End-of-Unit Assessment

1.

Which expression is equal to $6^4$ ?

A.

10

B.

24

C.

$4^6$

D.

$6 \boldcdot 6 \boldcdot 6 \boldcdot 6$

Answer:

$6 \boldcdot 6 \boldcdot 6 \boldcdot 6$

Teaching Notes

Students selecting A are mistaking exponentiation for addition, while 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.

2.

Select all the expressions that are equivalent to $3x^4$ .

A.

$3(x \boldcdot x \boldcdot x \boldcdot x)$

B.

$3+x^4$

C.

$3x + 3x + 3x + 3x$

D.

3(x^2 \boldcdot x^2)

E.

$3x \boldcdot 3x \boldcdot 3x \boldcdot 3x$

Answer: A, D

Teaching Notes

Students selecting B might be confusing the “next to” notation for multiplication with addition. Students selecting C might be confusing exponentiation with addition. Students selecting E may not understand that the exponent only applies to the variable in this expression: $3x^4=3(x^4)$ .

3.

Which expression is equivalent to $20c-8d$ ?

A.

$2(10c+4d)$

B.

$4(5c-8d)$

C.

$4(5c-2d)$

D.

$c(20-8d)$

Answer:

$4(5c-2d)$

Teaching Notes

Students selecting A did not notice the subtraction symbol in the original expression was changed to an addition symbol. Students selecting B have not distributed the 4 to the $8d$ term. Students selecting D have not distributed the $c$ to the $8d$ term.

4.

Here is an expression: $3 \boldcdot 2^t$

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

Answer:

6
48

Teaching Notes

Students evaluate an exponential expression for different values of the variable.

5.

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

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

Answer:

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

Teaching Notes

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

6.

Jada makes sparkling juice by mixing 2 cups of sparkling water with every 3 cups of apple juice.

How much sparkling water does Jada need if she uses 15 cups of apple juice? Explain or show your reasoning.
How much apple juice does Jada need if she uses 6 cups of sparkling water?
Plot these pairs of measurements as points on the graph.

A blank coordinate plane with the origin labeled "O." The j-axis is labeled “cups of apple juice” and the numbers 0 through 22, in increments of 2, are indicated. Vertical gridlines are drawn at each integer from 1 to 22. The s-axis is labeled “cups of sparkling water” and the numbers 0 through 22, in increments of 2, are indicated. Horizontal gridlines are drawn at each integer from 1 to 22.
The variable $s$ represents the number of cups of sparkling water, and the variable $j$ represents the number of cups of apple juice. Write an equation that shows how $s$ and $j$ are related.

Answer:

10 cups. Sample reasoning: For each cup of apple juice used, $\frac{2}{3}$ cup of water is needed. $15 \boldcdot \frac{2}{3} = 10$ .
9 cups
Graph shows points at $(9,6)$ and $(15,10)$ .
$j = \frac{3}{2} s$ (or equivalent)

Minimal Tier 1 response:

Work is complete and correct.
Sample:

10. There are 5 times as many cups of apple juice, so there needs to be 5 times as many cups of sparkling water.
9
Graph shows points at $(9,6)$ and $(15,10)$ .
$j=1.5s$

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 $\frac32$ or $\frac23$ 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 $\frac32$ or $\frac23$ (or their equivalents) in equation ; incorrect interpretation of the relationship—that it is additive (for instance, that there is always 1 more cup of apple juice than sparkling water), or that there is more sparkling water than apple juice.

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 36 units.

Complete the table to show the length and width of at least 3 different rectangles that also have a perimeter of 36 units.

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

12 6
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 your table as points on the graph. Make sure to label the axes.

Answer:

Sample response: See table.
Sample responses:
- If we know the length of the rectangle, we can find its width by subtracting the length from 18.
- The sum of the width and length is 18.
- If we know the width, we can find the length by subtracting the width from 18.
Sample responses:
- In the equation $w = 18 - \ell$ , the rectangle’s length is the independent variable and width is the dependent variable.
- In the equation $\ell = 18 - w$ , 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 ( $w$ )
12	6
10	8
12	6
4	14

Minimal Tier 1 response:

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

See table.
$w + \ell = 18$
$\ell = 18 - w$ . 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: the 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 to 36 instead of 18. Rectangles chosen have an area of 36 square units instead of a perimeter of 36 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.