037i - Consolidation Linear Expressions

Lesson 37 — Consolidation and Check: Expanding, Factorising and Simplifying

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

Learning Intentions

• To consolidate and connect expanding, factorising, simplifying and rearranging linear expressions.
• To identify and correct common errors from across the unit.

Success Criteria

I can:

1. Expand linear expressions accurately, including negative and multi-term brackets.
2. Factorise using the highest common factor, and verify by expanding back.
3. Simplify expressions containing more than one variable.
4. Rearrange a formula to change its subject.
5. Choose and sequence the right skills to solve a multi-step problem.

Warmup

(6 minutes — diagnostic relay, mini whiteboards)

One quick question per lesson in the block — answer as fast as you can, then check.

1. Name the property: . (Lesson 31)
2. Expand: . (Lesson 32)
3. Factorise fully: . (Lesson 33)
4. Simplify: . (Lesson 34)
5. Make the subject: . (Lesson 35)

Answers: commutative; ; ; ; .

Activities

Activity 1 — Error Clinic (14 min)

Explicit re-teaching, driven by the most common mistakes seen across Lessons 31–36.

Each “broken” solution below contains exactly one error. I do the first two; we do the third; you do the rest.

Broken solution 1:

Broken solution 2:

Broken solution 3 (we do): — “fully factorised.”

(Error: still shares a common factor of ; the true HCF is , giving .)

You do — find and correct the error in each broken solution:

6. Rearranging to make the subject: .

(Answers: 4. Error — the -terms were combined incorrectly (, not ); correct simplification is . 5. Error — the term was dropped entirely when collecting; correct simplification is . 6. Error — was isolated by subtracting instead of dividing by it, after subtracting ; correct rearrangement is .)

Teacher note: Write each broken solution on the board and have students diagnose the error verbally with a partner before revealing the correction — this rehearses the vocabulary of Lesson 31’s properties (“you haven’t distributed across every term,” “that’s not the highest common factor,” “you divided too early”).

Activity 2 — Consolidation Stations (12 min)

Four short stations, one skill each. Pairs rotate every 3 minutes; self-check against posted answers.

Station A — Expand: ; ; .

Station B — Factorise: ; ; .

Station C — Simplify: ; .

Station D — Rearrange: make the subject of ; make the subject of .

(Answers: A — ; ; . B — ; ; . C — ; . D — ; .)

Activity 3 — Applied Consolidation Task (13 min)

Pairs, then whole-class share. Combines every skill from the block in one multi-step problem.

A landscaper prices a paving job as dollars, where is the number of square metres paved.

1. Expand and fully simplify the expression for .
2. Factorise your simplified expression fully.
3. Rearrange the simplified formula to make the subject.
4. The landscaper’s invoice was 382$. How many square metres were paved?

Socratic scaffolding (Polya’s cycle):

Prompt	Purpose
Understand the problem.	You need , but the formula gives in terms of — and it’s not yet in a simplified, usable form.
Devise a plan.	Work through the skills in order: expand fully, factorise to check structure, then rearrange to isolate , and finally substitute the known invoice total.
Carry out the plan — expand.	.
Carry out the plan — factorise, to check for full simplification.	— confirms the HCF is , and the expression is genuinely as simple as it can be.
Carry out the plan — rearrange.	From : , so .
Carry out the plan — substitute.	.
Looking back — check.	Substitute into the original formula: ✓

Answers: 1. ; 2. ; 3. ; 4. .

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Expand: .
2. Factorise fully: .
3. Simplify: .
4. Make the subject of .
5. Reasoning. A job costs dollars for hours. If a customer paid 56x$, showing every step.

Answers: 1. ; 2. ; 3. ; 4. ; 5. hours; check: ✓

Common Misconceptions

Misconception	How to pre-empt it
Multiplying only one term when expanding a bracket.	Draw the arrow to every term; verify numerically before moving on.
Stopping factorising before reaching the true HCF.	Always test: can the bracket’s contents be factorised any further?
Combining terms from different variable “families.”	Colour-code or underline each family before collecting.
Applying an inverse operation to only one side, or in the wrong order, when rearranging.	Use the “last done, first undone” rule, and check both sides every time.
Treating expand, factorise, simplify and rearrange as unrelated topics rather than connected tools.	Use multi-step tasks (like Activity 3) that require choosing and sequencing several skills together.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Simplify , then factorise your answer fully.

Answer

has no common factor other than — it is already fully factorised (prime coefficients).

E2 (AMC Junior style). A rectangle’s perimeter is . Factorise this expression, then find the rectangle’s width and length if they are and one other linear expression.

Answer

Since and , width + length . With one side , the other side is also — the rectangle is a square.

E3 (Challenge). Show that and are identical expressions, using full working.

Answer

They are not identical — and differ by , which is zero only at .

E4 (Challenge). A formula is . Simplify, factorise, and then rearrange to make the subject. Find when .

Answer

Factorised: has no common factor ( is prime and shares no factor with ), so it stays as is.

At : .

Homework

1. Expand: (a) (b) (c) .
2. Factorise fully: (a) (b) (c) .
3. Simplify: (a) (b) .
4. Rearrange: (a) , make the subject (b) , make the subject.
5. A delivery service charges dollars for km. Expand and simplify , then find the cost for a km delivery.
6. Reasoning. Explain, using an example from this unit, why checking your answer (by expanding back, or by substitution) is an essential final step, not an optional extra.
7. Reasoning. A student says “factorising and expanding are opposite skills, so I only need to practise one of them.” Explain why this reasoning is flawed.
8. Challenge. A job costs dollars for hours of work. Simplify , rearrange to make the subject, and find if the job cost 88$.
9. Challenge. Show that and are identical expressions, with full working.

Answers: Q1 — (a) (b) (c) . Q2 — (a) (b) (c) . Q3 — (a) (b) . Q4 — (a) (b) . Q5 — ; at : 62-3(x-2)C=8x+12-x+5=7x+17x=\dfrac{C-17}{7}C=88x=\dfrac{71}{7}2(3x+4)+3(2x-1) = 6x+8+6x-3 = 12x+5$, matching exactly.