Module 15 of 45Lesson 1 of 40 · Unit 1 · From Numbers to Letters
Algebra · Unit 1
From Numbers to Letters
Algebra looks like a new language, but it is just arithmetic with one twist: a letter stands in for a number you do not know yet. Master that one twist and everything in this course opens up.
By the end of this unit you can
- Turn word descriptions like “5 more than n” into algebra
- Use algebra shorthand: write 3 × n as 3n and n ÷ 4 as n4
- Substitute a number into an expression and evaluate it
- Handle expressions with two different letters
- Substitute negative numbers without sign slips
- Use BIDMAS so 3n² and (3n)² never get confused
The one idea behind all of it
A letter in algebra is a box with a number hiding inside. The expression tells you what will happen to that number, whatever it turns out to be:
if n = 4, then 3n + 2 = 3 × 4 + 2 = 14
Swap the letter for the number, then calculate with BIDMAS as usual.
That swap is called substitution. Every formula you will ever use — areas, speeds, phone tariffs, exam grades — works exactly this way.
1
Saying it in symbols
Turn everyday words into an expression
Read the words slowly and build the expression piece by piece. “5 more than n” means start at n, add 5: n + 5. “3 less than x” means start at x, take 3 away: x − 3 — watch the order, it is the x that loses 3.
Worked example · Write “7 more than double m” as algebra
1
Find the pieces
“double m” and “7 more”
2
Double m means
2 × m = 2m
3
7 more than that means
add 7
4
Write it
2m + 7
Build in the order the words act: first the doubling, then the adding. 2m + 7 and 2(m + 7) are different expressions — the words tell you which.
Practice · Write as algebra · 10 questions
15 more than n
Answern + 5
23 less than x
Answerx − 3
3double m
Answer2m
4half of t
Answert2
54 times k
Answer4k
6the sum of a and b
Answera + b
710 minus y
Answer10 − y
87 more than double n
Answer2n + 7
93 less than 5 times x
Answer5x − 3
101 more than a quarter of p
Answerp4 + 1
2
Algebra shorthand
Why we never write the × sign
Algebra drops the multiplication sign: 3 × n is written 3n, a × b is written ab, and n × n is written n². Division becomes a fraction: k ÷ 4 is written k4. Numbers always go in front of letters: 3n, never n3.
Worked example · Write 2 × a × b + a × a in shorthand
1
Drop the × signs in 2 × a × b
2ab
2
a × a is a squared
a²
3
Put it together
2ab + a²
a × a = a², not 2a. Doubling and squaring are different: 2a means two copies added; a² means a multiplied by itself.
Practice · Write in shorthand · 10 questions
13 × n
Answer3n
2a × b
Answerab
3n × n
Answern²
4k ÷ 4
Answerk4
51 × x
Answerx
62 × a × b
Answer2ab
75 × m × m
Answer5m²
8(a + b) ÷ 2
Answera + b2
9x × y × 3
Answer3xy
10n × n × n
Answern³
3
Substituting a number
The letter leaves, the number takes its place
To substitute, rewrite the expression with the number in the letter’s place — brackets help — then calculate. If n = 4, then 3n + 2 becomes 3 × (4) + 2 = 14. Never glue digits together: 3n with n = 4 is 12, not 34.
A letter is just a number that has not arrived yet. When it arrives, the machine runs.
Worked example · Evaluate 5n − 3 when n = 6
1
Write the expression
5n − 3
2
Replace n with (6)
5 × (6) − 3
3
Multiply first (BIDMAS)
30 − 3
4
Finish
27
Always write the substitution line “5 × (6) − 3” before calculating. One extra line, and sign and order errors almost disappear.
Practice · Evaluate · 10 questions
13n + 2 when n = 4
Answer14
22x + 7 when x = 5
Answer17
35m − 3 when m = 6
Answer27
410 − 2k when k = 3
Answer4
54t + 1 when t = 0
Answer1
6n2 + 3 when n = 8
Answer7
77a − 10 when a = 2
Answer4
820 − 3y when y = 5
Answer5
96p + p when p = 3
Answer21
10x4 + 9 when x = 12
Answer12
4
Two letters at once
Each letter gets its own number
With two letters, substitute both at the same time and keep them apart: if a = 5 and b = 2, then 2a + 3b = 2 × (5) + 3 × (2) = 10 + 6 = 16. Write both substitutions before calculating anything.
Worked example · Evaluate 4x − 2y when x = 3, y = 5
1
Replace each letter
4 × (3) − 2 × (5)
2
Multiply both parts
12 − 10
3
Finish
2
Two letters, two substitutions, one line each. Do not calculate anything until every letter has been swapped for its number.
Practice · Evaluate · 10 questions
12a + 3b when a = 5, b = 2
Answer16
24x − 2y when x = 3, y = 5
Answer2
3a + 2b when a = 7, b = 4
Answer15
45m − n when m = 3, n = 8
Answer7
53p + 2q when p = 4, q = 5
Answer22
6xy when x = 6, y = 7
Answer42
72ab when a = 3, b = 4
Answer24
810 − a − b when a = 3, b = 5
Answer2
9ab − c when a = 4, b = 5, c = 9
Answer11
10a + b2 when a = 9, b = 5
Answer7
5
Negative numbers
Brackets are your seatbelt
When the number is negative, ALWAYS put it in brackets before calculating. If n = −3, then n² = (−3)² = 9, and 2n = 2 × (−3) = −6. Without the brackets, minus signs get lost — with them, the arithmetic runs itself.
