The “ah ha” moment
A.
Problem Solving Courses
B.
The UNC Statewide Mathematics
Contest: 7-12th graders
C.
Undergraduate Research Projects
RICHARD GRASSL - UNC
Problem Solving
Courses
-CAPSTONE COURSE FOR
ELEMENTARY TEACHERS
-MA LEVEL FOR INSERVICE
TEACHERS
“An empty mind cannot solve problems”
-Polya
Number themes:
1. Arithmetic growth
a.
Differencing
b.
Gauss forward and backward
sum
2. Geometric growth
a.
Geometric ratios
b.
Shift and subtract
3. Greatest common divisors
4. Least common multiple
5. Special sequences of numbers
a.
Odds and evens
b.
Squares
c.
Triangular numbers
d.
Prime and composite numbers
6. Parity
Algebraic themes:
7. Factoring
8. Factor theorem
9. Remainder theorem
10. Rational root theorem
11. Add-in and subtract-out
12. Telescoping or collapsing
sums/products
13. Averages
Geometric themes
14. Symmetry
15. Properties of diagonals in polygons
16. Pythagorean theorem
17. Congruent triangles
Counting themes
18. Binomial coefficients
19. Permutations
20. Compositions
21. Principle of inclusion-exclusion
22. Pigeonhole principle
23. Mutually exclusive and exhaustive
partitions of sets
The Polya Four Step
Understand
Restate it, do I need definitions, assumptions, what kind of answer
should I get? What skills do I need?
Strategy
List different types of heuristics to use (data collection-pictureformulas etc), create a plan of attack, list tasks, organize…
Implement
Execute your plan-keep a record to document successes and failures
Tie together
Restate the problem, doublecheck, search for essence of problem,
create extensions
Research suggests that a key difference between novice and expert
problem solvers is the amount of time devoted to considering
different strategies.
Across:
Down:
1. Square of a prime
1. Square of another prime
4. A prime number
2. A square
5. A square
3. A prime number
1
1
4
5
2
3
Choose two points.

What is the probability that the distance
between them is an integer?
m
n
How many fractions
can you make if m and n
are positive integers and the following hold?
(a) m < n
(b) m + n = 575
(c) Each fraction is reduced
Start making them:
1
2
574
573
575 = 52 x 23
665 = 5 x 7 x 19
3
572
4
5
571
570
...
287
288
How many positive integers n have
divisors?
2
3
4
5
6
7
8
9
10
1
1
1
1
1
1
1
1
1
2
3
2
5
2
7
2
3
2
4
3 6
4 8
9
5 10
SOLUTION: The number of divisors d(n) satisfies:
Now solve:
n
2
n
2
2 n
Only 8, 12 will work
d(n)  2 n
Overview of problems

Find positive integers n and a1, a2, a3, …,
an such that a1+a2+…+an=1000 and the
product a1a2a3…an is as large as possible.

How many rectangles of all sizes are
there in a subdivided 4 by 5 rectangle?

How many positive integers have their
digits in increasing order? Like 347.
Find positive integers n and a1, a2, a3…, an such
that a1+a2+a3+…+an = 1000 and the product
a1a2a3…an is as large as possible.
SUM
PRODUCT
2 + 8 = 10
2 * 8 = 16
5 + 5 = 10
5 * 5 = 25
2 + 4 + 4 = 10
2 * 4 * 4 = 32
2 + 2 + 3 + 3 = 10
2 * 2 * 3 * 3 = 36 BEST!
CONCLUSION
Have as many 3’s as possible with a few 2’s
Replace 2 + 2 + 2 with 3 + 3
Never use any:
4’s
5’s
6’s
7’s
8’s
2+3
3+3
3+4
4+4
9’s
3+3+3
EXTEND
Allow rational parts
Allow real numbers
How many rectangles are there in a
subdivided 4 by 5 rectangle?
2x1
4x5
1
1
1
1
1
x
x
x
x
x
1
2
3
4
5
20
16
12
8
4
2
2
2
2
2
x
x
x
x
x
1
2
3
4
5
15
12
9
6
3
3
3
3
3
3
x
x
x
x
x
1
2
3
4
5
10
8
6
4
2
4
4
4
4
4
x
x
x
x
x
1
2
3
4
5
5
4
3
2
1
(1 + 2 + 3 + 4 + 5) + 2(1 + 2 + 3 + 4 + 5) + 3(1 + 2 + 3 + 4 + 5)
+ 4(1 + 2 + 3 + 4 + 5) = (1 + 2 + 3 + 4 + 5)(1 + 2 + 3 + 4) = 150
5 .6 4 . 5
(1+2+3+4+5)(1+2+3+4)= ( 2 )( 2 ) = ( 62 ) ( 52 )
What do you hope
to hear when a
student gets to
this stage?
“ah ha”
How many positive integers have their
digits in increasing order? Like 347.
Start with an easier problem.
123
124
125
126
127
128
129
134
135
136
137
138
139
145
146
147
148
149
156
157
158
159
167
168
169
178
179
189
7 + 6 + 5 + 4 + 3 + 2 + 1 = 28
234
235
236
237
238
239
245
246
247
248
249
256
257
258
259
267
268
269
278
279
289
PROOF: Just
choose 3 of the 9
digits.
6 + 5 + 4 + 3 + 2 + 1 = 21
9
Continue: 28 + 21 + 15 + 10 + 6 + 3 + 1 = 84 = ( 3 )
…back to the original problem.
(
9
9
9
9
9
9
9
9
9
) + ( ) +( ) + ( ) + ( ) +( ) +( ) + ( ) + ( )
1
2
5
6
7
8
9
3
4
=29 -1
“Ah ha” moment
Just choose any subset of {1, 2, 3, 4, 5, 6, 7, 8, 9}
{7, 4, 5, 2}
2457
(
m+n
m
n
) - ( ) - ( ) = mn
2
2
2
-Algebraically
-How many man-woman dancing pairs?
m
-How many lines?
.
.
.
.
.
.
.
.
n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
..
...
....
Anatomy of a good problem.

