- Subject:
- Geometry
- Material Type:
- Lesson Plan
- Level:
- Middle School
- Grade:
- 7
- Provider:
- Pearson
- Tags:

- License:
- Creative Commons Attribution Non-Commercial
- Language:
- English
- Media Formats:
- Interactive, Text/HTML

# Education Standards

# Circle

# Find the Ratios in Polygons and Circles

# Measuring Circles

## Overview

Students will measure the circumference and diameter of round things in the classroom and discover the ratio pi. They will see that the ratio of a circle's circumference to its diameter can be used to solve for the circumference when the diameter is known.

# Key Concepts

Students have seen circles before, but have not analyzed the relationships between parts of a circle. The ratio of the circumference to the diameter is pi, about 3.14 or about $\frac{22}{7}$. Students see that all of the objects they measure have this ratio (or close, depending on accuracy) and that the ratio is true for all circles. Students also see that the ratio can be used to solve for the circumference of a circle if the diameter (or radius) is known.

# Goals

- Measure round things looking for similarities.
- Find the ratio of the circumference to the diameter of those round things.
- Find a formula to find the circumference of a circle.

SWD: Make sure students understand these domain-specific terms:

It may be helpful to preteach these terms to students with special needs. If possible, reinforce the definitions of these terms with visual supports (diagrams).

ELL: As new vocabulary is introduced, be sure to repeat it several times and to allow students to repeat after you as needed. Write the new words as they are introduced and allow enough time for ELLs to check their dictionaries or briefly consult with another student who shares the same primary language if they wish.

- ratio
- circumference
- circle
- diameter
- scatter plot

# Compare Diameter and Circumference

# Lesson Guide

Show students the bike wheel image and discuss the question about which is longer—the length of the diameter of the bike wheel or the distance around the bike wheel.

Remind students that the distance around a circle is called the *circumference*, and the distance across is called the *diameter*.

# Mathematics

Briefly discuss the opening questions, taking a few comments and observations. Students will revisit these questions at the end of the lesson and be able to answer correctly.

## Opening

# Compare Diameter and Circumference

Compare the length of the diameter of the bike wheel to the distance around the wheel (the circumference).

- Which value do you think is greater? About how much greater?

# Math Mission

# Lesson Guide

Discuss the Math Mission. Students will explore the relationship between the circumference and the diameter of a circle.

## Opening

Explore the relationship between the circumference and the diameter of a circle.

# Find Ratios in Polygons and Circles, Part 1

# Interventions

**Student has difficulty getting started. **

- What do you know about the side lengths in a square?
- How does the distance across compare to a side length?
- If you were to divide the hexagon into triangles, what kind of triangles would they be?

# Mathematical Practices

**Mathematical Practice 1: Make sense of problems and persevere in solving them.**

- Look for students who work out the ratio of the perimeter of each polygon to the distance across each polygon.

**Mathematical Practice 8: Look for and express regularity in repeated reasoning.**

- Some students will notice the similarities between the polygons and circles.
- Look for students who see that the ratio of circumference to diameter should be the same every time, and who understand that it isn’t only because of accuracy in measuring.

# Possible Answers

- Square: 2.83:1, Hexagon: 3:1, Octagon: 3.06:1, Decagon: 3.09:1, Dodecagon: 3.11:1, 14-gon: 3.12:1, 16-gon: 3.121:1, 18-gon: 3.126:1, 20-gon: 3.129:1
- The ratios do not change because the distance across is measured in terms of side length. As the figures change, they change proportionally. All regular polygons with any given number of sides are similar (as are all circles).
- The relationship between the diameter and the perimeter of regular polygons is about the same as (though slightly less than) the ratio between the diameter and the circumference of a circle. The ratio between the diameter and the perimeter of regular polygons increases toward the ratio between the diameter, and the circumference of a circle as the number of sides increases.

## Work Time

# Find Ratios in Polygons and Circles, Part 1

Explore the given regular polygons using the Circle interactive.

- What is the ratio of the perimeter of each polygon to the distance across the polygon?
- Press and drag the vertices to change the size of each polygon. Do you think the ratio of the perimeter to the distance across has changed? Explain.
- Based on your explorations, explain the relationship between the diameter and the perimeter of regular polygons.

INTERACTIVE: Circle

## Hint:

- To find the ratio of the perimeter of each polygon to the distance across the polygon, think about how the distance across the polygon is related to its side lengths.

# Find Ratios in Polygons and Circles, Part 2

# Lesson Guide

Have rulers ready for students to use, as well as scissors and string (consider cutting the string in advance to save time).

