Check Your Readiness

1.

An airplane flew across the Atlantic Ocean. The table shows the amount of time and the distance traveled when the airplane was traveling at a constant speed. Complete the table with the missing values. Explain or show your reasoning.

time (hours)	distance (miles)
2
3	1,230
6

Answer:

Explanations vary. Sample explanations:

Start with the last row. Find the missing value by doubling 1,230. Now find the distance corresponding to 2 hours by multiplying by $\frac{1}{3}$ or dividing by 3.

time (hours)	distance (miles)
2	$\frac{1}{3}\boldcdot(2,460)=820$
3	1,230
6	$2\boldcdot(1,230)=2,460$

Find the unit rate: that is, the number of miles the plane travels in 1 hour. Because 1 hour is $\frac{1}{3}$ of 3 hours, the plane will travel $\frac{1}{3}$ of 1,230 miles in 1 hour. This is 410 miles. This information can then be used to complete the table as shown.

time (hours)	distance (miles)
2	$2\boldcdot410=820$
3	$3\boldcdot410=1,230$
6	$6\boldcdot410=2,460$

Teaching Notes

In this unit, students are expected to use tables to solve problems involving constant speed. This context should be a familiar one from grade 6.

The numbers are constructed so that it is simple to scale up and then down, making scaling a more strategic choice than computing a unit rate. Look carefully at student work to see what methods are used to find the missing values. Two different methods are presented in the solution.

If most students struggle with this item, plan to first do the activity “Batches of Nihaizu Sauce” in Grade 6, Unit 2, Lesson 11 so that students can first see a simpler example of a table of equivalent ratios.

2.

Strawberries cost $3.00 per pound. For each question, explain or show your reasoning.

How many pounds of strawberries can you buy for $1.00?
How many pounds of strawberries can you buy for $11.00?

Answer:

$\frac{1}{3}$ or 0.33
Sample explanation:
$\frac{11}{3}$ or 3.67 (If students multiply 0.33 by 11, then they could also have 3.63.)
Sample explanation:

strawberries (pounds) cost (dollars)

$\frac{1}{3}$ 1

1 3

$\frac{11}{3}$ 11

strawberries (pounds)	cost (dollars)
$\frac{1}{3}$	1
1	3
$\frac{11}{3}$	11

Teaching Notes

The purpose of this assessment item is to see if students can find and use unit rates. Students are given one unit rate (number of dollars per pound), and then they need to use this to find the other unit rate (number of pounds per dollar). Teachers may wish to examine student work in the second question, which can be solved using either of the unit rates. If the unit rate for pounds per dollar is used, the second question is a multiplication problem. If the unit rate given in the task is used, the second question is a division problem. Either method works. Looking at student work also provides some insight into their thinking about ratios. For both questions, a double number line diagram or a table could be used strategically. If students use these representations correctly when not prompted, this foundational knowledge can be used as needed during the unit.

If most students struggle with this item, plan to emphasize techniques for finding both unit rates in the proportional relationship when solving problems in this unit. Show how both unit rates can be found when using representations like double number lines or tables of equivalent ratios.

3.

Clare made some lemonade by mixing 6 cups of water with 8 tablespoons of lemonade powder.

How many tablespoons of powder per cup of water is that?
How many cups of water per tablespoon of powder is that?

Answer:

$\frac{8}{6}$ (or equivalent)
$\frac{6}{8}$ (or equivalent)

Teaching Notes

This item assesses student understanding of unit rates. Although students are not asked to use the unit rates to find other values, they need to find both unit rates for the context: tablespoons per cup and cups per tablespoon. Students might use discrete diagrams, double number line diagrams, or tables to arrive at their responses.

If most students struggle with this item, plan to first do the activity "Cooking Oatmeal" in Grade 6, Unit 3, Lesson 5 to focus on ways to figure out both unit rates in a set of equivalent ratios.

4.

An area of 6 square yards is equal to 54 square feet. 9 square yards is equal to how many square feet? Explain or show your reasoning.

Answer:

81 square feet. Explanations vary. Sample explanations:

First find the unit rate (number of square feet per square yard) by dividing each number of square feet by the number of square yards: $54 \div 6=9$ . Then multiply the unit rate by 9 to yield 81 square feet in 9 square yards.
- area (square yards) area (square feet)
  
  3 27
  
  6 54
  
  9 81

area (square yards)	area (square feet)
3	27
6	54
9	81

Divide 6 and 54 by 2 to get the entries in the first row. Then multiply the entries in the first row by 3 to get the third row.

