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Sharon Tooney
MATHS PROGRAM : STAGE TWO
YEAR FOUR
WEEKLY ROUTINE
Monday Tuesday Wednesday Thursday Friday
Whole Number 2 Terms 1-4 Number & Algebra Terms 1-4:
Addition and Subtraction 2 Terms 1-4 : Multiplication &
Division 2 Terms 1 & 3: Patterns and Algebra 2 Terms 2 & 4:
Fractions and Decimals 2
Statistics & Probability Terms 1 & 3: Data 2 Terms 2
& 4: Chance 2
Measurement & Geometry Term 1: Length 2 / Time 2/ 2D 2 /
Position 2 Term 2: Mass 2 / 3D 2 / Angles 2 Term 3: Volume and
Capacity 2 / Time 2 / 2D 2 / Position 2 Term 4: Area 2 / 3D2 /
Angles 2
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Sharon Tooney
K-6 MATHEMATICS SCOPE AND SEQUENCE
NUMBER AND ALGEBRA MEASUREMENT AND GEOMETRY STATISTICS &
PROBABILITY
TERM
Whole Number
Addition & Subtraction
Multiplication & Division
Fractions & Decimals
Patterns & Algebra
Length Area Volume & Capacity
Mass Time 3D 2D Angles Position Data Chance
K 1 2 3 4
Yr 1 1 2 3 4
Yr 2 1 2 3 4
Yr 3 1 2 3 4
Yr 4 1 2 3 4
Yr 5 1 2 3 4
Yr 6 1 2 3 4
NB: Where a content strand has a level 1 & 2, the 1 refers
to the lower grade within the stage, eg. Whole Number 1 in S1 is
for Yr 1, Whole Number 2 is for Yr 2.
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Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: NUMBER AND ALGEBRA
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: Whole Number 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A
student: uses appropriate terminology to describe, and symbols to
represent, mathematical ideas MA2-1WM checks the accuracy of a
statement and explains the reasoning used MA2-3WM applies place
value to order, read and represent numbers of up to five digits
MA2-4NA
Background Information The convention for writing numbers of
more than four digits requires that numerals have a space (and not
a comma) to the left of each group of three digits when counting
from the units column, eg 16 234. No space is used in a four-digit
number, eg 6234. Language Students should be able to communicate
using the following language: largest number, smallest number,
ascending order, descending order, digit, ones, tens, hundreds,
thousands, tens of thousands, place value, expanded notation, round
to. Refer also to language in Whole Numbers 1.
Recognise, represent and order numbers to at least tens of
thousands apply an understanding of place value to read and write
numbers of up to five digits arrange numbers of up to five digits
in ascending and descending order state the place value of digits
in numbers of up to five digits - pose and answer questions that
extend understanding of numbers, eg 'What happens if I rearrange
the digits in the number 12 345?', 'How can I rearrange the digits
to make the largest number?' use place value to partition numbers
of up to five digits and recognise this as 'expanded notation', eg
67 012 is 60 000 + 7000 + 10 + 2 partition numbers of up to five
digits in non-standard forms, eg 67 000 as 50 000 + 17 000 round
numbers to the nearest ten, hundred, thousand or ten thousand
Learning Across The Curriculum Cross-curriculum priorities
Aboriginal &Torres Strait Islander histories & cultures
Asia & Australias engagement with Asia Sustainability General
capabilities Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
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Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Recognise, represent and order numbers to at least tens of
thousands
1
Doubling and Halving Write on the board a selection of whole
numbers between 20 and 50:
21 24 28 32 35 38 43 46 Ask students if they can double any of
the numbers straight away (e.g. 21, 32). Cross out these numbers
and record on the board, for example, 21 2 = 42, 32 2 = 64. Ask
students to use their books and to work in pairs to double the
remaining numbers. Go through the numbers one by one, inviting
students to the board to explain their method to the class. Look
for these methods: using known facts, for example 19 2 is 2 less
than double 20; splitting the number into tens and ones or units,
for example 28 2 is double 20 + double 8; splitting the number in
other ways, for example 38 2 is double 35 plus double 3. Use a
diagram to show students how they can always double a two -digit
number by doubling the tens and doubling the ones or units.
Ask students to use this method to double 28, then 36, doing as
much as possible mentally.
Support: provide concrete materials where appropriate Extension:
increase the complexity of the questions
Whiteboard, markers, paper and pencils
2
Doubling and Halving With Money Review previous lesson, show the
class how the method can be extended to doubling a sum of money
such as: $27.38 by splitting the dollars and the cents. Give one or
two examples to practise, such as $13.09 and $36.75. Repeat the
above for halving numbers, starting with some simple practice of
halving
numbers to 20, including odd numbers (e.g. half of 15 is 7
).
What do you think the answer to half of 120 will be? Why?
Establish that half of 120 is the same as half of 12 multiplied by
10, so the answer is 60. Write on the board: half of 120 = half of
(12 10) = (half of 12) 10 Now ask for: half of 80, half of 140,
half of 320. Get students to explain their answers. Practise
halving a few more multiples of 10 to 200, and multiples of 100 to
2000. Give the class some two-digit numbers under 100 to halve,
inviting them to explain their strategies. Show them how they can
always halve two -digit numbers by partitioning into tens and ones
or units, and how to halve sums of money by partitioning into
dollars and
Support: provide concrete materials where appropriate Extension:
increase the complexity of the questions
Whiteboard, markers, paper and pencils
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Sharon Tooney
cents, using diagrams similar to those for doubling. Give one or
two examples of amounts of money to halve, such as $8.26 and
$14.50.
3
Counting In Fours or Eights Use a counting stick.
Tell students that one end is zero. Count along the stick and
back again in fours. Point randomly at divisions on the stick,
saying: - What is this number? How do you know? Encourage students
to use multiplied by and divided by in their answers. Point out
that they can use the mid-point of the stick as a reference point.
For example: I know that halfway is 4 multiplied by 5, or 20, and
the next point is 4 more, or 24. Say that this is a good way to
remember awkward facts. To remember 10 times a number is always
easy. To find 5 times a number is also easy, as it is half of 10
times the number. For example, 10 times 4 is 40, so 5 times 4 is
half of 40, or 20. Repeat, this time counting in eights.
Support: multiplication tables for use as a direct reference
Extension: increase the complexity of the questions
Counting stick, paper and pencils
4
Recognising Multiples of 4 or 8 (for example) Using a 100s
chart. Highlight multiples of 4, for example. Ask students to
discuss the patterns that they can see, and then to describe them.
Cover part of the 100s chart with a square of paper and ask
students to identify which multiples of 4 are hidden. For each
multiple, ask one of these questions: - How many fours are in ? -
What is divided by 4? - Tell me two division facts that you know
for ? Move the paper square around to different positions on the
grid. Repeat with other multiples, for example multiples of 8,
etc.
