The Areas of Squares

5 min

Teacher Prep
Setup
Give students 2 minutes of quiet work time followed by a whole-class discussion.

Narrative

The purpose of this Warm-up is for students to review how to find the area of a region on a grid by decomposing and rearranging pieces. They will use these techniques in a later lesson to understand and explain a proof of the Pythagorean Theorem.

Monitor for students who use different strategies for finding area, including putting pieces together to make whole units in different ways and adding up the whole grid then subtracting the white space to indirectly figure out the area of the shaded region.

Launch

Display the image for all to see and ask students to vote on which of the shaded regions they predict has the larger area without doing any calculations. Record the results next to the displayed image. Then give students 2 minutes of quiet work time followed by a whole-class discussion.

Student Task

Which shaded region is larger? Explain your reasoning.

Two quadrilaterals, labeled “A” and “B,” on square grids. Both quadrilaterals are not aligned to the horizontal or vertical gridlines.
Two quadrilaterals, labeled “A” and “B,” on square grids. Both quadrilaterals are not aligned to the horizontal or vertical gridlines. Quadrilateral “A” is on a grid that has 2 rows of 4 squares. The quadrilateral is drawn starting at the left most vertex. The second vertex is 2 squares to the right and 1 square up from the first vertex. The third vertex is 2 squares to the right and 1 square down from the second vertex. The fourth vertex is 2 squares to the left and 1 square down from the third vertex. The first vertex is 2 squares to the left and 1 square up from the fourth vertex. Quadrilateral “B” is on a grid that has 3 rows of 3 squares. The quadrilateral is drawn starting at the left most vertex. The second vertex is 1 square to the right and 2 squares up from the first vertex. The third vertex is 2 squares to the right and 1 square down from the second vertex. The fourth vertex is 1 square to the left and 2 squares down from the third vertex. The first vertex is 2 squares to the left and 1 square up from the fourth vertex.

Sample Response

Figure B has the larger shaded region because Figure A is 4 square units and Figure B is 5 square units.

Activity Synthesis (Teacher Notes)

The goal of this discussion is to make sure students see different methods for determining the areas of different shapes. 

Invite students who used the strategies described in the Activity Narrative to share their methods for finding area. Record and display their responses for all to see.

Anticipated Misconceptions

If students think that the side length of either quadrilateral is 2 units because the sides are partitioned into two sub-pieces that look about 1 unit long, consider asking:

  • “How did you measure the side of each shape?”

  • “How could a compass or tracing paper be used to compare the side lengths to 2 units on the grid?”

Standards
Building On
  • 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
  • 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
Building Toward
  • 8.G.B·Understand and apply the Pythagorean Theorem.
  • 8.G.B·Understand and apply the Pythagorean Theorem.

15 min

15 min