com/-qLeYllmOBIw/W8tpanrBPkI/AAAAAAAAKxo/FaW-6zE3lzQfTSMtRv1V0EK_5PE7LHGgQCLcBGAs/s320/Comparison%2BFreebie%2BCover.jpg" width="320" /></a><span style="font-family: inherit;"><span style="background-color: white; color: #222222;">I have created an entirely FREE set of comparison activities for you! These activities will bring your students through from concrete comparison with cubes to representative modeling and even includes comparison word problems! </span></span></h4><!-- Begin Mailchimp Signup Form --><link href="//cdn-images.mailchimp.com/embedcode/classic-10_7.css" rel="stylesheet" type="text/css"></link><style type="text/css"> #mc_embed_signup{background:#fff; clear:left; font:14px Helvetica,Arial,sans-serif; } /* Add your own Mailchimp form style overrides in your site stylesheet or in this style block. 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color: #222222; font-family: inherit;"><br /></span><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><div style="margin-left: 1em; margin-right: 1em;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><div style="margin-left: 1em; margin-right: 1em;"><br /></div><div style="margin-left: 1em; margin-right: 1em;">Pencil Clip Art by Creative Clips </div><br /><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-53873331462243927932018-10-06T18:19:00.000-04:002018-10-06T18:46:26.367-04:00Adding Fractions with Unlike Denominators<a href="http://2.bp.blogspot.com/-jMYcUFgB5Og/W7k7CqdUphI/AAAAAAAAKRY/tzWB2GXKUDIPDznNMJSJexs94p5-hHQGACK4BGAYYCw/s1600/Unlike%2BDenominators%2BBlog.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-jMYcUFgB5Og/W7k7CqdUphI/AAAAAAAAKRY/tzWB2GXKUDIPDznNMJSJexs94p5-hHQGACK4BGAYYCw/s320/Unlike%2BDenominators%2BBlog.png" width="213" /></a>How do you add fractions with unlike denominators? Find common denominators of course! You know, however, that I'm a big proponent of students understanding why they are using a strategy. If our students don't know why they are using a strategy so many of our learners don't know <i>when </i>to apply that strategy. They become students who have "strong rote skills but struggle with application".<br /><br />An incredibly effective method of helping our students to understand strategies is to give them concrete and representative experiences to make sense of an abstract strategy. I want to share 2 representative strategies for adding fractions with unlike denominators.<br /><br /><h4 style="text-align: center;"><span style="font-weight: normal;">Representative Methods of Adding Fractions </span></h4><div><a href="https://2.bp.blogspot.com/-SJImgsI26zc/W7ku2brvPPI/AAAAAAAAKQY/89ZXYO215SQqrOJH_aQRIoHlgclcdRD2QCLcBGAs/s1600/adding%2Bfractions.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="830" data-original-width="895" height="296" src="https://2.bp.blogspot.com/-SJImgsI26zc/W7ku2brvPPI/AAAAAAAAKQY/89ZXYO215SQqrOJH_aQRIoHlgclcdRD2QCLcBGAs/s320/adding%2Bfractions.png" width="320" /></a>Begin by representing each fraction. Your students can see that the two fractions can not be combined as is because the pieces of each fraction are not the same size. </div><div><br /></div><div>Ask your students what they might be able to do to make it so that the fraction pieces <i>are </i>the same size.<br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-6H6B5reWhH0/W7ku2Xy9sQI/AAAAAAAAKQc/Fa8Q8fxOTTQStu4h5HPDPA8EnBm0hYsOwCLcBGAs/s1600/Adding%2BFractions%2BOverlay.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="848" data-original-width="896" height="302" src="https://1.bp.blogspot.com/-6H6B5reWhH0/W7ku2Xy9sQI/AAAAAAAAKQc/Fa8Q8fxOTTQStu4h5HPDPA8EnBm0hYsOwCLcBGAs/s320/Adding%2BFractions%2BOverlay.png" width="320" /></a></div><br /> Your students may suggest creating two like arrays. In the example to the right, You can easily draw eighths over top of the fourths and fourths over top of the eighths to create two arrays each with 32 pieces.<br /><br />Once the two arrays are equal it is easy to add the two fractions together because you have found a common denominator of 32! </div><h4 style="text-align: center;"></h4><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-iqhYej5CKB8/W7ku2UdHqNI/AAAAAAAAKQg/F2w401Nc9Xcf2c9YsXi9EfBHgaVeYDreACLcBGAs/s1600/Adding%2BFractions%2BLCD.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="841" data-original-width="896" height="300" src="https://2.bp.blogspot.com/-iqhYej5CKB8/W7ku2UdHqNI/AAAAAAAAKQg/F2w401Nc9Xcf2c9YsXi9EfBHgaVeYDreACLcBGAs/s320/Adding%2BFractions%2BLCD.png" width="320" /></a></div><div style="text-align: center;"><div style="text-align: left;">You may be thinking to yourself that 32nds aren't necessarily easy numbers to work with and that the method is less than efficient. That's ok! Once your students understand that they can combine fractions by making two arrays with the same number of pieces they can explore ways to become more efficient by finding the <i>least </i>common denominator! </div><div style="text-align: left;"><br /></div><div style="text-align: left;">Consider the graphic to the left. Post this picture for your students and ask if they could add these two fractions together using those models. Do your students recognize that both models show eighths even though the shape of the eighths is different in each picture? </div><div style="text-align: left;"><br /><br /></div><div style="text-align: left;"><hr /><b><u>Related Resources:</u></b></div><div class="separator" style="clear: both; text-align: center;"><a href="https://www.teacherspayteachers.com/Product/Adding-and-Subtracting-Fractions-with-Unlike-Denominators-Worksheets-4107505?aref=229v6xld" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://4.bp.blogspot.com/-JiIAkthpgbc/W7kzBkNsWCI/AAAAAAAAKRA/EJsYzmpDAx082F13UE2g_Eie2s3Pb3kywCLcBGAs/s200/Unlike%2BCover.jpg" width="200" /></a><a href="https://www.teacherspayteachers.com/Product/Adding-and-Subtracting-Fractions-with-Like-Denominators-Worksheets-4107499?aref=229v6xld" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://3.bp.blogspot.com/--hBpJ4SzHRM/W7kzBu3wBEI/AAAAAAAAKQ4/IyrAIM4LqIgcBeMLNqm_MLDKvRIVYj9EQCLcBGAs/s200/Like%2BCover.jpg" width="200" /></a><a href="https://www.teacherspayteachers.com/Product/Adding-and-Subtracting-Fractions-Worksheets-Bundle-4107532?aref=229v6xld" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://4.bp.blogspot.com/-TF7J1Pve5Ts/W7kzBruQajI/AAAAAAAAKQ8/KFxD1N6kwFIa027uS5FPYupRYhXVrKMLgCLcBGAs/s200/Cover%2BUpdate%2B2.jpg" width="200" /></a></div><br /><div style="text-align: left;"><b><u><br /></u></b></div><br /><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-54819548937946160412018-09-29T14:56:00.003-04:002018-09-29T15:23:58.298-04:00Teaching Math Strategies that Stick! <div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-CoP4MCLEBAw/W6_QpWkLBwI/AAAAAAAAKFA/82ygOhrqMMIzo1Lu1aago2kLBAbYgIlVgCLcBGAs/s1600/Strategies%2BThat%2BStick%2B%25281%2529.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Teaching math strategies for addition is commonplace in the first grade classroom. It's so important to be sure that you go beyond the anchor chart and teach strategies such as counting on in ways that allow your students to understand so they can apply the strategy in word problems and other contexts. Teaching math can be fun and productive when you use strategies that stick! " border="0" data-original-height="1200" data-original-width="800" height="320" src="https://4.bp.blogspot.com/-CoP4MCLEBAw/W6_QpWkLBwI/AAAAAAAAKFA/82ygOhrqMMIzo1Lu1aago2kLBAbYgIlVgCLcBGAs/s320/Strategies%2BThat%2BStick%2B%25281%2529.png" title="Teaching math strategies for addition is commonplace in the first grade classroom. It's so important to be sure that you go beyond the anchor chart and teach strategies such as counting on in ways that allow your students to understand so they can apply the strategy in word problems and other contexts. Teaching math can be fun and productive when you use strategies that stick! " width="213" /></a></div>We know that when we teach procedures in math that many of our students lack the ability to generalize those procedures and apply them when it's appropriate. How often have you heard this description of a struggling student:<br /><h4 style="text-align: center;"><i> "They have great rote procedures but they struggle when it comes to the application". </i></h4><div style="text-align: center;"><div style="text-align: left;">It's so common! There are steps we can take as teachers to help teach our students strategies in ways that allow the strategy to transcend "procedure status" and to stick! </div><h4 style="text-align: left;"></h4><h4 style="text-align: left;">1) Link It Back<br />2) Context Is King<br />3) Think CRA</h4><h3 style="text-align: center;"> Link It Back</h3><div style="font-size: medium; font-weight: 400; text-align: left;">If I am teaching a strategy such as counting on to add, I want immediately link back to what my students already know about addition. If my students know that addition puts parts together, when we talk about counting on we will be talking all about how this is a <i>strategy to put parts together. </i>Down the line when your students see an addition sign, they won't default to the counting on procedure because "the plus sign means I grab the bigger number and count on". They will default to counting on IF it is an appropriate strategy because it is the most efficient strategy to put parts together.<br /><br /></div><h3 style="text-align: center;"> Context Is King</h3><div style="font-size: medium; font-weight: 400; text-align: left;"><a href="https://1.bp.blogspot.com/-gNsICLsgZqs/W6_HIPIiKoI/AAAAAAAAKEc/wqkjqhvA4M4V_4l7UjZaOZt7ErPNJ0p7wCLcBGAs/s1600/Bears%2Bin%2BCave%2BContext.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="color: black;"><img border="0" data-original-height="563" data-original-width="656" height="273" src="https://1.bp.blogspot.com/-gNsICLsgZqs/W6_HIPIiKoI/AAAAAAAAKEc/wqkjqhvA4M4V_4l7UjZaOZt7ErPNJ0p7wCLcBGAs/s320/Bears%2Bin%2BCave%2BContext.png" width="320" /></span></a>So how do I teach my students to count on without outright stating "We can count on to add parts together. You count on by..."? I give my students a context that lends itself to counting on even though they don't yet know that strategy!<br />In this example, you can see that I have told my students there are 7 bears in the cave and 3 outside. I want them to figure out how many bears there are altogether without ever getting to see the bears inside the cave. This scenario lends itself <i>perfectly </i>to the counting on strategy. Once my students think they have solved the problem we can get out manipulatives to show 7 bears in the cave and 3 outside and discuss whether or not our counting on strategy was successful in putting the parts together.<br /><br /></div><h3 style="text-align: center;">Think CRA</h3><div style="font-size: medium; font-weight: 400; text-align: left;"><div><a href="https://4.bp.blogspot.com/-NVbGR26Ri-k/W6_HJhKh6-I/AAAAAAAAKEo/aIGlXmcs5lINyn7XZ1NK8G_i_egAmlynQCLcBGAs/s1600/Money%2BContext.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><span style="color: black;"><img border="0" data-original-height="1310" data-original-width="1090" height="320" src="https://4.bp.blogspot.com/-NVbGR26Ri-k/W6_HJhKh6-I/AAAAAAAAKEo/aIGlXmcs5lINyn7XZ1NK8G_i_egAmlynQCLcBGAs/s320/Money%2BContext.png" width="265" /></span></a>When teaching a new strategy that is more abstract in nature such as counting on, it's always helpful to think CRA. <a href="http://www.k5mathspot.com/2015/12/when-they-still-dont-get-it.html" target="_blank">Concrete, Representative, Abstract.</a> This means that when I begin teaching the strategy to my students I may start with an activity where I show them a wallet and tell them that there are $6 inside. Outside, there are $5 more. Students can try the counting on strategy but can touch, feel, and manipulate the dollars to solidify their thinking.<br />At some point, I would want to move to a more representative model such as the pictures of the bears in the cave shown above. In this way, our students are able to "trust" that there are a given number of bears in the cave. They are beginning to be comfortable manipulating the numbers without having to physically manipulate the scenario.</div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-4ZknhDxR0oE/W6_HI_pZ1NI/AAAAAAAAKEg/jfHVdDYbCWMHKy_wPpi1c5Fkeh2xy0hpACLcBGAs/s1600/Abstract%2Bcounting.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><span style="color: black;"><img border="0" data-original-height="844" data-original-width="1125" height="240" src="https://4.bp.blogspot.com/-4ZknhDxR0oE/W6_HI_pZ1NI/AAAAAAAAKEg/jfHVdDYbCWMHKy_wPpi1c5Fkeh2xy0hpACLcBGAs/s320/Abstract%2Bcounting.png" width="320" /></span></a></div>Ultimately, when teaching counting on or when teaching any variety of other math strategies, you will want your students to be able to perform the strategy given an equation with pencil and paper or mentally. This is the abstract level and is best reinforced once your students have solidified the strategy through concrete and representative means.<br /><br /><i>Following these three steps will help to ensure that the strategies you are teaching are sticking for your students. You are truly teaching strategies rather than a standalone procedure! </i></div><div style="text-align: left;">________________________________________________________________________________</div><div style="text-align: left;"><b>Related Resources: </b></div><div class="separator" style="clear: both; text-align: left;"><a href="https://www.teacherspayteachers.com/Product/Counting-On-Addition-Strategy-4094700?aref=lu0dk6w0" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://2.bp.blogspot.com/-6Ens_hN_XuQ/W6_HEDvLPXI/AAAAAAAAKEY/qIihfs6cAIIylNjt-376d-v5bKp6yQLmwCLcBGAs/s200/Cover%2BUpdate%2B2.jpg" width="200" /></a></div><br /><br /><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-18098258840276313402018-09-18T09:46:00.001-04:002018-09-18T09:46:14.546-04:00How To Teach Rounding to Struggling Learners<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-0RAMH_ehigA/W6EBlr-7hvI/AAAAAAAAJzQ/rKC6nNMRt8o4FMREFNpTlXlE3_MLG7fDwCLcBGAs/s1600/Rounding%2BActivities.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="800" height="320" src="https://2.bp.blogspot.com/-0RAMH_ehigA/W6EBlr-7hvI/AAAAAAAAJzQ/rKC6nNMRt8o4FMREFNpTlXlE3_MLG7fDwCLcBGAs/s320/Rounding%2BActivities.png" width="213" /></a></div>If your grade level has standards around rounding numbers then you have thought long and hard about how to teach rounding to students who are struggling with math. There are two main schools of thought here. 1) Teach the students HOW to round. This may include a rhyme, an anchor chart, or another method that teaches a procedure that, when applied correctly, will result in rounded numbers. 2) Conceptually based methods such as a number line.<br /><br />The thing is, rounding is actually a very simple skill. <i>Students who struggle to round numbers most often don't have a misconception about how to round, they lack an internal number line and the number sense that allows them to be able to round. </i><br /><i><br /></i><br /><div style="text-align: center;">In other words, it's not a rounding issue, it's a number sense issue. </div><div style="text-align: center;"><br /></div><div style="text-align: left;">To teach struggling learners to round you want to spend the majority of your time developing number sense and, when you do, your students will pick up the skill of rounding quickly as a related after thought. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">Two main activities you will want to focus on are: </div><div style="text-align: left;"></div><ol><li>Identifying benchmark "round" numbers when counting by 10, 100, 1,000, tenths or whatever you may be rounding to. </li><li>Knowing how to count between these benchmark numbers. </li></ol><div style="text-align: center;"><b><u>Activities for Identifying Benchmark Numbers</u></b></div><div style="text-align: center;"><br /></div><div style="text-align: left;">Play round robin skip counting games like "Buzz". Students stand in a circle and you will identify a start number (ex: 250) and a buzz number (ex: 430). Starting with the start number students will count round robin from the start number to the buzz number. Whoever says the buzz number sits down and the game continues. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">Play "Find the Hidden Number" by giving students a number line counting by a particular interval (ex: 100, 200, 300, etc.) Ask students to point to the number line to find the spot where numbers are "hiding" (ex: Where is 438 hiding?) </div><div style="text-align: left;"><br /></div><br /><div style="text-align: center;"><b><u>Activities for Counting Between Benchmark Numbers </u></b></div><div><div style="text-align: center;"><br /></div><div>Write the numbers between two benchmarks on post-it notes. Post the benchmark numbers on a classroom white board and draw a number line between the two numbers. Give students the remaining post-it notes and ask them, one at a time, to determine the spot where they think their number would go. (ex: 4,500 to 4,600. Post-it notes include 4,510/4,520/4,530, etc) </div></div><div><br /></div><div>Repeat the same activity as on the post-it notes, however, give students their own white board and ask them to write in the numbers between two benchmarks. They may write all of the numbers at once or you may ask them to plot specific numbers in your chosen order. </div><div><br /></div><hr /><div><b><u>Related Resources: </u></b><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://www.teacherspayteachers.com/Product/5th-Grade-Decimal-Rounding-Activity-4053362?aref=frbkflyc" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://1.bp.blogspot.com/-wb4WtJ22ah0/W6D8ieCdFWI/AAAAAAAAJy4/S4Cf8loUIvUqSJu-JHT_h-aXWVOqA-A9gCLcBGAs/s200/Cover%2BUpdate%2B2.jpg" width="200" /></a><a href="https://www.teacherspayteachers.com/Product/Rounding-to-the-Nearest-10-100-and-1000-Worksheets-4068043?aref=frbkflyc" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://4.bp.blogspot.com/-U0PJxYJjghk/W6D8YHngzJI/AAAAAAAAJyw/COEEorBCWDY_Q0LBJ61RGAGq6GL2eUcHwCLcBGAs/s200/Cover%2BUpdate%2B2.jpg" width="200" /></a><a href="https://www.teacherspayteachers.com/Product/Rounding-To-the-Nearest-10-1493414?aref=frbkflyc" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://1.bp.blogspot.com/-sJ2NWGB1fjc/W6D8cBNyyOI/AAAAAAAAJy0/Wk9scSizQOASSNnP50w2-KRRiv2v0nAswCLcBGAs/s200/Cover%2BUpdate.jpg" width="200" /></a></div><br /><br /></div><div><br /></div><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-83768187944395096262018-09-12T14:28:00.006-04:002018-09-12T14:39:53.399-04:00Differentiate Worksheets Without Breaking a Sweat<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-YTbhtHHkNk8/W5laXoLvGzI/AAAAAAAAJnw/CkYjY3TB4NU1aOJmt4N6rPeo5ET_4TUYQCLcBGAs/s1600/Differentiate%2BWorksheets%2B%25281%2529.