Algebra, Sequences & Function Machines — 11+ Topic Guide

Part of the GrammarMock topic guides. Drill with the Algebra & Sequences mini-tests.

What "algebra" means at 11+

Nobody expects a Year 5 child to expand brackets like a GCSE student. 11+ algebra is three specific skills: solving a simple equation, running a function machine forwards and backwards, and finding the rule in a sequence. All three are teachable to full fluency.

Solving simple equations

"If 4b − 5 = 23, what is b?"

The method to teach: undo operations in reverse order. The equation did "×4 then −5" to b; to get b back, do "+5 then ÷4". So 23 + 5 = 28, then 28 ÷ 4 = 7.

Say it as a story: "b was multiplied by 4 and then 5 was taken away, leaving 23. Put the 5 back: 28. Un-multiply: 7."

Harder variant with brackets: 3(x − 2) = 18. Undo the ×3 first: x − 2 = 6, so x = 8. The bracket means "this whole thing was multiplied by 3" — undo the outside before the inside.

Two-variable substitution: "If 5a + 3b = 41 and a = 4, find b." Substitute what you know: 20 + 3b = 41, then solve normally. The only skill is not panicking at seeing two letters.

Function machines

Forwards is easy (input 6 → ×3 → +4 → output 22). The exam money is in backwards:

"A machine multiplies by 5 then subtracts 12. The output is 43. What was the input?"

Rule: go backwards through the machine, inverting each operation. Last operation was −12, so first undo: 43 + 12 = 55. Then undo ×5: 55 ÷ 5 = 11.

The classic error is undoing in the forwards order (÷5 first, then +12: 43÷5 = 8.6 + 12 = 20.6 — nonsense, and the nonsense is the tell). Teach: "walk back through the machine from the exit."

Sequences: finding and using the rule

Term-to-term rules ("each term is double the previous, minus 2") — just run the rule carefully, writing every term. Errors here are pure arithmetic slips; writing each step kills them.

Position-to-term (nth term) — the stretch skill: "What is the 10th term of 6, 10, 14, 18...?" The difference is 4, so the rule is "4n plus-or-minus something". 4×1 = 4, but the first term is 6, so the rule is 4n + 2. Check with term 2: 4×2 + 2 = 10 ✓. Then the 10th term is 4×10 + 2 = 42 — no need to write out ten terms.

Teach the check step as compulsory: build the rule from term 1, verify on term 2, only then use it.

Missing middle terms: "4, ?, 16, 32, 64" — work from the ends. The right side doubles, so test doubling from the left: 4 → 8 → 16 ✓.

Consecutive number puzzles

"Two consecutive odd numbers add to 52. What is the larger?"

Method without algebra: half of 52 is 26, so the numbers straddle 26 → 25 and 27. Larger = 27.

Method with algebra (for children ready for it): n + (n + 2) = 52, so n = 25.

Three consecutive numbers summing to S: the middle one is always S ÷ 3. "Three consecutive odd numbers sum to 87" → middle = 29 → they're 27, 29, 31. This ÷3 shortcut turns a hard-looking question into a five-second one.

Age problems (the hard finisher)

"Adam is 3 times as old as Ben. In 7 years, Adam will be twice as old as Ben. How old is Adam?"

These need a table: ages NOW and ages LATER as separate columns. Now: Ben = b, Adam = 3b. In 7 years: Ben = b + 7, Adam = 3b + 7. The condition: 3b + 7 = 2(b + 7). Solve: 3b + 7 = 2b + 14, so b = 7, Adam = 21. Check in the story: now 21 and 7 (3× ✓); in seven years 28 and 14 (2× ✓). The check-in-the-story step is what separates a right answer from a plausible-looking wrong one.

One-week drill plan

• Days 1-2: equations (one-step, then two-step, then brackets)
• Day 3: function machines backwards
• Days 4-5: nth-term rules with the verify-on-term-2 habit
• Day 6: consecutive number shortcuts + one age problem
• Day 7: Algebra & Sequences mini-test, timed