Lesson 3

Reasoning to Find Area

Lesson Narrative

This lesson is the third of three lessons that use the following principles for reasoning about figures to find area:

  • 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 composed from pieces that don't overlap, the sum of the areas of the pieces is the area of the figure. If a given figure is decomposed into pieces, then the area of the given figure is the sum of the areas of the pieces.

Following these principles, students can use several strategies to find the area of a figure. They can:

  • Decompose it into shapes whose areas they can calculate.
  • Decompose and rearrange it into shapes whose areas they can calculate.
  • Consider it as a shape with one or more missing pieces, calculate the area of the shape, then subtract the areas of the missing pieces.
  • Enclose it with a figure whose area they can calculate, consider the result as a region with missing pieces, and find its area using the previous strategy.

Use of these strategies involves looking for and making use of structure (MP7); explaining them involves constructing logical arguments (MP3). For now, rectangles are the only shapes whose areas students know how to calculate, but the strategies will become more powerful as students’ repertoires grow. This lesson includes one figure for which the “enclosing” strategy is appropriate, however, that strategy is not the main focus of the lesson and is not included in the list of strategies at the end.

Note that these materials use the “dot” notation (for example \(2 \boldcdot 3\)) to represent multiplication instead of the “cross” notation (for example \(2 \times 3\)). This is because students will be writing many algebraic expressions and equations in this course, sometimes involving the letter \(x\) used as a variable. This notation will be new for many students, and they will need explicit guidance in using it.

Learning Goals

Teacher Facing

  • Compare and contrast (orally) different strategies for calculating the area of a polygon.
  • Find the area of a polygon by decomposing, rearranging, subtracting or enclosing shapes, and explain (orally and in writing) the solution method.
  • Include appropriate units (in spoken and written language) when stating the area of a polygon.

Student Facing

Let’s decompose and rearrange shapes to find their areas.

Required Preparation

Make sure students have access to items in their geometry toolkits: tracing paper, graph paper, colored pencils, scissors, and an index card to use as a straightedge or to mark right angles.

For the warm-up activity, prepare several copies of the pair of figures on the blackline master, in case students propose cutting them out to compare the areas.

Learning Targets

Student Facing

  • I can use different reasoning strategies to find the area of shapes.

CCSS Standards

Building On

Addressing

Glossary Entries

  • area

    Area is the number of square units that cover a two-dimensional region, without any gaps or overlaps.

    For example, the area of region A is 8 square units. The area of the shaded region of B is \(\frac12\) square unit.

    Figure A on the left composed of 8 identical shaded squares arranged in 3 rows. Figure B on the right consists of one square with a diagonal segment from corner to corner. Half of the square is shaded.
  • compose

    Compose means “put together.” We use the word compose to describe putting more than one figure together to make a new shape.

    Image on left shows three separate parts of a shape; image on right shows those three parts put together to create an oval.
  • decompose

    Decompose means “take apart.” We use the word decompose to describe taking a figure apart to make more than one new shape.

    Image on left shows three parts put together to create an oval; the image on the right shows the oval separated into the three parts.
  • region

    A region is the space inside of a shape. Some examples of two-dimensional regions are inside a circle or inside a polygon. Some examples of three-dimensional regions are the inside of a cube or the inside of a sphere.