Problem-Solution Essays

Problem-Solution Essays in mathematics are structured written responses where students identify a mathematical problem, explain its significance, and systematically demonstrate how to solve it. This form of mathematical communication is essential in Writing Task 2 for Lower Secondary Mathematics, as it develops critical thinking, logical reasoning, and the ability to articulate mathematical processes clearly.

In the context of mathematics education, problem-solution essays go beyond simply calculating answers. They require students to contextualize problems, explain their reasoning, show their working systematically, and verify their solutions. This skill is invaluable because mathematics is fundamentally about solving problems, whether they involve algebra, geometry, statistics, or real-world applications. Being able to communicate your mathematical thinking clearly demonstrates deep understanding and is a skill that examiners assess rigorously.

These essays also prepare students for higher-level mathematics where justification and proof become increasingly important. By mastering problem-solution essays now, students develop the foundation for mathematical argumentation, learn to identify when solutions are reasonable, and build confidence in tackling unfamiliar problems. This writing task assesses not just computational skills but also comprehension, analysis, and communication—all essential mathematical competencies.

Problem Statement: A clear description of the mathematical challenge or question that needs to be addressed, including all given information, constraints, and what needs to be found.

Solution Strategy: The planned approach or method chosen to solve the problem, which might involve selecting appropriate formulas, operations, or problem-solving techniques.

Working/Workings: The step-by-step calculations, transformations, or logical progressions shown in solving the problem, demonstrating how you moved from the given information to the final answer.

Mathematical Reasoning: The logical thinking process used to justify why a particular approach or step is valid, often involving mathematical rules, properties, or theorems.

Verification: The process of checking whether the obtained solution is correct by substituting back into the original problem, using alternative methods, or checking if the answer makes sense in context.

Context/Application: The real-world scenario or practical situation in which a mathematical problem is embedded, requiring interpretation and modeling skills.

Unknown/Variable: The quantity or value that needs to be determined in the problem, often represented by letters such as x, y, or specific descriptive terms.

Given Information: The facts, values, measurements, or conditions provided in the problem that serve as the starting point for the solution.

Structure of Problem-Solution Essays

A well-constructed mathematical problem-solution essay follows a clear logical structure consisting of four essential components. First, the problem identification section clearly restates what needs to be solved and identifies all given information. This demonstrates comprehension and ensures you're answering the correct question.

Second, the planning and strategy section explains which approach, formula, or method you will use and why it's appropriate. For example, if solving a word problem about consecutive numbers, you might explain: "I will use algebra, letting x represent the first number, because the relationship between the numbers can be expressed as an equation."

Third, the implementation and working section contains your systematic step-by-step solution. Each step should follow logically from the previous one, with clear mathematical operations shown. This is where calculations occur, equations are solved, or geometric constructions are completed. Clarity is paramount—an examiner should be able to follow your thinking without confusion.

Finally, the conclusion and verification section states your final answer clearly, checks its reasonableness, and may verify it through substitution or an alternative method. For instance, if you calculated that a rectangle's length is 12 cm, you might verify by checking that this length with the given width produces the stated area.

Mathematical Communication Principles

Precision in mathematical writing means using exact terminology and notation. Instead of writing "the big number," write "the larger value" or use proper notation like "x₂ > x₁." Every mathematical symbol should be used correctly: = means "equals," ≈ means "approximately equals," and these are not interchangeable.

Coherence ensures that your solution flows logically from one step to the next. Use transitional phrases like "therefore," "consequently," "substituting this value," or "applying the distributive property" to guide the reader through your reasoning. Each new line of working should clearly follow from what came before.

Completeness requires showing sufficient detail without unnecessary repetition. While you should show all significant steps, you don't need to write "2 + 2 = 4" if this is a minor part of a larger calculation. However, any step involving the main variable or a key transformation should be explicit.

