DEVELOPING PROBLEM-SOLVING ABILITIES FOR MIDDLE SCHOOL STUDENTS MAX WARSHAUER HIROKO WARSHAUER 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/(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, 3) (0, 2) (0, 1) (1, 2) (2, 2) (3, 2) (1, 1) (2, 1) (3, 1) (4, 1) (4, 2) (0, 0) 1 (2, 0) (3, 0) (4, 0)
GRID TO WORK ON HOW MANY PATHS TO (3, 3)? (0, 3) (1, 3) (2, 3) (3, 3) (4, 3) (0, 2) (0, 1) (1, 2) (2, 2) (3, 2) (1, 1) (2, 1) (3, 1) (0, 0) (1, 0) (2, 0) (3, 0)
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