Problem-solving strategies Notes

Why Problem-Solving Is A Skill (Not A Trick)

1. In mathematics, a problem is any task where the solution method is not immediately obvious.
2. That is why some questions feel easy to one student and challenging to another.
3. Often the difference is not "being good at math", it is having reliable strategies and knowing when to use them.

Pólya's Four-Step Framework Organizes Almost Any Problem

The mathematician George Pólya described a four-step approach that works in mathematics and many other subjects:

1. Understand the problem
2. Devise a plan
3. Carry out your plan
4. Look back

Step 1: Understand The Problem Before You Calculate

Understanding is active. Before you solve, you should be able to state:

1. What do I need to find (the unknown)?
2. What do I already know (the givens) and what are their units?
3. Are there conditions (whole number, positive, maximum, minimum)?
4. Can I draw a diagram, make a table, or label a sketch?
5. Can I restate the question in my own words?

Step 2: Devise A Plan Using A Strategy That Fits

1. A plan is a decision about which method to try first.
2. Common strategies include:
   1. Guess and check
   2. Look for a pattern
   3. Eliminate possibilities
   4. Use a formula
   5. Solve an equation

1. Choosing a plan is often the hardest part.
2. You can make it easier by asking: "What structure do I see?" For example:
   1. If an unknown is inside an expression, you may want to form an equation.
   2. If something changes repeatedly over time, look for a pattern.
   3. If a diagram is involved, draw and label it, then look for relationships.

Step 3: Carry Out The Plan Clearly And Monitor Your Work

When you execute the plan:

1. Show each step so someone else can follow.
2. Check intermediate results (for example, negative time or negative length signals an error).
3. If you get stuck, do not keep repeating the same move. Go back to Step 2 and choose another strategy.

Step 4: Look Back To Confirm Meaning (Not Just Arithmetic)

Looking back includes:

1. Does the answer make sense in the context?
2. Are the units correct?
3. Does the answer meet any conditions (for example, whole number, positive)?
4. Could there be another solution you missed?

Core Problem-Solving Strategies And When To Use Them

1. A good strategy toolbox helps you move forward when you feel stuck.
2. The goal is not to memorize a long list, but to recognize when each tool is useful.

Strategy 1: Represent And Simplify The Information

1. Many real-life problems are difficult because the information is messy.
2. Simplification means keeping the important structure while reducing clutter.
3. Useful representations include:
   1. a labeled diagram
   2. a table (for patterns or trials)
   3. a timeline (start and finish times)
   4. defining variables with meanings and units

Strategy 2: Take 1 (Solve A Simpler Version)

This is a powerful method for linear situations (constant rate or constant price).

Strategy 3: Guess And Check (Systematically)

Guess-and-check is valid if it is organized:

1. start with a reasonable guess
2. record results in a table
3. adjust the next guess based on what the previous result tells you

Strategy 4: Look For A Pattern (Especially In Growth Problems)

Pattern recognition is common in:

1. doubling and halving
2. repeated operations
3. sequences
4. counting shapes in diagrams

Strategy 5: Eliminate Possibilities Using Constraints

1. When there are multiple conditions, write them clearly and rule out options that violate them.
2. Typical constraints include:
   1. must be positive (lengths, time)
   2. must be an integer (number of items)
   3. must be between 0% and 100% (percentages)

Strategy 6: Use A Formula After You Identify Quantities

Formulas are shortcuts, but only after you know what each quantity represents. Common examples:

1. $v=\frac{d}{t}$ for average speed
2. percentage relationships (part, whole, percent)
3. area and volume formulas

Strategy 7: Solve An Equation By Defining Variables Clearly

Many word problems become straightforward when you translate relationships into algebra.

Common Causes Of Incorrect Solutions And How To Prevent Them

Two frequent causes of incorrect solutions are:

1. Starting calculations without a plan
2. Not checking whether the answer makes sense

A small routine helps prevent both.

A Mini-Checklist For Multi-Step Problems

1. Write what you must find (one sentence).
2. List the given information (with units).
3. Choose a strategy and name it (equation, pattern, formula, etc.).
4. Do the mathematics carefully.
5. Look back: units, size, and conditions.

Building Independence: What To Do When You Are Stuck

When you feel stuck, use a structured reset rather than guessing randomly:

1. Return to Step 1 and restate the problem.
2. Identify unknowns, knowns, and constraints.
3. Try a new representation (diagram, table, timeline).
4. Switch strategy (for example, from guessing to forming equations).