Mid-Unit Assessment

1.

Select all the equations where $x=3$ is a solution.

A.

$x-3=0$

B.

$x+1=2$

C.

$9-x=3$

D.

$6=2x$

E.

$\frac{1}{6}x=3$

F.

$3x=9$

Answer:

A, D, F

Teaching Notes

Students who select choice B are subtracting instead of adding. Students who select choices C and E might think that “solution to an equation” means “what comes after the = sign.” Students who do not select A may not understand how to substitute for a variable. Students who do not select choice D or F may have a misconception about the “next to” notation for multiplication.

2.

Which equation represents the hanger diagram?

A balanced hanger diagram with two circles on the left each labeled x, and 5 squares on the right each labeled 1.

A.

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

B.

$x = \frac25$

C.

$x+2 = 5$

D.

$2x = 5$

Answer:

$2x = 5$

Teaching Notes

Students who select choice A did not notice 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 C 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 $\frac{1}{3}(s+12)$ .

A.

$4s$

B.

$\frac{1}{3}s + 12$

C.

$\frac{1}{3} \boldcdot s + \frac{1}{3} \boldcdot 12$

D.

$\frac{1}{3}s + 4$

E.

$\frac{1}{3}s + 4s$

Answer: C, D

Teaching Notes

Students who do not select C and D may not understand both the $s$ and 12 are multiplied by $\frac{1}{3}$ .

4.

There are $x$ students in a school, and $\frac14$ 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 46 sixth graders in the school.
Solve the equation to find how many students are in the school.

Answer:

$\frac{1}{4}x$ (or equivalent)
$\frac{1}{4}x=46$ (or equivalent)
$x=184$ . There are 184 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: 30% of 140 is $x$ .
Write an equation that represents the statement: 64% of $y$ is 40.
45% of $z$ is 72. Find the value of $z$ .

Answer:

$(0.3) \boldcdot 140 = x$ (or equivalent)
$0.64y = 40$ (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.

Diego sells raffle tickets for a school fundraiser. He collects $1.75 for each ticket.

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

number of tickets sold 20 50 $r$

amount collected in dollars
How many tickets would Diego need to sell to collect $140? Explain your reasoning.

number of tickets sold	20	50	$r$
amount collected in dollars

Answer:

number of tickets sold 20 50 $r$

amount collected in dollars 35 87.50 $1.75r$
80 tickets. Sample reasoning: $1.75r=140$ and $140 \div 1.75 = 80$ .

number of tickets sold	20	50	$r$
amount collected in dollars	35	87.50	$1.75r$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

See table.
$1.75r=140$ , $r = 140 \div 1.75$ , $r=80$

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 $r$ is recorded in the last column of the table, but keeps the multiplicative relationship of 1.75.

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 raffle tickets sold and the amount of money collected. Then they determine the number of tickets it would take to collect $140.

7.

Mai’s plant was 2.6 centimeters tall when she bought it. Now the plant is 10.4 centimeters tall.

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

A
Tape diagram A, 2 parts, labeled 2 point 6, 10 point 4. Total, x.

B
Tape diagram B, 2 parts, x, 2 point 6. Total, 10 point 4.

C
Tape diagram C, 2 parts, labeled 10 point 4, x. Total, 2 point 6.
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:

B
$x + 2.6 = 10.4$ (or equivalent)
$x=7.8$ . ( $10.4 - 2.6 = 7.8$ )
The plant grew 7.8 centimeters since Mai 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 an incorrect solution to the equation.
Sample errors: Answer to part a is the only flawed response. An 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: Work shows reasonable responses to parts c and d based on an incorrect equation.
Sample errors: Equation in is incorrect. Part d contains 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 for 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.