Support: multiplication tables for use as a direct reference
Extension: increase the complexity of the questions
100s Chart, paper squares, paper and pencils
5
Using Addition and Subtraction to Solve Grid Puzzles Draw an
incomplete 3 by 3 grid on the board:
164 30 20 418
Ask students to complete the grid using addition down and
across. Repeat with other examples. When students are confident,
use this grid:
70 40
297 562 Point to the empty space at the top left and ask: - When
I add 40 to this number, I get the answer 297. What is the number?
How did you work it out?
Support: provide concrete materials where appropriate Extension:
increase the complexity of the questions
3x3 grids, paper and pencils
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Sharon Tooney
Repeat with the other empty spaces. Ask students to complete
more examples of the second type of grid.
6
Adding and Subtracting Mentally Pairs of Two -Digit Numbers Part
A Remind the class that an easy way to add or subtract 9 to or from
a number is to add or subtract 10 then adjust the answer by 1.
Reinforce that when adding, the answer is adjusted by subtracting
1, since an extra 1 has been added. Similarly, when subtracting,
the answer is adjusted by adding 1, since 1 more than needed has
been taken away. Support each explanation using an empty number
line:
Ask the class to count on in nines from 75. Stop them after
about ten steps, then ask them to count back in nines to 75.
Discuss strategies. - What is an easy way to add or subtract 19 to
or from a number? Agree it is adding or subtracting 20 then
adjusting by 1. Extend to adding or subtracting 29, 39, 49, by
adding or subtracting the nearest multiple of 10 and adjusting.
Include crossing the 100 boundary. Ask students to record their
answers. Encourage students to dispense with the support of the
empty number line. Get them to count on or back for the multiple of
10, and then do the adjustments. Repeat with adding or subtracting
11, 21, 31, - What is an easy way to add or subtract 18, 28,
58?
Support: provide concrete materials where appropriate Extension:
increase the complexity of the questions
Number lines, paper and pencils
7
Adding and Subtracting Mentally Pairs of Two -Digit Numbers Part
B Establish using the nearest multiple of 10 and adjusting by 2.
Provide a few practice examples as per previous lesson. For
example, use an interactive whiteboard Number spinner with a
spinner labelled 8, 9, 18, 19, 28, 29. Start with a score of 250.
Spin the spinner. Ask students to subtract the number rolled from
the score and to record their answers. The game ends when the score
becomes a one-digit number. Relate the strategies to the context of
money. Set a problem such as : - I bought a bag of apples for 75c
and a melon for 69c. How much did they cost altogether? - How can
we work this out mentally? Take feedback and jot on the board: 75c
+ 70c = 145c and 145c 1c = 144c. Establish that 144c is better
expressed as $1.44. Repeat with a problem such as: - Melons now
cost 85c. How much more do they cost? Give out copies of BLM: Shop
Prices to pairs of students. Explain that list A and list B show
the prices of items in two different shops. Students should select
one price from each list. Working mentally, Student A should find
the total of the two items, while Student B finds their difference.
They then check each others answers and discuss errors. On the next
turn, the students swap roles. Repeat several times.
Support: provide concrete materials where appropriate Extension:
increase the complexity of the questions
IWB, whiteboard, markers, paper and pencils, Shop Prices BLM
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Sharon Tooney
8
Review Ask students to explain how any errors in the sums and
differences activity from the previous lesson were made. Write on
the board: 53 + 24. Demonstrate how to do this calculation by
adding the tens first. Ask students to partition the numbers. 53 +
24 = (50 + 3) + (20 + 4) = (50 + 20) + (3 + 4) = 70 + 7 = 77 Work
through other examples with the class. Demonstrate an example which
crosses the tens boundary: 38 + 43 = (30 + 8) + (40 + 3) = (30 +
40) + (8 + 3) = 70 + 11 = 81
Support: provide concrete materials where appropriate Extension:
increase the complexity of the questions
Whiteboard, markers, paper and pencils
9
Revision
10
Assessment
ASSESSMENT OVERVIEW
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Sharon Tooney
SHOP PRICES
$2.75 86C
95C
80C
$2.50
$3.62
87C
98C
$1.43 84C
LIST A
LIST B
28C
61C
51C
49C 9C
19C
31C
78C
69C
37C
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Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: NUMBER AND ALGEBRA
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: Addition and Subtraction 2 KEY CONSIDERATIONS
OVERVIEW OUTCOMES A student: uses appropriate terminology to
describe, and symbols to represent, mathematical ideas MA2-1WM
selects and uses appropriate mental or written strategies, or
technology, to solve problems MA2-2WM checks the accuracy of a
statement and explains the reasoning used MA2-3WM uses mental and
written strategies for addition and subtraction involving two-,
three-, four and five-digit numbers MA2-5NA
Background Information Students should be encouraged to estimate
answers before attempting to solve problems in concrete or symbolic
form. There is still a need to emphasise mental computation, even
though students can now use a formal written method. When
developing a formal written algorithm, it will be necessary to
sequence the examples to cover the range of possibilities, which
include questions without trading, questions with trading in one or
more places, and questions with one or more zeros in the first
number. This example shows a suitable layout for the decomposition
method:
Language Students should be able to communicate using the
following language: plus, add, addition, minus, the difference
between, subtract, subtraction, equals, is equal to, empty number
line, strategy, digit, estimate, round to, change (noun, in
transactions of money). Word problems requiring subtraction usually
fall into two types either 'take away' or 'comparison'. Take away
How many remain after some are removed? eg 'I have 30 apples in a
box and give away 12. How many apples do I have left in the box?'
Comparison How many more need to be added to a group? What is the
difference between two groups? eg 'I have 18 apples. How many more
apples do I need to have 30 apples in total?', 'Mary has 30 apples
and I have 12 apples. How many more apples than me does Mary have?'
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. Difficulties could
arise for some students with phrasing in relation to subtraction
problems, eg '10 take away 9' should give a response different from
that for '10 was taken away from 9'.
Apply place value to partition, rearrange and regroup numbers to
at least tens of thousands to assist calculations and solve
problems select, use and record a variety of mental strategies to
solve addition and subtraction problems, including word problems,
with numbers of up to and including five digits, eg 159 + 23: 'I
added 20 to 159 to get 179, then I added 3 more to get 182', or use
an empty number line:
- pose simple addition and subtraction problems and apply
appropriate strategies to solve them use a formal written algorithm
to record addition and subtraction calculations involving two-,
three-, four- and five-digit numbers, eg
solve problems involving purchases and the calculation of change
to the nearest five cents, with and without the use of digital
technologies solve addition and subtraction problems involving
money, with and without the use of digital technologies -use a
variety of strategies to solve unfamiliar problems involving money
-reflect on their chosen method of solution for a money problem,
considering whether it can be improved calculate change and round
to the nearest five cents use estimation to check the
reasonableness of solutions to addition and subtraction problems,
including those involving money
Learning Across The Curriculum
Cross-curriculum priorities Aboriginal &Torres Strait
Islander histories & cultures Asia & Australias engagement
with Asia Sustainability General capabilities
Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
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Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Apply place value to partition, rearrange and regroup numbers to
at least tens of thousands to assist calculations and solve
problems
1
Doubles of Multiples Ask students: - What is double 3? What is
double 30? Continue with other pairs e.g. 8, 80; 6, 60 Repeat
asking students to find doubles of hundreds e.g. 300, 100, 200,
400. Record doubles for reference:
5 50
500
10 100
1000 etc
- How can we use these facts to double numbers like 320? Work
through double 300 and double 20 600 + 40 = 640 Repeat asking
students to double other three-digit numbers up to 500. Record for
reference:
120 230 180 90
240 460 360 180
Display the table below. Point to a number and ask students to
halve the number. Discuss the methods the students used.
