Illuminations: Corner to Corner

# Corner to Corner

## Exploring Diagonals and the Pythagorean Theorem

 Students further explore square roots using the diagonals of rectangles. Using measurement, students will discover a method for finding the diagonal of any rectangle when the length and width are known, which leads to the Pythagorean Theorem.

### Learning Objectives

 Students will: Measure the length, width, and diagonals of rectangles. Explore irrational numbers, namely square roots. Understand the Pythagorean Theorem.

### Materials

 Centimeter Rulers Calculators Rectangles and Diagonals Activity Sheet

### Instructional Plan

At the end of the Diagonals of Squares lesson, students were asked to describe a pattern that relates the length of a diagonal to the side length of a square. Students likely came up with the relation d = 1.4s, or something close to that. Now is an appropriate time to reveal that the actual formula is d ≈ s√2. (To convince them that this is the case, you may wish to have students use this formula to determine the length of the diagonal for several squares and compare the results to actual measurements. Or, you can simply have students use a calculator to find that √2 ≈ 1.14142135623…, which is close their estimate of 1.4. Regardless of the method of presentation, it is important that students see how square roots occur when finding the length of the diagonal, as that will form the basis of this lesson.

Begin this lesson by asking, "Previously, you found the relationship between side length and diagonal length for a square. Today, you’re going to examine the relationship between diagonal length and the length and width of a rectangle. What relationship do you predict between the length and width of a rectangle and its diagonal?" Students should individually record their predictions in Question 1 on the Rectangles and Diagonals activity sheet.

After students have recorded their predictions, display the Motivational Problem overhead sheet, which asks students to consider the following problem:

Jaime bikes 5 miles south and 12 miles east. He knows of a diagonal shortcut that he can take to get back home. If Jaime takes the shortcut, how many miles will he travel on the return trip?

Discuss this problem with the class, and allow students to estimate and record their answers on Question 2. Do not solve the problem now, as students will return to it at the end of the lesson.

Allow students time to explore the length, width, and diagonal of several rectangles in the classroom or on the school grounds. These rectangles could be floor tiles, chalkboards, windows, desktops, and so forth. In fact, a large rectangle could even be drawn in the parking lot using chalk. Such a rectangle should be large enough that students could pace out the number of steps. For example, the rectangle could measure 5 feet by 7 feet, which has a diagonal of approximately 9.2 feet.

As with the previous lesson, some students may "eyeball" their estimates, while others may estimate the distance using their shoes as a unit of measure. On a blank piece of paper, students should sketch the rectangle that they are considering and record the measurements.

When students reconvene in the classroom, ask what methods they used to estimate the length, width, and diagonal. If any students used their shoe size, ask if those with small feet had different answers than students with larger feet. What difference does this make?

Students may also use the Shape Cutter activity to estimate the length of a diagonal. As shown below, students can create a rectangle of any size, cut along the diagonal, and move one of the halves to compare the length of the diagonal to the side lengths.

After students have made their predictions, they should pair up and compare. Students should not alter their predictions based on this discussion, because their predictions will be revisited at the conclusion of the lesson.

Regardless of the warm‑up problem that students considered, they should recognize that the diagonal is always longer than either the width or the length of the rectangle. Ask students to make predictions about how much longer. That is, is the diagonal double the shorter side? Is it one‑and‑a‑half times the longer side? Is it longer or shorter than the sum of the length and the width?

Using the examples of rectangle on the second page of the activity sheet, each pair of students should measure the side length of each rectangle. Then, using a ruler, they should draw a diagonal and measure it. These three measurements should then be recorded on the activity sheet. (Although not a requirement, you may want to encourage students to use centimeters for their measurements. The rectangles on the activity sheet have dimensions that are whole numbers of centimeters.)

 Length Width Diagonal Length Ratio of Diagonal Length to Side Length 1 3 3.2 √10 2 3 3.6 √13 3 3 4.2 √18 4 3 5 √25 5 3 5.8 √34

Students will need some guidance with how to complete the fourth column. You could make the following statement: "Here’s something cool! Each of the numbers in the third column is equal to √n, for some value of n. Use your calculator to determine the value of n for each row." Once the square roots have been recorded in the fourth column, students will likely see a relationship among the numbers. [The relationship is that n = length2 + width2. Note that students can find the value of n by squaring the value in the third column and then rounding the answer to the nearest whole number. For instance, the diagonal length given in the third column of the second row is 3.6. When this value is squared and rounded to the nearest whole number, the result is 3.62 ≈ 13; and, the value in the fourth column of the second row is √13.]

For students who do not see the relationship, you may want to suggest that they cross out the third column and ask, "Do you see any relationship between the length and the width to the numbers in the fourth column?"

Ask students, "What happens to the length of the diagonal each time?" [The decimal value seems to start with a pattern, but then it changes. On the other hand, the difference between the numbers under the square root sign in the fourth column increase by 2 from row to row. That is, the difference from 10 to 13 is 3; then, the difference from 13 to 18 is 5; then, the difference from 18 to 25 is 7; and so on.]

