Preserving Equality — Algebraic Thinking Across Grade Levels

Equality is a fundamental concept of mathematics. The way that we can play around with values — adjusting numbers slightly — but maintain equality is fascinating, and differs between operations. Here is a look at how I’ve seen this play out recently in grades 1, 4, and 8.

Grade 1: Pass It To The Other Side (addition/Subtraction)

In a recent “Number of the Day” activity (Investigations Math), students brainstormed ways to make the number 10. I saw all sorts of solutions: some using addition, some using subtraction, some incorporating both operations.

Many students followed a pattern to generate more but different equations. This student built from the known fact 6 + 4, decomposing the number 6 into 2 + 2 + 2 and 3 + 3.

6 + 4 = 10

2 + 2 + 2 + 4 = 10

3 + 3 + 4 = 10

This student generated differences of ten by thinking about the place value.

70 - 60 = 10

95 - 85 = 10

101 - 91 = 10

Meanwhile, first grade student Sakura developed this list:

This idea had surfaced several times during our work on part-part-whole, how many of each problems (e.g. there are 7 vegetables: some are peas, some are carrots). One student had articulated this as “The peas are getting one more, and the carrots are counting down.” It’s an astute observation, but not yet at the point of a generalization.

To preserve the sum of two addends, add n to one addend and subtract n from the other addend. In first grade, we are often doing this for n=1. When you add one to an addend, subtracting one from the addend will keep the sum the same.

Created using Mathigon’s Polypad

This remains true for other values of n, but that involves a lot of working memory for young children. But examining that simple little relationship — one more, one less — sets an important numerical foundation for primary grade students. This seems to work with addition, but does it work with subtraction? Why or why not?

Grade 4: Double and Halve (Multiplication/Division)

Upper elementary students will often stretch this thinking to multiplication. If 5 x 5 = 25, shouldn’t 4 x 6 = 25?

created using Mathigon’s Polypad

Hmm… but it doesn’t.

n \times n = n^2

(n+1) \times (n-1) = n^2 + n - n - 1, so

(n+1) \times (n-1) = n^2 - 1

This will also hold true for numbers that aren’t square numbers. We could represent 8 x 6 as n \times (n-2) and 9 x 7 as (n+1) \times (n-3), and continue to prove it either algebraically or with the visual of the array.

created using Mathigon’s Polypad

n \times (n-2) = n^2 -2n

(n+1) \times (n-3) = n^2 + n - 3n - 3, so

(n+1) \times (n-3) = n^2 -2n -3

And you can see that the difference between the products is 3 when the two factors have a difference of 2 (2 + 1 = 3). But basically: this doesn’t work because we’re too busy thinking additively. We need to think multiplicatively.

We won’t preserve the equality if we add something to one factor and subtract it from the other. We need to multiply the factor by something and divide it by the other.

If you double one factor (multiply by 2), you halve the other factor (either divide by two or multiply by \frac{1}{2}. (Furthermore, to preserve equality with addition you add or subtract the opposite number, and with multiplication you multiply or divide by the reciprocal.)

This is a record from a fourth grade number talk. It moves slowly through the doubling and halving, adjusting one factor at a time. Students can see that 5 x 12 is half of 10 x 12, because it’s half as many groups. And then 5 x 24 is double 5 x 12 because there’s twice as many groups.

We actually have two doubling and halving relationships to explore there! There are two products of 60 and two products of 120.

10 \times 6
5 \times 12
10 \times 12
5 \times 24

You can even spot my crude open array diagrams attempting to show the relationship.

Grade 8: What happens with Volume? (Exponential Relationships)

In elementary grades, students study the area of rectangles, which is a multiplicative relationships. So what happens with volume?

Well, it depends on the shape. Last Friday, my 8th graders started working on finding the volume of cylinders. They are familiar with finding the area of a circle from 7th grade, as well as finding the volume of prisms (three dimensional solids with polygonal bases).

(In fact, I described a lot of the K-7 learning progression around geometric measurement in a blog post last April: “The Learning that Led to Today.”)

So quickly we were able to apply the 7th grade learning to determine the formula for the volume of a cylinder:

\pi r^2 \cdot h

bh, where b is the area of the base (\pi r^2)

Late in the lesson, students were asked create two different cylinders with the same volume. Which to say: we will use the same formula, but different values for the radius and the height, and preserve the equality.

