# 4.1 Factors and Multiples

## Unit Goals

• Students apply understanding of multiplication and area to work with factors and multiples.

### Section A Goals

• Determine if a number is prime or composite.
• Explain what it means to be a factor or a multiple of a whole number.
• Relate the side lengths and area of a rectangle to factors and multiples.

### Section B Goals

• Apply multiplication fluency within 100 and the relationship between multiplication and division to find factor pairs and multiples.

### Problem 1

#### Pre-unit

Practicing Standards:  3.MD.C.7.a

Find the area of each rectangle. Explain your reasoning.

1.

2.

### Solution

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### Problem 2

#### Pre-unit

Practicing Standards:  3.MD.C.7

On the grid, draw a rectangle whose area is represented by each expression.

1. $$3 \times 5$$
2. $$4 \times 8$$

### Solution

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### Problem 3

Tyler wants to build a rectangle with an area of 20 square units using square tiles.

1. Can Tyler build a rectangle with a width of 4 units? Explain or show your reasoning.
2. Can Tyler build a rectangle with a width of 6 units? Explain or show your reasoning.

### Solution

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### Problem 4

List the possible side lengths of rectangles with an area of 32 square units. Explain or show how you know your list is complete.

### Solution

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### Problem 5

List the factor pairs of each number. Is each number prime or composite? Explain or show your reasoning.

1. 37
2. 27
3. 77

### Solution

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### Problem 6

1. Calculate the area of each rectangle.

2. How did you use multiplication facts to calculate the areas?

### Solution

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### Problem 7

#### Exploration

1. You want to arrange all of the students in your class in equal rows.

1. How many rows can you have? How many students would be in each row?
2. What if you add the teacher to the arrangement? How would your rows change?
2. Find some objects at home (such as silverware, stuffed animals, cards from a game) and decide how many rows you can arrange them in and how many objects are in each row.

### Solution

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### Problem 8

#### Exploration

What is the largest prime number you can find? Explain or show why it is a prime number.

### Solution

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### Problem 1

Pens are sold in packages of 5 and also in packages of 6.

1. Jada wants to buy 60 pens for her class. Which packages of pens and how many should Jada buy if she doesn't want any extras? Explain or show your reasoning.
2. Han wants to buy 55 pens for his class. Which packages of pens and how many should Han buy? Explain or show your reasoning.

### Solution

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### Problem 2

1. Find the factor pairs of 36.
2. How many factors does 36 have?
3. List the factors of 15.

### Solution

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### Problem 3

Select all numbers that are multiples of 8.

A:  16
B:  28
C:  40
D:  54
E:  66
F:  72
G:  84
H:  96

### Solution

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### Problem 4

#### Exploration

1. List the multiples of 2 up through 30.
2. List the multiples of 3 up through 30.
3. What do you notice about the numbers in the two lists?

### Solution

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### Problem 5

#### Exploration

Which number(s) from 1 to 100 have the largest number of factors? Explain or show how you know.

### Solution

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