DEVELOPING PROBLEM-SOLVING
ABILITIES FOR
MIDDLE SCHOOL STUDENTS
M A X WA R S H A U E R
H I R O K O WA R S H A U E R
NAMA NAMAKSHI
NCTM REGIONAL CONFERENCE & EXPOSITION
CHICAGO, ILLINOIS
NOVEMBER 29, 2012
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 selfsustaining 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
Credit: Dr. M Warshauer
Strategy: Change focus in more ways
Same amount
At the end: still
same amt.
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!
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+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)
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.
12 oz.
34 oz.
0
0
12
2
12
0
0
14
0
12
12
14
12
12
0
26
0
24
12
26
12
24
4
34
2
34
4
0
2
0
0
2
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, 3)
(0, 2)
(0, 1)
(0, 0)
(1, 2) (2, 2) (3, 2)
(4, 2)
(1, 1) (2, 1) (3, 1) (4, 1)
1
(2, 0) (3, 0) (4, 0)
GRID TO WORK ON
HOW MANY PATHS TO
(3, 3)?
(0, 3)
(0, 2)
(0, 1)
(1, 3) (2, 3)
(3, 3)
(1, 2) (2, 2)
(3, 2)
(1, 1) (2, 1)
(3, 1)
(0, 0) (1, 0)
(2, 0) (3, 0)
(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
Hiroko Warshauer
Nama Namakshi
[email protected]
[email protected]
[email protected]