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Worksheets are not one size fits all when it comes to how to teach math in an elementary classroom. With little to no preparation, however, you can differentiate worksheets to fit the needs of all of your learners. Just think C-R-A and you'll be prepared for all of your students in no time flat! #differentiation #Worksheets #MathWorksheets #TheMathSpot " border="0" data-original-height="1200" data-original-width="800" height="320" src="https://1.bp.blogspot.com/-YTbhtHHkNk8/W5laXoLvGzI/AAAAAAAAJnw/CkYjY3TB4NU1aOJmt4N6rPeo5ET_4TUYQCLcBGAs/s320/Differentiate%2BWorksheets%2B%25281%2529.png" title="Worksheets are not one size fits all when it comes to how to teach math in an elementary classroom. With little to no preparation, however, you can differentiate worksheets to fit the needs of all of your learners. Just think C-R-A and you'll be prepared for all of your students in no time flat! #differentiation #Worksheets #MathWorksheets #TheMathSpot " width="213" /></a></div>What if I told you that you could differentiate your math worksheets, for any grade level, without getting out your computer, your white out, your pens or, really, doing any work at all? Would you believe me? What if I told you that, without any work at all, you could be differentiating in a <b>more </b>effective way?<br /><br />Read on, my friends.<br /><br />You've heard me go on and on about the benefits of a C-R-A (Concrete, Representative, Abstract) approach in the classroom and it's the perfect approach to differentiating independent practice without breaking a sweat! Take a look at how one worksheet can be used to reinforce place value by meeting the needs of (at least) 3 different types of learners.<br /><div style="text-align: center;"><br /><a href="https://2.bp.blogspot.com/-YqQVjDPjq8Q/W5lYZT-PDAI/AAAAAAAAJnM/wcQpNSpfhE4MstYNxc4wZjeiRfquX9G4QCEwYBhgL/s1600/Concretepv.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="468" data-original-width="633" height="295" src="https://2.bp.blogspot.com/-YqQVjDPjq8Q/W5lYZT-PDAI/AAAAAAAAJnM/wcQpNSpfhE4MstYNxc4wZjeiRfquX9G4QCEwYBhgL/s400/Concretepv.png" width="400" /></a><b><u>The Concrete Learner</u></b></div><div style="text-align: left;">For your learners who need the most support, give your students the worksheet along with a set of base ten blocks. Students can build each problem with blocks as they learn. </div><div style="text-align: left;"><br /></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><br /></div><div style="text-align: center;"><b><u>The Representative Learner</u></b></div><div style="text-align: left;"><div class="separator" style="clear: both; text-align: center;"></div><a href="https://2.bp.blogspot.com/-KXd6zFO6BrQ/W5lcx-KqU3I/AAAAAAAAJoA/Is4L_o-NcIwpZ58jCPxeGbrJSRezN2A1wCLcBGAs/s1600/reptogether.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1124" data-original-width="822" height="400" src="https://2.bp.blogspot.com/-KXd6zFO6BrQ/W5lcx-KqU3I/AAAAAAAAJoA/Is4L_o-NcIwpZ58jCPxeGbrJSRezN2A1wCLcBGAs/s400/reptogether.png" width="292" /></a>Your representative learner thinks that solving place value problems using base ten blocks is a breeze but isn't quite ready to use mental strategies to solve this type of problems. A place value chart may be just the scaffold they need to solve these problems.<br /><br />Earlier on students may be drawing dots on a place value chart but, as they progress, they may be able to simply write the numerals in each place to organize their thinking.<br /><br />A place value drawing would be an equally valid way to show this thinking in a representative way and is a more direct link to the base ten blocks for students who are just ready to dip their toe into a representative model.</div><div class="separator" style="clear: both; text-align: center;"></div><div style="text-align: left;"></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"></div><br /><a href="https://3.bp.blogspot.com/-0Y4iCnIuqeI/W5lYXQCKX2I/AAAAAAAAJnc/EnZD6bL7BiswNC4mzxpc4dKrXGf2a854wCEwYBhgL/s1600/abstractpv.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="633" data-original-width="848" height="297" src="https://3.bp.blogspot.com/-0Y4iCnIuqeI/W5lYXQCKX2I/AAAAAAAAJnc/EnZD6bL7BiswNC4mzxpc4dKrXGf2a854wCEwYBhgL/s400/abstractpv.png" width="400" /></a><b><u>The Abstract Learner </u></b></div><div style="text-align: left;">For your learners who are starting to be able to do this work mentally, you may challenge your students to write an equation that matches each problem. Linking their place value thinking to the equation without the use of manipulatives or representative models will allow them to become more fluent and automatic with both of these skills! </div><div style="text-align: left;"><br /></div><div style="text-align: left;"><div style="text-align: center;"><b><u>Resources shown in this entry: </u></b></div></div><div class="separator" style="clear: both; text-align: center;"><a href="https://www.teacherspayteachers.com/Product/2nd-Grade-Fall-Place-Value-Worksheets-4057289" style="margin-left: 1em; margin-right: 1em;" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="320" src="https://2.bp.blogspot.com/-mEZMFzPRXKA/W5lYVDEuO5I/AAAAAAAAJnE/LoSusLXw88oGt4nSRgf_CsPV55ECT065QCLcBGAs/s320/Cover%2BUpdate%2B2.jpg" width="320" /></a></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div style="text-align: left;"><b><u><br /></u></b></div><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-47132331833036211802018-09-09T15:00:00.000-04:002018-09-09T15:00:12.609-04:00How to Teach Rounding Decimal Numbers<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-LNLemL2d2s4/W5VtmMQWyjI/AAAAAAAAJfM/PYjfq2lKBqkVE_22Y-eArj-JUCEsFqJeQCLcBGAs/s1600/Rounding%2BDecimals%2BConcrete%2BBackground%2B%25281%2529.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Rounding decimal numbers is a skill all 5th grade students need to have. Unfortunately, an anchor chart isn't going to cut it! Your students need an activity that allows them to understand decimal numbers on the number line. When your students have robust number sense around decimal numbers, rounding decimals will be a breeze! Teaching rounding lessons using this method will ensure that every student in your class is a whiz when it comes to rounding! #Rounding #Decimals #5thGradeMath" border="0" data-original-height="1200" data-original-width="800" height="400" src="https://4.bp.blogspot.com/-LNLemL2d2s4/W5VtmMQWyjI/AAAAAAAAJfM/PYjfq2lKBqkVE_22Y-eArj-JUCEsFqJeQCLcBGAs/s400/Rounding%2BDecimals%2BConcrete%2BBackground%2B%25281%2529.png" title="Rounding decimal numbers is a skill all 5th grade students need to have. Unfortunately, an anchor chart isn't going to cut it! Your students need an activity that allows them to understand decimal numbers on the number line. When your students have robust number sense around decimal numbers, rounding decimals will be a breeze! Teaching rounding lessons using this method will ensure that every student in your class is a whiz when it comes to rounding! #Rounding #Decimals #5thGradeMath" width="266" /></a></div>A few years back I wrote the post <a href="http://www.k5mathspot.com/2014/10/5-and-above-give-it-shove-no-really.html" target="_blank">"5 And Above, Give It a Shove... No, Really..." </a> It's been a few years but when I type "rounding" into Pinterest, 3 of the top 5 results still involve a search for an anchor chart.<br /><br />Anchor charts are great, but when your students have an <b>understanding of what rounding means</b> and <b>number sense around the numbers they are rounding</b> an anchor chart with a "Rounding Roller Coaster" is just not necessary. At all.<br /><br />In the original post mentioned above, I describe a method for rounding numbers to the nearest ten. The thing is, that exact same method can be used to round to any place! The difficulty when teaching rounding of decimal numbers to 5th graders is that you need to know where to begin. Some students have no concept of what rounding is while others know what rounding means but don't understand decimal numbers. Some students are missing both of these pieces! Using a number line we can get out in front of both of these major misconceptions to help our students to be successful.<br /><br /><div style="text-align: center;"><b><u>Understanding Rounding</u></b> </div><div style="text-align: center;"><br /></div><div style="text-align: left;"><a href="https://4.bp.blogspot.com/-HgImXuT6zfA/W5Vp-sZEm-I/AAAAAAAAJes/1wHG2Q56X_EV7IN7p0Z0f05d3uIRFRTPgCLcBGAs/s1600/Hundreds%2BRounding%2BHand.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="410" data-original-width="493" height="266" src="https://4.bp.blogspot.com/-HgImXuT6zfA/W5Vp-sZEm-I/AAAAAAAAJes/1wHG2Q56X_EV7IN7p0Z0f05d3uIRFRTPgCLcBGAs/s320/Hundreds%2BRounding%2BHand.png" width="320" /></a>Step 1 to rounding decimals (or any numbers) is for students to understand that rounding means finding the closest "round" number based on the place being rounded to. I like to play a game called "Find the Hidden Number" to reinforce this idea. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">Even when teaching rounding decimals, I start this activity with whole numbers. I want students to understand rounding and using "easier" numbers allows them to free up the mental space they need to think about rounding- not the numbers they are rounding! </div><div style="text-align: left;"><br /></div><div style="text-align: left;">Here you can see I have a number line with numbers between 0 and 1,000 counting by hundreds. To play "Find the Hidden Number" you will give your students a number and they will point to the spot where that number would be "hiding" on the number line. If you asked your students to find 624 they would need to point between 600 and 700. Then, ask your students which of those hundreds 624 would be closer to. If you find your students can't quickly and automatically tell you the closest hundred, chances are they are nowhere near ready to start rounding decimals- their understanding of whole numbers isn't even solid yet! </div><div class="separator" style="clear: both; text-align: center;"></div><br /><div style="text-align: center;"> <b><u>Decimal Number Sense </u></b></div><div style="text-align: center;"><b><u><br /></u></b></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-5rbfzJ6kV-k/W5VqAvNWgWI/AAAAAAAAJew/Vjp1iGZ4FcgREKDjuCqZeRghYtIey90lgCLcBGAs/s1600/Ordering%2BDecimals%2BHand.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" data-original-height="330" data-original-width="371" height="284" src="https://3.bp.blogspot.com/-5rbfzJ6kV-k/W5VqAvNWgWI/AAAAAAAAJew/Vjp1iGZ4FcgREKDjuCqZeRghYtIey90lgCLcBGAs/s320/Ordering%2BDecimals%2BHand.png" width="320" /></a></div><div style="text-align: left;">Once your students understand what it means to round by finding the nearest benchmark number, you will want to do number sense building activities around decimal numbers. One activity I love is ordering decimal numbers. I give students a blank number line and the numbers 0 and 1. One at a time, I give students number tiles for the tenths that come between 0 and 1 and ask students where on the number line these numbers would go. One at a time they place these numbers thinking about the scale and placement of numbers between 0 and 1. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">The first time through the activity I do not correct students who place a tile incorrectly. I simply wait for them to get the tile that would go in that spot and let the students notice for themselves that there is a problem with their placement. When students finish, clear the board and play again giving the tiles in a different order. Look for students who start to notice their previous errors and who are able to put the tiles on the board without any corrections throughout the game. This game can be played over and over with numbers between 0 and 1 or even between 0.05 and 0.06. The range is endless and will allow your students to practice understanding decimal numbers. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">A variation on the game after your students are confident with the number tiles is to take the tiles away giving students only the end benchmark numbers and asking them to fill in the decimals that would come between by writing them in themselves. </div><div style="text-align: left;"><br /></div><div style="text-align: center;"><b><u>Putting It All Together </u></b></div><div style="text-align: center;"><b><u><br /></u></b></div><div style="text-align: left;">Now that your students understand what it means to round and they have number sense around decimal numbers, it is very simple to put the two skills together. Given a number such as 0.81 and asking students to round to the nearest tenth, here's how their thinking would play out. </div><div style="text-align: left;"></div><ol><li>Students would first know know "0.81 is between 8 tenths and 9 tenths" by playing "Find the Hidden Number". </li><li>Thanks to the number line activities, students would easily know that 0.81 is much closer to 8 tenths than 9 tenths. </li></ol><div><a href="https://www.teacherspayteachers.com/Product/Rounding-Decimal-Numbers-4053362" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="320" src="https://3.bp.blogspot.com/-LbNQfFqsscQ/W5VtONNxloI/AAAAAAAAJfE/qh_Menkty-gtj59Qai1Y4QXdo96C0ksIgCLcBGAs/s320/Cover%2BUpdate%2B2.jpg" width="320" /></a>No rhymes, no anchor charts with roller coasters, no tricks at all. An understanding of rounding and a strong internal number line are all your students need to be successful! </div><div><br /></div><div>If you are looking for <a href="https://www.teacherspayteachers.com/Product/Rounding-Decimal-Numbers-4053362" target="_blank">detailed lesson plans with pictures and examples</a> along with all of the number tiles needed to teach this lesson, I have compiled these resources for you! Head to my TPT store to check it out :) </div><br /><br /><br /><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-15228998274281050492018-08-17T09:19:00.000-04:002018-08-17T09:19:08.410-04:00How Early Can I Start Intervention Groups? <a href="https://1.bp.blogspot.com/-ypXpQFo7-54/W3bKxZfBGOI/AAAAAAAAIyI/gTlUAkXMc9AC9iXUSrLwdRdL0_mx2_nCwCLcBGAs/s1600/Math%2BIntervention%2BGroups.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="800" height="320" src="https://1.bp.blogspot.com/-ypXpQFo7-54/W3bKxZfBGOI/AAAAAAAAIyI/gTlUAkXMc9AC9iXUSrLwdRdL0_mx2_nCwCLcBGAs/s320/Math%2BIntervention%2BGroups.png" width="213" /></a>It goes without saying, but I will say it anyways, back to school is a crazy time of year! You are teaching your students routines and expectations setting the tone for the school year. On top of that, you are getting to know your students and beginning the process of building relationships with each of them.<br /><br />I find that I am able to build relationships when I am able to work in small groups with students. The extra attention students receive in a small group let them get to know me and lets me get to know them as a person but also how they function and think as a learner. I prioritize getting small groups up and running as soon as the routines in the classroom allow this to happen effectively!<br /><br />But what do you do in intervention groups in those early days? You likely haven't delivered a great deal of tier 1 instruction yet this year! The early weeks of the school year are the perfect time to get to know what your students have not mastered from the previous year and to get ahead of the game by tackling these deficits early on.<br /><br /><a href="https://www.teacherspayteachers.com/Product/Primary-Math-Intervention-Checklists-2062413?aref=a5cdraem" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://4.bp.blogspot.com/-KdGx7wavoD0/W3bK4KikXZI/AAAAAAAAIyM/lux1k1fjtDoF8Txb5cOIaPLSQbeNjWxtACLcBGAs/s200/Cover%2BUpdate.jpg" width="200" /></a>I have created these <a href="https://www.teacherspayteachers.com/Product/Primary-Math-Intervention-Checklists-2062413?aref=a5cdraem" target="_blank">*free* checklists</a> to be used in kindergarten, first grade and second grade classrooms. I picked 4-6 skills from the previous grade level that students MUST know in order to be successful this school year. Understanding where each of your students performs on each of these skills will help you to hit the ground running!<br /><br /><h4 style="text-align: center;">When Will I Get This Done??? </h4>You may be wondering, when will I possibly have time for another assessment or to use this checklist at all in the beginning of the year??? Here are some tips to help you out!<br /><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"></div><ul><li>The full checklist does not need to be conducted at once. Think about the most efficient way to learn about each item.</li><li>•<span style="font-family: inherit; text-indent: -0.31in;">Consider which items are rote skills that a parent or TA could assess.</span></li><li>•<span style="font-family: inherit; text-indent: -0.31in;">Does your district give a universal screener that addresses some of these items? Use those results to mark off the checklist!</span></li><li>•<span style="font-family: inherit; text-indent: -0.31in;">Is there a quick way to incorporate this assessment into your daily routines? For example, could you give 1 word problem each day as an introduction to your math block?</span></li><li>•<span style="font-family: inherit; text-indent: -0.31in;">Are there any items you want to be sure to conduct and observe yourself? Prioritize the items you want to see for yourself in a face to face interview. </span></li></ul><br /> <br /><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-12143599296586207712018-08-15T13:17:00.000-04:002018-09-02T20:05:28.919-04:00Teaching Place Value the CRA Way!<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-vPTIq8_gDNA/W4x6IcfIaBI/AAAAAAAAJRc/pF962-Gr7FojCkkmnC8JeHaISyOmXI17QCLcBGAs/s1600/Copy%2Bof%2BCopy%2Bof%2BPlace%2BValue%2BCRA.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1200" data-original-width="800" height="320" src="https://1.bp.blogspot.com/-vPTIq8_gDNA/W4x6IcfIaBI/AAAAAAAAJRc/pF962-Gr7FojCkkmnC8JeHaISyOmXI17QCLcBGAs/s320/Copy%2Bof%2BCopy%2Bof%2BPlace%2BValue%2BCRA.png" width="213" /></a></div>Place value comes up every year throughout elementary school. But what makes it so difficult for our students to understand and what can we do to make place value instruction more clear for our most struggling students.<br /><br />Quick and simple place value games can help your students to build understanding and gain fluency with place value concepts. But before I get to the games...