Equality: Part 2

In my last post, I suggested a way to build on what students already know to introduce the equal sign.

Let’s say you followed that plan and your students are now proficient in using symbols to describe a quantity as equal to, greater than or less than another quantity.

What do you do next?

Use numbers. Instead of writing “length of line A=length of line B,” have them measure the quantities and then write number sentences like 8=8 or 7<8.

After enough practice using numbers, give them tasks that invite them to use different operations to create equal quantities.

What might this look like?

Let’s start with addition.

Give students Unifix cubes and ask them to complete the following tasks:

  • If you have a bar that is 5 units long and you join it to a bar that is 10 units long, is it more than, less than or the same as a bar that is 15 units long?  Show with the cubes how you know.
  • Is a 13 unit long bar more than, less than or the same as a 6 unit long bar joined to 7 unit long bar?  How do you know?
  • Is a 2 unit long bar joined to a 6 unit long bar, more than, less than or the same as a 3 unit bar joined to a 5 unit bar?  How do you know?

In order to solve the tasks, students can join the bars, determine how large the new bar they created is and compare it to the other bar.   Also include tasks where the quantities aren’t equal.

You can then introduce notation to help students record these relationships in various ways:

  • 5 +10=15
  • 13=6+7
  • 2 +6 =3+5.

Eventually you can move to doing these tasks without the cubes and using larger numbers.

Can you see how this plan could be adapted for other operations?

The key is providing situations that allow students to create different quantities, explore what is the same about them and then record this relationship using mathematical notations.  It builds on what students know and formalizes it using symbols.

I’d love to hear what happens when you try it with students.

This plan was based on the Measure Up research that Barb Dougherty and her colleagues conducted in Hawaii.  It’s a really interesting project and I’ll write more about it some other time, but if you want to know more now, check out the research:

Dougherty, B. (2007). Measure up: A quantitative view of early algebra. Algebra in the early grades, 389-412.

Dougherty, B. J., & Venenciano, L. C. (2007). Measure Up for Understanding: Reflect and Discuss. Teaching Children Mathematics13(9), 452-456.