Problem-Solving Strategies in Math: Methods and Examples

What are problem-solving strategies in math?

Problem-solving strategies are organized frameworks used to approach and solve math word problems. Rather than guessing an answer, these strategies provide a logical path from reading the question to verifying the final result.

A widely used framework divides the problem-solving process into four distinct steps. Following these steps helps build confidence and accuracy in mathematics.

1. Understand the problem: Read carefully, identify what is known, and clarify what needs to be found.
2. Make a plan: Choose an appropriate mathematical strategy, such as drawing a diagram or writing an equation.
3. Carry out the plan: Perform the calculations carefully and record the steps.
4. Look back: Check the answer to ensure it makes sense in the context of the original question.

Understand the question

The first phase of problem solving is making sense of the information provided. Highlighting key phrases can be helpful, but learners must read the entire context before deciding what math to do.

It is common to look for isolated keywords to guess an operation, such as assuming that the word "more" always means addition. However, keywords can be misleading. Consider the phrase, "Alex has $10$ apples, which is $3$ more than Jamie." This actually requires subtraction to find Jamie's total.

To avoid mistakes, summarize the problem in your own words. Identify the starting values, the conditions, and the unknown quantity before translating words into math.

Make a plan

Once the problem is clear, the next step is selecting a strategy. Different types of problems respond better to different methods. Many multi-step word problems require using more than one strategy in sequence.

The table below outlines common problem characteristics and a recommended strategy for each:

Draw a representation

Visualizing a problem is often the most powerful way to solve it. Drawing a diagram, sketch, or formal model translates abstract text into a concrete image. When quantities are compared or partitioned, drawing bar models in math makes the relationships visible.

Example 1: Using a visual representation

Question: A piece of ribbon is $45 \text{ cm}$ long. It is cut into two pieces so that one piece is twice as long as the other. How long is the longer piece?

Method:

1. Draw a bar model showing the total divided into $3$ equal parts, because one piece is one part and the other is two parts.
2. Find the value of one part by dividing the total length by $3$. $45 \div 3 = 15$.
3. Multiply the value of one part by $2$ to find the length of the longer piece. $15 \times 2 = 30$.

Answer: The longer piece is $30 \text{ cm}$ long.

Check: The shorter piece is $15 \text{ cm}$. $30$ is twice as long as $15$, and $30 + 15 = 45$. The answer is correct.

Work forwards or backwards

Most problems are solved by working forwards: starting with known values and applying operations to reach an unknown final result. However, when a problem provides the final result and asks for the original starting value, it is usually more efficient to work backwards.

Working backwards means starting with the final answer and reversing every operation step-by-step until the original value is recovered. The inverse of addition is subtraction, and the inverse of multiplication is division.

Example 2: Working backwards to find a start value

Question: Maria spends $15 \text{ dollars}$ on a book. Then, she spends half of her remaining money on lunch. She has $12 \text{ dollars}$ left. How much money did she start with?

Method:

1. Identify the final amount: $12 \text{ dollars}$.
2. Reverse the last action. She spent half her money, so the inverse is multiplying the remainder by $2$. $12 \times 2 = 24$.
3. Reverse the first action. She spent $15 \text{ dollars}$, so the inverse is adding $15$. $24 + 15 = 39$.

Answer: Maria started with $39 \text{ dollars}$.

Check: Start with $39$. Subtract $15$ to get $24$. Take half of $24$ to get $12$. The final amount matches the problem.

Look for patterns and structure

When a problem describes a sequence of events or a growing shape, the best strategy is often to look for a pattern. Organizing the information into a table helps reveal how the numbers change from one step to the next.

Once the rule for the pattern is identified, it can be extended to find future values without drawing every intermediate step.

Example 3: Extending a sequence

Question: A theater has $10$ seats in the first row, $12$ seats in the second row, and $14$ seats in the third row. If this pattern continues, how many seats are in the eighth row?

Method:

1. Write down the sequence of seats for the first few rows: $10, 12, 14$.
2. Identify the pattern rule. Each row has $2$ more seats than the row before it.
3. Apply the pattern up to the eighth row. Row 4 is $16$, Row 5 is $18$, Row 6 is $20$, Row 7 is $22$, and Row 8 is $24$.

Answer: There are $24$ seats in the eighth row.

Check: The eighth row is $7$ rows after the first row. Adding $7$ increments of $2$ gives $14$. $10 + 14 = 24$.

Check and explain

The final step in any problem-solving process is verifying the solution. A completed calculation is not automatically correct; it must make sense in the real world. Checking reasonableness of answers ensures that minor arithmetic errors do not lead to impossible conclusions.

Common checking strategies include:

• Using estimation: Round the original numbers and quickly estimate the result. If the calculated exact answer is very far from the estimate, there is likely an error.
• Using inverse operations: Recalculate the problem backward to see if you return to the starting numbers.
• Plugging in values: For algebraic equations, substitute the found value back into the original equation to ensure both sides remain equal.

Frequently asked questions

What are the four steps of problem-solving?

The four standard steps are to understand the problem, make a plan, carry out the plan, and look back to check the answer.

Why is drawing a model an effective strategy?

Drawing a model translates abstract numbers and words into a visual representation. It makes the mathematical relationships clear, such as showing which parts make up a total or how quantities compare to one another.

When should I use the working backwards strategy?

Use the working backwards strategy when a problem gives you the final outcome after a series of changes and asks you to find the original starting value.