tag:blogger.com,1999:blog-62785404216866299192017-07-23T06:48:43.238-04:00The 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.comBlogger49125tag: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-6352606178303063692017-03-15T13:12:00.003-04:002017-03-15T13:14:18.437-04:00Have 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="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=f4c68b73ce" 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;">Sign Up For Your Free Learning Reflection</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-70142903174573214292017-03-12T09:08:00.002-04:002017-03-15T13:28:44.972-04:00The 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="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_11.html" target="_blank">Post 4: The Math Spot Compares Fractions- Abstract Thinking</a><br /><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=fc5ae4fa12" 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;">Sign Up For Your Free Learning Reflection</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-8592995040810950332017-03-09T21:03:00.000-05:002017-06-22T07:07:45.829-04:00The 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="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_11.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-90192688153861209782017-03-05T15:05:00.000-05:002017-06-22T07:08:39.441-04:00The 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="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_11.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-10067689854841084962017-03-02T21:18:00.002-05:002017-06-22T07:09:05.677-04:00The 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="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions_11.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-87815121207364555912016-08-16T12:57:00.001-04:002017-03-04T15:27:03.946-05:00Subitizing: Why, When and What Is It? <div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-HGgZdme_gcc/V7NFHd6sCxI/AAAAAAAAC60/izz3rd5IbRUF_tvUML3tkgCa6-CeIJTBgCLcB/s1600/Subitizing%2BBlog%2BHeader.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="330" src="https://3.bp.blogspot.com/-HGgZdme_gcc/V7NFHd6sCxI/AAAAAAAAC60/izz3rd5IbRUF_tvUML3tkgCa6-CeIJTBgCLcB/s640/Subitizing%2BBlog%2BHeader.jpg" width="640" /></a></div><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 /><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-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.com0tag:blogger.com,1999:blog-6278540421686629919.post-38804691746866311552016-06-05T17:26:00.002-04:002017-03-04T15:30:46.813-05:00Balancing the Equation<div style="text-align: center;"><span style="font-size: x-small;"><i>I was sent a copy of the book "Balancing the Equation" for the purpose of this review. An affiliate link has also been included.</i></span></div><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-nSAWnVt8dRI/V1SY14KZoYI/AAAAAAAAC3k/zDXWACQr2Cwnp6-qDnMYkWEUS0PXyqXEQCLcB/s1600/Balancing%2BThe%2BEquation%2BBook.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-nSAWnVt8dRI/V1SY14KZoYI/AAAAAAAAC3k/zDXWACQr2Cwnp6-qDnMYkWEUS0PXyqXEQCLcB/s400/Balancing%2BThe%2BEquation%2BBook.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div>There has long been a debate in mathematics instruction. Should students be required to develop <b>procedures, skills, memorization and fluency </b>or should <b>conceptual understanding, reasoning and student centered discovery </b>rule the day? <u>Balancing the Equation: A Guide to School Mathematics for Educators and Parents </u> sets out to settle the debate.<br /><br />As the title may suggest, this is not an either or argument and the two are certainly not mutually exclusive. Mathematics instruction should strive to find a balance between these two ideas. Not only are conceptual understanding and procedures and skills both valid points of view but they each support one another. Swing the pendulum too far in one direction or the other and, ultimately, students will not gain the level of proficiency needed to demonstrate mastery of math content at their grade level and beyond.<br /><br />This text is broken into two sections. The first section is focused around the history of math instruction, standards and political the initiatives that come into play. A strong argument is made for the need for a balanced approach to math instruction based on this history as the authors examine what happens when the pendulum swings too dramatically to one side of the "equation".<br /><br />If you have ever found yourself frustrated by current math instruction, assessment or standards or if you have found yourself engaged in a heated conversation with someone who is passionately against "Common Core" this section of the text is absolutely worth a read. You will be presented with facts and research based ideas which inform the current direction of math instruction. This section of the text is billed as being "for educators" with the caveat that parents would also benefit from the content. I, however, believe that every frustrated, confused or overwhelmed parent would benefit greatly from reading this section of the book. The argument for unified standards and balanced instruction- including not only the "how" of procedural fluency but also the "why" and "when" of conceptual understanding and application- is so clearly and concisely laid out that it would put any mind at ease.<br /><br />The second section of the text is billed as being "for parents". In this section of the text, parents will learn what they should expect to see in terms of their child's math instruction and what to advocate for if instruction seems to be lacking. This section of the text is also applicable to educators as it is a powerful self-assessment in terms of the instruction you are offering in your own classroom.<br /><br />Overall, the clear, concise, research based, practical nature of this text make it a worthwhile read for parents and educators alike. I would highly recommend Balancing the Equation as an addition to your summer reading list!<br /><br />The text is available on Amazon.com if you are interested in reading more:<br /><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=c25a03612a71bebb35cde53b92457aab">Balancing the Equation: A Guide to School Mathematics for Educators and Parents (Contexts for Effective Student Learning in the Common Core)</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" /><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-84119020610539562292016-06-01T20:47:00.000-04:002017-03-04T15:30:43.490-05:00My Most Frequently Asked Question<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-XwzCunah914/V0-B6itQrsI/AAAAAAAAC2c/wGrqPOh_fe0StPHnzwBvpJ-RxZW-p-plgCLcB/s1600/IMG_3182.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/-XwzCunah914/V0-B6itQrsI/AAAAAAAAC2c/wGrqPOh_fe0StPHnzwBvpJ-RxZW-p-plgCLcB/s320/IMG_3182.PNG" width="319" /></a></div><br />In my math room there is one question, far and away, that I ask more often than any other. I'm not talking about a question or prompt such as "Could you show that to me in another way?" or "Why do you think that works?" those questions could be used in any classroom at any grade level in any subject.<br /><br />I'm talking about a very specific MATH content question. I have been asking my first graders from the first day of school and I will continue up to the last.<br /><br />It's quite simple. I draw a plus sign (or a minus sign) on the board and I ask "What is this? What do we say when we read it? WHAT DOES IT DO TO NUMBERS?" I ask this question day in and day out. I ask it when we are learning a new strategy. I ask it when we are practicing an old strategy. I ask this question to death. And there is an important reason why.<br /><br /><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; text-align: center; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;"><b>When teaching a new strategy for addition students need to know, above all else, that they are learning a strategy for efficiently putting parts together. </b></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;"><br /></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;">How many times have you had a student who answers a question such as 8 + __ = 10 with the number 18? You ask if the answer makes sense and the student says “Yes! I started with the big number and counted 8 more and got 18!” The students have learned a strategy or procedure and think that the plus sign indicates that they should perform that procedure when really the plus sign means they need to find an efficient way to put parts together- in the case of the above example they don't have both <i>parts </i>so they must do something to find out what they can put together with 8 to make 10.