# Exploring Comparison Subtraction with a Life-Sized Number Line

### Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.
 I borrowed this photo from Linda at Kinder Doodles!  Be sure to check out this post for great ideas for building number sense!

Comparison subtraction is such a hard concept for kiddos!  Pose the following problem:

Susan has 8 marbles.  Lisa has 3 marbles.  How many more marbles does Susan have than Lisa?

If you asked this question to a classroom of 1st graders, what answer do you think you’d get most often?  Did you say 11?  Trust me, they would!  They might also say 8, because that’s how many Susan has. Way too often take-away subtraction is the only type of subtraction students see in our classrooms.  If you look at standardized tests, though, the type of subtraction most often tested is comparison subtraction.  Why is that?  In Planting the Seeds of Algebra, author Monica Neagoy quotes Bob Moses, founder of a movement called the Algebra Project, regarding subtraction:

“If students are only taught the take-away metaphor, they’ll have trouble in algebra.  In algebraic expressions involving subtraction, such as x2 – x1 or y2 – y1, the take-away metaphor is of no help.  The distance between two points, on the x-axis or y-axis, is the helpful metaphor.”

Now, if you don’t totally understand everything Mr. Moses said in this quote, don’t feel bad.  The point is that when an expert on algebra says we need to teach other structures besides take-away, we should listen. And if if take-away is not the helpful subtraction structure for algebra readiness, then it’s not going to be tested on standardized tests.

I blogged before about a kinesthetic approach to teaching comparison subtraction, and today I want to suggest using a floor-sized number line as an additional representation.  As pictured above, create a floor-sized number line in your classroom.  I’ve seen teachers put them by the door where students line up to leave the classroom and use it as a classroom management technique. Have two students start at 0, and then have one student walk 8 steps and the other walk 3.  The question becomes, “How many steps farther did Student 1 walk than Student 2?”  Or “How many more steps did Student 1 walk than Student 2?”  Or, and this one is important because it’s being used with great regularity on standardized tests, “What is the difference in the steps walked by Student 1 and Student 2?”  Mix up your questions as you repeat the activity, so students become familiar with comparison subtraction in all of its forms!

1. Hi Donna,
I’m a first grade teacher and , boy, I can relate to children struggling with this. My class is made up of almost 60% English learners which makes it even trickier because the language in the word problem are tricky. I have started using one trick that I learned from one of my students after I asked her to explain her thinking. She said ____ needs this many to catch up. For some reason, that made sense to the majority of my class, so that’s one of the phrases I through in there.
Julie [email protected]

• Great suggestion, Julie! Thanks! From the mouth of babes, right?

2. I love the floor size number line! I have used a floor line in a similar way to look at different concepts. The first time I did it I put numbers 1 to 25 (the number of students I had) on the number line on the floor. Then each day every child had to choose a different number from a hat, and for the entire day it was their line number. Because it changed every day they got lots of practice. This year I am working on ordinal numbers (1st, 2nd, 3rd, etc) with my kids, so I wrote these on the floor line. Then we review them each time we line up. For example, I might ask one child which place they are in line at any given time. I love the subtraction idea and plan to give it a try. I love that a basic concept can be modified slightly to cover more topics. The kids generally understand how it works, and the changes keep it interesting. Thanks!

• What great ideas, Karen! Thanks so much for sharing. 🙂

3. I used to find my students really struggled with comparism subtraction even up through second grade. As my teaching practices have changed and I am exposing kids to these ideas at a younger age, it has gotten much better. I think one thing that has helped is when I am passing out materials and I want each kid to have 5 or 10 or something, I always give them less than they need and go back around and ask them how many they have and how many more they need to make 10 (or whatever number).

Tara
The Math Maniac

• That is a GREAT idea, Tara!!