Check Your Readiness

1.

Noah is three years younger than four times the age of his brother. Select all the equations that correctly represent the relationship between Noah’s age $n$ and his brother’s age $b$ .

A.

$n=3-4b$

B.

$n=4b - 3$

C.

$n = 4(b-3)$

D.

$b=\frac{n+3}{4}$

E.

$b=4n−3$

F.

$b= \frac{n}{4}+ 3$

Answer: B, D

Teaching Notes

This problem is designed to expose some common errors that arise when writing equations to represent the relationship between quantities. Choices B and D are correct. Students who made other choices may have made these errors. Students who select choice A may have incorrectly wrote the terms of the expression in the order they were given. Choice C represents the statement “Noah is four times the age of someone three years younger than his brother.” Choice E switches the ages of Noah and his brother. Choice F is close but incorrect. The correct equation can be separated into two fractions to obtain $b = \frac{n}{4}+ \frac34$ .

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.

A rectangular prism has length $\ell$ , width $w$ , and height $h$ . Select all the statements that must be true.

A.

The volume is $(\ell w h)^2$ .

B.

The volume is $2(\ell w h)$ .

C.

The volume is $(\ell w h)$ .

D.

The surface area is $2(\ell wh)$ .

E.

The surface area is $2 (\ell w) +2(wh) + 2(\ell h)$ .

F.

The surface area is $2 (\ell w) \boldcdot 2(wh) \boldcdot 2(\ell h)$ .

Answer: C, E

Teaching Notes

If students have trouble here, ask them to explain their choices. Their difficulties may involve interpreting the variables rather than the geometry or the formulas.

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.

3.

Select all the proportional relationships.

A.

<p>A graph. Rides taken. Number of tickets. </p>

B.

$x$	$y$
0	6
3	9
6	12
9	15
12	18

C.

day	1	2	3	4
number of cell phones sold	5	10	15	20

D.

$y=\frac13 x$ , where $x$ and $y$ are both positive numbers.

E.

Clare borrowed $100 from her father. She pays him back $10 each week. $M$ is the amount of money Clare still owes, and $W$ is the number of weeks: $100 - 10W = M$ .

Answer: A, C, D

Teaching Notes

Work with linear functions, which begins in Lesson 8, builds off of students’ understanding of proportional relationships. Some students may think that choice A does not represent a proportional relationship since the point $(0,0)$ is not explicitly marked on the graph. Choices B and E are both examples of linear relationships that are not proportional. Students who do not select choice D may be thrown by the fact that the constant of proportionality is a unit fraction (because they know that $y=\frac{1}{x}$ is not 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.

4.

A recipe for salad dressing calls for $\frac13$ tablespoon of vinegar for every tablespoon of oil. The equation $V =\frac13L$ gives the amount of vinegar needed ( $V$ ) in terms of the amount of olive oil used ( $L$ ).

Write another equation describing the recipe, this time giving $L$ in terms of $V$ .

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

Answer:

$L=3V$ or $L = \frac{V}{\frac13}$

Teaching Notes

This problem gives a proportional relationship using an equation of the form $y=\frac{1}{k}x$ . In solving the problem, students may isolate the variable algebraically, or they may simply recall that an equation of this type can be rewritten in the equivalent form $x=ky$ .

If most students struggle with this item, during Activity 2 of Lesson 8, 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.

5.

Given the equation $y=\text-5x -(1.5)$ :

When $x$ is 2, what value of $y$ makes the equation true?
When $x$ is -2.5, what value of $y$ makes the equation true?
When $y$ is -16.5, what value of $x$ makes the equation true?

Answer:

-11.5
11
3

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 parts a and b ask for the value of $y$ , part c asks for the value of $x$ .

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.

6.

Here is a rectangle:

What is the area of the rectangle?
What is the perimeter of the rectangle?

Answer:

28 cm²
22 cm

Teaching Notes

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 units.

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.

7.

The area of a circular table is $225\pi$ in². The area, $A$ , of a circle with radius $r$ is $A=\pi r^2$ . The circumference, $C$ , of the circle is $C=2 \pi r$ .

What is the radius of the table?
What is the circumference of the table, to the nearest inch?
What is the diameter of the table?

Answer:

15 inches. (Solving for the radius, $r$ , in the equation $\pi r^2 = 225\pi$ gives $r^2=225$ and $r=15$ .)
94 square inches. ( $2\pi r = 30 \pi$ , then use an approximation for $\pi$ .)
30 inches. (twice the radius)

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 $\pi$ , the work is much harder.

If most students struggle with this item, spend additional time on Lesson 13, Activity 1, and revisit this item at the end of Activity 1. Students may be less familiar with using $\pi$ in a reported area (225 $\pi$ in²) 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.