Students learned previously how to find the area of a square given its side length. In this lesson, students investigate the areas of squares presented in different ways. The work of this lesson is intentionally focused on setting students up to see the geometric connection between a square’s area and its side length, which will happen in a following lesson.

Students begin by comparing figures whose areas are easily determined by either composing and counting square units or by decomposing and rearranging into simpler, familiar shapes (MP7). Next, students manipulate two sets of shapes to see that if one set of shapes completely covers another set of shapes with no gaps and no overlaps, then both sets must have the same area. This idea will help students understand and explain informal proofs of the Pythagorean Theorem in later lessons.

Lastly, students find areas of “tilted” squares using a strategy where the tilted square is surrounded by a larger square whose area can easily be determined. The area of the 4 triangles that are not part of the tilted square are then subtracted from the area of the larger square (MP7).

One of the activities in this lesson works best when each student has access to devices that can run the applet. The applet allows students to manipulate small pieces digitally as they consider the area of the three squares without having to keep track of multiple small pieces of paper.