Unit 3, Lesson 3: Exploring Circumference...circle A 3 circle B 10 circle C 24 circle D 18 circle E...

Unit 3, Lesson 3: Exploring Circumference

3.1: Which Is Greater? (5 minutes)

Setup:

Students in groups of 2. Display the image for all to see. 1 minute of quiet think time, followed by small-group and whole-class discussion.

Lesson GoalsUnderstand as the constant of proportionality in the relationship between thediameter and circumference of a circle.Recognize that the relationship between the diameter and circumference of a circleis proportional and the constant of proportionality is a little more than 3.

Required Materialsempty toilet paper rollmeasuring tapescylindrical household items

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GRADE 7 MATHEMATICS

Open Up Resources(openupresources.org)

Unit 3: Measuring Circles, Lesson 3: Exploring Circumference 1

Student task statementClare wonders if the height of the toilet paper tube or the distance around the tube is greater. Whatinformation would she need in order to solve the problem? How could she find this out?

Possible responses

Clare needs to measure the length anddistance around the tube (for example,with a flexible tape measure).

Anticipated misconceptions

Students may not understand what ismeant by the height of the tubebecause it can sit in two different ways.Point these students to the first pictureof the tube, and ask them to identifythe height as the tube is sitting in thatpicture.

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3.2: Measuring Circumference and Diameter (25 minutes)

Setup: Groups of 2–4. Distribute 3 circular objects and measuring tapes or string and rulers to each group.

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Student task statementYour teacher will give you several circular objects.

1. Measure the diameter and the circumference of the circle in each object to the nearest tenth of acentimeter. Record your measurements in the table.

object diameter (cm) circumference (cm)

2. Plot the diameter and circumference values from the table on the coordinate plane. What do you notice?

3. Plot the points from two other groups on the same coordinate plane. Do you see the same pattern that

Possible responses

1. Answers vary.

2. Answers vary. The graph includesthree points that are close tolying on a line that passes through

.

3. Answers vary. The graph includesmore points that are close tolying on a line that passes through

.


Students may try to measure thediameter without going across thewidest part of the circle, or maystruggle with measuring around thecircumference. Mentally check thattheir measurements divide to getapproximately 3 or compare with yourown prepared table of data and promptthem to re-measure when theirmeasurements are off by too much. Ifthe circular object has a rim or lip, this

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you noticed earlier? could help students keep themeasuring tape in place whilemeasuring the circumference.

If students are struggling to see theproportional relationship, remind themof recent examples where they haveseen similar graphs of proportionalrelationships. Ask them to estimateadditional diameter-circumferencepairs that would fit the pattern shownin the graph. Based on their graphs, dothe values of the circumferences seemto relate to those of the diameters in aparticular way? What seems to be thatrelationship?

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Are you ready for more?Suppose you had another circular object with a diameter that is half as long as the diameter of your largestcircle. What would its circumference be?

Possible Responses

Answers vary, depending on the size ofthe circular objects students measured.Possible solution:

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3.3: Calculating Circumference and Diameter (10 minutes)

Setup:

Ask students to label pictures with measurements. Remind students of the constant of proportionality from previous activity. Quiet worktime followed by whole-group discussion.

GRADE 7 MATHEMATICS



Student task statementHere are five circles. One measurement for each circle is given in the table.

Use the constant of proportionality estimated in the previous activity to complete the table.

diameter (cm) circumference (cm)

circle A 3

circle B 10

circle C 24

circle D 18

circle E 1

Possible responses

See lesson plan for completed table.


Some students may multiply thecircumference by the constant ofproportionality instead of dividing by it.Prompt them to consider whether thediameter can be longer than thecircumference of a circle.

Some students may struggle to divideby 3.1 if that is the constant ofproportionality decided on in theprevious lesson. Ask these students ifthey could use an easier number astheir constant, and allow them to divideby 3 instead. Then ask them how theiranswer would have changed if theydivided by 3.1.

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Lesson Synthesis (5 minutes)What is the relationship between the diameter and the circumference of a circle? How does this relate to the number ?

3.4: Identifying Circumference and Diameter (Cool-down, 5 minutes)

Setup: None.

Are you ready for more?The circumference of the Earth is approximately 40,000 km. If you made a circle of wire around the globe, thatis only 10 meters (0.01 km) longer than the circumference of the globe, could a flea, a mouse, or even a personcreep under it?

Possible Responses

Yes! Each meter added to the diameterof a circle adds about 3.1 meters to thecircumference of the circle. So if thecircumference of the Earth is increasedby 10 meters, this means that a littlemore than 3 meters have been addedto the diameter. So there would beabout 1.5 meters of distance betweenthe rope and the Earth, making it easyfor a flea, mouse, or person to go underthe rope!

GRADE 7 MATHEMATICS



Student task statementSelect all the pairs that could be reasonable approximations for the diameter and circumference of a circle.Explain your reasoning.

A. 5 meters and 22 meters.

B. 19 inches and 60 inches.

C. 33 centimeters and 80 centimeters.

Possible responses

Only B, because is the onlyquotient that is a little more than 3.

GRADE 7 MATHEMATICS



Unit 3, Lesson 3: Exploring Circumference...circle A 3 circle B 10 circle C 24 circle D 18 circle E...

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