Lesson 2

Finding Area by Decomposing and Rearranging

2.1: What is Area? (10 minutes)

Warm-up

This warm-up activates and refines students' prior knowledge of area. It prompts students to articulate a definitionof area that can be used for the rest of the unit. This definition of area is not new, but rather reiterates what students learned in grades 3–5.

Before this lesson, students explored tiling and tile patterns. Here, they analyze four ways a region is being tiled or otherwise fitted with squares. They decide which arrangements of squares can be used to find the area of the region and why, and use their analysis to write a definition of area. In identifying the most important aspects that should be included in the definition, students attend to precision (MP6).

Students' initial definitions may be incomplete. During partner discussions, note students who mention these components so they can share later:

  • Plane or two-dimensional region
  • Square units
  • Covering a region completely without gaps or overlaps

Limit the whole-class discussion to 5–7 minutes to leave enough time for the work that follows.

Launch

Arrange students in groups of 2. Give students 2 minutes of quiet think time for the first question, and ask them to be ready to explain their decision. Then, give partners 3–4 minutes to share their responses and to complete the second question together.

Student Facing

You may recall that the term area tells us something about the number of squares inside a two-dimensional shape.

  1. Here are four drawings that each show squares inside a shape. Select all drawings whose squares could be used to find the area of the shape. Be prepared to explain your reasoning.
    Four drawings that each show squares inside a shape.
  2. Write a definition of area that includes all the information that you think is important. 

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may focus on how they have typically found the area of a rectangle—by multiplying it side lengths—instead of thinking about what “the area of any region” means. Ask them to consider what the product of the side lengths of a rectangle actually tells us. (For example, if they say that the area of a 5-by-3 rectangle is 15, ask what the 15 means.)

Some students may think that none of the options, including A and D, could be used to find area because they involve partial squares, or because the partial squares do not appear to be familiar fractional parts. Use of benchmark fractions may help students see that the area of a region could be a non-whole number. For example, ask students if the area of a rectangle could be, say, \(8\frac12\) or \(2\frac{1}{4}\) square units. 

Activity Synthesis

For each drawing in the first question, ask students to indicate whether they think the squares could or couldn't be used to find the area. From their work in earlier grades students are likely to see that the number of squares in A and D can each tell us about the area. Given the recent work on tiling, students may decide that C is unhelpful. Discuss students' decisions and ask:

  • “What is it about A and D that can help us find the area?” (The squares are all the same size. They are unit squares.)
  • “What is it about C that might make it unhelpful for finding area?” (The squares overlap and do not cover the entire region, so counting the squares won't give us the area.)
  • “If you think B cannot be used to find area, why not?” (We can't just count the number of squares and say that the number is the area because the squares are not all the same size.)
  • “If you think we can use B to find area, how?” (Four small squares make a large square. If we count the number of large squares and the number of small squares separately, we can convert one to the other and find the area in terms of either one of them.)

If time permits, discuss:

  • “How are A and D different?” (A uses larger unit squares and D uses smaller ones. Each size represent a different unit.)
  • “Will they give us different areas?” (They will give us areas in different units, such as square inches and square centimeters.)

Select a few groups to share their definitions of area or what they think should be included in the class definition of area. The discussion should lead to a definition that conveys key aspects of area: The area of a two-dimensional region (in square units) is the number of unit squares that cover the region without gaps or overlaps.

Display the class definition and revisit as needed throughout this unit. Tell students this will be a working definition that can be revised as they continue their work in the unit.

2.2: Composing Shapes (25 minutes)

Activity

In grade 3, students recognized that area is additive. They learned to find the area of a rectilinear figure by decomposing it into non-overlapping rectangles and adding their areas. Here students extend that understanding to non-rectangular shapes. They compose tangram pieces—consisting of triangles and a square—into shapes with certain areas. The square serves as a unit square. Because students have only one square, they need to use these principles in their reasoning:

  • If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
  • If a figure is decomposed and rearranged to compose another figure, then its area is the same as the area of the original figure.

