Check Your Readiness

1.

Jada is three years older than twice her brother’s age. Select all the equations that correctly represent the relationship between Jada’s age $j$ and her brother’s age $b$ .

A.

$j=2b+3$

B.

$j=2(b+3)$

C.

$j=\frac b 2 -3$

D.

$b=2j+3$

E.

$b=\frac{j-3} {2}$

F.

$b=\frac j 2 - 3$

Answer: A, E

Teaching Notes

This problem is designed to expose some common errors that arise when writing equations to represent the relationship between quantities. Choices A and E are correct.

Students who made other choices may have made these errors. Choice B represents the statement “Jada is twice the age of someone three years older than her brother.” In choice C, the operations are done in the correct order, but they are the inverse of the operations required. Choice D incorrectly switches the ages of Jada and her brother. Choice F is close but incorrect. The correct equation, $b=\frac{j-3} 2$ , can be separated into two fractions to obtain $b = \frac{j}{2} - \frac{3}{2}$ .

If most students struggle with this item and Activity 3 of Lesson 3, make time to discuss the first and third practice problems of Lesson 3. Prior to this lesson, the third practice problem in Lesson 1 also provides an opportunity to review writing equations in two variables. Offer tape diagrams as a tool from previous work to help students who struggle to connect equations and situations.

2.

Select all the proportional relationships.

A.

Coordinate plane, x, 0 to 6 by ones, y, 0 to 6 by ones. Line begins at 0 comma 1 and extends through 2 comma 6.

B.

A train travels at a constant speed of 60 miles per hour. The number of hours the train traveling is $t$ . The number of miles the train travels is $d$ .

C.

The relationship is represented by this table:

x	y
3	6
4	12
5	24

D.

$y=3x$ , where $x$ and $y$ are both positive numbers

E.

$y = \frac 1 x$

Answer: B, D

Teaching Notes

Work with linear functions, which begins in Lesson 8, builds off of students’ understanding of proportional relationships. Choice A is worth special attention: It is an example of a linear function that does not represent a proportional relationship.

If most students struggle with this item, revisit the definition of proportional relationships at the start of Lesson 8, Activity 2. Note that the lesson "Connecting Representations to Functions" has an optional activity that reviews connecting tables, situations, and equations.

3.

There are 16 cups in a gallon. The equation $c = 16g$ gives the number of cups in terms of the number of gallons. Write another equation for this situation, giving the number of gallons in terms of the number of cups:

$\displaystyle g=\underline{\hspace{1in}}$

Answer:

$g = \frac{1}{16} c$ (or equivalent)

Teaching Notes

This problem gives a proportional relationship using an equation of the form $y=kx$ . In solving the problem, students are prompted to remember that this equation can be written in the equivalent form $x=\frac{1}{k} y$ .

If most students struggle with this item, during Lesson 8, Activity 2, discuss the connection between the two equations that are possible for each part of the activity. Note that working with equations to think about relationships between properties, for example, volume and height for a fixed radius, relies on being able to rewrite simpler proportions for a targeted value. Practice this in the work up to Lessons 14 and 16.

4.

Given the equation $y=\text-3x+2.5$ :

When $x$ is 1, what value of $y$ makes the equation true?
When $x$ is -1.5, what value of $y$ makes the equation true?
When $y$ is 8.5, what value of $x$ makes the equation true?

Answer:

-0.5
7
-2

Teaching Notes

When students use equations to find input or output values of functions, they will need to substitute numbers for variables. This problem also assesses signed number arithmetic. Check to make sure students notice that while the first two parts ask for values of $y$ given $x$ , the third part asks for the value of $x$ given $y$ .

If most students struggle with this item, make time before the Launch of Activity 3 in Lesson 3 to ensure that students understand the meaning in context when substituting for one variable to determine the value of the other.

5.

Here is a rectangular prism.

Rectangular prism, dimensions 8 inches, 1 point 5 inches, 2 point 4 inches.

What is the surface area of the prism?
What is the volume of the prism?

Answer:

69.6 in $^2$ . ( $2 \boldcdot (1.5) \boldcdot (2.4) + 2 \boldcdot (1.5) \boldcdot 8 + 2 \boldcdot (2.4) \boldcdot 8$ )
28.8 in $^3$ . ( $(1.5) \boldcdot (2.4) \boldcdot 8$ )

Teaching Notes

The content assessed in this problem is first encountered in Lesson 12: How Much Will Fit?. Verify that students perform the right calculation and also use the appropriate units. This is a good problem to remind students about the notation for square and cubic units.

If most students struggle with this item, after Lesson 12, use the first and second practice problems to review how to find the volume of a rectangular prism.

6.

A circular field has area $14,400\pi$ square feet. The area, $A$ , of a circle with radius $r$ is $A=\pi r^2$ . The circumference, $C$ , is $C=2 \pi r$ .

What is the radius of the field?
What is the diameter of the field?
What is the circumference of the field, to the nearest foot?

Answer:

120 feet. Solving for the radius, $r$ , in the equation $\pi r^2 = 14,400 \pi$ gives $r^2 = 14,400$ and $r = 120$ . The radius must be positive.)
240 feet. (twice the radius)
754 feet. ( $C = 240\pi$ , then use an approximation for $\pi$ .)

Teaching Notes

This problem is especially helpful for explaining why an approximation for $\pi$ should only be used when necessary. If students begin by approximating $\pi$ to rewrite $14,400\pi$ , the work is much harder.

If most students struggle with this item, revisit this item at the end of Lesson 13, Activity 1. Students be less familiar with using $\pi$ in a reported area ( $14,400\pi$ square feet) than giving an approximation of the area. Discuss how this is one way to report measures and how knowing that $\pi$ is a little more than 3 gives us an idea of which whole numbers the measure is close to.

7.

A rectangle has length $x$ and width $y$ .

Select all the statements that must be true.

A.

The perimeter is $x + y$ .

B.

The perimeter is $xy$ .

C.

The perimeter is $2(x+y)$ .

D.

The perimeter is $2xy$ .

E.

The perimeter is $2x + 2y$ .

F.

The area is $x + y$ .

G.

The area is $xy$ .

H.

The area is $2xy$ .

Answer: C, E, G

Teaching Notes

If most students struggle with this item, as opportunities arise in Lesson 3, spend additional time reviewing what students already know about calculating measures of geometric figures. This item assesses students' knowledge of both area and perimeter (useful when working on volume in the latter part of the unit) and reasoning about geometric formulas using variables.