<br /><br /><h3 style="text-align: center;"><b><u>Where do we start and where are we going? </u></b></h3><ul><li>In Kindergarten, students are building a foundation by noticing that when they have ten ones and some more ones they create a teen number. </li><li>In first grade, place value explodes with students understanding that ten ones can be put together to create a unit called a "ten" and that two digit numbers are really telling the number of "tens" and "ones". </li><li>In second grade this understanding is extended as students work their way up to 1,000 noticing that this pattern of grouping ten of a unit together yields a new type of unit. </li><li>In third grade students round and continue adding and subtracting multi-digit numbers... not a big place value year in terms of developing new ideas around the base ten system :) </li><li>In fourth grade students resume the extension of units all the way up to 1,000,000 and begin to notice that instead of just saying "ten of a unit yields a new unit" that they can describe the relationship of units next door do one another by saying 10 times more or 10 times less. </li><li>In fifth grade students tie together each of these previous understandings and apply them not only to whole numbers but to decimal numbers as well! </li></ul><div>It is clear that mastering the standard at each grade level is critical to moving on to the next grade level because each of these understandings is so closely related and builds so specifically on previous understandings. </div><h3 style="text-align: center;"><b><u>What does this mean for my instruction?</u></b> </h3><div>It means that your students can not leave your grade level with an emerging understanding of place value concepts! I would go as far as to say that in the world of "<a href="http://www.k5mathspot.com/2015/12/when-they-still-dont-get-it.html" target="_blank">Concrete-Representative-Abstract</a>" that students need to have moved as far down this continuum as possible before moving to the next grade level. </div><div><br /></div><div>That being said, I am a staunch advocate of letting students stay in the concrete and representative stages as long as is needed because it will help your students to be more solid in their abstract understandings and your students will be more efficient at mental math involving place value in general!</div><div><br /></div><h3 style="text-align: center;"><b><u>What will this look like on a daily basis?</u></b></h3><div><ol><li>Use SO MANY manipulatives when your students are learning about place value. </li><li>Use a variety of manipulatives- don't just get stuck using base ten blocks daily! When your students use a variety of manipulatives they are deepening their understanding and building connections between these models. </li><li>Be thoughtful about using pre-grouped, non-grouped, proportional and non-proportional models when choosing manipulatives and representations. </li><li>Link manipulatives, representative models and abstract equations together. For example, <i>if you can show your understanding using place value disks, that's great! Can you write an equation that matches what you just did with your disks? </i></li><li>Repetition, repetition, repetition. My third grade teacher used to say "Practice doesn't make perfect, practices makes permanent!" That means that when our students are practicing these skills it is important that they are practicing correctly and that they are practicing in great volume. What they do well and often will become permanent! </li></ol><div><a href="https://1.bp.blogspot.com/-epV95FNFRWE/W3MsCogGueI/AAAAAAAAItk/ZP9FKgAwd3Y8EihhFX0_F5OAWXwBXOwSQCLcBGAs/s1600/1%2Bor%2Bbust.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1138" data-original-width="1359" height="267" src="https://1.bp.blogspot.com/-epV95FNFRWE/W3MsCogGueI/AAAAAAAAItk/ZP9FKgAwd3Y8EihhFX0_F5OAWXwBXOwSQCLcBGAs/s320/1%2Bor%2Bbust.png" data-pin-nopin="nopin" width="320" /></a></div></div><h3 style="text-align: center;"><u>So, What About The Games? </u></h3><div>Play games that promote repetition with linking of hands on tools to abstract ideas. The game "One or Bust" is an example of this type of game. Students choose a card and add that block to their game board. Students link their understanding between the concrete and the abstract as they play by stating aloud the change in their numbers as they progress through the game. In this 5th grade version of the game students would say something like "I have sixty-three hundredths. One more hundredth is sixty-four hundredths." </div><div><br /></div><div>If you are interested in games like "One or Bust", I have created a version of the game for numbers to 100, 1000 and 1 (decimals). Click the pictures to check them out! </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://www.teacherspayteachers.com/Product/5th-Grade-Place-Value-Decimal-Game-3997461?utm_source=k5mathspot.com&utm_campaign=Place%20Value%20Games%20Blog%20Post" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://2.bp.blogspot.com/-q4j79f5YUw0/W3MtHGsuD0I/AAAAAAAAIts/oHtyhdXIsqcVy4s1NEXq5D8x88s77C8IwCLcBGAs/s200/Cover%2BUpdate%2B%255BAutosaved%255D.jpg" data-pin-nopin="nopin"width="200" /></a><a href="https://www.teacherspayteachers.com/Product/2nd-Grade-Place-Value-Game-to-1000-3994935?aref=7ubdijyo&utm_source=k5mathspot.com&utm_campaign=Place%20Value%20Games%20Blog%20Post" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://1.bp.blogspot.com/-twXJ-gVPYK4/W3MtJL0_gLI/AAAAAAAAItw/RWznGZShWsQY4KfvKeSSgj6BFOAqFPTVwCLcBGAs/s200/Cover%2BUpdate.jpg" data-pin-nopin="nopin"width="200" /></a><a href="https://www.teacherspayteachers.com/Product/First-Grade-Place-Value-Game-2546241?utm_source=k5mathspot.com&utm_campaign=Place%20Value%20Games%20Blog%20Post%20" target="_blank"><img border="0" data-original-height="960" data-original-width="960" height="200" src="https://1.bp.blogspot.com/-Vi4b-jm0BDw/W3MtLCFSBuI/AAAAAAAAIt0/GVbz98W6wFsGqe-1MxXtHAKQMIkQPakHACLcBGAs/s200/Cover%2BUpdate%2BSquare%2B.jpg" data-pin-nopin="nopin" width="200" /></a></div><br /><br /><div><br /></div><div><br /></div><div><br /></div><br /><div style="text-align: center;"><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-83150670525041053082018-04-30T21:35:00.000-04:002018-04-30T21:35:09.803-04:00Summer Money for Teachers <a href="https://1.bp.blogspot.com/-rX8S-a2uU_4/WufCygdn7hI/AAAAAAAAHiM/uA32F65fAYAP2eopEeS5rr8_PBceSEUfACLcBGAs/s1600/VIP%2BKid.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" data-original-height="1102" data-original-width="735" height="320" src="https://1.bp.blogspot.com/-rX8S-a2uU_4/WufCygdn7hI/AAAAAAAAHiM/uA32F65fAYAP2eopEeS5rr8_PBceSEUfACLcBGAs/s320/VIP%2BKid.png" width="213" /></a>Summer is right around the corner and it's that time of year when some of us start to think about how we are going to earn some "alternative to summer school" side money during the summer months. A few months back I was thinking that I needed something to supplement my income and came across VIPKID.<br /><br />At first, I wasn't sure if this company was for real, however, I was sold when I learned that VIPKID is Forbes' #1 online company to work for (<a href="https://www.forbes.com/sites/laurashin/2018/01/17/work-from-home-2018-the-top-100-companies-for-remote-jobs/#5b4ad0dc76f0" target="_blank">really!</a>). I have been so pleased that I took the plunge ever since!<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-KWIQFuBnDX0/WufAF2VuZKI/AAAAAAAAHho/QDyYKCLe8BAsg_nSRv1sqiDnEtU12X2qQCLcBGAs/s1600/How%2BDoes%2BVIPKid%2BWork_.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="100" data-original-width="500" src="https://1.bp.blogspot.com/-KWIQFuBnDX0/WufAF2VuZKI/AAAAAAAAHho/QDyYKCLe8BAsg_nSRv1sqiDnEtU12X2qQCLcBGAs/s1600/How%2BDoes%2BVIPKid%2BWork_.png" /></a></div><div class="separator" style="clear: both; text-align: center;"></div><br /><br /><ul><li>Students in China sign up for English lessons through the VIPKid platform- it is highly desirable to have students learn from native English speakers as it produces a more American accent when speaking. </li></ul><ul><li>Teachers sign up to teach these students. You will set your schedule entirely. There's basically a grid of times 7 days a week and you open 30 minute time slots when you are interested in teaching. <b>You will teach only 1 student at a time. </b>If a student books a slot, you have committed to teach them. If no student has booked a slot, you have the flexibility to open and close that slot or adjust that timing as much as you would like. **It does take some time to be hired and to get your bookings going. If you are looking to do this in the summer, you will want to sign up now so you can get the ball rolling.** </li></ul><ul><li>VIPKIDs provides the interface, coordinates all of these sign-ups, provides teachers with as much or as little training as you are comfortable with AND PROVIDES ALL OF THE LESSONS. Yes. You do not have to lesson plan at all! </li></ul><ul><li>The interface basically looks like a PowerPoint on the left side of the screen, Skype in the middle with video feeds for teacher and student, and old school AIM to the far right where you can send text messages if needed. </li></ul><ul><li>The Fireman handle any and all IT issues. If I sign into a class and can't hear the student I just hit a button to "contact fireman" and the fireman will quickly show up and troubleshoot the issue. </li></ul><div>That's pretty much it. Student's sign up, teachers teach them with the provided materials, VIPKID takes care of all of the details and the fireman will be sure that IT issues are not even on your radar. </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-myxsFTNxepc/WufAfSdADXI/AAAAAAAAHh4/PgbL9iLkBc8ZCit3I0OUI1zua3nPiT-3ACLcBGAs/s1600/How%2BMuch%2Bwill%2BI%2Breally%2Bmake.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="100" data-original-width="500" src="https://1.bp.blogspot.com/-myxsFTNxepc/WufAfSdADXI/AAAAAAAAHh4/PgbL9iLkBc8ZCit3I0OUI1zua3nPiT-3ACLcBGAs/s1600/How%2BMuch%2Bwill%2BI%2Breally%2Bmake.png" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div>The company will tell you that you will make $14-22 per hour. That range seems large but here's how it works: </div><div><ul><li>When you have your interview you will be offered a base pay of $7-9 per class. Classes are each only 30 minutes so that puts us at $14-18 right off the bat. </li></ul><ul><li>If you show up to your class on time (which seems like a given... right??) you will get an extra $1 per class. So, if you are able to turn your computer on on time, you will be earning $15-19 per hour. </li></ul><ul><li>If you teach 30 classes in a month, you earn an extra $0.50 per class. (So now we're up to $16-$20 per hour) </li></ul><ul><li>If you teach 45 classes in a month, you earn an extra $1.00 per class. So, if your base pay is $9, you show up on time and you teach 45 classes in a month, you will earn (9x2)+(1x2)+(1x2)= $22 per hour. </li></ul>I personally have a base pay of $8 per hour, always show up on time, and generally teach about 30-40 classes per month. This leaves me making $19/hour. I accomplish 30 classes per month by teaching 1 class before school on Monday, 1 class before school on Friday and 2-4 classes per day each on Saturday and Sunday morning. This is about $150-190 per week for a very minimal commitment. My personal plan is to open more slots during the summer but, for the school year, this is enough to keep me busy and satisfied on the side! </div><div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-1yfgX-gKkS4/WufAhZseOQI/AAAAAAAAHh8/Gfbfh2WV2PwosUQYXV26DGz1KtdubeFIwCEwYBhgL/s1600/Is%2BThere%2Ban%2BInvestment_.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="100" data-original-width="500" src="https://3.bp.blogspot.com/-1yfgX-gKkS4/WufAhZseOQI/AAAAAAAAHh8/Gfbfh2WV2PwosUQYXV26DGz1KtdubeFIwCEwYBhgL/s1600/Is%2BThere%2Ban%2BInvestment_.png" /></a></div><div><br /></div></div><div>Yes, but it is incredibly minimal. In order to get started you need a computer, a headphone with attached microphone, a classroom background and minimal prompts. I, personally, bought a set of headphones that cost about $10 and then went to the dollar store for everything else. </div><div>At the dollar store I bought: </div><div><ul><li>An alphabet border for $1 that I taped together so it looks like a poster and hung that in the corner of the room where I would be sitting. </li><li>A pack of bows that I could hang up as rewards.</li><li>2 stuffed animals that I could use as puppets (that I really don't use anymore). </li><li>A pack of alphabet flashcards with cute animals on them. </li><li>A pack of foam letters</li><li>A whiteboard with an attached marker. </li></ul><div>$10 for headphones. $7 at the dollar store. I was all set to go. This is a very low risk online business to try out! </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-CC7j8GbM-yQ/WufAjHNlvWI/AAAAAAAAHiA/6XqRUgqEdVgUFVzHXqnx-7wEsqUy7j4lACEwYBhgL/s1600/Is%2Bthat%2BAll%2Bthere%2Bis_.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="100" data-original-width="500" src="https://2.bp.blogspot.com/-CC7j8GbM-yQ/WufAjHNlvWI/AAAAAAAAHiA/6XqRUgqEdVgUFVzHXqnx-7wEsqUy7j4lACEwYBhgL/s1600/Is%2Bthat%2BAll%2Bthere%2Bis_.png" /></a></div><div><br /></div><div><br /></div><div>Pretty much! Could you spend way more? Oh, certainly, just check out Pinterest pictures of VIPKID classrooms. Some people go crazy in their space! In fact, if you jump on any VIPKID forum you will find that there is a whole range of people in terms of how much time, enthusiasm and investment that they want to put into this business. I am definitely on the minimalist side of the scale. </div><div><ul><li>I invest exactly 0 minutes in Chinese social media. </li><li>I do not wear an orange shirt every (or any?) time that I teach. </li><li>I do not decorate my "classroom" for holidays and events. </li><li>I do not wear a crazy headband that I can add bows and stars to as students earn rewards. </li></ul><div>Why? That's just not me or my personality! I have found that by being myself and teaching in a way that I feel comfortable I am able to fill all 8-10 slots that I would like to teach per week. I have regular students who come back to me lesson after lesson. Because I don't put on a show I attract the older students who I <i>really </i>enjoy teaching, talking to, and connecting with! I am NOT putting down teachers who put more into it than I do! I encourage everyone to find their style, find what they are comfortable with and enjoying and pursue that avenue fully! The same way that I have attracted my "ideal student" you will attract the students that you fit with best by being yourself as well! </div></div></div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-Q7kAhTvuIr0/WufD885MG9I/AAAAAAAAHiU/2EfpeJ4CZ1EK22s-5ZXbWhdAVUtL2Yn3wCLcBGAs/s1600/Want%2Bto%2Bgive%2Bit%2Ba%2Btry_.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="100" data-original-width="500" src="https://1.bp.blogspot.com/-Q7kAhTvuIr0/WufD885MG9I/AAAAAAAAHiU/2EfpeJ4CZ1EK22s-5ZXbWhdAVUtL2Yn3wCLcBGAs/s1600/Want%2Bto%2Bgive%2Bit%2Ba%2Btry_.png" /></a></div><div><br /></div><div>If you are interested in learning more, click <a href="https://t.vipkid.com.cn/?refereeId=9885981&refersourceid=a01" target="_blank">HERE</a> to use my link to the VIPKID website. If you like what you see and are interested in applying, please reach out to me and I will support you in your application and interview process. By using my link I become your referral and am invested in seeing you be successful- so take advantage of that fact and reach out!! I have sincerely loved teaching for VIPKID and I would love to see others have the same opportunity that I have had! </div><br /><div style="text-align: center;"><br /><img class="nopin" height="50" src="https://1.bp.blogspot.com/-iF93yAWRsQs/WUuigPmhuCI/AAAAAAAAL_g/pyHwSsJFEZ8Pvq1facQ-YnT8EoJmfF0YgCLcBGAs/s1600/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-54638439770603504622017-04-10T18:22:00.000-04:002017-06-22T07:06:51.539-04:00Why Teaching Line Plots is Harder Than it Looks<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-kcWPKxYnJv4/WOwFcFSOxcI/AAAAAAAAElA/RI9Yxe2ouGIxwbVwohn6UqwqU3U-Fyf8gCLcB/s1600/Line%2BPlots%2BInstagram.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-kcWPKxYnJv4/WOwFcFSOxcI/AAAAAAAAElA/RI9Yxe2ouGIxwbVwohn6UqwqU3U-Fyf8gCLcB/s320/Line%2BPlots%2BInstagram.png" width="320" /></a></div><br />Much of the "data and graphing" instruction comes quite easily to students. A bar graph or pictograph is really quite intuitive to read- the difficulty in instruction comes in when students are asked to solve comparison word problems based on the data in the charts. And we KNOW that comparison word problems are notoriously difficult for students to solve as word problem types go.<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><br />Why then are line plots... which are really just a combination of a bar graph and a pictograph... so difficult for students to read and answer questions about? If we really drill down, creating a line plot and answering questions about a line plot in 4th and 5th grade requires so many components. Understanding which of these components are strengths and areas of need for your students can help you to narrow in on the reason they might be having difficulty. To build and interpret a line plot, students need:<br /><br /><ol><li>An understanding of the conventions of putting the chart together.</li><li>A thorough understanding of what each part of the line plot represents for interpretation.</li><li>Last of all, but certainly not least of all, an extensive understanding of fractions on a number line.</li></ol><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-TXZBipBnaxY/WOv7HkDGfTI/AAAAAAAAEkE/wY5HP6ko3_wMvmRTFQABicL07x2Ch10zACLcB/s1600/Fractions%2Bon%2Ba%2BNumber%2BLine.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://3.bp.blogspot.com/-TXZBipBnaxY/WOv7HkDGfTI/AAAAAAAAEkE/wY5HP6ko3_wMvmRTFQABicL07x2Ch10zACLcB/s640/Fractions%2Bon%2Ba%2BNumber%2BLine.png" width="640" /></a></div><br />In creating a line plot students first need to create a number line. When looking at the data set, helpful questions to ask students would include:<br /><br /><br /><div><ul><li>What is the smallest fraction in the data set? How can this help us decide where to start the number line?</li><li>What is the largest fraction in the data set? How can this data point help us to decide where to end the number line? </li><li>Where would 1/2, 1/4 and 1/8 be place on the number line. Where do the 1/4 and 1/2 marks overlap? Where do the 1/8 marks overlap with the other fractions. If 2/4 and 1/2 are at the same point, what does that tell us about these fractions? </li></ul><div><b>So what can you do to help? Give students a hands on opportunity to build a line plot. Give them a set of pencils or clip art pictures all measured to different lengths. Have students place these items out on a life-sized line plot so that they can see where it would make sense to start a number line, end a number line and how the number line could be best labeled. </b></div></div><div><b><br /></b></div><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-RL6tZjWxolM/WOv7I6t1HfI/AAAAAAAAEkI/sFzw5kk8jTAYkkPQdKiqq0EXb9ZiM-_7wCLcB/s1600/Line%2BPlot%2BConventions.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://4.bp.blogspot.com/-RL6tZjWxolM/WOv7I6t1HfI/AAAAAAAAEkI/sFzw5kk8jTAYkkPQdKiqq0EXb9ZiM-_7wCLcB/s640/Line%2BPlot%2BConventions.png" width="640" /></a></div> The conventions of creating a line plot aren't all that different from creating a bar graph, pictograph, or any other type of graphing representation. Students need to be sure to include a title, a number line, a label for the number line, they need to mark out a scale on the number line, and finally to represent their data points.