Unit 1, Lesson 2
Finding Area by Decomposing and Rearranging
Accelerated 6
Preparation
Lesson Narrative
In this lesson, students learn to reason flexibly about two-dimensional figures to find their areas.
Students begin by revisiting the definitions for “area” that they learned in earlier grades. They refine these definitions and arrive at a definition that can be used by the class for the rest of the unit. Along the way, students practice attending to precision (MP6).
Next, students use tangram pieces to explore ways of reasoning about area. They compose and rearrange a square and some triangles to form figures of certain areas. Students see that the area of a two-dimensional figure can be determined in multiple ways:
- by composing that figure using smaller pieces with known areas
- by decomposing the figure into shapes whose areas we can determine and adding the areas of those shapes
- by decomposing it and rearranging the pieces into a different but familiar shape whose area can be found
In an optional activity, students use these strategies to reason about the area of individual tangram pieces and practice constructing logical arguments to justify their reasoning (MP3).
By the end of the lesson, two key principles about area are made explicit: Figures that match exactly have equal areas, and area is additive (in that the area of a figure is the sum of the areas of all non-overlapping pieces that compose it).
In this lesson, students review the themes that arose when they shared their initial thoughts in Exercise 1 about what they think it should look like and sound like to do math together as a community. Students then have a chance to both affirm and add to the ideas that were generated.
- Calculate the area of a region by decomposing it and rearranging the pieces, and explain (orally and in writing) the solution method.
- Recognize and explain (orally) that if two figures can be placed one on top of the other so that they match up exactly, they must have the same area.
- Show that area is additive by composing polygons with a given area.
Let’s create shapes and find their areas.
Required Materials
Activity 1
Required Preparation
Make sure that students have access to their geometry toolkits, which should include tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles.
For every group of 2 students, prepare 1 set of tangrams that contains 1 square, 4 small triangles, 1 medium triangle, and 2 large triangles. Print and cut out the blackline master (printing on card stock is recommended), or use commercially-available tangrams.
Note that the tangram pieces used here differ from a standard set in that two additional small triangles are used instead of a parallelogram.
For the digital version of the activity, acquire devices that can run the applet.
For the digital version of the activity, acquire devices that can run the applet.
Area is the number of square units that cover a two-dimensional region without gaps or overlaps.
- The area of region A is 8 square units.
- The area of the shaded region of B is square unit.