034i - Collecting Like Terms

Lesson 34 — Simplifying Expressions by Collecting Like Terms

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

Learning Intentions

• To simplify expressions with several variables by collecting like terms accurately.
• To simplify expressions that arise after expanding two or more brackets.

Success Criteria

I can:

1. Identify like terms among expressions with more than one variable.
2. Collect like terms involving , and constants in a single expression.
3. Simplify expressions after fully expanding multiple brackets.
4. Explain why terms with different variable parts (e.g. and ) cannot be combined.
5. Use the commutative and associative properties to justify each step of collecting like terms.

Warmup

(5 minutes — retrieval, mini whiteboards)

Simplify:

Answers: ; ; ; .

Bridging question: What happens when an expression has two different variables mixed together, like ? Today’s lesson is about sorting the mess.

Activities

Activity 1 — Explicit Instruction: Sorting Terms with Multiple Variables (14 min)

Definition, revisited. Like terms have exactly the same variable part (including the same powers). -terms, -terms and constants are three separate “families” that never mix.

I do — the sorting strategy. Use colour or underline coding: circle -terms, box -terms, underline constants — before combining anything.

I do — terms with two variables multiplied together (). These form their own family, separate from and alone.

Why and are unlike terms — check numerically at : but — clearly different quantities, so they cannot be added into a single term.

We do: ; ; .

You do: Simplify.

(Answers: ; ; ; ; ; .)

Activity 2 — Simplifying after Expanding Two Brackets (13 min)

Explicit instruction linking Lesson 32’s expanding to today’s collecting.

I do:

We do: ; .

You do: Expand and fully simplify.

(Answers: ; ; ; ; ; .)

Activity 3 — Inquiry: the Perimeter with Two Variables (13 min)

Pairs, then whole-class share.

A garden bed is L-shaped, made from a large rectangle of width and length , with a smaller rectangle of width and length … (too complex — use the version below)

A trapezium-shaped garden has a top edge of length , a bottom edge of , and two equal slanted sides each of length .

1. Write an expression for the perimeter and simplify it fully.
2. If and , find the perimeter.
3. A second garden has a top edge of , bottom edge , and two slanted sides each of . Is its perimeter ever equal to the first garden’s, for some and ? Investigate using and then .

Socratic scaffolding for Q3:

Prompt	Purpose
Understand the problem.	You must compare two perimeter expressions — both containing and — not just one pair of numbers.
Devise a plan.	Simplify both perimeter expressions fully in terms of and first, then compare the simplified forms.
Carry out the plan for garden 1.	.
Carry out the plan for garden 2.	.
Compare.	. They are equal only when .
Looking back — test with numbers.	At : — differ by ✓. At : — differ by ✓.

Answers: 1. ; 2. at : ; 3. ; the perimeters differ by exactly , so they are equal only when — never for a genuine trapezium with two different parallel sides.

Checks for Understanding

(5 minutes — exit ticket, collected)

1. Simplify: .
2. Simplify: .
3. Are and like terms? Justify your answer numerically using .
4. Expand and simplify: .
5. Reasoning. A student simplifies as . Identify the error and give the correct simplification.

Answers: 1. ; 2. ; 3. No — at : but , different values; 4. ; 5. The student combined unlike terms across families (-terms with -terms); correct simplification groups each family separately: .

Common Misconceptions

Misconception	How to pre-empt it
Combining -terms with -terms, e.g. .	Test numerically: at , but .
Treating and (or and ) as like terms.	Substitute values to show they measure different quantities.
Losing a term’s sign when reordering a long expression.	Underline or colour-code each “family” of term before moving anything.
Forgetting a term altogether in a long expression with several variables.	Count the terms before and after simplifying — the count of each family should match.
Believing simplifying always reduces the number of terms to one.	Show examples where the fully simplified answer still has two or three terms (different families cannot merge further).

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Simplify .

Answer

E2 (AMC Junior style). A triangle has sides , and . Find an expression for its perimeter, fully simplified. If and , evaluate the perimeter.

Answer

At : .

E3 (Challenge). Simplify , then evaluate at , .

Answer

At : .

E4 (Challenge). Two expressions are and . Find and , fully simplified.

Answer

Homework

1. Simplify: (a) (b) (c) .
2. Are the following like terms? Justify each with a substitution: (a) and (b) and (c) and .
3. Expand and simplify: (a) (b) (c) .
4. A rectangle has width and length , and a second rectangle has width and length . Write an expression, fully simplified, for the sum of both perimeters.
5. Reasoning. Explain why cannot be simplified any further.
6. Reasoning. A student says and are unlike terms because the letters are in a different order. Use the commutative property to explain why they are wrong.
7. Challenge. Simplify , then evaluate at .
8. Challenge. Simplify where and .

Answers: Q1 — (a) (b) (c) . Q2 — (a) unlike, e.g. : , (b) like, both equal (c) like, since by the commutative property. Q3 — (a) (b) (c) . Q4 — perimeters and ; sum . Q5 — and have different variable parts (one has , one doesn’t), so they are unlike terms. Q6 — by the commutative property of multiplication, , so and represent the exact same product and are like terms. Q7 — ; at : . Q8 — .