Algebra: A Balancing Act
GCSE MATHS RESIT · EDEXCEL FOUNDATION · 4 OF 7
Algebra: a balancing act
Letters are just numbers you have not met yet. Five routines that turn up on every paper.
Paper 1 · Paper 2 · Paper 3
Lesduur is: 60 minIntroductie
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GCSE MATHS RESIT · EDEXCEL FOUNDATION · 4 OF 7
Algebra: a balancing act
Letters are just numbers you have not met yet. Five routines that turn up on every paper.
Paper 1 · Paper 2 · Paper 3
Lesson 4 of 7, about 60 minutes. Timings: do now 5 min, vocabulary 3 min, like terms 10 min, substitution 8 min, solving 12 min, expanding 7 min, nth term 7 min, exam question and exit 8 min.
Many resit students have decided algebra "isn't for them". The routines today are short and mechanical on purpose: they are algebra marks students can rely on with practice.
Device-free fallback for the whole lesson: every activity works on mini whiteboards or paper; the notes on each slide say how. Image credit: Jeremy Bishop / Unsplash
DO NOW · 4 MINUTES
Four from last lesson
Share £50 in the ratio 2 : 3
£20 and £30
Work out 20% of £35
£7
Work out 3 × −4
−12
Work out 5²
25 (5 × 5, not 10)
Students answer on paper as they arrive. Click four times to reveal the answers.
3 × −4 and 5² are here on purpose: both come back in substitution today.
Device-free: questions on the board before the lesson.
THE WORDS THE EXAM USES
Four words, four different jobs
TERM
3x
One part on its own
EXPRESSION
3x + 2
Terms added or taken away. No equals sign
EQUATION
3x + 2 = 14
Has an equals sign. You can solve it
FORMULA
C = 3 + 2m
A rule linking letters
3x means 3 × x. x² means x × x. 2(x + 1) means 2 × (x + 1).
"Simplify" means collect terms. "Solve" means find the number. "Expand" means remove the brackets.
Two minutes. These command words are worth learning because students often do the wrong job: for example, trying to "solve" an expression that has no equals sign.
Ask: "Which of the four could you solve?" (Only the equation.)
Device-free: students copy the four boxes into their books.
Sort into like terms. One fits no basket.
x terms
y terms
Numbers on their own
4x
7
−y
2y
xy
−3x
−2
x
5y
The sign in front belongs to the term. Pick it up with the term.
Answers: x terms 4x, −3x, x. y terms −y, 2y, 5y. Numbers 7, −2.
xy fits no basket: it is not an x term or a y term, it is its own kind of term. Many students think "it has an x in it, so it goes with the x terms".
Point out that the minus sign travels with the term: −3x, not 3x.
Device-free: write the nine terms on the board; students sort them into three lists on whiteboards.
COLLECTING LIKE TERMS
Simplify by sorting first
4x − y + 7 + 2y − 3x − 2 + x + 5y
x terms: 4x − 3x + x = 2x
y terms: −y + 2y + 5y = 6y
numbers: 7 − 2 = 5
A letter on its own means one of it: x is 1x.
−y is −1y.
This is the expression from the drag-and-drop, without xy. Colours match the baskets.
Four clicks: x terms, y terms, numbers, the final answer.
Common slips: 4x − 3x = x is fine, but students often write "1" (dropping the letter). And 4x − 3x + x is 2x, not 0x (they subtract the last x too).
Device-free: underline like terms in three colours on the board.
SUBSTITUTION
Swap the letter for the number
A taxi charges £3 to start, plus £2 a mile. The cost in pounds is C and the number of miles is m.
A 7-mile trip, so m = 7
The fare is £17. Multiply before adding, just like BIDMAS.
With a negative number, use brackets: if x = −4, then 3x = 3 × (−4) = −12.
2m means 2 × m. With m = 7 that is 14, not 27 ("2 and 7 side by side").
Two clicks: the working line, then the answer and the tip about negatives.
Ask: "How much for a 20-mile trip?" (£43.)
Device-free: whiteboards. Image credit: Greg Mak / Unsplash
C = 3 + 2m. Find C when m = 12.
27
60
17
36
Answer: 27 (3 + 24).
Slips: 60 = (3 + 2) × 12, adding first. 17 = 3 + 2 + 12, ignoring that 2m means multiply. 36 = 3 × 12, using the wrong numbers.
Device-free: whiteboards, show on three.
SOLVING EQUATIONS
Keep it balanced
3x + 5
20
Whatever you do to one side, do to the other. Undo the + 5 first, then the × 3.
