Photograph by Tracy Cullen; all rights reserved
mathematical explorations
p
Ji-Eun Lee and Kyoung-Tae Kim
It Is a
Piece
of Cake:
Algebraic Thinking in a Real-Life Situation
Picture a family gathering on a nice
summer day; in addition to delicious
food, fun activities, and much laughter
and talking, what else could be included?
Although it might seem odd, our family
gathering resulted in an important mathematics teaching and learning lesson.
The finale of the day involved
sharing a big cake in the shape of a
rectangular prism. This activity raised
interesting mathematical questions
and led to many discoveries and much
discussion. Initially, we asked the children to cut the cake and serve it to the
guests. They talked about how many
pieces they needed and how big each
piece should be. As with many children
who choose a piece of cake, they wanted
more icing. Subsequently, the children
and adults began asking questions about
the method of cutting the cake, the
amount of icing all would get, and the
Ji-Eun Lee, [email protected], is
on the faculty at Oakland University in
Rochester, Michigan. Kyoung-Tae Kim,
[email protected], teaches at John Carroll
University in University Heights, Ohio.
52
relationship between the number of cuts
and the number of faces with icing. This
real-life context intrigued the children,
and they began thinking about solution
strategies and sharing their findings.
In retrospect, the goal of this activity was to facilitate students’ ability
to identify and analyze algebraic
relationships in a real-life context. We
share activities that can be done by
middle school students to explore patterns and investigate multiple conjectures. When completed, students can
present their solution strategies.
CONNECTING TO PRINCIPLES
AND STANDARDS
This activity relates to areas in both
the Content Standards and the
Process Standards, found in Principles
and Standards for School Mathematics
(NCTM 2000):
Edited by Denisse Thompson, [email protected], Mathematics Department, University of South Florida in Tampa, and Gwen Johnson, [email protected],
Secondary Education, University of South Florida. This department is designed to provide
activities appropriate for students in grades 5–9. The material may be reproduced by classroom teachers for use in their classes. Readers who have developed successful classroom
activities are encouraged to submit manuscripts in a format similar to this “Mathematics
Exploration.” Of particular interest are activities focusing on the NCTM’s Content and Process
Standards and Curriculum Focal Points. Send submissions by accessing mtms.msubmit.net.
Mathematics Teaching in the Middle School
●
Vol. 14, No. 1, August 2008
Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved.
This material may not be copied or distributed electronically or in any other format without written permission from NCTM.
• Understand patterns, relations, and
functions
• Represent and analyze mathematical situations and structures using
algebraic symbols
• Use mathematical models to represent and understand quantitative
relationships
• Analyze change in various contexts
(p. 222)
Problem Solving
• Solve problems that arise in mathematics and in other contexts
• Apply and adapt a variety of appropriate strategies to solve problems
• Monitor and reflect on the process
of mathematical problem solving
(p. 256)
Reasoning and Proof
• Make and investigate mathematical
conjectures
• Select and use various types of reasoning and methods of proof (p. 262)
Connections
• Recognize and use connections
among mathematical ideas
• Recognize and apply mathematics
in contexts outside of mathematics
(p. 274)
Representation
• Create and use representations to
organize, record, and communicate
mathematical ideas (p. 280)
MATHEMATICAL EXPLORATIONS
THROUGH SHARING A CAKE
Students can be asked to cut and serve
pieces of cake to a group of people.
Set the stage for the exploration by
discussing how big each piece of cake
would be and how many pieces are
needed. Students will likely agree
to make all the pieces equal. At this
point, little mathematics is involved
except for simple counting and measurement skills. However, as pieces
begin to be distributed, students will
likely want pieces with the most icing.
The stage is set for several questions:
Fig. 1 Children cut the cake in one
direction only
• Will everyone get the same
amount of icing?
• Once we cut the cake, what will
be the maximum number of faces
with icing?
• Will the number of faces with icing be different each time we cut?
• What is the relationship between
the number of cuts and the number
of faces with icing?
At our gathering, Soodong (3 years
old), Limmy (a second grader), Joel
(a sixth grader), and Lyta (an eighth
grader) were in charge of this work.
Everyone agreed that the amount of
icing per piece would differ once we
started cutting. For example, Limmy
wanted a corner piece rather than a
middle piece because three faces of the
corner would be covered with icing.