</span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><br /></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"></div><br /><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;">By starting a new strategy with a quick conversation about the meaning of a plus sign and anchoring back to this idea throughout the development of the strategy students learn that they are developing a method for parts together- not a procedure for what to do when they see a + sign. </span></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-67201610360646783212016-05-23T19:01:00.001-04:002017-03-04T15:30:39.443-05:00Adding and Subtracting Ten Mentally<div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-h__Egsrnf3Y/V0OJl32QAvI/AAAAAAAAC1o/BFqKqDMQZ_UOEg07JFzlEK0AQGVdDEO8wCLcB/s1600/IMG_3132.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-h__Egsrnf3Y/V0OJl32QAvI/AAAAAAAAC1o/BFqKqDMQZ_UOEg07JFzlEK0AQGVdDEO8wCLcB/s320/IMG_3132.PNG" width="320" /></a></div><br />For some first grade students, adding and subtracting ten to a given number comes naturally. They know that 23 has 2 tens and 3 ones so, naturally adding another ten gives the answer of 33! For other students, this skill isn't so intuitive. You may have taught the skill using a place value chart and, while 80% of the class picked it up using that representation, there are still a few students who will tell you, with confidence, that 23 + 10 = 24.<br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; text-align: right;"><tbody><tr><td style="text-align: center;"><a href="https://1.bp.blogspot.com/-n2hCf_Bk7hQ/V0OCVPJ0XsI/AAAAAAAAC1M/zvLaVL82fDUfNoPNTMCVQlu2ViQXpgb3wCLcB/s1600/Place%2BValue%2BChart.png" imageanchor="1" style="clear: right; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><img border="0" height="171" src="https://1.bp.blogspot.com/-n2hCf_Bk7hQ/V0OCVPJ0XsI/AAAAAAAAC1M/zvLaVL82fDUfNoPNTMCVQlu2ViQXpgb3wCLcB/s400/Place%2BValue%2BChart.png" width="400" /></a></td></tr><tr><td class="tr-caption" style="text-align: center;">A place value chart model for adding ten. </td></tr></tbody></table>I have written at length about the <a href="http://www.k5mathspot.com/2015/12/when-they-still-dont-get-it.html">C-R-A framework of teaching math</a>. When I think about the skill and understanding required to add or subtract ten and I think about using a place value chart to promote understanding I see that the reason why some kids are falling out behind the rest of the class is that <i>a place value chart alone is a representative (verging on abstract) model. </i>What about the students who needed to take this skill back to the concrete level?<br /><br /> A strong method for teaching addition and subtraction of 10 is to start with the concrete, link it to a representation and then move to a more abstract model. Read on to see how each of these steps play out in the classroom:<br /><br /><div style="text-align: center;"><b><u><span style="font-size: large;">Concrete</span></u></b></div><a href="https://3.bp.blogspot.com/-QglTmDXPgAA/V0OLKuDt2QI/AAAAAAAAC10/Usim8BfZc3IKzD38v12pw4cv0ZRbwj0AgCLcB/s1600/Base%2BTen%2BBlocks.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-QglTmDXPgAA/V0OLKuDt2QI/AAAAAAAAC10/Usim8BfZc3IKzD38v12pw4cv0ZRbwj0AgCLcB/s320/Base%2BTen%2BBlocks.png" width="224" /></a>I have a few preferred concrete models for teaching this skill. First, I love using <a href="http://www.k5mathspot.com/2014/08/i-love-base-ten-blocks.html">base ten blocks </a>. They are an accessible tool in the classroom, students likely have already gained familiarity with the tool and they adequately demonstrate the difference in magnitude between a one and a ten. When teaching this skill I ask students to build a number with ten sticks and ones. I then ask them to tell me about what block they would need to have to show ten more. We may have some discussion about <i>It is so important that students can articulate that when they added a ten the ones remained the same but the tens changed. </i>If they aren't noticing this pattern, help them to notice and then test the pattern. Without this understanding, they will have a very hard time generalizing the rule!<br /><br />how we could use either a ten stick or ten ones but ultimately, push them to use a ten stick as it is the more efficient way of counting. Students then add a ten stick to their original number and write an equation to demonstrate what happened. For example: 23 + 10 = 33. I then ask students to tell me what happened to the tens and what happened to the ones using place value language. <br /><br />You can also use $10 bills and $1 bills to model this relationship. If your students are <i style="font-weight: bold;">very </i>strong in coin names and values you can also use dimes an pennies to demonstrate this relationship. If your students aren't super solid in their understanding of coins, I would save that representation for another day. I am a huge proponent of <a href="http://www.k5mathspot.com/2014/03/money-money-money.html">using dimes and pennies to teach and reinforce place value concepts</a>, however, I am a bigger proponent of staying focused and I don't want to muddy the waters in terms of the focus of this lesson.<br /><br /><div style="text-align: center;"><b><u><span style="font-size: large;">Representative</span></u></b></div>As mentioned above, a strong model for representing the number of tens and ones before and after <br /><a href="https://4.bp.blogspot.com/-n2hCf_Bk7hQ/V0OCVPJ0XsI/AAAAAAAAC1Q/ISZHJenx_VYHP3v7Oloj_HxBrqtYKW6WQCKgB/s1600/Place%2BValue%2BChart.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="140" src="https://4.bp.blogspot.com/-n2hCf_Bk7hQ/V0OCVPJ0XsI/AAAAAAAAC1Q/ISZHJenx_VYHP3v7Oloj_HxBrqtYKW6WQCKgB/s320/Place%2BValue%2BChart.png" style="cursor: move;" width="320" /></a>ten is added or subtracted would be a place value chart. If your students are demonstrating that they are able to use concrete materials but that the place value chart is a bit of a reach for them at this time, consider having your students draw a representation of ten sticks and ones. They can do this parallel to their use of base ten blocks and then drop the manipulatives once they are able to perform the skill AND talk about the problem using base ten language using their drawing alone.<br /><br /><div style="text-align: center;"><b><u><span style="font-size: large;">Abstract</span></u></b></div>Just because you are teaching using the CRA framework <i>does not </i>mean that you are teaching each of these three skills in isolation. As I mentioned in the "Representative" paragraph, the best way into the representative model is to ask students to do so parallel to the "Concrete" model so that students can make explicit connections. The abstract model of this work being, for example, a number sentence, can be linked to both the representative and the abstract. I ask students on day 1 of this skill when they are building with blocks to write a number sentence that matches what they have built with blocks. As students are drawing place value drawings I also ask them to write an equation that represents what they have done.<br /><br />And always, always, always, promote math talk and math language when your students are working. Their ability to articulate their thinking and the math processes will help to solidify their understanding and build connections to future learning!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://www.teacherspayteachers.com/Product/Adding-and-Subtracting-Ten-Math-Intervention-2558891" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/-xn8oDItgX74/V0OHPtWOQ1I/AAAAAAAAC1c/07U_3ikyevU1ehLpUj9i_aRr6837YatkQCLcB/s320/Add%2Band%2BSubtract%2B10%2BMentally.jpg" width="240" /></a></div>If you are looking for a math resource that will better guide you and your students through the development of this skill (1.NBT.5) from concrete to mental math please take a look at my <a href="https://www.teacherspayteachers.com/Product/Adding-and-Subtracting-Ten-Math-Intervention-2558891">5 Day Focus: 10 More and 10 Less</a>. This resource includes a pre and post assessment, 5 days of <i>detailed </i>lesson plans, independent, hands on activities for each day and tickets out the door for each lesson.<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-88038991922779072582016-05-14T11:40:00.000-04:002017-03-04T15:30:35.890-05:00Money, Money, Money!! <div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-FYECIT5YaT8/VzcVPTM5W1I/AAAAAAAAIfk/xMF1hTVzXSIk57luoczJxJtEAO-1QEfSwCLcB/s1600/IMG_3078.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-FYECIT5YaT8/VzcVPTM5W1I/AAAAAAAAIfk/xMF1hTVzXSIk57luoczJxJtEAO-1QEfSwCLcB/s400/IMG_3078.PNG" width="400" /></a></div><br />Teaching first graders about money is a layered skill. Students need to know the coin names, coin values, how to count by ones, fives, tens and even groups of 25 and they need to know how to organize and approach their coins so that their counting can be done in a manageable way. Over the years I have developed some tried and true favorite methods for teaching money that pull each of these skills together.