The Quirky Musings of Jack Handy and The Enigma of The Horse Problem

1. DEEP THOUGHTS DEEP THOUGHTS Instead of having answers on math tests, cant we just have opinions, and if my opinion is different than that of everyone else, hey, cant we all just get along. Instead of having answers on math tests, cant we just have opinions, and if my opinion is different than that of everyone else, hey, cant we all just get along. Jack Handy Jack Handy

2. THE HORSE PROBLEM THE HORSE PROBLEM A man named Joe buys a horse named Ed for A man named Joe buys a horse named Ed for $60 from a woman named Flo. $60 from a woman named Flo. Joe then sells the horse for $70, buys it back Joe then sells the horse for $70, buys it back again for $80 and sells it again for $90. again for $80 and sells it again for $90. How much does Joe make or lose in the horse How much does Joe make or lose in the horse trading business? trading business?

3. POLYAS Problem-Solving Process: POLYAS Problem-Solving Process: Getting To Know The Problem: Getting To Know The Problem: This involves making sense of the context of the problem, what This involves making sense of the context of the problem, what information is given (needed and extraneous), and what it is that is being information is given (needed and extraneous), and what it is that is being asked? asked? Devising a Plan to Solve the Problem: Devising a Plan to Solve the Problem: Students should be encouraged to share possible plans and to discuss Students should be encouraged to share possible plans and to discuss alternative approaches. alternative approaches. ESTIMATION should be encouraged at this point. ESTIMATION should be encouraged at this point. Implementing a Solution Plan: Implementing a Solution Plan: (See strategies to follow) (See strategies to follow) Look Back and Beyond: Look Back and Beyond: Does the result correlate with the estimation? Were all conditions met and Does the result correlate with the estimation? Were all conditions met and accounted for? accounted for? Encourage students to extend the problem through What if questions. Encourage students to extend the problem through What if questions.

4. Problem: Problem: A family has three children, Adam, Beatrice and Caroline. Each child has been given a chore. Adam is to vacuum the carpets in the house every third day. Beatrice must take out the garbage every fourth day and Caroline must mow the lawn every sixth day. In August, on which day(s) will all the children be doing their chores together? A family has three children, Adam, Beatrice and Caroline. Each child has been given a chore. Adam is to vacuum the carpets in the house every third day. Beatrice must take out the garbage every fourth day and Caroline must mow the lawn every sixth day. In August, on which day(s) will all the children be doing their chores together? Problem: Problem: I have an unlimited supply of pennies, nickels and dimes in my pocket. If I take three coins out of my pocket at a time, how much could I have in my hand? I have an unlimited supply of pennies, nickels and dimes in my pocket. If I take three coins out of my pocket at a time, how much could I have in my hand? Problem: Problem: Roll the dice 20 times. Subtract the small number from the large number. If they are the same, count as zero. Plot your results on a line graph. Roll the dice 20 times. Subtract the small number from the large number. If they are the same, count as zero. Plot your results on a line graph.

5. Problem: Problem: You have been given a piece of land for a garden. You must fence this land in totally to keep rabbits out. Assuming you need 24 square metres for your garden, what is the least amount of fence you could use? You have been given a piece of land for a garden. You must fence this land in totally to keep rabbits out. Assuming you need 24 square metres for your garden, what is the least amount of fence you could use? Problem: Problem: You are working in a bicycle repair shop and are in charge of making tricycles and bicycles out of extra parts. You have 18 frames (some for tricycles, some for bicycles) and 46 wheels. Assuming that the wheels on bicycles and tricycles are the same size, how many of each type of cycle can you make (ensure you use all the parts you have). You are working in a bicycle repair shop and are in charge of making tricycles and bicycles out of extra parts. You have 18 frames (some for tricycles, some for bicycles) and 46 wheels. Assuming that the wheels on bicycles and tricycles are the same size, how many of each type of cycle can you make (ensure you use all the parts you have). Problem: Problem: In the bus loading area there are four school busses lined up, front to back, waiting for students to get on. Each school bus is three metres long and there is a two metre space between each one. How long is the loading area? In the bus loading area there are four school busses lined up, front to back, waiting for students to get on. Each school bus is three metres long and there is a two metre space between each one. How long is the loading area? Problem: Problem: A woman appeared on Who Wants to Be a Millionaire and won a large sum of money. When she arrived back home she met her three best friends, one at a time. To each friend she met she gave half of the money she had then. After all three meetings she had $8 000.00 remaining. How much did she win on the show? A woman appeared on Who Wants to Be a Millionaire and won a large sum of money. When she arrived back home she met her three best friends, one at a time. To each friend she met she gave half of the money she had then. After all three meetings she had $8 000.00 remaining. How much did she win on the show?

6. PROBLEM SOLVING STRATEGIES (Any one or more of these can, and should, be used for a variety of problems) PROBLEM SOLVING STRATEGIES (Any one or more of these can, and should, be used for a variety of problems) Dramatize (act it out) or create a model of the situation Dramatize (act it out) or create a model of the situation Draw a picture Draw a picture Construct a table or chart Construct a table or chart Find a pattern Find a pattern Solve a simpler problem Solve a simpler problem Guess and check Guess and check Work backwards Work backwards Consider all possibilities Consider all possibilities

7. LISTEN TO THEM LISTEN TO THEM TALK!!!!!!! TALK!!!!!!! LET STUDENTS LET STUDENTS DEVELOP THEIR DEVELOP THEIR OWN STRATEGIES! OWN STRATEGIES!

