032i - Expanding Linear Expressions

Lesson 32 — Expanding Linear Expressions with Brackets

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

Learning Intentions

• To expand linear expressions involving one or more brackets, including negative multipliers and three-term brackets.
• To simplify expanded expressions by collecting like terms.

Success Criteria

I can:

1. Expand a single bracket with a positive or negative multiplier.
2. Expand a bracket containing three or more terms.
3. Expand and simplify expressions containing two or more brackets.
4. Correctly apply the “subtracting a bracket” rule, distributing the negative sign across every term.
5. Verify an expansion is correct by substituting a numerical value.

Warmup

(6 minutes — retrieval from Lesson 31, mini whiteboards)

Expand quickly:

Answers: ; ; ; .

Bridging question: Which property have you used in every single one of these? (The distributive property, from Lesson 31 — today we push it further: more terms, more brackets, more negatives.)

Activities

Activity 1 — Explicit Instruction: Brackets with Three Terms and Negative Multipliers (14 min)

I do — three terms inside the bracket:

I do — negative multiplier, three terms. Watch every sign flip individually.

Check numerically with : LHS ; RHS ✓

We do:

You do: Expand.

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

Activity 2 — Two Brackets: Expand and Collect (12 min)

I do — adding two brackets:

I do — subtracting a bracket. The minus sign applies to the whole second bracket.

I do — negative coefficient outside, three terms:

Check at : LHS ; RHS ✓

We do:

You do: Expand and simplify.

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

Activity 3 — Inquiry: is it always Equal? (13 min)

Pairs, then whole-class share.

Freddy claims that is always equal to , for every value of .

Investigate: is he right?

Socratic scaffolding:

Prompt	Purpose
Understand: what is Freddy claiming?	That the two expressions are equal for every value of , not just one.
Devise a plan: how can you test “always”?	Expand and simplify both expressions fully; compare the resulting simplified forms — this is Polya’s “devise a plan.”
Carry out the plan for the left side.	.
Carry out the plan for the right side.	.
Compare the simplified forms.	and have different coefficients of — they are not identical expressions.
Looking back — but don’t they look equal for some ?	Test : LHS ; RHS . Equal at !
Looking back — how do you disprove “always equal” convincingly?	Find just one value of where they differ. Test : LHS ; RHS . Not equal — Freddy is wrong.

Follow-up (contrast case). Show students a genuine identity: expand and , and check they match for — every time, because they are the same simplified expression, not just coincidentally equal.

Answers: ; . Not identical — they only agree at . , which matches for every : a genuine identity.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Expand: (a) (b) (c) .
2. Expand and simplify: .
3. Expand and simplify: .
4. Reasoning. A student expands as . Identify and correct the error.
5. Verify, by substituting , that and are equivalent.

Answers: 1. (a) (b) (c) ; 2. ; 3. ; 4. The second term’s sign was not flipped: , so the correct expansion is ; 5. At : and ✓ — equal, consistent with for all .

Common Misconceptions

Misconception	How to pre-empt it
Multiplying only the first term inside the bracket.	Draw an arrow from the multiplier to every term, including the third term in longer brackets.
Sign error when the multiplier is negative: instead of .	Model each product’s sign separately; keep a “negative times negative is positive” reminder visible.
Not distributing the minus sign across the whole second bracket when subtracting brackets.	Require the fully expanded intermediate line before collecting, every time.
Dropping the third term when a bracket has three terms.	Underline each term inside the bracket before expanding, so none are missed.
Believing two expressions equal at one -value must be equal for all .	Activity 3 confronts this directly — always test at least two values, or fully simplify and compare.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Simplify .

Answer

E2 (AMC Junior style). A rectangle has width and length . Write an expanded expression for its perimeter.

Answer

E3 (Challenge). Show that simplifies to , without expanding first. Then confirm by expanding both forms.

Answer

Since both terms share the factor : — this uses the distributive property “in reverse,” previewing factorising.

Expanding to confirm: , and ✓

E4 (Challenge). Expand , then evaluate at , .

Answer

At : .

Homework

1. Expand: (a) (b) (c) (d) (e) (f) .
2. Expand and simplify: (a) (b) (c) (d) .
3. A rectangle has width and length . Write an expanded expression for its perimeter.
4. Expand and check your answer by substituting into both the original and simplified forms.
5. Reasoning. A classmate expands as . Identify the two errors and give the correct expansion.
6. Reasoning. Explain, using an example, why finding two expressions equal at does not prove they are equal for every .
7. Challenge. Show that simplifies to using the distributive property “in reverse,” then confirm by expanding both forms fully.
8. Challenge. Two expressions are and . Determine whether and are identical expressions, giving full working.

Answers: Q1 — (a) (b) (c) (d) (e) (f) . Q2 — (a) (b) (c) (d) . Q3 — . Q4 — ; at : original ; simplified ✓. Q5 — sign on should become , and is correct but written as if the middle term’s sign was untouched; correct expansion is . Q6 — e.g. and (from Activity 3) both equal at but differ at ( vs ), so equality at one point is not enough. Q7 — ; expanding: , and ✓. Q8 — ; ; not identical (differ by for every ).