# 3.2 Area and Multiplication

## Unit Goals

• Students learn about area concepts and relate area to multiplication and to addition.

### Section A Goals

• Describe area as the number of unit squares that cover a plane figure without gaps and overlaps.
• Measure the area of rectangles by counting unit squares.

### Section B Goals

• Explain why the area of a rectangle can be determined by multiplying the side lengths.
• Solve problems involving the area of rectangles.

### Section C Goals

• Find the area of figures composed of rectangles.

### Problem 1

#### Pre-unit

Practicing Standards:  2.G.A.2

1. Partition the rectangle into 4 equal rows and 5 equal columns.
2. How many small squares are there in the rectangle?

### Solution

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### Problem 2

#### Pre-unit

Practicing Standards:  2.OA.C.4

Is the number of dots in each image even or odd? Explain how you know.

1.

2.

3.

### Solution

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### Problem 3

#### Pre-unit

Practicing Standards:  2.OA.C.4

How many dots are in each array? Explain or show your reasoning.

1.

2.

3.

### Solution

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### Problem 4

#### Pre-unit

Practicing Standards:  2.MD.A

Use the centimeter ruler to find the lengths of the two line segments A and B. Explain your reasoning.

### Solution

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### Problem 5

Which shape is the largest? Which shape is the smallest? Explain your reasoning. You may trace and cut out the shapes if it is helpful.

### Solution

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### Problem 6

Lin, Han, and Elena made letters from squares. Put the letters in order from least area to greatest area. Explain your reasoning.

### Solution

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### Problem 7

1. Find the area of each rectangle.

2. Can rectangles with different shapes have the same area? Explain your reasoning.

### Solution

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### Problem 8

Find the area of the rectangle. Explain or show your reasoning.

### Solution

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### Problem 9

#### Exploration

Which shape has greater area, a green triangle pattern block or a tan rhombus pattern block? Explain your reasoning.

### Solution

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### Problem 10

#### Exploration

Here are two rectangles.

1. What is the area of the larger rectangle?
2. What is the area of 3 smaller rectangles?
3. Can you cover the first rectangle with 3 of the smaller rectangles without cutting them up? Explain or show your reasoning.

### Solution

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### Problem 11

#### Exploration

1. How many different rectangles can you make with 36 square tiles? Describe or draw the rectangles.
2. How are the rectangles the same? How are they different?

### Solution

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### Problem 1

1. Use the grid to create a rectangle whose area can be represented by $$5 \times 7$$.
2. How does your rectangle represent the expression $$5 \times 7$$?

### Solution

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### Problem 2

Here are two squares. One of the squares is a square centimeter and one of them is a square inch.

Which square is a square centimeter? Which square is a square inch? Explain how you know.

### Solution

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### Problem 3

For each object, decide if you would use square centimeters, square inches, square feet, or square meters to measure its area. Explain your reasoning.

1. a baseball field
2. a table top
3. a cell phone screen

### Solution

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### Problem 4

The sides of the rectangle are marked in centimeters.

What is the area of the rectangle? Explain your reasoning.

### Solution

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### Problem 5

1. Use your ruler to find the area of the rectangle in square centimeters.

2. Use your ruler to draw a rectangle whose area is 18 square centimeters.

### Solution

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### Problem 6

Tyler has 40 carpet squares that are 1 foot on each side. He wants to use all of them to make a rectangle-shaped carpet for his room.

For the carpet to fit in the room, the longest side cannot be more than 12 feet. What side lengths could Tyler's rectangle have?

### Solution

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### Problem 7

1. Describe some patterns that you see for the numbers in the table.

2. Describe one of the patterns you saw using an equation.

### Solution

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### Problem 8

#### Exploration

1. Find a rectangle in your classroom or at home. Describe the rectangle.
2. Would you use square centimeters, square inches, square feet, or square meters to measure the area of the rectangle? Explain your reasoning.

### Solution

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### Problem 9

#### Exploration

What patterns do you notice in the three columns of the multiplication table?

### Solution

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### Problem 10

#### Exploration

Mai picked a mystery number that is less than 30. She says that she can show 3 different rectangles on this grid whose area is the same as her mystery number.

What could be Mai’s mystery number? Explain or show your reasoning.

### Solution

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### Problem 1

What is the area of this figure in square units? Explain or show your reasoning.

### Solution

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### Problem 2

Find the area of this figure. Explain or show your reasoning.

### Solution

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### Problem 3

Find the area of this figure. Explain or show your reasoning.

### Solution

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### Problem 4

#### Exploration

Lin says that she knows how to find the area of the figure. Diego says there is not enough information to find the area.

Do you agree with Lin or with Diego? Explain your reasoning.

### Solution

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### Problem 5

#### Exploration

1. Each image shows part of a shape filled with squares.

For each image, which do you think is greater, the number of squares in the image or the number of squares missing in the middle?