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
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
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|
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
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.
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.
- 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.
Move on to the next lesson, Exploring Diagonals and the Pythagorean Theorem.
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.]
- 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?