End-of-Unit Assessment

1.

A circle has a radius of 50 cm. Which of these is closest to its area?

A.

157 cm $^2$

B.

314 cm $^2$

C.

7,854 cm $^2$

D.

15,708 cm $^2$

Answer:

7,854 cm $^2$

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 the correct choice (C).

Students who select choice A have calculated $\pi r$ but have not squared the radius. Students who select choice B have calculated the circumference, $2 \pi r$ . Students who select choice D have combined the formulas for circumference and area, calculating $2 \pi r^2$ .

2.

The shape is composed of three squares and two semicircles. Select all the expressions that correctly calculate the perimeter of the shape.

3 blue squares side by side. Semicircle across the top of two squares and another one along the bottom of two squares.

A.

$40 + 20\pi$

B.

$80 + 20\pi$

C.

$120 + 20\pi$

D.

$300 + 100\pi$

E.

$10+10+10\pi+10+10+10\pi$

Answer: A, E

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 select choice B have found the perimeter of the rectangle and added it to the circumference of the circle. Students who select choice C have found the perimeters of the circle and the three squares. Students who select choice D have calculated the area of the shape rather than the perimeter. Students who do not select choice E may have calculated the perimeter in a different way (for example, by realizing that they could treat the two semicircles as one full circle) not recognizing this approach, or they may simply have figured that choice A is the only correct answer.

3.

Select all of the true statements.

A.

$\pi$ is the area of a circle of radius 1.

B.

$\pi$ is the area of a circle of diameter 1.

C.

$\pi$ is the circumference of a circle of radius 1.

D.

$\pi$ is the circumference of a circle of diameter 1.

E.

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

F.

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

Answer: A, D, E

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 selecting both choice A and choice B, need a review of how the area of a circle is determined. Students who select choice C instead of choice D, or selecting both choice C and choice D, need a review of how the circumference of a circle is determined. Students who do not select choice E may need some additional practice with proportional relationships. Students who select choice F may be confusing area with circumference or may need additional practice with proportional relationships.

4.

A class measured the radius and circumference of various circular objects. The results are plotted on the graph.

Does there appear to be a proportional relationship between the radius and circumference of a circle? Explain or show your reasoning.
Why might the measured radii and circumferences not be exactly proportional?

Answer:

Yes. Explanations vary. Sample explanation: If you divide each circumference by its radius, you get the numbers 6, 6.25, approximately 6.33, and approximately 6.29. These numbers are close enough that they are evidence of a proportional relationship between circumference and radius.
The measurements were taken using rulers that have only 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.
Acceptable errors: in Part A, writing or implying that the points are collinear.
Sample:

(With accompanying line drawn in) Yes, because the points are on a line that goes through $(0,0)$ .
The points are not exact because of error in measurement.

Tier 2 response:

Work shows general conceptual understanding and mastery, with some errors.
Sample errors: Division errors make it look as if the ratios are not similar enough to indicate a constant of proportionality, or explanation in Part B does not appeal to measurement error in some way.

Tier 3 response:

Significant Errors in work demonstrate lack of conceptual understanding or mastery.
Sample errors: errors types from Tier 2 response on both problem parts, or explanation in Part A does not appeal to facts students know about proportional relationships.

Teaching Notes

This problem mimics the activity that students did in Lesson 3, when they discovered that circumference and diameter are related by a constant of proportionality called $\pi$ . There are a variety of approaches that students can take to argue that the circumference and radius are proportional: for example, draw a line through the plotted points to show that the line comes very close to passing through each point and through $(0,0)$ , or divide each of the circumferences by its corresponding radius. Students who take the latter approach will find constants of proportionality in the 6 to $6.4$ range, because the true constant of proportionality is $2 \pi$ .

The second part of the question gets at another issue that came up during that activity: measurement error. In fact, the measured points are not in a true proportional relationship, though they are close enough that they are good evidence for the proportionality of radius and circumference.

5.

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

The distance around a circular track.
The total number of equally-sized tiles on a circular floor.
The amount of oil it takes to cover the bottom of a frying pan.
The distance your car will go with one rotation of the wheels.

Answer:

Circumference. The distance around the track is the circumference of the circular track.
Area. The number of tiles it takes to cover the floor times the area of each tile is the area of the floor.
Area. The pan is circular and the entire circular surface is being covered in oil. To know how much oil is used, we need to know the area of the circle (as well as the thickness of the layer of oil).
Circumference. The distance the car goes in one rotation is the distance around (circumference of) the tires.

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 around the track means around the circle.
Area, because covering a surface is about area.
Area, because you cover the inside of the pan with oil, not just the rim.
Circumference, because when a tire rolls it’s only the outside that counts.

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.

6.

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

A figure composed of a part of a square and a part of a circle.

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

Answer:

38 units (The quarter-circle’s perimeter is $\frac 1 4 \boldcdot 2 \boldcdot \pi \boldcdot 5$ units. The rest of the perimeter is 30 units. The total perimeter is approximately 37.9 units.)
95 square units (The quarter-circle’s area is $\frac 1 4 \boldcdot \pi \boldcdot 5^2$ square units. The rest of the area is 75 square units. The total area is approximately 94.6 square units.)

Teaching Notes

Watch for students accidentally using the inner 5-unit lengths as part of the perimeter of the figure. Some students may have trouble recognizing the bottom right as a quarter-circle. Direct these students to the phrasing “part of . . . a circle” in the problem.

7.

A groundskeeper needs grass seed to cover a circular field that is 290 feet in diameter.

A store sells 50-pound bags of grass seed. One pound of grass seed covers about 400 square feet of field.

What is the smallest number of bags the groundskeeper must buy to cover the circular field? Explain or show your reasoning.

Answer:

4 bags. The field’s size is $\pi \boldcdot 145^2$ square feet, just over 66,000 square feet. Each pound of seed covers 400 square feet; each 50-pound bag covers 20,000 square feet. The number of bags needed is given by:

$\displaystyle \frac{\pi \boldcdot 145^2}{20,000} \approx 3.30$

It is not possible to purchase 3.3 bags, and 3 bags is not enough. It takes 4 bags of grass seed to cover the field.

Minimal Tier 1 response:

Work is complete and correct, with complete explanation or justification.
Sample: The area of the field is $\pi \boldcdot 145^2$ square feet, and each bag covers 20,000 square feet. 3 bags cover 60,000 square feet and that's not enough. 4 bags cover 80,000 square feet and that is enough.

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 3.3 bags) but incorrect use of context gives answers of 3 or 3.3 bags; incorrect calculation of the number of square feet per bag but otherwise correct work, including correct use of contextual rounding; calculation errors, but not errors in formula application, when determining the size of the field or the number of bags; incorrect calculations in determining the number of bags 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 field using circumference, or using 290 as the radius; incorrect type of calculation performed on the bags, 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 field earns at least a Tier 3 response, regardless of other work. Any response giving an incorrect area of the field, 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 field, 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 pounds of grass seed and square feet of grass. Then the formula for the area of a circle is needed to calculate how many bags of seed cover the field. Finally, the student must interpret the result in context to determine the correct number of bags.

The same answer will come from any approximation that students use for $\pi$ .