<br /><br /><b>#1: Coin Names and Values</b><br /><a href="https://drive.google.com/file/d/0B6fiBv3-SZg-TGo3dVRybmZNbjg/view?usp=sharing" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img alt="Money Worksheet, Money Poster" border="0" height="400" src="https://3.bp.blogspot.com/-SmECxm9-Z80/VzeNWa3MbBI/AAAAAAAAC0Y/62KNkHWw9Tov_AvEHV9zjz7JCqA_Wa6KgCLcB/s400/Slide1.JPG" title="Money Poster" width="300" /></a>To teach coin names I start by reading a coin poem with my students which then serves as an anchor chart for our time spent focusing on money. I don't know the origin of the poem I use so I am not going to post it here but I am sure you have heard it.<br /><br /><i>"Penny, Penny, easily spent. Copper brown and worth one cent! Nickel, nickel...." </i><br /><br />Following the poem, I give each student a tray with 4 compartments as seen pictured above and I ask them to sort a large pile of coins. The only rules are that they need to say the name of each coin as they sort and they can't sort two of the same coin in a row.<br /><br />Following their sort, my students make a "mini money poster" coloring in the heads side of each coin, and writing in the name and value of each coin. If you would like a copy of the mini-money poster I use you can grab it (free) by clicking the picture.<br /><br /><br /><br />This song is a fun (and addictive) way to practice coin names and values as well. For students who are working on coin names it's a fun way to see and say the names of the coins. For others it's a quick and easy way to remember some of the more common coin combinations.<br /><br /><b>#2: Counting Coins</b><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-1PX_LKkf05Q/VzeRJYpmAoI/AAAAAAAAC0k/ygL06laxY-oOjvWCkZoSNthIHIuUo5HwACLcB/s1600/IMG_3095%2B%25281%2529.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img alt="Counting Dimes and Pennies" border="0" height="320" src="https://1.bp.blogspot.com/-1PX_LKkf05Q/VzeRJYpmAoI/AAAAAAAAC0k/ygL06laxY-oOjvWCkZoSNthIHIuUo5HwACLcB/s320/IMG_3095%2B%25281%2529.PNG" title="Counting Money" width="320" /></a></div>When I first introduce counting coins, I jump head first into combinations of dimes and pennies. The reason for this is that my students have just come out of a place value unit in which they were focused on counting and recognizing the value of groups of tens and ones. After the "Mini Money Poster" above, I hold up a ten stick and ask students "Which of the coins is most like this ten stick?" I then repeat with the ones block. I have students build numbers out of the ten sticks and ones and then place dimes on the ten sticks and pennies on the ones. After a few examples they quickly understand how to count the value of the coins.<br /><br /><br /><a href="http://3.bp.blogspot.com/-6T3Q6Yhi9Z0/Uy2tMQj82HI/AAAAAAAAB18/BjM1KUYRnyM/s1600/counting+coins.jpg" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="400" src="https://3.bp.blogspot.com/-6T3Q6Yhi9Z0/Uy2tMQj82HI/AAAAAAAAB18/BjM1KUYRnyM/s1600/counting+coins.jpg" width="400" /></a><br />This counting by coins poster is another reference for my students to use. The kids love getting to be the teacher and pointing to the chart as the class practices together! When putting this poster together my students are easily able to count by pennies and dimes but they are generally not as fluent with nickels and they certainly aren't fluent with quarters. We create this poster by coloring in the multiples of 5 and 25 on a hundreds chart to find the count sequence for five.<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.com4tag:blogger.com,1999:blog-6278540421686629919.post-7007284330362085522016-05-02T18:54:00.002-04:002017-03-04T15:30:32.238-05:00I Wasn't A Great Math StudentI wasn't a great math student in school. In fact, to say I was a "good" math student after maybe 5th grade would be generous. In the beginning of middle school we had to take a test. The test would determine whether we would be on the "regular" math track or the accelerated course of study for the remainder of our school years. ONE TEST. I took the test and was borderline. They had me take the test again. I was still borderline. They held me back... to be safe.<br /><div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-8Q4E7N79Y84/VyfaQbl9k-I/AAAAAAAACzk/BGIPHcPBu9gYAsIla8r9TgLKQb3CGIu2wCLcB/s1600/image2.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/-8Q4E7N79Y84/VyfaQbl9k-I/AAAAAAAACzk/BGIPHcPBu9gYAsIla8r9TgLKQb3CGIu2wCLcB/s320/image2.PNG" width="320" /></a></div><br />You see, the problem was that I never <i>understood </i>the math that we were learning and while I do have quite a bit of grit and perseverance, grit doesn't matter when math is just about memorizing rules. And I never was great at memorization. All of the grit in the world doesn't matter when you are faced with 5.25 x 10 and you know that you either have to <i>move the decimal right... or left? And is it one spot because there is one zero or two because ten has two digits?? </i><br /><br />And so, I struggled. Math rule, after math rule, after math rule was piled on high. As soon as the math regents (the NYS HS test you need to take to get your diploma) was over I quit the traditional sequence of math courses. I got out of my "regular" track and went in a completely different direction taking an AP Statistics course. I LOVED it! You see, in statistics, we erased years and years of <i>math rules </i>and talked about what we wanted to know from numbers and what types of math we could apply to learn those things. We stopped talking about RULES and started THINKING about numbers.<br /><br />Fast forward a few years past college and into my first few years of teaching math. When the Common Core math standards came out there was so much talk about teaching <i>why </i>and <i>how </i>math works. Reading through the standards from first through fifth grade it became so apparent to me that if I could help students to master these standards, they would be better equipped than I had ever been as a student. We would teach them why and how and the challenge for students would then move away from a challenge of memorization to a challenge of persevering through problems requiring them to apply their knowledge.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-LDFmP8SdS0A/VyfaPorRIkI/AAAAAAAACzg/U6gBXce7hFQ29bu871zTUKrJsTP0s3HrACKgB/s1600/image1.PNG" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://4.bp.blogspot.com/-LDFmP8SdS0A/VyfaPorRIkI/AAAAAAAACzg/U6gBXce7hFQ29bu871zTUKrJsTP0s3HrACKgB/s320/image1.PNG" width="320" /></a></div>And so I looked to research based strategies. I went to a number of PD sessions and read a number of texts and, so often, I could see the <i>value </i>in the high level research based "conversations" that I was supposed to be having with my students but, day to day, there was no great guide on how to make "Research Based" REAL for the classroom. How to take these highly theoretical ideas and to make them work in a day to day math setting. How to use this research to make SMART instructional decisions. How to use this research to DIFFERENTIATE... really differentiate in a meaningful way! How to incorporate this research when creating assessments and when planning remediation.<br /><br />And that, my friends, is where The Math Spot came in. It is my mission through this blog to make "research based" real, manageable, and attainable for every elementary teacher. Even on top of the 2,000 other tasks you need to complete in a day :) Because if you know why and how students learn math you can help them understand why and how math works.<br /><br />And then, 15 years from now, our students won't be writing a blog post about how <i>they </i>weren't a great math student.<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.com2tag:blogger.com,1999:blog-6278540421686629919.post-19645472570376156052016-03-05T11:20:00.002-05:002017-03-04T15:30:27.783-05:00Understanding the Count Sequence<div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><div class="separator" style="clear: both; text-align: center;"><a href="https://4.bp.blogspot.com/-uaC0NXpDuU0/Vzux25iy1nI/AAAAAAAAC04/OaFVUKdc0toLt__MJTfvwF-P5HcAytinwCLcB/s1600/IMG_3103%2B%25281%2529.