Problem-Solving Strategies

The algebraic approach is powerful for problems involving unknowns and relationships. Define variables clearly, translate words into equations, and solve systematically. For example: "Let n be the number of students. Then 5n + 3 represents the total number of pencils needed."

The visual/diagrammatic approach uses drawings, charts, or geometric representations. This works excellently for geometry problems, ratio problems, or situations involving spatial relationships. Always label diagrams clearly with measurements, angles, or variables.

The working backwards strategy starts with the desired outcome and reverses operations. This is particularly useful for problems involving sequences of operations, such as: "If a number is doubled and then 7 is added to get 23, what is the original number?"

Example 1: Algebraic Word Problem

Problem: Sarah has three times as many books as Michael. Together they have 48 books. How many books does each person have?

Solution Essay:

Understanding the Problem: We need to find how many books Sarah and Michael each have. We know that Sarah has three times Michael's amount, and their combined total is 48 books. Let me define variables to represent these quantities.

Strategy: I will use algebra to solve this problem. Let m represent the number of books Michael has. Since Sarah has three times as many, Sarah has 3m books. Their total is 48 books, which gives me an equation I can solve.

Working: Let m = number of Michael's books Then 3m = number of Sarah's books

Setting up the equation: m + 3m = 48 4m = 48 m = 48 ÷ 4 m = 12

Therefore, Michael has 12 books. Sarah has 3m = 3 × 12 = 36 books

Verification: Let me check this answer. Michael has 12 books and Sarah has 36 books.

• Is Sarah's amount three times Michael's? 36 = 3 × 12 ✓
• Do they total 48? 12 + 36 = 48 ✓

Conclusion: Michael has 12 books and Sarah has 36 books. This solution satisfies both conditions given in the problem.

Example 2: Geometric Problem

Problem: A rectangular garden has a perimeter of 36 meters. Its length is 4 meters more than its width. Find the dimensions of the garden.

Solution Essay:

Problem Analysis: I need to find both the length and width of a rectangle. The given information includes the perimeter (36 m) and the relationship between length and width (length is 4 m greater than width).

Approach: Since the dimensions are related, I'll use algebra. Let w represent the width in meters. Then the length is (w + 4) meters. The perimeter formula for a rectangle is P = 2l + 2w, which I can use to create an equation.

Solution Steps: Let w = width (in meters) Then length = w + 4

Using the perimeter formula: Perimeter = 2 × length + 2 × width 36 = 2(w + 4) + 2w 36 = 2w + 8 + 2w 36 = 4w + 8 36 - 8 = 4w 28 = 4w w = 7

Therefore: width = 7 meters length = w + 4 = 7 + 4 = 11 meters

Verification:

• Perimeter check: 2(11) + 2(7) = 22 + 14 = 36 meters ✓
• Length-width relationship: 11 = 7 + 4 ✓

Answer: The garden is 11 meters long and 7 meters wide.

Example 3: Multi-Step Problem

Problem: A shop sells notebooks for £3 each and pens for £1.50 each. Emma spent £24 and bought twice as many pens as notebooks. How many of each item did she buy?

Solution Essay:

Understanding the Situation: This problem involves two items with different prices and a spending constraint. I need to find the quantities of notebooks and pens Emma purchased, knowing the relationship between these quantities and the total cost.

Planning My Solution: Let n represent the number of notebooks. Since Emma bought twice as many pens as notebooks, she bought 2n pens. The total cost equation will combine the cost of notebooks (3n) and the cost of pens (1.50 × 2n), which equals £24.

Detailed Working: Let n = number of notebooks Then 2n = number of pens

Cost equation: (Cost of notebooks) + (Cost of pens) = Total spent 3n + 1.50(2n) = 24 3n + 3n = 24 6n = 24 n = 4

Therefore: Number of notebooks = 4 Number of pens = 2n = 2 × 4 = 8

Checking the Solution:

• Cost of 4 notebooks: 4 × £3 = £12
• Cost of 8 pens: 8 × £1.50 = £12
• Total: £12 + £12 = £24 ✓
• Pens are twice notebooks: 8 = 2 × 4 ✓

Final Answer: Emma bought 4 notebooks and 8 pens, spending exactly £24.

Question Type 1: Number Problems

Typical Question: "The sum of three consecutive even numbers is 78. Find these numbers. Show all your working and explain your method."

How to Answer: Begin by clearly defining your variables. State: "Let n be the first even number. Then the next two consecutive even numbers are (n + 2) and (n + 4)." This shows you understand the structure of consecutive even numbers.

Set up your equation explicitly: "The sum equals 78, so: n + (n + 2) + (n + 4) = 78." Then solve systematically, showing every step: combine like terms (3n + 6 = 78), isolate the variable (3n = 72), and solve (n = 24).

State all three numbers clearly: "The three consecutive even numbers are 24, 26, and 28." Always verify by adding: 24 + 26 + 28 = 78. This verification demonstrates mathematical thinking and often earns additional marks.

Question Type 2: Ratio and Proportion Problems

Typical Question: "A mixture contains orange juice and water in the ratio 2:3. If there are 750 ml of mixture, how much orange juice is there? Explain your solution method."

How to Answer: Start by explaining what the ratio means: "The ratio 2:3 means that for every 2 parts of orange juice, there are 3 parts of water, making 5 parts total." This shows understanding beyond mere calculation.

Set up the solution: "If 5 parts = 750 ml, then 1 part = 750 ÷ 5 = 150 ml." Show this division clearly. Then find the orange juice: "Orange juice is 2 parts, so 2 × 150 = 300 ml."

Conclude with verification: "Check: Orange juice (300 ml) + water (450 ml) = 750 ml, and 300:450 simplifies to 2:3." This thorough approach demonstrates complete understanding and earns full marks.

Question Type 3: Perimeter and Area Problems

Typical Question: "A square and a rectangle have the same perimeter of 32 cm. The rectangle's length is 10 cm. Compare their areas and explain which shape has the greater area."

How to Answer: Address each shape systematically. For the square: "Since perimeter = 32 cm and a square has 4 equal sides, each side = 32 ÷ 4 = 8 cm. Area of square = 8 × 8 = 64 cm²."

For the rectangle: "Perimeter = 2l + 2w. Substituting: 32 = 2(10) + 2w, so 32 = 20 + 2w, giving 2w = 12, therefore w = 6 cm. Area of rectangle = 10 × 6 = 60 cm²."

Make the comparison explicit: "The square has an area of 64 cm² while the rectangle has 60 cm². The square has a greater area by 4 cm². This demonstrates that among shapes with equal perimeter, a square encloses the maximum area."

Question Type 4: Real-World Application Problems

Typical Question: "A mobile phone plan costs £15 per month plus £0.10 per text message. Last month, the total bill was £23.50. How many text messages were sent? Show your method clearly."

How to Answer: Explain your variable: "Let t represent the number of text messages sent." Then build the equation step by step: "The total cost is the fixed monthly cost plus the variable texting cost: 15 + 0.10t = 23.50."

Solve with clear steps: "0.10t = 23.50 - 15, so 0.10t = 8.50, therefore t = 8.50 ÷ 0.10 = 85 text messages."

Verify in context: "Check: Fixed cost £15 + (85 messages × £0.10) = £15 + £8.50 = £23.50 ✓" This real-world verification shows you understand the context, not just the mathematics.

Tip 1: Always Define Your Variables Clearly

Examiners specifically look for clear variable definitions. Write statements like "Let x = the number of apples" rather than just starting with "x + 5 = 12." This explicit definition earns communication marks and prevents confusion in complex problems. Many students lose marks simply because the examiner cannot follow which variable represents what quantity.

Tip 2: Show Every Significant Step

While you don't need to show that 2 + 3 = 5, you must show all algebraic manipulations, formula substitutions, and logical progressions. If you expand brackets, show the expansion