# Exploring, Building, and Discussing Equivalent Expressions

How might we facilitate experiences where learners write expressions in equivalent forms to build foundations for writing and solving equations? How might we concurrently facilitate meaningful mathematical discourse so that students develop confidence and flexibility when using expressions and equations?

Katie England and I recently led a webinar where we explored these important ideas. View the recording from that Bringing Expressions and Equations to Life webinar here.

We chose to start the webinar with a Math Talk—a structure where we (as a community of learners) share some of our mathematical thinking and then make connections between our various approaches.

Here’s the Math Talk prompt that we started with:

We asked participants to work on this task individually for two minutes and to dig deep with their thinking. Participants shared their expressions in the Chat as they thought of them.

Pause: Pause here for two minutes. Set a two-minute timer and engage in this activity yourself. Write as many expressions as you can that are equivalent to 10x + 24.

After two minutes of think time, we stepped back and reviewed our collective body of work in the Chat. Here is a summary of the expressions that were shared in the Chat in just two minutes of think time:

ten times a number and 24

5x + 5x + 30 ‒ 6

2(5x + 12)

6(x + 4) + 4x

5x + 5x + 12 + 12

.5(20x + 48)

(100x + 240)/10

1/2(20x + 48)

24 + 10x

10(x + 2.4)

(20x + 48)/2

10(x + 2) + 4

x + x + x + x + x + x + x + x + x + x + x + 24

7x + 3x + 24

x + 1 + 9x + 23

2(5x + 10) + 4

I received \$24 dollars to walk my first dog and after that I got \$10 an hour for each new dog.

2 + 2 + 6 + 14 + 5x + 5x

A plumber charges a flat fee of \$24, plus \$10 per hour.

‒2(‒5x ‒ 12)

2(x + x + x + x + x) + 24

11x ‒ x + 25 ‒ 1

x + 3x + 6(x + 4)

The product of two times the sum of five times a number and twelve

‒(24 ‒ 10x)

10(x + (12/5))

ten times x added to twenty-four

‒30x + 2 + 50x ‒ 10x + 2*22/2

5x + 5x + 20 + 4

x^lnx ‒ x^lnx + 10x + 62 ‒ (4×3)

2(5x) + 2(12)

2(2x + 3x) + 24

10000000 + 23 + 10x

dy/dx (5x^2 + 24x)

(100x^2 ‒ 576) / (10x ‒ 24), x not 24

10*x^1 + 24*x^0

2.5(4x + 9.6)

log(10^10) x + 24

2x(20 ‒ 15) + 24

4x + 6x + 6(4)

Wow!! What an amazing collective body of thinking!

Where did your mind go while reading this list? Did you start making connections—considering how different expressions in this list are related? Did you notice patterns between some of these expressions? What would you want to do next with this collective body of thinking?

In planning this Math Chat, Katie and I used and then sought to model the use of the 5 Practices for Orchestrating Productive Mathematics Discussions from Margaret (Peg) Smith & Mary Kay Stein.

In preparing for the webinar, we anticipated the responses our learners would share—we were delighted to find that while we anticipated many of the responses our colleagues would share, we certainly didn’t anticipate them all! This is the beauty of mathematics—the diversity of thought that we all bring to the same mathematics prompt.

Learners made their thinking visible in the Chat; meanwhile, Katie and I monitored the thinking that was shared in the Chat.

We selected and sequenced two particular expressions that we wanted to discuss further in whole group discussions (more on that in a little bit).

Then we considered how we would connect these ideas/expressions to expose some interesting mathematics.

This Math Chat was facilitated using an Open Share Structure from Elham Kazemi & Allison Hintz’s Intentional Talk: How to Structure and Lead Productive Mathematics Discussions. This discussion structure focuses on sharing as many different ideas as possible to see a range of possibilities in the mathematical thinking. Read more about the Open Share Structure in this blog post.

Next, we shifted the conversation to two particular expressions that we selected for a follow-up Math Talk. We asked our colleagues to consider: Are these expressions equivalent? How do you know?

Participants discussed a number of different ways to check for equivalency including: simplifying both expressions into the same expression; substituting values in for x and checking for the same resulting value for both expressions; graphing both expressions (as y = [expression]) and analyzing the graphs.

Given more time (and a format for sharing all of our work live), we might have pushed further here and asked participants to show what they know using words, pictures, graphs, numbers, symbols, and equations (inspired by the work of Jill Gough and Jennifer Wilson).

We used the Building Expressions file (from Texas Instruments’ Building Concepts series for Expressions & Equations) to explore this question of equivalent expressions further.

Here’s a video clip building out the second expression and then exploring the value of each expression for varying values of x. What do you notice? How does this relate to our question about the equivalence (or lack thereof) of these two expressions?

What if we made a quick change to one of these expressions and then explored again? Check out the change (and resulting impact) in the following video clip. What do you notice? How does this relate to our question about the equivalence (or lack thereof) of these two (new) expressions?

This exploration helped us uncover the idea that two expressions are equivalent if they produce the same output values for all input values of x. This file provides one way to model this definition of equivalence and use it to verify whether or not two expressions are equivalent.

After this exploration, we stepped back from the discussions we had engaged in and considered some of the facilitation moves that Katie and I had incorporated. We discussed our use of the 5 Practices.

We also discussed the use of different discussion structures from Intentional Talk. As mentioned earlier, we started the webinar with an Open Strategy Share discussion where we all shared as many different expressions as we could that were equivalent to 10x + 24. Then, later, when we were considering is 2(5x+12) equivalent to 2(5x+10)+4 (and how do you know) we were facilitating a Compare & Connect discussion that felt different. This discussion had a different intent than the previous one. Intentional Talk challenges us to think about how a focus on these different discussion structures helps create these distinct learning experiences with the same underlying content.

How else might we facilitate meaningful mathematical discourse so that students develop confidence and flexibility when using expressions and equations? How might we facilitate experiences where learners write expressions in equivalent forms to build foundations for writing and solving equations? More on this journey in an upcoming post…!