Worked example · Evaluate 10 − 2n when n = −4
1
Replace n with (−4)
10 − 2 × (−4)
2
Multiply first
2 × (−4) = −8
3
Subtracting −8 means adding 8
10 − (−8) = 10 + 8
4
Finish
18
Two minus signs met and became a plus. That only stays visible because the brackets kept the −4 in one piece.
Practice · Evaluate · 10 questions
13n when n = -2
Answer-6
2n + 7 when n = -3
Answer4
3n² when n = −3
Answer9
42x + 1 when x = -5
Answer-9
510 − n when n = -4
Answer14
64m − 3 when m = -2
Answer-11
7n² + 1 when n = −5
Answer26
86 − 2k when k = -3
Answer12
9ab when a = -4, b = 3
Answer-12
10n² − n when n = −2
Answer6
6
BIDMAS inside expressions
3n² and (3n)² are different animals
BIDMAS still rules after substitution. In 3n², only n is squared: if n = 2, that is 3 × 4 = 12. In (3n)², the whole 3n is squared: (6)² = 36. Brackets first, then indices, then multiply/divide, then add/subtract.
Worked example · Evaluate 2(n + 3) when n = 5
1
Replace n with (5)
2 × (5 + 3)
2
Brackets first
5 + 3 = 8
3
Multiply
2 × 8 = 16
The bracket is a package: everything inside gets finished before anything outside touches it.
Practice · Evaluate · 10 questions
12(n + 3) when n = 5
Answer16
23n² when n = 2
Answer12
3(3n)² when n = 2
Answer36
45(x − 1) when x = 4
Answer15
5n² + 2n when n = 3
Answer15
62m² when m = 3
Answer18
7(2m)² when m = 3
Answer36
810 − (k + 3) when k = 4
Answer3
94(t + 2) − 5 when t = 3
Answer15
10(n + 1)² when n = 4
Answer25
The substitution routine
Same four moves, every single time.
1
Write the expression firstCopy it down exactly. No shortcuts yet.
2
Swap every letter for its number — in bracketsOne line: 5n − 3 becomes 5 × (6) − 3. Negative numbers keep their brackets.
3
Calculate with BIDMASBrackets, indices, multiply/divide, add/subtract.
✓
Sense-check the answerPositive when you expected negative? Huge when inputs were small? Re-check step 2.
The mistakes that cost the most marks
1Gluing digits together
Wrongn = 4, so 2n = 24
Rightn = 4, so 2n = 2 × 4 = 8
2n always means 2 × n. The invisible multiplication sign is the single most important thing in algebra notation.
2Confusing 2a with a²
Wronga = 5, so a² = 10
Righta = 5, so a² = 5 × 5 = 25
2a is a doubled; a² is a times itself. For a = 5 they give 10 and 25 — not even close.
3Losing the minus when squaring
Wrongn = −3, so n² = −9
Rightn = −3, so n² = (−3)² = 9
Squaring a negative gives a positive: (−3) × (−3) = 9. Brackets around the substituted number make this automatic.
4Squaring more than you should
Wrongn = 2, so 3n² = 36
Rightn = 2, so 3n² = 3 × 2² = 3 × 4 = 12
The little ² touches only what it sits on. Only (3n)² squares the whole thing.
5Reversing “less than”
Wrong“3 less than x” → 3 − x
Right“3 less than x” → x − 3
It is the x that loses 3. Try it with numbers: 3 less than 10 is 7, which is 10 − 3.
Tasks to submit
Criterion B · Investigating patterns
The Number-Trick Machine
Here is a number trick. Think of a number. Add 4. Double the result. Subtract 8. Halve the result. Try it — you always end up back at your starting number.
- Run the trick for at least six starting numbers, including 0 and a negative number. Record the results in a table.
- Describe what you notice.
- Let the starting number be n. Write what each step does to n, one line per step.
- Use your algebra to prove the trick always returns n.
- Now design your own trick that always ends on 5, whatever number the player thinks of. Write it as instructions, test it on three numbers, and prove it with algebra.
Submit: photos or a scan of your working, or a typed document. Marked out of 8 against Criterion B. Testing many numbers is not enough for the top band — the algebra proof is what shows the trick can NEVER fail.
Criterion D · Applying maths in real-life contexts
The Gaborone Taxi
A Gaborone taxi driver charges a P30 pick-up fee plus P7 for every kilometre. The fare in pula for a journey of k kilometres is 30 + 7k.
- Explain what the 30 and the 7 each mean in real life.
- Use substitution to find the fare for journeys of 3 km, 8 km and 10 km.
- A passenger pays P100. Roughly how far did they travel? Show your reasoning.
- Write the fare expression for a different driver who charges P20 pick-up and P9 per kilometre. Which driver is cheaper for a 4 km trip? For a 12 km trip?
- Give two real-life situations where the model 30 + 7k would give the wrong fare, and explain why.
Submit: your working plus short written explanations. Marked out of 8 against Criterion D. The top band needs part 5 done properly — a real discussion of where the model breaks, not just correct arithmetic.
When you are ready
The mastery quiz is 10 questions drawn at random from a
bank of 25. You need 70% to pass and you have three attempts. Work through the practice questions
above first — especially any concept where you had to look at the answer more than twice.
Printable version
Download Unit 1 as a PDF — the same lesson, laid out for printing and for working offline.
Download Unit 1 as a PDF — the same lesson, laid out for printing and for working offline.