Desmos Math from Amplify: Grade 8, Unit 5, Lesson 11 (The Volume of a Cylinder)

Many students messed around with it, using the “check my work” function to adjust and revise their work. To anchor them on our thinking about the relationship between these variables, and preview a little bit of Monday’s lesson (“Scaling Cylinders with Functions”), I used a few student work samples that showed successful pairings of cylinders with the same volume.

I told students to think about scale factor — which indicates multiplication. I gave students a chance to think independently before sharing with their randomly assigned table groups, and then sharing out.

“You’re doubling and halving,” Emma at Table 1 said.

“Say more,” I prodded.

“You double one thing, and then you have the other, and it’ll stay the same,” Emma continued.

I recorded 8 \cdot 15 and 4 \cdot 30 as examples. “Oh, yes, we’ve definitely talked about this relationship over the years that I’ve known you,” I said. I’ve known some of these students since kindergarten.

“So let’s show how this works with Otto’s work in the top right. Where do we see the doubling, and where do we see the halving?”

“The doubling is 3 and 6,” called out David from Table 4. “And the halving… wait.”

“That’s not halving. That’s… the opposite of quadrupling?” said Sofia at Table 3.

“Okay, so the doubling has a scale factor of 2, and and opposite of a quadrupling has a scale factor of…”

“Dividing by 4?” asked Henry at Table 1.

“Scale factor is multiplication,” Otto said. “So… multiplying by \frac{1}{4}.”

“Precisely!” I recorded the thinking on the board.

“Oh, that keeps happening…” Sofia mused. “Doubling and one-fourth-ing.”

“That’s weird,” I said, hoping it would cause students to think. (I’m sure some of them were happy to accept the weirdness without any consideration. I think Inaya, who was sitting at Table 3, rolled her eyes. She hates when I play dumb.)

“I bet it has something to do with squaring,” Sadie at Table 2 said.

“YES! It does! But what?”

“The radius is squared… so… instead of halving we are \frac{1}{2} \cdot \frac{1}{2}-ing.”

“Ahh, there’s the \frac{1}{4}!” I recorded it on the board. “And what if the relationship wasn’t doubling and halving-halving? What if it were tripling, and…” I started to write on the board again. “So one of these has a radius of 3 and the other has a radius of 9. The one with a radius of 3 has a height of 24, and the one with a radius of 9 has a height of…?” I looked expectedly at these kids that were perilously close to the bell to end 3rd period.

“Is it 3? \frac{1}{2} \cdot  \frac{1}{2} \cdot  \frac{1}{2}…? And \frac{1}{8} of 24 is 8?” Otto asked.

Fascinating! I couldn’t tell if he thought it would involve scale factors of 3 and \frac{1}{2}^3 because it would work out nicely — another integer — or because he thought that the relationship was always scale factors of n and \frac{1}{2^n}. It’s actually \frac{1}{9}, or \frac{1}{3} \cdot \frac{1}{3}.

But we were almost out of time, and there would be time to explore this during the next lesson: Scaling Cylinders Using Functions.

“Preserving Equality”

With addition, you can add a number to an addend and its opposite to another addend while maintaining the same value.

With multiplication, you can multiply a factor by a number and another factor by that number’s reciprocal, and maintain the same value.

Then, what feels simple and pure in younger grades, gets thornier when we enter the world of middle school geometric measurement.

You may have noted that many of the examples in the post used expressions rather than equations. Like we looked at

8 \cdot 15

4 \cdot 30

horizontally, rather than vertically. I think it’s important to give students more experience with the equals sign — we recently did a few lessons about “true or false equations” from the Grade 1 edition of Investigations — and wrestling with the notion of equality and symbolic representation. We want students to see that

5 + 2 = 4 + 3

Not just because both sides evaluate to 7, but also because we have subtracted 1 from an addend and added 1 to the other addend. We will continue to make this connection across all of the classes. Watching these ideas surface in several different grade levels also reminds me of times — often in past tense — when we let lessons breathe more, and nudged students to develop conjectures and articulate generalizations. What opportunities can I orchestrate in upcoming weeks?

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