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Number and Algebra: Number strategies

Level 1

AO1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
This means students will use counting strategies including counting on and back, double counting, and skip counting. This corresponds to the counting stages of the number framework so achieving level one means that a student is at the Advanced Counting Stage. Examples of their strategies might be, to calculate 6 + 5 count 7, 8, 9, 10, 11, to calculate 12 – 3 count 11, 10, 9, or to calculate three groups of three double counting 1, 2, 3,...4, 5, 6,...7, 8, 9. Grouping and equal sharing strategies are simple ways to solve addition, subtraction, multiplication and division, and fractions of sets problems without counting every object. Examples of these strategies might be; knowing 4 + 4 equals 8, skip counting 5, 10, 15, 20 to count four groups of five, or sharing objects in ones, twos or threes to find one quarter of a set of 12 items.

Level 2

AO1: Use simple additive strategies with whole numbers and fractions.
This means students will learn to treat whole numbers as units of ones that can be split and recombined to make calculations easier. Additive strategies are about a type of thinking not the operation of addition. So additive strategies can be applied to addition, e.g. 47 + 38 is 50 + 40 – 5, subtraction, e.g. 74 – 8 = as 74 – 4 – 4 = , multiplication, e.g. 4 x 6 = as 4 + 4 + 4 + 4 = , which is 8 + 8 = , division, e.g. 18 ÷ 3 = , as 5 + 5 + 5 = 15 so 6 + 6 + 6 = 18. Additive strategies may also be applied to finding fractions of sets particularly halves, thirds, quarters, fifths, eighths and tenths. Level Two corresponds to students being proficient at the Early Additive stage of the number framework.

Level 3

AO1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
This means students will use a range of mental strategies based on partitioning and combining to solve addition and subtraction problems with multi-digit whole numbers and simple decimals (tenths). These strategies include, standard place value, e.g. 603 – 384 = box. as 60 – 38 tens less one (219), rounding and compensating, e.g. 923 – 587 = box. as 923 – 600 + 13 = box. , and reversing (applying inverse), e.g. 923 – 587 = box. as 587 + box. = 923. Students should also connect known multiplication facts to solve multiplication and division problems, e.g. 13 x 6 = box. as 10 x 6 + 3 x 6 = box. (distributive property), 14 x 9 = box. as 2 x (7 x 9) = box. (associative property) and 36 ÷ 9 = box. using 4 x 9 = 36 (inverse). This multiplicative understanding allows students at Level Three to find fractions of quantities, e.g. two-thirds of 24 as 24 ÷ 3 x 2 = 16, find simple equivalent fractions related to doubling and halving, e.g. 3/4 = 6/8 , to add and subtract fractions with the same denominators, e.g. 3/4 + 3/4 = 6/4 = 1 2/4, and to convert improper fractions to mixed numbers, e.g. 17/3 = 5 2/3. Students should know the decimals and percentage conversions of simple fractions (halves, quarters, fifths, tenths) and use these to solve simple percentage of amount problems, e.g. 50% is fifty out of one hundred. 50% is one half so 50% of 18 is 9 or five is half of ten so = 0.5 and = 0.5. Level Three corresponds to the Advanced Additive stage of the number framework.

Level 4 (strategy and knowledge)

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 = box. 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 = box. as 6 x box. = 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 box. = 24 (Four-sevenths of what number is twenty-four?) or box. % 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 = box. 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 = box. as 16 x 4 ÷ 100 = 0.64 and 24 ÷ 0.3 = box. 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:

  1. 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.
  2. 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).
  3. 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.
  4. 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.
  5. 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.
  6. 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.

numberline.

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 = box. 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 = box. and +3 + +2 = box. have the same answer, +5.

Level 5 (strategy and knowledge)

AO1: Reason with linear proportions.

This means students will explore linear proportions in a variety of contexts. Linear proportions apply to situations which can be modelled using equivalent fractions, i.e. a/b = c/d where a,b,c, and d are integers (usually whole numbers). So proportional reasoning pervades many of the outcomes in all three strands and many contexts including:

