M A X W A R S H A U E R

H I R O K O W A R S H A U E R

N A M A N A M A K S H I

NCTM REGIONAL CONFERENCE & EXPOSITION

CHICAGO, ILLINOIS

NOVEMBER 29, 2012

DEVELOPING PROBLEM-SOLVING

ABILITIES FOR

MIDDLE SCHOOL STUDENTS

OUTLINE

Introduction – Max

Problem Session 1 – Nama

Problem Session 2 – Hiroko

Problem Session 3 and Conclusion – Max

MATHWORKS

Center for innovation in mathematics education

Mission: To develop model programs and self-

sustaining learning communities that engage

K-12 students from all backgrounds in doing mathematics at a high level

Three Pillars

Summer Math Programs

Curriculum Development & Implementation

Teacher Professional Development

SUMMER MATH PROGRAMS

Honors Summer Math Camp Residential Program

60 students, 15 counselors

6 weeks

Junior Summer Math Camp

Commuter Program—200 students

Residential Program—36 students

Primary Math World Contest—Hong Kong

JUNIOR SUMMER MATH CAMP

Began 16 years ago

Grades 4-8

Laboratory for developing new ideas for teacher training and curriculum

development

STUDENTS COMMENTS ABOUT THE SUMMER MATH PROGRAMS

“This program helped me understand the rules that I followed. I liked this experience because

I didn’t just learn the rules, I understood why.”

“This program has taught me perseverance in

the face of difficult problems, a skill that is easily

applicable to fields in addition to math.”

COMMON CORE STATE STANDARDS AND PROBLEM SOLVING

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 sure of structure

Look for and express regularity in repeated reasoning

PROBLEM TYPES

Number theory

Algebra

Logic

Geometry

Combinatorics

COFFEE AND CREAM Adapted from Johnson, K., Herr, T., Kysh, J. (2004) Crossing the River with Dogs. Course Math 7378A @

txstate.edu Spring 2011

Suppose you have a cup of coffee and a cup of cream.

Take a spoonful of the cream, pour it into the coffee,

and stir it up. Then take a spoonful of the coffee and

cream mixture, pour it back into the cream, and again

mix it up. Determine if there is more coffee in the

cream cup, more cream in the coffee cup, or the same

amount of coffee in the cream cup as cream in the

coffee cup.

C & C: DISCUSSION

Is there enough information?

What strategies did you use?

Do you need algebra?

COFFEE & CREAM SOL 1

Strategy: Change focus in more ways

The cream in the coffee displaces some amount of coffee, and the coffee it displaces ends up in the cream cup. So the amount of

cream in the coffee cup is same as the amount of coffee in the

cream cup!

Same amount

At the end: still same amt.

Credit: Dr. M Warshauer

COFFEE & CREAM SOL 2 Credit: Blog Site: http://mrhonner.com

Strategy: Work with some concrete numbers. Try to keep it simple.

Lets assume there’s 1 spoonful liquid in each cup

Now the mixture has equal parts coffee and cream per spoonful:

1/2 part coffee & 1/2 part cream. Transferring a spoonful back into the empty cup doesn’t change the ratios.

COFFEE & CREAM SOL 3 credit: yours truly

Strategy: Covert to algebra and keep track in a table

Coffee Cup Cream cup Amt. of coffee in cream cup

Amt. of cream in coffee cup

T spoonfuls coffee T spoonfuls cream 0 spoons 0 spoons

T+1 spoonfuls T-1 spoonfuls 0 spoons 1 spoonful

Ratio per Spoon of

cream:

T+1 sp 1sp

1 sp 1/(T+1)

Ratio per spoon of

coffee: T/(T+1)

1/(T+1) T/(T+1)

T+1 – 1 T+1 – [1/(T+1) + T/(T+1)] T – T/(T+1) + 1–1/(T+1) T – T/(T+1) + [T/(T+1)]

T-1 + 1 T-1 + [1/(T+1) + T/(T+1)] T-1 + 1/(T+1) + [T/(T+1)]

T/(T+1)

T/(T+1)

JARS OF WATER

You have two unmarked water jars, one of capacity 12 ounces and the other of

capacity 34 ounces, determine how you

can use these containers to have a total

of exactly 4 ounces of water. You are

allowed to empty the jars or fill them up with water as many times as you wish.

You have unlimited supply of water.

JARS OF WATER

What strategies can we use?

Trial and Error

Converting to Algebra

Systematic List

Number Theory - Euclidean Algorithm

JARS OF WATER

Strategy: Systematic List

12 oz. 34 oz.

0 0

12 0

0 12

12 12

0 24

12 24

2 34

2 0

0 2

12 oz. 34 oz.

12 2

0 14

12 14

0 26

12 26

4 34

4 0

We filled up the 12 oz. container from

the pond 6 times and we emptied

the 34 oz. container into the

pond 2 times to

reach the desired configuration

JARS OF WATER

Strategy: Convert to algebra

34x + 12y = 4

Can we find two numbers x and y that satisfy this equation?

Can applying the Euclidean Algorithm help us here?

COUNTING THE WAYS

Archimedes, an ant, starts at the origin in the coordinate plane. Every minute he can crawl one

unit up or one unit to the right, thus increasing one of

his coordinates by 1. How many different paths can

Archimedes take to the point (4,3)?

Work individually to come up with 2 different ways to

solve the problem.

GRID TO WORK ON

HOW MANY PATHS TO (4, 2)?

(4, 2)

(0, 0) 1 (2, 0) (3, 0) (4, 0)

(0, 1)

(0, 2)

(1, 1) (2, 1) (3, 1) (4, 1)

(1, 2) (2, 2) (3, 2)

(4, 3)

GRID TO WORK ON

HOW MANY PATHS TO (3, 3)?

(3, 3)

(1, 0) (0, 0) (2, 0) (3, 0)

(0, 1)

(0, 2)

(0, 3) (1, 3) (2, 3)

(1, 1) (2, 1) (3, 1)

(1, 2) (2, 2) (3, 2)

(4, 3)

STUDENT APPROACHES

Brute force

Sum of two previous paths

Observe pattern of Pascal’s triangle

Counting argument C(n,m)

GENERAL CASE

How many different paths can Archimedes take to the point (n, m)?

THE DONUT PROBLEM

How many ways can one select a dozen donuts from glazed, chocolate, and jelly-filled?

SIMPLE CASES

Two types of donuts

Select only 2 donuts; 3 donuts; 4 donuts

“To think deeply of simple things”

-Arnold Ross

GENERAL CASE

Using intercell partitions

Place dividers between each type of donut.

There are 12 locations to place the donuts, and

we need to use two “intercell partitions.”

A GEOMETRY PROBLEM

Let AB = 10 units, AP = PD = 8 units. Parallelogram ABCD has the same area as

Triangle APR.

Compute the length of QB.

THANK YOU

Max Warshauer max@txstate.edu

Hiroko Warshauer hw02@txstate.edu

Nama Namakshi nn1052@txstate.edu