The Problem Solving and Communication Activity Series, 2008-2009

This Math Forum program is designed to help students expand their repertoire of ways to approach challenging problems. Over the course of the year, we covered many of the fundamental problem solving strategies in conjunction with the Problems of the Week. Please visit the Activity Series main page for an overview of the program.

This page shows the alignments of PoWs to problem-solving strategies during the first year we offered the activity series. Each problem focuses on a particular strategy in developmental sequence, and we provide a document describing activities to do with your students and examples of typical student responses to the problem. The problem-specific documents both illustrate the activities and help you anticipate ideas that might come up in your class. We cycle through the strategies multiple times during the year so that students can get better at problem solving over the course of the Activity Series and the school year.

The documents below are all in PDF format. If you have a Class membership or higher, you can see all of the supporting documents for all of the 2008-09 PoWs, including links to the problems, at the 2008-2009 PoW Teacher Resources Page, also known as the Teacher Support pages.

Round 0: Introducing the Activity Series

 

The introductory document contains a series of activities designed to get students thinking about what good problem solvers do, and how to communicate their thinking by writing (or talking) mathematically.

Problem Solving and Communication Activity Series: Program Description & Introduction

 

Round 1: Understanding the Problem

 

What does it mean to fully understand a problem, and how does it help students find solution paths and build confidence? Included in these documents are several activities that support students to develop strategies for understanding challenging math problems, along with facilitation suggestions for teachers.

  August 18 to August 31 August 25 to September 7
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Subway Tokens 
 Operation Warm Up 
 Cadence 
 Points, Lines, & Planes 

 

Round 2: Understanding the Problem II

 

We will continue to explore how to understand what the problem is asking by providing a second week's worth of activities for each of the four new PoWs.

  September 1 to September 14 September 8 to September 21
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Zelma's ZIP Code 
 Filling Glasses 
 Tiling Triangles 
 Divided Rectangle 

 

Round 3: Guess and Check

 

Guess and check is an important (and popular) problem-solving strategy, though it often gets a bad rap and may not be developed into the strong and powerful resource it could be. The guess and check strategy has at least three purposes: (1) to understand a problem thoroughly, (2) to home in on a solution, and (3) to discover efficient ways to jump to a solution by noticing patterns and developing related algebraic representations.

  September 15 to September 28 September 22 to October 5
  FunPoW AlgPoW PreAlgPoW GeoPoW
   You Think Your Teacher is Tough! 
 Eating Contest 
 Caliyah's Cart 
 Dilating a Quadrilateral 

 

Round 4: Solve a Simpler Problem

 

Solve a Simpler Problem is a technique that can be used in several ways to solve challenging problems. In some situations you can see how to work the problem with easier numbers. This may show you an approach that you can try with the more difficult numbers. Second, you can choose to break the original problem into smaller steps, finding answers for parts of the problem, and then putting those together for the whole solution. Finally, students may see a way to change this hard problem into one that they have solved before.

  September 29 to October 12 October 6 to October 19
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Order in the Court 
 Sierpinski Stages 
 Women's Walkway
 Where's Juanita Walking? 

 

Round 5: Making a Table

 

The Tables and Patterns strategy is a way to organize your problem solving that makes it easier to explore patterns in the calculations and results. It is often used after some initial work on the problem using Understanding the Problem or Guess and Check strategies. Tables can be used to efficiently home in on answers, or you can use tables to organize the logic of your calculations and make explicit the relationships between quantities in the problem. As you may have seen in the Simpler Problem strategy, tables can help put different iterations in order and compare them. Tables can take the form of simple t-tables to very complex spreadsheets. Spreadsheets and other related software are especially efficient because they can be used to rearrange your work for different comparisons without having to write it all over again.

  October 13 to October 26 October 20 to November 2
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Growing Worms 
 To Be or Not To Be 
 Measuring Melons 
 The Shortest Possible Side 

 

Round 6: Understanding the Problem (Revisited)

 

What does it mean to fully understand a problem, and how does it help students find solution paths and build confidence? This time around we have expanded activities to elicit relevant knowledge, recognize implications, and make drawings that focus on the key mathematical information.

  October 27 to November 9 November 3 to November 16
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Wooden Legs 
 Voter Turnout 
 Ostrich Llama Count 
 A Minor Problem 

 

Round 7: Guess and Check (Revisited)

 

In our previous round of Guess and Check, we focused on using guess and check to understand the problem and to home in on a solution. In this round we delve deeper into the uses of guess and check and present activities that help students:

  • Test their solutions in a variety of ways.
  • Use guess and check to ask good questions and get "unstuck."
  • Figure out good guesses or problem approaches.
  November 10 to November 23 November 17 to November 30
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Nuts in a Bowl 
 A Championship Season 
 Two Heads Are Better Than One 
 Triple Tango 

 

Round 8: Solve a Simpler Problem (Revisited)

 

Solve a Simpler Problem is a technique that can be used in several ways to solve challenging problems. In some situations you can see how to work the problem with easier numbers. This may show you an approach that you can try with the more difficult numbers. Second, you can choose to break the original problem into smaller steps, finding answers for parts of the problem, and then putting those together for the whole solution. Finally, students may see a way to change this hard problem into one that they have solved before.