  1. Using multiple ways to represent the same number in operator situations, for example 45% x 52 = box. can be seen as 0.45 x 52 and 45/100 x 52, and comparing the potential results of operator situations by suspending calculation and thinking relationally, 30% x 34 = 60% x 17 (by doubling and halving) or 1.3 x 3.3 < 3.9 x 1.2 (since 1.3 x 3.3 = 3.9 x 1.1).
  2. Comparing the results of sharing situations which involve fractional quotients, for example 3 pizzas shared among 5 boys (3/5 pizza each) results in a lesser share than 2 pizzas shared among 3 girls (2/3 pizza each), and find the difference in shares (2/3 -3/5=1/15)
    3. .
  3. Comparing the size of two fractions, decimals or percentages, using benchmark fractions or equivalence, give the difference between the fractions, and name a fraction between two fractions. For example, 4/7 > 5/9 since 4/7 is 1/14 greater than 1/2 and 5/9 is 1/18 greater than 1/2 or 4/7 = 36/63 and 5/9= 35/63 so the difference between 4/7and 5/9 is 1/63.
  4. Reasoning qualitatively about the size effect on a fraction as the numerator, denominator, or both numbers are changed. For example, given the fraction 5/11, reason that 5/10 and 6/11will be greater, 4/11 and 5/12 will be less, and comparing it with 4/10 and 6/12 will require further investigation.
  5. Measuring one fraction with another either by converting to equivalent forms or scaling the result of the same divisor acting on one. For example, each trip takes 3/4 of a full tank of petrol. You have 2/5 of a tank. What fraction of a trip can you make? as 3/4 = 15/20 and 2/5 = 8/20 so 8/20 is 8/15 of 15/20 (2/5 ÷3/4 = 8 /15) or 1 ÷3/4 = 4/3 (1 1/3 trips on a full tank) so 2/5 x 4/3 = 8/15 trips with two-fifths of a tank.
  6. Other examples of reasoning with linear proportions are discussed through the other achievement objectives.

AO2: Use prime numbers, common factors and multiples, and powers (including square roots).

This means students will know that prime numbers are numbers divisible by only themselves and one, and apply this to the fundamental law of arithmetic that every counting number has a unique prime factorisation, e.g. 36 = 2 x 2 x 3 x 3 = 22 32. They should apply prime factorisation to problems that involve factors and multiples, including finding the least common multiple or highest common factor. For example, “What sized cuboids can be made using 105 unit cubes?”, or “What is 105 out of 231 in simplest form?”

They should understand and use the additive law of exponents, that is ab x ac = ab+c and a b ÷ ac = a b - c and compare powers relationally (without calculation) where this is appropriate, e.g. 36 >63 because (3x3)x(3x3)x(3x3)>6x6x6. Students should understand the arithmetic and geometric origin of square roots. For example, a square of area 144cm2 has a side length of 12cm, and use common square roots to estimate the value of other square roots. For example, √36 = 6 and √49 = 7 so √42 ≈ 6.5. They should also understand the convention for negative exponents through pattern. For example 21= 2 so 20= 1 so 2 -1= 1/2 since the effect of decreasing the exponent by one is to divide the previous power by two.

AO3: Understand operations on fractions, decimals, percentages, and integers.

This means students will understand calculations involving fractions, decimals, percentages and integers assumes accuracy in calculation and the exercising of appropriate choice between mental, written and machine methods given the complexity of the numbers involved and the significance of the calculation in the context of the problem. Understanding also implied the prudent use of estimation to check the reasonableness of calculations and as an end in itself where approximations are sufficient.

Students should be able to explain the calculation steps (procedures) they followed and justify those steps by describing the quantities involved. For example, the calculation 1.4 x 0.6 = 0.84 might be justified as 14/10 x 6/10 = 84/100 or 14 x 6 = 84 and the size of answer being about half of 1.4.

The problems solved should involve result unknown, for example 56% of 38 = box. , change unknown, for example box. % of 38 = 21.28, or start unknown, for example 56% of box. = 21.28.

AO4: Use rates and ratios.

This means students will solve problems involving rates and ratios. In this curriculum rates are defined as a multiplicative relationship between different measures, for example, 24 litres per 60 minutes, while ratios are defined as a multiplicative relationship between identical measures, for example, 30 litres: 40 litres. This distinction is blurred where the measures are of the same attribute, for example, 10mL per 1 Litre, but problems involving unit conversion are delayed until Level Six. In terms of their behaviour problems involving both rates and ratios can be modelled by the equation a/b = c/d where one of the values, a, b, c, or d is unknown or as a situation where a/b and c/d must be compared. Rate and ratios can also be represented by ratio tables or double number lines. For example:

A wallpaper hanger mixes 300 grams of glue powder to every 4 litres of water. She wants to make up 25 litres of paste. How many grams of powder will she need?

ratios.

At Level Five students are expected to solve problems of this type in which the unknown can be in any of the four positions on the table and in which the scalar within (e.g. 4 x box. = 25) or between (e.g. 4 x box. = 300) operators are positive integers or fractions. Students should be able to use equivalent rates to compare two given rates and express the part-whole relationships in ratios as equivalent fractions to compare given ratios. For example, 3 litre orange: 5 litres apple has a stronger orange flavour than 4:6 because the part-whole fractions are 3/8 and 4/10 respectively which have equivalent forms of 15/40 and 16/40.

AO5: Know commonly used fraction, decimal, and percentage conversions.

This means students will be able to express any of the fractions (halves, quarters, thirds, fifths, eighths, tenths, hundredths and thousandths) as decimals and percentages. For example, 3/8 = 0.375 = 37.5% and use whatever form is easiest for a given calculation, e.g. 30% of $78 as 3/10 of 78 = box. . Students should also be able to give the fraction form of any decimals to three places and vice versa, e.g. 1.346 = 1346/1000, and express percentages, including those greater than one hundred, as decimals and vice versa, e.g. 1.75 as 175%.