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="318" src="https://4.bp.blogspot.com/-uaC0NXpDuU0/Vzux25iy1nI/AAAAAAAAC04/OaFVUKdc0toLt__MJTfvwF-P5HcAytinwCLcB/s400/IMG_3103%2B%25281%2529.PNG" width="400" /></a></div><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Your kindergartners can count to 10, or 20 or even 100. But you ask them what number comes after 8 and your most struggling learners start counting back at one to figure it out. They <b>know </b>the count sequence but they don't <b>understand </b>the count sequence. </span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Understanding the count sequence means more than being able to recite the numbers 1-10. The standards state that an understanding of the count sequence includes being able to count to 100 by ones and tens, counting forward from a given number and being able to write the numbers 0-20. </span><br /><span style="font-family: inherit;"><br /></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><br /></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;">One of my favorite kindergarten activities includes numbers 1-10 on simple bunting flags. I hang up a line of yarn about eight feet long across two file cabinets and allow my students to hang the numbers up on the line. By being strategic, you can level and differentiate this simple activity to allow your students to develop confidence and fluency where they are and slowly push them to a deeper level of understanding of numbers 1-10.</span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Level 1</span><span style="font-family: inherit;">* Lay out all flags on the ground and put ten clips on the line. Students then “find” the numbers 1-10 and clip them up onto the line in order. <span style="font-weight: bold;">Tell Me More: </span>Some students may be able to do this activity independently right away. Others may struggle and require prompts. One easy yet powerful prompt is to say <span style="font-style: italic;">“Could you use counting to see which number comes next?” </span>This allows students to relate the count sequence that they are familiar with to the numbers 1-10.<br /> </span><span style="font-family: inherit;"> </span><span style="font-family: inherit;">Level 2 </span><span style="font-family: inherit;">* Lay out all flags on the ground and keep a small bucket of pins near by. Students will then find the numbers 1-10 and clip them onto the line in order. <span style="font-weight: bold;">Why? </span>The change between this activity and the previous is subtle yet powerful. Students need to begin to think about the space between the numbers. If students leave gaps that are too large I ask them questions such as <span style="font-style: italic;">“What are you saving space for between the 3 and the 4? Is there a number that goes in there? Let’s put those numbers closer together so that all of the numbers are spaced out evenly.”</span></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><span style="font-family: inherit;"><span style="font-style: italic;"><br /></span></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><a href="https://1.bp.blogspot.com/-Z0pE9b2t7ec/VtsGs5_3SLI/AAAAAAAACsI/NRsaNqp_aS4/s1600/3%2Bor%2B8.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="296" src="https://1.bp.blogspot.com/-Z0pE9b2t7ec/VtsGs5_3SLI/AAAAAAAACsI/NRsaNqp_aS4/s400/3%2Bor%2B8.png" width="400" /></a><span style="font-family: inherit;"><span style="font-style: italic;"><br /></span></span><span style="font-family: inherit;">Level 3</span><span style="font-family: inherit;">*Put the bunting flags into a pile. Hang the 10 clothes pins on the line. Students would ten take the top flag off of the pile and determine where on the line it would go. Students can “use counting” to find the placement of each flag. <span style="font-weight: bold;">What should I look for? </span>Look for students who are starting to recognize where on the number line a number would be placed. Do they recognize that the 10 goes at the end without counting? If they have the number 5, for example, already hung on the number line do they begin counting from that number instead of beginning at one each time? If not, you can prompt and model what that might look like. Noticing students demonstrating these skills will let you know that your students are ready to move to a more difficult version of this skill.<br /> </span><span style="font-family: inherit;"><br /></span><span style="font-family: inherit;">Level 4</span><span style="font-family: inherit;">The most difficult version of this activity includes putting the flags into a pile and the pins in a small container. Students take the top flag off of the pile and grab a pin to put it on the line where they think it goes. <span style="font-weight: bold;">How does this help? </span>Numbers like 1 and 10 will be easier for your students to place. In thinking about the placement of the other numbers, your students will need to think about which numbers are closer to 1 or ten, which numbers they need to ‘save space for’ and about the spacing between numbers. This also leads to a natural conversation about comparing numbers.</span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;"><a href="https://3.bp.blogspot.com/-Kw_feHJ4lc0/VtsHK29Ft1I/AAAAAAAACsM/2sMjlK1aE0o/s1600/Missing%2B7.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="296" src="https://3.bp.blogspot.com/-Kw_feHJ4lc0/VtsHK29Ft1I/AAAAAAAACsM/2sMjlK1aE0o/s400/Missing%2B7.png" width="400" /></a><span style="font-family: inherit;"><br /></span></div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;">*Variation* </div><div style="direction: ltr; margin-bottom: 0pt; margin-left: 0in; margin-top: 0pt; unicode-bidi: embed; word-break: normal;">As a QUICK fluency activity, have your students close their eyes. Remove one of the banners. As soon as they open their eyes they should yell out... as quickly as they can... which flag is missing. Ask students to explain what they saw that let them know, so quickly, which number was missing. </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-13034089329754836122016-01-04T18:03:00.001-05:002017-03-04T15:30:22.597-05:00Double Ten Frames<div class="separator" style="clear: both; text-align: center;"><a href="https://3.bp.blogspot.com/-hAosQtnesYk/V1LWGiZLAUI/AAAAAAAAC2w/4cz8Y0RVKXcDswEiBGzzmnJ3ZrE8fRaowCLcB/s1600/IMG_3199.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="482" src="https://3.bp.blogspot.com/-hAosQtnesYk/V1LWGiZLAUI/AAAAAAAAC2w/4cz8Y0RVKXcDswEiBGzzmnJ3ZrE8fRaowCLcB/s640/IMG_3199.PNG" width="640" /></a></div><br />Double ten frames are such a strong visual representation of teen numbers. The best part? It is a versatile tool as well! You can use the double ten frame to develop a deep understanding of teen number but also to build an understanding of the benchmarks of five and ten, addition and subtraction strategies.<br /><br /><div style="text-align: center;"><b>#1: Discovering Teen Numbers</b></div><br />I have had my kids building teen numbers for over a month now. It's a staple in their math centers. I have them focus on different attributes of teen numbers each time they do the activity but it is really about understanding the power of the "ten" in teen numbers.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-mbgB_Y60w_A/Vor0h4sZRDI/AAAAAAAACqw/sWiecrtVzJ8/s1600/build.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="320" src="https://1.bp.blogspot.com/-mbgB_Y60w_A/Vor0h4sZRDI/AAAAAAAACqw/sWiecrtVzJ8/s320/build.png" width="244" /></a></div><br />Early on, I give my students a number and just ask them to put out that many counters on their double ten frame being sure to fill up the first ten frame before moving on to the next.<br /><br />After they build a number I ask about what they see and they tell me about the ten part and the "other" part. In building a significant amount of numbers my students begin to make discoveries about how the ten stays the same and how the teen number "tells" us how many ones there will be after the ten. <b>I have found that laying this foundation leads to a very strong understanding of larger 2 digit numbers in future months. </b><br /><b><br /></b><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: right; text-align: left;"><tbody><tr><td style="text-align: center;"><span style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><a href="https://www.teacherspayteachers.com/Product/Teen-Number-Centers-1682260"><img border="0" height="240" src="https://3.bp.blogspot.com/-4ABCYEkUKHA/Vor0yr_zEiI/AAAAAAAACq8/iIeBLnn_qA8/s320/Slide5.JPG" width="320" /></a></span></td></tr><tr><td class="tr-caption" style="text-align: center;"><a href="https://www.teacherspayteachers.com/Product/Teen-Number-Centers-1682260">Change the graphics and the kids think <br />it is a new and interesting game! </a></td></tr></tbody></table>Sometimes, I require that my students write a number bond with a ten part to describe their teen number. Sometimes, I ask that they write a number sentence that matches the teen number. Sometimes, I ask for both. Sometimes I even ask questions such as <i>If I had a full ten frame and then 6 more, what number would that make?