Multiplicative strategies progression

Learners solve simple multiplication problems by counting all the objects. Typically, learners will use fingers, counters or other objects. For example, a learner may solve 2 x 3 by counting two groups of three, each with three ones (1, 2, 3; 4, 5, 6).

Learners solve simple multiplication problems by skip-counting. For example, a learner may solve 4 x 5 by skip-counting in fives (5, 10, 15, 20).

Learners solve single-digit multiplication and division problems, using repeated addition or deriving unknown information from known multiplication and division facts. Examples can include the following:

• Repeated addition: A learner may solve 4 x 6 by adding 6 + 6 + 6 + 6 to reach 24.
• Deriving from known facts: A learner may solve 72 ÷ 8 = 9 by recalling that 8 x 8 = 64 and then adding one more 8 to get 72. Alternatively, a learner may solve this by recalling that 8 x 10 = 80 and then subtracting one 8 to get 72.

Learners use mental strategies to solve multiplication and division problems that have single-digit multipliers or divisors. Learners use mental strategies that are based on derivations from known multiplication or division facts. Partitioning strategies include the following:

• Deriving from known facts: Learners partition mentally to allow the use of known number facts first, for example, 10 x 13 = 130 so 9 x 13 = 130 – 13 = 117.
• Place value partitioning: Learners break numbers into tens, for example, 14 x 5 can be solved as (10 + 4) x 5. 10 x 5 = 50 and 4 x 5 = 20, so the solution is 50 + 20 = 70.
• Using reversibility: Learners change a division problem into a multiplication one and use known facts, for example, 72 ÷ 4 can be solved as 4 x ? = 72. The learner knows that 4 x 20 = 80, and 8 ÷ 4 is 2, so 4 x 18 = 72. (Note that this problem involves deriving from known facts as well as using reversibility.)

Learners solve multi-digit multiplication and division problems with whole numbers by using partitioning strategies. Partitioning strategies include the following:

• Deriving from known facts: Learners use known facts to reach a solution.
  4 x 25 = 100, so 8 x 25 = 100 + 100 = 200.
  72 ÷ 4 can be solved as 72 ÷ 2 = 36, 36 ÷ 2 = 18 (because dividing by 4 is the same as dividing by 2 twice).

• Using equivalent expressions:
  360 ÷ 5 can be solved as 720 ÷ 10 = 72.
  81 ÷ 3 can be solved as 9 x (9 ÷ 3) = 9 x 3 = 27.

• Place value partitioning: Learners partition numbers by using knowledge of place value. For example, 24 x 36 can be solved as 20 x 36 + 4 x 30 + 4 x 6
  = 720 + 120 + 24
  = 864.

Alternatively, learners may use a calculator or written algorithm to solve a problem. If so, they can justify the solution by demonstrating or explaining why it is reasonable.

• Calculator (with explanation): Learners use a calculator to find the answer and are able to explain why the answer obtained is reasonable. For example: 45 x 23 = 1,035 (using a calculator). “1,035 is a reasonable answer because I know that 45 x 20 is 900.”

Learners solve multiplication or division problems with decimals, fractions and percentages, using
partitioning strategies.

• Converting between fractions and percentages
  25% off $56 is $42. The learner knows that 25% is 1/4 and 1/4 of 56 is 14. 56 – 14 = 42.

Alternatively, learners may use a calculator or written algorithm to solve a problem. If so, they can justify the
solution by demonstrating or explaining why it is reasonable.

• Standard algorithm (with reasonableness)
  2/3 x 3/4 = 6/12 (by multiplying the 2 x 3 and the 3 x 4). “This is reasonable because I know that 2/3 of 1 is 2/3 and 2/3 is a little more than a half (6/12).”