<br /><br /><b>So what can you do to help? It may be helpful to students to link the conventions of a line plot to the conventions of other types of graphs that they already know about. We know that research tells us that whenever we can make links and connections information will stick more easily. Could students look at a bar graph of similar information and find each of these components on both the bar graph and the line plot? </b><br /><br /><ul><li>Where on the graphs do we learn what the graph is all about? </li><li>Where on the graphs do we find out, for example, the height of the smallest plant?</li><li>Where on the graphs do we find out if they are measuring, for example, in inches or centimeters? </li><li>Where on the graphs can we find out how many plants, for example, are 3 1/2 inches tall? </li></ul><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-8imxiaYI03I/WOv9Rd161bI/AAAAAAAAEkY/ZYoZtwoQfQQtTFo9U03wjUb6vcnZsQzgwCLcB/s1600/Line%2BPlot%2BInterpretation.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://3.bp.blogspot.com/-8imxiaYI03I/WOv9Rd161bI/AAAAAAAAEkY/ZYoZtwoQfQQtTFo9U03wjUb6vcnZsQzgwCLcB/s640/Line%2BPlot%2BInterpretation.png" width="640" /></a></div><div><br /></div><br />A number of difficulties are presented when students are asked to interpret the information on a line plot.<br /><br /><ul><li>If students are creating a number line about the height of a variety of plants- do they really recognize that each "X" on the line plot stands for it's own plant? If they don't, they are going to have a very difficult time in answering a question that asks, for example, "What is the total height of all of the plants measured?" <i>You will know if students aren't understanding the meaning of the "X" if students add up all of the fractions listed on the scale rather than adding up the total of all data points. </i></li><li>Students may need to add fractions with different denominators. If a set of data has pieces measured to the nearest 1/4 inch there is a good chance some pieces of data will be listed as 1/2. The same situation may occur if data points are measured to the nearest 1/8th. Students may have more difficulty performing operations on fractions with different denominators. </li><li>Students may have difficulty understanding that each piece of data has multiple labels. For example, consider the story <i>Mike road his bike every day for a week. On each day he wrote down the number of miles he traveled. In his journal his list said: 2 1/4, 3, 5 1/2, 3 1/4, 3 1/4, 5, 2. </i>In this example story, each of these pieces of data really has 2 labels. The first piece of data, 2 1/4, really represents that on <b>day 1, </b>Mike traveled 2 1/4 <b>miles. </b>Each of the pieces of data is a separate day and each of those days has been measured in miles. </li></ul><div><b>So what can you do to help? I stand by my suggestions in the first two paragraphs. If students are not making these connections when they are looking at a line plot on paper you can either take the line plot off of the paper and bring it into real, hands on life so students can understand the significance and meaning of each "x" on the line plot. You can also relate the line plot to other graphs and representations you use in the classroom to help students to build connections! </b></div><div><br /></div><div><a href="https://www.teacherspayteachers.com/Product/Line-Plots-Reading-and-Creating-1002321" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;" target="_blank"><img border="0" height="150" src="https://2.bp.blogspot.com/-q9NKYeqte-k/WOwBX8XS67I/AAAAAAAAEkk/M6VSJDYVA9w1VG678o9qSMXMv5CGrMWQQCLcB/s200/Line%2BPlot%2BWorksheet%2BCover.jpg" width="200" /></a><a href="https://www.teacherspayteachers.com/Product/Reading-and-Creating-Line-Plots-with-Fractions-1303404" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img border="0" height="150" src="https://3.bp.blogspot.com/-DQXTmPkOYBY/WOwBdmLbcQI/AAAAAAAAEks/La5KGc7TBM8cCh6pj5cV0I5la9QaI0w2wCLcB/s200/Slide1.JPG" width="200" /></a>If you are looking for further line plot resources, I have two resources that might be helpful. First is a <a href="https://www.teacherspayteachers.com/Product/Reading-and-Creating-Line-Plots-with-Fractions-1303404" target="_blank">hands-on line plots exploration. </a>Students work to measure the plants of a fictional kindergarten classroom and help them to best organize their information. Next is a set of <a href="https://www.teacherspayteachers.com/Product/Line-Plots-Reading-and-Creating-1002321" target="_blank">differentiated worksheets </a>for students to use in the classroom and for homework. An "easy" "medium" and "hard level" have been created for classwork and homework in both the areas of reading and creating line plots. Plus, I have included task cards perfect for math centers or small group instruction. </div><div><br /></div><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="92" data-original-width="245" src="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" /></a></div><br /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-6352606178303063692017-03-15T13:12:00.003-04:002018-09-18T14:08:02.471-04:00Have a Gritty Day! There has been quite a bit of research about the importance of grit in allowing students to learn and be successful both in school and beyond. I love research, it fuels the choices that I make in the classroom, but I'll be honest, if I can't figure out how a piece of research <i>looks </i>in my teaching, I have a hard time.<br /><br /><a href="https://4.bp.blogspot.com/-y_s4IgiBbyI/WMldbG_LtwI/AAAAAAAAEgQ/BIC5W8OK7xE1ZZxMGdExTT1gXWV5AjZlACLcB/s1600/It%2527s%2Btime%2Bto%2Bget.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-y_s4IgiBbyI/WMldbG_LtwI/AAAAAAAAEgQ/BIC5W8OK7xE1ZZxMGdExTT1gXWV5AjZlACLcB/s320/It%2527s%2Btime%2Bto%2Bget.png" width="213" /></a>The grit research, to me, was difficult because while I could understand the connection between grit and success I didn't think it was enough to be sure I was careful with the language that I was using so that I was using language that promoted a growth mindset and diminishing language that would reinforce a fixed mindset.<br /><br />I had a principal once that told us that we needed to think of our profession as being equal to the medical profession. That we were diagnosticians and practitioners and that we needed to keep up with current research. If we weren't implementing best practice that would be considered equal to a doctor committing malpractice. It sounds harsh but you really can't argue with the logic!<br /><br />And so I kept thinking about grit and how I could be more explicit- beyond being purposeful in my choice of language. I needed my students to know what they were working towards! <b>I have figured out how to make this connection more explicit and wanted to be sure to share this strategy with all of you.</b><br /><br />I have taught my students that, sometimes, at the end of a math lesson they will feel very confident. <i>This means that they have met the learning target, they might have gotten many questions correct on their exit ticket, they might even be thinking something like "this is easy!". </i>If they are feeling that way they can say that they are having a confident day in math. Other days we might be working on something that feels really tricky! <i>They may feel confused or they might feel like they are thinking really hard but it's still not making a lot of sense. They might start to feel frustrated because even though they are trying, they keep getting answers wrong. They might even see other students having a confident day and they might be wondering why it's not easy for them. </i>I tell my students that if they are feeling like they are having a day that is very tricky they have two choices. They can either get mad/sad/frustrated or they can <b>get gritty. </b>Gritty means that you hunker down and say "I can figure this out!" "I can work at this!" "I will be able to do it!!".<br /><br /><a href="https://2.bp.blogspot.com/-wNu7caLFhM0/WMlicrOJpvI/AAAAAAAAEgg/hnkrKLP-c-cSQUlNJBRMqNJg4JwKeacWgCLcB/s1600/Grit%2BMindset.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-wNu7caLFhM0/WMlicrOJpvI/AAAAAAAAEgg/hnkrKLP-c-cSQUlNJBRMqNJg4JwKeacWgCLcB/s320/Grit%2BMindset.png" width="213" /></a>When we are "getting gritty" we even go so far as to make a little fist and make a "tough" face and say "I can get this!!" I practice this with my kids outside of the context of math at the beginning of a lesson. I remind them by saying <i>"We're going to work on some pretty important stuff today! You might get it right away and have a confident day, but if it's feeling tough remember, you can always get gritty and we'll figure it out together. Show me what you will say when you're feeling gritty" </i>and they respond by all saying, emphatically, "I can get this!"<br /><br />During a lesson when a student is struggling I can then easily put their mind at ease by acknowledging that the work is difficult and that I know they can get gritty and figure it out.<br /><br />You see, my students know that those two choices they have when work gets hard lead to very different results. If they get mad/sad/frustrated they will still be mad/sad/frustrated at the end of the lesson. If they get gritty, they have a chance of moving over and becoming confident.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-pl-wDVSFkXk/WMlvpbqa8YI/AAAAAAAAEg0/PD_aqhZt3_k7g7ZJiDTw_FrIYqGoyypLgCLcB/s1600/Grit%2BMindset%2B2.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://2.bp.blogspot.com/-pl-wDVSFkXk/WMlvpbqa8YI/AAAAAAAAEg0/PD_aqhZt3_k7g7ZJiDTw_FrIYqGoyypLgCLcB/s320/Grit%2BMindset%2B2.png" width="320" /></a></div>Not every math day ends in success for all students. And that's okay. And I want my students to know this is okay too! At the end of a lesson we will often reflect and ask "Who had a confident day? Who had to get gritty today?" and I am able to praise them for sticking to it and working even when the work got tough. I remind them of opportunities they will have in the future to continue practicing this work.<br /><br />I am including a reflection to use in your classroom either after a math lesson or at any other time during the day when you are noticing that there are some students who could use some reassurance. If students were able to "get gritty" they had a successful learning day and their efforts should be celebrated!<br /><br /><div style="text-align: center;"></div><!-- Begin MailChimp Signup Form --> <br /><div id="mc_embed_signup"><form action="https://k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c13b65c715" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h3 style="text-align: center;">Sign Up Here to Grab Your Learning Reflection</h3><div class="indicates-required"><div style="text-align: center;"><label for="mce-EMAIL">Email Address <span class="asterisk">*</span></label> <input class="required email" id="mce-EMAIL" name="EMAIL" type="email" value="" /></div></div><div class="mc-field-group"><div style="text-align: center;"><label for="mce-FNAME">First Name </label> <input class="" id="mce-FNAME" name="FNAME" type="text" value="" /></div></div><div class="mc-field-group"><div style="text-align: center;"><label for="mce-LNAME">Last Name </label> <input class="" id="mce-LNAME" name="LNAME" type="text" value="" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c13b65c715" tabindex="-1" type="text" value="" /></div></div><div class="clear"><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe" name="subscribe" type="submit" value="Send Me My Freebie!" /></div></div></div></form></div>The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-70142903174573214292017-03-12T09:08:00.002-04:002018-10-21T14:36:23.297-04:00The Math Spot Compares Fractions: Abstract Thinking We've made it! We're at the last post in the 4 part fraction comparison series. If you haven't read parts <a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank">1</a>, <a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions.html" target="_blank">2,</a> and <a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_9.html" target="_blank">3</a>, go back and begin there. Up until this point we have focused on teaching fraction comparisons by allowing students to have experiences that would promote understanding of fractions and fractional parts. <br /><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">The concrete, representational, abstract (CRA) framework taught us to begin with concrete experiences and to link those experiences to both representations and abstract thinking. The focus of this post is concerned with promoting concrete thinking but, you may have noticed, that questions that point towards concrete thinking have been embedded in both the concrete and representational work that have been done thus far. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">This is really the heart of the CRA framework- you are making links to abstract thinking all along the way! <b>That is to say that abstract is not it's own isolated step.</b> I'm not saying students shouldn't explicitly practice at the abstract level- of course they should to build fluency with the skill they are navigating. Students benefit from opportunities to access and apply their previous knowledge in order to become fluent in a skill. But, generally, "abstract" is not a stand alone step. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="https://1.bp.blogspot.com/-4DGAaapDnDg/WMVIRQcR3GI/AAAAAAAAEcw/o8_6tJxUacsZJjp3Jln0agKozdee-n8IgCLcB/s1600/Depositphotos_19438669_original.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="152" src="https://1.bp.blogspot.com/-4DGAaapDnDg/WMVIRQcR3GI/AAAAAAAAEcw/o8_6tJxUacsZJjp3Jln0agKozdee-n8IgCLcB/s200/Depositphotos_19438669_original.jpg" width="200" /></a>Notice on day 1 when you introduced fractions with an "Epic" story problem. Right off the bat you <i>What happens to the size of the piece of the candy bar as it is broken up further? What happens to the denominator of the fraction as the candy bar is broken up further? What do you think you might know about fractions as the denominator gets larger? </i></div>begin to ask questions that promote abstract thinking. I suggested that as you go through the Epic story problem that you chart what happens to the candy bar as more friends show up. Looking at that chart, it would be very logical to ask students <br /><div class="separator" style="clear: both; text-align: left;"><i><br /></i></div><div class="separator" style="clear: both; text-align: left;">A powerful math lesson will tie together aspects of the concrete, representative and abstract within a lesson and will give students opportunities to make explicit connections. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><b>If you have stuck with me throughout this 4 part fraction series, you certainly deserve a freebie! I have created a CRA cheat sheet for you. The top half of the sheet will describe pathways from concrete through representative down to the abstract for fraction comparison models. The bottom half of the sheet lists sample linking questions you can use to be SURE that your students will be able to mentally, quickly, accurately, FLUENTLY compare fractions based on reasoning. Sign up in the box below and you will be brought to a *secret* page on my website where the cheat sheet is housed! </b> </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;"><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank">Post 1: The Math Spot Compares Fractions- Introduction</a></div><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions.html" target="_blank">Post 2: The Math Spot Compares Fractions- Comparison Tools</a><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_9.html" target="_blank">Post 3: The Math Spot Compares Fractions- Representative Models</a><br /><a href="https://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_12.html" target="_blank">Post 4: The Math Spot Compares Fractions- Abstract Thinking</a><br /><!-- Begin MailChimp Signup Form --> <br /><div id="mc_embed_signup"><form action="https://k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=7f41a3f743" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h3 style="text-align: center;">Sign Up Here For Your Cheat Sheet</h3><div class="indicates-required"><div style="text-align: center;"><label for="mce-EMAIL">Email Address <span class="asterisk">*</span></label> <input class="required email" id="mce-EMAIL" name="EMAIL" type="email" value="" /></div></div><div class="mc-field-group"><div style="text-align: center;"><label for="mce-FNAME">First Name </label> <input class="" id="mce-FNAME" name="FNAME" type="text" value="" /></div></div><div class="mc-field-group"><div style="text-align: center;"><label for="mce-LNAME">Last Name </label> <input class="" id="mce-LNAME" name="LNAME" type="text" value="" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c13b65c715" tabindex="-1" type="text" value="" /></div></div><div class="clear"><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe" name="subscribe" type="submit" value="Send Me My Freebie!" /></div></div></div></form></div>The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-8592995040810950332017-03-09T21:03:00.000-05:002018-10-21T14:36:01.421-04:00The Math Spot Compares Fractions: Representative Models<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-eJ_c1HN8IVo/WLjSdZUdmBI/AAAAAAAAEYA/GqMS0L9gTQ4uETTbaZgvAP7GH_No0FGgACPcB/s1600/Copy%2Bof%2B3%2BSteps%2BTo%2BA.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-eJ_c1HN8IVo/WLjSdZUdmBI/AAAAAAAAEYA/GqMS0L9gTQ4uETTbaZgvAP7GH_No0FGgACPcB/s320/Copy%2Bof%2B3%2BSteps%2BTo%2BA.png" width="320" /></a></div><br />We are heading into post #3 of my four part fraction comparison series. If you haven't read<a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank"> post #1 </a>and <a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions.html" target="_blank">post #2</a> take a quick break to go and read those now! <br /><br />Now that we are all caught up, let's do a quick recap of the big ideas:<br /><br /><ul><li>Students have had A LOT of experience in the whole number world. They WILL make generalizations that don't necessarily work for fractions if you don't ground them in concrete experiences. </li><li>Concrete experiences come in all sorts of shapes and sizes! There is no one perfect concrete model and the more concrete experiences you expose your students to the more anchors they have from which to draw future connections. </li></ul><div>But here's the thing. You've heard your colleagues say it and you have probably said it yourself... "In the real world they aren't going to be able to whip out their pattern blocks and say 'Hold on while I figure this out with my shapes!'". </div><div><br /></div><div>And you and your colleagues would be 100% right. This is exactly why I stress that it's a concrete, representative, abstract PROGRESSION. You start in concrete activities and begin to link those to representative models and abstract ideas. Depending on the topic you are teaching this may happen over the course of one lesson or over the course of multiple weeks! It is so important that you be in tune with your students and that you are able to move within the progression in a way that is responsive to their needs. </div><div><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-2W4DUVqM_AA/WLxHDwICUCI/AAAAAAAAEZE/R3lH-ZvkNCUxEq7YbH3NUKHFTUQ0STmBwCPcB/s1600/Fraction%2BBars.