Non-Calculator Strategies (Paper 1)

Paper 1 is the non-calculator paper. It carries the same weight (80 marks) as each calculator paper, so strong non-calculator skills are essential. This lesson covers mental and written techniques that will help you work efficiently and accurately without a calculator.

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Mental Arithmetic Fundamentals

Times Tables and Key Facts

You should know instantly:

• All times tables up to $12 \times 12$12×12.
• Squares of numbers 1–15 (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225).
• Cubes of 1–5 (1, 8, 27, 64, 125) and $10^{3} = 1000$103=1000.
• Powers of 2: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024.
• Key fraction–decimal–percentage equivalents.

Fraction–Decimal–Percentage Equivalents

Fraction	Decimal	Percentage
1/2	0.5	50%
1/4	0.25	25%
3/4	0.75	75%
1/5	0.2	20%
1/3	0.333...	33.3...%
2/3	0.666...	66.6...%
1/8	0.125	12.5%
1/10	0.1	10%
1/100	0.01	1%

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Efficient Written Methods

Long Multiplication

Use the grid (box) method or the column method — whichever you find more reliable.

Worked Example 1

$47 \times 36$47×36

Grid method:

	40	7
30	1200	210
6	240	42

Total = 1200 + 210 + 240 + 42 = 1692

Column method:

• $47 \times 6 = 282$47×6=282
• $47 \times 30 = 1410$47×30=1410
• 282 + 1410 = 1692

Long Division

Worked Example 2

$966 \div 23$966÷23

$23 \times 40 = 920 \to 966 - 920 = 46$23×40=920→966−920=46 $23 \times 2 = 46$23×2=46 Answer: 40 + 2 = 42

Adding and Subtracting Fractions

Step 1: Find the lowest common denominator (LCD). Step 2: Convert each fraction. Step 3: Add or subtract the numerators. Step 4: Simplify if possible.

Worked Example 3

3/4 − 2/5 LCD = 20 15/20 − 8/20 = 7/20

Multiplying Fractions

Multiply numerators together and denominators together. Cancel common factors first to keep numbers small.

Worked Example 4

$8/15 \times 5/12$8/15×5/12 Cancel: 8 and 12 share factor $4 \to 2/15 \times 5/3$4→2/15×5/3 Cancel: 15 and 5 share factor $5 \to 2/3 \times 1/3 =$5→2/3×1/3= 2/9

Dividing Fractions

Keep the first fraction, flip the second, then multiply.

Worked Example 5

$3/7 \div 9/14$3/7÷9/14 $= 3/7 \times 14/9$=3/7×14/9 Cancel: 3 and 9 share factor $3 \to 1/7 \times 14/3$3→1/7×14/3 Cancel: 7 and 14 share factor $7 \to 1/1 \times 2/3 =$7→1/1×2/3= 2/3

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Percentage Calculations Without a Calculator

Finding a Percentage of an Amount

Use the building-block method:

• $10$10 divide by 10
• $5$5 half of 10%
• $1$1 divide by 100
• Combine as needed

Worked Example 6

Find 17.5% of £240. 10% = £24 5% = £12 2.5% = £6 17.5% = £24 + £12 + £6 = £42

Percentage Increase/Decrease

Worked Example 7

Increase £360 by 15%. 10% = £36 5% = £18 15% = £54 New amount = £360 + £54 = £414

Reverse Percentages

Worked Example 8

After a 20% reduction, a jacket costs £56. Find the original price. £56 represents 80% (100% − 20% = 80%) $1$1 £0.70 $100$100 £70

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Working with Decimals

Multiplying Decimals

Remove the decimal points, multiply as whole numbers, then put the point back.

Worked Example 9

$0.3 \times 0.07$0.3×0.07 $3 \times 7 = 21$3×7=21 Count decimal places: 1 + 2 = 3 Answer: 0.021

Dividing Decimals

Make the divisor a whole number by multiplying both numbers by a power of 10.

Worked Example 10

$4.56 \div 0.8$4.56÷0.8 $= 45.6 \div 8 =$=45.6÷8= 5.7

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Common Non-Calculator Topics

Certain topics appear regularly on Paper 1:

1. Fractions, decimals and percentages — conversions and calculations.
2. Ratio and proportion — sharing in a ratio, scaling recipes.
3. Expanding and factorising — brackets, quadratic expressions.
4. Solving equations — linear and simple quadratics by factorising.
5. Pythagoras' theorem — with "nice" numbers (3-4-5, 5-12-13, 8-15-17 triangles).
6. Area and perimeter — including circles (often leave answers in terms of $\pi)$π).
7. Probability — tree diagrams and two-way tables without complex decimals.
8. Sequences — finding nth terms.
9. Surds — simplifying and rationalising (Higher).
10. Standard form — addition, subtraction, multiplication, division.

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Checking Your Answers

Estimation

Round each number to 1 significant figure and calculate a rough answer.

Example: Is $47 \times 36 = 1692$47×36=1692 reasonable? Estimate: $50 \times 40 = 2000$50×40=2000. The answer 1692 is close — it looks correct.

Inverse Operations

Check subtraction with addition, check division with multiplication.

Example: You calculated $966 \div 23 = 42$966÷23=42. Check: $42 \times 23 = 42 \times 20 + 42 \times 3 = 840 + 126 = 966$42×23=42×20+42×3=840+126=966. Correct.

Substitution

For algebra, substitute your answer back into the original equation.

Example: You solved 3x + 7 = 22 and got x = 5. Check: 3(5) + 7 = 15 + 7 = 22. Correct.

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Worked Practice

Question: Express 0.36̇ (0.3666...) as a fraction.

Let x = 0.3666... 10x = 3.666... 100x = 36.666...

100x − 10x = 36.666... − 3.666... 90x = 33 x = 33/90 = 11/30

Check: $11 \div 30 = 0.3666$11÷30=0.3666... Correct.

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Question: Work out $(2 + \sqrt{3})(5 - \sqrt{3})$(2+3​)(5−3​). Give your answer in the form $a + b\sqrt{3}$a+b3​.

$= 10 - 2\sqrt{3} + 5\sqrt{3} - (\sqrt{3})^{2}$=10−23​+53​−(3​)2 $= 10 + 3\sqrt{3} - 3$=10+33​−3 = $7 + 3\sqrt{3}$7+33​

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Time Management on Paper 1

• Do not spend excessive time on any single arithmetic calculation.
• If a calculation seems very messy, re-read the question — you may be approaching it incorrectly.
• Leave harder questions and return to them — there may be easier marks later in the paper.
• Use the last 5 minutes to check your answers using estimation and inverse operations.

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Summary Checklist

• I know my times tables, squares, cubes and powers of 2 fluently
• I can add, subtract, multiply and divide fractions efficiently
• I can calculate percentages using the building-block method
• I can handle decimal multiplication and division without a calculator
• I use estimation to check whether my answers are reasonable
• I know the common non-calculator topics and have practised them
• I leave time at the end of Paper 1 to check my work

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Deeper non-calculator techniques