Mathematical Reasoning

The FSCE 11+ exam stands apart from many other 11+ exams because it requires you to explain your mathematical reasoning in writing. This is not about showing your calculations (though that matters too) — it is about explaining WHY you did what you did and HOW you arrived at your conclusion. This lesson teaches you how to write clear, logical mathematical explanations that earn full marks.

Why Reasoning Matters

In many exams, you write a number and move on. In the FSCE, you might be asked:

• "Explain how you know..."
• "Show that..."
• "Give a reason for your answer."
• "Is this statement always true, sometimes true, or never true? Explain why."

These questions test whether you truly understand the mathematics, not just whether you can follow a procedure.

How to Show Your Working

Showing your working means writing down each calculation step. This is different from reasoning, but both are important.

Good Working vs Poor Working

Question: What is 15% of 240?

Poor Working	Good Working
36	10% of 240 = 24
	5% of 240 = 12
	15% = 24 + 12 = 36

The poor working gives the right answer but earns no method marks if the answer is wrong. The good working shows the examiner your thought process.

How to Explain Your Reasoning

Reasoning means explaining the mathematical logic behind your answer. Here is a framework:

1. State what you know (the mathematical fact or rule you are using).
2. Apply it to the question (show how it connects to this specific problem).
3. Draw your conclusion (what this tells you about the answer).

Before/After: "Just the Answer" vs "Answer with Reasoning"

Question: Is 147 divisible by 3? Explain how you know.

Just the Answer	Answer with Reasoning
"Yes"	"Yes, 147 is divisible by 3. I know this because the divisibility rule for 3 states that if the sum of the digits is divisible by 3, then the number is divisible by 3. The digits of 147 are 1 + 4 + 7 = 12, and 12 ÷ 3 = 4 with no remainder, so 147 is divisible by 3."

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Question: A triangle has angles of 50° and 70°. What is the third angle? Explain your reasoning.

Just the Answer	Answer with Reasoning
"60°"	"The third angle is 60°. I know that angles in a triangle always add up to 180°. So the third angle = 180° - 50° - 70° = 60°."

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Question: Tom says "If I multiply two odd numbers, I always get an odd number." Is he correct? Explain.

Just the Answer	Answer with Reasoning
"Yes"	"Tom is correct. When you multiply two odd numbers, the result is always odd. For example: 3 x 5 = 15 (odd), 7 x 9 = 63 (odd), 11 x 3 = 33 (odd). This is because odd numbers have no factor of 2, so the product cannot have a factor of 2 either, which means it must be odd."

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Question: Compare 0.5 and 0.45. Which is bigger? Explain your reasoning.

Just the Answer	Answer with Reasoning
"0.5 is bigger."	"0.5 is bigger than 0.45. To compare decimals, write them with the same number of decimal places: 0.5 = 0.50. Now compare digit by digit. The tenths digit is the same... no — 0.50 has 5 tenths and 0 hundredths (50 hundredths in total). 0.45 has 4 tenths and 5 hundredths (45 hundredths in total). Since 50 hundredths > 45 hundredths, 0.5 > 0.45."

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Question: A shop has a "Buy 3 get 1 free" offer on tins of beans costing 80p each. Jake needs 7 tins. How much will he pay?

Just the Answer	Answer with Reasoning
"£4.80"	"With 'Buy 3 get 1 free', every fourth tin is free. For 7 tins: tins 1, 2 and 3 are paid for; tin 4 is free; tins 5, 6 and 7 are paid for. So Jake pays for 6 tins. 6 x 80p = 480p = £4.80."

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Five Clear Before/After Comparisons

Example A — Divisibility

Question: Is 234 divisible by 9?

Just the Answer	Answer with Reasoning
"Yes"	"Yes. The sum of the digits is 2 + 3 + 4 = 9. Since 9 is divisible by 9, the original number 234 is also divisible by 9. To verify: 234 ÷ 9 = 26."

Example B — Angles

Question: Can a triangle have two obtuse angles? Explain.

Just the Answer	Answer with Reasoning
"No"	"No, a triangle cannot have two obtuse angles. An obtuse angle is greater than 90°. If a triangle had two obtuse angles, their total would already be more than 180°. But the angles in a triangle must add up to exactly 180°, so there would be no room for the third angle. Therefore, a triangle can have at most one obtuse angle."

Example C — Fractions

Question: Which is larger, 3/7 or 5/12?

Just the Answer	Answer with Reasoning
"3/7"	"To compare 3/7 and 5/12, I need a common denominator. The LCM of 7 and 12 is 84. 3/7 = 36/84. 5/12 = 35/84. Since 36/84 > 35/84, 3/7 is larger than 5/12."

Example D — Probability Reasoning

Question: A bag contains 3 red and 5 blue balls. One ball is picked at random. Sam says the probability of picking red is 3/5 because there are 3 red and 5 blue. Is Sam correct?

Just the Answer	Answer with Reasoning
"No"	"Sam is not correct. The probability of picking red is the number of red balls divided by the TOTAL number of balls. Total balls = 3 + 5 = 8. So P(red) = 3/8, not 3/5. Sam made the mistake of dividing by the number of blue balls instead of the total."

Example E — Always, Sometimes, Never

Question: "When you square a number, the result is always bigger than the original number." Is this always true, sometimes true, or never true?

Just the Answer	Answer with Reasoning
"Sometimes true"	"This is sometimes true. For numbers greater than 1, squaring does give a bigger result (e.g. 3² = 9 > 3). But for numbers between 0 and 1, squaring gives a SMALLER result (e.g. 0.5² = 0.25 < 0.5). And for 0 and 1, squaring gives the same number (0² = 0, 1² = 1). So the statement is only sometimes true."

The FSCE Reasoning Framework

Use this structure for reasoning answers: