Lesson 1

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. In the following activities, students will use these techniques to check area estimates made when approximating the value of the side length of a square used to calculate the area by squaring. They will also use these techniques to understand and explain a proof of the Pythagorean Theorem.

As students work, identify students who used 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.

Tell students they are comparing the area of regions on two grids with the same unit size. Poll students on which region they predict is the larger area. Give students 2 minutes of quiet work time followed by a whole-class discussion. 

Students may 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. If so, point out that they do not lay along grid lines, they can use a compass or tracing paper to compare the side lengths to 2 units on the grid.

Poll students on which shaded region is the larger area. Select previously identified students to share their strategies for finding area. Record and display their responses for all to see.

If time allows, ask students to estimate the side lengths of each shaded region to prepare them for the work they will do in the next activity.

The purpose of this task is for students to find the areas of squares whose side lengths are not easy to determine by inspection. The squares are represented in an increasingly abstract way: the first square shows all of the square units explicitly, making it easy to see that putting two “leftover” triangles together will make a rectangle with the same dimensions, or even piecing individual square units together that can be counted. The next only shows the grid units along the edges, providing a bridge to the third, which only provides labels for side lengths. The numbers are chosen to encourage the decomposing, rearranging, and subtracting strategy for finding the area, although decomposing and using the formulas for the area of a square and a triangle is also perfectly appropriate.

This sequence of tasks also provides practice for students in using the area and finding strategies they will need to understand and explain a proof of the Pythagorean Theorem in a later lesson.

Identify students who find the area of the extra area in the triangles by:

• composing two triangles to make a rectangle.
• using the formula for the area of a triangle and multiplying the result by four.

Be sure these strategies are made explicit for the first square to help students who are only able to count unit squares make a successful transition to the third square. While decomposing the square and rearranging the pieces is a correct strategy, students need the subtraction strategy for more abstract work later, so it is better not to highlight the decomposing strategy that does not generalize as well.

Give 2–3 minutes quiet work time and then ask students to pause and select 2–3 previously identified students to share how they found the area of the first square. Work time to finish the remaining questions.

Select students to explain how they found the area of the third square. Emphasize the strategies of decomposing and either rearranging or finding the areas of the triangles using the triangle area formula.

In this activity, students use what they know about finding the area of a square from a given side length to think about the converse problem: how do you find the side length of a square with a given area? Students solve this problem in general in the next lesson, but this lesson sets them up for that work.

• First, they work with squares whose side lengths lie along grid lines, so both the side lengths and areas can be found simply by counting length or area units.
• They also see a “tilted” square whose area can be determined by counting square units or by decomposing and rearranging the square or a related figure. In this case, the side lengths are not apparent, so students have to reason that since the area is the same as one of the earlier squares, the side lengths must also be the same. They should check that this appears to be true by tracing one of the squares on tracing paper and seeing that it does, in fact, line up with the other. 

The last question asks them to use the insights from the previous questions, tracing paper, or a ruler to estimate the side lengths of some other tilted squares (MP5). In the next lesson, they will see other techniques for estimating those side lengths as well.

Monitor for students who use these strategies for the fourth question:

• reason that the side lengths are between 5 and 6 because the areas are between 25 and 36 (MP3)
• measure with tracing paper or a ruler

Provide access to tracing paper. Students in groups of 2. 3 minutes of quiet work time followed by a partner discussion on the first three problems. Pause students and make sure everyone agrees on the answers for the first three questions. Students continue with the fourth question, then a whole-class discussion.

A key take-away from this activity is that if the area of a square is in between the areas of two other squares, then the side lengths are also in between, which in the case of the squares in this task can be reinforced by estimating with tracing paper.

Select previously identified students to share their solutions. Sequence with a reasoning strategy followed by a measuring strategy using a ruler or tracing paper. Use the measuring strategy to verify the reasoning strategy.

In this activity, students use squares with known areas to determine the total area of five shapes. How students determine the area of the shapes is left open-ended on purpose (MP1). Students may calculate each shape individually, or, with a bit of rearranging, they may “fit” the shapes into the squares. It is possible to fit all 5 shapes into the two smaller squares or into the one larger square. Identify groups who calculate the area of the shapes individually versus those fitting the shapes into squares.

While these shapes are likely to seem arbitrary to students in this lesson, they are actually part of a transformations-based proof of the Pythagorean Theorem that students will revisit in a future lesson. The five cut-out shapes in this activity should be reserved at the end of the activity for future use.

You will need the blackline master for this activity unless you are using the digital version.

The purpose of this discussion is for students to share the different ways they determined the total area of the shapes and to get students thinking about strategies for calculating area and rearranging figures in preparation for future lessons. Select previously identified groups to share, starting with those who calculated each shape individually. If any group fit the shapes into both the two smaller squares and then into the larger square, ask them to demonstrate how they did so for the class.