### Clarifications

*Clarification 1:*Instruction includes the exploration and analysis of circular objects to examine the proportional relationship between circumference and diameter and arrive at an approximation of pi (π) as the constant of proportionality.

*Clarification 2:* Solutions may be represented in terms of pi (π) or approximately.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**7

**Strand:**Geometric Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Circumference
- Constant of Proportionality
- Diameter
- Pi (π)
- Proportional Relationship

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students solved problems involving the perimeter and area of two-dimensional figures. In grade 7, students explore the proportional relationship between circumferences and diameters of circles and develop and learn a formula to solve circumference problems. In grade 8, students will learn and use the Pythagorean Theorem to find the distance between points in the coordinate plane, and this builds the foundation for the equation of a circle in high school geometry.- Instruction includes opportunities for students to see circular or cylindrical household objects of different sizes. Students will measure the diameter and the circumference of the circle in each object to the nearest tenth of a centimeter to arrive at an approximation of pi (π) as the constant of proportionality. Students can record the values in a table and plot the points on a coordinate plane to discover the pattern that arises
*(MTR.5.1)*. Students should complete multiple trials to best support their conclusions using both radius and diameter.

Trial #1Trial #2 - Instruction emphasizes the relationship between radius and diameter so students will easily move between the equivalent forms of the circumference formula
*(MTR.3.1).* - Instruction includes student understanding that circumference of a circle is the same as perimeter of any other two-dimensional figure.
- Students are expected to know approximations of pi ($\frac{\text{355}}{\text{113}}$, $\frac{\text{22}}{\text{7}}$ or 3.14).

### Common Misconceptions or Errors

- Students may invert the terms radius and diameter. To address this misconception, review parts of a circle including radii, diameters and chords.
- Students may incorrectly believe pi is a variable, rather than a constant for every circle.
- Students may confuse circumference and area. To address this misconception, help students connect circumference as perimeter of a circle.

### Strategies to Support Tiered Instruction

- Teacher provides opportunities for students to measure the radius and diameter of various circles and to explore and discuss the similarities and differences between radius and diameter.
- To clarify misconceptions between the relationship of the diameter and circumference, instruction includes solving for the constant of proportionality between a given diameter and circumference of a circle and discussing the patterns that arise. Teacher provides opportunities for students to solve for the circumference of a given circle in terms of pi before replacing the value of pi with an approximation to determine the estimated circumference.
- Teacher co-constructs a graphic organizer with students containing color-coded examples of circumference, area, diameter and radius.

### Instructional Tasks

*Instructional Task 1*

**(MTR.4.1, MTR.7.1)**Amy and Eunice are participating in a bike-a-thon this weekend. Amy has 29-inch road bike wheels and Eunice has 26-inch mountain bike wheels, where the bike wheel measurements are based on their diameter.

- Part A. If they choose a bike-a-thon distance of 5 miles, whose bike wheels will need to do the fewest revolutions to reach the finish line?
- Part B. How many more revolutions will the other bike need to make to reach the finish line? Explain your reasoning.

### Instructional Items

*Instructional Item 1*

Determine the circumference of the following circles.

*Instructional Item 2*

When baking an apple pie, a strip of aluminum foil needs to be placed around the edge of the crust until the last 20 minutes of baking so that it will not burn. If using a 9$\frac{\text{1}}{\text{2}}$-inch diameter pie pan, how long should the strip of foil be?

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Perspectives Video: Experts

## Perspectives Video: Professional/Enthusiasts

## Perspectives Video: Teaching Ideas

## Problem-Solving Tasks

## Tutorials

## MFAS Formative Assessments

Students are asked to write the formula for the circumference of a circle, explain what each symbol represents, and label the variables on a diagram.

Students are asked to solve a problem involving the circumference of a circle.

Students are asked to explain the relationship between the circumference and diameter of a circle in terms of pi.

## Original Student Tutorials Mathematics - Grades 6-8

Explore the origins of Pi as the ratio of Circumference to diameter of a circle. In this interactive tutorial you'll work with the circumference formula to determine the circumference of a circle and work backwards to determine the diameter and radius of a circle.

Learn to solve problems involving the circumference and area of circle-shaped pools in this interactive tutorial.

## Student Resources

## Original Student Tutorials

Explore the origins of Pi as the ratio of Circumference to diameter of a circle. In this interactive tutorial you'll work with the circumference formula to determine the circumference of a circle and work backwards to determine the diameter and radius of a circle.

Type: Original Student Tutorial

Learn to solve problems involving the circumference and area of circle-shaped pools in this interactive tutorial.

Type: Original Student Tutorial

## Perspectives Video: Expert

<p>A math teacher describes the relationship between area and circumference and gives examples in nature.</p>

Type: Perspectives Video: Expert

## Problem-Solving Tasks

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

Type: Problem-Solving Task

In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

Type: Problem-Solving Task

In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

Type: Problem-Solving Task

## Tutorials

This video shows how the area and circumference relate to each other and how changing the radius of a circle affects the area and circumference.

Type: Tutorial

In this video, students are shown the parts of a circle and how the radius, diameter, circumference and Pi relate to each other.

Type: Tutorial

This video shows how to find the circumference, the distance around a circle, given the area.

Type: Tutorial

## Parent Resources

## Perspectives Video: Expert

<p>A math teacher describes the relationship between area and circumference and gives examples in nature.</p>

Type: Perspectives Video: Expert

## Problem-Solving Tasks

The goal of this task is to model a familiar object, an Olympic track, using geometric shapes. Calculations of perimeters of these shapes explain the staggered start of runners in a 400 meter race.

Type: Problem-Solving Task

In this problem, geometry is applied to a 400 meter track to find the perimeter of the track.

Type: Problem-Solving Task

In this task, a typographic grid system serves as the background for a standard paper clip. A metric measurement scale is drawn across the bottom of the grid and the paper clip extends in both directions slightly beyond the grid. Students are given the approximate length of the paper clip and determine the number of like paper clips made from a given length of wire.

Type: Problem-Solving Task