Repeat.
60 190 490 180 240
460 230 90 300 470
120 70 480 30 360
380 250 500 270 150
What happens when we halve an odd multiple of ten? What is the
inverse operation to halving?
Support: provide concrete materials where appropriate Extension:
increase the complexity of the questions
Whiteboard, markers, paper and pencils
2
Addition Families Write on the board: 1 + 2 + 3 + 4 + 5 + 5 + 6
+ 7 + 8 + 9 Ask students to add these up. Agree on finding pairs
which sum to 10 and count up in 10s to get the answer. Write on
board 3 + 4 + 7. Remind students of the method of finding pairs
that sum to 10. Discuss responses and highlight the pair that sums
to 100. Give students similar lists of three multiples of 10 to
add. Discuss responses. Organise students into groups of 3 or 4 and
give each group the cards from Addition Families. (see attached)
The groups play a matching pairs activity. They place the cards
face down. In turn they turn two cards over and keep them if they
are equal e.g. 3 + 2 +7 12 When all pairs have been claimed
students ask each other for pairs which will complete their family.
e.g. If a student has 3 + 2 + 7 and 12 he/she could ask another
player have you got 30 + 20 + 70 and 120? if the player has the
cards, he/she must surrender them. At the
Support: concrete materials to support addition
Addition Families Cards, whiteboard, markers, paper and
pencils
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Sharon Tooney
end of game the winner is the student who has collected the most
families.
3 Addition Spiders Draw on the board the first empty-box
statement of a spider diagram: 140 + + 230 - What pairs of numbers
could complete this number sentence? - Which pair was the easiest
to find? Why? Make connections to previous lessons. Extend the
spider diagram by adding more empty-box statements:
Discuss efficient methods for completing the diagram. Display
the following table:
How many sets of four squares can you find that add up to 200.
Have students create addition spiders to demonstrate answers.
Support: provide Addition Spider BLM for students and concrete
materials to support addition Extension: increase the complexity of
the questions
Whiteboard, markers, paper and pencils, table of figures
4
Add or Subtract the Nearest Multiple Of 10, Then Adjust Part A
Introduce quick fire questions involving multiples of 10 e.g. 80
30, 20 + 40, 50 30. Extend to adding three multiples of 10 e.g. 20
+ 50 + 10 = or: 40 + 30 20 = Now consider 60 + 20. Identify answer.
What if the calculation were 60 + 19? Discuss. Repeat interactively
with a series of examples adding 9, 19, 29, 39 etc. Refine
explanations by modelling on a number line e.g. 60 + 19 =
Now consider 57 + 20. Identify answer. What if the calculation
were 57 + 19? Discuss. Repeat interactively with a series of
examples starting with any two-digit number, adding 9, 19, 29, 39
etc. Refine explanations by modelling on a number line. e.g. 47 +
39
Write: 24 + 9 = 86 + 9 =
Support: provide blank number lines for students to work from
Extension: increase the complexity of the questions. Encourage
working mentally.
Number lines, whiteboard, markers, paper and pencils
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Sharon Tooney
24 + 19 = 86 + 19 = 24 + 29 = 86 + 29 = Ask students for the
answers and discuss their methods and the pattern. Ask them to
extend the pattern. Discuss crossing the 100 boundary. Repeat for
subtraction. Write: 34 9 = 86 9 = 34 19 = 86 19 = 34 29 = 86 29
=
5
Add or Subtract the Nearest Multiple Of 10, Then Adjust Part B
Write on the board 56 30; ask students for the answer. Repeat for
56 29. Refer to previous lesson to establish prior knowledge. Give
children further examples to complete e.g. 63 19, 78 39 etc. Invite
children to explain their strategies. Refine explanations by
modelling on an empty number line. What is 56 28? Draw on the
board: 56 Invite a child to model on a number line e.g.
Establish the answer will be 28. Refine model to show the tens
jumps can be replaced by one jump to the nearest 10, and then
adjust with an addition. Play race to zero in pairs. Each child
starts by writing 250. Take it in turns to roll the 9, 9, 19, 19,
29, 29 dice. Subtract the dice roll from their number each time.
First to get down to a units number is the winner.
Support: provide blank number lines for students to work
from
Number lines, whiteboard, markers, paper and pencils
10
Revision and Assessment
ASSESSMENT OVERVIEW
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Sharon Tooney
ADDITION FAMILIES
9+6+1 90+60+10
9+8+4
16 160 21
6+5+4 60+50+40 90+80+40
15 150 210
4+5+3 40+50+30 9+7+2
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Sharon Tooney
12 120 18
4+9+7 40+90+70 90+70+20
20 200 180
5+8+6 50+80+60 3+6+8
19 190 17
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Sharon Tooney
30+60+80 5+8+4 50+80+40
170 17 190
5+7+4 50+70+20
14 140
2+4+5
11
20+40+50 110 ADDITION FAMILIES
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Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: NUMBER AND ALGEBRA
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: Multiplication and Subtraction 2 KEY CONSIDERATIONS
OVERVIEW OUTCOMES A student: uses appropriate terminology to
describe, and symbols to represent, mathematical ideas MA2-1WM
selects and uses appropriate mental or written strategies, or
technology, to solve problems MA2-2WM checks the accuracy of a
statement and explains the reasoning used MA2-3WM uses mental and
informal written strategies for multiplication and division
MA2-6NA
Background Information An inverse operation is an operation that
reverses the effect of the original operation. Addition and
subtraction are inverse operations; multiplication and division are
inverse operations. Linking multiplication and division is an
important understanding for students in Stage 2. They should come
to realise that division 'undoes' multiplication and multiplication
'undoes' division. Students should be encouraged to check the
answer to a division question by multiplying their answer by the
divisor. To divide, students may recall division facts or transform
the division into a multiplication and use multiplication facts, eg
is the same as . The use of digital technologies includes the use
of calculators. Language Students should be able to communicate
using the following language: multiply, multiplied by, product,
multiplication, multiplication facts, tens, ones, double, multiple,
factor, shared between, divide, divided by, division, halve,
remainder, equals, is the same as, strategy, digit. As students
become more confident with recalling multiplication facts, they may
use less language. For example, 'five rows (or groups) of three'
becomes 'five threes' with the 'rows of' or 'groups of' implied.
