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”

-PolyaNumber themes:1. Arithmetic growth

a. Differencingb. Gauss forward and backward sum

2. Geometric growtha. Geometric ratiosb. Shift and subtract

3. Greatest common divisors4. Least common multiple5. Special sequences of numbers

a. Odds and evensb. Squaresc. Triangular numbersd. Prime and composite numbers

6. Parity

Algebraic themes:7. Factoring8. Factor theorem9. Remainder theorem

10. Rational root theorem11. Add-in and subtract-out12. Telescoping or collapsing sums/products13. Averages

Geometric themes14. Symmetry15. Properties of diagonals in polygons16. Pythagorean theorem17. Congruent triangles

Counting themes18. Binomial coefficients19. Permutations20. Compositions21. Principle of inclusion-exclusion22. Pigeonhole principle23. Mutually exclusive and exhaustive partitions of sets

The Polya Four StepUnderstandRestate it, do I need definitions, assumptions, what kind of answer

should I get? What skills do I need?

StrategyList different types of heuristics to use (data collection-picture-

formulas etc), create a plan of attack, list tasks, organize…

ImplementExecute your plan-keep a record to document successes and failures

Tie togetherRestate 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:

1. Square of a prime

4. A prime number

5. A square

Down:

1. Square of another prime

2. A square

3. A prime number

1 2 3

1

4

5

Choose two points. What is the probability that the distance

between them is an integer?

How many fractions can you make if m and n are positive integers and the

following hold?

mn

(a) m < n

(b) m + n = 575

(c) Each fraction is reduced

Start making them:

1 2 3 4 5 287. . .574 573 572 571 570 288

575 = 52 x 23

665 = 5 x 7 x 19

How many positive integers n have divisors?2 1 23 1 34 1 2 45 1 56 1 2 3 67 1 78 1 2 4 89 1 3 910 1 2 5 10

nnd 2)(

n2

SOLUTION: The number of divisors d(n) satisfies:

Now solve: nn

22

Only 8, 12 will work

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

2 + 8 = 10

5 + 5 = 10

2 + 4 + 4 = 10

2 + 2 + 3 + 3 = 10

PRODUCT

2 * 8 = 16

5 * 5 = 25

2 * 4 * 4 = 32

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 9’s

2 + 3 3 + 3 3 + 4 4 + 4 3 + 3 + 3

EXTEND

Allow rational parts

Allow real numbers

How many rectangles are there in a subdivided 4 by 5 rectangle?

4 x 5

2 x 1

1 x 1 20 2 x 1 15 3 x 1 10 4 x 1 51 x 2 16 2 x 2 12 3 x 2 8 4 x 2 41 x 3 12 2 x 3 9 3 x 3 6 4 x 3 31 x 4 8 2 x 4 6 3 x 4 4 4 x 4 21 x 5 4 2 x 5 3 3 x 5 2 4 x 5 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

(1+2+3+4+5)(1+2+3+4)=5 . 6

2( )4 . 5

2( )= 62( ) 5

2( )

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.

123 134 145 156 167 178 189 124 135 146 157 168 179 125 136 147 158 169 126 137 148 159 127 138 149 128 139129

7 + 6 + 5 + 4 + 3 + 2 + 1 = 28

234 245 256 267 278 289 235 246 257 268 279 236 247 258 269237 248 259 238 249239

6 + 5 + 4 + 3 + 2 + 1 = 21

PROOF: Just choose 3 of the 9 digits.

Continue: 28 + 21 + 15 + 10 + 6 + 3 + 1 = 84 = 93

( )

…back to the original problem.

93( )

91( )

29

( ) ( )94

( )95

( )96

( )98

( )99

97

( )++ ++ ++ ++

=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+n2

( ) n2

( )m2

( )- - = mn

-Algebraically

-How many man-woman dancing pairs?

-How many lines?

. .

. . .

. . . .

. . . . .

. . . . . .

. . . . . . .

. . . . . . . .

m

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 + 1x4 + x2 + 1x6 + x4 + x2 + 1 = x4(x2+1)+(x2+1) = (x4+1)(x2+1)x8 + x6 + x4 + x2 + 1 – A geometric sum

x10-1

x2 - 1

x5 – 1 x5 + 1

x– 1 x + 1 = = (x4 + x3 + x2 + x + 1)(x4 – x3 + x2 – x + 1)

Generalize: 1001, 1001001, 1001001001, …

X15 – 1

X3 - 1

(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 Value1. 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?

SECOND ROUND

How many rectangles?

Express 83 as a difference of2 squares.

Example: 7 = 16 - 9

(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 140 students Schools:6 to over 75 2009 1800 students

- Of the top 25 winners about 19% were women

- In 2005, Olivia Bishop = First place - In 2007, Hannah Alpert = First place

- 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

MathematicsMechanical EngineeringElectrical EngineeringChemical EngineeringAerospace EngineeringComputer ScienceMedical SchoolLaw 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?

1

1 1

1 2 1

1 3 3 1

1 4 6 4 1

1 5 10 10 5 1

1 6 15 20 15 6 1

1 7 21 35 35 21 7 1

Pascal Triangle

HOCKEY STICK THEOREM

2

1

2

1

3

1

3

1

6

1

4

1

4

1

12

1

12

1

5

1

5

120

1

20

1

30

1

6

1

6

1

30

1

30

1

60

1

60

1

7

1

7

1

42

1

42

1

105

1

105

1

140

1

1

1

Leibnitz Harmonic Triangle

...30

1

20

1

12

1

6

1

2

1

0

1

...)6

1

5

1()

5

1

4

1()

4

1

3

1()

3

1

2

1(

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.

Look at row sums1, 2, 5, 15, 52, …

BELL NUMBERS

h’=f’(g(x))g’(x)Let h = f(g(x))

h1 = f1g1

h2 = f1g2 + f2g12

h3 = f1g3+3f2g1g2+f3g13

h4 = f1g4+f2[4g1g3+3g22]+f3[6g1

2g2]+f4[g1]4

h5= f1g5+ f2[5g1g4+10g2g3]+f3[10g1

2g3+15g1g22] +

f4[10g13g2]+

f5[g15]

Stirling numbers of 2nd kind1

1 11 3 1

1 7 6 11 15 25 10 1

…

=1=2=5=15=52…

1

1 1

1 3 1

1 4+3 6 1

1 5+10 10+15 10 1

…

In h4 4g1g3+3g22

In h5 10g12g3+15g1g2

2

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/facstaffGrassl/

It will be available after April 20, 2010