The children investigated the questions
by actually cutting the cake, drawing
pictures, constructing charts, and communicating verbally. Lyta and Joel led
the discussion while the adults asked
questions; Limmy and Soodong patiently waited for their pieces of cake.
The following activities explore relationships between the number of cuts
and the number of faces with icing.
Activity 1: Cutting the Cake
in One Direction Only
This activity helps students identify several initial conditions that set the stage
for in-depth investigation. Be sure to
explicitly define a cut as being vertical or
horizontal from edge to edge. Question
3 on activity sheet 1 leads to an investigation of the impact of the direction of
the cut on the number of faces with icing. For example, Lyta and Joel initially
made vertical cuts only (see fig. 1) and
found some patterns: (1) the number
of pieces is always one more than the
number of cuts; (2) after the first cut,
Vol. 14, No. 1, August 2008
●
Photograph by Kyoung-Tae Kim; all rights reserved
Algebra
Fig. 2 Joel’s chart illustrates the
possible cases when the total number
of cuts is given.
there are always two pieces with icing
on four faces; and (3) after the first cut,
the number of pieces with icing on
three faces is always one fewer than the
number of cuts. However, Lyta and Joel
easily realized that they would not cut
in one direction only but would make
both vertical and horizontal cuts when
they served the cake. At this point, Lyta
preferred to use drawings and charts,
and Joel explained his reasoning verbally
instead of cutting into the actual cake.
Activity 2: Cutting the Cake
Horizontally and Vertically
Part A of activity sheet 2 investigates
the different ways the cake can be cut
Mathematics Teaching in the Middle School
53
Fig. 3 Lyta’s chart shows a grid for the relationships among the number of cuts, the
methods of cutting, and the number of faces with icing.
swer is 100 divided by 2, which is 50,
plus 1. That is 51.” We asked, “If we
cut n times, how many possible ways
can we get?” Lyta wrote the following
formulas using the letter n:
hen n is an even number:
W
(n ÷ 2) + 1
W hen n is an odd number:
(n + 1) ÷ 2
both vertically and horizontally. Lyta
and Joel noticed two situations when
the cake is cut twice: (1) 2 vertical or
2 horizontal cuts are considered one
way in terms of the number of pieces
and the number of faces with icing
and (2) 1 vertical and 1 horizontal cut
with an intersection is a second way.
In figure 2, Joel charted all the possible ways to make cuts. For example,
the possibilities for 2 cuts are 2 vertical
and 0 horizontal, 1 vertical and 1 horizontal, or 0 vertical and 2 horizontal.
Joel eliminated the last choice because
it created the same results as the first
choice. After 3 cuts, he did not write
all the cutting choices that would
eventually be eliminated. After 5 cuts,
he was able to determine the total
number of cuts (see the circled numbers in fig. 2) based on the pattern he
discovered. Lyta had a similar process,
as shown in figure 3; rather than write
54
the number of ways the cake could be
cut, she listed the number of pieces
obtained for the given number of cuts.
When asked the number of ways
resulting from 12 cuts, Lyta and Joel
answered, “Seven.” When asked about
20, Joel kept counting: “Seven [for 13
cuts], 8 [for 14 cuts], 8 [for 15 cuts], 9
[for 16 cuts], 9 [for 17 cuts], 10 [for 18
cuts], 10 [for 19 cuts], 11 [for 20 cuts].
It is 11.” At this point, an adult asked
Joel and Lyta whether they could find
the number of cases for 100 cuts. After
some discussion, they determined that
the number for 100 cuts would be 51.
Joel explained: “I looked at the pattern.
I found that when the total number
of cuts is an even number, we need
to divide it by 2 and add 1 to get the
number of possible ways. If the total
number of cuts is an odd number, we
have to add 1 to it and then divide it
by 2; 100 is an even number, so the an-
Mathematics Teaching in the Middle School
●
Vol. 14, No. 1, August 2008
Part B of activity sheet 2 helps
students investigate how the number
and methods of cutting change the
number of faces with icing.
Students use the rectangular templates to explore the number of pieces
of cake and the number of faces with
icing when 2 cuts are made in different
ways. With 1 horizontal and 3 vertical
cuts, students get eight pieces of cake:
four with icing on three faces and four
with icing on two faces. With 2 vertical
and 2 horizontal cuts, students get nine
pieces: four with icing on three faces,
four with icing on two faces, and one
with icing on one face. Part C encourages students to illustrate other cases.
