7.1 Relationships, Patterns & Simple Algebra
Key Takeaways
- The same quantitative relationship can appear as a table, a graph, a sentence, or a simple algebraic expression — translate freely between forms before computing
- A linear pattern has a constant difference between successive terms; the nth term is built from that common difference and the starting value
- Simple algebra on QL uses the four arithmetic operations with a letter standing for an unknown or a varying quantity — substitute carefully and respect order of operations
- QL-level permutations count ordered arrangements of a few distinct items (for example 3 × 2 × 1 = 6); they are not advanced combinatorics formulae
Why relationships and patterns matter on QL
The CEA Quantitative Literacy (QL) construct includes relationships, pattern, and permutation: recognising how quantities depend on each other, extending numerical or spatial patterns, and counting simple ordered arrangements. On the Academic and Quantitative Literacy (AQL) test you work without a calculator, so every translation between a table, a sketch of a graph, a verbal rule, and a short algebraic expression must be done by hand.
This section is not school algebra for its own sake. It is tertiary-readiness literacy: can you see that “each extra tutorial costs R80 after a R25 booking fee” is the same story as a table of totals, a straight-line graph, and the expression C = 25 + 80n?
Four faces of the same relationship
| Form | What you see | What you do |
|---|---|---|
| Words | “Cost rises by R80 for each extra tutorial after a R25 fee.” | Underline the fixed part and the per-unit part |
| Table | Columns for n (tutorials) and C (rand) | Check the constant difference in C when n increases by 1 |
| Graph | Horizontal n, vertical C; points rise in equal steps | Steepness encodes the R80-per-tutorial rate |
| Symbols | C = 25 + 80n | Substitute a value of n and evaluate with order of operations |
If two forms disagree, one was misread — do not average them. Re-translate carefully.
Worked example — table → rule → expression
A writing-centre booking table for first-year commerce students:
| Tutorials booked (n) | Total fee (C in rand) |
|---|---|
| 0 | 25 |
| 1 | 105 |
| 2 | 185 |
| 3 | 265 |
Differences in C: 105 − 25 = 80, 185 − 105 = 80, 265 − 185 = 80. The change per tutorial is constant, so the relationship is linear.
- Fixed booking fee (when n = 0): R25
- Variable part: R80 per tutorial
- Expression: C = 25 + 80n
For six tutorials: C = 25 + 80 × 6. Multiply first: 80 × 6 = 480, then 25 + 480 = R505.
Check against the pattern: from n = 3 (C = 265) to n = 6 is three more steps of 80 → 265 + 240 = 505. Same answer.
Worked example — words → table → graph reading
A campus shuttle leaves the Science Faculty at 07:00. The verbal rule is: “Distance from campus increases by 5 km every 10 minutes until the first stop.”
| Time after departure (min) | Distance (km) |
|---|---|
| 0 | 0 |
| 10 | 5 |
| 20 | 10 |
| 30 | 15 |
On a distance–time sketch, the points (0, 0), (10, 5), (20, 10), (30, 15) lie on a straight line. Average speed for the first 30 minutes:
- Distance = 15 km; time = 30 min = 0.5 h
- Speed = 15 ÷ 0.5 = 30 km/h
Equivalently, 5 km / 10 min = 0.5 km per minute × 60 = 30 km/h. The steepness of that line is the rate; a flatter line would mean a slower shuttle.
Number patterns and the nth term (QL level)
A sequence has a common difference when each term minus the previous term is the same.
Sequence: 4, 7, 10, 13, …
- Common difference d = 7 − 4 = 3
- First term a₁ = 4
- General term for position n (starting at n = 1): aₙ = 4 + (n − 1) × 3 = 3n + 1
Checks: n = 1 → 3(1) + 1 = 4; n = 5 → 3(5) + 1 = 16. The 5th term is 16, not “add 3 five times to 4” without tracking position carefully.
Spatial pattern (tiles): A residence courtyard path uses square tiles. Row 1 has 1 tile, row 2 has 3 tiles, row 3 has 5 tiles (odd numbers). Row k has 2k − 1 tiles. Row 8 has 2(8) − 1 = 15 tiles. Total tiles in the first 4 rows: 1 + 3 + 5 + 7 = 16.
Simple algebraic expressions with arithmetic operations
Letters stand for numbers. Evaluate by substituting, then apply brackets → exponents (if any) → × and ÷ left to right → + and − left to right.
Example: A lab fee is F = 40 + 15h, where h is hours of after-hours access. For h = 4: F = 40 + 15 × 4 = 40 + 60 = R100.
Example with two operations: T = 3(m + 2) − 5, and m = 6.
- Bracket: m + 2 = 8
- Multiply: 3 × 8 = 24
- Subtract: 24 − 5 = 19
If you subtract before multiplying, you get a wrong 13 — classic order-of-operations trap.
Solving a one-step unknown (QL style): “Five times a number of modules, plus 2, equals 17.” Write 5x + 2 = 17 → 5x = 15 → x = 3. Keep steps reversible and check: 5(3) + 2 = 17.
Patterns and permutations at QL level
QL permutations are ordered arrangements of a handful of distinct items — count by multiplying choices, not by memorising advanced combinatorics.
Example — three SRC roles: Three distinct students (A, B, C) must be assigned as Chair, Secretary, and Treasurer (each role once).
- Choices for Chair: 3
- Then Secretary: 2
- Then Treasurer: 1
- Total ordered assignments: 3 × 2 × 1 = 6
Listing: ABC, ACB, BAC, BCA, CAB, CBA — six outcomes. This is not “choose any 3 from a large class” with binomial coefficients; if a stem only asks for orders of three named people into three roles, the product 3 × 2 × 1 is enough.
Example — timetable slots: Four elective codes (LAW, ECO, HIS, PSY) and two consecutive free periods — how many ordered pairs (Period 1, Period 2) with different codes?
- Period 1: 4 choices
- Period 2: 3 remaining choices
- Total: 4 × 3 = 12
If order did not matter, you would be counting unordered pairs (fewer results). Read whether the stem cares about order (permutation flavour) or only the set (combination flavour). QL usually signals order with words such as “first,” “next,” “sequence,” or named roles.
Common traps
| Trap | Wrong move | Correct move |
|---|---|---|
| Table with constant difference | Fit a multiplicative rule by force | Confirm equal steps; use C = start + d × n (with n defined carefully) |
| Graph vs table | Guess from axis labels only | Read paired values; check one calculated point |
| Expression a + b n | Add a + b then multiply by n | Multiply b × n first, then add a |
| “Next term” | Add the term number instead of the common difference | Always add d, or use the nth-term formula |
| Arrangements | Treat ordered roles as an unordered set | Multiply successive choices when order matters |
Practice habit
- Name the independent quantity (often n, time, or count) and the dependent quantity (cost, distance, score).
- Build or read a small table even if the stem only gives words.
- Write a compact rule and test it on one known pair.
- For arrangements, decide whether order matters, then multiply the shrinking choice counts.
These moves feed directly into change-and-rates items (next section), where the steepness of a graph is precisely the rate hidden inside a linear relationship.
A campus printing account charges a R20 activation fee plus R4 per page. Which expression gives the total cost C (in rand) for p pages?
The sequence 5, 9, 13, 17, … has a constant difference. What is the 6th term?
Three distinct first-years must speak first, second, and third in a tutorial. How many different speaking orders are possible?
A shuttle distance–time table shows 0 km at 0 min and 12 km at 24 min, with equal steps of 3 km every 6 minutes. What average speed does this linear pattern represent?