Calculating with Large Numbers - Progression Points

Number

1.0 Standard

At Level 1, students use materials to model addition and subtraction of subtraction by the aggregation (grouping together) and disaggregation (moving apart) of objects. They add and subtract by counting forward and backward using the numbers from 0 to 20.

1.5

• addition and subtraction of two-digit multiples of ten by counting on and counting back
• counting on from the larger of two collections to find their total
• use of the number properties (commutative and associative) of addition in mental computation, and recognition of complements to ten; for example, 3 + 4 + 7 + 6 = 3 + 7 + 4 + 6 = 10 + 10 = 20

1.75

• addition and subtraction of numbers less than 10 through recall and use of number facts

2.0 Standard

At Level 2 students mentally compute simple addition and subtraction calculations involving one- or two-digit natural numbers, using number facts such as complement to 10, doubles and near doubles. They describe and calculate simple multiplication as repeated addition, such as 3 × 5 = 5 + 5 + 5; and division as sharing, such as 8 shared between 4. They use commutative and associative properties of addition and multiplication in mental computation (for example, 3 + 4 = 4 + 3 and 3 + 4 + 5 can be done as 7 + 5 or 3 + 9)

2.25

• use of place value (as the idea that ‘ten of these is one of those’) to determine the size and order of whole numbers to hundreds

2.5

• automatic recall of number facts from 2, 5 and 10 multiplication tables
• use of strategies such as ‘near doubles’, ‘adding 9’ and ‘build to next 10’ to solve addition and subtraction problems
• use of written methods for whole number problems of addition and subtraction involving numbers up to 99

2.75

• use of algorithms for the addition and subtraction of numbers to two decimal places
• representation of multiplication as a rectangular array and as the area of a rectangle
• use of fact families (5 × 7 = 35, 35 ÷ 7 = 5) to solve division problems

3.0 Standard

At Level 3, students use place value (as the idea that ‘ten of these is one of those’) to determine the size and order of whole numbers to tens of thousands, and decimals to hundredths. They round numbers up and down to the nearest unit, ten, hundred, or thousand. They skip count forwards and backwards, from various starting points using multiples of 2, 3, 4, 5, 10 and 100.

They estimate the results of computations and recognise whether these are likely to be over-estimates or under-estimates. They compute with numbers up to 30 using all four operations. They provide automatic recall of multiplication facts up to 10 × 10.

They devise and use written methods for:

•         whole number problems of addition and subtraction involving numbers up to 999

•         multiplication by single digits (using recall of multiplication tables) and multiples and powers of ten (for example, 5 × 100, 5 × 70 )

•         division by a single-digit divisor (based on inverse relations in multiplication tables).

3.25

• appropriate selection and use of mental and written algorithms to add, subtract, multiply and divide (by single digits) natural numbers

3.75

• multiplication by increasing and decreasing by a factor of two; for example,

24 × 16 = 48 × 8

= 96 × 4 = 192 × 2

= 384 × 1 = 384

• recognition that multiplication can either enlarge or reduce the magnitude of a number (multiplication by fractions or decimals)
• use of inverse relationship between multiplication and division to validate calculations

4.0 Standard

At Level 4, students explain and use mental and written algorithms for the addition, subtraction, multiplication and division of natural numbers (positive whole numbers). They add, subtract, and multiply fractions and decimals (to two decimal places) and apply these operations in practical contexts, including the use of money. They use estimates for computations and apply criteria to determine if estimates are reasonable or not.

5.0 Standard

At Level 5, students identify complete factor sets for natural numbers and express these natural numbers as products of powers of primes (for example, 36 000 = 25 × 32 × 53).

Students use a range of strategies for approximating the results of computations, such as front-end estimation and rounding (for example, 925 ÷ 34 ˜ 900 ÷ 30 = 30).

Students use efficient mental and/or written methods for arithmetic computation involving rational numbers, including division of integers by two-digit divisors.