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://1.bp.blogspot.com/-2W4DUVqM_AA/WLxHDwICUCI/AAAAAAAAEZE/R3lH-ZvkNCUxEq7YbH3NUKHFTUQ0STmBwCPcB/s640/Fraction%2BBars.png" width="640" /></a></div> In post #2 I mentioned using fraction bars in a variety of forms including Cuisenaire Rods and Fraction bars or tiles. A logical representative step from these type of models would be to move towards a tape diagram. The prompt can be as simple as "Can you draw a tape diagram to match your fraction bars?" From there you will want to ask the students many questions to be sure they are drawing explicit connections between the concrete and the representative models.<br /><a href="https://2.bp.blogspot.com/-hfBuLlsXU9g/WMSoljvBe3I/AAAAAAAAEb4/34zppITyiQIgYgfMLqWYp5U89Vr94zY9gCLcB/s1600/IMG_4518.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://2.bp.blogspot.com/-hfBuLlsXU9g/WMSoljvBe3I/AAAAAAAAEb4/34zppITyiQIgYgfMLqWYp5U89Vr94zY9gCLcB/s200/IMG_4518.PNG" width="200" /></a><br />Consider giving students a task such as <i>Use fraction bars to show whether 1/4 or 1/2 is greater. </i>After the students had built models for 1/4 and 1/2 you would then say to them <i>Draw a tape diagram that matches your fraction bars to show whether 1/4 or 1/2 is greater. </i>Ask students to discuss with one another how their pictures match the bars or tiles that they built with. Ask students to point to the part in their drawing that matches the 1/2 bar. Ask students to point to the part in their drawing that matches the 1/4 tiles. Point out the discrepancies that you notice that a student doesn't! Are they not noticing that the whole needs to be the same size in their 1/2 and 1/4 tape diagrams? Ask students to describe, describe, describe how they came up with their representation. Before you know it, they will be able to use tape diagrams to show these comparisons without needing the materials first.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-pLPXeWT0FwE/WLxIfvABGRI/AAAAAAAAEZE/VlSL8-SI5WEE3gdbThABWQgAniQZe2bBACPcB/s1600/Epic%2BStory%2BProblems.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://3.bp.blogspot.com/-pLPXeWT0FwE/WLxIfvABGRI/AAAAAAAAEZE/VlSL8-SI5WEE3gdbThABWQgAniQZe2bBACPcB/s640/Epic%2BStory%2BProblems.png" width="640" /></a></div> All drawings don't need to be formal. I described using a tape diagram above but an informal drawing that represents what happened in a story problem will go very far in terms of allowing students to develop a spatial understanding of the fractions they are working with.<br /><br />Consider if you had told students an "epic" story problem about a round pizza. It would be a much larger leap for your students to make a matching drawing if you asked them to do so with a tape diagram. I am NOT saying that you shouldn't ask them to use a tape diagram to represent a "round" problem. But I am advocating that if you know moving to a drawing or representation will be a challenge for your students that you begin by asking them to make a representation that more closely aligns with the original problem. You can spend time in more than one concrete model or more than one representation. It would be beyond fair for you to ask students to draw circles showing what happened to a pizza and then asking them to draw a tape diagram that matches their circles.<br /><br />This is a progression, it is about being thoughtful, it is about knowing your students, it is about being as developmentally mindful as you can while building these big concepts!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-YNjw_E5gmLg/WLxIfsrt0qI/AAAAAAAAEZE/mQyNNjlz1sI0iAzLQLVsyLgEIOLKWRaIgCPcB/s1600/Rulers.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://3.bp.blogspot.com/-YNjw_E5gmLg/WLxIfsrt0qI/AAAAAAAAEZE/mQyNNjlz1sI0iAzLQLVsyLgEIOLKWRaIgCPcB/s640/Rulers.png" width="640" /></a></div> You may have heard that in the primary (K-2) world there is a BIG difference for students if they are counting to 10 on a number line vs. a number path. A number line marks the "end" of each period whereas a number path consists of adjoining boxes each representing a piece to the path. For primary students it is much easier to understand a number path than a number line because they can see that two, for instance, is two boxes on a number path. On a number line 2 looks like a finite point and they don't necessarily understand that everything up to that two constitutes the whole 2 we are talking about.<br /><br />In the intermediate world you can liken this to the difference between a tape diagram and a number line. A tape diagram is essentially a number path- you can see the full box that constitutes each period. A number line is much more difficult to interpret.<br /><br />A ruler, however, is a tool that many students are very familiar with and, if you use rulers with only specific fractions marked off (see post #2 for concrete examples) your students will be set up for success in number line work. Students can begin moving toward a number line by drawing and representing what they found on a ruler on a number line. To further ease the transition of using a number line to represent fractions, you may ask your students to shade in or highlight the number line up to a given number. This will create a visual that is almost a hybrid between a tape diagram and a number line.<br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-adAIxN7UgBQ/WMh_ecugOXI/AAAAAAAAEfU/7MnCXpsVhPYFb_ly7wonmAZNaJLoxNGQwCLcB/s1600/Paper%2BClips%2B%25283%2529.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://1.bp.blogspot.com/-adAIxN7UgBQ/WMh_ecugOXI/AAAAAAAAEfU/7MnCXpsVhPYFb_ly7wonmAZNaJLoxNGQwCLcB/s640/Paper%2BClips%2B%25283%2529.png" width="640" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div><div class="separator" style="clear: both; text-align: center;"></div><a href="https://4.bp.blogspot.com/-xFwEmbg0NRY/WMSo1JWhzBI/AAAAAAAAEb8/joziHJAqYtkOdyurKozc9KudDjp0823BQCLcB/s1600/IMG_4515.JPG" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://4.bp.blogspot.com/-xFwEmbg0NRY/WMSo1JWhzBI/AAAAAAAAEb8/joziHJAqYtkOdyurKozc9KudDjp0823BQCLcB/s200/IMG_4515.JPG" width="200" /></a>I described in the previous post how students could use pattern blocks to explore unit fractions and beyond when they are first comparing fractions. A simple transition between concrete and abstract is presented when your students are learning with pattern blocks. Students can simply trace the pattern blocks and label the unit fractions on their drawings. In this way, students can represent unit fractions while still having the opportunity to manipulate the materials first.<br /><br />As mentioned many times, throughout this post and the previous two, the real power in the C-R-A framework is asking the questions that join together the concrete models to the representations that students are drawing or interpreting. Ultimately, these connections will help students to begin to think in a more abstract way about these numbers and concepts.<br /><br />In my next post I will help you to tie it all together and to ask questions that are dangerously effective towards helping your students to achieve mastery in fraction comparison! </div><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank">Post 1: The Math Spot Compares Fractions- Introduction</a><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions.html" target="_blank">Post 2: The Math Spot Compares Fractions- Comparison Tools</a><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_9.html" target="_blank">Post 3: The Math Spot Compares Fractions- Representative Models</a><br /><a href="https://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_12.html" target="_blank">Post 4: The Math Spot Compares Fractions- Abstract Thinking</a><br /><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="92" data-original-width="245" src="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" /></a></div><br /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-90192688153861209782017-03-05T15:05:00.000-05:002018-10-21T14:35:30.386-04:00The Math Spot Compares Fractions- Comparison Tools <div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-eJ_c1HN8IVo/WLjSdZUdmBI/AAAAAAAAEYA/GqMS0L9gTQ4uETTbaZgvAP7GH_No0FGgACPcB/s1600/Copy%2Bof%2B3%2BSteps%2BTo%2BA.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-eJ_c1HN8IVo/WLjSdZUdmBI/AAAAAAAAEYA/GqMS0L9gTQ4uETTbaZgvAP7GH_No0FGgACPcB/s320/Copy%2Bof%2B3%2BSteps%2BTo%2BA.png" width="320" /></a></div><br />When I taught 4th grade special ed. I found that fractions in the context of a story made so much sense to my students. I would come up with elaborate scenarios - most involving food- that would start out by granting one student exclusive rights to an entire pan of brownies and would slowly erode their good fortune to the end of changing denominators and, ultimately, fraction learning. These were the days when I would hear cries of "You're the best teacher ever!" and "Math is so much fun!"... I get it, I was feeding them brownies. But they were learning so this was a true win-win scenario.<br /><br /><a href="https://4.bp.blogspot.com/-63L2FZVmqO0/WLx8CD5WU4I/AAAAAAAAEZQ/NDr-SUKlcOkSmjK83Jn7LLC5bZKdot-UgCEw/s1600/Comparing%2BFractions%2BManipulatives.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://4.bp.blogspot.com/-63L2FZVmqO0/WLx8CD5WU4I/AAAAAAAAEZQ/NDr-SUKlcOkSmjK83Jn7LLC5bZKdot-UgCEw/s200/Comparing%2BFractions%2BManipulatives.png" width="200" /></a>In my <a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank">first post of the comparing fractions series</a> I talked about how students make connections based on what they know. Coming out of the primary grade levels our students know a lot about whole numbers- but the rules of the whole number world don't always apply to fractions in the way that our kids think they will. It is so important to give our students opportunities to have concrete experiences with fractions so that they have solid anchors for their future fraction connections. My recommendation, however was not only to start with concrete experiences but to follow the <a href="http://www.k5mathspot.com/2015/12/when-they-still-dont-get-it.html" target="_blank">concrete-representative-abstract (CRA) model. </a><br /><br />This post will focus on a variety of concrete models and experiences that will help your students to form a strong anchor for their fraction learning. Anchors that you can easily connect to in representative and abstract ways through future lessons. I am recommending these in no particular order. Think about your students and the hands on experiences that you think might best resonate with them.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-2W4DUVqM_AA/WLxHDwICUCI/AAAAAAAAEYo/f9wU8aOdDFs4oXM1fBJRfvqaVv_ticnPACLcB/s1600/Fraction%2BBars.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://2.bp.blogspot.com/-2W4DUVqM_AA/WLxHDwICUCI/AAAAAAAAEYo/f9wU8aOdDFs4oXM1fBJRfvqaVv_ticnPACLcB/s640/Fraction%2BBars.png" width="640" /></a></div><a href="https://4.bp.blogspot.com/-LA2vpb53nII/WLx8CY_r-II/AAAAAAAAEZU/NmfXF2djYXUsl_phdYmCzuQpbZ72rsY0QCLcB/s1600/Fraction%2BBars%2BComparison.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://4.bp.blogspot.com/-LA2vpb53nII/WLx8CY_r-II/AAAAAAAAEZU/NmfXF2djYXUsl_phdYmCzuQpbZ72rsY0QCLcB/s200/Fraction%2BBars%2BComparison.png" width="200" /></a>A variety of fraction bars will be an indispensable tool when you are helping students to understand how to compare fractions. Cuisenaire Rods are a great first tool for exploration. Choosing one rod and naming it "the whole" ask your students to determine which color rod would represent 1/2, 1/3, 1/4 or 1/8. I love the connecting rods that I have listed under "tools" below because there is no risk of sloppy work leading to incorrect understandings.<br /><br />In the same vein, fraction bars will allow your students to see a similar relationship between the size of the fraction pieces. In this model you are losing the opportunity for your students to discover on their own the relationship between the size of the pieces, however, depending on the task your students have been given, this is just fine. For example, if your students already have an understanding of naming unit fractions and you are asking them to compare a variety of fractions and report what they are noticing about the pattern of denominators, fraction bars will be the perfect tool for your students to use.<br /><br />The last fraction bar model that I will mention- in this FAR from exhaustive list- is folded paper strips. Much like the Cuisenaire Rods, students will have an opportunity to reason about the size of the pieces as they are folding the strips into equal groups. I like this activity as more of an exploration because I will warn you that alternate conversations will emerge. Depending on the fine motor skills of your students, you will surely have students who do not fold into equal pieces.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-FTpNpCvJW7s/WLxIfh_k-_I/AAAAAAAAEYw/oCNwf2qVfTQu8ab-Bj4otuuc1h4bMngiwCLcB/s1600/Food.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://2.bp.blogspot.com/-FTpNpCvJW7s/WLxIfh_k-_I/AAAAAAAAEYw/oCNwf2qVfTQu8ab-Bj4otuuc1h4bMngiwCLcB/s640/Food.png" width="640" /></a></div>Food is a natural opportunity for students to experience fractions in a "real life" scenario. It also allows students to bring common sense into the equation. If I am sharing a pizza with 10 friends I will obviously get less pizza than if I am only sharing with one other person. The key here is to make sure that as you are using food like pizza, brownies, chocolate bars... <i>are you hungry yet?...</i>that you ask questions that will allow students to connect this understanding directly to fractions. Use fraction language. Chart out what happens as you cut the food into more and more pieces. Ask students what they notice about the denominators. Ask them to make predictions based on what they are noticing in the numeric patterns and then keep cutting the food up (or putting pieces back together) to test their conjectures.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-pLPXeWT0FwE/WLxIfvABGRI/AAAAAAAAEY0/vrsw0jyfKkIHZZE6rH6uXovJ8bm0d_nkgCLcB/s1600/Epic%2BStory%2BProblems.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://1.bp.blogspot.com/-pLPXeWT0FwE/WLxIfvABGRI/AAAAAAAAEY0/vrsw0jyfKkIHZZE6rH6uXovJ8bm0d_nkgCLcB/s640/Epic%2BStory%2BProblems.png" width="640" /></a></div> I'll be honest. Most of the "epic" story problems I have written are about food. I'm not apologetic though, is is a logical and very relatable context for students. I use epic story problems along side a prop to allow students to experience changing numerators and denominators for themselves. An epic story problem may take a good 30 minutes and is really more like a play. I might start out by picking a student and saying something like <i>"Julie woke up this morning and found that her mom had left her a chocolate bar with a note that said "Have a great day!"</i> I would have the student come over and hold the chocolate bar. "<i>Julie began walking to school and saw her friend Frank. Because Julie is SUCH A NICE FRIEND she shared the chocolate bar equally with Frank. Now Julie and Frank each have ______ of a chocolate bar." </i>I would have Frank come up to the front of the room and they would break the chocolate bar into 2 equal pieces each holding a half. I would continue this story, "<i>Julie and Frank came across Tanya and Priscella and, because they are SUCH NICE FRIENDS they decide to share the chocolate bar equally with them as well!" </i><br /><i><br /></i>You can see where I am going with this story. I always made a big play of sentences as "such a nice friend" and the whole class would say the line together like some sort of strange game show. I guarantee that if you try this strategy engagement will be high in your classroom. Just be sure to chart your findings as you go so that you have a record to debrief and talk about during and after the lesson.<br /><br />If you really want to go for it, you can turn the story around and have students deciding that they don't want to each chocolate and giving their portion to another friend. You will be able to work in adding unit fractions and comparing fractions with the same numerator in this way. Your students really won't think of this as "fraction work" until you chart the numbers to match because the context is so natural. This type of work helps to build the "right" kind of natural connections between what your students know and what you would like them to understand.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-YNjw_E5gmLg/WLxIfsrt0qI/AAAAAAAAEY8/k9qv9S9qm2YG5w2mW6d12xkySysguaebACLcB/s1600/Rulers.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://2.bp.blogspot.com/-YNjw_E5gmLg/WLxIfsrt0qI/AAAAAAAAEY8/k9qv9S9qm2YG5w2mW6d12xkySysguaebACLcB/s640/Rulers.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;">Rulers are another natural application of fractions. I love to use rulers that have specific fractions marked out on them and to work up to full rulers over time. Through School Specialty you can order rulers marked only to the inch, separate rulers marked to the 1/2 inch, and other rulers marked to the 1/4 inch. I have a link listed below so you can check them out but I am quite positive I didn't pay as much as Amazon is asking so I would shop around if I were you! </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">If you are teaching comparison of fractions on a number line, rulers are a concrete first step you can take. They take the principal of a number line and put it into a context that students are familiar with. I would talk about the different rulers and ask students to think about what labels might go on each of the different rulers and how they would label those rulers. </div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: left;">Another option is to take overhead projector pages - I'm sure you have some in a closet somewhere ;) - and to create a ruler with only inches marked, another with 1/2 inches in a different color, another with 1/4 inches in a different color and so on. You can then layer these rulers on top of one another so students can see how the full ruler is constructed. </div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-adAIxN7UgBQ/WMh_ecugOXI/AAAAAAAAEfU/mLqiRRTigWk29aV4D9p5HOmHbcvsSOiVACEw/s1600/Paper%2BClips%2B%25283%2529.