This mini-lesson walks you through the whole KS3 Algebra strand: using letters for numbers, simplifying and expanding, formulae, solving equations, coordinates & straight-line graphs, and sequences.
Make a guess: when a = 4, is 3a + 2 equal to 14 or 18?
Mission 1Is 3a + 2 equal to 14 or 18 when a is 4?
Mission 2Double it, add 3, get 11. What was my number?
Mission 3How many dots are in pattern 100?
Work through each screen, answer the questions as you go (some are calculations, some are ideas) and collect ⭐ stars. Press Start when you're ready.
Algebraic notation
Letters stand for numbers
In algebra a letter (a variable) stands for a number we don't know yet. There are shorthand rules for writing it:
3a means 3 × a — the number is written first, with no × sign.
ab means a × b.
a² means a × a (not a × 2).
a/2 or ½a means a ÷ 2.
Watch out:3a is not the same as 3 + a. 3a means three lots of a. If a = 5, then 3a = 15 but 3 + a = 8.
Quick check
What does it mean?
?Which expression means "double a number, then add 5"? Call the number n.
Substitution
Putting numbers in
Substitution means swapping a letter for its value and working out the answer. Remember to keep to the order of operations (BIDMAS).
Worked example — step by step
Find the value of 3a + 2 when a = 4.
Step 1: replace a with 4 → 3 × 4 + 2
Step 2: multiply first → 12 + 2
Step 3: add → 14
Watch out: do the multiplying before the adding. 3 × 4 + 2 = 14, not 3 × 6 = 18.
Calculate
Your turn — substitution
1Work out the value of 5x − 3 when x = 4.
Hint: 5x means 5 × 4. Do that first, then subtract 3.
Simplifying
Collecting like terms
To simplify an expression we collect like terms — terms with exactly the same letter part. You add or subtract their number parts (coefficients):
4a + 3a = 7a4 lots of a plus 3 lots of a make 7 lots of a
Unlike terms cannot be joined. 4a + 3b stays as 4a + 3b, because a and b are different.
Worked example
Simplify 5x + 2y + 3x − y.
x terms: 5x + 3x = 8x. y terms: 2y − y = y.
Answer: 8x + y
Watch out: a and a² are not like terms — a² means a × a. So a + a² cannot be simplified.
Quick check
Simplify it
?Simplify 7p + 4q − 2p + q.
Sort it
Simplify each expression
Tap the correct simplified form for each expression.
Mission 1 check
Is 3a + 2 equal to 14 or 18 when a is 4?
?Find the value of 5b + 3 when b = 2. Which answer and reason are right?
Mission 2 of 3
Double it, add 3, get 11. What was my number?
Make a guess: how do you undo 'double it, then add 3'?
Manipulating
Expanding brackets
To expand a bracket, multiply everything inside by the term outside:
3(x + 2) = 3x + 63 × x = 3x and 3 × 2 = 6
Worked example
Expand 5(2a − 3).
5 × 2a = 10a and 5 × (−3) = −15.
Answer: 10a − 15
Watch out: the outside number multiplies every term. 3(x + 2) = 3x + 6, not 3x + 2.
Quick check
Expand it
?Expand 4(3x − 5).
Manipulating
Factorising
Factorising is the reverse of expanding. You take out the highest common factor and put a bracket back:
6x + 9 = 3(2x + 3)3 is the common factor: 6x ÷ 3 = 2x, 9 ÷ 3 = 3
Worked example — step by step
Factorise 10a + 15.
Step 1: highest common factor of 10 and 15 is 5.
Step 2: divide each term by 5 → 2a and 3.
Step 3: write it as 5(2a + 3). Check: expand it back to 10a + 15. ✓
Tip: always check by expanding your answer — it should give the original expression.
Quick check
Factorise it
?Factorise fully 8x + 12.
Formulae
Using and rearranging formulae
A formula is a rule connecting quantities, e.g. the area of a rectangle: A = l × w. You can substitute values in, or change the subject to make a different letter the star of the show.
