# Lesson 1

The Areas of Squares and Their Side Lengths

### Lesson Narrative

Students know from work in previous grades how to find the area of a square given the side length. In this lesson, we lay the groundwork for thinking in the other direction: if we know the area of the square, what is the side length? Before students define this relationship formally in the next lesson, they estimate side lengths of squares with known areas using tools such as rulers and tracing paper (MP5). They also review key strategies for finding area that they encountered in earlier grades that they will use to understand and explain informal proofs of the Pythagorean Theorem.

In the warm-up, students compare the areas of figures that can easily be determined by either composing and counting square units or decomposing the figures into simple, familiar shapes. In the next activity, students find areas of “tilted” squares by enclosing them in larger squares whose areas can be determined and then subtracting the areas of the extra triangles. The next activity reinforces the relationship between the areas of squares and their side lengths, setting the stage for the definition of a square root in the next lesson.

### Learning Goals

Teacher Facing

• Calculate the area of a tilted square on a grid by using decomposition, and explain (orally) the solution method.
• Estimate the side length of a square by comparing it to squares with known areas, and explain (orally) the reasoning.

### Student Facing

Let’s investigate the squares and their side lengths.

### Required Preparation

1 copy of the Making Squares blackline master, pre-cut, for every 2 students. These pieces are used again in the lesson A Proof of the Pythagorean Theorem and should be saved.

### Student Facing

• I can find the area of a tilted square on a grid by using methods like “decompose and rearrange” and “surround and subtract.”
• I can find the area of a triangle.

Building On