  November 24 to December 7 December 1 to December 14
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Building Barns 
 How Many Ties? 
 Lillian's Lines 
 Pythagorean Circles 

 

Round 9: Making a Table (Revisited)

 

The Tables and Patterns strategy is a way to organize your problem solving that makes it easier to explore patterns in the calculations and results. It is often used after some initial work on the problem using Understanding the Problem or Guess and Check strategies. Tables can be used to efficiently home in on answers, or you can use tables to organize the logic of your calculations and make explicit the relationships between quantities in the problem. As you may have seen in the Simpler Problem strategy, tables can help put different iterations in order and compare them. Tables can take the form of simple t-tables to very complex spreadsheets. Spreadsheets and other related software are especially efficient because they can be used to rearrange your work for different comparisons without having to write it all over again.

  December 8 to December 28 December 15 to January 4
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Rugby Rules! 
 Two By Canoe 
 Gas Guzzlers 
 Circle Derby 

 

Round 10: Cases

 

Case-based reasoning helps problem solvers to understand the problem, work towards a solution, surface interesting mathematics, and verify the robustness of their solutions. To understand the problem, problem solvers might test interesting or representative cases and think about the different outcomes they see. When solving the problem, they might use cases to consider when certain outcomes will occur, or to narrow down the possibilities they have to investigate. Some problems have different answers for different cases. Exploring different cases can lead to questions that problem solvers might explore further, like, "what would happen if I used a negative number?" or, "would this work for obtuse triangles, too?" Finally, when determining whether a possible solution is correct, good problem solvers test their solution using multiple cases, especially cases that they know behave differently.

  December 29 to January 11 January 5 to January 18
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Change You Can't Count On 
 A Slippery Slope 
 Unknown Numbers 
 Making the Biggest Triangle 

 

Round 11: Logical Reasoning

 

Logic is an inherent part of the mathematical problem solving process and was used in some ways through our Activity Series already. However, some problems depend more on logic than on purely mathematical manipulations. Logic can help us find solutions when it looks as if we are unable to solve them based on our equations. Even with problems that are primarily solved through calculations, the questions and techniques of logical reasoning can help us organize and find efficient solution approaches to problems. In this sense Logical Reasoning is particularly useful in combination with approaches such as "Noticing and Wonderings" or "PoW IQ" from Understanding the Problem (Rounds 1, 2, or 6).

  January 12 to January 25 January 19 to February 1
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Dashing Through the Snow 
 Symbol Logic 
 Movie Seating 
 Two Rectangles 

 

Round 12: Change the Representation

 

All math problems, whether they are word problems, arithmetic problems, equations to solve, etc., come to us in a particular representation. Word problems are represented in story form, using words and often referencing a particular context. Arithmetic problems are represented numerically. Equations are represented using mathematical symbols. Each representation has benefits to the problem solver. For example, word problems allow students to apply their knowledge of the given context, which can allow them to check that their approaches are reasonable. Numeric and symbolic representations can make it easy for students to manipulate objects in the problem, and to quickly see patterns. Visual and physical representations, such as manipulatives, diagrams, and graphs, can often help students gain new insights into the problem and provide them with additional tools for solving it. Changing the representation can mean use of a different form of representation (e.g. using a line drawing for a word problem) or it can mean trying different ways of presenting the information in the same form (e.g. rewriting all of the numbers as fractions with a numerator of 1). Considering multiple representations and choosing representations that fit the problem well are important problem-solving skills.

  January 26 to February 8 February 2 to February 15
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Farmer in the Dilemma 
 Pressed for Time 
 Top Notch Nose Contest 
 Making a Six-Pointed Star 

 

Round 13: Make a Mathematical Model

 

A mathematical model is a way to describe a situation, usually real-world, using numeric and mathematical relationships. Mathematical models usually have inputs, operations on those inputs, certain parameters or constants that make the operations fit the particular situation, and outputs that result from performing the operations on the inputs. Sometimes in problem solving, coming up with the mathematical model to use is at the heart of the problem. Problem-solvers are engaged in noticing quantities and relationships, selecting operations to describe the relationships, and fitting those operations to the specific scenario by setting parameters. Other times, the operations and relationships are given in the problem, and the problem solver's job is to organize the information and apply it to determine a final answer. In either case, identifying quantities and relationships, and recording information as mathematically as possible are key components of making mathematical models.

