math by the month
S h u k - k wa n S . L e u n g a n d J a n e - J a n e L o
Sweet play
Robert Linton/iStockphoto.com
As part of her holiday preparations, Cathy decorates sugar cubes. Cathy’s aunt,
Ms. Betty, runs a bakery and likes to think of ways to make her store more attractive,
especially for holidays. She enjoys cutting, arranging, and designing seasonal displays with her famous brownies almost as much as she enjoys baking them. And
everyone loves the different sizes of Ms. Kate’s chocolate sticks.
The characters in our Sweet play math have never been scolded for playing with their food. However, you may want to substitute straws, paper
squares, alphabet blocks, or such commercially made manipulatives as
Unifix® cubes for the real sweets. Given no allergy concerns, teachers and
students alike would enjoy some sweet rewards after having a taste of
these mathematical explorations.
How many stickers? Cathy likes to turn her cube decorations to give them different
looks. After making her 4-unit solid designs, she dresses them up by putting pretty
stickers on all the faces that can be seen from the outside, including the faces that sit
on the glass display surface. Which designs use the fewest number of stickers? Which
designs use the most? Explain your answers.
Describing a cube’s top, front, and side views is one way to identify its design. For this solid
, the
top and front views look like this:
. A side view would show this: . Which of your 4-unit solid
designs has the following views? Top and front:
Side:
Is there more than 1 solution? Create a
design of your own, exchange with a partner, and solve each other’s puzzle.
VESTA HAN
February
15–19
Make a cube. Cathy’s 4-unit solid designs remind her friend Jenny of an old puzzle
called Soma cubes. Jenny must remove the long 1 × 4 solid and the flat 2 × 2 solid,
then use 3 cubes to build a 3-unit solid like this:
. This 3-unit solid piece and all
other 4-unit solids will form a Soma cube puzzle. Jenny challenges Cathy to use this
set of puzzle pieces to make a big 3 × 3 × 3 cube such as the one to the right. Build a
model with sugar cubes, blocks, or multilink cubes to show Cathy the solution.
February
22–26
VESTA HAN
February 1–5
Sugar cubes have been a part of Cathy’s holiday preparations ever since she found 100 of them in a box
she had long forgotten. Determine how many different solid designs Cathy can make by gluing 4 sugar
cubes together with 1 complete face connecting any 2 cubes. Count as “different” only the designs that
cannot be made into one another by turning and flipping. What will the designs look like? Compare your
results with your classmates’ results. How can you convince others that you found all the possibilities?
February 8–12
Grades 5–6
Shuk-kwan S. Leung and Jane-Jane Lo, who submitted these problems, wish to thank Vesta Han for the beautiful artwork and photographs.
Edited by Lynn Columba, [email protected], who teaches elementary and secondary school mathematics methods courses at Lehigh University’s
Department of Education and Human Services in Bethlehem, Pennsylvania. E-mail creative solutions and adapted problems to [email protected] for
potential publication, noting “readers exchange” in the subject line. Activities in “math by the month” are grouped by grade bands K­–2, 3–4, and
5–6; they feature a monthly theme. Specific math content varies by issue. See detailed submission guidelines at www.nctm.org/tcmdepartments.
Full-sized students’ activity sheets, by grade-band, accompany the online version of “math by the month” at www.nctm.org/tcm.
330
February 2010 • teaching children mathematics Copyright © 2010 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.
www.nctm.org
February
8–12
A 2-shelf display case shows off Ms. Betty’s various 4-unit brownies. She sorts her brownies into 2 different groups: the symmetrical ones and the nonsymmetrical ones. Draw what Ms. Betty’s shelf display
might look like. Can she make any other groupings?
February
15–19
During the Valentine season, Ms. Betty makes a fun display by arranging 4 brownies into different shapes where any 2 brownies are connected on an entire side. The only
shapes she considers as “different” are 2 shapes that cannot be made into each another by turning or flipping. For example, these 2 shapes are not considered different, because you can flip and turn the left one
to mirror the right one. Draw all the different shapes Ms. Betty can make using 4 brownies. How many are there?
