Mathematics Gr 10 · Exam RevisionDate: Top Journal

Maths is not about being fast. It is about being sure, one line of working at a time.The GOAT Notes™

THE PAPERS EXAM FORMAT

What you must be able to do

Say what each paper covers, how many marks it carries and how long you have for it
Place every topic on the paper map with its mark range
Name the four levels of thinking and the share of marks each one carries
Know what the information sheet holds and which formulae on it are Grade 10 work

Core content · Two papers, 200 marks

Grade 10 is examined by your school or province, never by a national paper, to the CAPS structure: two papers of 100 marks, 2 hours each. Every question is compulsory, with no multiple choice. The mark ranges below come from the three provincial November papers we read (Gauteng 2022, KwaZulu-Natal 2023 and 2024).

Mark ranges across the sittings we read.
Paper	Question	Topic	Marks
Paper 1	Q1	Algebra: factorising, surds, exponents, fractions	20–22
Paper 1	Q2	Equations, inequalities, word problems	12–14
Paper 1	Q3	Number patterns	10–12
Paper 1	Q4–5	Functions and graphs	29
Paper 1	Q6	Finance and growth, with exchange rates	10–13
Paper 1	Q7	Probability and Venn diagrams	13–15
Paper 2	Q1–2	Statistics	15–20
Paper 2	Q3	Analytical geometry	9–15
Paper 2	Q4–6	Trigonometry	39–41
Paper 2	Q7 onwards	Measurement, then Euclidean geometry riders	6–12, then 15–23

⭐ EXAM FAVOURITE Trigonometry is about 40% of Paper 2 and functions about 29% of Paper 1. Revise them first.

Core content · Four levels of thinking

CAPS cognitive split, the same for both papers.
Level	Share	Marks per paper	What it looks like
1 · Knowledge	20%	20	Write down, state, read off a graph
2 · Routine procedures	35%	35	Factorise, solve, find a gradient
3 · Complex procedures	30%	30	Several steps, or an unfamiliar setting
4 · Problem solving	15%	15	No set method: riders, proofs

Levels 1 and 2 are 55 marks a paper, before you touch a rider.

💡 EASY MARK Free marks: the first sub-question of most questions is level 1 or 2. Collect those first, then return to the harder ones.

Core content · The information sheet and the rules of the room

CAPS says a formula sheet must not be provided in Grades 10 and 11. Practice varies by province and year: the Gauteng November 2022 and KwaZulu-Natal November 2023 papers printed the full Grades 10 to 12 sheet, while the KwaZulu-Natal November 2024 papers and the June 2024 and 2025 tests we read printed none. Learn every Grade 10 formula, so a missing sheet costs you nothing.

⚠️ COMMON MISTAKE Where a sheet is printed, it holds far more than you need. Sequences and series, annuities, derivatives, the quadratic formula, decay formulae, the sine and cosine rules and the circle equation are Grade 11 or 12 work.

The formulae on the sheet that Grade 10 uses:

Simple growth: A = P(1 + in)
Compound growth: A = P(1 + i)ⁿ
Distance and midpoint formulae
Gradient m = (y₂ − y₁)/(x₂ − x₁), and y = mx + c
P(A) = n(A)/n(S) and P(A or B) = P(A) + P(B) − P(A and B)

📖 KEY DEFINITION Diagrams are not drawn to scale. Never measure; use only what is given.

⚡ MUST MEMORISE Calculator and rounding. Approved scientific calculator only, non-programmable and non-graphical. Round to 2 decimal places unless told otherwise.

🧠 ACTIVE RECALL

ONE-MINUTE SUMMARY Two papers, each 100 marks in 2 hours, every question compulsory. Paper 1: algebra about 21, equations 13, patterns 11, functions 29, finance 11, probability 14. Paper 2: statistics 18, analytical geometry 12, trigonometry 40, measurement and Euclidean geometry about 30. Levels split 20 / 35 / 30 / 15. Allow 1,2 minutes a mark.

