YEAR 5 - 2011 MATHS SCOPE AND SEQUENCE FOR ST ANTHONY’S (Adapted from the Australian Curriculum + Eva Devries scope and sequence.)
Year 5 National Curriculum Mathematics Scope and Sequence
Please note that the blue italics indicates content added to the original Australian Curriculum.
NUMBER & ALGEBRA
ContentStrand / CONTENT / ELABORATIONS / LEARNING EXPERIENCES / LANGUAGE / RESOURCES
NUMBER & PLACE VALUE
COUNTING / COUNTING
Count and order whole numbers beyond 100,000 and common and decimal fractions /
- working with numbers beyond 100,000 and decimal fractions to hundredths
- counting in common fractions beyond the 1 whole
Students begin working with numbers greater than 100 000
- use a number line (marked and unmarked) and hundreds boards to assist with countingforwards and backwards with large numbers
- use a number line (marked and unmarked) to assist counting in fractions beyond one whole
- use and interpret number lines and number boards from a range of starting numbers
- use mental groupings to count and to assist with estimating the number of items in large groups – group into 10’s, 100’s, 1000’s, etc.
- use the constant function of a calculator and to create and identify the counting number pattern to beyond 100,000
- count in whole number, common fractions and decimal fractions in appropriate contexts
- apply knowledge of counting patterns with smaller numbers to large whole numbers and decimals
number line
groups of
constant
number patterns decimal fraction
common fraction
sequence, order
NUMBER & PLACE VALUE
NUMBER SENSE / NUMBER SENSE
Describe, compare and order large numbers and numbers to 3 decimal places /
- using a variety of manipulatives and other materials to model and compare different representations of whole numbers and decimal fractions
- using place value to compare and order numbers, and locate them, relative to zero, on a number line
Students begin working with numbers greater than 100 000
- ask questions involving large numbers
- explain the position of any number on a number board in relation to other numbers
- recognise and describe subsets of numbers – prime and composite numbers, odd/even, square numbers
- interpret numerical information from factual texts
- place and justify large numbers in an appropriate position on a number line when none, some or all graduations have being identified
- place and justify agiven number on a number line where benchmark numbers have been identified– eg. 100, 10 000, etc
- adjust large numbers where appropriate in everyday situations
- place and justify common and decimal fractions in an appropriate position on a number line
odd, even
square numbers
benchmark
number tracks – linear model – number lines
common and decimal fractions
number position
round to, adjust to
greater/smaller than
FACTORS & MULTIPLES
Identify and describe factors and multiples of whole numbers and use them to solve problems /
- exploring factors and multiples
- investigate multiples and factors, using number sequences, and simple divisibility tests
- identify multiples and factors using basic facts and extensions (2x2=4: 20x20=400)
- identify factors and multiples of some two- and three-digit numbers (e.g. say that the factors of 80 are {1, 2, 4, 5, 8, 10, 16, 20, 40, 80}, say that the first six multiples of 25 are 25, 50, 75, 100, 125 and 150)
NUMBER & ALGEBRA
ContentStrand / CONTENT / ELABORATIONS / LEARNING EXPERIENCES / LANGUAGE / RESOURCES
NUMBER & PLACE VALUE / PLACE VALUE
Describe, compare, order and represent whole numbers beyond 100,000, decimals to thousandths, and common fractions /
- reading, recording, comparing and ordering numbers beyond 100 000 and numbers to 3 decimal places
- representing large numbers using place value materials, symbols, words and calculations
The word ‘and’ is used between the hundreds and the tens when reading a number, but not between other places e.g. seventy-three thousand, six hundred and sixty-three. Words such as ‘place’ and ‘round’ have different meanings in everyday language to those in a mathematics context. / PLACE VALUE
Students begin working with numbers greater than 100 000
- identify the number before and after - 10, 100, 1 000, etc before and after
- read, record, compare and orderwhole numbers larger than 100 000
- make,name and record large numbers and decimals to 3 places
- make the largest and smallest number given any 6/7 digits
- recognise the multiplicative nature of place value e.g. ones to tens is multiply by 10
- recognise the place value of each digit of numbers larger than 100,000 and decimals to 3 places
- represent numbers including using place value materials, symbols, words, place value charts and calculators
- represent numbers in expanded notation
e.g. 9 675.6 = 9 000 + 600 + 70 + 5 + 6/10 - use number expanders to explain the component parts of whole numbers and tenths, hundredthsand thousandths
- adjust numbers to the nearest 10, 100 or 1 000, etc. when estimating or to assist with calculations
- represent and record numbers including decimals to 3 places in context using different combinations of hundred and ten thousands, hundreds, tens, ones, tenths and hundredths
- compare whole numbers and numbers including decimals to 3 places in context, as ‘greater than’, ‘less than’ and ‘equal to’ including using the symbols (>, <, =)
ten, hundred, etc times larger (multiplicative nature of place value)
ten, hundred, etc times smaller (division linked to place value)
expanded notation
adjust
round off to
decimal and common fractions
less than, greater than, not equal to
NUMBER & ALGEBRA
ContentStrand / CONTENT / ELABORATIONS / LEARNING EXPERIENCES / LANGUAGE / RESOURCES
NUMBER & PLACE VALUE
