# Will you be celebrating August 15, 2017?

### Why, you may ask?

This question makes a great bell ringer or writing prompt for MS/HS math students –even for those who have not yet studied the Pythagorean Theorem. Of course, 8-15-17 is a Pythagorean Triple! This “holiday” does not happen every year. On this day, spend some time with your students discussing Pythagoras and his namesake theorem.

###### Picture from: http://www-history.mcs.st-and.ac.uk/Posters2/Pythagoras.html

* Pythagoras of Samos is often described as the first pure mathematician.* So begins the biography from the

**Mac Tutor History of Mathematics**website. But I don’t teach Geometry! So what? There is so much that can be discussed with regard to Pythagoras in all mathematics classes. Primary mathematicians, as well as intermediate and secondary, can discuss one of many

**quotes**attributed to Pythagoras:

*Number rules the universe.* Quoted in D MacHale, *Conic Sections *(Dublin 1993)

Mathematics is often defined as the study of patterns. What patterns can a young mathematician find by observing a table of Pythagorean triples? Below is a table of Primitive Pythagorean Triples. Share it with your students and ask them to be “pattern sniffers” (*Habits of Mind: An Organizing Principle for Mathematics Curricula* Cuoco, Goldenberg, & Mark, 1996) Then have them compare their patterns with those they find in this table of the Pythagorean Triples (which includes those made of multiples of the primitives).

### What about connecting fractions for students?

### Have students investigate these…

Method 1:

- Have students take any two odd numbers that differ by two, such as 3 and 5.
- Write the two numbers as the denominators of unit fractions:
^{1}/_{3 }and^{1}/_{5} - Find their sum:
^{1}/_{3}+^{1}/_{5}=^{8}/_{15} - The two numbers in the sum are always two sides of a primitive Pythagorean triangle!
- Hence, Happy Right Triangle Day! 8, 15, 17

Method 2:

- Have students take any two even numbers that differ by two, such as 2 and 4.
- Write the two numbers as the denominators of unit fractions: 1/2 and ¼
- Find their sum: 1/2 + 1/4 = ¾
- Observe that the two numbers in the reduced sum are always two sides of a primitive Pythagorean triangle!
- Hence, the basic 3, 4, 5 right triangle!

Note: neither using the two odds nor the two evens generate all the primitive Pythagorean triangles. See if your HS students can find a method using two fractions that does.

### For the Algebra teacher:

Have your students prove the two-fraction method that can be used to produce a Pythagorean triangle from any two initial numbers, *a* and *b*, given that a, b, and c are all whole numbers :

## OR

Given the figure to the right: Have students discuss and validate this method for generating Pythagorean Triples. They will need to apply the Pythagorean Theorem. This is good practice for applying the distributive property or for special products. Mathematics is beautiful! Mathematics if fun! Mathematics is full of interesting and surprising adventures! (Note: If you don’t believe the above statements – maybe you need to be teaching something else!) Here is a beginning list of other Resources for Pythagorean Triples and Pythagoras to engage and amuse your students: The Math Forum/NCTM: http://mathforum.org/dr.math/faq/faq.pythag.triples.html NASA: https://www.grc.nasa.gov/www/k-12/Numbers/Math/Mathematical_Thinking/pythtrip.htm NRich: https://nrich.maths.org/1309 http://nrich.maths.org/1311/1311 http://nrich.maths.org/1332/1332 Math is Fun: Interesting proofs for an infinite number of Pythagorean triples https://www.mathsisfun.com/numbers/pythagorean-triples.html Cut the Knot: A Collection of Proofs of the Pythagorean, even one by a former US President! http://www.cut-the-knot.org/pythagoras/index.shtml Texas Instruments: TI Nspire activities for MS & HS: https://education.ti.com/en/timathnspired/us/middle-grades-math/geometry https://education.ti.com/en/timathnspired/us/geometry/right-triangles-and-trig