Lyta excluded cutting in one direction only for a practical reason: She
would not cut the cake that way for
serving. After she completed the computations shown in figure 3, both she
and Joel started drawing the top view
of the cake (a two-dimensional shape)
to explore patterns.
Activity 3: Cutting the Cake:
Exploring More Patterns
Part A of activity sheet 3 helps
students find patterns to determine
the number of pieces of cake when 10
or more cuts are made. They stopped
constructing their charts (see figs.
2 and 3) at 9 cuts, so we wanted to
see how they would calculate 10 cuts
without using drawings.
Lyta and Joel identified different
patterns in the tables. Lyta added 1 to
the number of vertical cuts and the
number of horizontal cuts, respectively.
1. There are always four pieces (the four
corners) with icing on three faces.
2. The number of pieces with icing
on two faces will be the same for a
given number of cuts, regardless of
the way the cuts are made.
3. The number of pieces with icing
on two faces increases by two per
additional cut.
They concluded that for 10 cuts (1)
there are always four pieces with icing
on three faces and (2) there are always
sixteen pieces with icing on two faces.
Fig. 4 A cake containing 7 vertical cuts
and 3 horizontal cuts
For the number of pieces with icing
on one face, Lyta and Joel used different approaches to find patterns. Lyta
determined that the number of columns of pieces with icing on one face
is one fewer than the number of vertical cuts and that the number of rows
of pieces with icing on one face is one
fewer than the number of horizontal
cuts. The number of pieces with icing
on one face was (the number of vertical
cuts – 1) × (the number of horizontal
cuts – 1); for 4 vertical and 6 horizontal
cuts, (4 – 1) × (6 – 1) = 15 pieces that
have icing on only one face. Joel used
a pattern similar to the one he used
previously. He noticed that when only
1 cut is vertical or horizontal and the
rest are in the other direction, no pieces
have icing on only one face. Then, he
arranged the numbers of pieces with
icing on one face when he made 2
vertical and 1, 2, . . . , 7 horizontal cuts.
The number with icing on only one
face would be 1, 2, 3, 4, 5, and 6. In addition, 2 vertical and 8 horizontal cuts
would create seven pieces with icing on
one face, and 3 vertical and 7 horizontal cuts would produce twelve pieces
with icing on one face.
What happens if there are 5 vertical
and 5 horizontal cuts? Lyta simply applied her rule: (5 – 1) × (5 – 1) = 16. Joel
looked for a pattern when the number
of vertical cuts and the number of horizontal cuts were the same: zero pieces
with 1 vertical and 1 horizontal cut, one
piece with 2 vertical and 2 horizontal
cuts, four pieces with 3 vertical and 3
Vol. 14, No. 1, August 2008
●
Photograph by Kyoung-Tae Kim; all rights reserved
She found that the number of columns
of cake is always one more than the
number of vertical cuts, and the number
of rows is always one more than the
number of horizontal cuts. Then, she
multiplied these two numbers to find
the total number of pieces of cake. For
example, 2 horizontal and 8 vertical
cuts will make twenty-seven pieces of
cake: (2 + 1) (8 + 1) = 27.
Joel looked at his previous outcomes
when the cake had 1 vertical and up to
8 horizontal cuts. He found that the
number of pieces of cake increased by
two. Thus, the number of pieces for 1
vertical and 9 horizontal cuts would be
twenty: 4, 6, 8, 10, 12, 14, 16, 18, and
20. In the same way, the number of
pieces of cake for 2 vertical and up to 8
horizontal cuts increased by three: 6, 9,
12, 15, 18, 21, 24, and 27. Joel concluded that the number of pieces for 2
vertical and 8 horizontal cuts would be
twenty-seven.
After students complete the tables
in part A, ask them to discuss their
approaches. This provides an opportunity for students to realize that
different strategies can lead to the
same solution.
Part B of activity sheet 3 is
designed to help students look for
patterns in the number of faces with
icing when the cake is cut 10 or more
times. Lyta and Joel identified some
patterns jointly:
horizontal cuts, and nine pieces with
4 vertical and 4 horizontal cuts. Joel
concluded that the number of pieces
for 5 vertical and 5 horizontal cuts
would be sixteen. Additional cases can
be explored in part B of activity sheet
3. Figure 4 shows the thirty-two pieces
of cake made when there are 7 vertical
and 3 horizontal cuts: four with icing
on three faces, sixteen with icing on two
faces, and twelve with icing on one face.
