Mid-Unit Assessment

1.

Select all the equations where $x=5$ is a solution.

A.

$x-4=9$

B.

$1+x=6$

C.

$2=7-x$

D.

$10x=5$

E.

$\frac{1}{10}x=\frac12$

F.

$2x=5$

Answer:

B, C, E

Teaching Notes

Students who select choice A are likely adding instead of subtracting. Students who fail to select choice B may not understand how to substitute for a variable. Students who do not select choice C may think it cannot be true since the variable is on the right side. Students who select choices D and F might think that “solution to an equation” means “what comes after the = sign” or they may have a misconception about the “next to” notation for multiplication. Students who do not select choice E may have a misconception about evaluating expressions that involve fractions.

2.

Which equation represents the hanger diagram?

A balanced hanger diagram with three circles on the left each labeled x, and 4 squares on the right each labeled 1.

A.

x \boldcdot x  \boldcdot x = 1 \boldcdot 1 \boldcdot 1 \boldcdot 1

B.

x = \frac34

C.

3x = 4

D.

x+3 = 4

Answer: $3x = 4$

Teaching Notes

Students who select choice A did not notice that the equation involves multiplication instead of addition. Students who select choice B may have tried to solve the equation but made a mistake in the process. Students who select choice D may not understand the difference between an equation of the form

p+x=q

and

px=q

.

3.

Select all the expressions that are equivalent to $\frac{1}{5}(c+15)$ .

A.

\frac15 c +15

B.

\frac15 \boldcdot c + \frac15 \boldcdot 15

C.

\frac{1}{5} c+15c

D.

3c

E.

$\frac15 c+3$

Answer: B, E

Teaching Notes

Students who do not select B, C, and E may not understand that both the

c

and the

15

are multiplied by

\frac15

.

4.

There are $x$ students in a school, and $\frac13$ of them are in the sixth grade.

Write an expression with a variable that represents the number of sixth graders in the school.
Use the expression to write an equation that shows there are 52 sixth graders in the school.
Solve the equation to find how many students are in the school.

Answer:

$\frac{1}{3}x$ (or equivalent)
$\frac{1}{3}x = 52$ (or equivalent)
$x=156$ . There are 156 students in the school.

Teaching Notes

Students use a variable to write an expression and then an equation in the form $px=q$ to represent a situation. They solve the equation to solve a problem about the situation.

5.

Write an equation that represents the statement: $40\%$ of $120$ is $a$ .
Write an equation that represents the statement: $55\%$ of $b$ is $30$ .
$45\%$ of $c$ is $72$ . Find the value of $c$ .

Answer:

$(0.4) \boldcdot 120 = a$ (or equivalent)
$0.55b = 30$ (or equivalent)
160

Teaching Notes

Students write equations to represent relationships involving percentages. They may also use a tape diagram or double number line to answer the last question.

6.

During a school fundraiser, Noah sells gel pens for $1.25 each.

Complete the table to show how much money Noah would collect if he sold each number of pens.

number of pens sold 20 50 $p$

amount of money collected in dollars
How many pens would Noah need to sell to collect $120? Explain your reasoning.

number of pens sold	20	50	$p$
amount of money collected in dollars

Answer:

number of pens sold 20 50 $p$

amount of money collected in dollars 25 62.50 $1.25p$
96 pens. Sample reasoning: $1.25p=120$ and $120 \div 1.25 = 96$ .

number of pens sold	20	50	$p$
amount of money collected in dollars	25	62.50	$1.25p$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

See table.
$1.25p=120$ , $p=120 \div 1.25$ , $p=96$

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: Reasonable response to part b is based on an incorrect expression in the last cell of the table.
Sample errors: Substituted value for $p$ is recorded in the last column of the table, but keeps the multiplicative relationship of 1.25

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Table reflects a lack of understanding of the multiplicative relationship, which affects the equation in part b; work involves a misinterpretation of the situation that affects all or most problem parts, but work does show understanding of writing equations to represent situations and interpreting solutions to equations.

Teaching Notes

First, students complete a table to represent the relationship between the number of pens sold and the amount of money collected. Then they determine the number of pens it would take to collect $120.

7.

Jada’s plant was 3.7 centimeters tall when she bought it. Now the plant is 15.3 centimeters tall.

Which diagram (A, B, or C) represents this situation?

A

B

C
Write an equation with a variable that represents this situation.
Solve the equation you wrote.
Explain what the solution to the equation means in this situation.

Answer:

C
$3.7+x=15.3$ (or equivalent)
$x=11.6$ . ( $15.3-3.7=11.6$ )
The plant grew 11.6 centimeters since Jada bought it.

Minimal Tier 1 response:

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

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Acceptable errors: Reasonable response to part d is based on an incorrect solution to the equation.
Sample errors: Answer to part a is the only flawed response. Arithmetic error leads to an incorrect solution to the equation. Response to part d is incomplete.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Acceptable errors: Reasonable responses to the parts c and d based on an incorrect equation.
Sample errors: Equation is incorrect. There is an incorrect interpretation of the solution to the equation. Work involves a misinterpretation of the situation that affects all or most problem parts, but does show understanding of writing equations to represent situations and interpreting solutions to equations.

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. An incorrect diagram is chosen in part a.

Teaching Notes

This problem asks students to demonstrate their skill in writing and solving an equation of the form $x+p=q$ in a context. Students explain the meaning of the solution in the context.