Each question in the task aims to elicit discussions about these two principles. Though they may seem obvious, these principles still need to be stated explicitly (at the end of the lesson), as more-advanced understanding of the area of complex figures depends on them.

The terms compose, decompose, and rearrange will be formalized in an upcoming lesson, but throughout this lesson, look for opportunities to demonstrate their use as students describe their work with the tangram pieces. When students use “make” or “build,” “break,” and “move around,” recast their everyday terms using the more formal ones.

As students work, notice how they compose the pieces to create shapes with certain areas. Look for students whose reasoning illustrates the ideas outlined in the Activity Synthesis.

Demonstrate the use of the word “compose” by repeating students' everyday language use and then recast using the formal terms here.

Launch

Give each group of 2 students the following set of tangram pieces from the blackline master or from commercially available sets. Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram.

  • Square: 1
  • Small triangles: 4
  • Medium triangle: 1
  • Large triangles: 2

It is important not to give them more than these pieces.

Give students 2–3 minutes of quiet think time for the first three questions. Ask them to pause afterwards and compare their solutions to their partner's. If they created the same shape for each question, ask them to create a different shape that has the same given area before moving on. Then, ask them to work together to answer the remaining questions.

Classrooms using the digital activities can use physical tangram pieces or an applet with the same shapes to determine the relationships between the areas. Applet is adapted from the work of Harry Drew in GeoGebra.

Speaking, Conversing: MLR2 Collect and Display. Circulate and listen to the ways students describe composing, decomposing, and rearranging the shapes. On a display, write down common phrases you hear students say about each, such as “building,” “breaking apart,” “moving.” Include relevant pictures or drawings. Update the display as needed throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language during their paired and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

This applet has one square and some small, medium, and large right triangles. The area of the square is 1 square unit.

Click on a shape and drag to move it. Grab the point at the vertex and drag to turn it.

  1. Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.

  2. Use your shapes to create a new shape with an area of 1 square unit that is not a square. Draw your shape on paper and label it with its area.

  3. Use your shapes to create a new shape with an area of 2 square units. Draw your shape and label it with its area.

  4. Use your shapes to create a different shape with an area of 2 square units. Draw your shape and label it with its area.

  5. Use your shapes to create a new shape with an area of 4 square units. Draw your shape and label it with its area.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Find a way to use all of your pieces to compose a single large square. What is the area of this large square?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Give each group of 2 students the following set of tangram pieces from the blackline master or from commercially available sets. Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram.

  • Square: 1
  • Small triangles: 4
  • Medium triangle: 1
  • Large triangles: 2

It is important not to give them more than these pieces.

Give students 2–3 minutes of quiet think time for the first three questions. Ask them to pause afterwards and compare their solutions to their partner's. If they created the same shape for each question, ask them to create a different shape that has the same given area before moving on. Then, ask them to work together to answer the remaining questions.

Classrooms using the digital activities can use physical tangram pieces or an applet with the same shapes to determine the relationships between the areas. Applet is adapted from the work of Harry Drew in GeoGebra.

Speaking, Conversing: MLR2 Collect and Display. Circulate and listen to the ways students describe composing, decomposing, and rearranging the shapes. On a display, write down common phrases you hear students say about each, such as “building,” “breaking apart,” “moving.” Include relevant pictures or drawings. Update the display as needed throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students use mathematical language during their paired and whole-group discussions.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

Your teacher will give you one square and some small, medium, and large right triangles. The area of the square is 1 square unit.

  1. Notice that you can put together two small triangles to make a square. What is the area of the square composed of two small triangles? Be prepared to explain your reasoning.

  2. Use your shapes to create a new shape with an area of 1 square unit that is not a square. Trace your shape.
  3. Use your shapes to create a new shape with an area of 2 square units. Trace your shape.
  4. Use your shapes to create a different shape with an area of 2 square units. Trace your shape.
  5. Use your shapes to create a new shape with an area of 4 square units. Trace your shape.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Student Facing

Are you ready for more?

Find a way to use all of your pieces to compose a single large square. What is the area of this large square?

