# MA.7.GR.1.4

Explore and apply a formula to find the area of a circle to solve mathematical and real-world problems.

### Examples

If a 12-inch pizza is cut into 6 equal slices and Mikel ate 2 slices, how many square inches of pizza did he eat?

### Clarifications

Clarification 1: Instruction focuses on the connection between formulas for the area of a rectangle and the area of a circle.

Clarification 2: Problem types include finding areas of fractional parts of a circle.

Clarification 3: Solutions may be represented in terms of pi (π) or approximately.

General Information
Subject Area: Mathematics (B.E.S.T.)
Strand: Geometric Reasoning
Status: State Board Approved

## Benchmark Instructional Guide

• Area
• Circle
• Pi (π)
• Rectangle

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 6, students found the areas of rectangles and triangles, and solved problems involving the area of quadrilaterals and composite figures by decomposing them into triangles or rectangles. In grade 7, students find the area of a circles and other geometric figures. In both grade 7 and high school, students build on their knowledge of area to find the surface areas and volumes of various three-dimensional figures.
• Students are not expected to memorize the formula for the area of a circle (MTR.5.1).
• Students are expected to know approximations of pi ($\frac{\text{355}}{\text{113}}$, $\frac{\text{22}}{\text{7}}$ or 3.14).
• Instruction includes students exploring circles. Provide students with a circle and have them highlight the circumference. Students will then fold the circle in half, half again, and half once more to allow them to cut it into 8 wedges of equal size. Then arrange the wedges so they alternately point up and down, forming a rectangle, with the highlighted circumference being the bases. Ensure students realize that the length of the rectangle is approximately equal to half the circumference, or π$r$, of the circle and the height of the rectangle is equal to the radius, $r$, of the circle (MTR.4.1).
• For more accuracy, provide a circle with dashed lined for students to cut the circle into 16 equal-sized wedges.

• Have students describe the area of a circle and explain if the area of a circle changes if it is cut up and rearranged.
• Ask questions to elicit student thinking (MTR.4.1) such as:
• What formula was used to find the area of a circle?
• How is the formula for the area of a circle related to the formula for the area of a parallelogram?
• Instruction includes using circles on grid paper for students to estimate area before making precise calculations.
• The expectation of this benchmark is not to find the radius or diameter of a circle when given the area.

### 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 (π) is a variable, rather than a constant that does not change from one circle to the next. Review the development of pi in MA.7.GR.1.3.
• Students may confuse circumference and area. To address this misconception, help students connect area of a circle to area of a rectangle.
• Students may incorrectly double the radius (finding diameter), rather than squaring it, when finding an area. To address this misconception, review exponent rules from MA.6.NSO.3.3 and MA.7.NSO.1.1.

### Strategies to Support Tiered Instruction

• Teacher provides opportunities for students to measure the radius and diameter of various circles to explore and discuss the similarities and differences between radius and diameter.
• Instruction includes modeling the area of a given circle in terms of pi before replacing the value of pi with an approximation to determine the estimated area.
• Teacher co-constructs a graphic organizer with students containing color-coded examples of circumference, area, diameter and radius.
• Teacher co-constructs a table to find the constant proportionality between the diameter and circumference of a circle, allowing students to discover the pattern that represents pi (π).
• Instruction includes modeling the differences between doubling and squaring a radius. Doubling a radius would be represented by multiplying the given length by 2, whereas squaring a number would be represented by the area of a square with the given radius.
• For example, students can be given the table below to show how the left column doubles a length whereas the right column squares a length.

• Instruction includes rewriting the area formula for a circle in expanded form before evaluating.
• For example, the formula for the area of a circle, $A$ = π$r$², can be rewritten as $A$ = (π)($r$)($r$).
• Teacher helps students connect area of a circle to area of a rectangle.

The figure below is composed of eight circles, seven small circles and one large circle containing them all. Neighboring circles only share one point, and two regions between the smaller circles have been shaded. Each small circle has a radius of 5 centimeters.

• Part A. What is the area of the large circle?
• Part B. What is the area of the shaded part of the figure?

### Instructional Items

Instructional Item 1
What is the area of a circle whose radius is 4 centimeters? Round to the nearest hundredth.

Instructional Item 2
Find the exact area, in centimeters (cm), of each circle below.

Instructional Item 3
Jamilah wants to add to her kitchen countertop, which is currently in the shape of a rectangle. If she adds the solid, semicircular piece shown in the picture below, determine how many square feet, to the nearest tenth of a foot, of marble Jamilah will need for the addition.