Ribbons around the edges of Ms. Betty’s brownie designs make them even more attractive. Ms. Betty
notices that different designs may use different amounts of ribbon. Which designs use the most ribbon?
Which designs use the least? Predict which designs will need the least amount of ribbon if Ms. Betty
decides to make her designs with 9 brownies.
February
22–26
February 1–5
When Ms. Betty uses 1 or 2 brownies, she can make only 1 shape. These 2 shapes are considered the same:
.
No empty space, please. To save shelf space, Ms. Betty arranges the 4-unit brownies in such a way that
there is no empty space between any 2 designs. Can she do so with all her designs? Convince your friends
that your answer is correct by drawing the designs.
February
8–12
Sophia likes to make longer sticks, too, and she likes to use different numbers of sticks to make the same
length. She figures out that she can make a 6-unit stick by joining a 4-unit stick and a 2-unit stick. She can
also make a 6-unit stick by using six 1-unit sticks. How do you know if Sophia has figured out all the possible ways? How many different ways can she make a 12-unit stick? Try to find as many ways as possible.
Brian likes to make shorter sticks. He has chocolate sticks of 4 different lengths: 1 unit long, 4 units long,
7 units long, and 9 units long. By putting a 7-unit stick and a 9-unit stick next to each other and breaking
off the extra from the 9-unit stick, Brian makes a 2-unit stick. Using the same sticks in the same way that
Brian does, make an 8-unit stick. Now try to make 1 stick with the other length. Is there more than one
way to make certain lengths?
Pete wonders if it is possible for Brian to create any length of sticks between 1 unit long and 9 units long by
carefully using Brian’s ideas of breaking off sticks. What do you think? Tell Pete how to make each length.
www.nctm.org
teaching children mathematics • February 2010 VESTA HAN
February 1–5
Ms. Kate’s chocolate sticks come in many different lengths. Some are longer, and some are shorter. Think
about how to place the sticks together to make other lengths. Amy plays with chocolate sticks of 5 different lengths: 1 unit long, 2 units long, 3 units long, 4 units long, and 5 units long. She can make longer
ones by joining 2 sticks. How can Amy make a 6-unit stick? How many ways can she do that? What
is the shortest stick she can make by joining 2 sticks? What is the longest stick she can make by
joining 2 sticks? How many different lengths can she make by joining 2 sticks? List them all and
share your solutions with a partner.
February
15–19
Grades K–2
February
22–26
Melissa King/iStockphoto.com
Grades 3–4
331
➺ math by the month activity sheet
Grades 5–6
Name _____________________________
Sweet play
Feng Yu/iStockphoto.com
Cathy decorates sugar cubes as part of her holiday
preparations. She has never been scolded for
playing with her food. However, you may want to
substitute straws, paper squares, alphabet blocks,
®
or Unifix cubes for the real sweets. After having
a taste of these mathematical explorations, enjoy
some sweet rewards.
How many stickers? Cathy likes to turn her cube decorations to give them different looks. After
making her 4-unit solid designs, she dresses them up by putting pretty stickers on all the faces
that can be seen from the outside, including the faces that sit on the glass display surface. Which
designs use the fewest number of stickers? Which designs use the most? Explain your answers.
Make a cube. Cathy’s 4-unit solid designs remind her friend Jenny of an old puzzle called Soma
cubes. Jenny must remove the long 1 × 4 solid and the flat 2 × 2 solid, then use 3 cubes to build
a 3-unit solid like this:
. This 3-unit solid piece and all other 4-unit solids will form a Soma
cube puzzle. Jenny challenges Cathy to use this set of puzzle pieces to make a big 3 × 3 × 3 cube
such as the one to the right. Build a model with sugar cubes, blocks, or multilink cubes to show
Cathy the solution.
Describing a cube’s top, front, and side views is one way to identify its design. For this solid
, the top and
front views look like this:
. A side view would show this:
. Which of your 4-unit solid designs has the
following views? Top and front:
Side:
Is there more than 1 solution? Create a design of your own,
exchange with a partner, and solve each other’s.
