End-of-Unit Assessment

1.

A circle has a radius of 40 cm. Which of these is closest to its area?

A.

10,053 cm²

B.

5,026 cm²

C.

251 cm²

D.

126 cm²

Answer:

B

Teaching Notes

This question is meant to be a straightforward check that students can calculate the area of a circle. Because different classes may use different approximations for $\pi$ , students are not expected to find answers that precisely match B, the correct choice. Students who select choice A have combined the formulas for circumference and area, calculating $2 \pi r \boldcdot 2$ instead of squaring the radius. Students who select choice C have calculated the circumference, $2 \pi r$ . Students who select choice D have calculated $\pi r$ but have not squared the radius.

2.

The shape is composed of squares and quarter circles. Select all the expressions that represent its perimeter.

A.

$42+14\pi$

B.

$7 +7+7+ 7+ 7 + 7 + 3.5\pi + 3.5\pi + 3.5\pi + 3.5\pi$

C.

$91+14\pi$

D.

$147+49 \pi$

E.

$7 + 7+ 7+ 7 + 7+ 7 + 7 + 7 + 7 + 7$

Answer:

A, B

Teaching Notes

In addition to having students work with perimeter and circumference, this problem assesses students’ skill in partitioning a diagram into useful sections, another important strand in this unit. Students who do not select choice A may not recognize this answer choice as a simplified version of their own work. Students who do not select choice B may have calculated the perimeter in a different way (for example, by realizing they could treat the four quarter circles as one full circle) not recognizing this approach, or they may simply have figured that choice A is the only correct answer. Students who select choice C have found the sum of all the lengths shown, including those on the interior of the shape. Students who select choice D have calculated the area of the shape. Students who select choice E have mistakenly assumed that the arcs of the quarter-circles also have a length of 7 units.

3.

Select all true statements.

A.

Circle A has a circumference of $\pi$ .

B.

Circle B has a circumference of $\pi$ .

C.

Circle B has an area of $\pi$ .

D.

Circle C has an area of $\pi$ .

E.

$\pi$ is the constant of proportionality relating the radius of a circle to its circumference.

F.

$\pi$ is the constant of proportionality relating the diameter of a circle to its circumference.

Answer:

A, C, F

Teaching Notes

In this unit, $\pi$ is defined as the constant of proportionality relating the diameter to the circumference of a circle. But $\pi$ also appears in the formula for the area of a circle. This problem verifies students’ understanding of the dual roles played by $\pi$ in the study of circles. A student who cannot answer this question but can answer the previous two questions may be over-reliant on formula work. Students who select choice B instead of choice A, or who select both choice A and choice B, need a review of how circumference is determined. Students who select choice D instead of choice C, or who select both choice C and choice D, may need a review of how area is determined, or may be confusing area with circumference. Students who select choice E instead of choice F may be thinking of the area formula or may need additional practice with proportional relationships.

4.

For each quantity, decide whether circumference or area would be needed to calculate it. Explain or show your reasoning.

The distance around a Ferris wheel.
The amount of paint needed to paint a bullseye.
The amount of whipped cream needed to go around the edge of a pie.
The amount of material needed to make a drum head (the part of the drum you hit).

Answer:

Circumference. The distance around a Ferris wheel is the circumference of the wheel.
Area.The bullseye is a flat surface, so how much paint it takes depends on the area of that surface.
Circumference. The whipped cream goes only around the edge of the pie, which is the circumference of the pie.
Area. The drum head is a surface.

Minimal Tier 1 response:

Work is complete and correct.
Acceptable errors: An illuminating drawing can take the place of a verbal explanation.
Sample:

Circumference, because the distance around a figure is the definition of the circumference.
Area, because covering a surface is about area.
Circumference, whipped cream goes on the outer edges of the pie.
Area, because it’s covering the round surface of the drum.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Correct answers with no explanation or misguided explanation; one incorrect answer.

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Two or more incorrect answers; two or more answers with very poor explanation.

Teaching Notes

This problem has students distinguish area from circumference in various real-world contexts. It will likely be difficult for students to say precisely why each problem is about area or circumference. Look for responses that appeal to the exterior of a shape vs. the interior and to the fact that surfaces have to do with area.

5.

A class measured the radius $r$ , circumference $C$ , and area $A$ of various circular objects. The results are recorded in this table.

$r$ (cm)	$C$ (cm)	$A$ (cm²)
4	25	50
5	31.5	78.5
8	50	201
12	75	452

Does there appear to be a proportional relationship between circumference and area? Explain or show your reasoning.
Does the equation $A=\frac12 \boldcdot r \boldcdot C$ appear to describe the relationships in the table? Explain your reasoning.