Teaching Notes

This task assesses students’ ability to find equivalent ratios in a non-scaffolded but familiar context with friendly numbers. Because the context is familiar, students may already know that there are 9 square feet in a square yard and multiply 9 by the unit rate 9 to find out how many square feet there are in 9 square yards. Another method is to use the given information—54 square feet in 6 square yards—and multiply by a factor of $\frac{3}{2}.$ This multiplication might occur in two steps (halving and then multiplying by 3). Yet another method is to note that 9, the desired number of square yards, is $6+\frac{6}{2}.$ In square feet, that corresponds to $54+\frac{54}{2}$ .

As with previous assessment items, double number line diagrams, double tape diagrams, or ratio tables could be used to complete the task.

If most students struggle with this item, plan to spend time ensuring students understand the structure of representations like double number lines and tables of equivalent ratios when opportunities arise throughout the unit.

5.

The ratio of the number of ducks to the number of turtles in a pond is $5:4$ . Draw a double number line diagram that would show the number of turtles if there were 25 ducks.

Answer:

Teaching Notes

Double number line diagrams provide a transition to graphing points in the coordinate plane, which becomes very important in grade 7 and later grades. These diagrams offer a little more flexibility than double tape diagrams, in that a wider range of values can be easily plotted on the line segments.

For many of the previous assessment items, double number line diagrams could be effectively used but are not necessary. This task provides a means of directly assessing whether students can use these important representations.

If most students do well with this item, it may be possible to skip the Warm-up activity in Lesson 1 since students already have an understanding of double number lines.

6.

The table shows pairs of coordinates. Plot these in the coordinate plane. Be sure to label the axes.

$x$	$y$
3	2
4	5
0	6

Answer:

The points (3,2), (4,5), and (0,6) are plotted, a scale is indicated, and the axes are labeled with the appropriate variable.

Teaching Notes

This task assesses whether students can graph points in the coordinate plane. They need to make a scale and label the axes.

If most students do well with this item, it may be possible to skip the Warm-up activity in Lesson 10. Students who already have a solid understanding of plotting coordinates would be able to move right on to plotting and interpreting the graphs.

7.

If you mix blue and yellow paint in different ratios, you will get different shades of green paint. If the ratios are equivalent, the shades of green will be the same.

Clare mixed a batch of green paint using 8 cups of yellow paint and 5 cups of blue paint.
Jada mixed another batch of green paint using 6 cups of yellow paint and 4 cups of blue paint.

Are these two batches the same shade of green? Explain.

Answer:

No, they are different shades of green. Clare’s paint is more yellow than Jada’s paint. Explanations vary. Sample explanations:

Find unit rates for each. For Clare’s green paint, there are 8 cups of yellow paint for every 5 cups of blue paint, so this means that in the mixture there are $\frac{8}{5}$ cups of yellow for every cup of blue paint. For Jada’s mixture, there are 6 cups of yellow for every 4 cups of blue or $\frac{6}{4}$ cups of yellow for each cup of blue. Since the two unit rates are different the two shades of green are different.
Find a batch of each mixture with the same amounts of one type of paint. These tables show two batches with the same amount of yellow paint. Here is a table for Clare’s mixture.

cups of yellow paint cups of blue paint

8 5

32 20

Here is a table for Jada’s mixture.

cups of yellow paint cups of blue paint

6 4

30 20

Looking at the second row of each table, we see that Clare’s mixture has two extra cups of yellow paint for each 20 cups of blue paint. So Jada’s green paint is a little bit more yellow than Clare’s green paint.

cups of yellow paint	cups of blue paint
8	5
32	20

cups of yellow paint	cups of blue paint
6	4
30	20

Teaching Notes

This item has less scaffolding than earlier items and requires an explanation. Students may use a variety of methods, but given the numbers, there are two likely choices. One is to compute and compare the unit rates for each batch. The other is to find a batch of Clare’s mixture and a batch of Jada’s mixture that have the same amount of blue paint (or same amount of yellow paint) to find out if the two batches also have the same amount of the other color of paint.

Success with this type of problem may be a good indication that students are ready for the grade 7 material on ratios and proportional relationships.

If most students struggle with this item, plan to spend time ensuring students understand representations of sets of equivalent ratios as opportunities arise. Success with this type of problem may be a good indication that students are ready for the grade 7 material on ratios and proportional relationships.