Mid-Unit Assessment

1.

This hanger diagram is balanced. There are two labeled weights of 4 grams and 12 grams. The three circles each have the same weight.

What is the weight of each circle, in grams?

A balanced hanger diagram. There are 3 identical circles labeled x and a rectangle labeled 4 on the left side and 1 pentagon labeled 12 on the right side.

A.

$\frac38$

B.

1

C.

$\frac83$

D.

8

Answer:

$\frac83$

Teaching Notes

Many students will solve this problem by writing the equation $3x+4=12$ , though reasoning purely about the weights on the hanger diagram is also fine. Students selecting A likely made a division mistake in the last step of their algebra. Students selecting B probably divided by 4 instead of subtracting. Students selecting D may have stopped their equation solving at $3x=8$ , or they may have looked at the hanger and imagined isolating all the weights labeled $x$ , without dividing by 3.

2.

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

Tape diagram, 5 equal parts each labeled w, 1 small part labeled 4, total 99.

A.

Clare buys 4 bouquets, each with the same number of flowers. The florist puts an extra flower in each bouquet before she leaves. Clare leaves with a total of 99 flowers.

B.

Andre babysat 5 times this past month and earned the same amount each time. To thank him, the family gave him an extra $4 at the end of the month. Andre earned $99 from babysitting.

C.

A family of 5 drove to a concert. They paid $4 for parking, and all of their tickets were the same price. They paid $99 in total.

D.

There are 5 bags that each contain 4 large marbles and the same number of small marbles. Altogether, the bags contain 99 marbles.

E.

Han is baking five batches of muffins. Each batch needs the same amount of sugar in the muffins, and each batch needs four extra teaspoons of sugar for the topping. Han uses a total of 99 teaspoons of sugar.

Answer: B, C

Teaching Notes

The tape diagram represents the equation $5w+4=99$ .

Choice A would be correct if there were only four blocks of length $w$ in the tape diagram: students making this choice may have miscounted, since the four extra flowers fit with the block of length 4 at the end. Students failing to select B or C may be interpreting those situations in ways that can be described using the equation $5(w+4)=99$ , rather than $5w+4=99$ . Likewise, students selecting D or E may incorrectly believe that those situations are of a type that can be represented using the equation $5w+4=99$ .

3.

At practice, Mai does twice as many push-ups as Noah, and she also does 40 jumping jacks. Mai does 62 exercises in total. The equation $2x+40=62$ represents this situation. What does the variable $x$ represent?

A.

the number of jumping jacks Mai does

B.

the number of push-ups Mai does

C.

the number of jumping jacks Noah does

D.

the number of push-ups Noah does

Answer:

the number of push-ups Noah does

Teaching Notes

Students selecting B may be confused specifically about how to represent the statement “Mai does twice as many push-ups as Noah.” Students selecting A or C have made a mistake reading the problem: we already know Mai does 40 jumping jacks, and we do not know whether Noah does any push-ups at all.

4.

Solve each equation.

$25 - 3x = 40$
$\frac 1 3 (x - 10) = \text-4$

Answer:

$x = \text-5$
$x = \text-2$

Teaching Notes

Some students may struggle with the form of the equation in part a: After subtracting 25, is it $3x$ or $\text-3x$ that remains? The most likely error in part b is forgetting to properly distribute, though some students may multiply each side by 3 instead to get $x - 10 = \text{-}12$ .

5.

Andre tried to solve the equation $\frac 1 4 (x + 12) = 2$ .

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

$\frac 1 4 (x + 12) = 2$

$\frac{1}{4}x +12 -12 = 2 -12$

$\frac14x = \text-10$

$\frac14x \div \frac14= \text-10 \div \frac14$

$x = \text-40$

Answer:

Students circle $\frac{1}{4}x +12 -12 = 2 -12$ . Sample response: Andre subtracted 12 from each side, but on the left side of the original equation, the 12 is being multiplied by $\frac14$ . He needed to either first distribute the $\frac14$ or first divide each side by $\frac14$ .

Minimal Tier 1 response:

Work is complete and correct.
Sample: Andre subtracted 12 from each side, but that’s wrong because the 12 is in the parentheses. He should have done $12 \boldcdot \frac14$ and then subtracted.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Identifies the error correctly, but the explanation is either incorrect or vague; identifies the problem step by substituting $x=\text{-}40$ at each step, but no algebra mistake is identified; solves the equation correctly but does not identify Andre’s error.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Identifies an incorrect problem step as 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 food pantry is making packages. Each package weighs 64 pounds.

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

Each package contains 4 boxes. Each box contains a 7-pound bag of beans and a bag of rice. The bags of rice are all identical.
Each package contains 4 identical bags of rice and a 7-pound bag of beans.

Answer:

Sample response:

$4(x+7)=64$ (diagram shows 4 equal parts of $x+7$ with a total of 64)
$4x+7=64$ (diagram shows 4 equal boxes labeled $x$ and one box labeled 7, with a total of 64)

Minimal Tier 1 response:

Work is complete and correct.
Sample:

$4(x+7)=64$ .
$4x+7=64$ .

Tier 2 response:

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

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Writes both equations incorrectly or both responses are flawed in some way.

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.

Elena will make T-shirts for a school fundraiser. She will order T-shirts and print graphics on them. Elena must spend $349 on a printing machine and $4.80 per shirt for the blank shirts, ink, and other supplies.

Complete the table giving the total cost Elena will spend to make each specific number of T-shirts.
Write an expression for the cost of making $n$ T-shirts.
What is the maximum number of T-shirts Elena can make with a budget of $1,000?

number of shirts	cost in dollars
20
40
60

Answer:

number of shirts cost in dollars

20 445

40 541

60 637
$4.8n + 349$ (or equivalent)
135 T-shirts. Sample reasoning: Since the printing machine costs $349, Elena will have $651 left to spend on the shirts. Each shirt costs $4.80 to produce. Since $\frac{651}{4.8}$ is about 135.6, Elena can make a maximum of 135 T-shirts. She can’t make 136.

number of shirts	cost in dollars
20	445
40	541
60	637

Minimal Tier 1 response:

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

See table.
$4.8n + 349$ .
The equation is $4.8n + 349 = 1000$ . $4.8n = 651$ , so $n=135.625$ . She can make 135 shirts.

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 for part c but contains a calculation error; writes correct equation for part c but contains an arithmetic error; gives response of 135.6 or 136 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 number of shirts and cost (including part a); shows a consistent relationship between those values, but the relationship does not fit $px+q=r$ or $p(x+q)=r$ ; has three or more error types under Tier 3 response.

Teaching Notes

The expectation is for students to fill in the table using numeric evaluation, but some students may write the expression $4.8x + 350$ right away and use it. Similarly, some students will solve the equation $4.8x + 350 = 1,000$ in the last part, while others will work backward from the given information.

Watch for students answering 135.6 or 136 instead of 135. These students may not be taking the time to contextualize (MP2), failing to apply the equation’s mathematical solution to the context.