Grade 7

Readiness Check

Check Your Readiness

Jada volunteers at an animal shelter that holds picnics to introduce people to animals. For the first picnic, she brought 36 clementines for 16 people, which was the right amount. For the next picnic, there will be 40 people! How many clementines should Jada bring for all those people? Explain or show your reasoning. (Note: clementines are fruits that are like small oranges.)

Answer:

90 clementines. That is $\frac{36}{16}=2.25$ clementines per person. Sample reasoning: If $y$ is the number of clementines needed and $x$ is the number of people attending, then $y=2.25x$ . When $x=40$ , the equation $y=(2.25)\boldcdot40$ can be used to find that 90 clementines are needed.

Teaching Notes

In an earlier unit, students were introduced to proportional relationships, and they learned to use them to find equivalent ratios.

If most students struggle with this item, plan to use Lesson 2, Activity 2, to review strategies for working with ratios. While monitoring their work, notice which students are using each strategy listed in the lesson. During the Activity Synthesis, point out the connections between the shared strategies to support their understanding of the methods used. Students who struggle with the calculation may benefit from a more visual strategy, such as using a double number line or a table.

Solve each problem. Explain or show your reasoning.

Lin is baking cakes. Each cake needs $\frac34$ cup of brown sugar, and Lin uses $4\frac12$ cups of brown sugar for the entire batch of cakes. How many cakes does Lin bake?
$4\frac12$ cups of salad dressing fill $3\frac34$ identical jars. How many cups fill each jar?

Answer:

6 cakes. Sample reasoning: $4\frac12$ cups is the same as $\frac92$ cups (or equivalent), and $\frac{18}{4}\div\frac34=6$ .
$\frac65$ cups (or equivalent). Sample reasoning: $4\frac12\div3\frac34=\frac65$ .

Teaching Notes

In this unit, students will be asked to extend their understanding of ratios and unit rates to fractional quantities. This item assesses how well they are able to divide fractions, a grade 6 standard. In part a, students may find it easier to think visually or conceptually about how many times $\frac34$ goes into $4\frac12$ . The division in part b is harder to do without using the standard algorithm.

If most students struggle with this item, plan to use the Math Talk in Lesson 2, Activity 1, to review dividing a fraction by a fraction. This lesson also offers opportunities to practice this calculation within a context. If students who struggled did not use diagrams for making sense of the problem, monitor for and display the work of students who used diagrams.

Which of these expressions is equivalent to $0.04(x − 25)$ ?

$0.04x - 1$

$0.04x - 1.04$

$0.04x - 25$

$0.04x - 25.04$

Answer:

$0.04x - 1$

Teaching Notes

In this unit, students use the distributive property to analyze statements relating to percent increase and decrease, like “Kiran ran three miles, then half as much again.”

If most students struggle with this item, plan to use Lesson 4 and the lessons that follow to review and solidify using the distributive property in a variety of situations where students write equivalent expressions and find percentages.

Tyler uses 7.5 ounces of barbeque sauce from a full bottle. He estimates that he used about 25% of the barbeque sauce in the bottle. About how much barbecue sauce was in the full bottle? Explain or show your reasoning.
Tyler uses 20% of the remaining barbecue sauce. About how many ounces did he use the second time? Explain or show your reasoning.

Answer:

Answers vary due to the estimation described.

30 ounces. Sample reasoning: 7.5 is 25% of 30, so there are 30 ounces in a full bottle.
4.5 ounces. Sample reasoning: The remaining amount of BBQ sauce after the first pour is 22.5 ounces, and 20% of 22.5 is 4.5.

Teaching Notes

It is not required that students find exact solutions to this problem. However, the problem is designed to reveal the ways in which students are thinking about two common problem types: how to find a whole, given a part and a percentage, and how to find a given percentage of a given value. Students will study these types of questions more formally in this unit. Students who solve the problem exactly will discover that the two pours are the same amount of sauce. A tape diagram can help them understand why.

If most students struggle with this item, plan to use Lesson 6, and the lessons that follow to study these two types of percentage problems. In Lesson 6, Activity 2, students will begin thinking about percent increase and percent decrease situations.

To solve the problem “48 is what percent of 80?” Elena uses the double number line shown here. On the double number line, write the correct numbers at each unlabeled tick mark. Then solve the problem.

Answer:

The top tick marks are labeled 48 and 64, the bottom tick marks are labeled 60% and 80%. Sample response: 48 is 60% of 80. Because 16 is 20% of 80, tick marks on the top should be marked in multiples of 16, while tick marks on the bottom should be marked in multiples of 20. Working backwards from 80 and 100%, the first pair of marks are 64 and 80%, and the second pair are 48 and 60%. This means that 48 is 60% of 80.

Teaching Notes

A student may accidentally use a different but correct scale for the tick marks: one such example is to label the top tick marks 64 and 72, and the bottom tick marks 80 and 90. This is unlikely but correct.

If most students struggle with this item, plan to monitor for students who use a double number line to solve Lesson 6, Activity 2. Then use the Activity Synthesis of Lesson 6, Activity 2, to review using double number lines to calculate the percentage. These ideas will continue throughout the next several lessons.

Solve each problem. Explain or show your reasoning.

14 is 50% of what number?
24 is 120% of what number?
18 is what percent of 30?
What is 25% of 80?
What is 41% of 200?
What is 150% of 26?

Answer:

28. Sample reasoning: This question can be read as “14 is half of what number?” Doubling 14 gives the answer of 28.
20. Sample reasoning: Draw a double number line, with 4, 8, 12, 16, 20, 24 marked on the top line, and 20, 40, 60, 80, 100, 120 marked on the bottom line. 20 is marked above 100. Therefore, 24 is 120% of 20.
60%. Sample reasoning: $18\div30=0.6$ , so 18 is 60% of 30.
20. Sample reasoning: 25% of a number is $\frac14$ of that number. Then $80\boldcdot\frac14=20$ .
82. Sample reasoning: Multiply using a decimal: $200\boldcdot(0.41)=82$ . Alternatively: 41 is 41% of 100, so 82 must be 41% of 200.
39. Sample reasoning: 150% of a number is “the number, plus half again”: $26+13=39$ .

Teaching Notes

This problem provides an opportunity to assess students’ “go-to” strategies for solving percent problems. Some students who are able to reason their way through the previous two questions may have more difficulty with these problems, since no context is provided.

If most students struggle with this item, plan to review these strategies using this problem in the days leading up to Lesson 6. You can find similar problems in the practice problems of Grade 6, Unit 3, Lessons 10–16.