End-of-Unit Assessment

1.

Which expression is equal to $7^3$ ?

A.

$7 \boldcdot 7 \boldcdot 7$

B.

21

C.

$3^7$

D.

10

Answer:

A

Teaching Notes

Students selecting B are mistaking exponentiation for multiplication of the base and exponent instead of repeated multiplication of the exponent. Students selecting C switched the base and exponent, which almost never results in an equivalent expression. Students selecting D are mistaking exponentiation for addition.

2.

Select all the expressions that are equivalent to $5x^3$ .

A.

5+x^3

B.

5(x \boldcdot x \boldcdot x)

C.

5x \boldcdot 5x \boldcdot 5x

D.

5x +5x + 5x

E.

5(x^2 \boldcdot x)

Answer: B, E

Teaching Notes

Students apply the meaning of bases and exponents to conclude that B and E both have equivalent values of 81.

3.

Which expression is equivalent to $40a-10b$ ?

A.

$10(4a-10b)$

B.

$a(40-10b)$

C.

$10(4a+b)$

D.

$5(8a-2b)$

Answer:

D

Teaching Notes

Students selecting A have not distributed the 10 to the $10b$ term. Students selecting B have not distributed the $a$ to the $10b$ term. Students selecting C did notice that the subtraction symbol in the original expression was changed to an addition symbol.

4.

Here is an expression: $2 + 3^t$ .

Find the value of the expression when $t$ is 1.
Find the value of the expression when $t$ is 4.

Answer:

5
83

Teaching Notes

Students evaluate exponential expressions when given a specific value for the unknown.

5.

Write two expressions that are equivalent to $b+b+b+b+b$ .

The first expression should be a product of a coefficient and a variable.
The second expression should be a sum of two terms.

Answer:

$5b$ (or equivalent)
Sample responses: $2b+3b$ or $b+4b$

Teaching Notes

Students demonstrate their understanding of the words “sum,” “product,” “coefficient,” and “term” by generating equivalent expressions with a specified structure.

6.

Andre makes green paint by mixing 5 cups of yellow paint with 2 cups of blue paint.

To make the same shade of green, how much yellow paint does Andre need if he uses 8 cups of blue paint? Explain or show your reasoning.
How much blue paint does Andre need if he uses 15 cups of yellow paint?
Plot these pairs of measurements as points on the graph.
The variable $y$ represents the cups of yellow paint, and the variable $b$ represent the cups of blue paint. Write an equation that shows how $y$ and $b$ are related.

Answer:

20 cups. Sample reasoning: Each cup of blue paint means $\frac{5}{2}$ cups of yellow point, and $8 \boldcdot \frac{5}{2} = 20$ .
6 cups
Graph shows points at $(20,8)$ and $(15,6)$ .
$b = \frac{2}{5}y$ (or equivalent)

Minimal Tier 1 response:

Work is complete and correct.
Sample:

20. If Andre uses 4 times as much blue paint, he'll need to use 4 times as much yellow paint, and $4 \boldcdot 5 = 20$ .
6
Graph shows points at $(20,8)$ and $(15,6)$ .
$b=\frac{2}{5}y$

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: 1–2 incorrect answers but correct reasoning for the first question, or mostly correct answers but incomplete reasoning for the first question; the unit rate $\frac{2}{5}$ or $\frac52$ being multiplied by the incorrect variable in the equation.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: More than 2 incorrect problem parts; use of a unit rate other than $\frac{2}{5}$ or $\frac52$ (or their equivalents) in equation; incorrect interpretation of the relationship—that it is additive (for instance, that there are always 3 more cups of yellow paint than blue paint), or that there is more blue paint than yellow paint.

Teaching Notes

Students construct and evaluate an equation given a ratio. They then plot points to represent the relationship.

7.

This rectangle has a perimeter of 28 units.

Complete the table to show the length and width of at least 3 different rectangles that also have a perimeter of 28 units.

length ( $\ell$ ) width ( $w$ )

10 4
Describe the relationship between the values in the two columns of the table.
Write an equation to represent the relationship, where one variable is written in terms of the other. Identify the independent and dependent variables.
Plot the values in the table as points on the graph. Make sure to label the axes.

Answer:

Sample response: See table.
Sample responses:
- The width of the rectangle is the length subtracted from 14.
- The sum of the width and length is 14.
- If we know the length of the rectangle, we can find the width by subtracting the length from 14.
Sample responses:
- In the equation $w=14-l$ , the rectangle’s length is the independent variable and width is the dependent variable.
- In the equation $l=14-w$ , the rectangle’s width is the independent variable and length is the dependent variable.
Sample response: See graph. (Students may label the horizontal axis with "width" and vertical axis with "length" to match their equation.)

length ( $\ell$ )	width ( $w$ )
10	4
9	5
8	6
7	7

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample:

See table.
$w+l=14$
$l=14-w$ . The independent variable is the width, and the dependent variable is the length.
See graph.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: Equation correctly represents the situation, but is not structured to express the dependent variable in terms of the independent variable.
Sample errors: Arithmetic error leads to an incorrect solution. Axes on the graph are not labeled.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: Values chosen for the length and width add up to 28 instead of 14. Rectangles chosen have an area of 28 square units instead of a perimeter of 28 units.
Sample errors: Work involves a misinterpretation of the situation that affects all or most problem parts, but work does show understanding of using equations, tables, and graphs to represent a situation.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Work shows multiple Tier 3 errors and there are major omissions.

Teaching Notes

Students use a table, equation, and graph to represent a situation involving the perimeter of a rectangle. Depending on how they view the relationship, the length or width could be either the independent or dependent variables.