Number and Algebra
Level 1
Number strategies
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.
Number knowledge
AO1: Know the forward and backward counting sequences of whole numbers to 100. This means students will know the forward number word sequence to 100 is the counting pattern of words and symbols, 0, 1, 2, 3, 4,...; while the backward sequence is the pattern 100, 99, 98, 97,... Students will also be able to name the number before and after a given number since this relates to taking an item off or putting an item onto an existing set.
AO2: Know groupings with five, within ten, and with ten. N3. This means students will learn visual and symbolic patterns for the numbers to ten so they can be recognised without counting), groupings within and with five, for example 2 + 3, 5 + 4, names for ten for example 6 + 4 therefore 10 – 4, doubles to ten at least, for example 4 + 4, and groupings with ten, for example 10 + 6, 8 + 10 (teen numbers).
Equations and expressions
AO1: Communicate and explain counting, grouping, and equal-sharing strategies, using words, numbers, and pictures. This means students will explain the number strategies they use to others using a combination of words, numbers and pictures. This implies that students will learn to write equations to express their findings, for example 5 + 9 = 14, to express their ideas using their own language in conjunction with mathematical language, e.g. add, subtract, times, fraction, and to develop diagrams to represent their strategies, for example set diagrams or number lines.
Patterns and relationships
AO1: Generalise that the next counting number gives the result of adding one object to a set and that counting the number of objects in a set tells how many. This means students will understand the link between the cardinal and ordinal aspects of counting. The ordinal aspect refers to the fact that counting numbers have a conventional order. The last number in a count tells how many objects are in a set if all the objects are matched in one-to-one correspondence to the sequence of counting numbers. s. The next number in the counting sequence tells the result of adding an object while the number before in the sequence tells the count when an object is removed. The cardinal aspect involves knowing that when counting a set of items the last number describes all the items in the set, no matter their colour, size, arrangement or other attributes. This count can be trusted and built upon.
AO2: Create and continue sequential patterns.
This means the students will explore sequential patterns. A sequential pattern is one in which further members of that pattern can be predicted
from previous members. So 
..., and 1, 3, 5, 7, ... are sequential patterns. At
Level One students should be able to reproduce a given pattern using objects, drawings or symbols and continue the pattern on with justification,
e.g. It goes square, circle, star...They should also be able to invent their own patterns and communicate the “rule” for
their pattern to others.
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Level 2
Number strategies
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.
Number knowledge
AO1: Know forward and backward counting sequences with whole numbers to at least 1000. This means students will know the forward number word sequence to 1000 is the counting pattern of words and symbols, 0, 1, 2, 3, 4,...1000 while the backward sequence is the pattern 1000, 999, 998, 997, ... At level Two students should know these sequences in multiples of one, ten, e.g. 358, 348, 338,..., and one hundred, e.g. 247, 347, 447,... An important part of knowing these sequences is being able to name the number before and after a given number since this relates to taking an item off or putting an item onto an existing set, e.g. If a set contains 800 items, 799 items are left if one is removed. This also applies to the sequence in tens and hundreds, e.g. ten removed from a set of 503 results in 493 objects left..
AO2: Know the basic addition and subtraction facts.
This means students will know the basic addition facts from 0 + 0 = 0 to 9 + 9 = 18.
So 4 + 1 = 5, 8 + 6 = 14, and 9 + 3 = 12 are all basic addition facts. The basic subtraction
facts are the subtraction equivalent of the addition facts, so 5 – 1 = 4, 5 – 4 = 1, 12 -3 = 9 and
12 – 9 = 3 are all examples. It is important that students understand the commutative property of
addition, e.g. 4 + 7 = 7 + 4, and the inverse nature of addition and subtraction, e.g. 6 + 7 = 13 so
13 – 7 = 6, as a foundation for more difficult problems, as well as a way to connect basic facts.
Students also need to encounter the unknown in different positions within their basic facts, e.g. 4 + = 12
and – 5 = 8.
AO3: Know how many ones, tens, and hundreds are in whole numbers to at least
1000.
