Bridging topics (typical)
Problem Solving & Puzzles
Transition Year
- ✓By the end of this lesson students will be able to identify and apply various problem-solving strategies.
- ✓By the end of this lesson students will be able to develop critical thinking and logical reasoning skills.
- ✓By the end of this lesson students will be able to break down complex problems into manageable steps.
- ✓By the end of this lesson students will be able to communicate solutions clearly and justify their reasoning.
- ✓By the end of this lesson students will be able to appreciate the role of persistence in solving challenging problems.
Key concepts
Problem solving is the process of finding solutions to difficult or complex issues. In maths, it involves understanding the problem, devising a plan, carrying out the plan, and reviewing the solution. It's not just about getting the right answer, but also about the process of how you get there.
George Polya's four-step process is a widely recognised framework for approaching mathematical problems: 1. **Understand the Problem**: What is the unknown? What information is given? What are the conditions? Can you restate the problem in your own words? 2. **Devise a Plan**: How can you relate the given information to the unknown? Have you seen a similar problem before? Can you think of a strategy (e.g., draw a diagram, look for a pattern, work backwards)? 3. **Carry out the Plan**: Execute your plan carefully, checking each step. If your plan isn't working, don't be afraid to try a different approach. 4. **Look Back**: Examine your solution. Does it make sense? Can you check the result? Can you derive the result differently? Can you use the method for another problem?
There are many strategies you can employ: * **Draw a Diagram/Picture**: Visualise the problem to help understand relationships. * **Look for a Pattern**: Identify sequences, repetitions, or trends in data. * **Make a List/Table**: Organise information systematically to spot connections. * **Guess and Check (Trial and Error)**: Make an educated guess and refine it based on the outcome. * **Work Backwards**: Start from the desired end result and reverse the steps to find the beginning. * **Simplify the Problem**: Solve a smaller, easier version of the problem first to gain insight. * **Use Algebra/Equations**: Translate the problem into mathematical expressions and solve them. * **Logical Deduction**: Use given facts to infer new information and eliminate possibilities.
Puzzles are problems designed to test ingenuity, logic, and creative thinking rather than direct application of formulas. They are an excellent way to practise and develop your problem-solving skills in an engaging way, often requiring you to think 'outside the box'.
Key facts to remember
- 1Problem-solving is a fundamental skill that improves significantly with consistent practice.
- 2Polya's Four-Step Process (Understand, Plan, Do, Look Back) provides a structured and effective approach to any problem.
- 3There are multiple strategies for problem-solving; choose the one that best suits the specific problem.
- 4Don't be afraid to try different approaches if your initial plan doesn't yield a solution.
- 5Clear communication of your steps and reasoning is crucial, not just the final answer.
- 6Puzzles are excellent tools for developing logical thinking and creative problem-solving skills.
Worked examples
Example 1
Three friends, Aoife, Brian, and Ciara, each have a favourite colour: blue, green, or red. Aoife does not like red. Brian's favourite colour starts with the same letter as his name. What is Ciara's favourite colour?
Answer
Ciara's favourite colour is Red.
For logic puzzles, creating a table to track possibilities can be very helpful.
Example 2
A number is multiplied by 5, then 7 is subtracted from the result. If the final answer is 33, what was the original number?
Answer
The original number was 8.
When working backwards, remember to reverse the operation (e.g., addition becomes subtraction, multiplication becomes division).
Example 3
A sequence of patterns is made using matchsticks. Pattern 1 uses 3 matchsticks, Pattern 2 uses 5 matchsticks, and Pattern 3 uses 7 matchsticks. How many matchsticks would be needed for Pattern 10?
Answer
21 matchsticks.
Identifying the type of sequence (e.g., arithmetic, geometric) can help you find a general formula to solve for any term.
Common mistakes
- ✗Not fully understanding the question or misinterpreting key information before attempting to solve it.
- ✗Jumping straight to calculations without first devising a clear plan or strategy.
- ✗Giving up too quickly when a problem seems challenging, rather than trying alternative approaches.
- ✗Failing to check the final answer to ensure it is reasonable and makes sense in the context of the problem.
- ✗Ignoring specific conditions, units, or constraints mentioned in the problem statement.
Exam tips
- ★Read the question carefully and thoroughly. Highlight or underline key information, numbers, and what exactly is being asked.
- ★If you get stuck, don't panic. Take a deep breath, re-read the question, and try a different problem-solving strategy.
- ★Always show all your working steps clearly. This not only helps you organise your thoughts but also allows for partial credit if your final answer is incorrect.
- ★Allocate your time wisely. If a problem is proving very difficult, move on and come back to it later if time permits, rather than spending too long on one question.
- ★Practise a wide variety of problem types regularly to build confidence and familiarity with different strategies.
Ready to practise?
Try a problem on this topic
Snap a photo or type a question — get step-by-step working instantly.