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:
- Which cut is the easiest to start with?
[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.
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.
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.