Download Block 15Summary - MATH GR 3-5

Block 15: Least Common Multiple
Summary
All references are from Go Math! (Grade 5)
Factors and Multiples
Review the factors for 6 and 8.
Factors of 6: 1, 2, 3, 6
Factors of 8: 1, 2, 4, 8
List the multiples of 6 and 8.
Multiples of 6: 6 12 18 24 30 36 42 48
Multiples of 8: 8
16
24
32
40
48
Clarify the difference between factor and multiple
Emphasize that many students confuse the two.
Factor: A number multiplied by another number to find a product.
[Go Math! (Grade 5) Glossary, H5]
Multiple: The product of two counting numbers is a multiple of each of those numbers.
[Glossary, H8]
Using the list the multiples of 6 and 8.
Circle the common multiples.
Multiples of 6: 6 12 18 24 30 36 42 48
Multiples of 8: 8 16
24
32
40
48
Finding the Least Common Multiple (LCM)
Find the Least Common Multiple of 6 and 8.
The common multiples of 6 and 8 (in our list) are 24 and 48.
The smallest of these is 24.
We say that 24 is the Least Common Multiple of 6 and 8.
We use the abbreviation LCM for Least Common Multiple.
Find the LCM of 5 and 6.
Multiples of 5: 5 10 15 20 25 30
Multiples of 6:
6 12 18 24 30
Connections between LCM and product
The product of two numbers is always a common multiple.
In some cases:
The LCM is the product:
The LCM of 5 and 6 is 30
(which is 5 x 6 = 30)
In other cases,
The LCM is less than the product.
The LCM of 6 and 8 is 24 is the product.
(which is less than 6 x 8 = 48)
Discovering the connections between GCF, LCM, and product
Calculate the GCF of 5 and 6 (which is 1).
In that case, the LCM was equal to the product.
Ask: Is this always the case?
Demonstrate with other examples.
The GCF of 3 and 5 is 1
The LCM of 3 and 5 is 15 (which is 3 x 5)
Calculate the GCF of 6 and 8 (which is 2).
In this case, the LCM was 48 ÷ 2 = 24
Ask: Is this always the case?
Demonstrate with other examples.
The GCF of 4 and 6 is 2
The product: 4 x 6 = 24
The LCM of 4 and 6 is 12 (which is 24 ÷ 2)
The GCF of 6 and 9 is 3
The product: 6 x 9 = 54
The LCM of 6 and 9 is 18 (which is 54 ÷ 3)
Using LCM in the addition of fractions (Least Common Denominator)
Examine examples on pages 299 – 309 in Go Math! (Grade 5).
With each example, connect to the principle of LCM.
The Least Common Denominator (LCD) is always the LCM.
Special attention is given to examples on p. 305:
Will multiplying the denominators give the LCD?
Denominators of 11 and 44
No, the LCD is 44. [We see that 44 is a multiple of 11.]
Denominators of 3 and 11.
Yes, the GCF of 3 and 11 is 1, so the LCM is the product: 3 x 11 = 33.
Denominators of 4 and 6
No, the GCF of 4 and 6 is 2.
Therefore, the LCM will be the product divided by 2: 24 ÷ 12 = 2.
Denominators of 5 and 8.
Yes, the GCF of 5 and 8 is 1, so the LCM is the product: 5 x 8 = 40.
Summary
For any two numbers, the product of those numbers will always be a common multiple.
This comes from the Glossary of Go Math! (Grade 5):
Multiple: The product of two counting numbers is a multiple of each of those numbers.
Sometimes, the product is the least common multiple.
This occurs when the GCF is 1.
For example the GCF of 6 and 7 is 1.
In this case, the LCM is the product: 6 x 7 = 42.
In other cases, the LCM is less than the product.
This occurs when the GCF is greater than 1.
In these cases, we divide the product by the GCF. The result is the LCM.
For example the GCF of 8 and 12 is 4.
In this case, the LCM is less than the product.
8 x 12 = 96
LCM = 96 ÷ 4 = 24
It is important that students test these rules on any examples of their choice.