Check Your Readiness

1.

Diego made 6 boxes of spaghetti to feed the 20 people who attended his dinner party. Next time, 50 people are coming! How many boxes of spaghetti should Diego buy to feed all those people? Explain or show your reasoning.

Answer:

15 boxes.

Sample reasoning:

That is $\dfrac{6}{20} = 0.3$ box per person. So if $y$ is the number of boxes needed and $x$ is the number of people attending, then $y = 0.3x$ . When $x = 50$ , the equation $y = (0.3) \boldcdot 50$ can be used to find that 15 boxes are needed.

Teaching Notes

In an earlier unit, students were introduced to proportional relationships and 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.

2.

Solve each problem. Explain or show your reasoning.

Elena is feeding her neighbor’s dogs. Each dog gets $\frac23$ cup of dog food, and she uses $3\frac13$ cups of food. How many dogs does her neighbor have?
$5\frac58$ cups of water fill $4\frac12$ identical water bottles. How many cups fill each bottle?

Answer:

5 dogs. Sample reasoning: $3\frac13$ cups is the same as $\frac{10}{3}$ cups, and $\frac{10}{3} \div \frac23 = 5$ .
$\frac54$ cups (or equivalent). Sample reasoning: $5\frac58 \div 4\frac12 = \frac54$ .

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 $\frac23$ goes into $3\frac13$ . The division in part b is harder to do when the standard algorithm is not used.

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.

3.

Which of these expressions is equivalent to $0.05(x - 40)$ ?

A.

$0.05x - 40.05$

B.

$0.05x - 40$

C.

$0.05x - 2.05$

D.

$0.05x - 2$

Answer:

$0.05x - 2$

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.

4.

Diego pours 6.8 ounces of water from a full bottle. He estimates that he poured out 20% of the water in the bottle. About how much water was in the full bottle? Explain or show your reasoning.
Diego pours out 25% of the remaining water. About how many ounces did he pour the second time? Explain or show your reasoning.

Answer:

Answers vary due to the estimation described.

34 ounces. Sample reasoning: 6.8 is 20% of 34, so there are 34 ounces.
6.8 ounces. Sample reasoning: The remaining amount of water after the first pour is 27.2 ounces, and 25% of 27.2 is 6.8.

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 water. 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.

5.

To solve the problem “56 is what percent of 70?” Noah 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 56 and 63, the bottom tick marks are labeled 80% and 90%.

Sample response: 56 is 80% of 70.

Because 7 is 10% of 70, tick marks on the top should be marked in multiples of 7, while tick marks on the bottom should be marked in multiples of 10. Working backwards from 70 and 100%, the first pair of marks are 63 and 90%, and the second pair are 56 and 80%. This means that 56 is 80% of 70.

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 42 and 56, and the bottom tick marks 60 and 80. 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.

6.

Solve each problem. Explain or show your reasoning.

What is 25% of 160?
What is 39% of 200?
What is 150% of 32?
13 is 50% of what number?
18 is 120% of what number?
21 is what percent of 30?

Answer:

40. Sample reasoning: 25% of a number is $\frac14$ of that number. And $160\boldcdot \frac14=40$ .
78. Sample reasoning: Multiply using a decimal: $200 \boldcdot (0.39)=78$ . Alternatively, 39 is 39% of 100, so 78 must be 39% of 200.
48. Sample reasoning: 150% of a number is "the number plus half again": $32+16=48$ .
26. Sample reasoning: This question can be read as "13 is half of what number?" Doubling 13 gives 26.
15. Sample reasoning: Draw a double number line with 3, 6, 9, 12, 15, 18 marked on the top line and 20, 40, 60, 80, 100, 120 marked on the bottom line. 15 is marked above 100. Therefore, 18 is 120% of 15.
70%. Sample reasoning: $21 \div 30 = 0.7$ , so 21 is 70% of 30.

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 by 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.