SHOW
107
PROGRAM
SYNOPSIS
Segment 1 (5:59)
DIRK NIBLICK: TOO
MANY COOKOUTS,
PARTS 1 AND 2
Dirk Niblick, fearless leader
of the Math Brigade, helps
his neighbor Mr. Beazley plan
a barbecue. Together they
work out how to save money
on the party food by finding
the smallest number of
packages of hotdogs—12 to
a package—and buns—eight
to a package—needed to
feed his party guests.
Segment 2 (4:24)
COMMON
MULTIPLE MAN
When a couple has to figure
out how many hors d’oeuvres
to buy to serve either 12, 16,
or 24 guests equally, they call
on Common Multiple Man, a
superhero with a very
strange (but useful!)
super power.
You know,
0 is a multiple of every
number, because 0 times
any number is 0.
But when we talk
about a “least common
multiple” of two numbers,
we mean the smallest positive
multiple that the two numbers
have in common.
Factor ‘Em In: Exploring
Factors and Multiples
INTRODUCTION
F
actors and multiples have
traditionally been an important
part of the elementary school
mathematics curriculum, and for
good reason: Whenever
multiplication crops up, so do
multiples. Students who know
how to recognize and work
with multiples have a head
start on developing number
sense.
This module concentrates on
finding multiples that two or
more numbers have in common—that is, common multiples.
Multiples and factors appear in many parts of mathematics
beyond arithmetic (see CURRICULUM CONNECTIONS, page 49).
RE
BEFO
VIEWING
You can help your students become familiar with the idea of the
factors of a number, the multiples of a number, and the concept of
prime numbers (whole numbers that have exactly two factors) by
having regular practice with the concepts. Try offering some
informal challenges such as,“Name a multiple of 4 that’s bigger than
20. What could it be?” (Any number—such as 24, 28, 32, and so
on—that is a number multiplied by 4.) “What’s the smallest prime
number greater than 20?” (23) “I’m thinking of a number that is a
multiple of 5. It’s also 3 less than a multiple of 9. What could my
number be?” (15 is one possibility. 60 and 105—in fact,
any number that is 15 more than a multiple of 45—
will also work.)
Stretch your students’ awareness by including
impossible ones, too—but be sure to alert them
that there may be no solution—for example,“I’m
thinking of a number that is a multiple of 3 and is 1
less than a multiple of 12.” (That’s impossible,
because every multiple of 12 is also a multiple of
3. If a number is 1 less than a multiple of 12, it
can’t be a multiple of 3.)
45
MathTalk
R
AFTE
VIEWING
At the end of the last segment it seemed pretty clear that
Common Multiple Man wanted to be invited to the party.
Then instead of 12, 16, or 24 guests, how many guests
would there have been? (13, 17, or 25) Explore how many
hors d’oeuvres would have been needed then.
One way to approach finding a common multiple of 13, 17,
and 25 would be to write lists of the multiples of 13, of 17,
and of 25, looking for numbers that appear on
all three lists.
Note:
13, 26, 39, 52, . . . .
17, 34, 51, 68, . . .
25, 50, 75, 100, . . . .
A calculator is helpful here.
Press 1 3 – = = = = . . .
to get successive multiples of 13.
Just looking at these numbers, it isn’t immediately
clear that there will be any number that appears on all
three lists.
Some students may not have fully internalized the reasoning
here, and they will need to continue the lists for
a while to get a feeling for what is
You can simplify the
problem by first trying to
happening.
Ask students to work in
small groups to
continue the lists. Can
they think of any
shortcuts? Some may
suggest multiplying the
three factors (getting 5525).
That’s a lot of hors
d’oeuvres! How do you
know whether 5525 is
the lowest common
multiple or not?
find a number that is a
multiple of 13 and
of 17.
Is 221 a multiple of 25?
(No. Keep multiplying to find a multiple of both 221 and
25.) Be on the lookout for shortcuts.
It turns out that, by adding one more guest, the number of
hors d’oeuvres needed has jumped from 48 (which was the
least common multiple of 12, 16, and 24) all the way up to
5525. (Maybe that’s why the host and hostess didn’t want
to invite Common Multiple Man!)
46
221 is the first one!
N UM B E R S E N S E
activity
GAME: MULTIPLE MANIA
T
MATERIALS
his game helps players become more familiar with the idea of
multiples and common multiples of numbers in a context in which
strategy is important.
1.
Pass out a number cube and a copy of the reproducible page to
each pair of students. Give each student 10 chips or other small
objects of the same color,
with each partner in a pair
having a different color.
2. Review the rules of the
game on the reproducible
page with the class. You
may want to play one game
as a demonstration: Two
students can take turns
rolling the number cube,
while the whole class
discusses moves that could
be made.
3.
Now let pairs of students
play the game.
4. Encourage your students to
take the game board home
and play the game with their
parents or other family
members.
for each pair of students:
■ one number cube
(numbered 1–6)
■ copies of reproducible page 50
■ 20 chips—10 each of two
different colors
GAME::
SSA
MPPLLEE GAME
AM
Number Rolled
Number Covered on Game Board
Red rolls
5
15 (15 already contains an even number [0]
of Blue’s chips, so it can be covered. 20
is another possible play for Red.)