AO6: Know and apply standard form, significant figures, rounding, and decimal place value.

This means students will be able to express a given whole number or decimal measurements in standard form and vice versa and understand the potential rounding that may be involved. Standard form (scientific notation) at this level should involve integral exponents, e.g. 24 300 = 2.43 x 104 or 0.0243 = 2.43 x 10-2. This understanding of decimal place value and rounding should include interpretation of the potential value of a measurement when it is expressed using significant figures, e.g. 2.3m (2sf.) has a potential measurement of 2. 25≤m<2.35 whereas 2.30 (4sf.) has a potential measurement of 2.295≤m<2.305. Students should also apply decimal place value and sensible rounding through estimating in a way that is suitable to the context, and recognising the effects of that rounding on the accuracy of the estimation, e.g. 48.7 ÷ 2.13 = box. can be estimated by 50 ÷ 2 = 25 but the rounding results in an estimate that is too high. This does not include putting error bounds on estimations.

Level 6 (strategy and knowledge)

AO1: Apply direct and inverse relationships with linear proportions.
This means students will solve problems that linear proportions. The term "Linear proportions" encompasses a broad range of contexts including rates and ratios, scaling, probability, conversion between measures, derived measures such as density, conversion between numbers forms, partitioning and replicating, and using rational numbers in operations. Proportional thinking is applied across the objectives of all strands at Level Six. Structurally, problems involving direct linear proportions are of three main types:

  1. Find a missing value in an equality of the form a/b = c/d where one of the values a,b,c, or d is unknown, e.g. "Find 65% of 43", can be represented as 65/100 = x/43.
  2. Determine the size relationship between a/b and c/d, e.g. "which is the stronger concentration of syrup to water; 2:7 or 3:11?" can be represented as 2/9 > 3/14 (the fractions represent the part to whole relationships).
  3. Find values for a, b, c, or d that satisfy the inequality a/b< c/d, e.g. For what positive integer values of x is the following inequality true, 2/5 > 6/x?

Inverse relationships in this objective refer to two types:

  1. Apply an inverse operation where the given information requires it, e.g. "36% of what amount is $26.64?" can be represented as 36x/100 = 26.64.
  2. Solve problems with inverse proportions, e.g. "A car travels from A to B in 25 minutes at 100 kilometres per hour. How long will the trip take at 80 kilometres per hour?" can be represented as 25 x 100 = 80x.
  3. Both direct and inverse proportional relationships should be represented through equations (as above), tables (including spreadsheets) and graphs.

AO2: Extend powers to include integers and fractions.
This means students will extend their understanding or powers to include powers involving fractions and integers. The conventions for the meaning of negative and fractional exponents are derived from the preservation of the number laws for exponents, ab x a = ab + c and ab ÷ ac = ab - c, e.g. 42 x 43 = 45 so 45 ÷ 42 = 43. So the meaning of a-b must preserve the truth of a-b x ab = ab + -b. Since ab + -b = a0 = 1, a-b must be the reciprocal of ab ( 1/ab) since ab x 1/ab = 1. For example, 63 = 216 so 6-3 = 1/63 = 1/216. The meaning of a1/b must preserve the truth of (a1/b)b = 1, e.g. a1/2x a1/2 = a1 = a, so a1/2 must be the square root of a (√a) and a1/3x a1/3 x a1/3 = a11 = a, so a1/3 must be the cube root of a (3√a). So in general a1/b = b√a. At Level Six students should accept that negative and fractional exponents behave in the same way as positive integral exponents and use the number laws to solve problems, e.g. If 43 = 64 and √4 = 2, what is 43 1/2? Or If 3√8 = 2, what is 8 2/3? .

AO3: Apply everyday compounding rates.
This means students will solve problems that involve examples of everyday compounding rates. "Compounding rates" refers to situations in which a quantity is compounding, over time, as a fixed rate is applied cumulatively. The most common example of this is compound interest on bank deposits. In this situation the interest earned in one year becomes part of the total amount on which interest is calculated for the following year. Other examples include simple situations of growth and decay, e.g. inflation, bacterial growth, half-lives..

AO4: Find optimal solutions, using numerical approaches.
This means students will structure calculations to find optimal solutions. Optimal solutions are those that maximise or minimise a quantity of importance while meeting the constraints of a situation. For example, "find the square based prism with a volume of 700cm3 that has the minimum surface area", involves minimising surface area while meeting the constraints of given shape and volume. Numerical approaches involve structuring calculations in a systematic way so that the optimal solution is found, and considering the degree of accuracy required. Usually this involves constructing a table (spreadsheet is an example). For example:
At Level Six students should be able to:

  1. Construct a table that contains the relevant variables without unnecessary duplication, e.g. Edges A and B are defined by the same measure in this problem.
  2. Create formulae that calculate the quantities required by the problem, e.g. Length of Edge C and surface area in this problem.
  3. Recognise and act on the need to work within values initially selected to increase the level of accuracy, e.g. work with decimals values for Edge A and B between 8 and 10 for this problem
table.