Mid-Unit Assessment

1.

This hanger diagram is balanced. There are two labeled weights of 8 grams and 20 grams. The four circles each have the same weight.

What is the weight of each circle, in grams?

A balanced hanger diagram. There is 1 pentagon labeled 20 on the left side and 4 identical circles labeled x and a rectangle labeled 8 on the right side.

A.

3

B.

7

C.

5

D.

$\frac{20}{12}$

Answer:

A

Teaching Notes

Many students will solve this problem by writing the equation $4x+8=20$ , though reasoning purely about the weights on the hanger diagram is also fine. Students selecting B likely added 8 to 20 instead of subtracting. Students selecting C probably imagined isolating all the weights labeled x, without accounting for the 8. Students selecting D may have combined $4x$ and 8 to end of with $12x=20$ , then divided by 12.

2.

Select all the situations that can be represented by the tape diagram.

A.

Noah walked a family’s dog 5 times this past month and earned the same amount each time. To thank him, the family gave him an extra $6 at the end of the month. Noah earned $106 from dog walking.

B.

A family of 5 drove to a soccer game. They paid $6 for parking, and all of their tickets were the same price. They paid $106 in total.

C.

There are 5 bags that each contain 6 nickels and the same number of pennies. Altogether, the bags contain 106 coins.

D.

Kiran is baking 5 batches of muffins. Each batch needs the same amount of sugar in the muffins, and each batch needs six extra teaspoons of sugar for the topping. Kiran uses 106 total teaspoons of sugar.

E.

Priya buys 5 cases of water, each with the same number of bottles. The store clerk gives her an extra 6 bottles before she leaves. Priya leaves with a total of 106 bottles.

Answer:

C, D

Teaching Notes

The tape diagram represents the equation $5(s+6)=106$ .

Students failing to select C or D may be interpreting those situations in ways that can be described using the equation $5s+6=106$ , rather than $5(s+6)=106$ . Situations A and B describe 5 groups of $n$ with a constant of 6 which is not equivalent to $5s+30$ .

3.

Over the weekend, Jada made three times as much money babysitting than Han did mowing lawns. Jada made an additional $27 selling lemonade. From babysitting and selling lemonade, she has a total of $96. The equation $3x+27=96$ represents this situation. What does the variable $x$ represent?

A.

the amount in dollars that Jada made selling lemonade over the weekend

B.

the amount in dollars that Jada made babysitting over the weekend

C.

the amount in dollars that Han made mowing lawns over the weekend

D.

the amount in dollars that Han made babysitting over the weekend

Answer:

the amount in dollars that Han made mowing lawns over the weekend

Teaching Notes

Students selecting B may be confused specifically about how to represent the statement “Jada makes three times as much money babysitting as Han did mowing lawns.” Students selecting A or D have made a mistake reading the problem: we already know Jada makes $27 selling lemonade, and we do not know whether Han made any money babysitting over the weekend.

4.

Solve each equation.

$10-\frac14x=7$
$3(x+8)=21$

Answer:

$x=12$
$x=\text-1$

Teaching Notes

Some students may struggle with the form of the equation in part a: After subtracting 10, is it

\frac14

or

\text-\frac14x

that remains? The most likely error in part b is forgetting to properly distribute, though some students may multiply each side by

\frac13

instead to get

x+8=7

.

5.

Diego tried to solve the equation $\frac{1}{3}(x+15)=3$ .

Circle the step where Diego made a mistake. Then explain or show how to solve the equation correctly.

$\frac{1}{3}(x+15)=3$

$\frac{1}{3}x +15 -15=3 -15$

$\frac{1}{3}x=\text -12$

$\frac{1}{3}x \div \frac13=\text -12 \div \frac13$

$x= \text- 36$

Answer:

Students circle $\frac{1}{3}x +15 -15=3 -15$ . Sample reasoning: Diego tried to subtract 15 from each side, but on the left side of the original equation, the 15 is being multiplied by $\frac{1}{3}$ . He needed to either first distribute the $\frac{1}{3}$ or first divide each side by $\frac{1}{3}$ .