A = l w → w = A ÷ ldivide both sides by l to make w the subject
Worked example
A rectangle has area A = 24 and length l = 6. Find the width w.
w = A ÷ l = 24 ÷ 6 = 4
Golden rule: whatever you do to one side, do to the other — that keeps it balanced (more on that next).
Calculate
Your turn — using a formula
2The perimeter of a square is P = 4s, where s is the side length. If P = 36, find the side length s.
Hint: rearrange to s = P ÷ 4, then divide 36 by 4.
Equations
Solving linear equations
An equation says two things are equal, like a balance scale. To find the unknown, keep the scale balanced: do the same to both sides until the letter is alone.
Subtract 3 from both pans and the scale stays level: x = 5.
Worked example — step by step
Solve 2x + 3 = 11.
Step 1: subtract 3 from both sides → 2x = 8.
Step 2: divide both sides by 2 → x = 4.
Step 3: check by substituting: 2 × 4 + 3 = 11. ✓
Watch out: undo operations in reverse order — undo the "+3" before the "×2".
Calculate
Your turn — solve for x
3Solve 3x + 4 = 19. Type the value of x.
x =
Hint: subtract 4 from both sides (→ 3x = 15), then divide by 3.
Mission 2 check
Double it, add 3, get 11. What was my number?
?I think of a number. I multiply it by 4, then add 5. My answer is 33. What was my number, and what did I undo first?
Mission 3 of 3
How many dots are in pattern 100?
Make a guess: how can you find pattern 100 without drawing it?
Coordinates & graphs
Coordinates and functions
A point is written as (x, y) — go along the x-axis first, then up the y-axis. A function like y = x + 1 gives a y-value for every x, and plotting the points makes a graph.
When x = 1, y = 1 + 1 = 2, giving the point (1, 2) on the line.Quick check
Read the function
?For the function y = 2x + 1, what is y when x = 3?
Straight-line graphs
y = mx + c
Every straight line can be written y = mx + c:
y = m x + cm = gradient (steepness) · c = y-intercept (where it crosses the y-axis)
m is the gradient: how much y goes up for each 1 across. Bigger m = steeper.
c is the y-intercept: the y-value where the line crosses the y-axis (x = 0).
So for y = 3x + 2: gradient = 3, and it crosses the y-axis at 2. A simultaneous equations pair (two lines) is solved by the point where the lines cross.
Quick check
Gradient and intercept
?For the line y = 4x − 3, what are the gradient and y-intercept?
Sequences
Sequences and the nth term
A sequence is an ordered list of numbers following a rule. In a linear sequence the step (common difference) is the same each time:
The sequence 2, 5, 8, 11, … goes up in 3s. The rule is nth term = 3n − 1.
Finding the nth term
Step 1: the common difference is 3, so the rule starts with 3n.
Step 2: 3 × 1 = 3, but the first term is 2, so subtract 1.
Rule: nth term = 3n − 1. Check: n = 4 → 3 × 4 − 1 = 11. ✓
Calculate
Your turn — the nth term
4A sequence has nth term 4n + 1. Work out the 10th term (n = 10).
Hint: substitute n = 10 → 4 × 10 + 1.
Mission 3 check
How many dots are in pattern 100?
?Tiles make a pattern: 5, 9, 13, 17, … How many tiles are in pattern 50?
Sort it
Like or unlike?
Tap a term, then tap the box it belongs in. "Like 3x" means it can be added to 3x.
✅ Like 3x
🚫 Unlike 3x
Quick check
Spot the mistake
?Sam writes "3a + 2a = 5a²". What has gone wrong?
Recap
The key ideas to know
Notation: 3a = 3 × a; a² = a × a; 3a ≠ 3 + a
Substitution: swap the letter for its value, use BIDMAS
Equations: keep both sides balanced to find the unknown
Straight line: y = mx + c (m = gradient, c = y-intercept)
Sequences: nth term rule from the common difference
You've covered the key ideas in the KS3 Algebra strand — notation, manipulation, formulae, equations, graphs and sequences. Press Finish to see your score.
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Mini-lesson complete!
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You've worked through Algebra for KS3 Mathematics. 🎉