Number and Algebra: Number strategies and knowledge, Level 4
AO1: Use a range of multiplicative strategies when operating on whole numbers.
This means students will apply the properties of multiplication and division (commutative, distributive,
associative and inverse) to a range of number problems, particularly those requiring multiplication and division.
Students should exercise critical choice in their method of calculation - mental, machine or paper, and recognise
situations in which estimation should be used, including the checking of calculated answers. Strategies expected at
Level Four include; using common factors and multiples, e.g. 37 + 41 + 40 + 38 =
as 4 x 40 – 4,
using the distributive
property, e.g. 24 x 36 = 20 x 36 + 4 x 36, 9 x 78 = 9 x 80 – 9 x 2, or 276 ÷ 12 = 240 ÷ 12 + 36 ÷ 12,
using the
associative property, e.g. 12 x 33 = 4 x 99, or 216 ÷ 12 = 216 ÷ 2 ÷ 2 ÷ 3, and inverse operations (reversing),
e.g. 354 ÷ 6 = as 6 x
= 354. This objective also involves calculating powers,
e.g. 43 = 4 x 4 x 4 = 64, and
factorials, e.g. 4! = 1 x 2 x 3 x 4 = 24. Students should have strong mental strategies for operations on whole numbers
but also accurately carry out standard written algorithms, particularly for multi-digit multiplication and division.
Level Four corresponds to the Advanced Multiplicative stage of the number framework.
AO2: Understand addition and subtraction of fractions, decimals, and integers. This means students will understand decimals as fractions, and be able to express decimals in fraction form and vice versa, e.g. 2.47 = 2 + 4 tenths + 7 hundredths (2 + 4/10 + 7/100 ), or 247 hundredths (247/100). They should solve addition and subtraction problems with decimals and with fractions (denominators must be related multiples), e.g. 13.2 – 5.79 = 7.41 and 3/4 + 7/8 = 13/8 = 1 5/8 by choosing appropriately from mental, machine and paper methods. Students should apply the strategies used for mental calculation with whole numbers to addition and subtraction of decimals, including standard place value, compensation after rounding, and applying inverse (reversing). Formal written algorithms for decimal addition and subtraction should be taught at Level Four after students have the place value knowledge required to understand them..
AO3: Find fractions, decimals, and percentages of amounts expressed as whole numbers, simple fractions, and decimals.
This means students will understand that finding a decimal or percentage of an amount involves finding a fraction of
that amount, e.g. 40% of 56 = x 56 = 4 x 5.6 = 22.4. They should be able to solve problems of the form a/b x c = d
(a,b,c and d are whole numbers), where any one of the numbers is not known, e.g. 4/7 x
= 24 (Four-sevenths of what number is twenty-four?)
or % of 76 = 19 (What percentage of seventy-six is nineteen?).
Students should be able to multiply fractions with understanding, e.g. 2/3 x 4/5 =
as two-thirds of four-fifths, and use their multiplicative understanding of place value to solve multiplication and division problems with simple decimals,
e.g. 1.6 x 0.4 = as 16 x 4 ÷ 100 = 0.64 and
24 ÷ 0.3 = as 24 ÷ 3 x 10 = 80.
AO4: Apply simple linear proportions, including ordering fractions. This means students will solve problems involving linear proportions. “Linear proportion” is a term used to generalise situations that involve equivalent fractions. At Level Four students should be able to solve the following types of problems:
- Comparing the size of two fractions, by converting them to equivalent fractions with a common denominator, or with reference to benchmark fractions, e.g. 2/3 > 4/9 because 2/3 is greater than one half while 4/9 is less, or because 2/3 = 6/9.
- Finding equivalent ratios by either scaling up or down by a whole number multiplier, e.g. 2:5 is the same ratio as 8:20 (scaling up) or 12:18 is the same ratio as 2:3 (scaling down).
- Finding equivalent rates by either scaling up or down with the same measurement units, e.g. 18km in 15mins is the same speed as 72km in 60mins.
- Recognising when two “fraction of an amount” situations give equal or unequal answers, e.g 75% of $12 is the same as 25% of $36.
- Recognising when sharing division situations give equal or unequal shares, e.g. three pizzas shared between five people is a smaller share than two pizzas shared between three people.
- Finding how many measures of a fraction fit into one, e.g. A trip uses 2/5 of a tank of petrol. How many trips can be made on a full tank? (1 ÷ 2/5 = 5/2 = 1 1/2 ).
AO5: Know the equivalent decimal and percentage forms for everyday fractions. This means students will understand decimals and percentages as equivalent fractions, e.g. 3/8 = 375/1000 = 0.375 and 3/8 = 37.5/100 = 37.5%. They should know the fractions for halves, thirds, quarters, fifths, eighths, and tenths as decimals and percentages and be able to convert these decimals and percentages back to their simplest fraction form, e.g. 0.8 = 4/5. The fractions required also include those greater than one, e.g. 240% = 2.4 = 12/5.
AO6: Know the relative size and place value structure of positive and negative integers and decimals to three places. This means students will use a mental number line that includes the relative size of integers and decimals to three places and the whole numbers they know from previous levels. They should be able to locate the position of integers and decimals to three places on a given number line with adherence to scale, particularly where tenths and hundredths divisions are given, e.g.
Knowing decimal place value involves more than knowing the significance of the position of digits in a whole number, e.g.
in 24.671 the 7 means seven hundredths. Strategies for computation require a nested view of place value and understanding the
scaling effect as digits move to the right and left in place value. This means that nested in the ones are tenths, hundredths and
thousandths in the same way that nested in the hundreds are tens, ones, tenths, etc., e.g. 3.509 has 35.09 tenths, 350.9 hundredths,
3509 thousandths, etc... Understanding of nested place value is best demonstrated by calculations where place value units must be
constructed by combining or decomposing other place value units, e.g. 4.2 – 2.68 =
as the difference between 420 hundredths and 268 hundredths.
Students should known the multiplicative relationship between place values, e.g. one hundredth equals ten divided by one thousand, and the effect
of multiplying and dividing a given decimal by ten, one hundred, or one thousand, e.g. 30.4 divided by one hundred equals 0.304.
Students should know the effect of
adding and subtracting integers and be able to represent these operations on a number line,
e.g. +3 - -2 = and +3 + +2 =
have the same answer, +5.