</i><br /><br /><br /><br /><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; text-align: right;"><tbody><tr><td style="text-align: center;"><span style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><a href="https://www.teacherspayteachers.com/Product/First-Grade-Winter-Math-Centers-2277284"><img border="0" height="320" src="https://4.bp.blogspot.com/-vFJQIH_r1bc/Vor0g0z3caI/AAAAAAAACqs/qZ6YO-xsTyo/s320/teen.png" width="225" /></a></span></td></tr><tr><td class="tr-caption" style="text-align: center;"><a href="https://www.teacherspayteachers.com/Product/First-Grade-Winter-Math-Centers-2277284">Who says building teen numbers needs to be<br />boring? My students use blocks,<br />counters, even cotton balls to keep things interesting! </a></td></tr></tbody></table><br /><br /><br /><br /> I keep them on their toes and my students rise to the occasion with careful language and begin to make generalizations about teen numbers.<br /><br /><br /><br /><br /><br /><br /><br /><br /><div style="text-align: center;"><b>#2: Adding 3 Numbers Together</b></div><div style="text-align: center;"><b><br /></b></div><div style="text-align: left;">My students could tell you what addition means in their sleep. They clap their hands and cry out "Addition puts number parts together!". But throw a third number into the equation and they will look at you like <i>you </i>have a third head! After getting over the initial shock of an addition sentence with three addends, students can use a ten frame to practice a strategy such as looking for numbers that make a ten and then adding the third part on. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">No, not all addition equations with 3 addends <i>have </i>a pair of numbers that makes ten, but some do. And that is where you are going to start! </div><table cellpadding="0" cellspacing="0" class="tr-caption-container" style="float: left; margin-right: 1em; text-align: left;"><tbody><tr><td style="text-align: center;"><span style="clear: left; margin-bottom: 1em; margin-left: auto; margin-right: auto;"><a href="https://www.teacherspayteachers.com/Product/First-Grade-Winter-Math-Centers-2277284"><img border="0" height="320" src="https://2.bp.blogspot.com/-_trTdx_FfqQ/Vor3ukNp61I/AAAAAAAACrI/GXTonpq_c6Y/s320/3num.png" width="220" /></a></span></td></tr><tr><td class="tr-caption" style="text-align: center;"><a href="https://www.teacherspayteachers.com/Product/First-Grade-Winter-Math-Centers-2277284"><br /></a></td></tr></tbody></table><br />You can see how using a double ten frame allows students to experiment and figure out which number partners make a ten and then add in the final addend. If your students know their partners of ten this activity will be a breeze. If they are not yet fluent, this concrete activity will help them to understand how three addends come together as well as reinforcing partners of ten.<br /><br /><br /><br /><div style="text-align: center;"><br /></div><div style="text-align: center;"><b><br /></b></div><div style="text-align: center;"><b><br /></b></div><div style="text-align: center;"><a href="http://3.bp.blogspot.com/-1JA3TCnBtQg/Vor4NCWVIcI/AAAAAAAACrQ/ilBksiZQ9jA/s1600/Concrete.png" imageanchor="1" style="clear: right; display: inline !important; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-1JA3TCnBtQg/Vor4NCWVIcI/AAAAAAAACrQ/ilBksiZQ9jA/s320/Concrete.png" width="246" /></a><b>#3: Make a Ten Addition Strategy</b></div><div style="text-align: center;"><b><br /></b></div>Double ten frames can also be useful for illuminating the "make a ten" strategy for adding two numbers together. You can see in this example adding a 9 and a 5 together how the double ten frame helps to illuminate the related 10 + fact.<br /><br />A strong foundation in building teen numbers on the double ten frame as outlined in activity #1 will lead to success in this type of strategy.<br /><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.com0tag:blogger.com,1999:blog-6278540421686629919.post-53328397959336551002016-01-01T13:16:00.003-05:002017-03-04T15:30:17.937-05:00Comparison Conversations<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-GZLczbV2TLY/V1LfjzZj2_I/AAAAAAAAC3A/msVQbOAEG8QX4WV-KQ2bH0jCY_cUxRNYgCLcB/s1600/IMG_3201.PNG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="https://1.bp.blogspot.com/-GZLczbV2TLY/V1LfjzZj2_I/AAAAAAAAC3A/msVQbOAEG8QX4WV-KQ2bH0jCY_cUxRNYgCLcB/s640/IMG_3201.PNG" width="637" /></a></div><br />My kindergartners are beginning their formal exploration of number comparisons. Sure, we have been noticing that some numbers are larger than others and that some piles have more while others have fewer but they are ready to formally begin solidifying these understandings. The <i>most important </i>things I am keeping in the forefront of my mind when introducing comparison are to <b>keep it concrete </b>and <b>emphasize comparison conversation. </b><br /><b><br /></b>My students need to talk. A lot. It is one thing to be able to indicate which number or set of objects is more or fewer and another to be able to articulate it. And so, as my students explore numbers and counting we will talk, and talk, and talk some more to solidify the associated language!<br /><br /><u style="font-weight: bold;">Activity # 1: Counting & Comparing Sets</u><br /><u style="font-weight: bold;"><br /></u><i>Keep It Concrete: </i>Students will take a comparison card with two numbers. Students count out blocks or counters to represent each amount on the card and then match up the blocks to see which number is more and which number is less.<br /><i><br /></i><i>Comparison Conversation: </i>I like to put the numbers in a context on the cards. For example on the animal cards below the number corresponds to a type of animal. In this way, the students have more opportunities to talk about the comparison in a variety of ways. <i><a href="https://www.teacherspayteachers.com/Product/Comparison-Cards-Kindergarten-2280015">These cards are free in my TPT store if you would like to grab them :) </a></i><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-4Hx3eOblxLo/Voa-x4GH9eI/AAAAAAAACpQ/bJnX06g43Vg/s1600/compare2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="238" src="https://2.bp.blogspot.com/-4Hx3eOblxLo/Voa-x4GH9eI/AAAAAAAACpQ/bJnX06g43Vg/s320/compare2.png" width="320" /></a></div><br />When looking at this comparison card, students would lay out 2 counters to represent the 2 pandas and 6 counters to represent the 6 flamingos. Encourage the students to talk about and compare both the animals and the numbers in isolation.<br /><br /><i>"There are fewer pandas than flamingos."</i><br /><i>"Two is less than six." </i><br /><i>"There are more flamingos than pandas." </i><br /><i>"Six is more than two." </i><br /><br /><br /><br /><u style="font-weight: bold;">Activity # 2: Sorting Sets</u><br /><u style="font-weight: bold;"><br /></u><i>Keep It Concrete: </i>Students will have a target set of blocks with a given number such as 7. Have a student take a small handful or scoop out of a bin of blocks. They will count their set and then sort the amount under headings that say "More" "Less" or "Same" in relation to the target set. This activity can be used in a whole group setting taking turns scooping and counting and then doing a quick "Think, Pair, Share" before sorting, in a small group moving around the circle and giving each student a turn to scoop and count or as a partner or individual activity. You can also modify this activity by giving students cards with dots or pictures on them that they can count and sort. I would do that only after the students are becoming proficient with concrete materials because in a still picture students would not have the opportunity to move and match up the blocks to compare.<br /><i><br /></i><i>Comparison Conversation: </i>Every time a student sorts and places a card have them say aloud a full sentence explaining why they are sorting in the way that they are. "My scoop is more because 9 is more than 7." "My scoop is the same because they both have 7" or "My scoop is less because 2 is fewer than 7".<br /><br />Don't forget to grab your <a href="https://www.teacherspayteachers.com/Product/Comparison-Cards-Kindergarten-2280015">free comparison cards</a> and happy comparing! <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-39236107649937015822015-12-31T17:03:00.001-05:002017-03-04T15:30:12.980-05:00The Strange Way I Teach Fluency for "5" Facts<div class="separator" style="clear: both; text-align: center;"><a href="https://2.bp.blogspot.com/-2NoVXwh1et8/V1Ml4t8hjkI/AAAAAAAAC3Q/7mIgpxUOfNUr8-e48M92cz7UMu5UwCABACLcB/s1600/design.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="327" src="https://2.bp.blogspot.com/-2NoVXwh1et8/V1Ml4t8hjkI/AAAAAAAAC3Q/7mIgpxUOfNUr8-e48M92cz7UMu5UwCABACLcB/s640/design.png" width="640" /></a></div><br /><br />Last summer, along with a few other math bloggers, I did a <a href="http://k5mathspot.blogspot.com/2015/06/chapter-1-2-teaching-student-centered.html">book study on the book </a><i><a href="http://k5mathspot.blogspot.com/2015/06/chapter-1-2-teaching-student-centered.html">Teaching Student Centered Mathematics</a> </i>By John Van de Walle, LouAnn Lovin, Karen Karp, and Jennifer Bay-Williams. Chapter 8 revolved around developing number sense and outlined a series of important number relationships. One of these number relationships includes benchmarks of 5 and 10. When we return from break I will be focusing on benchmarks of 5 and fact fluency with 5+ facts... but maybe not in the way you would expect.