## Squares, Diagonals, and Square Roots

Students explore the relationship between the lengths of the sides and diagonals of a square. Students will use their discoveries to predict the diagonal length of any square.

If there is a baseball field near your school, take your students to it, and pose the following problem:

The infield in a baseball is a square, 90 feet on a side. The catcher, who plays behind home plate, has to throw the ball to second base if a runner is trying to steal. How far is it from home plate to second base?

If some students in your class play baseball or softball, you might want to have them help explain the problem to the rest of the class. If not, then you can personalize the problem by referring to a catcher from a nearby team.

Alternatively, you could use another scenario that is more interesting to your students. For the purpose of this lesson, the important part is that students should estimate the length of a diagonal that, because of the size of the square, would be difficult to measure directly. Any large square in a nearby location would be suitable. You could even use chalk to draw a large square in the parking lot.

Regardless of the square you use to introduce this topic, say to students, "Estimate the length of the diagonal of this square." Allow students to discuss their ideas with a partner. Observe their behaviors and listen to their predictions. Some students may "eyeball" the length and estimate the distance visually. Other students may actually pace off the lengths using their shoes as a unit of measurement. All students should record their ideas in Questions 1 and 2 on the Square Sides and Diagonals Activity Sheet.

Square Sides and Diagonals Activity Sheet

Once all students have formed an estimate, then ask, "Do you think there’s a rule you could use to determine the length of the diagonal if you knew the side length?"

To gain an intuitive feel for how the side length and the length of the diagonal are related, students can use the Shape Cutter activity. Students can create a square, divide it along the diagonal, and then rotate one of the pieces to compare the diagonal to the side. The picture below shows how this tool was used to create a 10 × 10 square, divide it into two triangles, and compare the diagonal to the side.

Students can repeat this process for squares of various sizes. They should start to see that the length of the diagonal is a little less than 1½ times the length of the side, regardless of the size of the square. (As shown above, the diagonal of a 10 × 10 square appears to be just a little more than 14 units.)

The main investigation for this lesson involves measuring and recording the side length and length of the diagonal for various squares. In pairs, have students measure the side lengths of the squares on the second page of the Square Sides and Diagonals activity sheet. Then, using a ruler, they should draw a diagonal and measure it as well. Finally, they should record the measurements in the table on the first sheet.

As students work, circulate to ensure that students are consistent in their units of measure. For instance, if they measure the side in centimeters, then the diagonal should be measured in centimeters also. (Note that the side lengths of the squares on the activity sheet are whole numbers of centimeters. It is therefore recommended that students measure the squares in centimeters.)

As students record their data, a pattern should emerge. Students should notice that the ratio in the third column of the table hovers around 1.4. (The value will vary slightly because of measurement accuracy.) Students’ tables should look something like this:

Side Length | Diagonal Length | Ratio of Diagonal Length to Side Length |

3 | 4.3 | 1.43 |

4 | 5.5 | 1.38 |

5 | 7.1 | 1.42 |

Discuss the results with the class. Students should be able to generate the following approximate rule for the length of the diagonal:

Length of Diagonal = 1.4 × Side Length

If your students are comfortable with algebraic notation, this could be written as:

d= 1.4s

Students should notice that this rule does not work exactly. In
fact, the exact rule that relates the diagonal to the side length is *d* = *s*√2,
and √2 ≈ 1.4142135623…. You may have students attempt to discover this
rule on their own, and you might ask students to think about this
question overnight.

To conclude the lesson, have students complete all of the questions on the activity sheet. In addition to using their rule to determine the distance from home plate to second base, students should answer the questions at the end of the activity sheet to indicate how their thoughts changed.

- Metric Rulers
- Square Sides and Diagonals Activity Sheet

**Assessments**

- Students should submit their activity sheet. Their answers to the last several questions, in particular, will provide a good assessment of their level of understanding. In addition, a class discussion about these questions and about student’s rules for finding the diagonal will also be enlightening in determining which students met the objectives for this lesson.

**Extensions**

- Students can investigate the length of the diagonal of other regular polygons (pentagons, hexagons, octagons) and compare it to the side length. Do similar patterns hold for these shapes?
- Ask students to explore the relationship between sides and diagonals of rhombi. Does the same pattern hold true? What similarities are there between diagonals of squares and rhombi? Are there any differences? Explain.
- Students can create a spreadsheet to record the side lengths (of squares or other polygons) and then to calculate the lengths of the diagonals. Students would need to develop the formula for performing this calculation. Alternatively, students could use a spreadsheet to record the length of the diagonal and the side length for similar figures, and the ratio of diagonal length to side length could be automatically calculated.

**Questions for Students**

1. Compare your prediction for how to determine the length of the diagonal with the rule you discovered during the investigation. What rule can be used to find the length of the diagonal of any square if you know the side length?

[Based on their measurements, students should notice that the length of the diagonal is approximately equal to 1.4 times the side length. The difference between this rule and their original prediction will vary, depending on their first estimate.]

2. The infield in baseball is a square with side length 90 feet. Determine the distance from home plate to second base. How does your answer compare to your prediction from earlier in the lesson? Explain why your estimate was too high or too low.

[The exact distance from home to second is 90√2 ≈ 127.3 feet. Students might estimate the answer to be 90 × 1.4 = 126 feet. Students should recognize that their predictions may have been inexact, because various methods for estimating will give a close approximation but not an exact measurement.]

**Teacher Reflection**

- As the students were measuring the sides and diagonals of their squares, what did you observe about accuracy and precision? Did you have to assist students with their measurements? If students had trouble measuring their shapes, what can you do in the future to improve this skill?
- What alternative patterns or methods did students discover that you did not anticipate? If the students did not discover alternate patterns, do you think there are any? Could you have led the students in another direction?
- How did the students demonstrate understanding of the materials presented?
- Were concepts presented too abstractly? Too concretely? How will you change the lesson if you teach it in the future?
- How did students illustrate that they were actively engaged in the learning process?

### Corner to Corner

Use pattern recognition to determine the length of a diagonal of a square, and attempt to discover the Pythagorean Theorem.

### Exploring Diagonals and the Pythagorean Theorem

### Learning Objectives

Students will:

- Measure the sides and diagonals of squares.
- Make predictions about, and explore the relationship between, side lengths and diagonals.
- Formulate a rule for finding the length of a diagonal based on the side length.

### Common Core State Standards – Mathematics

Grade 8, Geometry

- CCSS.Math.Content.8.G.B.7

Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.

### Common Core State Standards – Practice

- CCSS.Math.Practice.MP1

Make sense of problems and persevere in solving them.

- CCSS.Math.Practice.MP4

Model with mathematics.

- CCSS.Math.Practice.MP5

Use appropriate tools strategically.

- CCSS.Math.Practice.MP7

Look for and make use of structure.