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

## Related Courses

This benchmark is part of these courses.
1205040: M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812020: Access M/J Grade 7 Mathematics (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))

## Related Access Points

Alternate version of this benchmark for students with significant cognitive disabilities.
MA.7.GR.1.AP.4: Apply a given formula to find the area of a circle to solve mathematical problems.

## Related Resources

Vetted resources educators can use to teach the concepts and skills in this benchmark.

## Formative Assessments

Circle Area Formula:

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

Type: Formative Assessment

Center Circle Area:

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

Type: Formative Assessment

Broken Circles:

Students are asked to complete and explain an informal derivation of the relationship between the circumference and area of a circle.

Type: Formative Assessment

## Lesson Plans

Clean It Up:

Students will help a volunteer coordinator choose cleanup projects that will have the greatest positive impact on the environment and the community.  They will apply their knowledge of how litter can impact ecosystems along with some math skills to make recommendations for cleanup zones to prioritize.  Students will explore the responsibilities of citizens to maintain a clean environment and the impact that litter can have on society in this integrated Model Eliciting Activity.

Type: Lesson Plan

How Fast Can One Travel on a Bicycle?:

Students investigate how the pedal and rear wheel gears affect the speed of a bicycle. A GeoGebra sketch is included that allows a simulation of the turning of the pedal and the rear wheel. A key goal is to provide an experience for the students to apply and integrate the key concepts in seventh-grade mathematics in a familiar context.

Type: Lesson Plan

My Favorite Slice:

The lesson introduces students to sectors of circles and illustrates ways to calculate their areas. The lesson uses pizzas to incorporate a real-world application for the of area of a sector. Students should already know the parts of a circle, how to find the circumference and area of a circle, how to find an arc length, and be familiar with ratios and percentages.

Type: Lesson Plan

## Original Student Tutorials

Pizza Pi: Area:

Explore how to calculate the area of circles in terms of pi and with pi approximations in this interactive tutorial. You will also experience irregular area situations that require the use of the area of a circle formula.

Type: Original Student Tutorial

Swimming in Circles:

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

Type: Original Student Tutorial

## Perspectives Video: Expert

B.E.S.T. Journey:

What roles do exploration, procedural reliability, automaticity, and procedural fluency play in a student's journey through the B.E.S.T. benchmarks? Dr. Lawrence Gray explains the path through the B.E.S.T. mathematics benchmarks in this Expert Perspectives video.

Type: Perspectives Video: Expert

## Perspectives Video: Professional/Enthusiasts

Pizza Pi: Area, Circumference & Unit Rate:

How many times larger is the area of a large pizza compared to a small pizza? Which pizza is the better deal? Michael McKinnon of Gaines Street Pies talks about how the area, circumference and price per square inch is different depending on the size of the pizza.

Type: Perspectives Video: Professional/Enthusiast

Using Geometry for Interior Design and Architecture:

<p>An architect discusses how he uses circumference and area calculations to accurately create designs and plans.</p>

Type: Perspectives Video: Professional/Enthusiast

Eight Circles:

Students are asked to find the area of a shaded region using a diagram and the information provided. The purpose of this task is to strengthen student understanding of area.

## Tutorials

Impact of a Radius Change on the Area of a Circle:

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

Area of a Circle:

In this video, watch as we find the area of a circle when given the diameter.

Type: Tutorial

## MFAS Formative Assessments

Broken Circles:

Students are asked to complete and explain an informal derivation of the relationship between the circumference and area of a circle.

Center Circle Area:

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

Circle Area Formula:

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

## Original Student Tutorials Mathematics - Grades 6-8

Pizza Pi: Area:

Explore how to calculate the area of circles in terms of pi and with pi approximations in this interactive tutorial. You will also experience irregular area situations that require the use of the area of a circle formula.

Swimming in Circles:

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

## Student Resources

Vetted resources students can use to learn the concepts and skills in this benchmark.

## Original Student Tutorials

Pizza Pi: Area:

Explore how to calculate the area of circles in terms of pi and with pi approximations in this interactive tutorial. You will also experience irregular area situations that require the use of the area of a circle formula.

Type: Original Student Tutorial

Swimming in Circles:

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

Type: Original Student Tutorial

Eight Circles:

Students are asked to find the area of a shaded region using a diagram and the information provided. The purpose of this task is to strengthen student understanding of area.

## Tutorials

Impact of a Radius Change on the Area of a Circle:

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

Area of a Circle:

In this video, watch as we find the area of a circle when given the diameter.

Type: Tutorial

## Parent Resources

Vetted resources caregivers can use to help students learn the concepts and skills in this benchmark.