<br /><div style="text-align: center;"><span style="font-size: large;"><br /></span></div><div style="text-align: center;"><span style="font-size: large;">Take a <b><u>quick</u></b> look at this picture: </span></div><br /><div class="separator" style="clear: both; text-align: center;"></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-0naJQefKEcY/VoWhhOuNmRI/AAAAAAAACow/msYfEtWFzQc/s1600/snow2.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="245" src="https://3.bp.blogspot.com/-0naJQefKEcY/VoWhhOuNmRI/AAAAAAAACow/msYfEtWFzQc/s320/snow2.png" width="320" /></a></div><br /><div style="text-align: left;">How many snowflakes did you see? Chances are you knew that there were 7 without having to count each individual snowflake. You saw a group of 5 and 2 more and knew that there were 7. My students are able to do this as well. They see the picture and say "7" and can even articulate "I saw a 5 and then I know 2 more is 7" or "I saw a 5 and a 2 and that makes 7". </div><div style="text-align: left;"><br /></div><div style="text-align: left;">But for my tier 3 students if I were to show them the equation 5 + 2 they would NOT automatically say 7. <i>Until this next week that is :) </i></div><div style="text-align: left;"><i><br /></i></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-HG23Ckae4WQ/VoWhhJRZm7I/AAAAAAAACo4/Z9qKA6datUo/s1600/snow1.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="237" src="https://3.bp.blogspot.com/-HG23Ckae4WQ/VoWhhJRZm7I/AAAAAAAACo4/Z9qKA6datUo/s320/snow1.png" width="320" /></a></div><div style="text-align: left;">Sometimes all it takes is a little push to make what is intuitive in one setting -visual dot patterns- a piece of knowledge that can be generalized into other settings. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">In order to make the 5+ facts automatic for my students I will go back to flashing dot pictures (or snowflake pictures :) ) asking how many and having explicit conversations about what they knew, how they knew it and what this might look like in other forms. </div><br /><a href="http://2.bp.blogspot.com/-slUTFPDJJxc/VoWhhGr9qyI/AAAAAAAACo0/pR1-tctnq30/s1600/snow3.png" imageanchor="1" style="clear: left; float: left; margin-bottom: 1em; margin-right: 1em;"><img border="0" height="244" src="https://2.bp.blogspot.com/-slUTFPDJJxc/VoWhhGr9qyI/AAAAAAAACo0/pR1-tctnq30/s320/snow3.png" width="320" /></a><br /><br />Questions and prompts will include:<br /><i>*How many snowflakes do you see? </i><br /><i>*How do you know? </i><br /><i>*What parts do you see in the picture?</i><br /><i>*Could you write a number bond that shows those parts? </i><br /><i>*What would the total be? </i><br /><i>*What equation matches the snowflakes and your number bond? </i><br /><i><br /></i>Another variation of this activity will include flashing a dot picture and then asking students to reproduce the picture using bingo dabbers or Q-tips and paint. They could then create a number bond and number sentence that match what they know.<br /><br />Finally, I would end a lesson by throwing in some 5+ flashcards <b>mixed with other facts that they know</b> such as 1+ or 0+ facts. I <b>do </b>want my students to become automatic in their facts and these activities will allow them to be automatic, however, I <b>never </b>want my students to overgeneralize a new understanding.<a href="http://k5mathspot.blogspot.com/2015/12/rekenrek-one-moreone-less.html"> If you want to read more on how I strategically use flash cards you can read this post here. </a><br /><br />**<i>The snowflake 5 facts activity highlighted in this post is a part of my winter themed math centers for first grade students <a href="https://www.teacherspayteachers.com/Product/First-Grade-Winter-Math-Centers-2277284">found HERE</a>**</i><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.com1tag:blogger.com,1999:blog-6278540421686629919.post-45080281187779357722015-12-30T19:54:00.002-05:002016-08-12T17:14:56.546-04:00Independence in Math Class<div style="text-align: center;"><div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-skqpv0XraNw/V648Q4ffZ0I/AAAAAAAAC6Q/xRFDekZfHMIzR8uggZDNOzCgAjF8Ih3XwCLcB/s1600/Independence%2Bin%2BMath.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="330" src="https://1.bp.blogspot.com/-skqpv0XraNw/V648Q4ffZ0I/AAAAAAAAC6Q/xRFDekZfHMIzR8uggZDNOzCgAjF8Ih3XwCLcB/s640/Independence%2Bin%2BMath.jpg" width="640" /></a></div><b><i><br /></i></b><b><i>"But they could do it at the side table 10 minutes ago... why can't they do it now???"</i></b></div><div style="text-align: center;"><b><i><br /></i></b></div><div style="text-align: left;">Sound familiar? You have a handful of students who you have been pulling aside for additional help in math. You are even <a href="http://k5mathspot.blogspot.com/2015/12/when-they-still-dont-get-it.html">stepping back and thinking C-R-A</a> and in the small group setting your kids are really starting to come along. But then you send them off on their own and they are not able to perform the way you would have expected. </div><div style="text-align: left;"><br /></div><div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-3gbCV5pktjU/VoR74JyHk1I/AAAAAAAACoI/dkHp1v9_6Zw/s1600/independent.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="72" src="https://1.bp.blogspot.com/-3gbCV5pktjU/VoR74JyHk1I/AAAAAAAACoI/dkHp1v9_6Zw/s640/independent.png" width="640" /></a></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><br /></div><div style="text-align: left;">For students to be independent in their application of mathematics they need frequent opportunities to practice math independently. If I "hold their hands" during math lessons they will then require that hand holding to perform on a math task. </div><div style="text-align: left;"><br /></div><div style="text-align: left;">I have found that strategically planning my math lesson with opportunities for independence in mind, I can combat this problem. I work with my students in a pull out intervention group for 30 minutes each day but if you are a classroom teacher you can easily use this format in less face to face time. I break my lessons down into 3 main parts so that students are able to receive new instruction but then practice skills- both new and old- independently. </div><div style="text-align: left;"><br /></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-2W2sUXaplXQ/VoR8d7J3CzI/AAAAAAAACoQ/cY5TF6aap1c/s1600/1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="45" src="https://4.bp.blogspot.com/-2W2sUXaplXQ/VoR8d7J3CzI/AAAAAAAACoQ/cY5TF6aap1c/s400/1.png" width="400" /></a></div><div style="text-align: left;"><br /></div><div style="text-align: left;"><b><u>Concept Development:</u></b> We begin our lesson with some sort of concept development. Concept development lasts only about 10 minutes at most. Concept development may be a word problem, an introduction to a math tool, a puzzle or whatever experience I can give my students that will help them to construct meaning around our lesson goal. I put a particular focus on math language and math talk during concept development. Students need to be talking to one another using math language during this portion of the lesson. </div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-4Fr_J75_uKU/VoR8eFrYAqI/AAAAAAAACoU/LukQdShgOtU/s1600/2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="36" src="https://2.bp.blogspot.com/-4Fr_J75_uKU/VoR8eFrYAqI/AAAAAAAACoU/LukQdShgOtU/s320/2.png" width="320" /></a></div><b><u>Concept Practice:</u></b> Concept development then spills into concept practice. This will take the remainder of the first 15 minutes. This may include some whiteboard practice or a short problem set. This time allows me to get my first glimpse of students who may have misconceptions or difficulties when performing independently.<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-cVHoYgIUvn4/VoR8eAdtJ2I/AAAAAAAACoY/TLfg_doZRMI/s1600/3.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="36" src="https://4.bp.blogspot.com/-cVHoYgIUvn4/VoR8eAdtJ2I/AAAAAAAACoY/TLfg_doZRMI/s320/3.png" width="320" /></a></div><b><u>Independent Practice:</u></b> The entire second half of my math lesson is devoted to independent practice. If you are in the classroom and don't have a full 30 minutes to devote to tier 2 or tier 3 groups, this is where you would send your students off to practice on their own and grab another group. I run independent practice much like a classroom teacher would run math centers. I include centers that review concepts and skills my students have previously practiced, centers which allow the students more time to explore math patterns using hands on materials and centers which will allow students to make generalizations or break over generalizations.