Lesson 13

Find the Area of Figures

Warm-up: Number Talk: Extend Make a Ten (10 minutes)

Narrative

The purpose of this Number Talk is to elicit strategies students have for adding two numbers when one number is close to a whole number of tens. These understandings help students develop fluency in addition. Students may look for and make use of structure (MP7) in a number of ways. For example, they may add 1 to the first addend to make a full ten and subtract 1 from the second addend to find each sum. They may also notice how the addends compare to those in the previous expression and use the change to find the new sum.

In this string, students may also add the tens and ones separately to find the sum. Adding by place value is the focus of upcoming work. This Number Talk also enables the teacher to learn the strategies students currently have for addition.

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

  • Record answers and strategy.
  • Keep expressions and work displayed.
  • Repeat with each expression.

Student Facing

Find the value of each expression mentally.

  • \(109 + 4\)
  • \(109 + 14\)
  • \(209 + 34\)
  • \(219 + 34\)

Student Response

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Activity Synthesis

  • “What do you notice about how the first two expressions are related? What about how the last two expressions are related?” (The second sum is 10 more than the first. The fourth sum is 10 more than the third.)

Activity 1: Bye-Bye Squares (20 minutes)

Narrative

The purpose of this activity is for students to find the area of figures that are composed of rectangles but are not fully gridded with squares. Partially gridded figures help to prepare students to find the area of figures with only side length measurements. Students should be encouraged to find side lengths and multiply, rather than rely on counting, as the grids disappear. If students continue to draw in the squares, ask them if there is another way to find the area.

Engagement: Develop Effort and Persistence: Differentiate the degree of difficulty or complexity. Some students may benefit from starting with a smaller figure, one with more accessible values.
Supports accessibility for: Social-Emotional Functioning, Visual-Spatial Processing

Launch

  • Groups of 2
  • Sketch or display a rotated L-shape figure as shown.
  • “What do you notice? What do you wonder?” (Students may notice: The figure is not a rectangle. It could be split into smaller rectangles. Students may wonder: Why are there no squares inside? How can I find out how many squares will cover that shape?)
  • 1 minute: quiet think time
  • Share and record responses.
  • “What information would help you find the area of this figure?” (The side lengths. Being able to see the squares inside the figure.)
  • 1 minute: quiet think time
  • Share responses.
  • Display image from the first problem.
  • “What information is given in this figure that could help you find the area?” (Grid lines. The side lengths. Some of the squares.)
  • Share responses.

Activity

  • “Now work with your partner to find the area of this figure.”
  • 5 minutes: partner work time
  • Monitor for strategies for finding the side lengths and decomposing into rectangles.
  • “Let's look at the first figure.”
  • Have students share strategies for finding the side lengths and area of figures with a partial grid.
  • “Take a look at the next figure. Think about how you could find the area of this figure.”
  • 1 minute: quiet think time
  • “Work with your partner to find the area of this figure.”
  • 5 minutes: partner work time
  • Monitor for strategies for finding the side lengths.

Student Facing

What do you notice? What do you wonder?

A figure for finding area.

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

  1.  
    A figure for finding area.

  2.  
    T-shaped figure. If split horizontally, 2 rectangles. Top, 8 tick marks by 3 tick marks. Bottom, 4 tick marks by 2 tick marks. 

Student Response

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Activity Synthesis

  • “What was your strategy for finding the area of the second figure?”
  • “What helps you find the area of figures like these where the shape is not fully covered with squares?” (Imagining where the squares would be so I can count them or find the side lengths. Finding the side lengths and multiplying to find the area of rectangles in the figure.)

Activity 2: How Many Pavers Do We Need? (15 minutes)

Narrative

The purpose of this activity is for students to find the area of a figure composed of rectangles given only their side lengths. The context of paving a patio provides students a link to their experience with squares of various sizes and should help them imagine how the diagram of the patio could be covered with squares. Students decompose the patio into rectangles and can multiply to find the area of the patio, but they should make the connection that the number of pavers needed to cover the patio is the same as the area of the patio. When students connect the quantities in the story problem to an equation, they reason abstractly and quantitatively (MP2).

MLR7 Compare and Connect. Synthesis: Invite groups to prepare a visual display that shows the strategy they used to figure out the number of tiles and the area of the floor. Encourage students to include details that will help others interpret their thinking. Give students time to investigate each others’ work. During the whole-class discussion, ask students, “How did the same area show up in each method?” “Why did the different approaches lead to the same outcome?” “Did anyone solve the problem the same way, but would explain it differently?”
Advances: Representing, Conversing

Launch

  • Groups of 2
  • Display the image.
  • “This problem is about making a patio using pavers. Pavers are stones, bricks, or blocks that are put on the ground to make a path or paved area. This is a patio thatrsquo;s made of pavers.”
  • “Earlier, we’ve talked about several different types of units that we can use to measure area. Here, Noah is using pavers that are 1 square foot. That means the paver is 1 foot by 1 foot. Help him figure out how many pavers he needs to cover the patio. Take a minute to think about the situation.”
  • 1 minute: quiet think time

Activity

  • “Now work with your partner to answer the questions.”
  • 5 minutes: partner work time
  • Monitor for multiplication expressions students use to represent the area of the rectangles within the figure, such as:
    • \(9 \times 4 = 36\), \(3 \times 2 = 6\), \(36 + 6 = 42\)
    • \(6 \times 3 = 18\), \(6 \times 4 = 24\), \(18 + 24 = 42\)
    • \(9 \times 4 + 3 \times 2\) or \((9 \times 4) + (3 \times 2)\)
    • \(6 \times 3 + 6 \times 4\) or \((6 \times 3) + (6 \times 4)\)

Student Facing

Noah wants to use square pavers that are 1 square foot each to create a small patio in the community garden. A diagram of the patio is shown.

6-sided shape. Straight sides. All side lengths meet at right angles. Bottom, 3 ft. Right side rises 2 ft, then goes right 6 ft, up 4 ft. Top side length, 9 ft. Left side length, 6 ft.  

  1. How many 1 square foot pavers will Noah need to cover the whole patio?
  2. What is the area of the patio? Explain or show your reasoning.

    Photograph of patio. Floor paved with square tiles. 

Student Response

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Advancing Student Thinking

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 found the area of the figure?”
  • “How would overlapping the rectangles affects the number of squares it would take to cover the figure?”

Activity Synthesis

  • “How did you figure out how many pavers Noah would need?”
  • Select students to share a variety of strategies. Display any expressions students used in their explanations.
  • During each explanation, ask the class which measurements were and were not used and why.
  • “How does each expression represent a way of finding the area of the figure?” (The expression \((9 \times 4) + (3 \times 2)\) shows the figure decomposed into a big rectangle across the top and a small rectangle below. Each part of the expression in parentheses represents one of the smaller rectangles in the figure. The expressions \(6 \times 3 = 18\), \(6 \times 4 = 24\), and \(18 + 24 = 42\) show how the figure was decomposed into a 6-by-3 rectangle on the left and a 4-by-6 rectangle on the right, then the areas were added.)

Lesson Synthesis

Lesson Synthesis

“In this lesson, we found the area of figures even if they were not fully gridded with squares. What did you need to think about when finding the area of a figure with just side length measurements?” (I can imagine it being filled with squares and count them. I can break the shape into rectangles and multiply the side lengths and then add those areas together.)

Cool-down: Find the Area (5 minutes)

Cool-Down

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