Check Your Readiness

1.

A kids’ movie ticket costs $5.25.

One day, 200 kids’ tickets were purchased. What was the total cost of those tickets?
If $t$ is the number of kids’ tickets purchased, and $c$ is the total cost of those tickets, write an equation that relates $c$ to $t$ .
Another day, the total cost of kids’ tickets was $813.75. How many tickets were purchased that day?

Answer:

$1050
$c = 5.25t$ (or equivalent)
155 tickets ( $813.75 \div 5.25 = 155$ )

Teaching Notes

Students find a linear equation to represent a ratio. Students may or may not use the equation in order to answer the question in Part C. A ratio table would also be effective, but because the numbers are large, the equation may be more efficient. Because the unit rate is given, students might also answer Part C by doing the relevant arithmetic without referring to the equation.

If most students do well with this item, plan to emphasize, in Lesson 3, the value of knowing that the circumference and diameter of a circle have a proportional relationship. Ask students, "If I know that the circumference is always about three times the diameter, what else can I figure out? What if the diameter is 10 units? 100 units? 1 unit? How do you know?" "What if I know that the circumference is 60 units?" Make sure that students understand the relationship between the patterns they noticed and the equation $C = \pi d$ . The warm-up in Lesson 4 gives another opportunity to assess students' understanding of the value of knowing the relationship. Optional Lesson 5 gives students more opportunities to practice using the proportional relationship.

2.

The sides of a square have length $s$ .

Write a formula for the perimeter $P$ of the square.
Write a formula for the area $A$ of the square.

Answer:

$P = 4s$ (or equivalent)
$A = s^2$ (or equivalent)

Teaching Notes

The formulas for the circumference and area of a circle build naturally on simpler formulas for perimeters and areas of polygons. This item recalls what students have done with perimeter and area of a square.

If most students struggle with this item, plan to incorporate these expressions into Activities 2 and 3 to connect how students used the squares' side lengths to find the perimeters and areas.

3.

Priya measured the perimeter of a square desk and found it to be 248 cm. Noah measured the perimeter of the same desk and found it to be 2,468 mm.

By how many millimeters do these measurements differ?
Why do you think that Priya and Noah may have found different measurements for the same desk?

Answer:

12 because 248 cm is 2,480 mm and $2480 - 2468 = 12$ .
Answers vary. Sample response 1: Priya may have rounded her measurements of each side of the square to the nearest cm while Noah may have rounded to the nearest mm. Sample response 2: The measurements are not exact and so there may have been some error in Priya’s measurement, Noah’s measurement, or both.

Teaching Notes

In this unit, students will measure circles in order to discover a relationship between the circumference and the diameter. As a result, they will need to deal with measurement error. This problem asks students to explain a discrepancy in measurement.

If most students struggle with this item, plan to incorporate reporting the measures in Activity 2 as millimeters as well as to the nearest centimeter. Ask how using different levels of precision changes how we report measurements. This will be important as students measure parts of a circle. A lack of precision may hide the relationships that students need to notice and use, and not understanding that some error is inherent in human measuring may mean that students discount the overall patterns.

4.

Each small square in the graph paper represents 1 square unit. Find the area of the given figure in square units. Explain your reasoning.

Answer:

22 square units. Possible strategy: Draw a 10 x 6 box that just encloses the triangle. That area is 60 square units. The three triangles that are inside the rectangle but outside the blue triangle are each right triangles with areas 24, 4, and 10. So, $60 - (24 + 4 + 10) = 22$ .

Teaching Notes

In the second section of this unit, students decompose, rearrange, and enclose shapes while exploring different methods to find the area of a circle. If most students struggle with this item, plan to use it as the Warm-up for Lesson 6. Students learned a variety of strategies for finding areas in Unit 1 of Grade 6, and revisiting the strategies may be enough support. Be sure to call on students who used different strategies.

5.

A map of Utah is shown. Which area is closest to the area of Utah?

A.

1,240 square miles

B.

87,150 square miles

C.

94,500 square miles

D.

119,000 square miles

Answer:

87,150 square miles

Teaching Notes

Students who select choice A have calculated the perimeter instead of the area. Students who select choice C calculated the area of the larger rectangle containing the state ( $350 \boldcdot 270$ ). Students who select choice D may have treated the sides of length 165, 70, and 105 miles as one long horizontal side, and multiplied their sum by 350.

If most students struggle with this item, plan to use it as part of the Warm-up for Lesson 6 as well. Ask about this item before asking about Item 4. In the other activities in the lesson, students practice decomposing figures to find areas. If students need more support, consider using activities from Unit 1 in Grade 6.

6.

Here is a picture of a circle. Each square represents 1 square unit.

Explain why the area of the circle is less than 4 square units.
Explain why the area of the circle is more than 2 square units.
Do you think the area of the circle is more or less than 3 square units?

Answer:

The circle fits inside a 2-by-2 square, so its area is less than 4 square units.
Answers vary. Sample response: Decompose two square units into 4 triangles that fit inside the circle.
Answers vary. Sample response: It is difficult to tell from the picture. It looks like the area of the circle is close to 3 square units.

Teaching Notes

Students estimate the area of an irregular shape by comparing it with shapes composed of rectangles and triangles. Some students may have already encountered the formula for the area of a circle before and may use that as a response in Part C. If not, students might cut up unit squares to see how many fit in the circle—but it will still be close! Encourage students who want to spend a long time on Part C to do the rest of the assessment first, and then come back to it. The idea of this problem is to get a sense of what strategies students use to compare areas.

If most students struggle with this item, it's not necessary to take action, but note their strategies as you prepare to teach Lesson 7.