Blue rolls
4
12
Red rolls
4
16 (12 is blocked by Blue’s single chip.)
Blue rolls
1
13 (Rolling 1 is the only way to cover 13 or
any other prime number.)
Red rolls
2
18
Blue rolls
6
12 (18 is blocked by Red, so this is the only
play. It unblocks 12, because 12 now has
an even number of chips. Until Blue gets
a third chip on 12, Red can capture both
of Blue’s chips by rolling 1, 2, 3, 4, or 6.)
Red rolls
3
12 (Red captures Blue’s two chips on 12,
removing them from play. Red’s chip
remains on 12.)
Blue rolls
3
— (Each multiple of 3 is covered by one
Red chip, so Blue cannot play. Therefore
Red wins.)
In the course of discussing the rules of the game
you can ask questions like these:
■ If it’s the first roll of the game and you roll a 3, where could you put
your chip? (On any multiple of 3—12, 15, or 18.)
■ What must you roll to be able to put a chip on 18? (1, 2, 3, or 6;
note that the other factors of 18—namely 9 and 18—aren’t on the
number cube.)
■ What must you roll to be able to put a chip on 19? (Only 1.)
47
MathTalk
keep
thinking
VARIANTS OF “MULTIPLE MANIA”
You can discuss variants of the game as a group, and then assign teams of
two students to explore how the changes affect the game. For example,
the game board could have the numbers 21 through 30, or 100 through
109. The numbers don’t even have to be consecutive whole numbers.
Or you might change the numbers on the number cube.
PARTY PLANNING WITH MULTIPLES
Explore how the least common multiple changes
as guests are added or subtracted. The
information can be displayed in a table like this:
Some students may need to start with simpler
examples. Suppose that the number of guests
could have been 4, 8, or 12. The least common
multiple of 4, 8, and 12 is 24. What if each number
was increased by 1, so that 5, 9, or 13 guests might
show up? Start to build a chart like the one below.
This can be done in small groups with a class
discussion at the end. Students should be asking
themselves and each other two questions: Is 210 (for
example) a multiple of 6, 10, and 14? Is it the smallest
multiple of 6, 10, and 14?
ests
of gu e
r
e
b
Num ight com
m
who
12
8
13
4
9
14
5
10
15
6
11
16
7
12
8
48
on
omm
c
t
s
a
Le
ple
multi
24
585
210
1155
48
R. S. V. P.
Number o
f
who migh guests
t come
12
16
24
13
17
25
14
18
26
Least com
mon
multiple
48
5525
1638
R.
S.
V.
P.
N UM B E R S E N S E
FOR THE PORTFOLIO
1.
Suggest that students make a more complete list of least common multiples for the
chart on page 48, subtracting or adding guests. Examine the list for patterns to try
to determine why sometimes the least common multiple is fairly small, like 48, and
sometimes much larger, like 5525. (12, 16, and 24 have several factors in common,
while 13, 17, and 25 have only one common factor—1.) Encourage students to try
lots of examples, including smaller numbers, and only two at a time instead of
three. (It turns out that the product of two numbers is the same as their least
common multiple times their greatest common factor. For example, 8 x12 is the
same as 24 x 4 [and note that 24 is the least common multiple of 8 and 12;
4 is their greatest common factor.]) Have students explain in writing their
reasoning.
2. Students may explore and write up strategies for the MULTIPLE MANIA game.
What is a good first move for each of the possible rolls of the number cube? Why?
CURRICULUM CONNECTIONS
Multiples are important when calculating with fractions, since multiplying
the numerator and denominator of a fraction by the same positive whole
number gives a fraction with the same value.
The concept of multiples underlies proportional reasoning. For instance,
the ratio of 6 to 5 is the same as the ratio of 12 to 10 because 12 and
10 are the same multiple of 6 and 5, respectively.
Each side is
two times as long,
so the shapes are
similar.
This idea of proportion permeates geometry and
measurement, too. Two geometric figures are similar if all
the measurements of one are the same multiple of the
corresponding measurements of the other.
Math Talk
NUMBER
SENSE
GEOMETRY
CONNECTIONS
Let Me Count
the Ways:
Counting with
Combinatorics
What Shape Is
Your Number?
Finding Number
Patterns in Squares
and Triangles
Finding
multiples
Shape–by–Number:
Building Rectangles
49
MathTalk
NAME
Family Page
MULTIPLE
MANIA
NUMBER OF PLAYERS: Two
WHAT YOU NEED:
■ a number cube
■ 10 chips each
(a different color for each player)
RULES
OF
THE
GAME
1. On each turn a player rolls the
number cube and places a chip on
any number that is a
multiple of the number
rolled, as long as the
space already contains
an even number of the
opponent’s chips.
Remember that zero
The opponent’s
is an even number!
chips on that
space (if any) are
removed. Any of your own chips
already on that space stay there.
The only case in which a player may
not place a chip on a multiple of the
number rolled is if that space is
blocked by an odd number of the
opponent’s chips.
2. The game ends when:
a. each player has played all 10
chips. The player with more chips
on the board is the winner. It’s a
tie if both players have the same
number of chips.
or
b. one player has no legal move.
The other player is the winner.
50
©1995 Children’s Television Workshop