Activity 1: Customers Cut the Cake
Each day the local baker makes several rectangular sheet cakes,
which he cuts into eighths. He sells 1/8 of a sheet cake for $1.59. As
part of a new promotional campaign for his store, he wants to cut his
sheet cakes into eighths a different way each day. Customers who
suggest a new way to cut the cakes into eighths win a free piece of
cake each day for a week. What are some of the different ways to cut
Some questions to ask students include:
- What is the shape of the baker's cakes? [Rectangular cakes.]
- What are the restrictions on the ways the cakes can be cut?
[Pieces must contain the same amount of cake; they do not have to be
the same shape.]
- How can we verify that pieces that are not the same shape
contain the same amount of cake? [Cut the pieces into smaller parts and
lay one on top of the other.]
This activity gives students opportunities to represent parts of a
whole by using an area model of fractions. Many students know that
fractions often refer to equal-sized parts of a unit, but they
frequently overgeneralize and believe that the pieces have to be
congruent rather than merely contain the same area.
Encourage students to solve the problem in pairs or in small groups by using the Cakes Cut Into Eighths activity sheet.
Alternatively, students can use rectangular pieces of paper to
model the cakes, sketching the shapes or cutting the paper into eight
pieces and verifying the equivalence of the pieces by cutting and
overlapping. Some students may have a limited view and think that all
the cuts must be parallel to one side of the rectangle. Challenge them
to think of other ways to make the cuts. (It is important for students to know that each rectangle on the activity sheet represents a full cake.)
Have students place their designs on the chalkboard or the
overhead projector. Ask the students to decide which designs are the
same and which are different. Examples of some diagrams are shown
below. Have the students discuss whether the two rectangles in this
figure are cut differently.
What factors should be considered when deciding whether the two
designs are different? (The number of pieces, the equivalence of the
pieces, and whether the location of the pieces makes a difference
should be considered.) The two rectangles in the above figure contain
the same eight pieces, but the pieces are arranged differently. The
students can decide whether they want to consider these as two
Another way that must be considered is using cuts that are curves or
combinations of line segments, such as the examples in the figure
The equivalence of shapes formed by cuts that are curves is
difficult to determine but is a good investigation in itself. The
equivalence of shapes that are formed by cuts that are combinations of
straight line segments is easier to determine. Including these types of
shapes, however, again greatly increases the number of possibilities.
By interacting with students, decide which designs to include in the
Ask each group to choose one design and either to post it on the
chalkboard or bulletin board or to draw it on the overhead projector.
Students in each group should be prepared to explain how they know that
their method shows eighths. One way to verify that a solution does in
fact result in eighths is to cut up the individual pieces further and
lay them on top of each other to verify the equivalence of their areas.
As a follow-up activity,
teachers may choose to discuss with students why or why not each cake on below is cut into eighths.
Activity 2: You Can Eat Your Cake and Have It, Too!
The baker is conducting a second contest, this time for his
employees. As part of a new promotional campaign for his store, each
day he wants to feature sheet cakes that have been cut into four pieces
in a different way. The pieces do not have to be equal for this
promotion. The baker has challenged his employees to suggest
interesting ways to cut the cakes into four pieces. The employees must
also determine the price for each piece. The bakery sells 1/8 of a
sheet cake for $1.59. What are some of the different ways the cakes can
be cut, and how much should each piece cost?
Some questions for the students to discuss include:
- What are the restrictions on the ways the cakes are cut? [Each cake
must be cut into four pieces, not necessarily equal-sized pieces.]
- How can we determine the fractional parts of the pieces we
cut? [We can use equivalences we know, such as 2/4 is equivalent
to 1/2, to find the value of each part, if we partition by finding
parts of parts.]
- What will happen if we just cut four pieces at random? [It will be difficult to determine the size of each piece.]
Instead of focusing on making equal-sized pieces as in the previous
activity, this activity explores determining the fractional parts and
cost of pieces when a unit is cut into four unequal parts.
Encourage students to solve the problem in pairs or small groups by
using the Cakes Cut Into Fourths activity sheet or rectangular paper to
model the cakes. They sketch the pieces or cut the rectangles into four
pieces and determine the value of each piece.
One way to find the value of each piece is to add partitioning
lines so that the whole is partitioned into equal-sized pieces.
Students may remember some of the ways they cut the cakes into eighths
in the first activity, which may help them. Once again, weighing could
solve this problem.
The following overhead can be projected after students have had time to create their designs.
Have students share and discuss their designs. The rectangles
shown below show a few possible ways to cut the rectangular cakes into
four parts. The pieces have been labeled to show the fraction of the
cake they represent.
Students must then find the cost for each piece, if 1/8 of a cake costs $1.59. What should the total cost of one whole cake be? (If 1/8 of a cake costs $1.59, then a whole cake should cost eight times as much, or $12.72.)
Challenge students to explain and verify their solutions.
Activity 3: That's the Way the Cookie Crumbles!
You bought a baker's dozen (13) of cookies that you want to
share equally with your family. How many cookies will each person get?
Ask students to compare this problem with the one posed in the
first activity. Give them time to think about the similarities and
differences between this problem and the problem posed in Activity 1.
[This problem is similar to the first problem, that of
cutting rectangular sheet cakes into eighths in different ways, because
they both involve partitioning a unit into parts. This problem is
different in several ways: The whole or unit in this group of
13 cookies; the problem does not specify exactly into how many pieces
to cut each cookie or how many people are sharing the cookies; the
problem has different solutions for students who have different-sized
families; and this problem involves a different interpretation of
rational numbers — the quotient interpretation. The quotient
interpretation refers to the fact that in this problem, in which
thirteen cookies are being shared by n people (n is the number of
people in the family), the number of cookies each person receives is
the quotient, 13 ÷ n.]
Consider grouping students according to the number of people in
their families. Students can draw 13 circles on a piece of paper to
practice dividing the cookies. After each group has solved the problem,
share the solution processes with the whole class. Have each student
complete a table similar to the sample below.
The table should include all the different-sized families of
students in the class and also contain a few other family sizes,
including one or two families that are larger than the largest family
in the class. Students should begin completing the table by including
solutions from the groups in the class and then working on solving the
problems for other family sizes.
Discuss the solutions, focusing on the patterns students see
in the number of cookies for each person as the size of the family
changes. The goal for students is to generalize that the number of
cookies for each person is equal to the number of cookies divided by
the number of people sharing them.
Ask students to write a rule that represents the number of cookies each person receives if thirteen cookies are shared by n people (where n is the number of people in the family).
[The number of cookies each person receives is the quotient, 13 ÷ n.]
Ask students to describe the way they solved the problem in their
groups (e.g., by drawing a picture, using long division, and so on) so
that a student in another class would understand what problem was
solved and how it was solved.