1. Facilitator: David Brown
From Triangles to Circles and Back - Exploring Connections among Common Core Standards
Facilitator: David Brown
May 3, 2014

2. Workshop Goals Setting the stage: Standards for Mathematical Practices
Hands-on exploration of Pythagorean triples incorporating NYS Secondary CCLS-M
Discuss geometry and algebra connections
Digging Deeper

3. Standards for Mathematical Practice
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6

4. Motivation from Homer Simpson
Clip – Homer3 (Tree House of Horror VI)
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6

5. A Surprising Equation? 178212 + 184112 = 192212
Check on TI84-Plus: (1782^ ^12)^(1/12) =
1922
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6
Verification!!
Maybe??
How do we know this is FALSE?

6. Fermat’s Last Theorem
an + bn = cn has no positive integer solutions if n>2.
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6
Pierre de Fermat,
Contrast: Rich structure if n=2.
Pythagorean Theorem

7. Pythagorean Theorem On to Part I of today’s Activity.
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6
If a and b are the legs of a right triangle and c is the hypotenuse, then a2 + b2 = c2.
Pythagorean Triples
Algebraic View: Integers (a, b, c) that satisfy a2 + b2 = c2
Geometric View: Integers (a, b, c) that are the side lengths of a right triangle.

8. Pythagorean Triples Are there infinitely many Pythagorean triples?
How many entries can be even?
Can the hypotenuse ever be the only even side?
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6

9. Pythagorean Triples
Are there infinitely many primitive Pythagorean triples?
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6 a b c 3 4 5 12 13 7 24 25 9 40 41 11 60 61 84 85
PATTERNS?
FORMULA(S)?
Have we found ALL triples now?
Well…no!

10. Pythagorean Triples
Are there infinitely many primitive Pythagorean triples?
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6 a b c 4 3 5 8 15 17 12 35 37 16 63 65 20 99
101 24
143
145
PATTERNS?
FORMULA(S)?
NOW have we found ALL triples?
WELL…

11. Pythagorean Triples
General formula: If p and q are positive integers with q>p, then
a = q2 – p2
b = 2pq
c = p2 + q2
always yields a Pythagorean triple!
Every Pythagorean triple is of this form or a “dilation” of this form.

12. Pythagorean Triples a = q2 – p2 b = 2pq c = p2 + q2
Find a triple not on any of the previous lists.
a = b = c = 65
Now we have new number theory question!
For what integers p, q does q2 – p2 = 33?

13. Pythagorean Triples a = q2 – p2 b = 2pq c = p2 + q2
How do we derive this general formula for triples?
More geometry - Look to the circle!
The rational parameterization of the unit circle gives rise to Pythagorean triples!

14. Pythagorean Triples
Exploring triangles within circles - GeoGebra

15. Pythagorean Triples Draw line between (-1,0) and (x,y) on unit circle.
If (x,y) is rational, then slope (m) is also rational. Why?
If m is rational then so is (x,y).
The line between (-1,0) and (x,y) is given by y=m(x+1)

16. Pythagorean Triples
If (a,b,c) is a Pythagorean triple, then (a/c,b/c) is . . .
A rational point on the unit circle!
a2 + b2 = c2 implies
(a2/c2) + (b2/c2) = (c2/c2)
(a/c)2 + (b/c)2 = 1

17. Pythagorean Triples Intersect y=m(x+1) and x2 + y2 = 1
x2 + (m(x+1))2 =1
Yields x and y in terms of m:
x = (1-m2)/(1+m2) y = (2m)/(1+m2)
Set m = p/q, with q>p
Substitute and simplify.

18. Pythagorean Triples x = (1-(p/q)2)/(1+p/q2) y = (2(p/q))/(1+(p/q)2)
x = (q2–p2)/(p2+q2) y = 2pq/(p2+q2)
a = q2 – p2
b = 2pq
c = p2+q2

19. Which Practice Standards Did We Use?
Make sense of problems and persevere in solving them.
Reason abstractly and quantitatively.
Construct viable arguments and critique the reasoning of others.
Model with mathematics.
Use appropriate tools strategically.
Attend to precision.
Look for and make use of structure.
Look for and express regularity in repeated reasoning.
Handout Practice Standards.
“Focus” is also a way to allow time for the kinds of rich classroom discussion and interaction that support the Standards for Mathematical Practice. PARCC MCF Page 6

20. CCSSM Content Standards
Grade 8 Geometry (8.G)
Understand and apply the Pythagorean Theorem.
6. Explain a proof of the Pythagorean Theorem and its converse.
7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

21. CCSSM Content Standards
HS Algebra
Arithmetic with Polynomials & Rational Expressions
A-APR
Use polynomial identities to solve problems.
4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 – y2)2 + (2xy)2 can be used to generate Pythagorean triples.

22. CCSSM Content Standards
HS Algebra
Creating Equations A-CED
Create equations that describe numbers or relationships
1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

23. CCSSM Content Standards
HS Algebra
Reasoning with Equations & Inequalities A-REI
Understand solving equations as a process of reasoning and explain the reasoning
1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

24. CCSSM Content Standards
HS Algebra
Reasoning with Equations & Inequalities A-REI
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.

25. CCSSM Content Standards
HS Geometry
Expressing Geometric Properties with Equations G-GPE
Translate between the geometric description and the equation for a conic section
1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

26. Digging Deeper Complex Numbers
If x and y are integers and we form a+bi=(x+iy)2,
then a2+b2 is a perfect square. So, a and b are legs of an
integer-sided right triangle.
60 Degree Triples
If a, b, and c are whole-number sides of a triangle with a
60 degree angle, then c2 = a2-2ab+b2 and
a = n2 – nd + d2
b = 2nd - d2
c = n2 – nd +d2

27. Digging Deeper Fermat’s Last Theorem
If a, b, and c are whole-numbers, then the equation
an + bn = cn
has no solution.