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

Students may consider the area to be the number of pieces in the compositions, instead of the number of square units. Remind them of the meaning of area, or prompt them to review the definition of area discussed in the warm-up activity.

Because the 2 large triangles in the tangram set can be arranged to form a square, students may consider that square to be the square unit rather than the smaller square composed of 2 small triangles. Ask students to review the task statement and verify the size of the unit square.

Activity Synthesis

Invite previously identified students (whose work illustrates the ideas shown here) to share. Name these moves explicitly as they come up: compose, decompose, and rearrange.

  • First question: Two small triangles can be composed into a square that matches up exactly with the given square piece. This means that the two squares—the composite and the unit square—have the same area.

Tell students, “We say that if a region can be placed on top of another region so that they match up exactly, then they have the same area.” 

  • Second question: Two small triangles can be rearranged to compose a different figure but the area of that composite is still 1 square unit. These three shapes—each composed of two triangles—have the same area. If we rotate the first figure, it can be placed on top of the second so that they match up exactly. The third one has a different shape than the other two, but because it is made up of the same two triangles, it has the same area.
    Three figures, each a different arrangement to two small tangram triangles.

Emphasize: “If a figure is decomposed and rearranged as a new figure, the area of the new figure is the same as the area of the original figure.”

  • Third and fourth questions: The composite figures could be formed in several ways: with only small triangles, with two triangles and a medium triangle, or with two small triangles and a square.
  • Last question: A large triangle is needed here.  To find its area, we need to either compose 4 smaller triangles into a large triangle, or to see that the large triangle could be decomposed into 4 smaller triangles, which can then be composed into 2 unit squares.
Representation: Develop Language and Symbols. Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following terms and maintain the display for reference throughout the unit: area, compose, decompose, and rearrange.
Supports accessibility for: Conceptual processing; Language

2.3: Tangram Triangles (15 minutes)

Optional activity

In this activity, students use the areas of composite shapes from the previous activity to reason about the area of each tangram shape. Students may have recognized previously that the area of one small triangle is \(\frac12\) square unit, the area of one medium triangle is 1 square unit, and the area of one large triangle is 2 square units. Here they practice articulating how they know that these observations are true (MP3). The explanations could be written in words, or as clearly-labeled illustrations that support their answers.

As partners discuss, look for two ways of thinking about the area of each assigned triangle: by composing copies of the triangle into a square or a larger triangle, or by decomposing the triangle or the unit square into smaller pieces and rearranging the pieces. Identify at least one student who uses each approach.

Launch

Arrange students in groups of 2. Assign the first and second questions to one partner and the second and third questions to the other partner. Give each group access to the geometry toolkits and the same set of tangram pieces as used in the earlier activity.

Give students 3–4 minutes of quiet time to find the areas of their assigned triangles and to construct their explanations, followed by a few minutes to share their responses with their partner. Tell students that as one partner explains, the other should listen carefully and either agree or disagree with the explanation. They should then come to an agreement about the answers and explanations.

Classrooms using the digital activities can use an applet to assist in determining the areas of the triangles.

Speaking, Writing: MLR8 Discussion Supports. Use this routine when students compare areas of triangles and squares to support the use of mathematical language. As students share their responses with their partner, circulate and encourage listeners to push speakers to use the language “compose,” “decompose,” or “rearrange” in their explanations. Look for students who name the square or larger triangles “composite figures” and amplify this language. Encourage students to borrow words and phrases from each other and to use this language in their written responses.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)

Student Facing

Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.

  1. The area of the small triangle is ____________ square units. I know this because . . .
  2. The area of the medium triangle is ____________ square units. I know this because . . .
  3. The area of the large triangle is ____________ square units. I know this because . . .

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Launch

Arrange students in groups of 2. Assign the first and second questions to one partner and the second and third questions to the other partner. Give each group access to the geometry toolkits and the same set of tangram pieces as used in the earlier activity.

Give students 3–4 minutes of quiet time to find the areas of their assigned triangles and to construct their explanations, followed by a few minutes to share their responses with their partner. Tell students that as one partner explains, the other should listen carefully and either agree or disagree with the explanation. They should then come to an agreement about the answers and explanations.

