# Geometry

## Objective

Determine the relationship between the circumference and diameter of a circle and use it to solve problems.

## Common Core Standards

### Core Standards

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• 7.G.B.4 — Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

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• 7.RP.A.2

## Criteria for Success

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1. Understand there is a proportional relationship between the circumference of any circle and its diameter.
2. Determine that the ratio of the circumference of a circle to its diameter is equivalent to ${\pi}$.
3. Know the formula that relates the circumference and diameter of a circle: $C={\pi} d$.
4. Use the formula $C={\pi} d$  to solve problems.

## Tips for Teachers

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• The following materials are needed for this lesson: rulers, string, circles handout, and graph paper.
• Lessons 6 and 7 focus on the relationship between a circle’s circumference and its diameter. In Lesson 6, students determine this relationship and the formula that represents it, making connections to proportional relationships and constant of proportionality. In Lesson 7, students will solve real-world and mathematical problems involving circumference, including problems involving semi-circles.
• Consider having students create their own reference sheet to add to and refer back to throughout the unit as needed. This can be placed in a sheet protector or created on cardstock. Alternatively, once both formulas for circles have been determined, students may be given a state-provided or pre-created reference sheet to refer to as needed. Here is the Massachusetts Comprehensive Assessment System Grade 7 Mathematics Reference Sheet.

#### Remote Learning Guidance

If you need to adapt or shorten this lesson for remote learning, we suggest prioritizing Anchor Problems 1 and 2 (benefit from worked examples). Students can also use the interactive geogebra tool (linked to the right) to discover the relationship As a discovery problem, if live discussion or reflection of the problem were possible, it would allow for students to arrive at the conclusion on their own. Find more guidance on adapting our math curriculum for remote learning here.

#### Fishtank Plus

• Problem Set
• Student Handout Editor
• Vocabulary Package

## Anchor Problems

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### Problem 1

Using the circles handout, measure the circumference and diameter of each circle and record your results in the table below.

 Circle Diameter ($d$) Circumference ($C$) Ratio of $C/d$ A B C D E F G
1. Create a graph to show the relationship between circumference and diameter. Place the values for diameter along the $x$-axis and the values for circumference along the $y$-axis.
2. What is the constant of proportionality? Write an equation to relate the circumference and diameter of any circle.

### Problem 2

Use the relationship between circumference and diameter to solve the two problems below.

1. The diameter of a circular clock is 12 inches. What is the circumference of the clock?
2. The distance around a circular cookie is approximately 18.84 cm. What is the diameter of the cookie?

## Problem Set

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The following resources include problems and activities aligned to the objective of the lesson that can be used to create your own problem set.

• Include problems where students are given either the circumference or the diameter and asked to find the other value.

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### Problem 1

Describe the relationship between the circumference of a circle and its diameter.

### Problem 2

The top of a can of tuna is in the shape of a circle. If the distance around the top is approximately 251.2 mm, what is the diameter of the top of the can of tuna? What is the radius of the top of the can of tuna?

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