Identify Equivalent Expressions

Students determine whether two expressions are equivalent by applying properties of operations, evaluating both at several values, and reasoning about their structure. They justify their conclusions using mathematical properties and counter-examples.

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Grade 6 · Expressions and Properties · question 1 of 3

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5(3n + 1) = n + 5

How it is taught

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Two expressions are equivalent only if they match for every value.

One matching input is not enough; a counter-example kills equivalence.

Read it: Multiply both parts

3(a + 8) is a rectangle 3 tall and a + 8 wide. Cut it at the +: one part is 3 × a = 3a, the other 3 × 8 = 24. So 3(a + 8) = 3a + 24.

Draw the rectangle: the number outside is its height, the parts inside are its widths.
Multiply the height by each part: 3 × a = 3a and 3 × 8 = 24.
Add the two areas: 3(a + 8) = 3a + 24.

The 3 multiplies everything inside the parentheses, so it multiplies both parts.

Read backwards, the same picture factors 3a + 24 into 3(a + 8).

The rectangle reads both ways, so 3(a + 8) and 3a + 24 are one area; and terms with the same letter, like 2a and 5a, join into 7a.

The number outside the parentheses multiplies every part inside.

Writing 3(a + 8) = 3a + 8.

The 3 multiplies the 8 too: 3 × 8 = 24, so 3(a + 8) = 3a + 24.

14(x + 10) = 14x + ___ — 140. The 14 multiplies both parts: 14 × 10 = 140.

Is 3(x + 4) equivalent to 3x + 12?

Two expressions are equivalent if they give the same value for every x. Let's test these two.

On the left, three times the quantity x plus four. On the right, three x plus twelve. Are they the same?

Let's use the distributive property to rewrite the left side. The three multiplies everything inside the parentheses.

First, three times x is three x.

Next, three times four is twelve.

So the left side becomes three x plus twelve — exactly the right side!

Let's double-check by plugging in a number. Try x equals two.

Left side: three times two plus four is three times six, which is eighteen.

Right side: three times two is six, plus twelve also makes eighteen.

Both sides give eighteen, and the distributive property proves it always works. The expressions are equivalent.

Grade 6 · Using the distributive property to test equivalence

What you see

While they climb, your page writes one true line to ask about tonight, and a short note each Sunday.

MayaGrade 3 · Mathematics

Did you know?

A chessboard has 64 squares — share them out between the 8 rows and each row holds exactly 8.

Today’s goal: 1 star — one skill mastered, every rung of its ladder climbed
Climbing now: Divide within 100

An example student.

MayaGrade 3 · Mathematics

Maya earned 2 stars this week — Interpret multiplication as equal groups, which took 6 answers over 3 days before its star landed 3 days ago and Multiply by 6, 7, 8 and 9. Next up: Divide within 100.

An example week.

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