Understanding Comparision Word Problems

Why is it that comparison word problems are particularly difficult for elementary school kiddos? There are a number of challenges present in these specific types of word problems that we don't have to contend with when we are doing more simple problem types such as put together, take apart, add to, or take from word problems! 

**Hint: Read through to the bottom of this post- there's a HUGE freebie waiting for you!** 

Challenge #1:

There are some specific vocabulary challenges around comparison word problems. A bit of vocabulary work with your students pointing out these challenges will help them go far when solving comparison word problems.
Show a picture of two different pencils, each with a measurement written alongside. Ask your students "How much longer is pencil B than pencil A?"  I will bet that a large percentage of your class will tell you the length of pencil B rather than how much longer pencil B is! 

Now try to ask this question in a different way. First, ask the students 
How long is pencil A?
How long is pencil B?
...I'll bet that all of your students get these answers correct! 

Now, ask your students which pencil is longer. Once they identify that pencil B is longer, ask How much longer

More of your students will get this question correct than the original question because you have walked them through the difference between long and longer. 

Now, vocabulary challenges aside, this math concept is difficult and I am still sure that there are a group of students who we still need to help.

Challenge #2:

The second main difficulty with this type of problem is that a comparison is an abstract amount that doesn't necessarily exist! The number of centimeters fewer is not something our students can see or touch, it's an abstract concept. 
Lay centimeter cubes along the length of pencil A and pencil B. Ask your students again, 
How long is pencil A?
How long is pencil B? 
Which pencil is longer? 
How much longer? 

This time, with the concrete modeling, many more of your students will be able to see how much longer pencil B is then pencil A!

If you need to make this even more clear, push the two pencils together so that the centimeter cubes are right alongside one another. Your students can see pencil A and pencil B are the same up to 5 cm but pencil B has one, two, three more centimeters than pencil A. Pencil B is 3 centimeters longer!

At this point many or most of your students will be very successful with the comparison language, however... You don't want them to be at the concrete stage forever so, what can we do? We can use a part part whole diagram to represent the same thinking. You can see below that when we draw a pencil A as a box and pencil B as a box we can use a dotted line to show the difference between the two. At this point, the diagram makes it very clear that pencil A and the difference put together are the same as the length as pencil B. Your students can write a missing addend equation to solve for the part that shows the difference in the length of the pencils.
This will still be a difficult concept for your students! Don't expect them to learn the skill overnight. Most of your students will need extensive hands-on practice linked to the part-part-whole diagram many, many times before this will be an internalized understanding.

 At this point, your students have done vocabulary work and have expanded their math understanding and they will be much more successful with comparison word problems! 

I have created an entirely FREE set of comparison activities for you! These activities will bring your students through from concrete comparison with cubes to representative modeling and even includes comparison word problems! 

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Related Resources: 

Pencil Clip Art by Creative Clips 

Adding Fractions with Unlike Denominators

How do you add fractions with unlike denominators? Find common denominators of course! You know, however, that I'm a big proponent of students understanding why they are using a strategy. If our students don't know why they are using a strategy so many of our learners don't know when to apply that strategy. They become students who have "strong rote skills but struggle with application".

An incredibly effective method of helping our students to understand strategies is to give them concrete and representative experiences to make sense of an abstract strategy. I want to share 2 representative strategies for adding fractions with unlike denominators.

Representative Methods of Adding Fractions 

Begin by representing each fraction. Your students can see that the two fractions can not be combined as is because the pieces of each fraction are not the same size. 

Ask your students what they might be able to do to make it so that the fraction pieces are the same size.

 Your students may suggest creating two like arrays. In the example to the right, You can easily draw eighths over top of the fourths and fourths over top of the eighths to create two arrays each with 32 pieces.

Once the two arrays are equal it is easy to add the two fractions together because you have found a common denominator of 32! 

You may be thinking to yourself that 32nds aren't necessarily easy numbers to work with and that the method is less than efficient. That's ok! Once your students understand that they can combine fractions by making two arrays with the same number of pieces they can explore ways to become more efficient by finding the least common denominator! 

Consider the graphic to the left. Post this picture for your students and ask if they could add these two fractions together using those models. Do your students recognize that both models show eighths even though the shape of the eighths is different in each picture? 

Related Resources:

Research Based Strategies 

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