ESTIMATION / Use estimation and rounding to check the reasonableness of answers to calculations /
- recognising the usefulness of estimation to check calculations
- applying mental strategies to estimate the result of calculations, such as estimating thecost of a supermarket trolley load
- use estimation in addition, subtraction, multiplication and division to check for the reasonableness of answers
- identify situations in which estimating amounts assists in quick calculations – eg. adjusting the cost of shopping items to ensure you have enough money
reasonableness
calculate
NUMBER & PLACE VALUE
ADDITION & SUBTRACTION / Identify and solve problems involving addition and subtraction of large numbers and decimals fractions to 3 places ( in context), selecting from a range of computation methods, strategies, known basic facts and appropriate digital technologies /
- using mental, written and technology assisted methods and choosing the most appropriate strategy to solve the problem
- creating and solving addition and subtraction problems involving whole and decimal numbers
- use computation strategies to solve problems involving large whole numbers including decimals to three places in context by:
- counting on or back,
- recalling all addition and subtraction basic facts and their extensions
- breaking one or more numbers into place value or other compatible parts, working with the parts, and putting these parts back together
- using doubling and / or halving
- adjusting one or both numbers and where necessary compensating for the adjustments
- using place value knowledge
- apply the inverse relationship of addition and subtraction to check solutions
- solve one and two-step problems involving whole numbers and decimals, money and measures, in familiar contexts
- align places and including zeros when using tradition written methods to add and subtract whole numbers and decimals to 3 places
- record addition and subtraction equations ( number sentences) using drawings, tables, numbers, words and symbols and combinations of these
BACKGROUND INFORMATION
Word problems requiring subtraction usually fall into two types – either ‘take away’ or ‘comparison’. The comparison type of subtraction involves finding how many more need to be added to a group to make it equivalent to a second group, or finding the difference between two groups. Students need to be able to translate from these different language contexts into a subtraction calculation.
The word ‘difference’ has a specific meaning in a subtraction context. In everyday situations, if asked the difference between two groups, a student may say that a particular group is bigger than another, but they may be referring to the attribute of size rather than the numeric value of the group e.g. one group may be comprised of larger buttons.
NUMBER & ALGEBRA
ContentStrand / CONTENT / ELABORATIONS / LEARNING EXPERIENCES / LANGUAGE / RESOURCES
NUMBER & PLACE VALUE
MULTIPLICATION AND DIVISION / MULTIPLICATION & DIVISION
Solve problems involving multiplication of large numbers by one- or two digit numbers using efficient mental, written strategies and appropriate digital technologies
Solve problems involving division by a one digit number, including those that result in a remainder
Use efficient mental and written strategies and apply appropriate digital technologies to
solve problems /
- interpreting and representing the remainder in division calculations sensibly for the context
- exploring techniques for multiplication such as the area and partition models
- using calculators to check the reasonableness of answers
- applying the commutative, associative and distributive laws
- recall multiplication and division facts to 10 x 10
- relatemultiplication and division facts to at least 10x10 e.g. 6x4 = 24, 244 = 6, also 4x6 = 24, 246 = 4
- determine whether a problem needs multiplication or division to be solved
- pose and solve multiplication and division problems involving numbers beyond 100,000, including decimals to 3 places
- select and use mental, written or calculator strategies to solve multiplication and division problems
- record remainders to division problems
- apply the inverse relationship of multiplication and division to check answers
- record remainders to division problems in symbols
- explain the relationship between subtraction and division as repeated subtraction & the relationship between addition and multiplication
- apply the commutative, associative and distributive properties to assist calculations and choose whether to use mental, written or technology assisted methods or a combination of these (e.g. 47 + 95 + 13 = 47 + 13 + 95 = 60 + 95 = 155, calculate 4 lengths of 3.25 metres as 3.25 m × 4 by writing 3 m × 4 makes 12 m and 0.25 m × 4 makes 1m more, total 13 m) and discuss the method used
factor
square number,
approximately, divisor,
division, multiplication, estimate,
number facts, product,
remainder, remaining,
inverse.
array
communicative
associative
distributive
lots of, groups of, etc.
times
multiplied by
divided into
shared between
BACKGROUND INFORMATION
When beginning to build and read multiplication tables aloud, it is best to use a language pattern of words that relates back to materials such as arrays. As students become more confident with recalling multiplication number facts, they may use less language. For example, ‘seven rows (or groups) of three’ becomes ‘seven threes’ with the ‘rows’ or ‘groups of’ implied.
Division problems may be of different types. For example sharing problems such as share $45 between three friends and grouping or repeated subtraction type problems, such as how many dance tickets costing $3 each may be bought for $45.
