Check Your Readiness

1.

Select all the tables that could represent proportional relationships.

A.

x	y
2	3
5	7.5
10	15

B.

x	y
0	0
3	7
6	14

C.

x	y
0	2
2	4
4	6

Answer: A, B

Teaching Notes

In this unit, students review previous work with proportional relationships as a lead-in to linear equations.

If most students struggle with this item, plan to use this problem or a similar one as an additional warm-up activity. During Lessons 1 and 2 plan to emphasize multiple ways to identify whether a relationship is proportional, such as finding a constant of proportionality using a table of values and using coordinates of points on the graph.

2.

To mix a particular shade of purple paint, red paint and blue paint are mixed in the ratio $5:3$ . To make 20 gallons of this shade of purple paint, how many gallons of red and blue paint should be used?

Answer:

12.5 gallons red, 7.5 gallons blue

Teaching Notes

Students move from scale factors to proportional relationships in preparation for linear relationships. If most students struggle with this item, before beginning Lesson 1, do Grade 6, Unit 2, Lesson 16, Activity 2 to practice the concept of generating equivalent ratios.

3.

At one gas station, gas costs $2.75 per gallon. Write an equation that relates the total cost, $C$ , to the number of gallons of gas purchased, $g$ .

Answer:

$C=2.75g$ (or equivalent)

Teaching Notes

In grade 7, students wrote equations to describe proportional relationships. The graphs of these equations are lines through the origin. In this unit, students will write equations for proportional relationships as well as other linear relationships.

If most students struggle with this item, plan to do Lesson 1, Activity 3. During the Activity Synthesis spend some extra time sharing student equations and making connections to the tick-mark diagram.

4.

Plot and label 3 different points with $x$ -coordinate 3.
Plot and label 3 different points with $y$ -coordinate -5.

Answer:

Any 3 points with $x$ -coordinate 3 plotted and labeled. Sample responses: $(3,0), (3,\text -2), (3,4)$
Any 3 points with $y$ -coordinate -5 plotted and labeled. Sample responses: $(0,\text-5), (3,\text-5), (\text-4,\text-5)$

Teaching Notes

Students will need to be familiar with the coordinate plane to graph lines.

If most students struggle with this item, plan to pause students as they are working on Lesson 1, Activity 2, Question 4 to ensure that they can plot and mark points once they have identified the bug's location at the given time. If students need additional practice, refer to Grade 6, Unit 7, Lesson 11, Activity 1.

5.

On the coordinate plane, draw:

A line $m$ that is a translation of line $\ell$ .
A line $n$ that is a rotation of line $\ell$ , using the origin as the center of rotation.

Answer:

Any line parallel to $\ell$
Any line through the origin

Sample response:

Teaching Notes

In this unit, students are presented with various forms of linear equations and various ways of thinking about those forms. One interpretation is to consider nonproportional linear equations as vertical translations of the line $y=mx$ .

If most students struggle with this item, plan to use Activity 1 in Lesson 8 to review translations. If students need additional practice recalling translations, especially translations of lines, refer to Unit 1, Lesson 9, Activity 2.

6.

A store sells ice cream with assorted toppings. They charge $3.00 for an ice cream, plus $0.50 per ounce of toppings.

How much does an ice cream cost with 4 ounces of toppings?
How much does an ice cream cost with 11 ounces of toppings?
If Elena’s ice cream costs $1.50 more than Jada’s ice cream, how much more did Elena’s toppings weigh?

Answer:

$5.00
$8.50
3 ounces

Teaching Notes

Another interpretation of a linear equation is to start with a given amount and thereafter increase the amount at a constant rate. Students are asked to engage in repeated reasoning in anticipation of this way of thinking.

If most students struggle with this item, plan to review it with students before beginning Lesson 6, Activity 2 and amplify vocabulary such as "constant of proportionality" and "rate of change" starting in Lesson 3 Activity 1.