3x + 5 = 20
− 5 both sides: 3x = 15
÷ 3 both sides: x = 5
Check: 3 × 5 + 5 = 20
The balance is level because both sides are equal. Taking 5 off one side only would tip it.
Three clicks: subtract 5, divide by 3, check.
The check is free marks insurance: substitute the answer back in. If it does not give 20, something slipped.
Common slip: dividing first (3x + 5 ÷ 3) or doing −5 on one side only.
Device-free: draw the see-saw on the board.
Solve each one
1. 2x + 3 = 11, so 2x = 8 and x = 4.
2. x ÷ 5 = 6, so x = 30.
3. 4x − 7 = 13, so 4x = 20 and x = 5.
Answers: 8 and 4; 30; 20 and 5.
Distractors match slips: 14 (11 + 3, adding instead of subtracting), 1.2 (6 ÷ 5, dividing instead of multiplying), 6 (13 − 7), 16 (a near miss).
The ÷ 5 one needs the inverse: undo dividing by multiplying.
Device-free: board and whiteboards.
EXPANDING BRACKETS
The bracket is a rectangle
x
+ 4
3
3x
12
Multiply everything inside the bracket by the number outside. Signs come too.
5(2x − 1) = 10x − 5
−2(x + 3) = −2x − 6
Height 3, width x + 4. Area = 3 × x plus 3 × 4.
The area model: a rectangle 3 high and (x + 4) wide, split into two parts. The x part is drawn wider than the 4 part because x is unknown; only the 3 and the 4 are to scale with each other.
Three clicks: 3x, 12, the full line.
The classic slip: 3(x + 4) = 3x + 4, multiplying only the first term. The rectangle makes it obvious that the 4 part has area too.
Device-free: draw the rectangle on the board.
Expand 4(x + 6)
Answer: 4x + 24. Accepted with or without spaces, either order.
If anyone writes 4x + 6, draw the rectangle again.
Device-free: whiteboards.
SEQUENCES · NTH TERM
5, 8, 11, 14, … What is the rule?
Term
5
8
11
14
+3
+3
+3
3n
3
6
9
12
each term is 2 more than 3n
1 It goes up by 3, so start with 3n.
2 Write out 3n: 3, 6, 9, 12.
3 How far off is it? +2 each time.
Four clicks: the +3 jumps, the 3n row, the "+2" link, the rule.
Check: n = 1 gives 3 × 1 + 2 = 5. n = 4 gives 14. The 100th term is 302.
Common slip: writing n + 3 (the "add 3" rule). That gives 4, 5, 6, … which goes up by 1, not 3.
Device-free: write the two rows on the board.
Find the nth term of 7, 11, 15, 19, …
4n + 3
n + 4
4n + 7
7n + 4
Answer: 4n + 3. It goes up by 4, so 4n: 4, 8, 12, 16. Each term is 3 more.
Slips: n + 4 (the "add 4" rule). 4n + 7 (using the first term as the number). 7n + 4 (numbers swapped).
EXAM-STYLE · PAPER 3 · CALCULATOR
Bike hire costs £8, plus £3 for each hour.
Ella paid £29.
How many hours did Ella hire the bike for?
Cost in £
8 + 3 × hours
(3 marks)
3
minute
00
second
3 minutes on the timer.
Mark scheme style: M1 for 29 − 8 (= 21) or forming 8 + 3h = 29. M1 for 21 ÷ 3. A1 7 hours.
Trial and improvement also works and earns marks if shown: 5 hours = £23, 6 hours = £26, 7 hours = £29.
Common slip: 29 ÷ 3 ≈ 9.67, forgetting the £8 start charge.
Device-free: same question on paper.
How many hours did Ella hire the bike for?
Answer: 7 hours. Check: 8 + 3 × 7 = 29.
Ask two students to show different methods: an equation, and trial and improvement.
Which algebra skill do you trust most now?
Collecting like terms
Substitution
Solving equations
Expanding brackets
Exit poll. The least-trusted skill goes into next lesson's do now.
Device-free: sticky notes on the way out.
BEFORE NEXT LESSON · 10 MINUTES
Three to try at home
1
Simplify 5a + 2b − 3a + b
2
Solve 3x − 4 = 11
3
Expand 6(x − 2)
Check question 2 by putting your answer back in. Corbettmaths 5-a-day (Foundation) mixes algebra into every set.
Answers: 1) 2a + 3b. 2) x = 5. 3) 6x − 12.
Next lesson: shape and space. Area and perimeter formulas are substitution in disguise.