Within each pair, one student could tackle all of the "3‑by" rectangles, and another student could tackle all of the "4‑by" rectangles and compare the results after measuring.

 Length Width Diagonal Length Ratio of Diagonal Length to Side Length 1 4 4.1 √17 2 4 4.5 √20 3 4 5 √25 4 4 5.7 √32 5 4 6.4 √41

You may want to point out that a rectangle with side lengths of 3 and 4 has a diagonal length of 5. Ask students, "What makes 3, 4, and 5 so special? What can you do mathematically to rewrite this in a different way, not using square roots?"

Students may not immediately see a relationship between the values of the first two columns and the value of the fourth column. If that happens, allow students to work in groups of 3‑4 to discuss any patterns that they see. After several minutes of discussion, bring the class together to share their ideas. At this point, it is likely that at least one group will see a relationship between the length, width, and diagonal. Some possible relationships are:

l2 + w2 = d2

When students have found the pattern, they should return to their predictions and compare them with the relationship they discovered. Students should reflect upon the activity by answering the questions on the activity sheet.

Before the end of the lesson, be certain to discuss the motivational problem about Jaime. Students may not realize that the original problem relates to the lesson, because it uses a triangle, not a rectangle. Point out that the shortcut is actually the diagonal of a rectangle with side lengths of 5 miles and 12 miles. Students should be able to then determine the length of the shortcut [13 miles]. Also, allow students to examine their measurements of the various objects that they measured at the beginning of class. How did their measurements of the diagonal compare to the actual length?

### Questions for Students

 How quickly did you notice a pattern when recording the lengths of the diagonals? Was there anything that you did or saw that helped you recognize this pattern? How close were your measurements to the actual measurement of the diagonal? Why is this value often an estimate? When you measured the length and width of the rectangle, was the diagonal ever a whole number, or was it always a decimal number? Why do you think this is?

### Assessment Options

 Have students measure the length and width of common rectangles (a sheet of paper, a book cover, etc.). Based on those measurements, have them predict the length of the diagonal, and then have them measure to confirm their predictions. Students should have some experience in measuring and calculating the diagonal when the length and width are known, and calculating the width or length when the other and the diagonal are known. For example: Jaime is creating a fence for his backyard. He and his sister will split the backyard into two equal sections, so that each gets a triangular patch in which to create a garden. The length of the backyard is 7 yards, and the width is 10 yards. How long will the fence be that splits this rectangular backyard into two congruent triangles? [√149 &asymp 12.2 yards.] Ms. Suarez created a running game for her P.E. students. On a rectangular field, students have two options. They can either run along the diagonal, or they can run one length and one width instead. The length of the field is 20 meters, and the diagonal is 45 meters. How long is the width? [Approximately 40.3 meters.] Which would be the shorter distance for students to run, and why? [The diagonal, because it is approximately 45 meters, whereas the combined length and width is approximately 60.3 meters.] In addition to the above problems, the teacher should also check for understanding: As students complete the activity sheets. As students complete the in-class activity. As students discuss the problems with their partners.

### Extensions

 There are many examples on the Web that demonstrate the Pythagorean theorem. Many are interactive. Allow students to explore by searching for the Pythagorean theorem on their own, or allow them to visit some of the following sites: Pythagorean Theorem - MathWorld Pythagorean Proof - Nova Online Pythagoras Page - Jim Loy 45 (and Counting!) Proofs of the Pythagorean Theorem - Cut-The-Knot Shortest Proof of the Pythagorean Theorem - Mudd Math Fun Facts Animated Proof of the Pythagorean Theorem - Mark D. Meyerson Students should create similar rectangles and investigate. For example, students could create rectangles that measure 6 × 8, 9 × 12, 12 × 18, and so on, all of which are similar to the 3 × 4 rectangle examined during this lesson. Ask students to note any similarities and differences between the measurements of the original rectangle and the measurements of the new rectangle. If the length and width are doubled, is the diagonal also doubled? Ask students to think of some real-world situations in which they might need to find the length of the diagonal of a rectangle. What are some practical uses of the Pythagorean theorem?

### Teacher Reflection

 Did you have to assist students with their measurements? If so, what can you do in the future to improve this skill? Did students discover patterns or methods that you did not anticipate for finding the length of a diagonal? What were they? Are there any other methods that you could lead students to discover? What was the students’ motivation level during the lesson? How could you have increased motivation? How did the students demonstrate understanding of the material? What were some of the ways that the students illustrated that they were actively engaged in the learning process?

### NCTM Standards and Expectations

 Geometry 6-8Understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar objects. Measurement 6-8Understand both metric and customary systems of measurement. Select and apply techniques and tools to accurately find length, area, volume, and angle measures to appropriate levels of precision.
 This lesson prepared by Christopher Johnston.

1 period

### NCTM Resources

 More and Better Mathematics for All Students
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