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://4.bp.blogspot.com/-adAIxN7UgBQ/WMh_ecugOXI/AAAAAAAAEfU/mLqiRRTigWk29aV4D9p5HOmHbcvsSOiVACEw/s640/Paper%2BClips%2B%25283%2529.png" width="640" /></a></div><div class="separator" style="clear: both; text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"></div><a href="https://3.bp.blogspot.com/-YDwd5TZnjkw/WLx8CQmQ2yI/AAAAAAAAEZY/bJ1AaCrnh1wYJlUbXYpKjOjPBitY1YQxQCEw/s1600/Comparing%2BFractions%2BTangrams.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="200" src="https://3.bp.blogspot.com/-YDwd5TZnjkw/WLx8CQmQ2yI/AAAAAAAAEZY/bJ1AaCrnh1wYJlUbXYpKjOjPBitY1YQxQCEw/s200/Comparing%2BFractions%2BTangrams.png" width="200" /></a> Pattern blocks can be used much in the same way as the variety of fraction bars listed above. In fact, I would use them in almost exactly the same way that I described using Cuisenaire rods. In my next post I am going to talk about how to link these concrete models to representations in an effort to move towards more abstract and fluent thinking, but I will make the point now that pattern blocks are an easy tool to use to begin to make this transition because students can use pattern blocks to represent a comparison (one whole built with 1/2 pieces vs. one whole built with 1/6 pieces) and they can then trace those pieces to show their representation.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-VeHh8HUalVw/WLxIfipVDzI/AAAAAAAAEY4/keU-17ZmdisYfMKXi-UZ7VvtwAMYzjnRwCLcB/s1600/Paper%2BClips.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://2.bp.blogspot.com/-VeHh8HUalVw/WLxIfipVDzI/AAAAAAAAEY4/keU-17ZmdisYfMKXi-UZ7VvtwAMYzjnRwCLcB/s640/Paper%2BClips.png" width="640" /></a></div> I will touch on paper clips only briefly because they are the least connected fraction model, however, you can promote an understanding that it takes fewer large paperclips to fill a space than it does small paper clips. This understanding is found within a second grade geometry standard that I have <a href="http://www.k5mathspot.com/2015/02/fly-on-math-teachers-wall-fractions.html" target="_blank">written about here</a>. <-- That post also includes a free 2nd grade math activity that may be useful if you are a 3rd grade teacher just starting your fraction unit. This may be an unnecessary model if your students are able to grasp an understanding that lower denominators yield larger pieces, however, it is a tool to use if you have any students who still can't quite make that connection for themselves!<br /><a href="https://1.bp.blogspot.com/-LUz7n3yluIk/WLx8Ev7NG0I/AAAAAAAAEZc/kmX3hA5ZKZ0vyv_jhv4nJJUF9GAVmWaawCEw/s1600/Fraction%2BCuisenaire%2BRods.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="https://1.bp.blogspot.com/-LUz7n3yluIk/WLx8Ev7NG0I/AAAAAAAAEZc/kmX3hA5ZKZ0vyv_jhv4nJJUF9GAVmWaawCEw/s200/Fraction%2BCuisenaire%2BRods.png" width="200" /></a><br /><br />The concrete tools that can be used to model fractions for comparison are really endless. The main <br />takeaway here is that these models are so very necessary to build strong anchors for your students. So, what can you do now that these anchors are built? Follow me into post #3 as we work towards fluency and number sense around fractions.<br /><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank">Post 1: The Math Spot Compares Fractions- Introduction</a><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions.html" target="_blank">Post 2: The Math Spot Compares Fractions- Comparison Tools</a><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_9.html" target="_blank">Post 3: The Math Spot Compares Fractions- Representative Models</a><br /><a href="https://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_12.html" target="_blank">Post 4: The Math Spot Compares Fractions- Abstract Thinking</a><br /><div style="text-align: center;"><u><span style="font-size: x-large;">Tools</span></u></div><div style="text-align: center;"><span style="font-size: x-small;">A portion of the links and recommendations below are affiliate links. This means that at no additional cost to you I will earn a commission if you choose to make a purchase.</span></div><br />If you need any of the materials described in the post above, I have listed them below. I really recommend the "Connecting Rods". I love them because, unlike many other Cuisenaire Rods I have used, these link together. I can also recommend the fraction bars and the pattern blocks. I do NOT recommend the last link for the ruler. $7.23 for one ruler is crazy!! I bought them for much less from the School Smart catalog and bought a few of each of the 1 inch, 1/2 inch, 1/4 inch and fully marked rulers so that I have a continuum to work from. The link is there just so you can see what I was describing :) <br /><div style="text-align: center;"><br /></div><div style="text-align: center;"><iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=the0c319-20&marketplace=amazon&region=US&placement=B000F8XFB8&asins=B000F8XFB8&linkId=4f99db741d84443479eca1e7e5b8c0b9&show_border=true&link_opens_in_new_window=true" style="height: 240px; width: 120px;"></iframe><iframe frameborder="0" marginheight="0" marginwidth="0" scrolling="no" src="//ws-na.amazon-adsystem.com/widgets/q?ServiceVersion=20070822&OneJS=1&Operation=GetAdHtml&MarketPlace=US&source=ss&ref=as_ss_li_til&ad_type=product_link&tracking_id=the0c319-20&marketplace=amazon&region=US&placement=B000QDTYMQ&asins=B000QDTYMQ&linkId=106efe14e265c9bf7b51aac03ce25a42&show_border=true&link_opens_in_new_window=true" style="height: 240px; 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text-align: center;"><a href="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="92" data-original-width="245" src="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" /></a></div><br /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-10067689854841084962017-03-02T21:18:00.002-05:002018-10-21T14:34:55.047-04:00The Math Spot Compares Fractions: A Four Part Series <div class="MsoNormal" style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-eJ_c1HN8IVo/WLjSdZUdmBI/AAAAAAAAEX0/6p_TyAwBqFcscG5dl19mFdkwMVhGaB6wACLcB/s1600/Copy%2Bof%2B3%2BSteps%2BTo%2BA.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-eJ_c1HN8IVo/WLjSdZUdmBI/AAAAAAAAEX0/6p_TyAwBqFcscG5dl19mFdkwMVhGaB6wACLcB/s320/Copy%2Bof%2B3%2BSteps%2BTo%2BA.png" width="320" /></a></div><i><br /></i><i>Armando eats 1/8 of his box of chocolates in the morning and ¼ of his box of chocolates in the afternoon. Did Armando eat more chocolate in the morning or the afternoon? How do you know?</i></div><div class="MsoNormal"><span style="text-indent: 0.5in;"><br /></span></div><div class="MsoNormal"><span style="text-indent: 0.5in;">Comparing fractions <groan> you know that your students will come in with all sorts of “tricks” and misconceptions up their sleeves. But, never fear, because if you can predict it, you can prevent it. If you know your student will think that an eighth is larger than a third because 8 is bigger than 3 we just need to be sure that you are being strategic in your instruction and getting out ahead of these predictable misconceptions.</span><br /><span style="text-indent: 0.5in;"><br /></span></div><div class="MsoNormal">These common errors, so often, are born out of the idea that students are handed <i>numbers</i> and asked to reason with them. They draw from what they know the best. And in this case, students have spent the vast majority of their primary years working with whole numbers. When third grade rolls around and students are asked to compare these fractions it is entirely understandable that they draw on their knowledge of whole number comparisons. Who could blame them? <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">It is then imperative that we consider methods of building understanding of fractions and comparison for students before they overgeneralize and draw their own “connections” that are doing little more than building confusion. Using the concrete, representative, abstract model for instruction you can build understanding in the concrete and representative stages and slowly fold in opportunities for abstract thinking. Students will then have anchors in their concrete and representative work to pull from rather than drawing on whole number generalizations. <o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">The concrete, representative abstract (C-R-A) model calls for instruction to be built from concrete hands-on experiences, linked to visual representations and ultimately these experience allow students to generalize their understanding through purely numeric or mental work. </div><div class="MsoNormal"><o:p></o:p></div><div class="MsoNormal"><br /></div><div class="MsoNormal">So what, exactly, will the C-R-A framework look like at each step of the way? Follow me into the next three posts of this comparing fractions series. We will look to the 3rd and 4th grade fraction comparison standards and the most effective and efficient ways to promote understanding. <br /><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank">Post 1: The Math Spot Compares Fractions- Introduction</a><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions.html" target="_blank">Post 2: The Math Spot Compares Fractions- Comparison Tools</a><br /><a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_9.html" target="_blank">Post 3: The Math Spot Compares Fractions- Representative Models</a><br /><a href="https://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_12.html" target="_blank">Post 4: The Math Spot Compares Fractions- Abstract Thinking</a></div><br /><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="92" data-original-width="245" src="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" /></a></div><br /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-5661064873758083232016-08-30T18:56:00.000-04:002017-06-22T07:09:26.004-04:00I Have a Teaching Assistant for Math! ...but how can we best serve our students? <div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-VOVd-0uBqds/V7OIkZ6Ra8I/AAAAAAAAC78/OfTfAqb4siMAEPevry5UkSOk82xVtnEEQCLcB/s1600/Teaching%2BAssistants%2B%25281%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="330" src="https://1.bp.blogspot.com/-VOVd-0uBqds/V7OIkZ6Ra8I/AAAAAAAAC78/OfTfAqb4siMAEPevry5UkSOk82xVtnEEQCLcB/s640/Teaching%2BAssistants%2B%25281%2529.jpg" width="640" /></a></div><br /><br />You just found out that you have a teaching assistant during math for the school year. Awesome!! But you may be asking yourself, what is the BEST way to work together to help your students to grow? A few simple "do"s and "don't"s will get you on the right track.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-6WJWoWVzouw/V7OLar74a2I/AAAAAAAAC8E/qCdLs1MoXYwzPTL6IYiNI_cMbIjELWyWgCLcB/s1600/TA%2B1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://2.bp.blogspot.com/-6WJWoWVzouw/V7OLar74a2I/AAAAAAAAC8E/qCdLs1MoXYwzPTL6IYiNI_cMbIjELWyWgCLcB/s640/TA%2B1.png" width="640" /></a></div><br />It can be so tempting to quickly describe the activity you are planning on doing as your TA runs into the room (and let's be real, they are running because TAs are amazing people who are spread sooo thin!) This method sells your plans short and will impact your students in a negative way. This means time taken away from your students as you have a quick chat and it also means that your TA will not have the opportunity to clarify and really understand your plans to the fullest. Plus, you have to go through this same routine each time your TA comes in.<br /><br />Instead? Find time for a 10 minute appointment once a week. Honestly, that's all the time it will take. You will have peace of mind that your TA knows what is expected in the classroom and your TA will feel comfortable coming into a room knowing just what is in store for them.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-8ua4OyLfWZw/V7OLbAnHMCI/AAAAAAAAC8I/elkgdTYMcXcdBxXqEi6YGdrgqF_4DwtfACLcB/s1600/TA%2B2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://4.bp.blogspot.com/-8ua4OyLfWZw/V7OLbAnHMCI/AAAAAAAAC8I/elkgdTYMcXcdBxXqEi6YGdrgqF_4DwtfACLcB/s640/TA%2B2.png" width="640" /></a></div><br />Asking a TA to deliver new instruction is a big no-no. First off, your TA is not necessarily a certified teacher. They have not been specifically trained to understand what to look for in students as they are gaining a new skill, how to cut off misconceptions as they emerge and how to connect this new learning to prior instruction you have given. Don't get me wrong, the TAs I have worked with have been amazing and very well could teach the lesson but this is an unfair expectation to have.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-0WW2OyBixmA/V7OVNME-muI/AAAAAAAAC8s/8k7cY6ARx4EBpzFnXEj-Bgj3dYTxXxb1ACLcB/s1600/Partners%2Bof%2B10-%2B5%2BDay%2BFocus.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-0WW2OyBixmA/V7OVNME-muI/AAAAAAAAC8s/8k7cY6ARx4EBpzFnXEj-Bgj3dYTxXxb1ACLcB/s400/Partners%2Bof%2B10-%2B5%2BDay%2BFocus.jpg" width="300" /></a></div>Instead? Plan activities, practice pages or math centers that build upon the initial instruction you have given. As an example, take a look at an outline for a sample week of instruction in my math intervention room.<br /><br />The activities listed as "Instructional Activity" would be the initial instruction delivered by the classroom teacher. The next column over, "Independent Activity", is an activity which reinforces the instructional activity.<br /><br />When students complete the independent activity in my intervention room, I monitor their work, look for misconceptions and support... but only if needed.<br /><br />If you have a TA in your room, your students can still enjoy this level of support but you could be freed up to pull another small group, to pull for an assessment, or to teach students in any way necessary.<br /><br />If you click <a href="https://drive.google.com/file/d/0B6fiBv3-SZg-T3loWjhLWHFORGM/view?usp=sharing" target="_blank">HERE</a> , you will go to my Google Drive where you will find a free sample of the outline above. You will receive the lesson and all necessary materials for the day 1 instructional activity, independent activity and ticket out the door listed above so that you can try out this format in your classroom.<br /><br /><div class="separator" style="clear: both; text-align: center;"></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-_dDLRd3CbWc/V7OLa0XMmGI/AAAAAAAAC8M/kFIZcDmdEDcrDrxxkjWkImzSlZSYyWJkACLcB/s1600/TA%2B3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://4.bp.blogspot.com/-_dDLRd3CbWc/V7OLa0XMmGI/AAAAAAAAC8M/kFIZcDmdEDcrDrxxkjWkImzSlZSYyWJkACLcB/s640/TA%2B3.png" width="640" /></a></div><br />When there are multiple adults in the classroom it can be very easy for students to look for adult support... and sometimes that means too much adult support! Make sure that you and your TA have a conversation at your weekly meeting in terms of what you would expect your students to try independently and what you anticipate they may need more support on. You may even give your TA some prompts based on the anticipated road blocks in the activity you are assigning. I wrote much more on the topic of independence in <a href="http://www.k5mathspot.com/2015/12/independence-in-math-class.html" target="_blank">THIS</a> blog entry. Allowing and expecting independence is often an overlooked strategy for our learners with the greatest area of need.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-tTx6fIl2mGQ/V7OLbLstp7I/AAAAAAAAC8Q/Af3vlzhOIWcRdw98Lph7l7EjrLNKxeFqwCLcB/s1600/TA%2B4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="80" src="https://1.bp.blogspot.com/-tTx6fIl2mGQ/V7OLbLstp7I/AAAAAAAAC8Q/Af3vlzhOIWcRdw98Lph7l7EjrLNKxeFqwCLcB/s640/TA%2B4.png" width="640" /></a></div><br />And last, but certainly not least, at your weekly meeting make sure to spend 2-3 minutes talking about which students you and your TA noticed performing well on their own and who they noticed might need some additional support. The success that one of you noticed might be directly attributed to something that the other did and it is awesome to have that affirmation! In terms of need, your TA will have unique insight into how your students are doing when they are expected to work independently. Take advantage of this window of insight!! Share what you notice and celebrate, celebrate, celebrate the success together!<br /><br /><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" data-original-height="92" data-original-width="245" src="https://2.bp.blogspot.com/-c7NJQIXls8w/VzcbEKvXhII/AAAAAAAAIgQ/QeUVKZ3VCwwtLDo9-Ol9_3ckjdzU6gHegCPcBGAYYCw/s1600/signoff.png" /></a></div><br /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-12069311573367186592016-08-23T17:13:00.000-04:002017-03-04T15:23:27.303-05:00I'm Just Not a Sticker Teacher <div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-zf-jf73K2y0/V7N9QvpPAKI/AAAAAAAAC7Y/_Bs9dwzXQvsqOEbpa_ZsORKo5Xnizj0LwCLcB/s1600/Not%2Ba%2BSticker%2BTeacher.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="331" src="https://1.bp.blogspot.com/-zf-jf73K2y0/V7N9QvpPAKI/AAAAAAAAC7Y/_Bs9dwzXQvsqOEbpa_ZsORKo5Xnizj0LwCLcB/s640/Not%2Ba%2BSticker%2BTeacher.jpg" width="640" /></a></div><br />To be clear, I have no problem with stickers at all, I just don't use them! I found that it was more trouble than it was worth. We would get to the end of our short small group math block and the sticker fiasco would ensue. One student would get the last sparkle sticker, another student would decide that they wanted the sparkle sticker (mind you they never wanted a sparkle sticker before but, supply and demand...) their "reward" now turning into a sour experience. And then there would be the student who would take 10 minutes to decide which sticker they wanted.<br /><br />And does anyone really know why they earned a sticker and which of their learning behaviors earned them this reward? And what if a student's learning behaviors <i>weren't </i>worthy of the reward? Do you deny them the sticker? And if you do, did the student know that they weren't getting a sticker because their learning behaviors didn't warrant a treat? Probably not, I work with 5 and 6 year-olds. Their take away on a day they don't earn a sticker is that you. are. mean!<br /><br />And so I don't do stickers, stars, tokens, clip charts, or anything of the like. And yet my student's small group math behaviors are to be commended... so what is my trick?<br /><br />Individual bar graphs that are not at all explicitly tied to behavior.