Interesting and challenging

Open-ended (opportunity for extension)

A surprise occurs somewhere

A discovery can be made-leads to “ah ha”

Solutions involve understanding of distinct mathematical
concepts, skills

Problem and solution provides connections

Various representations allowed
Which of these numbers are prime?
101, 10101, 1010101, 101010101, …
STRATEGY: Place in a more general setting
x2
x4
x6
x8
+
+
+
+
1
x2 + 1
x 4 + x2 + 1
x 6 + x4 + x2 + 1
x10-1
x2 - 1
=
x5 – 1
x5 + 1
x– 1
x+1
Generalize:
X15 – 1
X3
-1
=
= x4(x2+1)+(x2+1) = (x4+1)(x2+1)
– A geometric sum
= (x4 + x3 + x2 + x + 1)(x4 – x3 + x2 – x + 1)
1001, 1001001, 1001001001, …
(x5 – 1)(x10 + x5 + 1)
(x – 1)(x2 + x + 1)
The UNC statewide
Mathematics Contest
7th-12th graders
Mathematics Contests
Eötvos competitions – Hungary,
1894-1905
 Polya competitions – Stanford, 1950’s
 Santa Clara Contest – Abe Hillman, 1960’s
 University of New Mexico – Hillman, Grassl
1970-1990
 University of Northern Colorado –
1992-2010

Goals – Educational Value
1.
Offer a unique educational challenge to all interested
students grades 7-12
2.
Recognize and reward talented students for their
extraordinary achievements
3.
Provide an opportunity for university faculty to
cooperatively engage in an educational endeavor involving
secondary school teachers, parents, and students
4.
Recruit talented mathematics students to major in
mathematics and the sciences
5.
Draw attention to basic themes in the secondary
curriculum that we think are important
What makes this contest different?

All students in grades 7-12 in Colorado are eligible. A student need
not be selected or prescreened.

All students in grades 7-12 take the same exam.

The contest is in two rounds:


First round (November) – at school site
Final round (February) – at UNC

First round is jointly graded by secondary teachers and UNC staff

Each round consists of 10 or 11 essay type questions

Certain problems are paired.


A theme is introduced in the FIRST ROUND and is built on in the FINAL
ROUND.
A solutions seminar for teachers and parents is offered.
Examples of paired problems…
FIRST ROUND
How many rectangles?
Express 83 as a difference of
2 squares.
Example: 7 = 16 - 9
SECOND ROUND
How many rectangles?
(a) Demonstrate that every
odd number 2n + 1 can
be expressed as a
difference of two squares
(b) Which even numbers can
be so expressed?
Some data…

1992
2009

- Of the top 25 winners about 19% were women

- In 2005, Olivia Bishop = First place
- In 2007, Hannah Alpert = First place



140 students
1800 students
Schools:6 to over 75
- 38% of the time First Place was achieved by
someone in 8th, 9th or 10th grade
How many positive integers have their
digits in strictly increasing order?
One of the contest winners zeroed in on the
following very succinct and beautiful solution:
“Since any subset of {1, 2, 3, 4, 5, 6, 7, 8, 9}
except the empty set will correspond to an
‘increasing’ integer, the answer is 29 -1.”
Our admonition to be creative echoes what Albert
Einstein once implied: We all have a brain. It’s
what we do with it that matters.
Where are they now?


