<br /><br /><i>Later this week I will do a post about how I strategically choose the independent practice stations because in order to get the most bang for your buck you want to choose centers that decompose the skills your students are working on.</i><br /><div><br /></div><div><br /></div><br /><a href="http://www.bloglovin.com/en/blog/4046831" title="Follow Polka Dots & Teaching Tots on Bloglovin"><img alt="Follow on Bloglovin" border="0" src="http://www.bloglovin.com/widget/bilder/en/widget.gif?id=4046831" /></a>The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com1tag:blogger.com,1999:blog-6278540421686629919.post-51429749485186696492015-12-29T12:45:00.000-05:002017-03-08T20:02:31.538-05:00When They STILL Don't Get It<div class="separator" style="clear: both; text-align: center;"><a href="https://1.bp.blogspot.com/-IWznF148TlA/WL9eESFszJI/AAAAAAAAEbY/7Y3M5XNn1Kg2v8BqsF2deHkiFwjMZ_l2QCEw/s1600/CRA%2B%25281%2529.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="400" src="https://1.bp.blogspot.com/-IWznF148TlA/WL9eESFszJI/AAAAAAAAEbY/7Y3M5XNn1Kg2v8BqsF2deHkiFwjMZ_l2QCEw/s400/CRA%2B%25281%2529.png" width="400" /></a></div><div class="separator" style="clear: both; text-align: center;"><br /></div><a href="http://3.bp.blogspot.com/-kLEAL5Jo0ys/VoK7RLdG-HI/AAAAAAAACng/JuGhq1Q-Loo/s1600/Infographic-CRA.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/-kLEAL5Jo0ys/VoK7RLdG-HI/AAAAAAAACng/JuGhq1Q-Loo/s320/Infographic-CRA.jpg" width="320" /></a><br /><br />You're teaching a new math strategy and many of your students are finding success... but then there is that handful. The handful of kids that are continuing to struggle and just aren't picking it up like the rest. It happens in all classrooms whether it be the make a ten strategy for addition in 1st grade, <a href="http://www.k5mathspot.com/2017/03/the-math-spot-compares-fractions-four.html" target="_blank">comparing fractions in 3rd and 4th grade</a>, multiplying two digit by two digit numbers in 4th or dividing decimals in 5th.<br /><br />The problem is universal and, to some degree, the solution is universal as well. When a student <br />is struggling you need to think C-R-A.<br /><br /><br /><a href="http://2.bp.blogspot.com/-ubWneEPf-88/VoK9GsIqNtI/AAAAAAAACno/_iC9YpPpIoQ/s1600/Abstract.png" imageanchor="1" style="clear: left; display: inline !important; float: left; margin-bottom: 1em; margin-right: 1em; text-align: center;"><img border="0" height="320" src="https://2.bp.blogspot.com/-ubWneEPf-88/VoK9GsIqNtI/AAAAAAAACno/_iC9YpPpIoQ/s320/Abstract.png" width="244" /></a> Take a look, for example, at this make a ten strategy found in many textbooks. The strategy is quite straight forward and requires that the students break up the 5 through the use of a number bond. For many students, this strategy is just fine and because they are flexible with numbers it will quickly turn into a mental strategy. However, if you are working with this type of strategy and you have a group of students who just aren't "getting it", take a step back and ask yourself if you have spent enough time working in concrete or representative methods.<br /><br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-LBI7Gm__eWY/VoK9Gk6fAZI/AAAAAAAACns/HAWuDNIBNXk/s1600/Concrete.png" imageanchor="1" style="clear: right; float: right; margin-bottom: 1em; margin-left: 1em;"><img border="0" height="320" src="https://3.bp.blogspot.com/-LBI7Gm__eWY/VoK9Gk6fAZI/AAAAAAAACns/HAWuDNIBNXk/s320/Concrete.png" width="246" /></a></div> If I wanted to teach the above make a ten strategy with concrete materials, I might get out blocks and have the students build towers out of the numbers they are adding together and then place these blocks into a double ten frame. The difference between this concrete strategy and the above abstract strategy is that students can physically break apart the tower of 5 and can easily see how a ten is made with the other part of the 5 left over. Students don't need to fluently know their decompositions of numbers to ten to practice the strategy when you use concrete materials. You don't want to be bogged down in these concrete strategies. When your students have mastered this strategy MOVE THEM ON. In fact, link the concrete strategy to a representation (shown below) and/or the abstract form of the strategy so that they can easily transition to more efficient methods.<br /><div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-m48g83FkAeY/VoK9GpD-6DI/AAAAAAAACnw/9tRK2mGpJHs/s1600/Representative.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/-m48g83FkAeY/VoK9GpD-6DI/AAAAAAAACnw/9tRK2mGpJHs/s320/Representative.png" width="241" /></a></div><br />Once your students are proficient using concrete materials, transition them over to a drawing or representative model. You can see how the model to the left is very similar to the concrete model above, however, the students don't have the benefit of physically breaking the 5 apart and seeing how it makes a complete ten. The difference is subtle as an adult but powerful for your students.<br /><br /><b>So that's it. No more pulling your hair out, no more frustrated students. If they are not "getting it" just take a step back and think C-R-A. Meet your students where they are and you will be so pleased with the results! </b><br /><br /><!-- Begin MailChimp Signup Form --> <br /><div id="mailchimp"><form action="//k5mathspot.us13.list-manage.com/subscribe/post?u=4b7c3426552111fed54c8f2d7&id=fc5ae4fa12" 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;"> Please send me the Fraction C-R-A Cheat Sheet!</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--> <br /><div style="text-align: center;"><img class="nopin" height="50" src="https://dl.dropboxusercontent.com/u/324927842/TheMathSpot/signoff.png" width="150" /></div>The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-52429403640467921022015-12-04T17:00:00.003-05:002015-12-22T10:05:19.121-05:00Rekenrek One More/One Less<div class="_209g _2vxa" data-block="true" data-offset-key="fed23-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span data-offset-key="fed23-0-0" style="font-family: inherit; font-size: large;">If you are not using a Rekenrek in your classroom you NEED to go and purchase one or make one ASAP. It is such an excellent tool when it comes to building number sense and fact fluency with your students. </span></div><div class="_209g _2vxa" data-block="true" data-offset-key="fed23-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span data-offset-key="fed23-0-0" style="font-family: inherit; font-size: large;"><br /></span></div><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-YLCbyH32Zd4/VmIIyFeXqSI/AAAAAAAACmM/CkOqHvCu2Mg/s1600/Rekenrek.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><span style="font-family: inherit; font-size: large;"><img border="0" height="258" src="http://3.bp.blogspot.com/-YLCbyH32Zd4/VmIIyFeXqSI/AAAAAAAACmM/CkOqHvCu2Mg/s320/Rekenrek.png" width="320" /></span></a></div><div class="separator" style="clear: both; text-align: center;"><span style="font-family: inherit; font-size: large;"><br /></span></div><div class="_209g _2vxa" data-block="true" data-offset-key="fed23-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span style="font-family: inherit; font-size: large;"><span data-offset-key="fed23-0-0">Currently, my students are working on their understanding of the "one more" and "one less" relationship. I would show a number of beads on my wooden Rekenrek that my students would then quickly report back to me. I would then slowly slide another bead over saying "one more is..." and the students would report the new number. We did this as a warm up fluency for a number of days. When they were quite automatic, I was ready to link this understanding to the equations. </span>On whiteboards, students would write equations related to what they saw and counted on the Rekenrek. </span></div><div class="_209g _2vxa" data-block="true" data-offset-key="fed23-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span style="font-family: inherit; font-size: large;"><br /></span></div><div class="_209g _2vxa" data-block="true" data-offset-key="fed23-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span style="font-family: inherit; font-size: large;">When the students were quite proficient with this skill, I moved this practice over to their math centers for mastery. I created this "Rekenrek board" out of a page protector, pipe cleaner and a few beads so that my students could practice this skill on their own. I can easily slip in a page with number bonds, addition equations, missing parts, etc. and they can use the Rekenrek tool to