SWD: Consider ways to optimize engagement and participation of all students. Students with motor difficulties may need support with the physical task of measuring round objects. It may be helpful to provide students with different responsibilities during the measuring task: measurement, recording of data, observation, and so on.

Consider ways to help students with special needs make connections between what they have learned about regular polygons and what they are learning about circles.

It may be helpful to provide students with concept maps or visual illustrations of the relationships between the different measurements of shapes (perimeter, side length, apothem, diameter and circumference, and so on).

ELL: When eliciting answers, be aware of the difficulties some ELLs encounter when they have to express themselves in a foreign language. If you hear them say the right things but use the wrong grammatical structure, show signs of agreement and softly rephrase using the correct grammar and the student’s words as much as possible. For example, if they say, “Across is like length,” you could rephrase by saying, “Yes, the distance across is the same as a side length.” Words in bold should be pronounced slowly so that students could repeat as necessary.

# Interventions

**Student has difficulty finding the ratios for each object.**

- Did you get the same ratio each time? Were the ratios close?
- Why do you think the ratios were close but different?

**Student does not see how to use the ratio.**

- What is the ratio of the circumference to the diameter?
- If the ratio was exactly 3 and you knew the diameter was 2, what would the circumference be?
- So, what calculation should you do if you know the diameter?

**Student has a solution.**

- If you could measure pi more accurately, how much difference do you think that would make in the answer?

**Student does not think that the ratio of circumference to the diameter is the same for all circles.**

- When you changed the size of the square and hexagon, did the ratio change? Why?
- Open the sketch again and change the size of the polygons with the circles showing. What do you notice?

# Mathematical Practices

**Mathematical Practice 6: Attend to precision.**

- Students use appropriate tools accurately as they measure carefully with the string and ruler. Notice students who choose to use metric measures (centimeters and millimeters), which would be much more accurate and easier to divide, versus standard (inches and fractions of inches). Some students may ask about using a sketch to model and measure a circle.

# Possible Answers

- Measurements will vary.
- The ratios should all be close to 3:1 with some variation due to measuring error. Students should realize that the ratio would always be the same if measurements were more accurate.
- Students should notice that the scatter plot is basically linear. The scatter plot shows that the relationship between the circumference and diameter is constant, as each point represents the same proportion, and pi is the slope of the line of best fit.

## Work Time

# Find Ratios in Polygons and Circles, Part 2

Find five round objects of different sizes in the classroom. Use string to find the distance around the object (circumference), and then use a ruler to measure the length of the string. Also measure the distance across each object (diameter).

- Record the circumference and diameter of each object in a table.
- For each object, find the ratio of the circumference to the diameter by dividing the circumference by the diameter. Record the ratio to the nearest tenth in your table. Is the ratio the same for all of the objects?
- Graph the diameter and circumference of each object on the same grid. Use the
*x*-axis to represent diameter and the*y*-axis to represent circumference. What do you notice about the graph?

HANDOUT: Find the Ratios in Polygons and Circles

# Find Ratios in Polygons and Circles, Part 3

# Possible Answers

- The ratio for the square is approximately 2.8:1, but for the circle it is slightly more than 3:1. This comparison demonstrates that even if the circle ratio was not known, we can see that it is more than 2.8:1 because the circle is outside the square, which has an equal distance across (the diagonal of the square is equal to the diameter of the circle). Likewise, the ratio for the hexagon is 3:1, but the circle is just outside, showing that the ratio is a little more than 3:1. Seeing these ratios gives some insight into how pi was derived, as Archimedes used a polygon with 96 sides and a circumscribed circle to approximate pi.

SWD: Students with disabilities may have difficulty generalizing their understanding of ratios using the ratio pi to make calculations. Direct instruction of this skill may be needed to support students.

## Work Time

# Find Ratios in Polygons and Circles, Part 3

Explore the polygons as they get closer to the circle.

- How do the ratios of perimeter to distance across for each polygon compare to the ratio of the circumference of the circle to the diameter of the circle?

INTERACTIVE: Circle

# Prepare a Presentation

# Preparing for Ways of Thinking

Look for these types of responses to be shared during the class discussion:

- Students who see the relationship between the side length of the hexagon and the distance across, seeing that the distance is two side lengths because the hexagon divides into equilateral triangles
- Students who understand why the ratio in the polygons won’t change (and why no side lengths are given)
- Students who understand that the ratio should be constant and can explain how to use it to find the circumference, given the diameter
- Students who understand that the variation in the line plot and scatterplot is due to measuring accurately

# Answers

Presentations will vary.

# Challenge Problem

### Answer

The length of the diameter is about 12 inches.