This means students will develop an additive view of whole number place value by knowing the significance of the
position of digits in a whole number, e.g. In 456 the 5 means five tens. However, many strategies for computation
require a nested view of place value. This means that nested in the hundreds are tens in the same way that nested in
the hundreds and tens are ones, e.g. 456 has 45 tens and 456 ones. An understanding of nested place value is best
demonstrated by calculations where tens must be constructed from ones, hundreds constructed from tens, tens created from breaking hundreds and
ones created from breaking tens. For example, calculations like 456 + 70 = or
456 - = 396, show whether students can apply place value in this way.
AO4 Know simple fractions in everyday use. This means students will understand the meaning of the digits in a fraction, how the fraction can be written in numerals and words, or said, and the relative order and size of fractions with common denominators (bottom numbers). Fundamental concepts are that fractions are iterations (repeats) of a unit fraction, e.g. 3/4 = 1/4 + 1/4 + 1/4 and 4/3 = 1/3 + 1/3 + 1/3 + 1/3 . This means the numerator (top number) is a count and the denominator tells the size of the parts, e.g. In 4/3 there are four parts. The parts are thirds created by splitting one into three equal parts. This means that fractions can be greater than one, e.g. 4/3 = 1 1/3, and that fractions have a counting order if the denominators are the same, e.g. 1/3, 2/3, 3/3, 4/3,... Note that whole numbers can be written as fractions, e.g. = 1. Fractions in everyday usage include halves, thirds, quarters (fourths), fifths, eighths, and tenths..
Equations and expressions
AO1: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols. This means students will be able to use words, symbols and diagrams to explain their number strategies to others. Recording also allows students to think through solutions to problems and allows them to reduce their working memory load by storing information in written form. This is particularly important for the solving of complex, multi-step problems. Students should be able to write the numerals for whole numbers, to 1000, and simple fractions. They should also be able to write addition, subtraction, multiplication and division equations with understanding of the meaning of these operations and of the equals sign as meaning “equal to”. Similarly they should know which operation to perform on a calculator if the numbers are beyond their mental range. Students should also be familiar with using empty number lines to record addition and subtraction strategies and of drawing arrays to record simple multiplication and division strategies. Formal written algorithms for multi-digit addition and subtraction should not be taught at Level Two until students have the place value knowledge required to understand them.
Patterns and relationships
AO1: Generalise that whole numbers can be partitioned in many ways.
Students at level two should understand that numbers are counts that can be split in ways that make the operations of addition, subtraction, multiplication and division easier. From Level One students understand that counting a set tells how many objects are in the set. An advance on this thinking is to realise that the count of a set can be partitioned and that the count of each subset tells how many objects are in that subset. Also required is understanding that partitions of a count can be recombined. For example, a count of ten can be partitioned into 1 and 9, 2 and 8, 3 and 7, etc. This objective also involves critical choice of partitioning. For example, 8 + 6 = can be solved by partitioning 6
into 1 and 5, 2 and 4, 3 and 3. Of these partitions 2 and 4 is the best strategic choice since it recombines into a “ten and...” fact, i.e. 8 + 6 = 8 + 2 + 4 = 10 + 4. At Level Two students are expected to understand the strategic importance of using place value as a way to partition numbers. Students should apply their partitioning generalisation to many problem types including combining (e.g. 27 + 9 = ) ,
separating (e.g. 105 - 19 = ), comparing (e.g. 45 +
= 106), duplicating (e.g. 8 x 5 = ) and sharing (e.g. 20 ÷ 4 = ).
AO2: Find rules for the next member in a sequential pattern.
This means students will explore sequential patterns, both can be either spatial,
e.g. ,.. or numeric, e.g. 1, 3, 5, 7, ... A pattern has consistency so further ter
ms of it can be anticipated from those already known. In spatial patterns students should be able to
identify the repeating element, e.g. 
in that above,
and use this to predict the shape in a given
ordinal position, e.g. the next shape is 
,
the eleventh shape will be 
. For simple number patterns
students should identify the consistent “gap” between the terms, e.g. 1, 3, 5, 7,... two is added each
time, and use this additive difference to find further terms. Students should also develop their concept
of relations between variables using
spatial patterns that can be represented using numeric tables of values, e.g. For this pattern, how many squares make 7 crosses?
 |
|
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| 1 cross 5 squares |
2 crosses 10 squares |
3 crosses 15 squares |
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Level 3
Number strategies
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 = as 60 – 38 tens less one (219), rounding and compensating,
e.g. 923 – 587 = as 923 – 600 + 13 = ,
and reversing (applying inverse), e.g. 923 – 587 = as 587 +
= 923. Students should also connect known multiplication facts to solve
multiplication and division problems, e.g.