<br /><br />Exciting, I know :)<br /><br />But this is the single most effective behavior plan I have employed in my intervention space. So here's how it works. Each week my students do "8 Minute Math" which is actually an AIMS Web progress monitor. If you aren't familiar with AIMS Web MComp, it's a front and back page with a mix of addition and subtraction questions. On the front the problems are mostly single digit addition but as students move down the sheet the problems increase in complexity. I love this 8 minute assessment because I can mark down the strategies my students are using (are they drawing dots? finger counting? counting on? drawing ten sticks and ones? are they confused about subtraction?). My students love the 8 Minute Math because they get to color in their sheets. <i>As a side note, if you would like a copy of the graphing form, click on the graphic below and it will bring you to my Google Drive. </i><br /><br /><a href="https://drive.google.com/file/d/0B6fiBv3-SZg-VE9aeVFMV0xPd3c/view?usp=sharing" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img border="0" height="320" src="https://2.bp.blogspot.com/-yOk-IwvmeDw/V7OBNMfkuvI/AAAAAAAAC7k/Ghx8oHD3jYgSPr0NoyaCaqI2llX4Fw5VgCLcB/s320/My%2BMath%2BProgress.jpg" width="240" /></a>You see, as soon as the 8 minutes is up, I grade the student's papers (I usually have 4-5 students in a group) on the spot. They immediately get to color in a new bar on their bar graph to show how many points they earned in a given week. And then we compare to the previous week. Now remember, I am working with 5 and 6 year old students so the comparison is general but they notice if their graph went up or down or stayed the same. They are then responsible for telling me why it is that their graph changed in the way that it did. Their answer for a graph going up may include "I worked hard this week and my brain got smarter" or "I practiced a strategy and it worked!" or any other statement that attributes their growth to the work that they put in. If students say "I don't know" or "I must have eaten smart flakes" or some other answer that doesn't illustrate the work they put in I will continue prompting until they realize that it is their work that allowed them to grow.<br /><br />And, of course, there are the not so fun conversations as well. Why did your graph go down this week? "Well, I rolled around on the floor and didn't work very hard on my 8 Minute Math..." And how do you feel about coloring in your graph when it goes down like that? "Not proud!" "What can I help you do next week to be sure you are on track to making your brain smarter?"<br /><br />And, at times, there are the conversations that break your heart because a student did try their best and their graph just isn't budging yet. But their classmates encourage them and we talk about how sometimes we don't master a strategy in a week. Sometimes, we just need a bit more time and we might not be there yet.<br /><br />Throughout the week as students are working I constantly prompt and praise them in ways that promote that their growth and learning is a result of the work that they are putting in. I say things such as "Wow, I see you are working really hard on that math center... what do you think will happen when you do your 8 Minute Math this week?"<br /><br />And so gone are the days of the stickers and I am currently in the land of "Your work today is really helping your brain to grow!" And the graph gives the students a visual of their brain growing over time. It is the single most amount of buy in I have ever had on a "behavior management system" but it is really so much more than that because my students have bought in for all of the right reasons.<br /><!--End mc_embed_signup--><br /><div style="text-align: center;"><img class="nopin" height="50" src="https://dl.dropboxusercontent.com/u/324927842/TheMathSpot/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-27909632864860771902016-08-22T16:22:00.000-04:002017-03-04T15:24:00.776-05:00YouTube Counting Song Roundup<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-3jpcVrMi9A0/V7tgdUPO07I/AAAAAAAAC90/fPzyz2fdSzcSjiv0J4YChTXC75CS-w2rACLcB/s1600/Teaching%2BAssistants%2B%25282%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="https://1.bp.blogspot.com/-3jpcVrMi9A0/V7tgdUPO07I/AAAAAAAAC90/fPzyz2fdSzcSjiv0J4YChTXC75CS-w2rACLcB/s1600/Teaching%2BAssistants%2B%25282%2529.jpg" /></a></div><br />YouTube is such an excellent resource when it comes to songs and videos. There is an absolute wealth of options on the site especially when it comes to number and counting songs. Sometimes there are almost too many options and crawling through can be very time consuming.<br /><br />Lucky for you I consumed my time on this task so you wouldn't have to. I spent waaay too much time watching cartoons singing about the numbers 1 - 100. I chose videos that have strong visual cues whether it be the objects counted, the numbers themselves or a combination of the two. I have done my best to sort down below so that this post can be an easy to use resource for you!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-MVKinvvR_Qo/V7tKvzIQE8I/AAAAAAAAC9A/sk2-r9ArWCMFAecupr1sZjXuEMuLFOZogCLcB/s1600/You1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="88" src="https://3.bp.blogspot.com/-MVKinvvR_Qo/V7tKvzIQE8I/AAAAAAAAC9A/sk2-r9ArWCMFAecupr1sZjXuEMuLFOZogCLcB/s640/You1.png" width="640" /></a></div><div style="text-align: center;"><br /><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/diMJIlv-4N0" width="560"></iframe> <br /><br /></div><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/7gm5usaEac0?list=PL0VE_cI7-AYQ9A2qqZ3IxL7P8mLBQDOoD" width="560"></iframe> <br /><br /></div><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/6poj1mR_cRY" width="560"></iframe> <br /><br />**This next one is great when you are ready to start the skill from counting beginning at a given number.<br /><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/ZLTO38Wy4Qc" width="560"></iframe><br /><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-0DgpmnAAfu4/V7tKv-bl3TI/AAAAAAAAC9E/2CPEpjLRDlsT2pMH-FiU5vQRjKgUnKUCwCLcB/s1600/You2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="88" src="https://3.bp.blogspot.com/-0DgpmnAAfu4/V7tKv-bl3TI/AAAAAAAAC9E/2CPEpjLRDlsT2pMH-FiU5vQRjKgUnKUCwCLcB/s640/You2.png" width="640" /></a></div><div style="text-align: left;"><br /></div><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/E88NeoaU-To" width="560"></iframe></div><div style="text-align: center;"><br /></div><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/rkd-Vs_gTyc" width="560"></iframe> </div><div style="text-align: center;"><br /></div><div style="text-align: center;">Both of the companies above "<a href="https://www.youtube.com/user/LittleBabyBum" target="_blank">LittleBabyBum</a>" and "<a href="https://www.youtube.com/user/storybots" target="_blank">StoryBots</a>" both have songs for each individual number from 1 - 10<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-oEFFNehzq2E/V7tU8OWWx8I/AAAAAAAAC9Y/4TcI4sllhjoQ5_QVezB1m6fKSX_T593iwCLcB/s1600/You5.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="88" src="https://4.bp.blogspot.com/-oEFFNehzq2E/V7tU8OWWx8I/AAAAAAAAC9Y/4TcI4sllhjoQ5_QVezB1m6fKSX_T593iwCLcB/s640/You5.png" width="640" /></a></div><br /><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/kEJP1myKkUc" width="560"></iframe><br /><br /><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/Sqm-kFNaw8c" width="560"></iframe><br /><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/--FGMreoq_hY/V7tKv7kLonI/AAAAAAAAC88/NCY4KqCFnQEPAnVa066tvtp8WBDpmbdEgCLcB/s1600/You3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="88" src="https://3.bp.blogspot.com/--FGMreoq_hY/V7tKv7kLonI/AAAAAAAAC88/NCY4KqCFnQEPAnVa066tvtp8WBDpmbdEgCLcB/s640/You3.png" width="640" /></a></div><br /><div style="text-align: center;">This next song should come with a warning. It WILL get stuck in your head. You have been warned...</div><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/e0dJWfQHF8Y" width="560"></iframe></div><br /><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/yTeUqWGCKjA" width="420"></iframe></div><br />**I'm sneaking in this "Count to 20" song as well :)<br /><br /><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/0VLxWIHRD4E" width="560"></iframe></div><div style="text-align: center;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-f6WHXJc8_gk/V7tKwW1HP9I/AAAAAAAAC9I/BtC0k80MdaM-zqGxRJWBg5kltooWybVhACLcB/s1600/You4.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="88" src="https://2.bp.blogspot.com/-f6WHXJc8_gk/V7tKwW1HP9I/AAAAAAAAC9I/BtC0k80MdaM-zqGxRJWBg5kltooWybVhACLcB/s640/You4.png" width="640" /></a></div><br /><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/yQSdKlNvrmw" width="560"></iframe></div><br /><div style="text-align: center;"><iframe allowfullscreen="" frameborder="0" height="315" src="https://www.youtube.com/embed/-gmEe0-_ex8" width="560"></iframe><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://www.teacherspayteachers.com/Product/Counting-and-Number-Math-Centers-2045643" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;" target="_blank"><img border="0" height="240" src="https://3.bp.blogspot.com/-aLPq3AGHr0c/V7tevrAy7YI/AAAAAAAAC9o/7KowzeMwxg0DTJt1itwf_QcIo0dtNpSNgCLcB/s320/Slide1.JPG" width="320" /></a></div><br /><div style="text-align: left;">If you are looking for additional kindergarten counting resources, I have a set of math centers and print and go pages devoted to learning the numbers and count sequence to 10. Easy to set up and your students will love them! <a href="https://www.teacherspayteachers.com/Product/Counting-and-Number-Math-Centers-2045643" target="_blank">Click HERE to see more. </a></div></div><br /><div style="text-align: center;"><img class="nopin" height="50" src="https://dl.dropboxusercontent.com/u/324927842/TheMathSpot/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-87815121207364555912016-08-16T12:57:00.001-04:002018-09-03T17:55:41.218-04:00How Will Subitizing Help My Students? <div class="separator" style="clear: both; text-align: center;"></div><a href="https://2.bp.blogspot.com/-kvW_EsqtsK0/W42sGFJkEGI/AAAAAAAAJTo/924atqEzEz0cW6xJENBPqwBpk3aYC6jlgCEwYBhgL/s1600/Subitizing%2BPin.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="If you teach kindergarten or first grade, understanding subitizing is a must! Activities that have your students looking for patterns such as using subitizing cards, dot cards or dice games will all support their number sense. #Subitizing #SubitizingCards #NumberTalks" border="0" data-original-height="1200" data-original-width="800" height="400" src="https://2.bp.blogspot.com/-kvW_EsqtsK0/W42sGFJkEGI/AAAAAAAAJTo/924atqEzEz0cW6xJENBPqwBpk3aYC6jlgCEwYBhgL/s400/Subitizing%2BPin.png" title="If you teach kindergarten or first grade, understanding subitizing is a must! Activities that have your students looking for patterns such as using subitizing cards, dot cards or dice games will all support their number sense. #Subitizing #SubitizingCards #NumberTalks" width="266" /></a><br />Subitizing can be one of those skills that we know our students need to have but we aren't sure when or how to go about making this happen. I want to clear subitizing up for you and to give you some "I can implement this tomorrow" activities to use in your classroom.<br /><br />Let's start with a definition: Subitizing is having the ability to look at a picture or a set of objects and to know how many their are, almost instantly, without counting. Playing a dice game would be a most basic example of subitizing. When you roll a die, you look down and know immediately that you rolled, for example, a 3. You didn't need to count the pips to know, without a doubt, how many dots their are. <i>Subitizing is a foundation upon which number sense is developed. </i><br /><br />Subitizing, however, goes beyond this initial ability. The type of subitizing I am referring to is called <b>perceptual subitizing</b>. You are perceiving "how many" almost instantly through sight. A second type of subitizing is called <b>conceptual subitizing. </b>Conceptual subitizing is better explained through the example of rolling two dice. You roll a pair of 5s and know, almost immediately, that you have rolled a 10. You didn't count each of the pips on the dice but you know the pattern of 5 and 5 making ten and therefore can quickly and accurately report "how many".<br /><br /><a href="https://3.bp.blogspot.com/-_anzxhe2rVQ/V7Mz8ErHp-I/AAAAAAAAC6k/MnHUiDpr4poBTXxS4bjPX1D5IuRc90enQCLcB/s1600/Subitizing.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="640" src="https://3.bp.blogspot.com/-_anzxhe2rVQ/V7Mz8ErHp-I/AAAAAAAAC6k/MnHUiDpr4poBTXxS4bjPX1D5IuRc90enQCLcB/s640/Subitizing.jpg" width="249" /></a>So, how can you make these two types of subitizing work for you and your students? In terms of perceptual subitizing, expose students to a number of "regular" representations of amounts. I have listed 6 suggestions off to the side but they are really just the tip of the iceberg.<br /><br />Students can work with these representations in a variety of ways and that is really the focus, students experiencing the representations over and over and over. You can try card flashes with ten frames or fingers having students answer as rapidly as they can. You could ask students to match a numeral to the ten frame it represents in a game of memory. Students can play a game of dominoes stating aloud the numbers they are matching together, students can play board games that require them to roll dice, you can practice counting aloud to ten with students holding up their fingers as they go, you can have students play memory with a deck of cards either with or without the face cards.<br /><br />Obviously there are SO many options!! As for when to use these activities and how often you should be working on subitizing with students I would say that you could do the full group activities as a math warm up, when your students are in line waiting to go somewhere or really any time you have 1-3 minutes to fill. Many of the other activities make a quick and easy math center that students can work on until these regular patterns start to become second nature to them.<br /><br />Once students are solid subitizing regular patterns to 5, I am sure to incorporate activities that focus on conceptual subitizing. The type of activity requiring conceptual subitizing that I focus on depends on the math skill I am developing.<br /><br />We know from the work of <a href="http://www.k5mathspot.com/p/books-i-love.html">John Van de Walle</a> in his book "Teaching Student Centered Mathematics" that number sense is built on a foundation of spatial recognition, the relationship of one more and one less, part part whole relationships and benchmarks of 5 and ten. Conceptual subitizing activities are obviously related to spatial recognition but can boost each of the other 3 understandings as well. In terms of a few examples, students can solidify their understanding of <a href="http://www.k5mathspot.com/2015/12/rekenrek-one-moreone-less.html">one more and one less </a> through practicing thinking about patterns they know and what that number would be if you added one or took one away. <a href="http://www.k5mathspot.com/2015/12/the-strange-way-i-teach-fluency-for-5.html">Fact fluency for the 5+ facts</a> can be grounded in a foundation of spatial understanding of dot patterns as well.<br /><br />Comment down below with either an activity you could do to boost your students subitizing skills using one of the representations listed- let's get a bank of quick activities going! <br /><br /><div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"></div><br /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--> <br /><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-82432157959065828062016-07-31T15:58:00.000-04:002017-03-04T15:29:39.677-05:003 Steps To A Successful Math Year<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-s94cDEb2FN0/V55nycLtSQI/AAAAAAAAC54/YeVI0SOEOp8OkZiJZrlZ1Y35Eah3F3FgACLcB/s1600/3%2BSteps%2BTo%2BA%2B%25281%2529.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://4.bp.blogspot.com/-s94cDEb2FN0/V55nycLtSQI/AAAAAAAAC54/YeVI0SOEOp8OkZiJZrlZ1Y35Eah3F3FgACLcB/s640/3%2BSteps%2BTo%2BA%2B%25281%2529.jpg" width="424" /></a></div><br />The beginning of the year is completely and utterly overwhelming. You have to rebuild your classroom, prepare for a new crew of kids, wrap your mind around beginning of the year professional development and prepare materials for those first few <b>exhausting </b>days.<br /><br />Oh, and did I mention curriculum?<br /><br />Yikes!<br /><br />Luckily, there are some steps you can take now to set yourself up for math success this school year. In the past there have been years I have gone in with the frame of mind that I am was prepping the first math unit. I thought about the pre and post assessment and what it might look like to differentiate throughout that unit- and these were not "bad" or "wrong" thoughts. I was on the right track. But what happens when that first unit ends and all of your students haven't mastered the standards? What do you do when you are looking at your tier 2 and tier 3 students before, during and after that first unit? I wasn't thinking big enough and I certainly wasn't setting myself up for year long success!<br /><br />As a math interventionist, I don't necessarily have a set math curriculum or program that I follow. My students have a variety of levels and needs and so I needed a new system. These are the steps that I follow to set myself up for a calm, focused school year. I have included templates that you can use so that you can set yourself up for success as well!<br /><br /><b>1) Think Long-Term </b><br />The first step to planning needs to be a year long plan. You need to know where you are going in terms of the big picture in order to assess if your students are on the right path. For me, there are two ways to make this work. Work either in months or in quarters. When writing this long term plan you may choose to write the standards across the year or, if you have a program that you follow, you may choose to write the names of the units across the year.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-qIr7bzGSUK0/V5vKUvD_wgI/AAAAAAAAC5Y/dKeQTQtW-osWsSVtfk3wMGQKPS6i24G_gCLcB/s1600/Primary%2BTiered%2BGoals.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="240" src="https://2.bp.blogspot.com/-qIr7bzGSUK0/V5vKUvD_wgI/AAAAAAAAC5Y/dKeQTQtW-osWsSVtfk3wMGQKPS6i24G_gCLcB/s320/Primary%2BTiered%2BGoals.jpg" width="320" /></a></div>In the case that you do follow a program and you are writing the names of units, look inside of the unit and find the skills or standards addressed and list those in your long term plan as well. <a href="https://drive.google.com/file/d/0B7gi7ieV54Vhb05iTFhGWE1YRGM/view?usp=sharing">HERE is a link</a> to my personal year long plan filled in for kindergarten, first and second grade. My tier 1 is based off of the Engage NY math modules. As an interventionist, it was important for me to back map and look at what tier 2 and tier 3 might look like across the year as well. It is so important to know that there will be students who do NOT follow your plan. The long term plan is still incredibly helpful because, for those outliers, it gives you a map to plot where they might be and the logical steps you could take to move the student closer to grade level expectations.