IDEAS FOR EXTENDING
THE EXPLORATION
The activities could be extended to a
study of graphic representations and
sequences so that students “identify
functions as linear or nonlinear and
contrast their properties from tables,
graphs, or equations” and “use graphs
to analyze the nature of changes in
quantities in linear relationships”
(NCTM 2000, p. 222).
REFERENCE
National Council of Teachers of Mathematics (NCTM). Principles and Standards for School Mathematics. Reston,
VA: NCTM, 2000. l
Mathematics Teaching in the Middle School
55
Appendix: Solutions to the Activity Sheets
Activity Sheet 1
1. a. rectangle; b. 6
2. a. 5; b. Answers will vary.
3. There will be two pieces, each with icing on 4 faces.
4. b.
Total No. of Cuts
No. of Pieces
No. of Pieces with—
5 Iced Faces 4 Iced Faces 3 Iced Faces 2 Iced Faces
0
1
2
3
4
1
2
3
4
5
1
0
0
0
0
0
2
2
2
2
0
0
1
2
3
0
0
0
0
0
1 Iced Face
0
0
0
0
0
5. Answers will vary but may include that there is only one way to get a piece of cake with 5 iced faces and 0 ways to get
pieces with 1 or 2 iced faces. With 1 or more cuts, you will only get two pieces with 4 iced faces. The number of pieces
with 3 iced faces will always be 1 fewer than the number of cuts.
Activity Sheet 2
1.
Total No. of Cuts
No. of Vertical Cuts
No. of Horizontal Cuts
1
0
2
1
0
3
2
1
0
4
3
2
5
4
3
6
5
4
3
7
6
5
4
8
7
6
5
4
0
1
0
1
2
0
1
2
3
0
1
2
0
1
2
0
1
2
3
0
1
2
3
0
1
2
3
4
1
2
3
4*
5*
6*
7*
8*
* Duplicates are not listed.
56
Mathematics Teaching in the Middle School
●
Vol. 14, No. 1, August 2008
Total No. of Ways Cut
(Vertically and/or Horizontally)
1
2
2
3
3
4
4
5
2. a. 7 ways; b. 12 ways; c. 51. Explanations will vary
but may include that when the number of cuts is even,
divide the number of cuts by 2 and add 1. When the
number of cuts is odd, add 1 and take half.
3. If n is even: (n ÷ 2) + 1; if n is odd: (n + 1) ÷ 2.
Explanations will vary.
4.
3
2
2
3
3
2
2
3
2
3
2
1
2
3
2
3
No. of
horizontal cuts
No. of pieces with
icing on 1 face
No. of faces with icing
3
2
1
No. of pieces
4
6
10
2
3
4
5
6 …
n
8 12 16 20 24 30 … (n + 1) • 4
No. of faces with icing
3
2
1
No. of pieces
4
8
4
No. of
horizontal cuts
No. of pieces with
icing on 1 face
No. of
horizontal cuts
No. of pieces with
icing on 1 face
Activity Sheet 3
1.
1
4
5
2
3
4
6
8 10 12 14 … (n + 1) • 2
No. of vertical
cuts
6 …
n
1
n
1
2
3
4
5
6 …
0
1
2
3
4
5 … (n – 1)
3
6 …
n
1
2
3
4
5
0
2
4
6
8 10 … (n – 1) • 2
No. of vertical
cuts
8.–9. Answers will vary.
No. of vertical
cuts
2
No. of vertical
cuts
7. 18
No. of
horizontal cuts
No. of cake
pieces
1
No. of vertical
cuts
6. 10
No. of
horizontal cuts
No. of cake
pieces
No. of
horizontal cuts
No. of cake
pieces
3
2. 45
3. (a + 1) • (b + 1)
4.
5.
3
No. of vertical
cuts
4
5
6 …
n
1
2
3
4
0
3
6
9 12 15 … (n – 1) • 3
5. Answers will vary.
6. We worked through all parallel cuts (0 vertical cuts) on
activity sheet 1. With only one vertical cut, all pieces of
cake will have two or three sides with icing.
7. (7 – 2) • 6 = 36 pieces
8. (a + 1) • (b – 1) pieces. Explanations will vary.