13 x 6 = as 10 x 6 + 3 x 6 =
(distributive property), 14 x 9 = as 2 x (7 x 9) =
(associative property)
and 36 ÷ 9 =
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.
Number knowledge
AO1: Know basic multiplication and division facts. This means students will know basic multiplication facts are those that range from 0 x 0 = 0 to 9 x 9 = 81. The division basic facts are the inverse of the multiplication facts. So 6 x 4 = 24, 4 x 6 = 24, 24 ÷ 6 = 4 and 24 ÷ 4 = 6 are all basic facts. Students should commit their basic facts to memory as soon as they understand the meaning of the equations and can use number properties to work them out, 8 x 7 = 56 means "eight sets of seven" and can be worked out by doubling 4 x 7 = 28. Note that 56 ÷ 7 can mean "fifty-six shared among seven" or "how many sevens are in fifty-six".
AO2: Know counting sequences for whole numbers. This means students will know the forward number word sequence for whole numbers is the counting pattern of words and symbols, 0, 1, 2, 3, 4,... ∞ (infinity) while the backward sequence is the pattern 1000 000, 999 999, 999 998, 999 997, ...beginning with any whole number. At Level Three students should know these sequences in multiples of one, ten, e.g. 358, 348, 338,..., one hundred, e.g. 247, 347, 447,..., one thousand, etc. An important part of knowing these sequences is being able to name the number before and after a given number since this relates to taking an item off or putting an item onto an existing set, e.g. If a set contains 43 560 items, 43 559 items are left if one is removed and 43 561 items are in the set if one is added. This also applies to the sequence in tens, hundreds, thousands, etc. e.g. ten thousand removed from a set of 701 000 results in 691 000 objects left. At Level Three students should also have experience with counting sequences in tenths, e.g. 4.6, 4.7, 4.8, 4.9, 5 ,...
AO3: Know how many tenths, tens, hundreds, and thousands are in whole
numbers.
This means students will develop a multiplicative view of whole number place value involves more than
knowing the significance of the position of digits in a whole number, e.g. In 239 456 the 3 means three ten thousands.
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 thousands are hundreds, tens and ones in the same way that
nested in the tens are ones and tenths, e.g. 239 456 has 23 ten thousands, 2394 hundreds, and 23 945 tens, etc.
An 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. For example, calculations like 2 004 - 700 = may require students to think of 1000 as ten hundreds. At Level Three students should connect the multiplicative value of the places, e.g.
one hundred thousand is ten times as much as ten thousand, and one hundred is the result of dividing one thousand by ten.
They should know the effect of multiplying and dividing by ten, 4200 is ten times more than 420 and 43 divided by ten is 4.3.
AO4: Know fractions and percentages in everyday use. This means students will understand the meaning of the digits in a fraction, how the fraction can be written in numerals and words, or said, and the relative order and size of fractions with common denominators (bottom numbers) or common numerators (top numbers). Fundamental concepts are that fractions are iterations (repeats) of a unit fraction, e.g. 3/5 = 1/5 + 1/5 + 1/5 and 5/3 = 1/3 + 1/3 + 1/3 + 1/3 + 1/3. This means the numerator (top number) is a count and the denominator tells the size of the parts, e.g. in 5/3 there are five parts. The parts are thirds created by splitting one into three equal parts. This means that fractions can be greater than one, e.g. 4/3 = 1 1/3, and that fractions have a counting order if the denominators are the same, e.g. 1/3, 2/3, 3/3, 4/3,... The size of the denominator also affects the size of the parts being counted in a fraction. For example, thirds of the same whole are smaller than halves of the same whole. So fractions with common numerators have an order of size based on the size of the parts, e.g. 2/7 < 2/5 < 2/3 (< means “less than”). Students at Level Three should know simple common fraction-percentage relationships, including 1/2 = 50%, 1/4 = 25%, 1/10 = 10%, 1/5 = 20%, and use this knowledge to work out non-unit fractions as percentages, e.g. 3/4 = 75%.
