031i - Properties of Operations

Lesson 31 — Naming and Using the Properties of Operations

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

Learning Intentions

• To name and correctly apply the commutative, associative, identity, distributive and inverse properties.
• To use these properties to justify why two expressions are equivalent.

Success Criteria

I can:

1. State the commutative property for addition and multiplication, and show it fails for subtraction and division.
2. State the associative property for addition and multiplication, and show it fails for subtraction and division.
3. Identify the identity elements for addition and multiplication.
4. Identify additive and multiplicative inverses.
5. Apply the distributive property to expand a product over a sum or difference.
6. Name the property used to justify each step of a simplification.

Warmup

(6 minutes — numeric investigation, mini whiteboards)

Evaluate each pair and decide whether order matters.

1. and
2. and
3. and
4. and

Answers: 1. — equal; 2. — not equal; 3. — equal; 4. — not equal.

Key observation: addition and multiplication “don’t care about order.” Subtraction and division do. This is the seed of today’s lesson: mathematics has names for these behaviours, and the names let us justify our algebra instead of just trusting it.

Activities

Activity 1 — Explicit Instruction: Five Properties, Five Definitions (14 min)

Build a reference table on the board as each property is introduced. Students copy it into their books — it becomes the lesson’s anchor for the rest of the week.

Commutative property — order doesn’t matter.

I do: ; .

Fails for subtraction/division: but .

Associative property — grouping doesn’t matter.

I do: ; .

Fails for subtraction: but .

Identity property — the “do nothing” number for each operation.

Distributive property — multiplication spreads over addition/subtraction.

I do: .

Inverse property — every number has an “undoer.”

We do: Classify each statement as true or false, and name the property (or identify the misuse):

You do: Match each statement to its property, and correct any that are false.

(Answers: 1. commutative; 2. false — distributive misapplied, correct is ; 3. associative; 4. inverse; 5. identity; 6. false — subtraction is not commutative; 7. distributive; 8. inverse.)

Activity 2 — Using Properties for Smart Calculation (10 min)

Explicit instruction: properties aren’t just for proofs — they make arithmetic faster.

I do:

We do:

You do: Calculate mentally, and state which property helped.

(Answers: 1. — commutative/associative to group first; 2. — distributive; 3. — commutative/associative; 4. — distributive.)

Activity 3 — Inquiry: Justify the Steps (10 min)

Pairs, then whole-class share.

Task 1. Simplify , writing every intermediate line and naming the property used at each step.

Task 2 — extending the distributive property. The distributive property was defined for two terms inside the bracket: . Does it still work for three terms, e.g. ?

Socratic scaffolding for Task 2:

Prompt	Purpose
Understand: what are you trying to show?	That expands to , not just the two-term case.
What property do you already trust, for two terms?	The distributive property, .
Devise a plan: can you rewrite three terms as two groups?	Associative property: .
Carry out the plan: apply the two-term rule you trust.	.
Apply it again to the remaining bracket.	.
Combine.	.
Looking back — does this always work, for any number of terms?	Yes — the distributive property extends to any number of terms by repeating the two-term case.

Task 3. Is subtraction distributive? Test by rewriting as (using the inverse property) and applying distribution to the sum.

(Answers: Task 1 — (commutative, reordering) (associative, regrouping) . Task 3 — , so yes: .)

Checks for Understanding

(5 minutes — exit ticket, collected)

1. State the commutative property for multiplication and illustrate it with and .
2. Show numerically that division is not commutative.
3. What is the additive inverse of ? What is the multiplicative inverse of ?
4. Simplify , naming the commutative and associative steps separately.
5. Reasoning. A student claims . Which property have they misapplied, and what is the correct expansion?

Answers: 1. ; ; 2. e.g. but — not equal; 3. additive inverse of is ; multiplicative inverse of is ; 4. (commutative) (associative) ; 5. Distributive property misapplied — the must multiply both terms: correct expansion is .

Common Misconceptions

Misconception	How to pre-empt it
Subtraction and division are commutative, like addition and multiplication.	Test numerically every time: . Keep a permanent counterexample on the wall.
— the distributive property applied to only one term.	Draw an arrow from the multiplier to each term; verify with substitution.
The identity element for multiplication is .	Contrast (annihilator) with (identity). Test both explicitly.
The associative property changes the order of terms.	Clarify: associative regroups (moves brackets); commutative reorders. Show both side by side on the same expression.
”Inverse” always means “negative.”	Distinguish the additive inverse (opposite sign) from the multiplicative inverse (reciprocal) with paired examples.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). Which of the following is not always true for real numbers : (a) (b) (c) (d) ?

Answer

(c) — subtraction is not commutative. A quick counterexample: but .

E2 (AMC Junior style). Use the distributive property to calculate mentally.

Answer

E3 (Challenge). For which value of does hold true for every and ?

Answer

Subtraction is associative only in the trivial case — for every other value the two sides differ.

E4 (Challenge). Solve by adding the additive inverse of to both sides. Then solve by multiplying both sides by the multiplicative inverse of .

Answer

This is exactly how “solving an equation” works — you are applying inverses to undo operations.

Homework

1. State whether each is true or false, naming the property (or explaining why it is false): (a) (b) (c) (d) .
2. Use the distributive property to expand: (a) (b) .
3. Find (a) the additive inverse of (b) the multiplicative inverse of (c) the additive inverse of .
4. Use properties to calculate mentally, and state which property helped: (a) (b) .
5. Simplify , showing the commutative and associative steps separately.
6. Reasoning. Explain why is called the “identity” for addition, using an example with a variable.
7. Reasoning. A classmate says “associative and commutative mean the same thing.” Explain the difference using an example of each.
8. Challenge. Show algebraically that , by applying the distributive property twice (as in Activity 3, Task 2).

Answers: Q1 — (a) true, commutative (b) false, subtraction not commutative: (c) true, associative (d) false, division not commutative: . Q2 — (a) (b) . Q3 — (a) (b) (c) . Q4 — (a) , commutative/associative (b) , commutative/associative. Q5 — . Q6 — adding never changes the value, e.g. for any , just as . Q7 — commutative reorders (e.g. ); associative regroups (e.g. ) — order of the numbers stays the same, only the brackets move. Q8 — .