Check Your Readiness

1.

The rectangle has sides measuring 7 cm and 4 cm. What is the area of this rectangle? Explain your reasoning.

Answer:

28 square centimeters. Sample reasoning:

Use a formula like $l\boldcdot w$ .
Draw and count unit squares.
Think in terms of tiling, but multiply to find the area.

Teaching Notes

This item assesses how students approach finding the area of a rectangle with whole-number side lengths. The purpose of including tick marks is to give assistance to students who wish to draw a grid of unit squares. Responses that show drawings with the incorrect number of unit squares, irregular rows, or irregular columns may indicate that students have not yet learned to structure two dimensional space, that is, to see a rectangle with whole-number side lengths as composed of unit squares, or composed of iterated rows or columns of unit squares.

If most students struggle with this item, plan to use the activities in Lesson 1 to support their understanding of area. The Practice Problems in Lesson 1 can be used for extra practice in calculating area. In Lesson 2, students will decompose and rearrange shapes to find their areas. Plan to emphasize tiling and square units in Activity 2 of Lesson 2 if students struggle to make sense of tiling the rectangle with 30 squares to find its area.

2.

Select all the expressions that give the area of the figure in square units.

A.

$7 \times 5$

B.

$(7 + 5) \times (5 + 2)$

C.

$(7 \times 3) + (5 \times 2)$

D.

$7 \times 5 \times 5 \times 2 \times 2 \times 3$

E.

$(7 \times 5) - (2 \times 2)$

Answer: C, E

Teaching Notes

This problem assesses two different prerequisite skills. To reason about the area of the shape, students will need to decompose the shape into two rectangles. Interpreting the expressions in the answer choices may also pose a challenge for some students. Note that Choices C and E represent two different methods. Choice C involves decomposing the figure into smaller rectangles. Choice E involves enclosing the figure with a larger rectangle and subtracting the extra rectangle.

If most students struggle with this item, plan to begin this lesson with a few examples of rectilinear figures whose areas can be found by decomposing rectangles. The Practice Problems in Lesson 1 can be used for this. Students will also get many opportunities in the first several lessons of this unit to decompose shapes and compare different ways to decompose the same shape.

3.

Each small square in the graph paper represents 1 square unit.

Which expression is closest to the area of the shaded rectangle, in square units?

A.

$6 \times 3$

B.

$5 \times 2$

C.

$5\frac12 \times 2\frac12$

D.

$6\frac12 \times 3\frac14$

Answer:

$5\frac12 \times 2\frac12$

Teaching Notes

This problem requires students to identify an expression that gives the area of a rectangle that is on a grid and whose side lengths are not whole numbers. Rather than relying on counting, students will need to recognize that the product of the side lengths gives the area of the rectangle and to estimate fractional units visually. In fifth grade, students learned to calculate the area of a rectangle by multiplying fractional side lengths.

If most students struggle with this item, plan to use this problem to draw out and support any misconceptions during the synthesis of Lesson 3 Activity 2. When the idea of multiplying side lengths to find the area of a rectangle comes up, ask students if it can also be done when the side lengths are not whole numbers.

4.

Select all the line segments that appear to be parallel to $g$ .

A.

Segment $a$

B.

Segment $b$

C.

Segment $c$

D.

Segment $d$

E.

Segment $e$

F.

Segment $f$

Answer: B, F

Teaching Notes

Students will need to be comfortable recognizing parallel lines before beginning their work with parallelograms later in the unit. Some students may correctly select segment b as being parallel to segment g, but not notice segment f because that segment is farther away.

If most students struggle with this item, plan to start with Lesson 4 Activity 1 Launch with an emphasis on defining the term “parallel.”

5.

Select all the figures that have sides that appear to be perpendicular.

A.

B.

C.

D.

E.

Answer: A, B

Teaching Notes

In this unit, students will find the area of parallelograms and triangles by decomposing them into shapes with perpendicular sides and rearranging the pieces. Students will need to be familiar with perpendicular lines in order to make sense of the term “height” of a parallelogram or triangle.

If most students struggle with this item, plan to start Lesson 5 Activity 2 by amplifying the term “perpendicular” for the students. Students may need some visual cues to support this concept.

6.

Select all the expressions that are equal to $10^4$ .

A.

10,000

B.

4,000

C.

$1,000 \times 4$

D.

$10 \times 4$

E.

$10 \times 10 \times 10 \times 10$

F.

1,000

G.

40

Answer: A, E

Teaching Notes

Exponential notation is introduced in Lesson 17 of this unit, in the context of calculating the surface area and volume of cubes. Students may have prior knowledge of exponents from their work with place value in fifth grade.

7.

Select all triangles that appear to be a right triangle.

A.

B.

C.

D.

E.

F.

Answer: A, C, F

Teaching Notes

This problem assesses whether students understand the term “right triangle.” This problem will also reveal whether students can picture a 90-degree angle.

If most students struggle with this item, plan to focus on this concept during Lesson 3 Activity 3, Off the Grid. In the Launch of this activity, include a discussion about the term “right angle” and point out the symbol used to identify it in the shapes. This concept will continue to be reinforced in the next several lessons.