You may be asking yourself what a Pythagorean Triple is but it is 3 numbers that fit the a^2 + b^2 = c^2 pattern. Today, December 16, 2020, is just such a day! 12/16/20 fits this pattern. Are there other days coming up that fit this pattern? Will we be alive when another happens?
Teaching adults how to identify this pattern 3-4-5 supports a deeper understanding of this theorem, in addition, to teaching them the formula. Do you have other ways of teaching the Pythagorean Theorem?
I would love to hear them!
Happy Pythagorean Triple Day!
I teach them 3 4 5, but I wish I had made a realization that today was the day of one. I also teach 5 -12 -13 as it is the other pretty one. I always have them discover the 3, 4, 5 with blocks. they use unit blocks and make a 3 - 4 right angle and then complete the triangle with 5 that they discover. Then we create squares with more blocks - creating the classic pythagorean picture. then I have them take the 3 square blocks and the 4 square blocks and place them one on one on the 5 square blocks and they see they equal one another. Just by doing that it creates a picture. Then we grow our triangles and shrink them. Haven't figured out what I can use to do this online except by making a video of me doing it but I would rather they do it.
The most fun I've ever had noodling around with pythagorean triples was in an exploration of families of triples at the NYC Community of Adult Math Instructors (CAMI) math teachers circle. Here are the notes and materials and some of the thinking that came out of our observations.
yours in productive struggle,
I tried a new lesson last winter that went pretty well and avoided memorizing the Pythagorean formula until students had explored making triangles that were or weren't right triangles.
I started by showing students an image of a triangle with a few measurements and asked them to write everything they knew about the figure. When it was time to discuss as a group, I tried not to add too much and mostly just took notes on what they said.
I then gave them a bunch of squares and asked them to choose three that would make a triangle with the negative space between. Once they could create a triangle, I asked them to find right triangles. They used protractors to check if there was a 90 degree angle.
I then asked them to see if they could find all combinations of three squares that would produce a right triangle.
They eventually used the formula as a way to find other combinations or check combinations that were very close to having a 90 degree angle, _____, for example.
Write everything you know about this figure. 20 min.
- Draw right triangle on the board. Label 3” and 4” for two lengths. Write everything you know about this figure.
- Individual, then turn to a partner. Share what you know.
- Take notes on the board.
- Hand out rulers. What else can you tell us? Make sure everyone is measuring from the right part of the ruler. ( “ means inches, the missing edge is 5 inches long)
- Make sure these ideas are discussed: triangle, length, inches, 90 degrees, hypotenuse, legs
- Maybe: area, square inches
- How big are the little squares? 1 square inch. How long are they on one side? 1 inch.
90 angles - 10 min.
- What is a 90 degree angle? What examples can you give?
- Do you see any examples of 90 degree angles in the room?
Door frame is a good example. (Return to this later to check to see if it is 90 degrees.)
- Why are 90° angles important? In construction? In manufacturing?
- How can we measure to make sure an angle is 90°? A protractor.
- Hand out protractors. Check to see if the angle is 90°.
- Make sure everyone knows how to center the protractor.
- (Other angles are about 54° and 37°)
- In a diagram, you may see 90° or a square □ in the corner. The square means 90°.
What do you notice? What do you wonder? - 15 min.
- Draw 3, 4, 5 triangle with squares off of the sides of the triangle. Draw grid within squares.
- Write what you notice. Write what you wonder.
- Share with a partner.
- Take notes as a group.
- Make sure these things are noticed: There are 9 squares, 16 squares, and 25 squares. The height of the “9 square” is 3. The height of the “16 square” is 4.
- How big are the little squares? 1 square centimeter. How long are they? 1 centimeter.
- Questions: Can you make triangles with any three squares? Can you make right triangles with any three squares?
Making triangles with squares - 20 min.
- Hand out squares. One set for each group of 3 students.
- What do you notice? Take notes.
- It’s possible to make a triangle with three squares. Demonstrate on the board. Trace the triangle on a blank piece of paper.
- Challenge: How many triangles can you make with these squares?
- Trace and label as many triangles as you can. Label the length of the sides.
- Use protractors to measure the angles. Add angle measurements to your triangles.
- What kinds of triangles did you make? Use meter stick to draw triangles on the board with approximate lengths and angles.
- Was anyone able to make a right triangle? Draw them on the board.
- If students don’t find any, ask them to try 5, 12, 13.
Pythagorean theorem - 10 min.
- In a right triangle, the combined areas of squares of the legs are equal to the square of the hypotenuse.
- What does this mean?
- Add the square of one leg to the square of the other leg to get the square of the hypotenuse.
- Who can explain the Pythagorean formula using the cut-up squares?
Sorry, I meant to add 8, 9, 12 as an example of a triangle that looked like it would be a right triangle. It produced an angle that was so close to 90 degrees that you couldn't tell the difference with a protractor. It makes me wonder about triples that are more Pythagorean than others. This one is about as close as you can get.