Section

Teacher-facing learning goals

Circumference of a Circle

Lesson 1

Create and describe (in writing) graphs that show measurements of squares.
Justify (orally and in writing) whether the relationship shown on a graph is close enough to a straight line through the origin that it might be a proportional relationship with some measurement error.
Recognize that when we measure the quantities in a proportional relationship, measurement error can cause the graph to be not perfectly straight and the quotients to be not exactly constant.

Lesson 2

Compare (orally) different ways to measure a circle, and generalize the relationship between radius and diameter.
Comprehend the terms “diameter,” “center,” “radius,” and “circumference” in reference to parts of a circle.
Describe (orally and in writing) the defining characteristics of a circle.

Lesson 3

Comprehend the word “pi” and the symbol \(\pi\) to refer to the constant of proportionality between the diameter and circumference of a circle.
Create and describe (in writing) graphs that show measurements of circles.
Generalize that the relationship between diameter and circumference is proportional and that the constant of proportionality is a little more than 3.

Lesson 4

Apply understanding of circumference to calculate the perimeter of a shape that includes circular parts, and explain (orally and in writing) the solution method.
Compare and contrast (orally) values for the same measurements that were calculated using different approximations for \(\pi\).
Explain (orally) how to calculate the radius, diameter, or circumference of a circle, given one of these three measurements.

Lesson 5 - Optional

Compare wheels of different sizes and explain (orally) why a larger wheel needs fewer rotations to travel the same distance.
Generalize that the distance a wheel rolls in one rotation is equal to the circumference of the wheel.
Write an equation to represent the proportional relationship between the number of rotations and the distance a wheel travels.

Area of a Circle

Lesson 6

Estimate the area of a complex, real-world region, e.g., a state or province, by approximating it with an irregular polygon, and indicate that it is an approximation when expressing the answer (orally and in writing).
Explain (orally and in writing) how to calculate the area of an irregular polygon by decomposing it.
Interpret floor plans and maps in order to identify the information needed to calculate area.

Lesson 7

Create a table and a graph that represent the relationship between the diameter and area of circles of various sizes, and justify (using words and other representations) that this relationship is not proportional.
Estimate the area of a circle on a grid by decomposing and approximating it with polygons.

Lesson 8

Generalize a process for finding the area of a circle, and justify (orally) why this can be abstracted as \(\pi r^2\).
Show how a circle can be decomposed and rearranged to approximate a polygon, and justify (orally and in writing) that the area of this polygon is equal to half of the circle’s circumference multiplied by its radius.

Lesson 9

Calculate the area of a shape that includes circular or semi-circular parts, and explain (orally and in writing) the solution method.
Comprehend and generate expressions in terms of \(\pi\) to express exact measurements related to a circle.

Let's Put it to Work

Lesson 10

Critique (orally and in writing) claims about the radius, diameter, circumference, or area of a circle in a real-world situation.
Decide whether to calculate the circumference or area of a circle to solve a problem in a real-world situation, and justify (orally) the decision.
Estimate measurements of a circle in a real-world situation, and explain (orally and in writing) the estimation strategy.

Lesson 11 - Optional

Apply circumference and area of circles to calculate the cost of a stained-glass window, and explain (orally and in writing) the solution method.
Design a stained-glass window that could be built for a given dollar amount, and present (orally, in writing, and through other representations) a justification that it costs less than the limit.
Make simplifying assumptions to solve problems about a stained-glass window.

7.3 Measuring Circles