This then leads to 'one three is three', 'two threes are six',
'three threes are nine', and so on. The term 'product' has a
meaning in mathematics that is different from its everyday usage.
In mathematics, 'product' refers to the result of multiplying two
or more numbers together. Students need to understand the different
uses for the = sign, eg 4 3 = 12, where the = sign indicates that
the right side of the number sentence contains 'the answer' and
Recall multiplication facts up to 10 10 and related division
facts count by fours, sixes, sevens, eights and nines using skip
counting use the term 'product' to describe the result of
multiplying two or more numbers, eg 'The product of 5 and 6 is 30'
use mental strategies to build multiplication facts to at least 10
10, including: using the commutative property of multiplication, eg
7 9 = 9 7 using known facts to work out unknown facts, eg 5 7 is
35, so 6 7 is 7 more, which is 42 using doubling and repeated
doubling as a strategy to multiply by 2, 4 and 8, eg 7 8 is double
7, double again and then double again using the relationship
between multiplication facts, eg the multiplication facts for 6 are
double the multiplication facts for 3 factorising one number, eg 5
8 is the same as 5 2 4, which becomes 10 4 recall multiplication
facts up to 10 10, including zero facts, with automaticity find
'multiples' for a given whole number, eg the multiples of 4 are 4,
8, 12, 16, relate multiplication facts to their inverse division
facts, eg 6 4 = 24, so 24 6 = 4 and 24 4 = 6 determine 'factors'
for a given whole number, eg the factors of 12 are 1, 2, 3, 4, 6,
12 use the equals sign to record equivalent number relationships
involving multiplication, and to mean 'is the same as', rather than
to mean to perform an operation, eg 4 3 = 6 2 - connect number
relationships involving multiplication to factors of a number, eg
'Since 4 3 = 6 2, then 4, 3, 2 and 6 are factors of 12' - check
number sentences to determine if they are true or false and explain
why, eg 'Is 7 5 = 8 4 true? Why or why not?' Develop efficient
mental and written strategies, and use appropriate digital
technologies, for multiplication and for division where there is no
remainder multiply three or more single-digit numbers, eg 5 3 6
model and apply the associative property of multiplication to aid
mental computation, eg 2 3 5 = 2 5 3 = 10 3 = 30 - make
generalisations about numbers and number relationships, eg 'It
doesn't matter what order you multiply two numbers in because the
answer is always the same' use mental and informal written
strategies to multiply a two-digit
Learning Across The Curriculum
Cross-curriculum priorities Aboriginal &Torres Strait
Islander histories & cultures Asia & Australias engagement
with Asia Sustainability General capabilities
Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
-
Sharon Tooney
should be read to mean 'equals', compared to a statement of
equality such as 4 3 = 6 2, where the = sign should be read to mean
'is the same as'.
number by a one-digit number, including: using known facts, eg
10 9 = 90, so 13 9 = 90 + 9 + 9 + 9 = 90 + 27 = 117 multiplying the
tens and then the units, eg 7 19: 7 tens + 7 nines is 70 + 63,
which is 133 using an area model, eg 27 8
using doubling and repeated doubling to multiply by 2, 4 and 8,
eg 23 4 is double 23 and then double again using the relationship
between multiplication facts, eg 41 6 is 41 3, which is 123, and
then double to obtain 246 factorising the larger number, eg 18 5 =
9 2 5 = 9 10 = 90 - create a table or simple spreadsheet to record
multiplication facts, eg a 10 10 grid showing multiplication facts
use mental strategies to divide a two-digit number by a one-digit
number where there is no remainder, including: using the inverse
relationship of multiplication and division, eg 63 9 = 7 because 7
9 = 63 recalling known division facts using halving and repeated
halving to divide by 2, 4 and 8, eg 36 4: halve 36 and then halve
again using the relationship between division facts, eg to divide
by 5, first divide by 10 and then multiply by 2 - apply the inverse
relationship of multiplication and division to justify answers, eg
56 8 = 7 because 7 8 = 56 record mental strategies used for
multiplication and division select and use a variety of mental and
informal written strategies to solve multiplication and division
problems - check the answer to a word problem using digital
technologies Use mental strategies and informal recording methods
for division with remainders model division, including where the
answer involves a remainder, using concrete materials - explain why
a remainder is obtained in answers to some division problems use
mental strategies to divide a two-digit number by a one-digit
number in problems for which answers include a remainder, eg 27 6:
if 4 6 = 24 and 5 6 = 30, the answer is 4 remainder 3 record
remainders to division problems in words, eg 17 4 = 4 remainder 1
interpret the remainder in the context of a word problem, eg 'If a
car can safely hold 5 people, how many cars are needed to carry 41
people?'; the answer of 8 remainder 1 means that 9 cars will be
needed
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Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Recall multiplication facts up to 10 10 and related division
facts Develop efficient mental and written strategies, and use
appropriate digital technologies, for multiplication and for
division where there is no remainder Use mental strategies and
informal recording methods for division with remainders
5
Remainders Students explore division problems involving
remainders, using counters eg We have to put the class into four
even teams but we have 29 students. What can we do? Students make
an array to model the solution and record their answer to show the
connection with multiplication eg 29 = 4 7 + 1. Students could
interpret the remainder in the context of a word problem eg Each
team would have 7 students and one student could umpire. Students
could record the answer showing the remainder eg 29 4 = 7 remainder
1. The teacher could model recording the students solutions, using
both forms of recording division number sentences. The teacher sets
further problems that involve remainders eg A school wins 125
computers. If there are seven classes, how many computers would
each class receive? Since only whole objects are involved, students
discuss possible alternatives for sharing remainders. Students
write their own division problems, with answers involving
remainders.
Support/Extension: adjust complexity of questions
accordingly
Counters, paper and pencils
6
Ancient Egyptian Long Multiplication The teacher explains to the
students that the Ancient Egyptians had a different number system
and did calculations in a different way. They used doubling to
solve long multiplication problems eg for 11 23 they would double,
and double again, 1 23 = 23 2 23 = 46 4 23 = 92 8 23 = 184 1+ 2 + 8
= 11, so they added the answers to 1 23, 2 23 and 8 23 to find 11
23. 23 46 184 + 253. Students are encouraged to make up their own
two-digit multiplication problems and use the Egyptian method to
solve them.
Support: concrete materials, multiplication tables to support
answering questions
Paper and pencils
7
Factors Game The teacher prepares two dice, one with faces
numbered 1 to 6 and the other with faces numbered 5 to 10. Each
student is given a blank 6 6 grid on which to record factors from 1
to 60. Students work in groups and take turns to roll the two dice
and multiply the numbers obtained. For example, if a student rolls
5 and 8, they multiply the numbers together to obtain 40 and each
student in the group places counters on all of the factors of 40 on
their individual grid ie 1 and 40, 2 and 20, 4 and 10, 5 and 8. The
winner is the first student to put three counters in a straight
line, horizontally or vertically.
