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Equivalent Expressions


Alignments to Content Standards: 6.EE.A.4

Task

Which of the following expressions are equivalent? Why? If an expression has no match, write 2 equivalent expressions to match it.

  1. $2(x+4)$
  2. $8+2x$
  3. $2x+4$
  4. $3(x+4)-(4+x)$
  5. $x+4$

IM Commentary

In this problem we have to transform expressions using the commutative, associative, and distributive properties to decide which expressions are equivalent. Common mistakes are addressed, such as not distributing the 2 correctly. This task also addresses 6.EE.3.

Solution

First, we notice that the expressions in (a) and (d) can be rewritten so that they do not contain parentheses.

  • To rewrite (a), we can use the distributive property: $2(x + 4) = 2x + 8$

  • To rewrite (d):

    $$ \begin{alignat}{2} 3(x+4)-(4+x) &= 3(x+4) - (x+4) &\qquad & \text{using the commutative property of addition}\\ &= (3-1)(x+4) &\qquad&\text{using the distributive property}\\ &= 2(x+4) &\qquad & \\ &=2x+8 &\qquad &\text{using the distributive property again} \end{alignat} $$

So the five expressions are equivalent to:

  1. $2x + 8$
  2. $8+2x$
  3. $2x+4$
  4. $2x + 8$
  5. $x+4$

Right away we see that expressions (a), (b), and (d) are equivalent, since addition is commutative: $2x + 8 = 8 + 2x$. We now only have to check (c) and (e). If $x=0$, (c) and (e) have the value 4, whereas (a), (b), and (d) have the value 8. So (c) and (e) are not equivalent to (a), (b), and (d). Moreover, if $x=1$, then (c) has the value 6 and (e) has the value 5, so those two expressions are not equivalent either.

For (c), we can use the distributive property and decompose one of the numbers to write two equivalent expressions:

$$ \begin{align} &2x+4 \\ &2(x+2) \\ &2(x+1+1) \end{align} $$

Many other equivalent expressions are possible.

For (e), we can add and subtract the same number or the same term to write two equivalent expressions:

$$ \begin{align} &x+4 \\ &x+4 + 7 - 7 \\ &3x -3x +x +4 \end{align} $$

Many other equivalent expressions are possible.

Ms.Bliss says:

10 months

I think building all of these with Algebra tiles would also help students create a solid foundation for understanding. As for (d) 3(x+4) - (4+x)... it's never to early to encourage students to view the negative as the opposite. My students will read this as, "3 times the quantity of x plus four, and the opposite of the quantity 4 plus x." With Algebra tiles, they will model the opposite as turning the tiles over from the positive to the negative. My two cents. Thanks for a great task.

lizyockey says:

over 5 years

This task could also be an illustration of 6.EE.3, generating equivalent expressions.

Kristin says:

over 5 years

Thanks for pointing this out--I added this to the commentary.

leannef says:

over 5 years

d) has the students doing the distributive property with subtraction -(4+x) = -4-x, I did not think we would be teaching that until multiplication by -1, which is not in 6th grade.

Could it be done as 3(x+4)-(x+4) = (3-1)(x+4) = 2(x+4) then distribute?

Bill says:

over 5 years

Leanne, excellent suggestion, I will edit this task now the change.

Bill McCallum

Shaun says:

over 3 years

I was wondering if the order of operations would force us to use the distributive property on the first parenthesis and then the second parenthesis and then complete the subtraction. I was not sure if it is ok to subtract the 3 and the 1 first as multiplication comes before subtraction. Please advise

Cam says:

over 3 years

Thanks for the question! Yes, it is definitely okay to subtract first. The distributive law (or rather, one of the two distributive laws) states that for any values of $a$, $b$, and $c$, we have $$ ac-bc=(a-b)c $$ The solution here applies this law to the values $a=3$, $b=1$, and $c=x+4$. Order of operations doesn't come in to play -- order of operations tells us how to interpret the notation in the expressions, but doesn't preclude us from applying valid algebraic manipulations in any order we so choose.