In this activity, students recall and refine their prior knowledge of area. They articulate a definition of “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.
 
In the Warm-up, students analyzed four ways that a region was tiled or otherwise fitted with squares. Here, students revisit the same images and decide which arrangements of squares can be used to find the area of the region and why. Students 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, monitor for students who mention the following aspects so they can share later:

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

Students may focus on how they have typically found the area of a rectangle—by multiplying its 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 Options A and D, could be used to find the area of the region 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  or  square units. 

The purpose of this discussion is to elicit key points to include in a class definition of "area." Invite students to share their responses to the first question and to explain their reasoning. Ask questions such as:

• “What is it about Shapes 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 Shape C that might make it unhelpful for finding the area?” (The squares overlap and do not cover the entire region, so counting the squares won't give us the area.)
• “If you think Shape B cannot be used to find the area, why not?” (We can't just count the number of squares and say that this number is the area because the squares are not all the same size.)
• “If you think we can use Shape B to find the 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 Shapes A and D different?” (Shape A uses larger unit squares and Shape D uses smaller ones. Each size represents a different unit.)
• “Will Shapes A and D  give us different areas?” (They will give us the same area, but in different units—for example, square inches and square centimeters.)

Select 2–3 groups previously identified 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 class 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 that this will be a working definition that can be revised as they continue their work in the unit.

In this activity, students further develop their understanding that area is additive. Students 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 the area of the new figure is the same as the area of the original figure.

Each question in the activity aims to elicit discussions about those two principles. Though they may seem obvious, these principles still need to be stated explicitly (at the end of the lesson) because a more-advanced understanding of the area of complex figures depends on them. As students work, look for those whose reasoning illustrates the principles.

This is the first time Math Language Routine 2: Collect and Display is suggested in this course. In this routine, the teacher circulates and listens to student talk while jotting down words, phrases, drawings, or writing that students use. The language collected is displayed visually for the whole class to use throughout the lesson and unit. The purpose of this routine is to capture a variety of students’ words and phrases—especially including everyday or social language and non-English—in a display that students can refer to, build on, or make connections with during future discussions, and to increase students’ awareness of language used in mathematics conversations.

In the digital version of the activity, students use an applet showing 8 tangram pieces to determine the relationships between the areas. Consider using the applet if physical tangram pieces are not available. The applet is adapted from the work of Harry Drew in GeoGebra.

The purpose of the discussion is to make two principles explicit: Two regions have the same area if they match up exactly when superimposed, and the area of a region is preserved even when the region is decomposed and rearranged.

Direct students' attention to the reference created using Collect and Display. Ask previously identified students to share the shapes they created and to explain how they knew those shapes have certain areas. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. 

After each student shares an explanation, use the terms “compose,” “decompose,” and “rearrange” to name these moves. Clarify that to compose means to join or put together and decompose means to break apart or separate.

Highlight the following points:

• 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, “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. The following three shapes—each composed of 2 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 figure has a different shape than the other two, but because it is made up of the same 2 triangles, it has the same area.

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 the four small triangles, with two small 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, either compose four smaller triangles into a large triangle, or recognize that the large triangle could be decomposed into four smaller triangles, which can then be composed into 2 unit squares.

In this activity, students reason about the area of each tangram triangle based on the areas of composite shapes that they previously created. Students may have recognized that the area of one small triangle is  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 articulate how they know that these conclusions are true, using written words or illustrations that are clearly labeled. They also listen to their partner’s explanations. In doing so, students practice constructing logical arguments and critiquing the reasoning of others (MP3).

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. 

In the digital version of the activity, students can use the same applet as in the previous activity to support their reasoning.

After partners share and agree 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 the assigned triangle into another shape, and one who reasoned in terms of decomposing the triangle or the unit square into smaller pieces and rearranging them. If these strategies are not brought up by students, be sure to make them explicit at the end of the lesson.