Mathematics 8 · Handsworth 2026–27 · Unit 4
Patterns, Expressions & Two-Step Equations
Ms. Ras Mulinta
Handsworth Secondary
Guided notes
Algebra is pattern-spotting written down. When something grows by the same amount each step, you can describe the whole pattern with one short rule — and then jump straight to the 100th step without listing the first 99. In this unit you read patterns from tables and graphs, write rules as expressions, plug numbers in to evaluate them, and finally solve two-step equations to find an unknown. We warm up with a quick integer review, because the patterns here can dip below zero. Each idea has a worked example, then a few problems to try. Answers sit at the end of each section so you can check yourself right away.
Name:Block:Date:
Integer warm-up signs we will need
Patterns and equations here use integers — whole numbers and their negatives. Two quick rules carry most of the work. Adding: same signs add and keep the sign (−3 + −5 = −8); different signs subtract and take the bigger size's sign (−7 + 4 = −3). Multiplying or dividing: same signs give a positive, different signs give a negative (−4 × 3 = −12, but −4 × −3 = 12).
Subtracting is just adding the opposite: 5 − 8 = 5 + (−8) = −3. And subtracting a negative adds: 6 − (−2) = 6 + 2 = 8.
Worked example
Find −2 × 5 + 3.
Multiply first: −2 × 5 = −10.
Then add: −10 + 3 = −7.
= −7
Check yourself
- −6 + 9
- 4 − 11
- −3 × −7
- −20 ÷ 5
Answers
1. 3 2. −7 3. 21 4. −4
1 · Discrete linear relations tables, graphs, a rule
A relation is linear when it changes by the same amount each step — a constant jump. It is discrete when only whole-number steps make sense (you can have 3 tables but not 31⁄2), so the graph is a set of separate points, not a joined line. To find the rule, ask two things: how much does the output jump each step (that is the multiplier), and where would it start at step 0 (the constant you add).
Worked example — read a table, write the rule
A pattern of tile figures uses this many tiles:
| Figure number (n) | 1 | 2 | 3 | 4 |
|---|
| Tiles (t) | 5 | 8 | 11 | 14 |
Each step the tiles jump by +3, so the rule multiplies n by 3.
3 × 1 = 3, but we have 5 — so we add 2 to land on the values: t = 3n + 2.
Check figure 4: 3 × 4 + 2 = 14. ✓
Rule: t = 3n + 2
Graphing it: plot each pair (n, t) as a single point — (1, 5), (2, 8), (3, 11), (4, 14). Because the jump is constant, the points line up straight. We leave them as dots, not a solid line, because only whole figure numbers exist.
Worked example — extend the pattern
Using t = 3n + 2, how many tiles in figure 10?
t = 3 × 10 + 2 = 30 + 2 = 32.
= 32 tiles
Check yourself
- A table shows: n = 1, 2, 3, 4 gives y = 7, 9, 11, 13. What is the jump each step, and what is the rule?
- Use your rule to find y when n = 8.
Answers
5. The jump is +2 each step, so multiply n by 2. 2 × 1 = 2 but we have 7, so add 5: y = 2n + 5. Check n = 4: 2 × 4 + 5 = 13. ✓
6. y = 2 × 8 + 5 = 16 + 5 = 21
2 · Writing & evaluating expressions substitution
An expression is a rule built from numbers, a variable (a letter standing for an unknown), and operations — like 3n + 2. It has no equals sign; it just describes a value. Evaluating means swapping the variable for a number (substitution) and working it out. Watch the order of operations: multiply or divide before you add or subtract.
Worked example — write an expression
A taxi charges a $4 flat fee plus $2 per kilometre. Write the cost for k kilometres.
Per-km part is 2k; the flat fee adds 4.
Cost = 2k + 4
Worked example — evaluate by substitution
Evaluate 2k + 4 when k = 6.
Substitute: 2 × 6 + 4.
Multiply first: 12 + 4 = 16.
= 16
Worked example — a negative value
Evaluate 5x − 3 when x = −2.
Substitute: 5 × (−2) − 3.
Multiply first: −10 − 3 = −13.
= −13
Check yourself
- Write an expression for "7 more than triple a number n."
- Evaluate 4a − 5 when a = 3.
- Evaluate 6 + 2m when m = −4.
Answers
7. 3n + 7 8. 4 × 3 − 5 = 12 − 5 = 7 9. 6 + 2 × (−4) = 6 + (−8) = −2
3 · Solving two-step equations undo in reverse order
An equation has an equals sign: the two sides balance. To solve it, find the value of the variable that makes the sides equal. A two-step equation like 3x + 5 = 20 was built by two operations on x (times 3, then add 5). To undo it, work backwards with the opposite operations: first undo the adding, then undo the multiplying. Whatever you do to one side, do to the other — that keeps the balance.
Worked example
Solve 3x + 5 = 20.
Undo the +5 first — subtract 5 from both sides: 3x = 15.
Undo the ×3 — divide both sides by 3: x = 5.
Check: 3 × 5 + 5 = 15 + 5 = 20. ✓
x = 5
Worked example — a negative coefficient
Solve −2x + 7 = 1.
Subtract 7 from both sides: −2x = −6.
Divide both sides by −2: x = 3 (negative ÷ negative is positive).
Check: −2 × 3 + 7 = −6 + 7 = 1. ✓
x = 3
Worked example — a negative solution
Solve 4x + 18 = 2.
Subtract 18 from both sides: 4x = −16.
Divide both sides by 4: x = −4.
Check: 4 × (−4) + 18 = −16 + 18 = 2. ✓
x = −4
Worked example — division form
Solve x/3 − 2 = 4.
Add 2 to both sides: x/3 = 6.
Multiply both sides by 3: x = 18.
Check: 18 ÷ 3 − 2 = 6 − 2 = 4. ✓
x = 18
Check yourself
- Solve 2x + 9 = 17.
- Solve 5x − 4 = 31.
- Solve −3x + 1 = 10.
Answers
10. 2x = 8, so x = 4. Check: 2 × 4 + 9 = 17. ✓
11. 5x = 35, so x = 7. Check: 5 × 7 − 4 = 31. ✓
12. −3x = 9, so x = −3. Check: −3 × (−3) + 1 = 9 + 1 = 10. ✓
If you want to go further
A pattern's rule and an equation are two sides of the same coin. The rule t = 3n + 2 tells you the output for any input. Turn it around — "which figure uses 32 tiles?" — and you get the equation 3n + 2 = 32, which you solve the same two-step way: 3n = 30, n = 10. Writing the rule and solving the equation are the same algebra, just asked in opposite directions.