Transcript Exploring Pythagorean Triples

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
Standards for Mathematical Practice
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.
Motivation from Homer Simpson
Clip – Homer3 (Tree House of Horror VI)
A Surprising Equation?
178212 + 184112 = 192212
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
3
5
7
9
11
13
b
4
12
24
40
60
84
c
5
13
25
41
61
85
PATTERNS?
FORMULA(S)?
Have we found ALL triples now?
Well…no!
Pythagorean Triples
• Are there infinitely many primitive Pythagorean
triples?
a
4
8
12
16
20
24
b
3
15
35
63
99
143
c
5
17
37
65
101
145
PATTERNS?
FORMULA(S)?
NOW have we found ALL triples?
WELL…
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.
Pythagorean Triples
a = q2 – p2
b = 2pq
c = p2 + q2
Find a triple not on any of the previous lists.
a = 33
b = 56
c = 65
Now we have new number theory question!
For what integers p, q does q2 – p2 = 33?
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!
Pythagorean Triples
Exploring 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
•
2
x
+
2
(m(x+1))
=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
(2(p/q))/(1+(p/q) )
• x = (q2–p2)/(p2+q2)
• a = q2 – p2
• b = 2pq
• c = p2+q2
y = 2pq/(p2+q2)
Which Practice Standards Did We Use?
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.
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
Solve equations and inequalities in one variable.
4. Solve quadratic equations in one variable.
A-REI
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 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
Digging Deeper
Fermat’s Last Theorem
If a, b, and c are whole-numbers, then the equation
an + bn = cn
has no solution.