Additive Strategies progression

Learners solve simple addition and subtraction problems by counting all of the objects. Typically, a learner will use fingers, counters or other objects.

For example, a learner may add 8 + 7 by starting from 1, counting 8 objects, then continuing to count 7 more objects to reach 15.

Learners use “in the head” (mental) strategies. They can count on (for addition) or back (for subtraction) from the first number given. They do not rely on fingers or other objects, and they can count in ones and in tens and in combinations of ones and tens.

• Counting in ones: A learner may solve 8 + 5 by starting at 8, then mentally counting 5 more, by ones, to reach 13.
• Counting in tens and ones: A learner may solve 46 – 23 by starting at 46, then mentally counting 20 back, by tens, to reach 26 and then counting 3 back, by ones, to reach 23.

Learners use mental strategies that require them to partition numbers (that is, to split numbers into parts). Partitioning strategies include the following:

• Deriving from known facts: Learners derive unknown information from a known fact. A learner may solve 26 + 6 by using what they know (6 + 6 = 12), then adding 20 + 12 to reach 32.
• Making tens: Learners partition numbers in order to be able to work from the nearest ten. A learner may solve 16 + 7 by splitting the 7 into 4 and 3, adding 16 + 4 to reach 20, then adding the 3 to reach 23.
• Using tidy numbers with compensation: Learners round a number to the nearest ten or hundred, then compensate for what has been added or subtracted. Learners know that, in the addition process, if they add something to a number, they must take it away again at the end (26 + 9 can be solved as 26 + 10 – 1 = 35) and that, in subtraction, if they take something away from a number, they must add it back on at the end (53 – 9 can be solved as 53 – 10 + 1 = 44).

Learners use partitioning strategies to solve more complex addition or subtraction problems. Partitioning strategies include the following:

• Deriving from known facts: Learners derive unknown information from a known fact, for example, solving 25 + 26 by first adding 25 + 25 to get 50, then adding 1 more to get 51.
• Making tens: Learners partition numbers in order to be able to work from the nearest ten. For example, a learner can solve 45 + 37 by splitting the 37 into 5 and 32, then adding 45 + 5 to get 50 and then adding 50 + 32 to get 82. Or the learner can solve 64 – 37 by splitting the 37 into 4 and 33, subtracting 64 – 4 to get 60 and then subtracting 33 from 60 to get 27.
• Place value partitioning: The learner breaks the numbers into ones, tens and hundreds, adds numbers of the same place value together and then combines these numbers. For example:
  23 + 34 can be solved as (20 + 30) + (3 + 4) = 50 + 7 = 57.
  657 – 234 can be solved as (600 – 200) + (50 – 30) + (7 – 4) = 400 + 20 + 3 = 423.
• Using tidy numbers with compensation: The learner rounds a number to the nearest ten or hundred, then compensates for what has been added or subtracted. For example, 46 + 19 can be solved as 46 + 20 – 1 = 65. Alternatively, learners may use a calculator or written algorithm to solve a problem. If so, they are able to justify the solution by demonstrating or explaining why it is reasonable.

• Standard algorithm explanation:

  

  65 is reasonable because 37 + 28 is a bit less than 70 (40 + 30).

  

  36 is reasonable because 74 - 38 is a bit more than 34 (74 - 40).

Learners use partitioning strategies to solve addition and subtraction problems involving decimals and integers. Partitioning strategies include the following:

• Using tidy numbers with compensation: For example, a learner solves 3.2 + 1.95 as 3.2 + 2 – 0.05 = 5.2 – 0.05 = 5.15.
• Place value partitioning: Learners partition numbers by place value. For example, a learner can solve 6.03 – 5.8 by subtracting 6.03 – 5 to get 1.03, then subtracting 1.03 – 0.8 to get 0.23.
• Using reversibility: Learners change a subtraction problem into an addition problem. For example, 6.03 – 5.8 becomes 5.8 + ? = 6.03, and -15 + 64 becomes 64 – 15.

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

Learners use partitioning strategies to solve addition and subtraction problems involving fractions.
Partitioning strategies include the following:

• Using equivalent fractions: Learners use knowledge of equivalent fractions to solve addition and
  subtraction problems. For example,

5/6 + 7/4 = 10/12 + 21/12 = 31/12.

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