2
3
4
5
6 …
n
1
2
6
9 12 15 18 21 … (n + 1) • 3
Vol. 14, No. 1, August 2008
●
Mathematics Teaching in the Middle School
57
activity sheet 1
Name ______________________________
Cutting the Cake in One Direction Only
1.a. What is the shape of the cake? ____________________
b. How many faces does the cake have? _______________
2.a. Before making any cuts, how many faces of the cake
have icing? _________
b. Explain your answer.
3.When you make 1 cut, determine the number of pieces and the number of faces with icing.
4.a. Record your results from problems 2 and 3 in the table below.
b. Complete the table if you continue to make cuts in just one direction.
Total Number
of Cuts
Total Number
of Pieces
5 Iced
Faces
Number of Pieces with—
4 Iced
3 Iced
2 Iced
Faces
Faces
Faces
0
1
2
3
4
5.What patterns do you notice in the table?
from the August 2008 issue of
1 Iced
Face
activity sheet 2
Name ______________________________
Cutting the Cake Horizontally and Vertically
Part A
1.Complete the table below to show the number of ways that a rectangular cake can be cut for the specific number of
cuts allowed. Be sure to cross out and discount any pairs that are equivalent. For example, 3 rows and 1 column =
1 row and 3 columns; you will just count these two as one way to cut the cake.
Total Number
of Cuts
1
2
Number of Vertical
Cuts
1
0
Number of Horizontal
Cuts
0
1
2
1
0
0
1
2
Total Number of Ways Cut
(Vertically and/or Horizontally)
3
4
5
6
7
8
2.Use the results in the table to find the number of ways for each total number of cuts. Explain your thinking.
a. 12 cuts
b. 25 cuts
c. 100 cuts
3.How many ways will you have if you are allowed n cuts? Explain.
from the August 2008 issue of
activity sheet 2
(continued)
Part B
4.Use the rectangle to represent the top of the cake. Make 1 horizontal and
3 vertical cuts. Show the number of faces with icing for each piece of cake.
5.Use the rectangle to represent the cake. Make 2 horizontal and 2 vertical
cuts. Show the number of faces with icing for each piece of cake.
Part C
In questions 6–7, make the cuts as described; in questions 8–9, choose your own number of vertical and horizontal cuts.
Use the drawings to find the number of pieces with icing on 1, 2, or 3 faces.
6.1 vertical cut and 4 horizontal cuts (a total of 5 cuts)
No. of faces with icing
No. of pieces
3
2
1
3
2
1
3
2
1
3
2
1
7.2 vertical cuts and 5 horizontal cuts (a total of 7 cuts)
No. of faces with icing
No. of pieces
8.___ vertical cut(s) and ___ horizontal cut(s) (a total of ___ cuts)
No. of faces with icing
No. of pieces
9.___ vertical cut(s) and ___ horizontal cut(s) (a total of ___ cuts)
No. of faces with icing
No. of pieces
from the August 2008 issue of
activity sheet 3
Name ______________________________
Cutting the Cake: Exploring More Patterns
Part A
1.Fill in each table. Describe any patterns you find.
No. of vertical cuts
1
No. of horizontal cuts
1
2
No. of cake pieces
4
6
3
4
6
...
n
...
No. of vertical cuts
2
No. of horizontal cuts
1
2
No. of cake pieces
6
9
3
4
5
6
...
n
...
No. of vertical cuts
No. of horizontal cuts
5
3
1
2
3
4
5
6
...
n
No. of cake pieces
2.How many pieces of cake will result if you make 4 vertical and 8 horizontal cuts? How do you know?
3. How many pieces of cake will result if you make n cuts, a of them vertical and b of them horizontal? Explain your thinking.
from the August 2008 issue of
activity sheet 3
(continued)
Cutting the Cake: Exploring More Patterns
Part B
4.Fill in each table.
No. of vertical cuts
2
No. of horizontal cuts
1
2
3
No. of pieces with icing on
1 face
0
1
2
4
No. of vertical cuts
No. of horizontal cuts
5
6
...
n
5
6
...
n
5
6
...
n
3
1
2
3
4
No. of pieces with icing on
1 face
No. of vertical cuts
No. of horizontal cuts
4
1
2
3
4
No. of pieces with icing on
1 face
5.Describe any patterns you find.
6.Why does this series of tables begin with 2 cuts?
7.How many pieces of cake have icing on 1 face if there are 5 vertical and 7 horizontal cuts?
8.How many pieces of cake have icing on 1 face if there are n cuts, a of them vertical and b of them horizontal?
Explain your thinking.
from the August 2008 issue of