Rice
UT Austin
Stanford
MIT
Cornell
U. Michigan
Harvard
Columbia
U. Wisconsin
CU Boulder
ASU
U. Chicago
AFA
Cal Tech
Harvey Mudd
Lawrence University
Wartburg College
UC Davis
Majors and PhD programs
Mathematics
Mechanical Engineering
Electrical Engineering
Chemical Engineering
Aerospace Engineering
Computer Science
Medical School
Law School
Undergraduate Research
Projects
1.
Leibnitz Harmonic Triangle
2.
For which n is Vn, the invertibles
in Zn, cyclic?
3.
What does [f(g(x))](n) look like?
Pascal Triangle
HOCKEY STICK THEOREM
1
1
1
1
1
1
1
1
7
2
3
4
5
6
1
3
6
10
15
21
1
4
10
20
35
1
1
5
15
35
1
6
21
1
7
1
1
0
Leibnitz Harmonic Triangle
1
1
1
2
1
3
1
4
1
6
1
7
1
12
1
30
1
42
1
6
1
20
1
5
1
2
1
12
1
30
1
60
1
105
1
3
1
4
1
60
1
140
1
5
1
20
1
6
1
30
1
105
1
42
1
7
1 1
1 1
1 1
1 1
1 1 1
1
1
  

 ...  (  )  (  )  (  )  (  )  ...
2 3
3 4
4 5
5 6
2 6 12 20 30
For which n is Vn cyclic?

V9 is but V8 is not

V9 = {1, 2, 4, 5, 7, 8} is generated by 2

V8 = {1, 3, 5, 7} is not cyclic
since 32 ≡ 52 ≡ 72 ≡ 1
Lots of references – start with Gallian
Leibnitz Rule for Differentiating a Product
(f g)’ = f g’ + f’g
(f g)’’ = f g’’ + 2 f’g’ + f’’g
(f g)’’’ = f g’’’ + 3f’g’’ + 3f’’g’+ f’’’g
What happens with [f(g(x))](n) ?
The n-th derivative of a composite function.
Let h = f(g(x))
h’=f’(g(x))g’(x)
h1 = f1g1
h2 = f1g2 + f2g12
h3 = f1g3+3f2g1g2+f3g13
h4 = f1g4+f2[4g1g3+3g22]+f3[6g12g2]+f4[g1]4
h5=
f1 g 5 +
f2[5g1g4+10g2g3]+
f3[10g12g3+15g1g22] +
f4[10g13g2]+
f5[g15]
Look at row sums
1, 2, 5, 15, 52, …
BELL NUMBERS
Stirling numbers of 2nd kind
=1
=2
=5
=15
=52
…
1
1
1
1
1
1
3
7
15
1
6
1
25
…
10
1
1
1
1
1
1
5+10
1
3
4+3
1
6
10+15
…
1
10
1
In h4
4g1g3+3g22
In h5
10g12g3+15g1g22
What did we learn?

Teachers need experiences constructing the same
mathematics that they will be teaching.

We should teach through exploration

True problem-solving episodes are a rarity in the teaching
of school or collegiate mathematics and we should do
everything we can to foster them when they do occur.

A child’s mind is a fire to be ignited, not a pot to be filled.

The curriculum should engage students in some problems
that demand extended effort to solve so they develop
persistance and a strong self-image.
...

Problem solving is not passive – students construct their
own solutions, their own problems.

A certain amount of struggle and frustration is natural,
expected, desired.

Care must be taken not to frustrate students to the point
where they might become disillusioned and disinterested –
students need to be exposed to learning situations where
their problem solving ability may be enhanced.

Leave questions open enough that students can extend
themselves, the problem, and utilize any technology they
may think is helpful and appropriate.

Allow students to go out and get the tools that they need to
solve the problems.
…continued.

Don’t impose your idea of a solution on the students, they
may come up with a better way.

Be ready to provide structure and leading questions when
called upon.

Be ready to learn in your classroom.

Expose students to a variety of technologies so they can
pick and choose which works best for them on each
problem they are given to solve.
A final thought.
“Teaching to solve problems is education of
the will. Solving problems which are not
too easy for him, the student learns to
persevere through unsuccess, to
appreciate small advances, to wait for the
essential idea, to concentrate with all his
might when it appears. If the student had
no opportunity in school to familiarize
himself with the varying emotions of the
struggle for the solution his mathematical
education failed in the most vital point.”
-George Polya
You can find this presentation at:
http://www.unco.edu/NHS/mathsci/facstaff
Grassl/
It will be available after April 20, 2010