033i - Factorising Linear Expressions

Lesson 33 — Factorising Linear Expressions Using the Highest Common Factor

Strand: Algebra | Descriptor: AC9M8A01 | Duration: 45 minutes

Learning Intentions

• To factorise linear expressions by identifying the highest common factor (HCF) of their terms.
• To recognise factorising as the reverse of expanding, and to verify factorisations by re-expanding.

Success Criteria

I can:

1. Find the HCF of two or more numeric terms.
2. Find the HCF of two or more algebraic terms, including a shared variable.
3. Factorise a linear expression fully, taking out the HCF.
4. Verify a factorisation is correct by expanding it back.
5. Factorise expressions with a negative HCF where appropriate.

Warmup

(6 minutes — retrieval, mini whiteboards)

1. Find the HCF of and .
2. Find the HCF of and .
3. Expand .
4. Expand .

Answers: ; ; ; .

The hook: “In Question 3, you turned into . Today you learn to run that machine in reverse — starting at and finding your way back to . This is called factorising.”

Activities

Activity 1 — Explicit Instruction: Factorising Using the HCF (16 min)

The idea. Factorising is expanding in reverse: it uses the distributive property read from right to left.

I do — numeric terms only.

Check by expanding back: ✓

I do — HCF includes a shared variable.

I do — the variable itself is common.

The four-step protocol:

1. Find the HCF of the coefficients.
2. Check whether every term shares a common variable.
3. Write the HCF (numeric variable) outside the brackets.
4. Divide each term by the HCF to find what goes inside. Check by expanding back.

I do — negative HCF. When the leading term is negative, it is often cleaner to factor out a negative:

Check: ✓

We do: ; ; .

You do: Factorise fully, and check one answer by expanding.

(Answers: ; ; ; ; ; ; ; .)

Activity 2 — Spotting the highest Common Factor (10 min)

The trap. Factorising is only fully done when the HCF — not just a common factor — has been taken out.

I do — incomplete vs complete:

Test for “fully factorised”: can the terms inside the bracket be factorised any further? If yes, you haven’t found the HCF.

You do: For each expression, factorise fully. If you first find only a partial factor, redo it with the full HCF.

(Answers: ; ; ; .)

Activity 3 — Inquiry: Factorising to Reveal Structure (9 min)

Pairs, then whole-class share.

A gym charges a 20$8nC = 20 + 8n$.

1. Factorise the expression .
2. What does the factorised form tell you about the total cost that the un-factorised form doesn’t make obvious?
3. A rival gym’s cost is . Expand this. Which gym is cheaper for visits?

Socratic scaffolding for Q2–3:

Prompt	Purpose
Understand the problem.	You need to compare two gyms’ costs, given in different forms (one factorised, one not).
Devise a plan.	Put both expressions into the same form — either both expanded, or both factorised — before comparing.
Carry out the plan for gym 1.	.
Carry out the plan for gym 2.	.
Compare the two expanded forms.	versus — gym 2 always costs exactly 4n$.
Looking back — could you see this from the factorised forms instead?	vs — both share the common factor ; the difference is entirely inside the bracket ( vs ), confirming the constant 4$ gap.

Answers: 1. ; 2. the factorised form shows every cost is a multiple of , and separates the fixed “per-block” structure; 3. — gym 2 is always 420+8nnn=5$60$64$.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Find the HCF of and .
2. Factorise fully: (a) (b) (c) .
3. Is fully factorised? Explain, and give the fully factorised form if not.
4. Reasoning. A student factorises as . Verify this is correct by expanding.
5. Factorise , taking out a negative HCF.

Answers: 1. ; 2. (a) (b) (c) ; 3. No — shares a further factor of ; fully factorised, ; 4. ✓ correct; 5. .

Common Misconceptions

Misconception	How to pre-empt it
Taking out a common factor that is not the highest one.	Always test: can what’s left inside the bracket be factorised further? If yes, redo with a larger factor.
Forgetting to divide every term by the HCF, e.g. .	Insist on the check: expand the answer back and compare with the original.
Treating as a common factor when one term has no , e.g. factorising out of .	Require every term to contain the variable before it can be factored out.
Sign errors when factoring out a negative HCF.	Expand back immediately — a sign slip shows up instantly.
Believing factorising and expanding are unrelated skills.	Pair every factorising question with an “expand to check” step, reinforcing they are inverse processes.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Factorise fully, and state the HCF used.

Answer

HCF of and is : .

E2 (AMC Junior style). The expression is factorised as where is as large as possible. Find .

Answer

, so , , , giving .

E3 (Challenge). Show that and share the same factorised structure in a certain sense — factorise both and compare.

Answer

. For : HCF is , giving . Both share the identical bracket — the second is exactly times the first, term for term.

E4 (Challenge). A number is always divisible by which whole number, for every whole number ? Justify using factorising.

Answer

— since is a factor for every value of , is always divisible by .

E5 (Investigation). Factorise and . What do you notice about their factorised forms? For which value of (other than ) does each expression equal zero?

Answer

, zero at or . , zero at or . Factorising reveals the “zero” values directly — a first taste of solving equations by factorising, which returns in Year 9.

Homework

1. Find the HCF of: (a) and (b) and (c) and .
2. Factorise fully: (a) (b) (c) (d) (e) (f) .
3. Is fully factorised? If not, give the fully factorised form.
4. Factorise , taking out a negative HCF.
5. A phone plan costs dollars for GB of extra data. Factorise this expression and explain what the factorised form reveals about the cost structure.
6. Reasoning. Explain why cannot be factored out of , using a substitution to justify your answer.
7. Reasoning. A student factorises as . Explain why this is correct but not fully factorised, and give the complete factorisation.
8. Challenge. Show that is always divisible by , for every whole number , using factorising.
9. Challenge. Two expressions are and . Factorise each fully. What do you notice about the relationship between the two brackets?

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) (d) (e) (f) . Q3 — no, shares a further factor of ; fully factorised, . Q4 — . Q5 — ; shows every cost is a multiple of . Q6 — substituting : , which does not divide evenly by in general, so is not a common factor — only terms that all contain can have it factored out, and the constant does not. Q7 — still shares a common factor of ; fully factorised, . Q8 — , and is a factor for every . Q9 — , — both share the identical bracket .