L4 · Rearranging expressions: commutative, associative, distributive

Learning goal

“I can rearrange an expression using the commutative, associative, and distributive properties to make calculations easier - and expand a bracket like 3(x + 4) into 3x + 12 without changing the value.”

Commutative property — order can swap in + or x: 4 + 9 = 9 + 4 and 3 x 7 = 7 x 3.

Associative property — grouping can move in + or x: (2 + 3) + 4 = 2 + (3 + 4).

Distributive property — a factor outside brackets multiplies EVERY term inside: 3(x + 4) = 3x + 12.

Expand — rewrite a bracketed expression without brackets, using the distributive property.

Worked example

Compute 4 x 13 x 25. Step 1: Scan for a friendly pair: 4 x 25 = 100. The commutative property lets the 25 move next to the 4: 4 x 25 x 13. Step 2: The associative property lets us group that pair first: (4 x 25) x 13. Step 3: 100 x 13 = 1300. Same product - far easier path. The original tree joins 4 x 13 first (a hard branch, 52, then x 25); the rearranged tree joins 4 x 25 first (100), then x 13. Swapping and regrouping redraw the tree but never change its value.

Check yourself — explain it

Swapping the order of two factors, as in 4 x 13 x 25 -> 4 x 25 x 13, uses the...

Paired problem — example still visible

Compute by rearranging: 18 + 45 + 82 -> 145 ((18 + 82) + 45 = 100 + 45)

Faded completion — last step is yours

"5 x 9 x 2 = (5 x ___) x 9 = ___" -> 2; 90

Ready to practise?

10 scored questions with a 3-rung hint ladder. Mastery needs 8/10 plus both gate questions.

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