IN THE EXAM

Trigonometry and functions are the two largest blocks. Do not leave either to the end.
The last questions of Paper 2 are geometry riders at level 3 and 4. Attempt them after the marks you can already see.

Paper 1 · ~21 marks · 21% of Paper 1

PAPER 1 · QUESTION 1 · EXPRESSIONS & EXPONENTS ALGEBRA · Q1

What you must be able to do

Identify irrational numbers, place a surd between consecutive integers and round as asked.
Expand a binomial × trinomial; factorise fully, including grouping and cubes.
Simplify algebraic fractions with cubes in the denominator.
Simplify with integer and rational exponents, and solve x^p/q = k.

Core content · Real numbers, surds and rounding

Question 1 often opens with quick marks here: classify numbers, place a surd without a calculator, or round to suit a context.

📖 KEY DEFINITION Rational: a/b with a, b integers, b ≠ 0; the decimal ends or recurs. Irrational: the decimal never ends or recurs (π, √5).

Given: −√(49/4) ; ∛9 ; 0,1̇6̇ ; 22/7 ; π − 1. Write down the irrational numbers. (2)

✓ A: ∛9: 9 is not a perfect cube, so the decimal neither ends nor recurs
✓ A: π − 1: an irrational number minus a rational number stays irrational

−√(49/4) = −7/2, 0,1̇6̇ = 16/99 and 22/7 are rational. An extra wrong number costs a mark.

⚠️ COMMON MISTAKE 22/7 is not π. It is only an approximation, and it is rational because it is a fraction of integers. The reverse trap: a root sign does not make a number irrational. √(49/4) = 7/2, so test for a perfect square or cube before you decide.

Without using a calculator, determine between which two consecutive integers 3√7 lies. (2)

✓ A: Move the 3 under the root: 3√7 = √(9 × 7) = √63
✓ A: 49 < 63 < 64, so √49 < √63 < √64, and 7 < 3√7 < 8

Show the perfect squares: the working earns the marks.

A square floor tile has an area of 1 200 cm². Calculate the length of one side, rounded to the nearest millimetre. (2)

✓ A: side = √1 200 = 34,641… cm
✓ A: The nearest millimetre is one decimal place in centimetres: 34,6 cm

Round once, at the end. With no instruction, give two decimal places: 34,64 cm.

Core content · Multiplying out: binomial × trinomial

Asked as “expand and simplify”: 2 × 3 = 6 products, then collect like terms.

Expand and simplify: (2a − 3b)(4a² + 6ab + 9b²) (3)

✓ A: 2a(4a² + 6ab + 9b²) = 8a³ + 12a²b + 18ab²
✓ A: −3b(4a² + 6ab + 9b²) = −12a²b − 18ab² − 27b³
✓ CA: The middle terms cancel: 8a³ − 27b³

Vanishing middle terms mean a difference of cubes: read backwards, it is the factorising formula.

Core content · Factorising fully

Asked as “factorise fully”. Common factor first; then count terms (two: squares or cubes; three: trinomial; four: group in pairs); then recheck every bracket.

⚡ MUST MEMORISE Cubes on sight: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1 000.
a³ + b³ = (a + b)(a² − ab + b²)
a³ − b³ = (a − b)(a² + ab + b²)

Factorise fully: 2x²y − 7xy − 15y (2)

✓ A: Common factor first: y(2x² − 7x − 15)
✓ A: = y(2x + 3)(x − 5)

Check the middle term: 2x × (−5) + 3 × x = −7x.

Factorise fully: x² − y² + 3x − 3y (3)

✓ A: Group in pairs: (x² − y²) + (3x − 3y)
✓ A: = (x − y)(x + y) + 3(x − y)
✓ CA: = (x − y)(x + y + 3)

Pair terms that share a factor: squares together, 3s together.