Support: concrete materials to aide answering questions
Dice, 6x6 grids, counters, paper and pencils
8
Tag Students find a space to stand in the classroom. The teacher
asks students in turn to answer questions eg What are the factors
of 16? If the student is incorrect they sit down.
Support: multiplication tables as direct reference
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Sharon Tooney
The teacher continues to ask the same question until a correct
answer is given. When a student gives a correct answer, they take a
step closer to another student and may tip them if within reach.
The tipped student sits down. The question is then changed. Play
continues until one student remains, who then becomes the
questioner. This game is designed for quick responses and repeated
games.
9 New From Old Students are asked to write a multiplication and
a division number fact. Each student uses these facts to build new
number facts eg Starting with12 3 = 4 Starting with 3 2 = 6 24 3 =
8 6 2 = 12 48 3 = 16 12 2 = 24 96 3 = 32 24 2 = 48 Possible
questions include: - what strategy did you use? - what other
strategies could you use? - what strategy did you use? - did you
use the relationship between multiplication and division facts?
Support: concrete materials where needed
paper and pencils
10
Revision and Assessment
ASSESSMENT OVERVIEW
-
Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: NUMBER AND ALGEBRA
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: Fractions and Decimals 2 KEY CONSIDERATIONS OVERVIEW
OUTCOMES A student: uses appropriate terminology to describe, and
symbols to represent, mathematical ideas MA2-1WM checks the
accuracy of a statement and explains the reasoning used MA2-3WM
represents, models and compares commonly used fractions and
decimals MA2-7NA
Background Information In Stage 2 Fractions and Decimals 2,
fractions with denominators of 2, 3, 4, 5, 6, 8, 10 and 100 are
studied. Denominators of 2, 3, 4, 5 and 8 were introduced in Stage
2 Fractions and Decimals 1. Fractions are used in different ways:
to describe equal parts of a whole; to describe equal parts of a
collection of objects;
to denote numbers (eg is midway between 0 and 1 on the
number line); and as operators related to division (eg dividing
a number in half). Money is an application of decimals to two
decimal places. Refer also to background information in Fractions
and
Decimals 1. Language Students should be able to communicate
using the following language: whole, part, equal parts, half,
quarter, eighth, third, sixth, fifth, tenth, hundredth, one-sixth,
one-tenth, one hundredth, fraction, numerator, denominator, whole
number, number line, is equal to, equivalent fractions, decimal,
decimal point, digit, place value, round to, decimal places,
dollars, cents. The decimal 1.12 is read as 'one point one two' and
not 'one point twelve'.
Refer also to language in Fractions and Decimals 1.
Investigate equivalent fractions used in contexts (ACMNA077)
model, compare and represent fractions with denominators of 2, 4
and 8; 3 and 6; and 5, 10 and 100 model, compare and represent the
equivalence of fractions with related denominators by redividing
the whole, using concrete materials, diagrams and number lines
record equivalent fractions using diagrams and numerals Recognise
that the place value system can be extended to tenths and
hundredths, and make connections between fractions and decimal
notation (ACMNA079) recognise and apply decimal notation to express
whole numbers, tenths and hundredths as decimals investigate
equivalences using various methods identify and interpret the
everyday use of fractions and
decimals, such as those in advertisements state the place value
of digits in decimal numbers of up to two decimal places use place
value to partition decimals of up to 2 decimal places partition
decimals of up to two decimal places in non-standard forms apply
knowledge of hundredths to represent amounts of
money in decimal form model, compare and represent decimals of
up to two decimal places apply knowledge of decimals to record
measurements, interpret zero digit(s) at the end of a decimal
recognise that amounts of money are written with two
decimal places use one of the symbols for dollars ($) and cents
(c)
correctly when expressing amounts of money use a calculator to
create patterns involving decimal
numbers place decimals of up to 2 decimal places on a number
line round a number with one or two decimal places to the nearest
whole number
Learning Across The Curriculum
Cross-curriculum priorities Aboriginal &Torres Strait
Islander histories & cultures Asia & Australias engagement
with Asia Sustainability General capabilities
Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
-
Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Investigate equivalent fractions used in contexts Recognise that
the place value system can be extended to tenths and hundredths,
and make connections between fractions and decimal notation
4
A Pikelet Recipe Students explore dividing wholes into equal
parts and use sharing diagrams to divide by fractions. The activity
aims to promote partwhole conceptual understanding and to assist
students perform fraction computations based on using a sound
understanding of the fraction concept. 1. Place 4 identical empty
cylindrical clear plastic tumblers near each other on a table. - I
want to pour half a glass of drink. Who can show me where about on
the glass I would need to fill it to? Provide the student with a
thin piece of masking tape to record his or her answer. A marking
pen can be used to identify the exact level. - Who thinks that this
is the place we should fill the tumbler to get half a glass? Allow
an opportunity for class discussion and if the student wishes, he
or she can move the tape. - How can we know if we are right? 2. Put
out another transparent tumbler with vertical sides. - Can you show
me where I would have to fill this glass to get one-quarter of a
glass? Attach a small piece of thin black tape at the indicated
location. - Does this look correct? (Adjust as directed.) Draw a
sketch of the tumbler on the board. Ask one student to add a line
to your diagram on the board to show one-quarter of a glass. 3. Put
out three empty transparent tumblers with vertical sides and one
tumbler full of water. - By pouring, and using any of these other
glasses, show me exactly a third of a glass of water? What fraction
remains in the glass? Draw a sketch of the three tumblers on the
board. Ask one student to add a line to your diagram on the board
to show one-third of a glass. - Who can show me two-thirds of a
glass by drawing a line on the glass I have drawn on the board?
4. I have 6 cups of milk. A recipe needs of a cup of milk. How
many times can I make the
recipe before I run out of milk? Can you draw your answer?
5. I have 6 cups of milk. A recipe needs one-quarter ( ) of a
cup of milk. How many times
can I make the recipe before I run out of milk? Can you draw
your answer? 6. Draw what would happen if I have 6 cups of milk and
a recipe needs three-quarters
( ) of a cup of milk. How many times can I make the recipe
before I run out of milk?
7. Who can draw what would happen if I have 6 cups of milk and a
recipe needs one-third
( ) of a cup of milk? How many times can I make the recipe
before I run out of milk?
8. I have 6 cups of milk. A recipe needs two-thirds ( ) of a cup
of milk. How many times can I
Support: representations of fractions as a reference
A pouring jug full of water (food colouring or cordial,
optional), 4 cylindrical clear plastic tumblers, thin strips of
masking tape or similar.
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Sharon Tooney
make the recipe before I run out of milk? Can you draw your
answer?
