Scottish learning platformNational 5 Mathematics · Free revision
National 5 Maths non-calculator practice
Practise fractions, surds and algebra without a calculator. Original National 5 Maths examples, worked answers and ways to check your working.
What are you practising?
National 5 Mathematics Paper 1 is non-calculator. Paper 2 permits a calculator. These contexts use the same course: avoid treating a topic as belonging exclusively to one paper. This short activity concentrates on exact arithmetic and algebra; it is not a complete Paper 1 checklist.
Have paper and a pen ready. Try each question before opening its worked answer. Write one step per line so you can see where a sign, fraction or factor has changed.
1. Add fractions using a common denominator
For 5/6 + 3/8, a common denominator is 24. Rewrite the fractions as 20/24 and 9/24, then add: 29/24 = 1 5/24. You add the numerators after making the denominators equal.
Try: calculate 7/10 − 1/4. Give your answer in its simplest form.
Show the worked answer for fractions
Use denominator 20: 7/10 = 14/20 and 1/4 = 5/20. Therefore 14/20 − 5/20 = 9/20. The numerator and denominator have no common factor greater than 1.
Check the size: 7/10 is 0.7 and 1/4 is 0.25, so the difference must be less than one half and greater than zero. This does not replace the exact calculation, but can reveal a mistake.
2. Simplify a surd using a square factor
To simplify √72, write 72 = 36 × 2. Then √72 = √36 × √2 = 6√2. Look for a square factor; do not split a square root across addition.
Try: simplify √75 + √12.
Show the worked answer for surds
√75 = √(25 × 3) = 5√3. √12 = √(4 × 3) = 2√3. Adding like surds gives 7√3.
Check by simplifying each term first. √75 + √12 is not √87: square roots do not distribute over a sum.
3. Keep an equation balanced
To solve 4(x − 2) = 2x + 10, expand to 4x − 8 = 2x + 10. Subtract 2x from both sides, then add 8: 2x = 18, so x = 9. Substitution gives 4(9 − 2) = 28 and 2 × 9 + 10 = 28.
Try: solve 3(2x − 1) = 4x + 9.
Show the worked answer for the equation
Expand: 6x − 3 = 4x + 9. Subtract 4x: 2x − 3 = 9. Add 3: 2x = 12. Divide by 2: x = 6.
Check both original sides: 3(12 − 1) = 33 and 24 + 9 = 33. Equal sides confirm the solution.
4. Factorise before solving a quadratic
For x² + 5x + 6 = 0, find two numbers with product 6 and sum 5: 2 and 3. Thus (x + 2)(x + 3) = 0. At least one factor must be zero, giving x = −2 or x = −3.
Try: solve x² − x − 12 = 0.
Show the worked answer for the quadratic
The numbers are −4 and 3: their product is −12 and their sum is −1. So (x − 4)(x + 3) = 0, giving x = 4 or x = −3.
Check the expansion: x² + 3x − 4x − 12 = x² − x − 12. Substitute both solutions into the original equation.
Turn a mistake into a useful next step
If an answer differs, find the first line where your working changes from the explanation. Name the difficulty: common denominators, square factors, bracket expansion or signs. Practise that step in a fresh example, then return to the original question without looking.
Use the current National 5 Mathematics course specification for the full course and assessment structure. Link to official papers for timed practice; these activities do not reproduce them.
Continue your practice
Choose another short activity, try it yourself and use the explanation to decide what to revisit.
Explore National 5 Mathematics on Tutor Scotland. Maths is included with Essential when it is available. Check current plans and availability.