7.1 Relationships, Patterns and Simple Algebra

Four clothes for one relationship

Quantitative Literacy (QL) is 50 multiple-choice questions inside the combined Academic and Quantitative Literacy (AQL) paper. Published NBT Test Content asks writers to formulate and apply simple formulae, identify trends and patterns, and interpret quantitative information in words, symbols, and graphs. The QL subdomain Relationships, pattern, permutation is the same job in more detail: recognise, interpret, and represent a relationship in graphs, tables, words, and symbols, then manipulate simple algebraic expressions with ordinary arithmetic. There is no calculator. If a formula is needed, the item provides it. Independent OpenExamPrep pattern-and-algebra practice uses residence, copy-shop, and teaching-laboratory stories. It is not MAT domain-and-range work, not a confidential NBTP paper, and not a claim of official sponsorship.

MAT is a different afternoon paper for faculties that require mathematics. A QL incubator table wants the same story in four clothes, a substitution, or a small order count. Factorising a cubic is the wrong exam.

Same relationship, four representations

Worked example: teaching incubator. A biology laboratory notice says: “The incubator starts at 18 °C and rises 2 °C each hour.”

Words: start at 18 °C, plus 2 °C per hour. Symbols: T = 18 + 2t. Graph: horizontal axis t in hours, vertical axis T in °C; the points (0, 18), (1, 20), …, (5, 28) lie on a straight line, because every hour adds the same 2 °C.

After 5 hours: T = 18 + 2 × 5 = 18 + 10 = 28 °C. After 7 hours: 18 + 14 = 32 °C. After 0 hours the formula still returns 18 °C, which matches the table’s first row. The sentence, the table, the formula, and the line are one relationship. A trap is 18 + 2 + 5 = 25, adding the three printed numbers instead of multiplying the hourly rise by the number of hours.

Worked example: residence shuttle. A notice says a shuttle leaves the gate every 12 minutes, starting at 07:00. Clock times: 07:00, 07:12, 07:24, 07:36, 07:48. The pattern is “add 12 minutes.” If n = 0 at 07:00, departure n is 07:00 plus 12n minutes. The departure with n = 5 is 12 × 5 = 60 minutes later, so 08:00. Listing five times and then adding 12 once more must give the same 08:00 as the formula. A trap is treating n as the clock hour.

Simple formulae: substitute, then invert

QL formulae in this subdomain are usually linear: a fixed start plus a constant rate times a count.

Worked example: faculty copy card. The shop charges R15 for the card plus R4 per colour page. The item gives C = 15 + 4n, with C in rand and n the page count.

• For 8 pages: C = 15 + 4 × 8 = 15 + 32 = R47.
• For C = R47: 15 + 4n = 47; 4n = 32; n = 8 pages.
• For C = R63: 63 − 15 = 48; 48 ÷ 4 = 12 pages.

Forward work substitutes. Reverse work subtracts the fixed fee, then divides by the rate. Trap readings: n = 47 (the total treated as a page count), n = 15 (the card fee treated as pages), n = 4 (the rand-per-page figure copied as a count).

Worked example: extra samples. A protocol gives volume V = 8 + 3s millilitres, where s is the number of extra samples after the first tray. For s = 6: V = 8 + 18 = 26 mL. If the bottle holds 41 mL: 41 − 8 = 33; 33 ÷ 3 = 11 extra samples. Keep millilitres on every line. 26 mL is not 26 L.

Manipulate simple expressions

“Manipulate” in QL means expand, substitute, and tidy with +, −, ×, and ÷. It does not mean MAT factorisation drills.

Worked example. Expression 2(t + 5) − 4. 2(t + 5) = 2t + 10. 2t + 10 − 4 = 2t + 6. If t = 7: 2 × 7 + 6 = 14 + 6 = 20. Direct substitution: 2(7 + 5) − 4 = 2 × 12 − 4 = 24 − 4 = 20. Same value. The expanded form is easier to graph as “start at 6, rise 2 per hour.”

Worked example. 3(x − 2) + 5 = 3x − 6 + 5 = 3x − 1. At x = 4: 12 − 1 = 11. Expanding first and substituting first must agree, or an option that dropped the minus sign is waiting.

Number patterns and order patterns

This subdomain names pattern and permutation. Treat them as two jobs.

Number pattern. Residence tutorial attendance: 12, 15, 18, 21. Common difference +3. Next term 24. If the first term is k = 1, a matching formula is 9 + 3k: k = 1 gives 12; k = 4 gives 21. The tenth term: 9 + 3 × 10 = 9 + 30 = 39, or 12 + 3 × 9 = 39. A doubling pattern is different: 2, 4, 8, 16 next 32, because each term is ×2, not +2. Mixing “add 2” with “multiply by 2” is a classic trap when the stem shows powers of two.

Order pattern (permutation). How many different orders can a small set occupy? Four lab groups present one after another. The first slot has 4 choices, then 3, then 2, then 1: 4 × 3 × 2 × 1 = 24 orders. That product is 4!, read “four factorial.” QL expects the multiplication, not a calculator key. Three students at three benches: 3 × 2 × 1 = 6 orders. Five posters on one rail: 5 × 4 × 3 × 2 × 1 = 120 orders. Trap answers: 4 (counting the groups instead of the orders), 8 (4 × 2), 16 (4 × 4, which would allow a group to present twice in the same line-up).

Worked example: three posters. Posters A, B, and C on one rail, left to right. The six orders are ABC, ACB, BAC, BCA, CAB, CBA. That list is the same 6 as 3 × 2 × 1. If the rail has four hooks and four posters, jump to 24 rather than trying to list them in the venue.

A method for mixed clothes

1. Name the independent quantity (hours, pages, group number) and the dependent quantity (temperature, cost, volume).
2. Check that the table, sentence, formula, and sketch tell the same story.
3. Substitute to go forwards; subtract then divide to go backwards.
4. For an order count, multiply the remaining choices; do not add them.
5. Interpret the last number back in the laboratory or campus story.

Traps

• Adding 18 + 2 + 5 instead of 18 + 2 × 5
• Treating the total cost C as the page count n
• Calling four groups “four orders”
• Drawing a curve when every increment is constant, or a straight line when the rule is doubling
• Reaching for MAT quadratic language on a linear campus table
• Using a calculator; QL is calculator-free

Independent OpenExamPrep QL teaching on relationships stays in those four clothes plus a small factorial: the same campus rule, written so you can still name the incubator, the copy card, or the presentation order after the last arithmetic step.