help them visualize the results. </span></div><div class="_209g _2vxa" data-block="true" data-offset-key="82ke0-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span data-offset-key="82ke0-0-0" style="font-family: inherit; font-size: large;"><br data-text="true" /></span></div><div class="_209g _2vxa" data-block="true" data-offset-key="df8vh-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span data-offset-key="df8vh-0-0" style="font-family: inherit; font-size: large;">This is NOT about finding a procedure for adding or subtracting one. This is about giving a visual cue to understand what it means to add or subtract one. As they are working on their centers, I am quick to check in with the student who is "playing" this game and I push them and ask questions which will allow them to internalize the skill as they demonstrate proficiency. These are questions like "You built 7, can you think about what one more will be without even pushing over another bead?" or "Look at your next problem, can you picture what 5 might look like in your head? Think about what one more will look like. What is 5 + 1? Now, build it with beads to check your thinking". </span></div><div class="_209g _2vxa" data-block="true" data-offset-key="df8vh-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span data-offset-key="df8vh-0-0" style="font-family: inherit; font-size: large;"><br /></span></div><div class="_209g _2vxa" data-block="true" data-offset-key="df8vh-0-0" style="background-color: white; color: #141823; direction: ltr; line-height: 18px; position: relative; white-space: pre-wrap;"><span style="font-family: inherit; font-size: large;">After my students were quickly able to answer these facts I made what may be considered a controversial move. I put the facts onto <b>flash cards</b>. GASP! But hear me out here. How many times have your students learned a new skill and then over generalized that skill to #AllTheNumbers!!! I know this will be a problem so I have just decided to be proactive about it. Each time my students learn a new set of facts I write them onto flashcards in a colored Sharpie marker. All of their +1 facts are on flashcards written in blue marker. My students also know their 0+ and +0 facts as well as their 10+ facts. The 0 facts are all written in a magenta marker and the 10+ facts are written in green marker. My students have one math center they can go to where they can individually or "race" a friend to practice their facts. My sole purpose in doing that activity is for them to realize that their new understanding about how to "think one more" can NOT be applied to all equations. Just the +1 equations. When they go through the flashcards and see the different colors their brain is triggered that this is a different type of fact and they need to change their thinking. <i>I have had MINIMAL over generalization errors since employing this technique. </i>Not to mention that my students feel SUPER successful when going through a stack of facts that they know. They love it. And I love that they know 30 of their addition facts without "memorizing" a thing. </span></div><span style="font-size: large;"><span style="font-family: inherit;"><br /></span></span><br /><a href="http://www.bloglovin.com/en/blog/4046831" title="Follow Polka Dots & Teaching Tots on Bloglovin"><span style="font-family: inherit; font-size: large;"><img alt="Follow on Bloglovin" border="0" src="http://www.bloglovin.com/widget/bilder/en/widget.gif?id=4046831" /></span></a><br />The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com0tag:blogger.com,1999:blog-6278540421686629919.post-78548468292430814682015-08-29T20:45:00.000-04:002015-08-29T22:57:49.865-04:00Buzzworthy Back to School Ideas<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-rZcUpPKM6lE/VeIVbeH2kmI/AAAAAAAAChk/s2t6d_snvNY/s1600/11059978_10100670978545487_3714611679297649384_n.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="319" src="http://2.bp.blogspot.com/-rZcUpPKM6lE/VeIVbeH2kmI/AAAAAAAAChk/s2t6d_snvNY/s320/11059978_10100670978545487_3714611679297649384_n.jpg" width="320" /></a></div><br />Back to school time is quickly approaching. One more week until we have students in my building! I would guess that you are either on a similar timeline, that is, if you haven't started back already! If you have found me through the blog hop, welcome! If you are here on your own, you are in for a treat! You have stumbled upon a blog hop focused on ideas that will help to get your math instruction ready for the school year ahead.<br /><br />Beginning a new year of math is always a bit mind blowing to me. The stark difference between an end of year kindergarten student and a beginning of the year kindergarten student can be SO very drastic. At the end of last year I was helping students to understand the meaning of addition and subtraction in simple word problems, decompose numbers to ten and explore teen numbers as a ten and some ones. Next week I will be helping a new crew to recognize and count to 10! Only two months ago my first graders were able to use multiple strategies to add and subtract with sums to 20 and could draw a simple part part whole to represent a word problem. My new first graders are still grasping the concept of addition.<br /><br />All of this being said, there is certainly a range of understanding and ability even within a class of students at the beginning of the school year. In order to start the year off right, it is important to do some sort of basic assessment that tells you who is where they need to be at the beginning of the school year and who is lacking foundational skills needed to start the year off right.<br /><br />Intervention groups can be started as early as the second week of school. Small groups with a targeted goal will help to get your whole class on track ASAP so that all students can get the most out of grade level instruction. Below, I have included three basic checklists. If any of your students can NOT complete the skills on the check list, you may consider starting some extra small group time with these students. <a href="https://www.teacherspayteachers.com/Product/Primary-Intervention-Checklists-2062413">CLICK HERE</a> for a (free) printable version of these checklists!<br /><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-sKdCAJSQsqY/VeImn-zJlLI/AAAAAAAACh4/cYnuzw3LCdk/s1600/Slide3.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://3.bp.blogspot.com/-sKdCAJSQsqY/VeImn-zJlLI/AAAAAAAACh4/cYnuzw3LCdk/s640/Slide3.JPG" width="480" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-AmFoDTDSqKc/VeImn4iQRLI/AAAAAAAACh0/V2LfNC8fFnY/s1600/Slide4.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://4.bp.blogspot.com/-AmFoDTDSqKc/VeImn4iQRLI/AAAAAAAACh0/V2LfNC8fFnY/s640/Slide4.JPG" width="480" /></a></div><br /><div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-DqcgE3icR78/VeImn5At2BI/AAAAAAAACh8/_tEdXdC168c/s1600/Slide5.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="640" src="http://3.bp.blogspot.com/-DqcgE3icR78/VeImn5At2BI/AAAAAAAACh8/_tEdXdC168c/s640/Slide5.JPG" width="480" /></a></div><br /><br />Small group intervention at the beginning of the year could look different from class to class. Some ideas you might consider would be:<br /><br /><ul><li>Pulling students for 5 minutes in the morning while others are unpacking and doing morning work. </li><li>Pulling a small group at the very end of math to review a concept while other students are completing an independent practice activity. </li><li>Providing a basket of hands on math centers to reinforce the target concept that can be done any time a student finished work early. </li><li>Providing an alternate homework that focuses on the target skill or concept. </li><li>Having a parent volunteer oversee students working on a pre-taught hands on center. **Parent volunteers should not be delivering intervention- the most highly qualified teachers should be working with the students with the most needs. But, after initial instruction parents could oversee independent practice. </li></ul><div>Good luck with your new students this year! The sooner you solidify their foundations, the sooner they will be ready to go with your new and exciting content!<br /><br /><div style="text-align: center;">Head on over to <a href="http://www.guided-math-adventures.com/?p=1417">Adventures in Guided Math</a> to continue the blog hop! <a href="http://www.guided-math-adventures.com/?p=1417"><img src="http://www.guided-math-adventures.com/wp-content/uploads/2015/05/button2.fw_.png" /></a></div></div><div><br /></div><div><br /></div><br /><a href="http://www.bloglovin.com/en/blog/4046831" title="Follow Polka Dots & Teaching Tots on Bloglovin"><img alt="Follow on Bloglovin" border="0" src="http://www.bloglovin.com/widget/bilder/en/widget.gif?id=4046831" /></a>The Math Spothttp://www.blogger.com/profile/09845707996634790905noreply@blogger.com7