Multiplication • Topic 2 of 4

Properties

The properties of multiplication are the rules that let children rearrange a problem to make it easier, and CTET asks about them in application form, usually by describing a child's strategy and asking which property is at work. The commutative property says the order of the two numbers does not change the product: a x b = b x a, so 3 x 5 = 15 is the same as 5 x 3 = 15. An array makes this visible, since 3 rows of 5 holds the same 15 dots as 5 rows of 3. The associative property says that when three or more numbers are multiplied, the grouping does not change the answer: (a x b) x c = a x (b x c). For example (2 x 3) x 4 = 6 x 4 = 24, and 2 x (3 x 4) = 2 x 12 = 24, the same total. The identity property says any number times 1 stays itself: a x 1 = a, so 25 x 1 = 25, and 1 is called the multiplicative identity. Set against this is the zero property: any number times 0 is 0, so 47 x 0 = 0. A common misconception, which CTET likes to probe, is that children assume these rules also apply to subtraction and division. They do not: 10 - 5 is not 5 - 10, and (10 - 5) - 2 is not 10 - (5 - 2).

✅ Solved examples

1. A child solves 8 x 5 by recalling 5 x 8 = 40 instead. Which property allows this, and what is the answer?
The commutative property, since a x b = b x a. So 8 x 5 = 5 x 8 = 40.
2. Show that (2 x 5) x 3 and 2 x (5 x 3) give the same product, and name the property.
(2 x 5) x 3 = 10 x 3 = 30, and 2 x (5 x 3) = 2 x 15 = 30. Both equal 30, illustrating the associative property of multiplication.
3. Find 63 x 1 and 84 x 0, and name the property each one uses.
63 x 1 = 63 by the identity property (any number times 1 is itself; 1 is the multiplicative identity). 84 x 0 = 0 by the zero property (any number times 0 is 0).
4. A pupil writes 6 - 2 = 2 - 6 and claims multiplication taught her this. What is the teaching point?
The commutative property holds for multiplication (and addition) but not for subtraction. 6 - 2 = 4 while 2 - 6 = -4, so order matters in subtraction. The pupil has over-generalised a multiplication rule.

✏️ Practice — try these, take hints as needed

1. Which property of multiplication is shown by 7 x 9 = 9 x 7?
The order of the numbers is swapped.
a x b = b x a.
Commutative property
2. What is 156 x 1, and which property gives the answer instantly?
Multiplying by 1 changes nothing.
1 is the multiplicative identity.
156 (identity property)
3. Use the associative property to compute (4 x 2) x 5 in two groupings.
Group one pair first, then multiply.
(4 x 2) x 5 versus 4 x (2 x 5).
(4 x 2) x 5 = 8 x 5 = 40, and 4 x (2 x 5) = 4 x 10 = 40
4. A number multiplied by 0 always gives what result?
This is the zero property.
It does not matter how large the number is.
0

📝 Topic test — 8 questions

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