Factorise fully: 16a³ + 2b³ (3)

✓ A: Common factor: 2(8a³ + b³)
✓ A: 8a³ + b³ = (2a)³ + b³, a sum of cubes
✓ CA: = 2(2a + b)(4a² − 2ab + b²)

⚠️ COMMON MISTAKE Wrong sign in the long bracket. The middle sign of the trinomial is always the opposite of the sign in the short bracket, and the last term is always +. Writing 2(2a + b)(4a² + 2ab + b²) loses the last mark. That trinomial never factorises further, so do not try.

Core content · Algebraic fractions

Asked as “simplify fully”. Factorise every numerator and denominator first, then cancel factors or use the LCD.

Simplify fully: (x² − 4)/(x³ − 8) ÷ (x² + 4x + 4)/(x² + 2x + 4) (4)

✓ A: x² − 4 = (x − 2)(x + 2) and x³ − 8 = (x − 2)(x² + 2x + 4)
✓ A: x² + 4x + 4 = (x + 2)²
✓ A: Multiply by the reciprocal: (x − 2)(x + 2)/[(x − 2)(x² + 2x + 4)] × (x² + 2x + 4)/(x + 2)²
✓ CA: = 1/(x + 2)

Simplify: 2/(x − 2) − (x² + 20)/(x³ − 8) (5)

✓ A: LCD: x³ − 8 = (x − 2)(x² + 2x + 4)
✓ A: = [2(x² + 2x + 4) − (x² + 20)] / [(x − 2)(x² + 2x + 4)]
✓ A: Numerator: 2x² + 4x + 8 − x² − 20 = x² + 4x − 12
✓ A: = (x + 6)(x − 2) / [(x − 2)(x² + 2x + 4)]
✓ CA: = (x + 6)/(x² + 2x + 4)

⚠️ COMMON MISTAKE The minus belongs to the whole numerator. −(x² + 20) is −x² − 20. Write −x² + 20 and the numerator becomes x² + 4x + 28, which will not factorise, so nothing cancels and the last marks go.

Core content · Laws of exponents

Asked as “simplify fully”, often with prime bases; answers use positive exponents.

⚖️ LAWS / FRAMEWORKS x^m · xⁿ = x^m+n
x^m ÷ xⁿ = x^m−n
(x^m)ⁿ = x^mn
(xy)^m = x^my^m
x⁰ = 1 and x⁻ⁿ = 1/xⁿ (x ≠ 0)

Simplify fully, with positive exponents: (−3a²b⁻¹)³ ÷ 9a⁴b⁻² (3)

✓ A: (−3a²b⁻¹)³ = −27a⁶b⁻³
✓ A: −27a⁶b⁻³ ÷ 9a⁴b⁻² = −3a²b⁻¹
✓ CA: = −3a²/b

An odd power keeps the negative sign: (−3)³ = −27.

⚠️ COMMON MISTAKE An exponent only touches what it sits on. 3x⁻² = 3/x², not 1/(3x²); (3x)⁻² = 1/(9x²). And a negative exponent never makes the answer negative: 2⁻³ = ⅛, not −8.

🔥 FREQUENTLY TESTED Prime bases. Unknown exponent on composite bases: write each base as primes (18 = 2 · 3², 6 = 2 · 3, 4 = 2²), then add and subtract exponents base by base.

Simplify fully. (4)

(18^x · 4^x+1) / (6^2x · 2^x−1)

✓ A: 18^x = 2^x · 3^2x and 4^x+1 = 2^2x+2
✓ A: 6^2x = 2^2x · 3^2x
✓ A: 2^{x + 2x + 2 − 2x − (x − 1)} · 3^{2x − 2x}
✓ CA: = 2³ · 3⁰ = 8

Bracket the exponent you subtract: −(x − 1) = −x + 1.

Without using a calculator, determine which is larger: 2⁹⁰ or 3⁶⁰. (3)

✓ A: Common exponent: 90 and 60 share the factor 30
✓ A: 2⁹⁰ = (2³)³⁰ = 8³⁰ and 3⁶⁰ = (3²)³⁰ = 9³⁰
✓ CA: Same exponent, larger base wins: 9³⁰ > 8³⁰, so 3⁶⁰ is larger

Neither the bigger exponent nor the bigger base decides alone.

