8th grade Instructional Unit
Unit 2 – Exponents & Square Roots Review
Unit 3 – Pythagorean Theorem Instructional Unit
by
Nancy Perzel
Day 4 - Sum of the Squares in a Right Triangle
Warm-Up:
Question to ponder:
"Suppose these three squares were made of
pure gold, and you were offered either the one square labeled ÔcÕ or the two
squares labeled ÔaÕ and ÔbÕ. Which would you choose? Why?"
Students could write a short response in their daily
notebook. After they have been
given enough time to respond to the prompt, students can share their answers
with the classÉ (Walk around the room while they are working to try and
identify students with different answers to offer a variety of perspectives
with the class.)
Activity 1:
In groups, ask students to construct a right triangle with a base of 2 and a height of 3 with a square on each side of the triangle. If your students are using a geoboard, they will need a total of 4 geoboards pushed together to create a (10x10) large geoboard. They can also complete these activities on GSP. Pythagorean 2x3 GSP file
Prompt the students to look for connections (working towards the Pythagorean Theorem).
We already know the lengths of the base (2) and the height (3), what are the areas of their squares? What is the area of the square of the hypotenuse? Focus on just the area of the squaresÉ students should see a relationship here. Hopefully they will be able to see that the sum of the areas of the two smaller triangles is equal to the area of the third.
** Revisit the warm-up question hereÉ what square(s) would you choose if they were made of pure gold? Does it matter??? Hopefully your students are a little amazed that it would have made no difference!
Activity 2:
After each group has completed the task create a table to record your data from Activity 1, leaving the last column blank for now. (They should have an understanding on how to estimate its value by taking the square root of the area of the square, you can address this for the first example, but the initial focus is for students to observe that the sum of the areas of the square on the base and the area of the square on the height is equal to the area of the square of the hypotenuse.)
|
Length of Base of Triangle (a) |
Area of the Square of the Base (a2) |
Length of Height of Triangle (b) |
Area of the Square of the Height (b2) |
Area of the Square of the Hypotenuse (c2) |
Length of the Hypotenuse (c) |
|
2 |
4 |
3 |
9 |
13 |
- |
|
|
|
|
|
|
- |
|
|
|
|
|
|
- |
|
|
|
|
|
|
- |
The students should proceed to test their hypothesis by completing a few additional examplesÉ
Additional
examples:
Depending on the level of your students, these examples can be completed together as a class, or students can complete them on their own, in groups or you can have different groups each work on one example and then share their results with the class. I chose small numbers for the dimensions of the triangles (made possible for geoboards), however if youÕre using GSP you can modify the numbers and create more variety.
|
Length of Base of Triangle (a) |
Area of the Square of the Base (a2) |
Length of Height of Triangle (b) |
Area of the Square of the Height (b2) |
Area of the Square of the Hypotenuse (c2) |
Length of the Hypotenuse (c) |
|
2 |
4 |
3 |
9 |
13 |
- |
|
1 |
|
4 |
|
|
- |
|
2 |
|
2 |
|
|
- |
|
1 |
|
3 |
|
|
- |
|
2 |
|
4 |
|
|
- |
|
3 |
|
4 |
|
|
- |
At this point, students should be able to see that the sum of the areas of the two smaller triangles is equal to the area of the third, based on a generalization of the data from several cases. Use the following GSP file to stretch and move the original 2x3 right triangle for additional support. Save a formal proof for tomorrow when the formula is revealed.
|
Length of Base of Triangle (a) |
Area of the Square of the Base (a2) |
Length of Height of Triangle (b) |
Area of the Square of the Height (b2) |
Area of the Square of the Hypotenuse (c2) |
Length of the Hypotenuse (c) |
|
2 |
4 |
3 |
9 |
13 |
- |
|
1 |
1 |
4 |
16 |
17 |
- |
|
2 |
4 |
2 |
4 |
8 |
- |
|
1 |
1 |
3 |
9 |
10 |
- |
|
2 |
4 |
4 |
16 |
20 |
- |
|
3 |
9 |
4 |
16 |
25 |
- |
Pose additional questions for students to identify the relationship between the length of the side and the area of the square, and vice versa, so they are sure to see the squared/ square root relationship hereÉ Now ask them to fill in the missing values of the table – find the length of the hypotenuse (estimation, calculator).
End
class by starting to move beyond the dependence on manipulatives, by drawing a
right triangle on the board. Give the
triangle larger dimensions, so it would be impossible to construct on their
geoboards. With a base length (7)
and the height (5) ask students to imagine a square on the baseÉ what would be
the area? (49) Now imagine a square
on the heightÉ what would be the area? (25) Now imagine a square on the hypotenuseÉ
what would be the area? (Sum of the first two squares = 74) You can draw a quick sketch for students
who might still need a visual aid.
Extra Practice/ Homework:
Have
students practice with a few more examples like this on their own, or in small
groups. They can include a sketch (using
dot/graph paper) or GSP if itÕs available.
|
Length of Base of Triangle (a) |
Area of the Square of the Base (a2) |
Length of Height of Triangle (b) |
Area of the Square of the Height (b2) |
Area of the Square of the Hypotenuse (c2) |
Length of the Hypotenuse (c) |
|
3 |
|
5 |
|
|
- |
|
6 |
|
8 |
|
|
- |
|
5 |
|
9 |
|
|
- |
|
10 |
|
7 |
|
|
- |
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