028i - Expressions with Brackets

Lesson 28 — Expressions with Brackets and Multiple Operations

Strand: Algebra | Descriptor: AC9M7A02 | Duration: 45 minutes

Learning Intentions

• To formulate expressions involving brackets and several operations.
• To expand brackets and collect like terms.

Success Criteria

I can:

1. Explain what brackets group together in an expression.
2. Expand a single bracket, e.g. .
3. Identify and collect like terms.
4. Simplify an expression involving both brackets and like terms.

Warmup

(6 minutes — numerical check, mini whiteboards)

Evaluate each pair and comment on what you notice.

1. and
2. and
3. and

(Answers: vs ; vs ; vs .)

Key observation from Q3: multiplying the bracket distributes across both terms inside. This numerical fact is exactly what expanding brackets does algebraically.

Activities

Activity 1 — Explicit Instruction: Expanding a Single Bracket (12 min)

The distributive law:

Area model — make it visual. A rectangle of height and width splits into two rectangles of areas and .

I do — four worked examples, each with the arrow annotation drawn in:

The most common error — the second term. Students frequently write . Emphasise: the multiplier reaches every term inside. Verify numerically with : , and ✓, whereas ✗.

Negative multipliers — model carefully:

Note the sign change in the second: .

You do: Expand.

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

Activity 2 — Like Terms and Collecting (10 min)

Definition. Like terms contain exactly the same variable part. Only like terms can be added or subtracted.

Like	Unlike
and	and
and	and
and	and

The fruit analogy — use, then retire. Three apples plus five apples is eight apples; three apples plus five bananas cannot be combined. Useful for a first grasp, but retire it quickly: it encourages reading as an object rather than a number.

I do:

Warning: and are not like terms. Test with : and — clearly different quantities.

You do: Simplify.

(Answers: ; ; ; ; ; .)

Activity 3 — Combining: Expand then Collect (12 min)

I do:

Emphasise the third example. The minus sign in front of the second bracket applies to both terms inside. Model the intermediate line explicitly, and check numerically with : , and ✓

Inquiry task — the perimeter puzzle. Pairs.

A rectangle has width cm and length cm.

1. Write an expression for its perimeter and simplify.
2. Write an expression for its area.
3. A second rectangle has width and length . Compare the two perimeters.
4. Are the two areas equal? Test with .

Socratic scaffolding for Q3–4:

Prompt	Purpose
Write each perimeter before simplifying.	First: . Second: .
Expand both.	and .
What do you notice?	The perimeters are identical for every .
Now predict: will the areas also match?	Most students say yes. Test it.
Test with .	First: . Second: . Different!
Looking back — what does this show?	Equal perimeters do not force equal areas. The same lesson as L15 and L18, now in algebra.

Answers: 1. ; 2. ; 3. both — equal; 4. no — versus at .

Checks for Understanding

(5 minutes — exit ticket)

1. Expand: (a) (b) (c) .
2. Simplify: (a) (b) (c) .
3. Expand and simplify: .
4. Reasoning. A student writes . Explain the error and check your correction with .
5. Are and like terms? Justify.

Answers: 1. (a) (b) (c) ; 2. (a) (b) (c) ; 3. ; 4. The must multiply both terms: . Check at : and ✓ (whereas ✗); 5. No — the variable parts differ ( versus ). At they give and .

Common Misconceptions

Misconception	How to pre-empt it
(multiplying only the first term).	Draw the arrow to each term every time. Verify numerically as a routine final check.
(sign error on the second term).	Model the sign of each product separately: and .
Subtracting a bracket without distributing the minus: .	Require the fully expanded intermediate line before collecting.
Combining unlike terms: .	Test numerically with , : versus .
Treating and as like terms.	Substitute a value to show they measure different things.
Dropping a term when collecting across a long expression.	Have students underline or colour like terms before combining.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Simplify .

Answer

E2 (AMC Junior style). The perimeter of a rectangle is . Its width is . Find an expression for its length.

Answer

E3 (Challenge). Show that simplifies to , and find its value when .

Answer

At : .

E4 (Challenge). A square has side . A rectangle has width and length . Show that the square’s area always exceeds the rectangle’s, and by how much.

Answer

The square always has exactly more square units, whatever is.

E5 (Challenge). Simplify , then evaluate at , .

Answer

At , : .

Homework

1. Expand: (a) (b) (c) (d) (e) (f) .
2. Simplify by collecting like terms: (a) (b) (c) (d) (e) (f) .
3. Expand and simplify: (a) (b) (c) (d) .
4. A rectangle has width and length . Write and simplify expressions for its perimeter and its area.
5. A triangle has sides , and . Write and simplify an expression for its perimeter.
6. Reasoning. Explain why cannot be simplified further. Test with , .
7. Reasoning. Show that always equals , whatever the value of .
8. Challenge. A rectangle has perimeter and width . Find an expression for its length.

Answers: Q1 — (a) (b) (c) (d) (e) (f) . Q2 — (a) (b) (c) (d) (e) (f) . Q3 — (a) (b) (c) (d) . Q4 — , . Q5 — . Q6 — the variable parts differ, so the terms are unlike; at the value is , which is not . Q7 — . Q8 — gives , so .