From Triangles to Circles and Back - Exploring Connections among

Common Core Standards

Facilitator: David Brown May 3, 2014

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

1.Make sense of problems and persevere in solving them.

2.Reason abstractly and quantitatively.

3.Construct viable arguments and critique the reasoning of others.

4.Model with mathematics.

5.Use appropriate tools strategically.

6.Attend to precision.

7.Look for and make use of structure.

8.Look for and express regularity in repeated reasoning.

Standards for Mathematical Practice

Clip – Homer3 (Tree House of Horror VI)

Motivation from Homer Simpson

178212 + 184112 = 192212

A Surprising Equation?

Check on TI84-Plus: (1782^12+1841^12)^(1/12) = 1922

Verification!!

Maybe??

How do we know this is FALSE?

Fermat’s Last Theorem

an + bn = cn has no positive integer solutions if n>2.

Pierre de Fermat, 1601-1665.

Contrast: Rich structure if n=2.

Pythagorean Theorem

Pythagorean Theorem

On to Part I of today’s Activity.

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.

Pythagorean Triples

• Are there infinitely many Pythagorean triples?

• How many entries can be even?

• Can the hypotenuse ever be the only even side?

Pythagorean Triples

• Are there infinitely many primitive Pythagorean triples?a b c3 4 55 12 137 24 259 40 4111 60 6113 84 85

PATTERNS?

FORMULA(S)?

Have we found ALL triples now? Well…no!

Pythagorean Triples

• Are there infinitely many primitive Pythagorean triples?a b c4 3 58 15 1712 35 3716 63 6520 99 10124 143 145

PATTERNS?

NOW have we found ALL triples?

FORMULA(S)?

WELL…

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.

Pythagorean Triples

a = q2 – p2 b = 2pq c = p2 + q2

Find a triple not on any of the previous lists.

Pythagorean Triples

a = 33 b = 56 c = 65

Now we have new number theory question!

For what integers p, q does q2 – p2 = 33?

a = q2 – p2 b = 2pq c = p2 + q2

How do we derive this general formula for triples?

Pythagorean Triples

More geometry - Look to the circle!

The rational parameterization of the unit circle gives rise to Pythagorean triples!

Pythagorean TriplesExploring triangles within circles - GeoGebra

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)

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

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.

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

1.Make sense of problems and persevere in solving them.

2.Reason abstractly and quantitatively.

3.Construct viable arguments and critique the reasoning of others.

4.Model with mathematics.

5.Use appropriate tools strategically.

6.Attend to precision.

7.Look for and make use of structure.

8.Look for and express regularity in repeated reasoning.

Which Practice Standards Did We Use?

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.

 

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.

  

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.

 

  

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.

  

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.

  

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.

  

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 aninteger-sided right triangle. 

60 Degree Triples

If a, b, and c are whole-number sides of a triangle with a60 degree angle, then c2 = a2-2ab+b2 anda = n2 – nd + d2

b = 2nd - d2

c = n2 – nd +d2

 

Digging Deeper

Fermat’s Last Theorem

If a, b, and c are whole-numbers, then the equation

an + bn = cn

has no solution.