• Lesson
6-8
1

Students explore two different methods for dividing the area of a circle in half, one of which uses concentric circles. The first assumption that many students make is that half of the radius will yield a circle with half the area. This is not true, and it surprises students. In this lesson, students investigate the optimal radius length to divide the area of a circle evenly between an inner circle and an outer ring.

Present the problem: A student has brought a giant cookie in for a celebration, and the cookie is as big as a large pizza. How could the cookie be cut so that everyone in the class gets their fair share? Note: You want students to think about cutting the circle in half, so make sure your fair share number is even—if there are an odd number of students in your class, include yourself.

Have students consider the following questions:

• Is the cookie big enough for the whole class to share?
• What fraction of the cookie should each student get?
• How would you divide the cookie?
• What is the first cut you would make? Why?

Suppose there are 22 students in a class. Ask:

[Many students will suggest cutting the cookie in half.]
• How should we divide the circle in half?
[Many students will suggest using the diameter.]

Pass out the Fair Share activity sheet. Direct students to draw lines in the first circle to indicate how they would partition it into 22 equal pieces. Specify that they must start by drawing a diameter. Observe the difficulties they encounter in drawing the rest of their lines—the middle of the circle gets filled with lines and becomes difficult to see.

 Fair Share Activity Sheet

Ask students whether this is the best way to divide up the cookie. Because of the large number of sectors, students will probably agree that the partitioning gets clumsy.

In the next problem, students will explore other ways to make the first partition—ways that will not create so much crowding at the center. The person who cuts the cookie will still need to cut 11 equal pieces from each half, but these pieces will be easier to make and share. Model the fraction problem on the board for students:

1/2 × 1/11 = 1/22

Challenge students to come up with another way to divide the circle’s area in half in Question 2 of the activity sheet. As you observe students working, ask those who have unique solutions to draw their solution on the board. This is an opportunity for differentiation. Some students will create elaborate half-cuts while others may still be struggling to understand how the diameter cuts the circle in half. If a student finishes quickly, ask them to think of other ways to divide the circle in half.

Students will probably have many ideas for ways to divide the area of the circle into two equal pieces, and some may be interesting to look at in the future. If no one mentions the concentric circle cut, draw the figure shown below on the board. Ask students where the inner circle should be drawn in order to get half of the total area in the inner circle and half the total area in the outer ring. This can also be posed as dividing the circle so the inner circle has an area equal to half the area of the outer circle.

Pass out different colors of construction paper and compasses. Have students make a point on the paper that is the center of the circle, construct a circle of any size, and then cut it out. Pass out the Giant Cookie Dilemma activity sheet. To complete Question 1 on the activity sheet, students will measure the radius of their circle, calculate the area, and then find half of the area. Have students estimate a radius for the inner circle that will result in the area of the inncer circle being equal to half the area of the outer circle. For Question 2, they should record this radius and construct it on the circle they cut out. Students should continue filling in the table in Question 3 until they have a good estimate of the inner radius, and then complete the remaining questions. At the end of the activity, students should have a good estimate for the radius of the inner circle and a drawing of the concentric circles based on their best radius estimate.

 The Giant Cookie Dilemma Activity Sheet

Ask students which cuts would make a better piece to eat: their original circle with 22 radial cuts? or the new concentric circles with 11 radial cuts? Have students compare their models and consider what it would be like to eat a cookie of that size and shape.

After students have worked with the paper copy and arrived at a final radius for the inner circle, ask them to calculate the ratio of the inner circle's radius to that of the outer circle. Students can then record their ratios on the Giant Cookie Ratios overhead. As data are written on the overhead, students should begin to see that the decimal ratio is close to 0.71, no matter how big the original circle is. Calculated algebraically, the exact ratio is 1/√2. Middle school students will probably not figure out the answer algebraically, but they should recognize that all the answers are approximately the same.

If time permits, or as an alternative to using construction paper circles, use the Cookie Cutter Applet to give students the opportunity to test different values of the smaller radius. As students change the length of the smaller radius, the applet shows the change in the area of the inner circle and the ratio of the smaller to the larger radius. As with the overhead sheet, students should see that the ratio approaches approximately 0.71.

Assessments

1. Have students construct the two concentric circles on grid paper. They can then verify that the area of the inner circle is equal to that of the outer ring by counting the squares.
2. Present students with the problem using a fixed value for the radius of the cookie. Tell them you baked a cookie on a pizza pan that is 16 inches in diameter. Ask students to write a paragraph explaining how they would use the concentric circles method to divide the cookie in half. They should provide their calculations with their written explanation.
3. Have students create a two-color target game so that there is an equal chance of landing on each color.

Extensions

1. If some students still don't believe that their inner cirlce has the same area as the outer ring, you could make a giant cookie, pressed onto and baked on a pizza tray, and bring it in to do an actual demonstration of the problem. Weigh the inner circle and outer ring separately. If they are the same size, then they will also be the same weight.
2. Present this situation to students: A bathroom tissue company has manufactured a roll of tissue that is "twice as big" as a standard roll. What will the roll look like compared to the original? Will it look twice as big? Students can write an explanation to a friend who doesn't believe that the new roll is really twice the size of the original.
3. Have students consider paper towels, toilet paper, or thread. What percentage of the radius represents halfway through a roll? Is this half the same as the half in the cookie problem?
4. Students who understand the relationship involved in dividing a circle in half can attempt to draw concentric circles that divide the area into 3 or 4 equal parts. Again, it is important for them to be able to generalize the rule for any circle.
5. Have students construct 2 concentric circles, in which the inner radius is 1/√2 time the size of the larger radius. Have them cut out the inner circle and then cut the ring into pieces. Challenge them to fit the pieces into the inner circle so that it is filled up.
6. Have students use algebra to verify the ratio of 1/√2. The area of the inner circle, radius r1 is equal to half of the area of the larger circle r2. That is,

1/2r22 = r12.

Solving this equation leaves 1/√2 = r1/r2.

Questions for Students

1. How many ways are there to divide the area of a circle in half?

[There are limitless ways to do this. However, in the context of further subdividing the cookie so 22 people can have equal shares of it, there are fewer practical ways to make the first cut.]

2. How long must the radius of the inner circle be in order to create an inner circle and an outer ring that have equal areas?

[1/√2]

3. If you wanted to double the area of a circle, how much bigger should you make the radius of a circle?

[The radius should be √2 (or about 1.41) times the size of the original radius. This question could help students understand what is happening mathematically in the cookie problem.]

Teacher Reflection

• Did students connect the number calculations to the geometric model?
• Did they discover the algebraic general rule?
• Did students have the opportunity to address twice as big and half as big in a real world context?
• Are students confident that half of the radius creates less than half the area?
• What difficulties did your students have? How did you resolve them?

### Learning Objectives

Students will:

• Find the areas of circles and rings
• Determine the ratio of the outer ring area to the inner circle area
• Find the optimal radius to use to create the inner circle in order for the areas to be equal

### Common Core State Standards – Mathematics

• CCSS.Math.Content.6.RP.A.1
Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

• CCSS.Math.Content.7.NS.A.3
Solve real-world and mathematical problems involving the four operations with rational numbers.

• CCSS.Math.Content.7.EE.B.3
Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or$2.50, for a new salary of \$27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

• CCSS.Math.Content.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.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP4
Model with mathematics.
• CCSS.Math.Practice.MP5
Use appropriate tools strategically.
• CCSS.Math.Practice.MP7
Look for and make use of structure.