Pre- and Post-Assessments & Answer Keys - Equivalent Expressions

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Equivalent Expressions Pre-Assessment

1. Leo and Silvia are looking at the following problem:

How does the product of 60x225 compare to the product of 30x225?

Silvia says she can compare these products without multiplying the numbers out. Explain how she might do this. Draw pictures to illustrate your explanation.

2. Evaluate the following numerical expression:

4-3(7-10)-(-6+14)

3. Below is a picture that represents 9+2.

Eleven squares are in a row touching each other. Nine of the squares are light gray. Two squares are dark gray.

a. Draw a picture that represents 4x(9+2).

b. How many times bigger is the value of 4x(9+2) than 9 + 2? Explain your reasoning.

4. To compute the perimeter of a rectangle you add the length, l and width, w and double this sum.

a. Write an expression for the perimeter of a rectangle.

b. Use the expression to find the perimeter of a rectangle with length 30 and width 75.

Equivalent Expressions Pre-Assessment -- ANSWER KEY

1. Leo and Silvia are looking at the following problem:

How does the product of 60x225 compare to the product of 30x225?

Silvia says she can compare these products without multiplying the numbers out. Explain how she might do this. Draw pictures to illustrate your explanation.

Solution:

Since 60 is twice 30, the product 60x225 is twice the product 30x225. We can write this as an equation:

60x225=(2x30)x225=2x(30x225)

The above explanation corresponds to the following picture.

A rectangle measuring 60 by 225 cut into two 30 by 225 rectangles.
Or it can be viewed as:
Two 30 by 225 unit rectangles stacked on top of each other.

The area of a 225 by 60 rectangle (60x225) is double that of a 225 by 30 rectangle (30x225). If we scale the width of the rectangle by a factor of 2, then the area of the resulting rectangle doubles. In other words, if one of the factors of the product 30x225 is scaled by a factor of 2 then the product is scaled by a factor of 2.

This Illustrative Mathematics question is licensed under CC BY-NC-SA 4.0.

2. Evaluate the following numerical expression:

4-3(7-10)-(-6+14)

Solution:

Correct response is 5. The common incorrect responses are -13, 15, 33.

3. Below is a picture that represents 9+2.

Eleven squares are in a row touching each other. Nine of the squares are light gray. Two squares are dark gray.

a. Draw a picture that represents 4x(9+2).

b. How many times bigger is the value of 4x(9+2) than 9 + 2? Explain your reasoning.

Solution:

a. A grid of squares is 4 rows by 11 columns. Nine of the squares in each row are light gray. Two of the squares in each row are dark gray, making two dark gray columns to the far right of the model.

b. The value of 4x(9+2) is four times the value of 9+2. We can see it in the picture since 4x(9+2) is visually represented as 4 equal rows with 9+2 squares in each row. Without even simplifying each expression, we can see that 4x(9+2) must be four times the value of 9+2.

We can also verify it by calculating the values of the expressions: 9+2=11 and 4x(9+2)=4x11=44 and then noting that 44÷11=4

But really all we have to do is notice at the step 4x(9+2)=4x11 that 4x(9+2) is 4 times 9+2.

This Illustrative Mathematics question is licensed under CC BY-NC-SA 4.0.

4. To compute the perimeter of a rectangle you add the length, l and width, w and double this sum.

a. Write an expression for the perimeter of a rectangle.

b. Use the expression to find the perimeter of a rectangle with length 30 and width 75.

Solution:

a. The description for computing the perimeter of a rectangle first adds the length l and the width w of the rectangle l+w.

Then it asks us to double this sum, that means we are taking the sum and multiplying it by 2: perimeter of a rectangle = 2(l+w)

b. Letting l=30 and w=75, we have: perimeter of this rectangle = 2(30+75)=2(105)=210.

This Illustrative Mathematics question is licensed under CC BY-NC-SA 4.0.

Equivalent Expressions Post-Assessment

1. Select all of the expressions which are equivalent to 18x+36.

a. 6(3x+6)

b. 18x+12-48

c. Find the area of the rectangle:

A rectangle model with the width being 9 and the length being 2x+4

d. 6x(3+6)

e. 3(6x+12)

f. 12x+6(3)-6x

2. Students were asked to write an expression equivalent to this one:

3(4x - 2) + 5(4x - 2)

Student A's expression

8(8x - 4)

Student B's expression

12x - 2 + 20x - 2

Is Student A's expression correct or incorrect? Explain your reasoning.

Is Student B's expression correct or incorrect? Explain your reasoning.

3. Sara bought 60 tickets for tickets to use at the fair. Each ride cost 3 tickets. Sara has played x games so far. Which expression best models the number of tickets Sara has left?

a. 3x+60

b. 60-3x

c. 60x-3x

d. 3x-60

4. Write 2 equivalent expressions that represent the perimeter of the rectangle.

A rectangle model with the width being 5x and the length being 2x+8.

Equivalent Expressions Post-Assessment -- ANSWER KEY

1. Select all of the expressions which are equivalent to 18x+36.

a. 6(3x+6) [correct]

b. 18x+12-48 [incorrect, integer operations result in -36]

c. Find the area of the rectangle: [correct, area is 9(2x+4)]

A rectangle model with the width being 9 and the length being 2x+4

d. 6x(3+6) [incorrect, simplifies to one variable term]

e. 3(6x+12) [correct]

f. 12x+6(3)-6x [incorrect, integer operations on coefficients result in 6x]

2. Students were asked to write an expression equivalent to this one:

3(4x - 2) + 5(4x - 2)

Student A's expression

8(8x - 4)

Student B's expression

12x - 2 + 20x - 2

Is Student A's expression correct or incorrect? Explain your reasoning.

Sample response:

"Student A is wrong because they added the number outside but they should not have added the [terms] inside the [parentheses]. The [expression] in the [grouping symbols] is the same, so the [expression] should have been 8(4x - 2)."

Is Student B's expression correct or incorrect? Explain your reasoning.

Sample response:

"The student did not [distribute] correctly. The [coefficient] should be [multiplied] by all of the [terms] inside the [parentheses], including the [constant terms]. The first [constant] should be [negative] 6 instead of negative 2 and the second [constant] should be [negative] 10 instead of [negative] 2."