8. CHARACTERISTICS OF A GOOD (RICH) LESSON: (see: Flewelling (2002). Realizing a Vision of Tomorrows Classroom, Rich Tasks) CHARACTERISTICS OF A GOOD (RICH) LESSON: (see: Flewelling (2002). Realizing a Vision of Tomorrows Classroom, Rich Tasks) Curriculum relevance Curriculum relevance Student relevance Student relevance Authentic content and structure Authentic content and structure Flexible- for different levels Flexible- for different levels Problem solving and question posing Problem solving and question posing Inquiry/exploration/investigation/experimentation Inquiry/exploration/investigation/experimentation Communication Communication Reflect on learning Reflect on learning Creative Creative Go to: http://math.unipa.it/~grim/AFlewelling70-72 Go to: http://math.unipa.it/~grim/AFlewelling70-72

9. PROBLEM OF THE DAY/ WEEK/ MONTH/ YEAR? PROBLEM OF THE DAY/ WEEK/ MONTH/ YEAR? Assuming you have access to pennies, nickels, dimes and quarters, how many combinations of change can you make for a dollar? Assuming you have access to pennies, nickels, dimes and quarters, how many combinations of change can you make for a dollar? Come ready to talk about your method of solving this problem and your thinking while you did so! Come ready to talk about your method of solving this problem and your thinking while you did so!

10. NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS (NCTM) Ontario Association of Mathematics Educators (OAME) NATIONAL COUNCIL OF TEACHERS OF MATHEMATICS (NCTM) Ontario Association of Mathematics Educators (OAME) The primary professional organization for teacher The primary professional organization for teacher of mathematics of grades K-12 of mathematics of grades K-12 1989- released Curriculum and Evaluation 1989- released Curriculum and Evaluation Standards Standards 1991- released Professional Standards for 1991- released Professional Standards for Teaching Mathematics Teaching Mathematics

11. CURRICULUM STRANDS CURRICULUM STRANDS Number Sense and Numeration Number Sense and Numeration Geometry and Spatial Sense Geometry and Spatial Sense Measurement Measurement Patterning Patterning Data Management and Probability Data Management and Probability

12. THE ONTARIO CURRICULUM GRADES 1-8 (read p 1-9 of the mathematics curriculum ) THE ONTARIO CURRICULUM GRADES 1-8 (read p 1-9 of the mathematics curriculum ) FIVE STRANDS: FIVE STRANDS: Number Sense and Numeration Number Sense and Numeration Measurement Measurement Geometry and Spatial Sense Geometry and Spatial Sense Patterning and Algebra Patterning and Algebra Data Management and Probability Data Management and Probability

13. Number Sense and Numeration Number Sense and Numeration Counting, numeral representation, more Counting, numeral representation, more and/or less than, equal to, part-whole and/or less than, equal to, part-whole relationships, base ten relationships, base ten

14. Measurement Measurement Linear measure, perimeter, area, volume, Linear measure, perimeter, area, volume, mass, time, money, comparing sizes of mass, time, money, comparing sizes of objects, non-standard and standard units objects, non-standard and standard units of measure of measure

15. Geometry and Spatial Sense Geometry and Spatial Sense Simple and complex shapes (two and Simple and complex shapes (two and three dimensional), transformational three dimensional), transformational geometry (flips, slides, turns), attributes geometry (flips, slides, turns), attributes of shapes (vertices, sides, faces), of shapes (vertices, sides, faces), graphing coordinates graphing coordinates

16. Patterning and Algebra Patterning and Algebra Simple repeating patterns, growing Simple repeating patterns, growing patterns, shape designs, sets of patterns, shape designs, sets of numbers, patterns in art, graphs, data numbers, patterns in art, graphs, data collection, equations, relationships, collection, equations, relationships, variables variables

17. Data Management and Probability Data Management and Probability Describing and organizing graphs, Describing and organizing graphs, statistics, trends, estimations, rations, statistics, trends, estimations, rations, fractions, collecting, presenting and fractions, collecting, presenting and comparing data comparing data

18. MATHEMATICS EXPECTATIONS EXPECTATION: What a child should be able to demonstrate ACTIONS: OBJECTS: This is what a child should This is the content that do to demonstrate their learning, will be demonstrated, e.g. e.g. model, compare, build, equivalent fractions, place connect value

19. FIVE PROCESS STANDARDS FIVE PROCESS STANDARDS 1. Problem Solving 1. Problem Solving 2. Reasoning and Proof (conceptual vs. procedural) 2. Reasoning and Proof (conceptual vs. procedural) 3. Communication (oral, written, drawn, kinesthetic 3. Communication (oral, written, drawn, kinesthetic 4. Connections (within and outside mathematics 4. Connections (within and outside mathematics 5. Representation (symbols, diagrams, graphs, charts, pictures) 5. Representation (symbols, diagrams, graphs, charts, pictures)

20. SHIFTS IN CLASSROOM ENVIRONMENT SHIFTS IN CLASSROOM ENVIRONMENT Classrooms as communities and not just collection of individuals Classrooms as communities and not just collection of individuals Toward logic and mathematical evidence as Toward logic and mathematical evidence as verification (away from teacher as authority) verification (away from teacher as authority) Toward mathematical reasoning (concepts) and Toward mathematical reasoning (concepts) and away from memorization away from memorization Toward conjecturing, inventing, problem solving Toward conjecturing, inventing, problem solving and creating and away from mechanics of getting and creating and away from mechanics of getting the right answer the right answer Toward connecting mathematics to other disciplines Toward connecting mathematics to other disciplines