Core content · Simplifying with rational exponents

x^p/q = (^q√x)^p: denominator is the root, numerator the power. Without a calculator, take the root first.

Without using a calculator, determine the value of: (3)

16^0,75 × (27/8)^−2/3

✓ A: 16^0,75 = 16^3/4 = (⁴√16)³ = 2³ = 8
✓ A: (27/8)^−2/3 = (8/27)^2/3 = (2/3)² = 4/9
✓ CA: 8 × 4/9 = 32/9

A negative exponent flips the fraction. Change a decimal exponent to a fraction first.

Simplify fully. (3)

∛(8x⁶) · √x ÷ x^5/2

✓ A: ∛(8x⁶) = 2x²
✓ A: √x = x^1/2
✓ CA: 2x^{2 + 1/2 − 5/2} = 2x⁰ = 2

2x⁰ = 2, but (2x)⁰ = 1.

Solve for x, where x > 0. (3)

4x^−2/3 = ¼

✓ A: x^−2/3 = 1/16, so x^2/3 = 16
✓ A: Raise both sides to the power 3/2: x = 16^3/2
✓ CA: x = (√16)³ = 64

Check: 64^2/3 = 16, and 4 ÷ 16 = ¼.

🧠 ACTIVE RECALL

Given: √(25/4) ; 3,14 ; ∛12 ; 2π ; 0,4̇
Which option lists ALL the irrational numbers, and only those?

Given: √0,49 ; 1,0̇5̇ ; ∜17 ; 3,1416 ; π + 2
Which option lists ALL the irrational numbers, and only those?

Given: √(36/49) ; √12 ; 0,2̇7̇ ; 1 − π ; ∛(−64)
Which option lists ALL the rational numbers, and only those?

Which ONE of the following numbers is rational?

Which ONE of the following statements is TRUE?

Without using a calculator, determine between which two consecutive integers 2√11 lies.

Without using a calculator, determine between which two consecutive integers 5√2 lies.

Without using a calculator, determine between which two consecutive integers 2∛7 lies.

Without using a calculator, determine between which two consecutive integers −3√6 lies.

Without using a calculator, determine between which two consecutive integers 3√10 lies.

Expand and simplify: (3x − 2)(x² − 4x + 5)

Expand and simplify: (2a + b)(a² − 3ab + 2b²)

Expand and simplify: (4 − x)(x² − 2x + 3)

Expand and simplify: (2p − 5)(p² + 3p − 1)

Expand and simplify: (y + 3)(2y² − y − 6)

Factorise fully: 3x³ − 81

Factorise fully: 54a³ + 16b³

Factorise fully: 250 − 2y³

Factorise fully: x⁴y + 64xy

Factorise fully: 5m³ − 40n³

Solve for x, where x > 0: 2x^3/4 = 54

Solve for x, where x > 0: 5x^−3/2 = 40

Solve for x, where x > 0: x^4/3 − 2 = 14

Solve for x, where x > 0: 3x^2/5 = 12

Solve for x, where x > 0: 2x^−3/4 = 1/4

Without using a calculator, determine which of 2⁶⁰, 3⁴⁰ and 5²⁰ is the largest.

Without using a calculator, determine which of 2¹⁰⁰, 3⁶⁰ and 7⁴⁰ is the smallest.

Without using a calculator, determine which of √3, ∛5 and ⁶√20 is the largest.

Without using a calculator, determine which of 2⁴⁸, 3³⁶, 5²⁴ and 10¹² is the largest.

Without using a calculator, arrange 2⁷⁰, 3⁴⁰ and 5³⁰ in increasing order.

