Problem-Solving Strategies in Math

Problem-Solving Strategies

• Read the question carefully: Identify the key information, the unknowns, and the relationships between them.
• Draw a diagram: Visual representations can clarify complex relationships and help to identify connections.
• Use variables: Represent unknown quantities with letters (e.g., 'x'). Formulate equations based on the problem's conditions.
• Break down the problem: Divide the problem into smaller, more manageable parts.
• Look for patterns: Recurring elements or trends can lead to solutions.
• Check your answer: Ensure it makes sense in the context of the problem. Verify the units and the reasonableness of the result.

Problem Types & Examples

• Simultaneous Equations:
  • Solve a system of two or more equations with multiple variables.
  • Example: "Two apples and three bananas cost £2.50. Three apples and two bananas cost £2.00. Find the cost of one apple and one banana."
• Linear Sequences and their Applications: -Identify the term-to-term rule or the nth term of a sequence.
  • Example: "A sequence starts with 3, 7, 11, 15... Find the 20th term and determine how many terms are needed to reach 100."
• Quadratic Equations: -Formulate and solve quadratic equations, which involve squared terms.
  • Example: "The area of a rectangle is 24cm². If the length is 2cm more than the width, determine the length and width."
• Inequalities: -Solve and represent inequalities on number lines including compound inequalities, and their applications
  • Example: "A farmer has a budget of £100 to buy feed for chickens and pigs. Chickens feed costs £2 per bag and pig feed costs £5 per bag. Write an inequality to represent how much of each type the farmer can buy."
• Ratio and Proportion: -Determine relationships between quantities through proportion or ratio and their applications to word problems.
  • Example: "A recipe for cookies requires 2 cups of flour and 1 cup of sugar. Sarah wants to make a larger batch using 6 cups of flour. How much sugar will she need?"
• Trigonometry: -Use trigonometric ratios (sin, cos, tan) to find missing sides or angles in right-angled triangles.
  • Example: "A ladder leaning against a wall makes a 70-degree angle with the ground. If the ladder is 5 meters long, how high up the wall does it reach?"
• Geometry: -Apply geometrical properties (angles, areas, volumes) to solve word problems.
  • Example: "A triangle has sides of length 5, 12, and 13. Determine if it is a right-angled triangle. What is its area?"
• Indices & Surds: -Use laws of indices and surds in problem solving.
  • Example: "The area of a square is √48 cm². Determine the side length"
• Algebraic Fractions: -Manipulating algebraic fractions in problem-solving scenarios.
  • Example "Given the time taken for car A to go a distance 'x'. Using a constant speed. Calculate the time taken for car B to travel the distance 'x', given the speed of car B is different to car A"
• Compound Interest: -Word problems related to calculating interest over time compounding periods
  • Example The principle sum of £1000 gains interest compounded yearly. Calculate the amount after 3 years, given an interest rate of 5%"
• Reverse Percentages: -Reverse percentage problems including profit, loss, increase decrease
  • Example A shop discounts a product by 20%. The new selling price is £40. Determine the original price.
• Combined Topics: -Problems combining multiple topics for a more complex challenge.
  • Example: "A farmer wants to fence a rectangular field with 100 metres of fencing. What dimensions give the maximum area?"

Key Skills

• Equation Formation: Translating word descriptions into algebraic expressions and equations.
• Substitution: Substituting numerical values into algebraic expressions and equations.
• Simplification: Simplifying algebraic expressions.
• Equation Solving: Using appropriate methods (e.g., solving simultaneous equations).

Higher Tier Considerations

• More challenging problem structures: Problems may require multiple steps or the combination of several mathematical concepts.
• Abstract reasoning: Problems might demand identifying hidden relationships or patterns.
• Greater emphasis on modelling and application: Problems might involve real-world contexts or situations requiring thoughtful application of mathematical concepts.

This quiz focuses on problem-solving strategies used in mathematics, particularly through techniques like reading questions carefully, breaking down problems, and using variables. It includes examples of simultaneous equations and linear sequences to illustrate these strategies in action.