Minimal Tier 1 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample: Diego subtracted 15 from each side, but that’s wrong because the 15 is in the parentheses. He should have done $\frac{1}{3} \boldcdot x$ and $\frac{1}{3} \boldcdot 15$ before subtracting.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Identifies the correct error, but the explanation is either incorrect or vague; identifies the problem step by substituting $x=\text-6$ at each step, but no algebra mistake is identified; solves the equation correctly but does not identify Diego’s error.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Identifies a step that does not contain the error; points to a different “error” in the problem step.

Teaching Notes

This problem points to a common error in solving equations of the form $p(x+q)=r$ .

6.

A farmer is packing fruit and vegetable crates. Each crate weighs 30 pounds.

Here are two situations. Write an equation to represent each situation. If you get stuck, consider drawing a diagram.

Each crate contains 3 bags of fruit with the same weight and a 6-pound bag of vegetables.
Each create contains 3 boxes. Each box contains a 6-pound bag of vegetables and a bag of fruit. The bags of fruit have the same weight.

Answer:

$3x+6=30$
$3(x+6)=30$

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$3x+6=30$ .
$3(x+6)=30$ .

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Writes one of the equations correctly with errors in the other.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Writes both equations incorrectly, has flaws in both responses.

Teaching Notes

Students need to distinguish a situation leading to an equation of the form $p(x+q)=r$ from a situation leading to an equation of the form $px+q=r$ . The instructions encourage using a tape diagram, but the diagram is not required.

7.

Mai will make school spirit bracelets for a school fundraiser. She will order bead wiring and alphabet beads to create the school name on the bracelets. Mai must spend $250 on wire and $5.30 per bracelet for beads.

Complete the table giving the total cost Mai will spend to make each specific number of bracelets.
Write an expression for the cost of making $n$ bracelets.
What is the maximum number of bracelets Mai can make with a budget of $1,500?

number of bracelets	cost in dollars
20
40
60

Answer:

number of bracelets cost in dollars

20 $356

40 $462

60 $568
$5.3n+250$ (or equivalent)
235 bracelets. Sample reasoning: Since the wire costs $250, Mai will have $1,250 left to spend on the bracelets. Each bracelet uses $5.30 of beads. Since $\frac{1250}{5.3}$ is about 235.8, Mai can make a maximum of 235 bracelets. She can’t make 236.

number of bracelets	cost in dollars
20	$356
40	$462
60	$568

Minimal Tier 1 response:

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

See table.
$5.3n+250$ .
The equation is $5.3n+250=1500$ . $5.3n=1250$ , so $n=235.849$ . She can make 235 bracelets.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Writes one incorrect entry in the table; uses good reasoning in part c but contains a calculation error; writes a correct equation in part c but contains an arithmetic error; gives response of 235.8 or 236 to part c.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Writes more than one incorrect entry in the table; writes an incorrect equation in part b that is more than a transcription error; writes a correct equation for part c, but the work to solve that equation contains an algebraic error.
Acceptable errors: Bases work for parts b and c on incorrect table entries or on a misunderstanding of the situation that nonetheless leads to an equation of the form $px+q=r$ or $p(x+q)=r$ .

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Shows an inconsistent relationship between the number of bracelets and the cost (including part a); shows a consistent relationship between those values, but the relationship does not fit $px+q=r$ or $p(x+q)=r$ ; three or more error types under Tier 3 response.

Teaching Notes

The expectation is for students to complete the table using numeric evaluation, but some students may write the expression $5.3x+250$ right away and use it. Similarly, some students will solve the equation $5.3x+250=1,500$ in the last part, while others will work backward from the given information.

Watch for students answering 235.8 or 236 instead of 235. These students may not be taking the time to contextualize (MP2), failing to apply the equation’s mathematical solution to the context.