Factorise fully: x³ − 8 + 3x² − 6x. Show every line of working and name the method you use on each line. (4)

Simplify fully: 3/(x + 3) − (2x² + 63)/(x³ + 27). Show every line of working and say in words what each line does. (5)

Show, without using a calculator, that (12ⁿ · 3ⁿ⁺² − 36ⁿ) ÷ 6²ⁿ⁻¹ has the same value for every natural number n, and determine that value. Show every line and say in words what each line does. (5)

Explain to a Grade 8 learner why 2³⁰ is smaller than 3²⁰, even though 30 is bigger than 20, and how you can prove it without a calculator.

Explain to a Grade 8 learner why 10 − (x + 4) is not equal to 10 − x + 4, and why this matters when you subtract one algebraic fraction from another.

Explain to a Grade 8 learner what 125^2/3 means, which part of the exponent is the root and which is the power, and why it helps to take the root first.

ONE-MINUTE SUMMARY Question 1 is about 21 marks of technique. Bracket surds between perfect squares or cubes; round once, at the end. Expand all six products. Factorise in order and watch the middle sign in a cube's long bracket. In fractions, factorise first and cancel factors only. With exponents, use prime bases, root before power, and positive exponents.

IN THE EXAM

Paper 1, Question 1 in all three November sittings read: 20–22 of 100 marks, compulsory.
Expect surds, expanding, factorising, one algebraic fraction and exponents, with and without a calculator.
Commands: “factorise fully”, “simplify fully”, “determine”, “round off”. “Without using a calculator” means show every line.
Marks are per step (A); CA carries a slip forward, so keep working from your own line.
Question 2 reuses this toolkit: quadratics by factorising and exponential equations.

🔒 EQUATIONS

🔒 PATTERNS

🔒 FUNCTIONS

🔒 FINANCE

🔒 PROBABILITY

🔒 ANALYTIC GEO

🔒

9 more topics locked

This is a free preview of Topic 1. The full guide has every topic, the self-marking quizzes, mock exams, planner, journal and wellbeing tools.

Unlock the full guide on WhatsApp See pricing

Daily
Journal

Date

SMTWTFS

I am becoming the best version of myself

Things I accomplished today

I am thankful for

Something that inspired me today

Today I felt

Daily affirmations

What can I do to make tomorrow better

Saved!

Your journalled days

Your journal is saved only on this device and browser (it does not sync across devices and is lost if you clear browser data). Use "Export" to keep a copy.

Feeling the pressure? Take a breath.

Exam stress is completely normal. Here are a few things that actually work. None of this replaces a doctor or a trusted adult.

Breathe in

Box breathing: 4 counts in, 4 hold, 4 out, 4 hold. Do it four times.

Anxiety

Grounding: name 5 things you see, 4 you hear, 3 you can touch.
Exam morning: eat, drink water, arrive early, avoid last-minute cramming.
Reframe "I am going to fail" into "I have prepared, I will do my best".

Concentration

Work in 25-minute blocks, then a 5-minute break (Pomodoro).
Put the phone in another room while you study.
One task at a time. Multitasking costs more than it saves.

Sleep

Cramming all night backfires: sleep is when the brain locks in memory.
No screens for 30 minutes before bed.
The same bedtime every night keeps the brain sharp.

Study habits

Active recall (testing yourself) beats passive re-reading by far.
Space your study across days; do not cram it all into one.
Practise past papers under time. That is exactly how these notes are built.

If it gets too much, talk to a parent, teacher or a trusted adult. You do not have to carry it alone. SADAG (free, 24 hours): 0800 567 567, or SMS 31393.

Use the daily journal to clear your head →

Build your own revision plan

Look at the worked example below first, then make your own. Tick off each topic as you finish it.

Example (already filled in)

Date	Subject	Topic	Done
Mon 8 Jul	Life Sciences	Your weakest topic, notes + summary	✓
Tue 9 Jul	Life Sciences	Next topic + 10 practice questions	✓
Wed 10 Jul	Life Sciences	A full past paper under time	–
Thu 11 Jul	Life Sciences	Mark the paper, revise weak spots	–

Your plan

Saved!

Your plan is saved only on this device. Use "Export plan" to keep a copy.

Every mark you drop in Grade 10 is one you already know how to get. Go and get it.