Equations and expressions
AO1: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality. This means students will use words, symbols and diagrams to explain their number strategies to others. Recording also allows students to think through solutions to problems and allows them to reduce their working memory load by storing information in written form. This is particularly important for the solving of complex, multi-step problems. Students should be able to write the numerals for whole numbers to 1 000 000 at least, simple fractions, percentages and decimals. They should also be able to write addition, subtraction, multiplication and division equations with understanding of the meaning of these operations and of the equals sign as meaning “equal to”. Similarly, they should know which operation to perform on a calculator if the numbers are beyond their mental range. Students should also be familiar with using empty number lines to record addition and subtraction strategies, arrays to record multiplication and division strategies, and strip diagrams or double number lines to solve problems with fractions and percentages. Formal written algorithms for multi-digit addition and subtraction should be taught at Level Three after students have the nested place value knowledge required to understand them.
Patterns and relationships
AO1: Generalise the properties of addition and subtraction with whole numbers.
This means students will generalise, which means to establish properties that hold for all instances.
Generalisation begins with noticing patterns and relationships in a few specific instances, defining the
variables involved, noticing the relationships between the variables, then using appropriate mathematical
terminology and symbols to describe the relationships. At Level Three students develop many generalisations
that allow them to perform mental strategies effectively. These generalisations include, the commutative property of addition and multiplication, e.g. 7 x 8 = 8 x 7, the associative property of addition and multiplication, e.g. (2 x 3) x 4 = 2 x (3 x 4), the distributive property of multiplication, e.g. 8 x 7 = 8 x 5 + 8 x 2, the inverse relationships of addition and subtraction, and of multiplication and division, e.g. 6 x 7 = 42 so 42 ÷ 7 = 6, and identities for all four operations, e.g. 17 x 1 = 17, 17 ÷ 1 = 17. It is not expected that students use algebraic symbols to express these generalisations. However, students should be able to look for
relationships across the equals sign in equations to determine missing numbers, e.g. 4 x 12 =
x 6 without calculating 4 x 12.
AO2: Connect members of sequential patterns with their ordinal position and use
tables, graphs, and diagrams to find relationships between successive
elements of number and spatial patterns.
This means students will recognise that a sequential pattern can be either spatial,
e.g. ..,
or numeric, e.g. 1, 3, 5, 7, ... A pattern has consistency so further terms of it can be anticipated from those already known.
The focus in this thread is that students become increasingly sophisticated at describing the relationships between variables found
in sequential patterns. With spatial patterns, students at Level Three should be able to identify the repeating element, e.g.
, and use simple multiplicative thinking
to predict the shape in a given ordinal position, e.g.
Every third shape is 
so the thirtieth shape will be
so the thirty-second shape will be
With number patterns students should identify the consistent relationship between variables in simple multiple situations,
e.g. 4, 8, 12, 16,... are all multiples of four, or identify the additive “gap” between the terms,
e.g. 4, 7, 10, 13,... three is added each time.
They should be able to describe these rules in their own words and use their rules to find further terms.
Students also use tables, graphs, diagrams and word rules to find and describe relationships in patterns, e.g.
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“There is always one more peg that the number of towels. The first towel took two pegs.”
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Level 4
Number strategies 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 =
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.
Equations and expressions
AO1: Form and solve simple linear equations. This means students will solve simple linear equations in the form y = mx + c, where x and y are related variables and where m is a whole number and c is an integer, e.g. q = 3p – c, or a + 5 = 4b. When the value of one variable is given the value of the other can be found by solving the equation, e.g. 3p – 6 = 18. Students should understand the equals sign as a statement of balance and know what operations to both sides of an equation preserve that balance, e.g. take off the same number from both sides. At Level Four students should be able to find the required value using both sensible estimation and improvement, and by formal methods of applying inverse operations, e.g. 3p – 6 = 18 so 3p = 24 (adding six to both sides) so p = 8 (dividing both sides by three).