  February 9 to February 22 February 16 to March 1
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Toys to Go 
 A Phone-y Deal? 
 Lincoln's Top Hat 
The Coordinated Quadrilateral

 

Round 14: Working Backwards

 

Working backwards is a particularly useful problem-solving strategy when you can clearly define the goal or end state of the problem, and you know a sequence of operations that were used in the problem. Reversing the operations and working backwards from the goal helps problem-solvers to describe the initial conditions or the most efficient path to the goal state. Working backwards is often applied to logic problems, like the famous one about crossing the river with a cabbage, a goat, and a wolf, in which you know the goal state (everything on the opposite side of the river), and you know what the legal moves are (rowing one animal across the river without leaving the cabbage with the goat or the wolf with the goat). You can work backwards from the goal, asking yourself, what must have been the last animal rowed across the river? What must have happened just before that?

Another sort of problem that working backwards can be applied to is a problem involving operations on a quantity, in which you know the final outcome after all the operations have been applied, and you need to find out the initial quantity. In these sorts of problems, it's useful to play the situation backwards, performing the inverse of each operation on the known, ending quantity until you are left with the initial quantity. This process of inverting operations is very similar to the process of solving an algebraic equation by "undoing" what's been done to the variable.

  February 23 to March 8 March 2 to March 15
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Chicken Little's Eggs 
 Swapping Marbles 
 Mixing Marbles 
Snail's Trail

 

Round 15: Planning and Reflecting

 

As students get comfortable with more strategies, they begin to recognize and use multiple approaches in their problem-solving process. Following Polya (1945), it is common to break the problem-solving process down into four phases: Understanding the Problem, Making a Plan, Carrying Out the Plan, and Checking/Reflecting. Two of the most common issues in problem solving are (1) forgetting part of the problem or the ideas you have discovered that might be useful and (2) getting stuck, trying the same thing over and over without making new progress. Planning and reflecting can help you both solve problems and learn from your experience. What does it mean to get good at making a plan? How do I know when to start carrying out the plan? How do I know if I'm on a dead-end path? How do I effectively check my work?

  March 9 to March 22 March 16 to March 29
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Sharing Birthdays 
 Fraction Debate 
 Peeling Potatoes 
 Building a Regular Hexagon 

 

Round 16: Getting Unstuck

 

If you never get stuck, then you are not solving interesting enough problems. Getting stuck (and, we hope, getting "unstuck"), is at the heart of problem solving. Challenging problems require that they be represented in many ways, approached with a variety of strategies, and checked again and again and again.

  March 23 to April 5 March 30 to April 12
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Zoo Train 
 Polynomial Power 
 Dollar Dealing 
 Pondering Pantographs 

 

Round 17: Play

 

When students do the Noticing/Wondering activity, we often have them try to group their noticings into "quantities" and "relationships". With a little practice, students get adept at finding the quantities and the relationships that are explicitly stated in the problem. However, interesting math problems usually have deeper layers of relationships that only emerge as problem solvers "play" with the relationships and quantities.

In the recent activities focusing on Planning and Getting Unstuck, we began to highlight some of the phases of problem solving, and to show how many of the activities in this series can be used to explore relationships as you begin problem solving or if you get stuck along the way.

Continuing in this vein, Round 17 focuses on some of the ways problem solvers play with relationships and explore patterns before they delve deeply into a single problem-solving strategy. In order to make clear different aspects of problem solving, we've broken the "play" process out somewhat artificially - expert problem solvers move back and forth fluidly between understanding the problem, playing with relationships, and carrying out strategies. However, for purposes of illustration, we think it will be useful to focus on those phases separately.

  April 6 to April 19 April 13 to April 26
  FunPoW AlgPoW PreAlgPoW GeoPoW
   She Counts Seashells 
 The Bouncing Ball 
 Bartering for Bananas 
 Regional Ratios 

 

Round 18: Change the Representation (Revisited)

 

We return to the strategy explored in Round 12.

  April 20 to May 3 April 27 to May 10
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Sports Weigh In 
 Field Trip 
 Suli's Sense 
 Flying High 

 

Round 19: Make a Mathematical Model (Revisited)

 

We return to the strategy explored in Round 13.

  May 4 to May 17 May 11 to May 24
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Four Dates 
 The ABCs of Exponents 
 Waste Not, Want Not 
 A Circle Inscribed in an Isosceles Triangle 

 

Round 20: Understanding the Problem - Wondering

 

In our third time through Understanding the Problem, we focus on wondering - on those questions problem solvers ask themselves as they try to understand the problem and find a solution path. It is usually the wondering that leads us to a solution. First we notice. Then we wonder about meaning and implications and possibilities. Then we pursue those questions and work our way to a solution.

As we've worked on noticing and wondering with students, we've noticed that their wondering is often interesting but could be more mathematical. Persistence is an important part of the process, for instance investigating aspects we wonder about, even if they don't seem promising. It helps to have a repertoire of reflective questions (wonderings) that help us get unstuck, connect to prior knowledge, and use our noticings to solve the problem. So, this round we have shared a lot of questions that have been useful to us.

  May 18 to May 31 May 25 to June 7
  FunPoW AlgPoW PreAlgPoW GeoPoW
   Moons, By Jove! 
 Factor Bingo 
 Sing a Song of Sunscreen 
 In a Pickle 

 



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