5 Lamington Bars : Forming Equivalent Fractions Students
encounter partitioning a rectangle in two directions. The activity
aims to promote partwhole conceptual understanding leading to
simple fraction multiplication. 1. Lamingtons are pieces of sponge
cake covered in chocolate icing and dipped in shredded coconut. Mrs
Packer makes excellent lamingtons and she likes to put a layer of
whipped cream in the middle of her lamingtons. Mrs Packer starts
with a large rectangular sponge cake. 2. Distribute rectangular
sheets of brown paper. Show by folding the piece of paper how Mrs
Packer could make four lamington bars.
Check to see which way the paper has been divided. If your
students use different methods to form quarters ask them if each
person would still get the same cut of cake. If all students create
quarters by folding in the same direction take your piece of paper
and fold it a different way to the direction the class has chosen.
Compare the different ways of forming quarters shown above. Ask
your students to show how the pieces of cake are equal. 3. I am
going to make eight smaller lamington bars. Fold the rectangle into
eighths as below.
If I wanted to eat this much (show three-quarters of the
horizontally divided rectangle) how many of the smaller lamington
bars would this be equal to? Remember that you have to explain your
answer.
Support: representations of fractions as a reference
Brown paper, paper and pencils
6
Mrs Packers Visitors : Comparing Fractions Students encounter
partitioning a rectangle into different amounts and comparing the
resulting fractions. 1. Mrs Packer was expecting guests. She made
five lamington bars and put them on two tables ready for the
guests. As each guest arrived Mrs Packer asked the guest to choose
a table. Once seated, the guests cannot change tables but must
equally share the lamington bars with all the guests at the table.
2. Mrs Packer has placed one lamington bar on one table and four
lamington bars on the other table.
Place one rectangular sheet of brown paper on one table and four
rectangles of brown paper on another table. 3. Mrs Packer is
expecting eight guests. I want eight of you to play the part of the
guests. The aim is to get as much of the lamington bars as you can
but you cannot change tables after you sit down and everyone must
wait until the last person sits down to share the
Support: representations of fractions as a reference
Tables, brown paper, paper and pencils
-
Sharon Tooney
lamington bars at their table. Send eight students out of the
class and give each one a number to represent the order in which
they should return. As each student comes in and sits down, ask the
class to record how much each person at that table will receive.
Remember that as you sit down you will have to explain why you
chose the table you sat at. 4. Show by folding the piece of paper
how much each person on your table receives. What would be the best
solution? Record your answer. 5. Repeat the activity with two
lamington bars on one able and three on the other.
7
Related Fractions 1 : One-Half, One-Quarter and One-Eighth
Students explore the relationships between the unit fractions ,
and through dividing a
continuous unit. They then express the equivalence between
various units, as well as the relationship between the unit
fraction and the whole. The activity aims to promote an
understanding of the relationship between unit fractions with
related denominators.
1. Write the fractions one-half ( ), one-quarter ( ) and
one-eighth ( ) on the board. Hold
up a paper streamer approximately 90 cm long. Using this paper
streamer, how could you make one of these fractions? Allow the
students some time to think about the question. Which of these
fractions will be the easiest to make? Why? Focus the questions on:
How do you know that you have one-half (or one-quarter or
one-eighth)? 2. Fold the paper streamer in half and then fold one
half in half. Unfold the streamer and display it to the class.
Point to each part in turn and ask: - What fraction of the streamer
is this part? How do you know? 3. If I fold one-quarter in half,
what will I have? Fold the quarter in half and, as before, point to
each part in turn and ask: - What fraction of the streamer is this
part? How do you know? Emphasise reversibility: If I fold the
quarter in half I get two-eighths and two eighths is the same as
one quarter. 4. Which is the biggest part? Which is the smallest
part? Can anyone see two fractions that would be the same as
another fraction? 5. Show me two-eighths. Show me two-quarters.
Show me two-halves. 6. Draw the streamer and show how halves,
quarters and eighths are related to each other.
Support: representations of fractions as a reference
Whiteboard, markers, paper and pencils, paper streamers
10
Revision and Assessment
ASSESSMENT OVERVIEW
-
Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: MEASUREMENT AND GEOMETRY
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: Mass 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A
student: uses appropriate terminology to describe, and symbols to
represent, mathematical ideas MA2-1WM selects and uses appropriate
mental or written strategies, or technology, to solve problems
MA2-2WM measures, records, compares and estimates the masses of
objects using kilograms and grams MA2-12MG
Background Information In Stage 2, students should appreciate
that formal units allow for easier and more accurate communication
of measures. Students are introduced to the kilogram and gram. They
should develop an understanding of the size of these units, and use
them to measure and estimate. Language Students should be able to
communicate using the following language: mass, measure, scales,
kilogram, gram. The term 'scales', as in a set of scales, may be
confusing for some students who associate it with other uses of the
word 'scales', eg fish scales, scales on a map, or musical
scales.
These other meanings should be discussed with students.
Use scaled instruments to measure and compare masses (ACMMG084)
recognise the need for a formal unit smaller than the kilogram
recognise that there are 1000 grams in one kilogram, ie 1000 grams
= 1 kilogram use the gram as a unit to measure mass, using a scaled
instrument associate gram measures with familiar objects, eg a
standard egg has a mass of about 60 grams (Reasoning) record
masses using the abbreviation for grams (g) compare two or more
objects by mass measured in kilograms and grams, using a set of
scales interpret statements, and discuss the use of kilograms
and grams, on commercial packaging (Communicating, Problem
Solving)
interpret commonly used fractions of a kilogram, including
, , , and relate these to the number of grams
solve problems, including those involving commonly used
fractions of a kilogram (Problem Solving)
record masses using kilograms and grams, eg 1 kg 200 g
Learning Across The Curriculum
Cross-curriculum priorities Aboriginal &Torres Strait
Islander histories & cultures Asia & Australias engagement
with Asia Sustainability General capabilities
Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
-
Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Use scaled instruments to measure and compare masses
1
Calibrated Elastic Band Students work in a small group to hang
known masses (in grams) from a large, thick elastic band. Masses
will need to be large enough to stretch the elastic band or use
thinner elastic bands for smaller masses. Students calibrate the
stretch by marking and labelling the levels on a backboard of
cardboard or paper. Students use the scale to measure and record
the mass of other objects from the classroom. It may be necessary
to have a supply of elastic bands available as the elastic band may
not return to its original length if used repeatedly or with
objects of a large mass. Check the mass of the objects by measuring
with a set of scales.
Support: peer tutor grouping strategies
Large elastic bands, nail or hook, cardboard, paper clips,
objects to compare, paper and pencils
2
How Heavy Are My Books? Students work individually or in pairs
to select six books and estimate the mass of the books in kilograms
and grams. Students select appropriate scales to weigh the books.
Students find and record the mass of each individual book and then
calculate the mass of the six books by adding the six results.