From the February 2010 issue of
VESTA HAN
VESTA HAN
Sugar cubes have been a part of Cathy’s holiday preparations ever since she found 100 of them in a box she had
long forgotten. Determine how many different solid designs Cathy can make by gluing 4 sugar cubes together
with 1 complete face connecting any 2 cubes. Count as “different ” only the designs that cannot be made into
one another by turning and flipping. What will the designs look like? Compare your results with your classmates’
results. How can you convince others that you found all the possibilities?
➺ math by the month activity sheet
Grades 3–4
Name _____________________________
Melissa King/iStockphoto.com
Sweet play
Ms. Betty’s famous brownies are everyone’s favorite dessert. She sells a lot of them
in her bakery every day. Ms. Betty likes to think of ways to make her store more
attractive, especially for holidays. She enjoys cutting, arranging, and designing seasonal displays with her brownies almost as much as she enjoys baking them. After
having a taste of these mathematical explorations, enjoy some sweet rewards.
When Ms. Betty uses 1 or 2 brownies, she can make only 1 shape. These 2 shapes are considered the same: .
During the Valentine season, Ms. Betty makes a fun display by arranging 4 brownies into different
shapes where any 2 brownies are connected on an entire side. The only shapes she considers as “different”
are 2 shapes that cannot be made into each another by turning or flipping. For example, these 2 shapes are
not considered different, because you can flip and turn the left one to mirror the right one. Draw all the
different shapes Ms. Betty can make using 4 brownies. How many are there?
A 2-shelf display case shows off Ms. Betty’s various 4-unit brownies. She sorts her brownies into 2 different groups: the
symmetrical ones and the nonsymmetrical ones. Draw what Ms. Betty’s shelf display might look like. Can she make any
other groupings?
Ribbons around the edges of Ms. Betty’s brownie designs make them even more attractive. She notices that different
designs may use different amounts of ribbon. Which designs use the most ribbon? Which designs use the least? Predict
which designs will need the least amount of ribbon if Ms. Betty decides to make her designs with 9 brownies.
No empty space, please. To save shelf space, Ms. Betty arranges the 4-unit brownies in such a way that there is no empty
space between any 2 designs. Can she do so with all her designs? Convince your friends that your answer is correct by
drawing the designs.
From the February 2010 issue of
➺ math by the month activity sheet
Grades K–2
Name _____________________________
Ms. Kate’s chocolate sticks come in many different lengths. Some are longer, and some are shorter. Think about how to place the sticks together to
make other lengths. After having a taste of these mathematical explorations, enjoy some sweet rewards.
VESTA HAN
Sweet play
Amy plays with chocolate sticks of 5 different lengths: 1 unit long, 2 units long, 3 units long, 4 units long, and 5 units long.
She can make even longer ones by joining 2 sticks. How can Amy make a 6-unit stick? How many ways can she do that?
What is the shortest stick she can make by joining 2 sticks? What is the longest stick she can make by joining 2 sticks? How
many different lengths can she make by joining 2 sticks? List them all and share your solutions with a partner.
Sophia likes to make longer sticks, too, and she likes to use different numbers of sticks to make the same length. She
figures out that she can make a 6-unit stick by joining a 4-unit stick and a 2-unit stick. She can also make a 6-unit stick
by using six 1-unit sticks. How do you know if Sophia has figured out all the possible ways? How many different ways can
she make a 12-unit stick? Try to find as many ways as possible.
Brian likes to make shorter sticks. He has chocolate sticks of 4 different lengths: 1 unit long, 4 units long, 7 units long, and
9 units long. By putting a 7-unit stick and a 9-unit stick next to each other and breaking off the extra from the 9-unit stick,
Brian makes a 2-unit stick. Using the same sticks in the same way that Brian does, make an 8-unit stick. Now try to make
1 stick with the other length. Is there more than one way to make certain lengths?
Pete wonders if it is possible for Brian to create any length of sticks between 1 unit long and 9 units long by carefully using
Brian’s ideas of breaking off sticks. What do you think? Tell Pete how to make each length.
From the February 2010 issue of