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Guided Practice

When you multiply two different numbers to get a product, the numbers you multiply are called the factors of the product.

For example, 6 × 5 = 30, so 5 and 6 are factors of the number 30.

To list all the factors of a product, list all the different ways you can multiply two whole numbers to get that product.

1 × 30 = 30, 2 × 15 = 30

3 × 10 = 30, 15 × 6 = 30

The factors of 30 are 1, 2, 3, 5, 6, 10, 15, 30.
To determine the greatest common factor of two numbers, you can first list the factors of both numbers.

  95 : 1, 5, 19, 95
  25 : 1, 5, 25

Identify the common factors. The greatest of the common factors is the greatest common factor (GCF).

The common factors of 95 and 25 are 1 and 5.

So, the GCF of 95 and 25 is 5.

To write a ratio in simplest form, you can use the GCF of the numerator and denominator.

Write 60/45 in simplest form.

60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
45: 1, 3, 5, 9, 15, 45

GCF(60, 45) = 15

What is the greatest common factor (GCF) of 36 and 48?

GCF(36, 48) =

List all the factors of 24 in numerical order:

1, , , ,
   , , , 24

List all the factors of 50 in numerical order:

1, , ,
   , , 50

List all the factors of 100 in numerical order:

1, , , , ,
   , , , 100

List all the factors of 31 in numerical order:
,

Determine the GCF of 16 and 40.

GCF(16, 40) =

Determine the GCF of 80 and 90.

GCF(80, 90) =

Determine the GCF of 54 and 29.

GCF(54, 29) =

Determine the GCF of 33 and 75.

GCF(33, 75) =

Use the GCF to write 74/88 in simplest form.

      GCF(74, 88) =

74 ÷ =

88 ÷ =

74/88 =

Use the GCF to write 96/45 in simplest form.

      GCF(96, 45) =

96 ÷ =

45 ÷ =

96/45 =

Use the GCF to write 35/70 in simplest form.

      GCF(35, 70) =

35 ÷ =

70 ÷ =

35/70 =

When you multiply two different numbers to get a product, the product is a multiple of each number.

For example, 6 × 5 = 30, so 30 is a multiple of 6, and 30 is a multiple of 5.

To list multiples of a number, multiply that number by 1, 2, 3, and so on. The products are the multiples:

1 × 5 = 5,    2 × 5 = 10

3 × 5 = 15, 14 × 5 = 20

The numbers 5, 10, 15, and 20 are all multiples of 5. There are an infinite number of multiples of 5.
To determine the least common multiple of two numbers, first multiply the two numbers. Then divide the product by the GCF of the numbers.

To determine the LCM of 12 and 8, first multiply the two numbers: 12 × 8 = 96.

Then, determine the GCF. Divide the product, 96, by the GCF: 96 ÷ 4 = 24.

The least common multiple of 12 and 8 is 24.

The product of the GCF and LCM of two numbers is equal to the product of the numbers.

GCF(12, 8) = 4; LCM(12, 8) = 24.
GCF(12, 8) × LCM(12, 8) = 24 × 4
GCF(12, 8) × LCM(12, 8) =    12 × 8

(a × b)/GCF(a, b) = LCM(a, b)

(a × b)/LCM(a, b) = GCF(a, b)

What is the greatest common factor (GCF) of 4 and 10?

GCF(4, 10) =

What is the least common multiple (LCM) of 4 and 10?

LCM(4, 10) =

List four multiples of 4:

, , ,

List four multiples of 7:

, , ,

List four multiples of 9:

, , ,

Determine the LCM of 9 and 5.

LCM(9, 5) =

Determine the LCM of 6 and 10.

LCM(6, 10) =

Determine the LCM of 1 and 9.

LCM(1, 9) =

Determine the GCF and LCM of 14 and 30.

GCF(14, 30) =
LCM(14, 30) =

Determine the GCF and LCM of 9 and 27.