NUMBER & ALGEBRA
ContentStrand / CONTENT / ELABORATIONS / LEARNING EXPERIENCES / LANGUAGE / RESOURCES
FRACTIONS & DECIMALS
COMMAN FRACTIONS / Compare and order common unit fractions and locate and represent them on a number line
Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator /
- reading, representing & recording common fractions , equivalent, proper, improper fractions and mixed numbers
- recognising the connection between the value of a unit fraction and its denominator
- modelling and solving addition and subtraction problems involving fractions by using jumps on a number line, or making diagrams of fractions as parts of shapes
- investigate common fractions using different models i.e. set model, linear model, area model
- count forwards and backwards using common fractions and counting beyond 1
- pose and solve problems, one and two step, problems involving fractions in everyday contexts e.g. ‘If one-quarter of the class is going to the computer room, how many students will be left in the classroom?
- Rename 2/2, 4/4, 8/8 = 1
- investigate why 1/5 < 1/3
e.g. if the cake is divided between five people, the slices are smaller than if the cake is shared between three people - order unit fractions by referring to the denominator or diagrams e.g. 1/5 < 1/4 < 1/3
- explain and demonstrate that the denominators of fractions represent the number of equal pieces into which the whole was divided
- explain and demonstrate that the larger the denominator of a fraction, the smaller the fraction
- compare and order common fractions (1/2, 1/3, 1/4, 1/5, 1/8 and 1/10) and justify decisions by using a number line
- identify related common fractions – eg. ½, ¼, 1/8, or 1/5 and 1/10
- compare, order and rename mixed numbers & improper fractions,and justify decisions by using a number line and making the mixed numbers using materials such as card to show wholes and fractions of shapes
- determine equivalent fractions using diagrams, paper folding, number lines and proportional thinking
e.g. ½ = 2/4 = 4/8
- enter common fractions (1/2, 1/3, 1/4, and 1/5) into a calculator based on interpreting the vinculum as “divide”
- model and solve addition and subtraction problems involving fractions by using jumps on a number line, or making diagrams of fractions as parts of shapes
numerator, denominator,
equivalent,
decimal places, decimal point.
mixed numbers
proper and improper fractions
BACKGROUND INFORMATION
When students create their own area models encourage the use of rectangular shapes rather than circular, which are difficult to represent accurately.
When using discrete materials to model fractions, students may not appreciate ‘the part of a whole’
e.g. when circling three counters to show a quarter of twelve.
In developing the concept of a fraction, students come to understand:
- fraction involves ‘equal parts’ illustrated by area models and parts of a collection
- fraction as an operator (of 20 = 10)
- fraction as a number on a number line
- fractions and decimals are representations of the same number.
NUMBER & ALGEBRA
ContentStrand / CONTENT / ELABORATIONS / LEARNING EXPERIENCES / LANGUAGE / RESOURCES
FRACTIONS & DECIMALS
DECIMALS / Recognise that the number system can be extended beyond hundredths
Compare, order and represent decimals /
- using knowledge of place value and division by 10 to extend the number system to thousandths and beyond
- recognising the equivalence of thousandths and 0.001
- recognising that the number of digits after the decimal place is not equivalent to the value of the fraction
- locating decimals on a number line
- compare and classify decimal numbers as ‘smaller than 1’, ‘and ‘greater than 1’
- explain that when a whole is divided into 10 equal parts, these parts are called ’tenths’, 100 equal parts, these parts are called ‘hundredths’, etc.
- read and record decimals in context to 3 places e.g. 2.5 is the same as ‘two wholes and five-tenths’
- recognise the decimal point as marking the end of the whole number representation and the beginning of the decimal representation
- recognise the multiplicative and ‘division’ nature of place value e.g. ones to tenths is divide by 10
- recognise place value of each digit of a number including thousands, hundreds, tens, ones, tenths, thousandths, hundred thousands
- describe the place-value of large numbers including decimals to 3 places in context using a number expander
- compare and order decimal numbers using place value
- explain the connection between one-tenth as a common fraction and one-tenth as a decimal fraction, etc.
- make the connection between tenths and hundredths e.g. ten hundredths is the same as one tenth, etc
- count forwards (to ten) and backwards (to zero) using tenths and hundredths and thousandths starting at 0 to start with and then from any number
- express and interpret tenths as decimals e.g. 1/10 = 0.1 and vice versa and hundredths and thousandths
- record and interpret numbers expressed as decimals in terms of wholes and parts e.g. 4.5m is half way between 4m and 5m, it is 4 whole metres and half a metre
- check whether a calculation is reasonable by using an alternative method e.g. use a number line or calculator to show that ½ is the same as 0.5 and 5/10
- explain the relationship between fractions and decimals e.g. ½, 2/4, 5/10 is the same as 0.5
- interpret and represent the number displayed on a calculator by rounding to one or two decimal places
hundredths
part
whole
ten tenths,
hundred hundredths
expanded notation
decimal
fraction
number line
digit
value
equivalent, equal
round off to
adjust
calculate
NUMBER & ALGEBRA