Patterns and relationships
AO1: Generalise properties of multiplication and division with whole numbers. This means students will generalise, which means to establish properties that hold for all occurrences. This involves the ability to look at several examples, notice what changes (variables) and what does not, use appropriate mathematical terminology and symbols to describe the pattern, and apply the generalisation to other examples. At Level Four students should be able to describe and apply the properties of multiplication and division as these operations apply to whole numbers. These properties include commutativity, distributivity, associativity, inverse and identity. This includes the ability to express generalisations using words and symbols, e.g. 4 x 6 = 24 so 24 ÷ 6 = 4 and 24 ÷ 4 = 6 (example) leading to a x b = c so c ÷ b = a and c ÷ a = b. This is the inverse relationship of multiplication and division.
AO2: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns. This means students will describe the function rule for a linear relationship as well as recognise recursive relationships where more complex relationships are involved. For example, given the pattern of fish made with matchsticks and counters below, students should be able to represent the relationships in a table and graph and use these representations to predict the terms in the sequence:
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Counters |
1 |
2 |
3 |
4 |
5 |
Matchsticks |
8 |
14 |
20 |
26 |
32 |
Level Four students should be able to:
- Give linear rules connecting the variables, e.g. "the number of matchsticks is the six times the number of counters plus two", or "take one off the number of fish, multiply that number by six then add eight".
- Extend the graph or table of a linear relationship to predict further co-ordinate pairs, recognising that constant difference (add six in the fish pattern) is associated with points that lay on a line.
- Use recursive methods to predict further members of a sequence where the relationship is non-linear, e.g. The sequence of triangular numbers:
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| +2 | +3 | +4 | +5 | ||||||||||||||||||
Recursive means finding what is added to or subtracted from one term to get the next.
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Level 5
Number strategies 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:
- Using multiple ways to represent the same number in operator situations, for example 45% x 52 = 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).
- 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) .
- 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.
- 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.
- 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.
- 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 = , change unknown, for example % of 38 = 21.28, or start unknown, for example 56% of = 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?
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 = 25) or between (e.g. 4 x = 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 = . 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 = 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.
Equations and expressions
AO1: Form and solve linear and simple quadratic equations.
Students should be able to form the linear equation or simple quadratic (y = ax2 or y = x2 ± c, a and c are integers) to model a given situation (see patterns and relationships). They should understand that solving an equation involves finding the value of a variable when the other variable is defined, and interpret how the solution relates to the original context. Students should be able to solve linear and simple quadratic equations by applying inverse operations with an understanding of the equals sign as a statement of transitive balance, for example (3q + 7)/4 = 16, by multiplying both sides by four, subtracting seven, etc. They should also recognise where it is appropriate to solve an equation through trial and improvement, and find the missing value by systematic calculation.
Patterns and relationships
AO1: Generalise the properties of operations with fractional numbers and integers.
This means students will understand that to generalise means to establish properties that hold for all occurrences.
This involves the ability to examine a number of cases, define the variables involved, use appropriate mathematical terminology and symbols,
and ultimately reason with the properties themselves. Fractional numbers, for the purpose of this objective, are defined as rational numbers in the form a/b,
where a and b are whole numbers and b ? 0. At Level Five students should be able to demonstrate their understanding of the properties of addition, subtraction,
multiplication and division as these operations apply to fractional numbers and integers. These properties include commutativity, distributivity, associativity,
inverse and identity. Demonstration of understanding should involve applying these properties in solving a variety of problems, using the properties to solve
equations without calculating both sides, for example, 6 x = 3 x 70 + 3 x 9, justifying their responses to conjectures such as true/false statements, and expressing the generalisations algebraically, for example, the commutative property for multiplication of integers may be represented by a x b = b x a, where a and b are integers. Students should be able to express the operations on fractional numbers algebraically, for example, a/b + c/d = (ad + cb)/bd, and substitute number values into the equation to confirm that it holds for all addition examples they attempt.
AO2: Relate tables, graphs, and equations to linear and simple quadratic relationships found in number and spatial patterns.
This means students will recognise the features of tabular, graph and equation representations of linear and quadratic relationships. This includes connecting constant first or second order difference in tables with linear and quadratic relations respectively, with the graph (linear and parabolic) and standard equation forms (y = mx + c and y = a x 2 + bx + c) for such relations. For example, given the spatial pattern below students can use a table, graph or equation to represent the relation and solve problems.
This includes finding both a recursive and direct (functional) rules and using them to find further terms using a spreadsheet or calculator, for example:
Students should also use these tabular and graphic representations for other relationships, such as simple exponential and step relations, but it is acceptable for them to use recursive rules for these more difficult relations.
