Check Your Readiness

1.

A ribbon is 24 inches long. You need to cut it into pieces so that all the pieces are the same length and each piece is a whole number of inches.

Can you cut the ribbon into 5-inch pieces?
What are all the possible whole-number lengths of the equal-length pieces?

Answer:

No.
1, 2, 3, 4, 6, 8, and 12 inches.

Teaching Notes

In this unit, students use contextual situations to develop strategies and definitions for common factors, the greatest common factor, common multiples, and the least common multiple. This problem assesses how they use the factors of one number to solve a real-world situation.

If most students struggle with this item, plan to spend additional time on Activity 2, and provide manipulatives to help students act out the problem to understand the context.

2.

Select all the numbers that point $A$ could reasonably represent on this number line.

A number line with 7 evenly spaced tick marks, labeled 0 through 6 are indicated. Point A is plotted on the number line, more than halfway between 1 and 2.

A.

0.65

B.

1.3

C.

1.5

D.

1.6

E.

1.65

Answer: D, E

Teaching Notes

In this unit, students plot signed rational numbers on number lines. This problem requires understanding that the value must be larger than 1.5 because it is closer to 2 than to 1, but also that it is not possible to identify, by sight, the exact position of a point on the number line.

If most students struggle with this item, plan to allow for more practice with number lines without tick marks. Activity 1 provides an opportunity to use landmark numbers in reasoning about placement.

3.

Here is a number line with some points labeled.

<p>Number line, tick marks labeled 0, blank, A, blank, 1, B, blank, blank, 2, blank, blank, C, 3.</p>

Write the number at $A$ as a fraction.
Write the number at $B$ as a decimal.
Write the number at $C$ in two different ways: as a fraction and as a decimal.

Answer:

$\frac{1}{2}$ (or equivalent)
$1.25$
$2.75$ or $\frac{11}{4}$ (or equivalent)

Teaching Notes

In this unit, students plot signed rational numbers on number lines. This problem assesses whether students can identify fractions and decimals at tick marks.

If most students struggle with this item, plan for more practice with number lines, including lines with and lines with tick marks. Activity 1 provides an opportunity to use landmark numbers in reasoning about placement.

4.

For each pair of numbers, fill in the blank with $<$ , $=$ , or $>$ .

$1.41 \underline{\hspace{.5in}} 1.397$
$1.5 \underline{\hspace{.5in}} \frac{3}{2}$
$\frac{4}{3} \underline{\hspace{.5in}} \frac{5}{3}$
$\frac{4}{5} \underline{\hspace{.5in}} \frac{4}{7}$
$\frac{3}{4} \underline{\hspace{.5in}} \frac{6}{8}$
$\frac{4}{9} \underline{\hspace{.5in}} \frac{3}{5}$

Answer:

$1.41 > 1.397$
$1.5 = \frac{3}{2}$
$\frac{4}{3}$ < $\frac{5}{3}$
$\frac{4}{5}$ > $\frac{4}{7}$
$\frac{3}{4}$ = $\frac{6}{8}$
$\frac{4}{9}$ < $\frac{3}{5}$

Teaching Notes

In this unit, students determine the relative order and magnitude of signed rational numbers and then express this with inequality statements. This problem assesses whether students can use the structure of decimals and fractions to determine which of two values is “less than” or “greater than.”

If most students struggle with this item, plan to share examples of common incorrect responses, and analyze the errors after doing Activity 1.

5.

Andre and Mai leave the post office at the same time. They walk in opposite directions. Andre walks 50 feet, and Mai walks 30 feet. When they stop walking, how far apart are they? Explain or show your reasoning.

Answer:

80 feet. Sample reasoning: Since they walk in opposite directions, we need to add the distances they each travel.

Teaching Notes

In this unit, students reason about the meaning of positive and negative numbers in context. This problem assesses students’ capacity to reason about distances in opposite directions.

If most students struggle with this item, plan to revisit this item during the Activity Synthesis of Activity 3. Invite students to share how they could draw a diagram to illustrate this situation. Invite students to share how they connect “opposite direction“ to “opposites“ and to share why a number line is helpful for thinking about this situation. Students will have additional opportunities throughout the unit to practice illustrating situations and finding distances using number lines.

6.

Find the coordinates of each point.

<p>Coordinate plane, origin O, marked by ones. Point A is 3 right, 3 up. Point B is 1 right, 4 up. Point C is 4 right, 1 up. Point D is 2 right. Point E is on the veritcal axis halfway between 1 and 2.</p>

Answer:

$A (3,3), B (1,4), C (4,1), D (2,0), E (0,1.5)$

Teaching Notes

In this unit, students plot signed coordinate pairs in all four quadrants of the coordinate plane. This problem assesses if students can identify the coordinates of points in the first quadrant.

If most students struggle with this item, plan to address any misconceptions during Activity 1. During the Activity Synthesis, share several student responses with an emphasis on the order of an ordered pair.

7.

On a coordinate grid, the $x$ -axis represents the age of a plant in weeks, and the $y$ -axis represents the height of the plant in inches.

Blank coordinate plane, origin O, horizontal axis, age of plant in weeks, 0 to 9 by ones, vertical axis, height of plant in inches, 0 to 7 by ones.<br>

After 1 week, the plant was 2 inches tall. Plot and label this point $A$ .
After 2 weeks, the plant was 4 inches tall. Plot and label this point $B$ .
After 3 weeks, the plant was 4.5 inches tall. Plot and label this point $C$ .
Plot a point $D$ that you think could represent the plant’s age and height after 4 weeks. Give the plant’s age and height at that time.

Answer:

See image.
See image.
Sample response: $D (4,5)$ indicates that after 4 weeks, the plant is 5 inches tall. Point $D$ ’s $x$ -coordinate must be 4.

Teaching Notes

In this unit, students plot points in all four quadrants of a coordinate plane to represent a situation. This problem assesses if students can interpret information from a written context as coordinate pairs in the first quadrant of the coordinate plane.

If most students struggle with this item, plan to spend additional time after Activity 1 asking students where the points would be plotted on the axes they chose, and ensuring that students understand how to plot a point from a context on a coordinate plane.