$\frac{C}{d}$ ≈ 3.14

$\frac{\mathrm{37}}{d}$ = 3.14

$\frac{\mathrm{37}}{\mathrm{3.14}}$ = d

d ≈ 12

## Work Time

# Prepare a Presentation

Prepare a presentation about the circumference of a circle. Use examples of your work to illustrate your explanation.

# Challenge Problem

If the circumference of a circle is approximately 37 inches, what is the length of the diameter?

# Make Connections

# Mathematics

Facilitate the discussion to help students understand the mathematics of the lesson, making sure to address any questions students have from their work. Ask questions such as these:

- What ideas did you have for finding the ratios in the polygons?
- How were these ratios similar to the ratio in the circles?
- What does the scatterplot show?
- What does the line plot show?
- How did [student names] organize their thoughts differently? Which way of thinking makes more sense to you? Which way of thinking brought out the structure of the mathematics?
- How did [student names] make sense of the problem?
- Could you state what [student names] said in a different way?
- What is the ratio of the circumference to the diameter? Will it always be the same? Why wasn’t it when you measured?
- How could you measure more accurately?

SWD: For students with language-based learning vulnerabilities or learning challenges participating in a whole class discussion, Ways of Thinking can be intimidating for a variety of reasons. However, it is important for students to work on the speaking and listening skills implicit to this portion of the lesson. Possible supports for students prior to and during this portion of the lesson include: a few minutes to discuss their ideas, the questions posed, and what has taken place during the lesson with a partner or small group before sharing in the whole-class setting.

Conferencing with individual students prior to the discussion to ascertain what they might be able to successfully accomplish.

ELL: As with other discussions, consider presenting some of the questions in writing to support ELLs. Also consider providing sentence frames, such as the following (in the order the questions were given):

Also, make sure all students understand the terms plot and scatterplot. Allow them to use the dictionary if needed.

- “The ideas I have are…”
- “These ratios were similar to the ratios in the polygons in that…”
- “The scatterplot shows…”

## Performance Task

# Ways of Thinking: Make Connections

Take notes about other students’ understandings of the circumference of a circle.

## Hint:

As your classmates present, ask questions such as:

- How do the ratios for the polygons compare to the ratio for the circle?
- What do you think the formula for the circumference of a circle might be? How is this formula similar to the formula for the perimeter of a regular polygon?
- Can you explain how you made your graph?
- What unit did you use to measure the objects in the classroom? Why did you use that unit?

# Find Circumference

# Answers

- The circumference is about 31 inches.

*C* = *d*(3.14)

*C* = 10(3.14) = 31.4

Remind students that the radius is half of the diameter. Circles are often defined in terms of the radius, because a circle consists of all the points that are a given distance (the radius) from another point. A bicycle wheel and its spokes is a good analogy for students.

- The circumference is about 94 inches.

*C* = *d*(3.14)

*C* = (15 ⋅ 2)(3.14)

*C* = 30(3.14) = 94.2

## Work Time

# Apply the Learning: Find Circumference

The ratio of the circumference of a circle to the diameter of a circle is called pi (π ), which is a letter of the Greek alphabet.

π = $\frac{C}{d}$

The value of pi is approximately 3.14159, but the actual value is a decimal that goes on forever and has no pattern, making pi an irrational number. We generally use the value 3.14 to represent pi, because this value is close enough to the real value to be useful in most situations. We use this value of pi to represent a ratio, but it cannot truly be written as a ratio of rational numbers.

The circumference of a circle is equal to pi times the diameter.

*C* = π*d*

We can also say that the circumference of a circle is equal to pi times twice the radius.

*C* = 2π*r*

Use the formulas to find the circumferences of the following circles.

- A circle with a diameter of 10 inches
- A circle with a radius of 15 centimeters

## Hint:

- When you use the formula, think about whether you are given the radius or the diameter.

# Circumference

# A Possible Summary

The ratio of the circumference to the diameter is the same for any circle. The ratio is called pi (π) and is approximately 3.14. The ratio is most often used to find the circumference of a circle for a given diameter.

Because $\frac{C}{d}=\pi $, it is true that π*d* = *C*.

## Formative Assessment

# Summary of the Math: Circumference

Write a summary about the circumference of a circle.

## Hint:

Check your summary.

- Do you explain what you know about the ratio of the circumference of a circle to the diameter of a circle?
- Do you explain how you can use this ratio to find measures in a circle?

# Reflect On Your Work

# Lesson Guide

Have each student write a brief reflection before the end of class. Review the reflections to see students’ strategies for measuring circumference. If you notice insightful comments about how to measure circumference, plan to share them with the class in the next lesson.

## Work Time

# Reflection

Write a reflection about the ideas discussed in class today. Use the sentence starter below if you find it to be helpful.

**One way that a circle is like a regular polygon is …**