Answer:

No. Dividing the area by the circumference gives a different answer every time.
Yes. When I substitute in values of $r$ , $C$ , and $A$ from the same row of the table, it makes the equation true, or at least it's very close. The measurements were made with rulers that only have so much accuracy. Students needed to round their answers to the nearest ruler marking, or perhaps rounded even less accurately than that. Also, students probably didn’t hold the rulers perfectly still or perfectly straight.

Minimal Tier 1 response:

Work is complete and correct.
In verifying the equation, at least two sets of values are checked, or an argument is made about why $\frac12 \boldcdot r \boldcdot C$ is equivalent to $\pi r^2$
Sample:

No. $50 \div 25 = 2$ , $201 \div 50 \approx 4$ .
Yes. Trying different sets of numbers, I get $50 = \frac12 \boldcdot 4 \boldcdot 25$ , which is $50=50$ , and $201 = \frac12 \boldcdot 8 \boldcdot 50$ which is $201=200$ . Even though the two sides aren't exactly equal, this can be explained by slight measurement error.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Reasoning in Part A is not sufficiently explained; concluding that the equation does not describe the relationship because correctly substituting values results in equations that are technically not true like $201=200$ .

Tier 3 response:

Significant errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: Conclusion that the area and circumference are in a proportional relationship; no visible effort to check whether the given equation described the relationship.

Teaching Notes

This problem hearkens back to an the activity that students did in Lesson 3, when they discovered that circumference and diameter are related by a constant of proportionality called $\pi$ . In this problem, students examine the relationship between circumference and area, which is not proportional. Some students may remember the relationship $\text{Area} = \frac12 \text{circumference} \boldcdot \text{radius}$ from Lesson 8, but the expectation is that students will use the values in the tables to verify it.

6.

This figure is made from part of a circle and part of a square.

What is the perimeter of this figure, to the nearest unit?
What is the area of this figure, to the nearest square unit?

Answer:

19 units (The quarter-circle’s perimeter is $\frac14 \boldcdot 2 \boldcdot \pi \boldcdot 3$ units. The rest of the perimeter is 14 units. The total perimeter, rounded to the nearest unit, is 19 units.)
23 square units (The quarter-circle’s area is $\frac14 \boldcdot \pi \boldcdot 3^2$ square units. The rest of the area is 16 square units. The total area is approximately 23 square units.)

Teaching Notes

Watch for students accidentally using the inner 3-unit lengths as part of the perimeter of the figure. Some students may have trouble recognizing the top left as a quarter-circle. Direct these students to the phrasing “part of a circle” in the problem.

7.

A painter needs to paint the bottom of a circular pool. The pool has a radius of 30 feet.

A store sells 5-gallon cans of paint. One gallon of paint covers 300 square feet.

What is the smallest number of 5-gallon cans the painter must buy to cover the bottom of the pool? Explain or show your reasoning.

Answer:

2 cans. The pool’s size is $\pi \boldcdot 30^2$ , just over 2,827 square feet. Each gallon of paint covers 300 square feet; each can of paint covers 1,500 square feet. The number of cans needed is given by $\displaystyle \frac{\pi \boldcdot 30^2}{1,500} \approx 1.9$ Because you cannot buy 0.9 of a can, 2 cans will be needed.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: The area of the pool bottom is $\pi \boldcdot 30^2$ . Each can of paint covers 1,500 square feet, and that is not enough. 2 cans cover 3,000 square feet.

Tier 2 response:

Work shows good conceptual understanding and mastery, with either minor errors or correct work with insufficient explanation or justification.
Sample errors: Correct calculation (about 1.9 cans) but incorrect use of context gives answers of 1 or 1.9 cans; incorrect calculation of the number of square feet per gallon but otherwise correct work, including correct use of contextual rounding; calculation errors, but not errors in formula application, when determining the size of the pool bottom or the number of cans; incorrect calculations in determining the number of cans when using a strategy that does not involve dividing.

Tier 3 response:

Work shows a developing but incomplete conceptual understanding, with significant errors.
Sample errors: Incorrectly determining the area of the pool bottom using circumference, or using 60 as the radius; incorrect type of calculation performed on the cans, such as dividing in reverse order; two or more error types from Tier 2 response.
Acceptable errors: Any response giving the correct area of the pool earns at least a Tier 3 response, regardless of other work. Any response giving an incorrect area of the pool, but correct work on the proportional relationship based on that incorrect area, earns at least a Tier 3 response.

Tier 4 response:

Work includes major errors or omissions that demonstrate a lack of conceptual understanding and mastery.
Sample errors: Incorrectly determining the area of the pool, along with incorrect work on the proportional or contextual relationship; answer without explanation, regardless of accuracy.

Teaching Notes

In this problem, students must identify the proportional relationship between gallons of paint and area of a circular pool. Then the formula for the area of a circle is needed to calculate how much paint is needed to paint the floor of the pool. Finally, the student must interpret the result in context to determine the correct number of cans needed. The same answer will come from any approximation that students use for $\pi$ .