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Multiplicative strategies progression


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Last updated 10 January 2013 11:05 by NZTecAdmin

The multiplicative strategies progression describes the processes that learners use to solve problems involving multiplication and division. The emphasis in the multiplicative strategy progression is on understanding. Learners who use calculators or traditional algorithms to solve problems also need to be able to decide if the answers they obtain are reasonable.

Most adults will be able to:
Activities
1.
  • solve multiplication problems by counting all the objects.

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).

 
  • solve multiplication problems by counting all the objects.
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2.
  • solve multiplication problems by skip-counting, often in conjunction with one-to-one counting and often keeping track of the repeated counts by using materials (for example, fingers) or mental images.

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).

Skip-counting

Learners develop their understanding of multiplication by skip-counting in twos, threes and fives to solve simple problems.

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3.
  • solve single-digit multiplication and division problems mentally, using known multiplication facts and repeated addition.

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.

Understanding multiplication

Learners use already-known multiplication facts to develop quick recall of unknown facts.

Deriving multiplication and division facts

Learners extend their repertoire of multiplication and division facts by using already-known facts to derive unknown facts.

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4.
  • solve multiplication and division problems with single digit multipliers or divisors mentally, using partitioning strategies and deriving from known multiplication facts.

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.)

Division strategies

Learners develop mental strategies for solving division problems with single-digit divisors.

Multiplication strategies

Learners develop mental strategies for solving multiplication problems with single-digit multipliers.

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5.
  • solve multiplication or division problems with multidigit whole numbers, using partitioning strategies

or alternatively

  • justify the reasonableness of answers to problems solved, using a calculator or algorithm.

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.”

Dividing options

Learners use strategies, traditional written methods and calculators to solve division problems.

Multiplying options

Learners use strategies, traditional written methods and calculators to solve multiplication problems.

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6.
  • solve multiplication or division problems with decimals, fractions and percentages, using partitioning strategies

or alternatively

  • justify the reasonableness of answers to problems solved, using a calculator or algorithm.

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).”

Dividing with decimals

Learners use estimation strategies and calculators to solve division problems involving decimals.

Multiplying with decimals

Learners use estimation strategies and calculators to solve multiplication problems involving decimals.

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