GCF(9, 27) =
LCM(9, 27) =

Determine the GCF and LCM of 49 and 28.

GCF(49, 28) =
LCM(49, 28) =

You can write a simple sum or difference as a product, using the GCF.

The GCF of 36 and 8 is 4. And 4 × 9 = 36, and 4 × 2 = 8. So, 4 × 9 + 4 × 2 = 36 + 8. This is the Distributive Property in reverse.

The same works for subtraction.

Rewrite each sum or difference as a product using the GCF.

36 + 8 = 4 × + 4 ×

27 – 9 =        × 3 – × 1

Rewrite the sum or difference as a product using the GCF.

56 + 16 = ( + )

Rewrite the sum or difference as a product using the GCF.

77 + 44 = ( + )

Rewrite the sum or difference as a product using the GCF.

81 – 18 = ()

Rewrite the sum or difference as a product using the GCF.

66 – 39 = ()

Rewrite the sum or difference as a product using the GCF.

12 + 84 = ( + )

Rewrite the sum or difference as a product using the GCF.

96 – 18 = ()

Rewrite the sum or difference as a product using the GCF.

132 – 55 = ()

Rewrite the sum or difference as a product using the GCF.

115 – 75 = ()

Rewrite the sum or difference as a product using the GCF.

234 – 90 = ()

Rewrite the sum or difference as a product using the GCF.

255 + 272 = ( + )

A histogram counts data collected in equal intervals. For example, suppose you measured the heights of your classmates to the nearest inch. You can display the results in a histogram:

There is 1 student who has a height that is 50, 51, 52, or 53 inches, rounded to the nearest inch.

There are students who have heights between 58 and 61 in.

A total of students were counted.

A total of student(s) could have a height of 54 inches.

A total of student(s) could have a height of 62 inches.

The histogram shows the number of steps your friends took in one day.

student(s) took fewer than 3955 steps.

student(s) took fewer than 7355 steps.

The histogram shows the ages of many U.S. presidents when they took office.

presidents were between 60 and 71 years old.

The histogram shows data for U.S. presidents.

These data show flights cancelled in one day for 23 different airlines.

airline(s) had more than 92 cancelled flights.

airline(s) had more than 362 cancelled flights.

The histogram shows the number of people who watch a certain TV show.

Most people who watch the show are between the ages of and .

A total of people were surveyed.

The histogram shows the results of rolling two dice 33 times.

An 8, 9, or 10 was rolled times.

An 11 or 12 was rolled times.

On the model at the right, you can drag the dot on the grid to create different multiplication sentences below the grid. Then you can slide the dot on the bottom slider to break up the multiplication in different ways. Try it out!

Model the multiplication 10 × (8 + 7) = 150. How is the 10 represented in the model? How are 8, 7, and 150 represented in the model? How is 8 + 7, or 15, represented in the model?

What is the area of each of the two rectangles you created in the model? How can you find these areas using the number sentence? (Scroll down for more.)

Think about 14 × 15. You may not be able to do that in your head. But you can decompose one of the factors to make it simpler to multiply. For example, 14 × 15 is equal to 15 × 14. This is also equal to 15 × (4 + 10), which is equal to (15 × 4) + (15 × 10).

15 × 4 is equal to 60, and 15 × 10 is equal to 150, so the product is 150 + 60, or 210. Show this product on the model at the right.

Try this method with 11 × 14. Figure out how to break that up to simplify it. Then show your product on the model.

When you think about it, you can probably simplify all the 11 multiplication facts in the same way. Try it out. Then check your work on the model. Explain how the model represents your product.

  • 12 × 11 = ?
  • 13 × 11 = ?
  • 14 × 11 = ?
  • 15 × 11 = ?
  • 16 × 11 = ?

Try the 12 facts, 13 facts, and so on. Think of a way to simplify those products. Then check your work on the model and explain how the model represents your product.