Students check their calculation by weighing the six books and
commenting on any variation from their calculation. The final
report should include the reasons for the selection of the
measuring device.
Support: peer tutor grouping strategies
Assorted scales, books, pencils and paper
3
Which Scales? Students work in small groups to trial and record
the smallest and largest masses that can be accurately measured on
various measuring devices. The devices may include bathroom scales,
kitchen scales, balances, etc. Ensure that the students are
conversant in how to read the scale on each measuring device and
that the scales are set at zero. Students may need reminding to
handle the equipment carefully and to check that scales have been
placed on a firm, flat surface.
Support: peer tutor grouping strategies
Different measuring devices, different objects to weigh, paper
and pencils
4
Toy Story Students bring a toy from home to be weighed. In small
groups, students weigh and record the mass of each toy. Students
find the total mass of all toys in the group by weighing, by also
by adding the masses.
Extension: students graph the mass of each toy in the group.
Toys, scales, paper and pencils
5
Fruit Salad Students work in pairs or small groups to select a
measuring device and then measure the mass of individual pieces of
fruit, or vegetables. Students estimate then calculate how many
pieces would be needed to make a kilogram. Students check their
calculations by working with other groups to weigh and count 1
kilogram of the fruit or vegetables.
Support: peer tutor grouping strategies
Fruit, vegetables, kitchen scales, pencils and paper
6
Aussies Abroad Students work in small groups to investigate the
gross and net weights of small plastic jars and large glass
containers of vegemite. If several different examples are used,
each container can be examined by a small group and then rotated to
the next group. Students determine which containers would hold the
greater volume of vegemite and find how many of each container
would fit into a 10kilogram carton
Support: use of calculators Extension: compare the vegemite
containers by finding the best value for money.
Different size jars of vegemite, scales, calculators, paper and
pencils
-
Sharon Tooney
7
Mass Measurement Story Problem Provide students with a variety
of problems involving mass, in which they need to determine the
operation required to solve the problem. Examples include: 1.
Selmas body weight is 22 kilograms, while Kiaras body weight is 3
kilograms heavier than Selma. How heavy is Kiara? 2. Aquilas mum
wants to make a cake. She bought 585 grams of flour, 250 grams of
eggs, and 150 grams of sugar. What is the total weight of the
things that Aquila bought? 3. Andi had 1 kilogram of candy. After
she gave some to Nadia, she still has 290 grams left. How heavy was
the candy that Andi gave to Nadia? 4. The limit of the baggage that
each person can bring in the aeroplane is 20 kilograms. Mitchells
baggage weighs 24000 gram. How much over the limit is this? 5.
Zandas mum bought 17 kg of rice, while Wendy and Cassies mum bought
15 kg and 22 kg. What is the total weight of rice that was bought?
Discuss how the students solved each problem and their results.
8
Making Chocolate Cake Present the following recipe to the class:
Recipe for Chocolate Cake: 4 eggs (1 egg is about 75 gram) 150 gram
of sugar 100 gram of hazelnut, finely ground 5 tablespoons cocoa
powder (1 table spoon is about 10 gram) 300 g dark chocolate 100ml
whipping cream (50 ml is about 50 gram) Possible questions: - What
is the total mass of this chocolate cake? - What is the total mass
of 5 chocolate cakes? - If I ate half of the cake, what would be
the mass of the part I ate?
Support: concrete materials to answer questions
Chocolate cake recipe, paper and pencils
9
Light Challenges Students use the "feel" of 10 grams to make
some guesses about light objects. They are not allowed to use any
measuring scales to help with Their guesses. Students put each of
Their guesses on named pieces of paper in the challenge containers
to be checked at the end of the lesson. Possible challenge stations
include: Challenge 1: How many paper clips in 10 grams? Challenge
2: How many drawing pins 20 grams? Challenge 3: How many cm cubes
in 50 grams? Challenge 4: How many marbles in 40 grams? Challenge
5: How many teaspoons of rice in 30 grams? etc After students have
moved through the challenges and posted their prediction, determine
the answer to each as a class, using scales to measure. Check
student predictions and have students make generalisations about
their prediction
10 gram weights, variety of materials, paper and pencils
-
Sharon Tooney
verses the correct answer for each challenge.
10 Revision and Assessment
ASSESSMENT OVERVIEW
-
Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: MEASUREMENT AND GEOMETRY
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: Angles 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A
student: uses appropriate terminology to describe, and symbols to
represent, mathematical ideas MA2-1WM checks the accuracy of a
statement and explains the reasoning used MA2-3WM identifies,
describes, compares and classifies angles MA2-16MG
Background Information A simple 'angle tester' can be made by
placing a pipe-cleaner inside a straw and bending the straw to form
two arms. Another angle tester can be made by joining two narrow
straight pieces of card with a split-pin to form the rotatable arms
of an angle. Language Students should be able to communicate using
the following language: angle, arm, vertex, right angle, acute
angle, obtuse angle, straight angle, reflex angle, angle of
revolution. The use of the terms 'sharp' and 'blunt' to describe
acute and obtuse angles, respectively, is counter-productive in
identifying the nature of angles. Such terms should not be used
with students as they focus attention on the external points of an
angle, rather than on the amount of turning between the arms of the
angle.
Compare angles and classify them as equal to, greater than or
less than a right angle (ACMMG089) compare angles using informal
means, such as by using an 'angle tester' recognise and describe
angles as 'less than', 'equal to', 'about the same as' or 'greater
than' a right angle classify angles as acute, right, obtuse,
straight, reflex or a revolution describe the size of different
types of angles in relation
to a right angle, eg acute angles are less than a right angle
(Communicating)
relate the turn of the hour hand on a clock through a right
angle or straight angle to the number of hours elapsed, eg a turn
through a right angle represents the passing of three hours
(Reasoning)
identify the arms and vertex of the angle in an opening, a slope
and/or a turn where one arm is visible and the other arm is
invisible, eg the bottom of an open door is the visible arm and the
imaginary line on the floor across the doorway is the other arm
create, draw and classify angles of various sizes, eg by tracing
along the adjacent sides of shapes draw and classify the angle
through which the minute
hand of a clock turns from various starting points
(Communicating, Reasoning)
Learning Across The Curriculum
Cross-curriculum priorities Aboriginal &Torres Strait
Islander histories & cultures Asia & Australias engagement
with Asia Sustainability General capabilities
Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
-
Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Compare angles and classify them as equal to, greater than or
less than a right angle
1
Measuring Angles in the Classroom Students use the windmill
pattern as an angle tester to measure and record at least three
different angles found in the classroom. Students record an acute,
an obtuse and a right angle. Discuss how the angles on the windmill
sheet can be used as informal units to measure other angles. - What
have we learnt about angles? Discuss strategies that students might
use to copy and measure angles in the classroom in terms of
windmill units. - How would I measure an angle? - How have we used
the windmill angle tester to measure other angles? Students
measure, draw and label at least three corners (including an acute,
an obtuse and a right angle) in the classroom. Discuss students
responses and the range of angle sizes found. - Who had the
smallest angle? - Who had the largest angle? - How big are these
angles? - How many windmill angles fit into an acute angle? - How
many windmill angles fit into an obtuse angle?