<br /><b><br /></b><b>2) Break It Down</b><br />Whether you chose to break your year into months or quarters, it's time to dive into each of those pieces and to think about the intermediate steps needed to meet the goals in that particular month or quarter. At this point, I like to plan in one or two week periods of time. This can become a bit tricky because some skills or standards are better suited to be long term goals while other skills could be a focus for a shorter period of time and then moved to independent practice. When I look at the first quarter of first grade for students in tier 3, students need to be able to<br /><br /><ul><li>Count to 100 by 1s and 10s <i>This is an ongoing skill. I think of this as a fluency focus and would probably play counting games as a lesson opener for the first few weeks of school to reinforce mastery. </i></li><li>Represent addition and subtraction with objects and drawings <i>This is a LOADED skill! For quarter 1 I broke this skill down into understanding what addition means in terms of both putting together and adding to, the meaning of subtraction (take away only for quarter 1) and then mixing the two together to be sure students understand the meaning of each operation and aren't just memorizing "what they are doing" in a given week. </i></li><li>Add and subtract within 5 fluently- <i>I didn't even list this as a focus this quarter once I broke the weeks down. I would absolutely do a pre and post assessment for this skill, however, for quarter 1 I am more focused on the students' understanding of addition and subtraction, the development of their spatial understanding of amounts and the relationship of 1 more and 1 less. If students are successful with those skills they will certainly improve in their fact fluency to 5. </i></li><li>Count forward from a given number- <i>This is another ongoing goal. This fluency focus would be appropriate once students are able to count to 20 without prompting. </i></li><li>Write numbers to 20- <i>Writing numbers to 10 would be an immediate focus if they fell down in this area on a pre assessement. Numbers to 20? I listed this as a later fluency skill. Even so, I would likely put emphasis on other skills first knowing that in quarter 2 when students are composing and decomposing teen numbers that this skill will continue to develop. </i></li><li>Spatial relationships of numbers 1-10- <i>Fluency, fluency, fluency. This includes dot patterns, Rekenrek work, ten frames and general subitizing activities. </i></li><li>1 more/1 less numbers 1-20- <i>While these are number sense skills, I do think they warrant some direct instruction. Students often understand one more and one less in a conversational sense but I chose to spend a week on each of these skills to explicitly link the understanding to a student's understanding of addition and subtraction. </i></li></ul><div>"Breaking it Down" over the course of the first quarter might look like the following:</div><div><br /></div><div><a href="https://4.bp.blogspot.com/-SNlDOMbwxiI/V5l4hBLtReI/AAAAAAAAC5E/URO4O7MxIf4dDVsnRqni9Id4j4i9sejYQCLcB/s1600/Slide1.JPG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="360" src="https://4.bp.blogspot.com/-SNlDOMbwxiI/V5l4hBLtReI/AAAAAAAAC5E/URO4O7MxIf4dDVsnRqni9Id4j4i9sejYQCLcB/s640/Slide1.JPG" width="640" /></a>Take note that the "Week Focus" moves week by week but that the fluency focus may be less linear. Some of the skill may (and will) overlap. I listed general starting spots but this focus is much more flexible!<br /><br /><b>3) Zoom in Close</b><br /><a href="https://3.bp.blogspot.com/-fyPve99vT2k/V5vJ-6V6VKI/AAAAAAAAC5U/lOAis1IyAwQ_SecHKshMn-jIIe-H2YbkQCLcB/s1600/Add%2Band%2BSubtract%2B10%2BMentally.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-fyPve99vT2k/V5vJ-6V6VKI/AAAAAAAAC5U/lOAis1IyAwQ_SecHKshMn-jIIe-H2YbkQCLcB/s320/Add%2Band%2BSubtract%2B10%2BMentally.jpg" style="cursor: move;" width="240" /></a>The last step in the process would be to take each week and break it down in terms of what your lessons might look like on a daily basis. I would NOT do this ahead of the school year. You know that your weekly plan might need to be flexible and your daily lessons, even more so. I generally plan a few weeks at a time taking into account specific student needs, formative assessment data and taking into account the general "things that pop up" that might disturb math on any given day like an assembly or a birthday party.... these things happen :)<br /><br />If you have used any of my <a href="https://www.teacherspayteachers.com/Store/The-Math-Spot/Category/5-Day-Focus-258099">5 Day Focus resources</a>, this is what you find towards the front of the document on the "Overview" page. When I plan the weekly overview I am <i>not</i> going in and writing the details of each day. I <i>am</i> writing down the big ideas for the day along with any centers or assessment materials I might need.<br /><br /><b>Your Turn! </b><br />I have put together planning templates for your year, months or quarters and weeks. I am not promising this work is easy. And I am certainly not implying that this work will go quickly. But, I promise, if you dedicate 2-3 hours now to complete step 1 and step 2, you will thank yourself later because your weekly and daily planning will <i>fly</i>. If you are shorter on time, an hour spent on step 1 and maybe the first quarter or the first few months of step 2 will still set you up for success. You are building your road map and will save yourself frustrating and unnecessary detours later. <a href="https://drive.google.com/file/d/0B7gi7ieV54Vhd18wLVhBM1BMcmM/view?usp=sharing">Click HERE to grab your free planning templates. </a><br /><br /><b>And last but not least... </b><br />I have a $10 TPT gift card to give out so you can kick your back to school off in the right direction! There are a number of ways you can enter in the Rafflecopter below. I will chose a winner on Monday night so that you are able to get your shop on in the sale Tuesday. </div><div><a class="rcptr" data-raflid="14333f291" data-template="" data-theme="classic" href="http://www.rafflecopter.com/rafl/display/14333f291/" id="rcwidget_ghae4ay5" rel="nofollow">Click HERE to Enter the Giveaway</a><script src="https://widget-prime.rafflecopter.com/launch.js"></script><br /><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-exywJNL0qIk/V55i9VeERCI/AAAAAAAAC5s/I_eE60c-hTYVFrYga_IPdq6stmJbWVoZACLcB/s1600/13680026_10208169538568654_1913472043498685975_o.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="201" src="https://4.bp.blogspot.com/-exywJNL0qIk/V55i9VeERCI/AAAAAAAAC5s/I_eE60c-hTYVFrYga_IPdq6stmJbWVoZACLcB/s320/13680026_10208169538568654_1913472043498685975_o.jpg" width="320" /></a></div><div class="" style="clear: both; text-align: center;">Many of my favorite math bloggers are linking up to bring you additional back to school tips, tricks, freebies and even a few more chances to win TPT gift cards. </div> <br /><div style="text-align: center;"><img class="nopin" height="50" src="https://dl.dropboxusercontent.com/u/324927842/TheMathSpot/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com5tag:blogger.com,1999:blog-6278540421686629919.post-16983291992993265072016-07-07T06:58:00.000-04:002017-03-04T15:30:51.099-05:00Balancing The Equation Blog Hop: Chapter 2 <div class="separator" style="clear: both; text-align: center;"><i><span style="font-size: x-small;">I was sent a copy of the book "Balancing the Equation" for the purpose of review. An affiliate link has also been included in this post. </span></i></div><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-WYyhvD-BBuU/V3UqvHQtSkI/AAAAAAAAC4Q/pt50o5ur_ggBICdIsysYzmlcs_6Wy8FKwCLcB/s1600/Balancing%2BThe%2BEquation.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="228" src="https://2.bp.blogspot.com/-WYyhvD-BBuU/V3UqvHQtSkI/AAAAAAAAC4Q/pt50o5ur_ggBICdIsysYzmlcs_6Wy8FKwCLcB/s400/Balancing%2BThe%2BEquation.jpg" width="400" /></a></div><div class="MsoNormal"><br /></div><div class="MsoNormal"><a href="https://www.amazon.com/gp/product/1936763680/ref=as_li_tl?ie=UTF8&camp=1789&creative=9325&creativeASIN=1936763680&linkCode=as2&tag=the0c319-20&linkId=8f79ba22143632fcb062cac07e822132" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="200" src="//ws-na.amazon-adsystem.com/widgets/q?_encoding=UTF8&MarketPlace=US&ASIN=1936763680&ServiceVersion=20070822&ID=AsinImage&WS=1&Format=_SL110_&tag=the0c319-20" width="130" /></a><img alt="" border="0" height="1" src="//ir-na.amazon-adsystem.com/e/ir?t=the0c319-20&l=am2&o=1&a=1936763680" style="border: none !important; margin: 0px !important;" width="1" /><span style="font-family: inherit;">Thank you for joining in the “Balancing the Equation” book study. I am joining in a group of dedicated math bloggers to take a closer look at this text. My opinion? You want a copy for yourself. I consider myself to be quite well versed in understanding math education, where it falls relative to the current political arena (and yes, math education is very tied up in politics!) but this book has given me a more robust understanding of the very specific twists and turns that have taken place that bring us to the space in math education that we occupy today. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">If you are just joining in the book study, you may want to go back (links at the bottom of this post) to begin the study from the beginning. Each day a new blogger has added their chapter to the conversation so, depending on the date you are joining us, you may want to grab a coffee (or your beverage of choice), sit back, and hop on through. Alright, enough chit chat, on to Chapter 2! <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Throughout the text the authors continue to revisit the pendulum swinging between developing conceptual understanding and developing procedural skills. Chapter 2 of <i>Balancing the Equation </i>seeks to examine the historical context behind these pendulum swings. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Upon first glance, it seems like there is a simple answer and that balance needs to be struck between conceptual understanding and procedural skills. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Major spoiler: Yes, balance is the answer. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">In fact, it’s the name of the book J And yet, that is not how math education is handled in all districts, buildings and classrooms. So the question then becomes <b>why? </b><o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;">The answer, my friends, lies in a historical context that is littered with political tension and decisions being informed by groups of people who have little to no background in child development. Not shocked by this answer? Neither was I. And yet I found this chapter to be wildly interesting to read in terms of understanding how entire generations could be effected so profoundly by a pendulum swing in one direction or another. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Kanhold and Larson argue that, initially, math education was not a mandate in our country in terms of childhood education. As a need arose for citizens to know basic math, basic math skills were introduced in classrooms. And these skills were taught rotely. This makes sense. Students needed to know how to “do specific things” with numbers and they were taught how to perform those specific skills, provided with examples and then allowed to practice those skills. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">As time went on and as the scope of mathematical skills broadened, theories on more effective math teaching practices began to emerge. A man by the name of Warren Colburn introduced the idea that perhaps if students were given math tasks which allowed them to develop an understanding of math concepts that they may be able to more easily and effectively understand mathematics and apply their understandings. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">It was only 4 short years before Colburn’s theories received significant backlash. The reasoning? Parents would rather that math be taught the “old fashioned way”. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Really? <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">Parent preference is not grounded in research on how students learn best and yet the tides turned back to the “old fashioned” approach. To be fair here, Larson and Kanhold do point out that the preference of parents is often grounded in their perception of the current math teaching approach and that perception is often derived from what is seen in curriculum or how curriculum is interpreted and implemented by individual teachers. This constant tug between educational theory and </span></div><div class="MsoNormal"><span style="font-family: inherit;">implementation is another common theme throughout the history of math education.</span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">This same pattern continued in turn with the pendulum swinging back and forth every few decades (the text goes into detail here outlining a number of shifts and the forces at work). Sometimes the swing happened at the hand of research as it did in the 1950s and 1960 when “New Math” was introduced. Sometimes, the swing occurred as a result of parents or legislators. Legislation introducing state testing for students at a variety of grade levels was the cause of one such swing. Despite a 2008 study commissioned by the national government that stated that math education must include a balance of conceptual understanding and procedural fluency, a drive towards procedural skills occurred when NCLB (No Child Left Behind) was introduced. High stakes testing quickly drove education at the classroom level to focus on a more narrow set of standards that would be tested at a given grade level. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal"><span style="font-family: inherit;">CCLS (Common Core Learning Standards) have now been introduced and embedded within the standards themselves you will find very specific language that aims to convey the balance required for highly effective math instruction. But, if we have learned anything in chapter 2 of this book, we know that an attempt to shift the landscape of math education will surely be met with backlash and a pull back to what is traditional. Continue on the blog hop to continue the book study with a review of chapter 3 to see how the most current shift is playing out. <o:p></o:p></span></div><div class="MsoNormal"><span style="font-family: inherit;"><br /></span></div><div class="MsoNormal">If you would like to keep up with this blog hop... and why wouldn't you :) Check out the line up below!<br /><br /><ul style="background-color: white; border: 0px; color: #222222; font-family: Verdana, Arial, Tahoma; font-size: 14.4px; line-height: 21.6px; list-style: square inside; margin: 0px 0px 20px 1.5em; outline: 0px; padding: 0px; vertical-align: baseline;"><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><a href="http://www.kidsmathteacher.com/2016/07/balancing-equation-book-study-blog-hop.html" style="border: 0px; color: #555555; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; text-decoration: none; vertical-align: baseline;">Table of Contents, About the Authors, and Introduction</a></li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;"><a href="http://wp.me/p37wzq-xo" style="border: 0px; color: #555555; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;" target="_blank">Evil Math Wizard — Chapter 1</a>: Why Mathematics Education Needs to Improve</li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">7/7/16 <a href="http://www.k5mathspot.com/2016/07/balancing-equation-blog-hop-chapter-2.html" style="border: 0px; color: #555555; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;" target="_blank">The Math Spot — Chapter 2</a>: A Brief History of Mathematics Education</li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">7/8/16 <a href="http://www.theresearchbasedclassroom.com/2016/07/balancing-equation.html" style="border: 0px; color: #555555; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;" target="_blank">The Research Based Classroom — Chapter 3</a>: The Common Core Mathematics Debate</li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">7/9/16 Math Coach’s Corner — First half Chapter 4: The Equilibrium Position and Effective Mathematics Instruction</li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">7/9/16 LIVE WEBINAR with Matthew Larson (<a href="http://%28http//k-5mathacademy.com/balancing-the-equation)" style="border: 0px; color: #555555; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">register here</a>)</li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">7/10/16 The Recovering Traditionalist — Second half Chapter 4: The Equilibrium Position and Effective Mathematics Instruction</li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">7/11/16 <a href="http://www.guided-math-adventures.com/?p=1824" style="border: 0px; color: #555555; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;" target="_blank">Guided Math Adventures — Chapter 5</a>: How to Help Your Child Learn Mathematics</li><li style="border: 0px; font-family: inherit; font-size: 14.4px; font-style: inherit; font-weight: inherit; margin: 0px; outline: 0px; padding: 0px; vertical-align: baseline;">7/12/16 Kids Math Teacher — Epilogue, Appendix, and Recap</li></ul></div><div class="separator" style="clear: both; text-align: center;"></div><br /><div style="text-align: center;"><img class="nopin" height="50" src="https://dl.dropboxusercontent.com/u/324927842/TheMathSpot/signoff.png" width="150" /></div><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=c0c9055c1f" class="validate" id="mc-embedded-subscribe-form" method="post" name="mc-embedded-subscribe-form" novalidate="" target="_blank"><div id="mc_embed_signup_scroll"><h4 style="text-align: center;"> Research Based Strategies </h4><h4 style="text-align: center;">& Exclusive Freebies In Your Inbox</h4><div style="text-align: center;"><input class="" id="mce-FNAME" name="FNAME" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="text" value="First Name" /></div></div><div style="text-align: center;"><div style="text-align: center;"><input class="required email" id="mce-EMAIL" name="EMAIL" onblur="if(this.value=='')this.value=this.defaultValue;" onfocus="if(this.value==this.defaultValue)this.value='';" type="email" value="Email Address" /></div></div><div class="clear" id="mce-responses"><div class="response" id="mce-error-response" style="display: none;"></div><div class="response" id="mce-success-response" style="display: none;"></div><!-- real people should not fill this in and expect good things - do not remove this or risk form bot signups--><div aria-hidden="true" style="left: -5000px; position: absolute;"><div style="text-align: center;"><input name="b_4b7c3426552111fed54c8f2d7_c0c9055c1f" tabindex="-1" type="text" value="" /></div></div><div style="text-align: center;"><input class="button" id="mc-embedded-subscribe;" name="subscribe" type="submit" value="Sign Me Up!" /></div></div></form></div><!--End mc_embed_signup-->The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0