Click to download a PDF of second-tier material relating to Level 5 Patterns and Relationships (527KB)
Level 6
Number strategies 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:
- 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.
- 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).
- 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:
- 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.
- 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.
- 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:
- 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.
- Create formulae that calculate the quantities required by the problem, e.g. Length of Edge C and surface area in this problem.
- 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
Equations and expressions
AO1: Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns. This means students will create equations to model everyday situations, e.g. express a taxi charge as a linear equation (flagfall and kilometre rate) or the exponential relationship between the number of repeated folds (in thirds) of a paper strip and the number of sections formed. This includes forming pairs of simultaneous linear equations. Students should be able to form equations from tables of values, using differences between terms, constant first order for linear relations, constant second order differences for quadratic relations and constant ratio for simple exponentials. They should use algebraic manipulation skills to simplify expressions, including rational expressions involving exponents, e.g. 9n4 / 6n3. Students should apply their manipulations skills to solve linear and quadratic equations by applying inverse operations with an appreciation of equality and connect their solutions to corresponding situations of inequality, e.g. If (6x - 8)/4 = 10 has the solution x = 8 then (6x - 8) /4 < 10 has the solution x < 8. They should be able to solve quadratic equations by factorising and have the disposition and capability to check all of their algebraic solutions by substituting values. Solving simple exponential equations should be done by inspection at this level, e.g. 3x = 81 by recognising 34 = 81 so x = 4. Pairs of simultaneous equations may be solved by substitution, elimination and by intercept of graphs.
Patterns and relationships
AO1: Generalise the properties of operations with rational numbers, including the properties of exponents. This means students will generalise, which means to establish properties that hold for all occurrences. This involves the ability to examine a number of cases, define the variables involved, use appropriate mathematical terminology and symbols, and ultimately reason with the properties themselves. Rational numbers are defined as those that can be expressed in the form a/b, where a and b are integers and b ? 0. At Level Six students should be able to describe and apply the properties of addition, subtraction, multiplication and division as these operations apply to rational numbers and exponents. These properties include commutativity, distributivity, associativity, inverse and identity. This includes the ability to express the generalisations algebraically, e.g. the commutative property of addition may be represented by a/b + c/d = c/d + a/b. Students should be able to express the multiplication and division of exponents with common bases algebraically, e.g. (an)/ am = an - m, and derive other properties of exponents by applying first principles to specific cases, e.g. 32 x 42 = 3 x 3 x 4 x 4 = 12 x 12 = 122 leading to anbn = (ab)n .
AO2: Relate graphs, tables, and equations to linear, quadratic, and simple exponential relationships found in number and spatial patterns. This means students will relate the shape of graphs to the type of relation they show; lines for linear, parabolas for quadratics and exponential curves. They should also be able to find the equations using the slope and intercept for linear equations (relating slope to the constant between y-values), and using vertex, orientation and specific ordered pairs of parabolas for quadratics. They should recognise the relationship involved considering the difference and ratios between terms in tabular form. The spatial and number patterns involved are those that yield appropriate co-variation between variables within the pattern. The advantage of spatial patterns is that students are able to generate potential relationships by attending to the spatial elements within the pattern and validate their findings through mapping back to the pattern itself. Students at Level Six should connect the spatial, tabular, graphic and equation representations of a relationship and choose which representation they see as most useful to solve a given problem. For example, consider the growing diamond pattern:
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Year One |
Year Two |
Year Three |
The relationship between years and matches could be expressed as:
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...or as an equation m = 2y2 + 2y where m represents the numbers of matches and y the number of years.
AO3: Relate rate of change to the gradient of a graph. This means students will connect the difference between successive terms in a relationship as the rate of change and know how this shows in graphical representation. They should do so through interpreting everyday contexts such as the speed of falling objects (e.g. parachutists), growth of organisms (e.g. algae) or compound growth (e.g. debt if unpaid). This includes knowing that constant differences between terms result in linear graphs and equations, constant second order differences and ratio suggest different models of varying rate of change that can be presented as quadratic or exponential equations. Students should be able to map from a graph to the situation that produced the graph, e.g. describe the speed of a car from a time and distance graph.
