Download GCF and LCM

Cuisenaire® Trains
Exploration # 1
a) Take out a brown rod and place it horizontally in front of you. Make all possible samecolored trains underneath the brown rod.
b) Take out an orange rod and place it horizontally in front of you. Make all possible samecolored trains underneath the orange rod.
c) Take out a blue rod and place it horizontally in front of you. Make all possible samecolored trains underneath the blue rod.
d) Take out a dark green rod and place it horizontally in front of you. Make all possible
same-colored trains underneath the dark green rod.
©Dr Barbara Boschmans
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If we number the cuisenaire rods from 1 to 10, white being 1, orange 10, label all the rods you
found in exploration # 1.
a) Brown Rod =
All the trains underneath brown
b) Orange =
All the trains underneath orange
c) Blue Rod =
All the trains underneath blue
White =
Red =
Purple =
Brown =
White =
Red =
Yellow =
Orange =
White =
Light Green =
Blue =
d) Dark Green Rod =
All the trains underneath dark green White =
Red =
Light Green =
Dark Green =
All the colors represent the ____________ of the top rod.
Exploration # 2
a) Compare the brown rod and the orange rod and their corresponding trains. Which is the
biggest color rod that fits underneath BOTH?
______________
This rod is worth: _______________
We call this the ________________________________ of 8 (brown) and 10 (orange).
b) Compare the blue rod and the dark green rod and their corresponding trains. Which is the
biggest color rod that fits underneath BOTH?
______________
This rod is worth: _______________
We call this the ________________________________ of 9 (blue) and 6 (dark green).
c) Compare the blue rod and the orange rod and their corresponding trains. Which is the
biggest color rod that fits underneath BOTH?
______________
This rod is worth: _______________
We call this the ________________________________ of 9 (blue) and 10 (orange).
©Dr Barbara Boschmans
Page 2 of 7
Exploration # 3
a) Take out a purple and dark green rod and place them horizontally in front of you. Make
two trains of equal length.
It will take ___ dark green rods and ___ purple rods until they are equal in length.
b) Take out a blue and light green rod and place them horizontally in front of you. Make two
trains of equal length.
It will take ___ light green rods and ___ blue rods until they are equal in length.
c) Take out a black and red rod and place them horizontally in front of you. Make two trains
of equal length.
It will take ___ red rods and ___ black rods until they are equal in length.
Exploration # 4
If we number the cuisenaire rods from 1 to 10, white being 1, orange 10, label all the rods you
found in exploration # 3.
a) Purple =
Dark Green =
Total train = ____ which is the ______________ of 4 (purple) and 6 (dark green).
b) Blue =
Light Green =
Total train = ____ which is the ______________ of 9 (blue) and 3 (light green).
c) Black =
Red =
Total train = ____ which is the ______________ of 7 (black) and 2 (red).
©Dr Barbara Boschmans
Page 3 of 7
Greatest Common Factor in Venn Diagrams
Example: Find the greatest common factor of 24 and 30, GCF(24,30)
Prime factorizations:
24
24  2 3  3
30
30  2  3  5
2
2
The prime factorizations of 24 and 30
have two numbers in common; one 2
and one 3. These go in the intersection
3
of the two circles on the Venn diagram.
2
The “other” factors are placed as shown
on the right. The greatest common
factor will be the product of the
numbers in the intersection.
By looking at the Venn Diagram, we can see that the GCF(24,30) = 2  3 = 6.
5
Practice Problems: find the prime factorization of each number. Then place the prime factors in the
appropriate part of the Venn diagram and find the GCF of the numbers.
GCF(72, 54) =
GCF(84, 56) =
GCF(21,10) =
GCF(18,45,60) =
©Dr Barbara Boschmans
Page 4 of 7
Least Common Multiple in Venn Diagrams
Example: Find the least common multiple of 24 and 30, LCM(24,30)
Prime factorizations:
24
24  2 3  3
30  2  3  5
30
2
The prime factorizations of 24 and 30
2
have two numbers in common; one 2
5
and one 3. These go in the intersection
3
of the two circles on the Venn diagram.
2
The “other” factors are placed as shown
on the right. The least common multiple
will be will be the product of the
numbers in the Venn diagram.
By looking at the Venn Diagram, we can see that the LCM(24,30) = 2  2  2  3  5 = 120.
Practice Problems: find the prime factorization of each number. Then place the prime factors in the
appropriate part of the Venn diagram and find the LCM of the numbers.
LCM(72, 54) =
LCM(27, 36) =
LCM(9,16) =
LCM(24,90,100) =
©Dr Barbara Boschmans
Page 5 of 7
The Slide Method
Let us start with an example. Find the greatest common factor and the least common multiple of 150 and 4000.
Step 1:
10
Step 2:
10
5
150
15
4000
400
150
15
3
4000
400
80
GCF = 10  5  50
LCM = 10  5  3  80  12000
Using the Slide Method you repeatedly find common factors. You write these common factors to the left, and
you write the quotients that result from dividing by the common factors on the right. You stop when the resulting
quotients no longer have any common factors except 1. The GCF is then the product of the factors down the
left-hand side and the LCM is the product of the factors down the left-hand side AND the numbers in the last
row.
Use the Slide Method to find the GCF and LMC of:
1. 360 and 1344
2. 144 and 2240
3. 2880 and 2400
©Dr Barbara Boschmans
Page 6 of 7
Applications GCF, LCM
1) Marc vacuums the rugs every 18 days, mows the grass every 12 days, and pays the bills every 15 days.
Today he did all three.
a) What is the earliest time he will have to do all three again at once?
b) What is the earliest time he will have to do two again at once?
2) A machine has three colored lights each of which flashes on for an instant at regular intervals. The
yellow one flashes 15 times per minute, the blue one flashes 6 times per minute and the red one flashes
on 12 times per minute. You watch the machine and see all three lights flash on simultaneously. How
much longer must you wait before three lights flash on again simultaneously?
3) Mary and Sandy are on the same swim team for the summer. The coach placed each team member
on a different training schedule. As a result, Mary will swim laps every other day and Sandy will swim laps
every three days. The schedule begins on the first day of summer, with both girls swimming laps. How
many times will Mary and Sandy swim laps on the same day during the first ten weeks of summer?
4) Lisa Marie and Jenni want to make pies. Lisa Marie likes strawberry pie and Jenni likes peach pie. A
quart of fruit contains either 15 strawberries or 6 peaches. When they finish shopping, they discover that
they had the same number of strawberries and peaches. What is the least number of fruit that Lisa
Marie can buy?
5) As the owner of Bryce’s Super Sports, Bryce purchases 245 white golf balls, 238 yellow golf balls, and 84
orange golf balls. He asked his stock boy, Chad, to divide them up into bags, so that each bag of balls
had at least one of each color in them, and that each bag had the same number of total balls, and
that there were no balls left over. What is the largest number of balls he can put into each bag?
©Dr Barbara Boschmans
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