Classrooms using the digital activities can use an applet to assist in determining the areas of the triangles.

Speaking, Writing: MLR8 Discussion Supports. Use this routine when students compare areas of triangles and squares to support the use of mathematical language. As students share their responses with their partner, circulate and encourage listeners to push speakers to use the language “compose,” “decompose,” or “rearrange” in their explanations. Look for students who name the square or larger triangles “composite figures” and amplify this language. Encourage students to borrow words and phrases from each other and to use this language in their written responses.
Design Principle(s): Cultivate conversation; Optimize output (for explanation)

Student Facing

Recall that the area of the square you saw earlier is 1 square unit. Complete each statement and explain your reasoning.

  1. The area of the small triangle is _________ square units. I know this because . . .
  2. The area of the medium triangle is _________ square units. I know this because . . .
  3. The area of the large triangle is _________ square units. I know this because . . .

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Anticipated Misconceptions

If students initially have trouble determining the areas of the shapes, ask how they reasoned about areas in the previous activity. Have samples of composed and decomposed shapes that form one square unit available for students to reference.

Activity Synthesis

After partners shared and agreed on the correct areas and explanations, discuss with the class:

  • “Did you and your partner use the same strategy to find the area of each triangle?”
  • “How were your explanations similar? How were they different?”

Select two previously identified students to share their explanations: one who reasoned in terms of composing copies of their assigned triangle into another shape, and one who reasoned in terms of decomposing their triangle or the unit square into smaller pieces and rearranging them. If these approaches are not brought up by students, be sure to make them explicit at the end of the lesson.

Lesson Synthesis

Lesson Synthesis

There are two principles that can help us reason about area:

  1. If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.
  2. The area of a figure can be found by adding the areas of its parts. If we compose (put together) a new figure from smaller pieces without overlapping them, then the sum of the areas of the pieces is the area of the new figure. Likewise, if we decompose (cut or break apart) a given figure into pieces, then the area of the given figure is the sum of the areas of the pieces. Even if we rearrange the pieces, the overall area does not change. 

Here is an example. Suppose we know the area of a small triangle and wish to find the area of a large triangle. Demonstrate the following (using the tangram pieces, if possible):

  • We can use 4 small triangles to compose a large triangle. Here are two ways to do so. If we place a large triangle on top of a composition of 4 small triangles and they match up exactly, we know that the area of the large triangle is equal to the combined area of 4 small triangles. 
    2 triangles composed of smaller triangles 
  • We can decompose the large triangle into 4 small triangles. Again, we can reason that the area of one large triangle is equal to the combined area of 4 small triangles. 
    Three figures: A large tangram triangle; A large tangram triangle decomposed into 4 small tangram triangles.
  • Suppose we don't know the area of a small triangle, but we do know the area of a square that is composed of 2 small triangles. We can decompose the large triangle into 4 small triangles and then rearrange them into 2 squares. We can reason that the area of the large triangle is equal to the combined area of 2 squares. This is because when the 4 rearranged small triangles are placed on top of two squares, they match up exactly.
    Three figures: A large tangram triangle; A large tangram triangle decomposed into 4 small tangram triangles; 4 small tangram triangles arranged as two tangram squares.

We will look more deeply into these strategies in the next lesson.

2.4: Cool-down - Tangram Rectangle (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

Here are two important principles for finding area:

  1. If two figures can be placed one on top of the other so that they match up exactly, then they have the same area.

  2. We can decompose a figure (break a figure into pieces) and rearrange the pieces (move the pieces around) to find its area.

Here are illustrations of the two principles.

Images of squares and triangles being decomposed and composed to form different shapes. 

  • Each square on the left can be decomposed into 2 triangles. These triangles can be rearranged into a large triangle. So the large triangle has the same area as the 2 squares.
  • Similarly, the large triangle on the right can be decomposed into 4 equal triangles. The triangles can be rearranged to form 2 squares. If each square has an area of 1 square unit, then the area of the large triangle is 2 square units. We also can say that each small triangle has an area of \(\frac12\) square unit.