# Lesson 12

## Warm-up: Number Talk: So Close (10 minutes)

### Narrative

The purpose of this Number Talk is to elicit strategies and understandings students have for adding two numbers when one number is close to a multiple of ten. These understandings help students develop fluency and will be helpful later in this lesson when students add the area of parts of a figure to determine the area of the whole figure.

When students use the fact that one number is close to 10 to find the sum, they look for and make use of structure (MP7).

### Launch

• Display one expression.
• “Give me a signal when you have an answer and can explain how you got it.”
• 1 minute: quiet think time

### Activity

• Keep expressions and work displayed.
• Repeat with each expression.

### Student Facing

Find the value of each expression mentally.

• $$9 + 6$$
• $$29 + 6$$
• $$59 + 6$$
• $$49 + 8$$

### Activity Synthesis

• “How are the numbers in the four expressions alike?” (The first number has two digits and is 1 away from a whole ten. The second number is a single-digit number.)
• “How did these features help you find each sum?” (We could add the second number to the whole ten closest to the first number and then take away 1.)
• “Who can restate _______ ’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone approach the problem in a different way?”
• “Does anyone want to add on to____’s strategy?”

## Activity 1: Rectangles in Rectangles (15 minutes)

### Narrative

The purpose of this activity is for students to learn that area is additive. Students decompose a rectangle into two smaller ones and find the sum of their areas in order to find the area of the whole rectangle. They can find the area of the two smaller rectangles by counting or by multiplying the side lengths.

When students consider how to decompose a larger rectangle into smaller ones to facilitate the process of finding area, they look for and make use of structure (MP7).

Action and Expression: Develop Expression and Communication: Identify connections between strategies that result in the same outcomes but use differing approaches. When students share different ways that they found the area, elicit from students how the strategies vary but result in the same answer.
Supports accessibility for: Conceptual Processing

### Launch

• Groups of 2
• “You are going to answer some questions about planting vegetables and flowers in this garden. Take a minute to think about the vegetables and flowers you would plant in a garden space.”
• 1 minute: quiet think time
• Share responses.

### Activity

• “Now work with your partner on the first problem.”
• 2–3 minutes: partner work time
• Monitor for students who found the area by:
• using counting strategies
• multiplying the side lengths of the entire rectangle
• adding the areas of the smaller rectangles
• Select previously identified students to share how they found the area.
• “How do you know that adding the area of the parts is the same as finding the area of the whole garden?” (The part covered with vegetables and the part covered with flowers make up the whole garden. If I add the area of both parts, I get the same number if I counted all the squares in the whole garden.)
• “Now you are going to design your own garden. Be sure to explain the area of each part of your garden and the area of the whole garden.”
• 5 minutes: independent work time
• Monitor for students who create rectangular gardens and gardens composed of rectangles.

### Student Facing

1. This rectangle represents space in a community garden. The shaded part is covered with vegetables and the unshaded part is covered with flowers. Each square represents 1 square foot.

What is the area of the whole space?

2. Design your own garden. Find the area of each part of the garden and the area of the whole garden.

### Student Response

If students count one by one to find the area of their gardens, consider asking:

• “Tell me how you found the area of your garden.”
• “How could you use multiplication to find the area of your garden?”

### Activity Synthesis

• Select 2–3 students who created a rectangular garden that is decomposed into rectangles to share.
• “How many square units does each part of the garden cover?”
• “How many square units does the garden cover?”
• “What equation would represent how you found the area of the rectangle?”

## Activity 2: Find the Rectangles (20 minutes)

### Narrative

The purpose of this activity is for students to find the area of a figure by decomposing it into two non-overlapping rectangles. The synthesis should emphasize different strategies and also encourage students to directly link expressions and the use of parentheses to the way they decomposed the figure. If students drew gardens in the shape of the image in the launch, display those drawings as well during the notice and wonder.

Some students may partition diagonally to split the figure into what looks like 2 symmetrical parts, or cut the figure up into more than 2 parts. These are both acceptable ways of finding the area. Ask students who partition diagonally to find the area in the way they partitioned, but then encourage them to find a second way that has partitions on one of the grid lines. As students look through each others' work, they discuss how the representations are the same and different and can defend different points of view (MP3).

When students notice that the smaller parts of the figure can be added to find the total area of the figure they are looking for and making use of structure (MP7).

This activity uses MLR7 Compare and Connect. Advances: representing, conversing

### Launch

• Groups of 2
• Display the image of the gridded figure.
• “What do you notice? What do you wonder?” (Students may notice: It looks like 2 rectangles. It looks like a big rectangle with a chunk missing. There are squares. Students may wonder: What is this shape called? Could we find the area of the shape? How would we find the area?)
• 1 minute: quiet think time
• Share responses.
• “This isn’t a shape that we have a name for like a square or triangle. Because of this, we’ll call it a ‘figure’ as we work with it in this activity. This word will be helpful in describing other shapes that we don’t have a name for.”
• “Talk with your partner about different ways you could find the area of this figure.”
• 1 minute: partner discussion

### Activity

• “There are many ways to find the area of this figure. Take some time to find the area. We are going to share these with the class, so you may want to include details such as shading, notes, and labels to help others understand your thinking.”
• 5 minutes: independent work time

### Student Facing

What do you notice? What do you wonder?

Find the area of this figure. Explain or show your reasoning. Organize it so it can be followed by others.

### Student Response

If students add numbers that indicate they tried to find the area by adding the areas of rectangles that overlap, consider asking:

• “Tell me about how you broke the figure apart into rectangles to find the area.”
• “How would overlapping the rectangles affect the number of squares it would take to cover the figure?”

### Activity Synthesis

MLR7 Compare and Connect

• Have students display their work.
• 5–7 minutes: gallery walk
• “What is the same and what is different about how other students found the area of the figure?” (Some students broke the rectangle apart vertically and some students broke it apart horizontally. Others imagined a missing part that wasn’t there. They all found the same area. They all broke the rectangle into smaller parts and then added the parts to find the area.)
• 30 seconds quiet think time
• 1 minute: partner discussion
• Display an expression that reflects how students found the area, such as:

$$\displaystyle 4 \times 5 + 5 \times 10$$

• “How does this expression show how to find the area?” (The $$4 \times 5$$ represents the rectangle on the top of the figure if we break it into 2 rectangles. The $$5 \times 10$$ represents the bottom rectangle.)
• Add parentheses to create the expression:

$$\displaystyle (4 \times 5) + (5 \times 10)$$

• Parentheses are grouping symbols that can be used in expressions or equations. To show how you saw the figure we can show the first rectangle with $$4 \times 5$$ and the second rectangle with $$5 \times 10$$. Parentheses let us put both rectangles in the same expression like $$(4 \times 5) + (5 \times 10)$$ and see which part of the expression represents each rectangle.”
• Reinforce the meaning of parentheses in a similar way with other ways students decomposed the figure.

## Lesson Synthesis

### Lesson Synthesis

Display the figure from the last activity.

“Today we learned that we can decompose figures into rectangles to find the area. Why does it make sense that we can decompose a figure in many ways, but still get the same area for it?” (No matter how we decompose the figure, as long as we include all the squares, we are getting the total area.)