Support: individual assistance as required
Bent straws, Windmill BLM or transparencies, pencils and
paper
2
Measuring Body Angles Students investigate and record angles
made by parts of their body, using the windmill angle tester to
measure the angles. Discuss different angles that can be made with
the human body. Stand with one arm straight out to the side, then
bend your arm at the elbow. Have students do the same, and discuss
the angles they can make. Revise the use of terminology arms and
vertex, before discussing the arms and vertex in the body angles.
Ask a student to hold her arm out straight and ask what the angle
is at the elbow. Introduce the term straight angle. - Can you show
us how to make an angle with a part of your body? - What angles can
you see when I bend my arm like this? - How would you describe
these angles? - What angles can you make with your elbow? - When I
make an angle with my elbow, where are the arms and the vertex of
the angle? - How big is the angle at the elbow when an arm is held
straight out from the body? - What could we call this angle? Have
your students work in pairs to: make different body angles and
discuss these with their partner complete the body angles sheet.
Discuss students answers to the body angles questions. Focus on the
largest and smallest
Extension: Point out that some angles go beyond a straight
angle, e.g. most people can bend their wrist more than six windmill
angles. Such angles are called reflex angles. Find some more
examples.
Windmill BLM, Body Angles BLM, pencils and paper
-
Sharon Tooney
angles which students can make by bending their wrists. What
were the easiest angles to find or make? Can anybody tell us about
body angles which we havent already discussed? What are the largest
and smallest angles you can make with your wrist? Can you estimate
the size of these in windmill angles?
3
Drawing Two-Line Angles Students draw diagrams that can
represent angles in any situation. They investigate the similarity
between two-line angles in different locations. Revise and discuss
situations in which the size of an angle may change. These may
include body angles, the hands of a clock, or scissors. Discuss how
angles on objects or in different situations can be fixed or
changeable. - We have discussed how the angles on some objects are
fixed or dont change, and angles on other objects can change by
opening or turning. - Tell us about some angles in this room that
are fixed. - Tell us about objects in this room that have
changeable angles Discuss how to draw an angle diagram that could
represent any of these situations and ask students to demonstrate
on the board. - How can you draw an angle so that it can look like
either a fixed angle or one that can be changed? Ask students to
suggest the angles on objects or shapes that could be represented
by the angle diagrams on the chalkboard. Introduce and discuss the
drawing two-line angles sheet. Have your students complete the
drawing two-line angles sheet. Discuss students answers to the
worksheet questions. Review the different types of angles students
have identified. Review the different parts of angles on a variety
of objects. - What is the same about all the angles you have found?
- What can you tell us about the parts of these angles? - What have
you learnt about angles?
Support: have students work in pairs to complete the drawing
two-line angles sheet. extension: Discuss what it means to say that
angle is an abstract concept (Angle is an abstract concept because
it represents the same idea occurring in different situations; it
is abstracted from all those contexts. Similarly, the angle
diagrams above are called abstract diagrams because they do not
represent any particular angle but what is common to all angles of
that size, in different situations.)
Objects with movable arms, Drawing Two-Line Angles BLM, pencils
and paper, access to angle testers and pattern blocks
4
Measuring The Angle Of Opening Of Doors Students are introduced
to the concept of a one-line angle by measuring the angle of
opening of a door. Students measure the angle of opening of a door
using the house activity sheet and a floating door, using pattern
block corners. Open and close the classroom door slowly. Discuss
how the door turns or pivots on the hinges. Discuss the angle of
opening of the door by looking at the top edge and then the bottom
edge. Discuss how to visualise the arm formed by the doorway at the
bottom edge. Demonstrate opening the classroom door to about 45 and
the door on the house worksheet or the model house to about 45 and
use a bent straw to check that the angles are equal. Discuss how
the angle could be measured with pattern block corners. - Describe
what is happening when this door opens and closes? - What allows
the door to swing this way? - How could we describe this in
mathematical terms? - How could I measure the angle of opening?
Support: individual assistance as required, peer tutor grouping
strategies
House BLM, A5 card, pattern blocks, scissors and bent straw;
optional model house for teachers demonstration
-
Sharon Tooney
- How could I make the same angle of opening with the model door
or house worksheet door? How could I measure this angle? Activity A
Have pairs of students prepare their house worksheets and lay the
sheets on their desks. Explain how Student A will select a pattern
block angle and open the house door to match the angle without
their partner seeing. Student B will estimate which pattern block
angle was chosen. The players measure the angle and then reverse
the roles. Activity B Demonstrate to the students how to fold the
A5 card to make a floating door. Hold the floating door upright on
a desk. Discuss how one arm of the angle must be imagined when the
door is opened. Ask your students to make a floating door and
repeat the activity of measuring the opening with a pattern block.
Discuss the different angles that can be made when the door is
opened. Ensure students understand that part of the angle when a
door is opened needs to be imagined or remembered, as it cannot be
seen. - What are the largest and smallest angles you can make when
you open the door? - In an angle of opening, where is the vertex?
Where are the arms of the angle?
10
Revision and Assessment
ASSESSMENT OVERVIEW
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Sharon Tooney
Windmill
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Sharon Tooney
Body Angles
Raise one arm at your side, like this:
What angle sizes can you make?
Draw the smallest angle and the largest angle.
Make your hand flat and then make an angle
At your wrist, like this:
What angle sizes can you make?
Draw the smallest angle and the largest angle.
Complete the drawing of a school student to
make the following angles:
angle right arm raised = 3 windmill angles
angle at right elbow = 2 windmill angles
angle left arm raised = 5 windmill angles
angle at left elbow = 4 windmill angles
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Sharon Tooney
Drawing Two-Line Angles
Part 1. Each of these objects makes an angle. Draw the angles on
each object.
Part 2. Draw the three angles separately here:
Part 3. Find a way to check that the angles you drew in Part 2
are the same size as the angles you found in Part 1. Write how you
measured the angles.
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Sharon Tooney
Part 4. Here is another angle:
Draw and label three different objects that make an angle this
size:
Part 5. What is an angle?
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Sharon Tooney
House
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Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: MEASUREMENT AND GEOMETRY
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: 3D 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A student:
uses appropriate terminology to describe, and symbols to represent,
mathematical ideas MA2-1WM checks the accuracy of a statement and
explains the reasoning used MA2-3WM makes, compares, sketches and
names three-dimensional objects, including prisms, pyramids,
cylinders, cones and spheres, and describes their features
MA2-14MG
Background Information When using examples of Aboriginal rock
carvings and other Aboriginal art, it is recommended that local
examples be used wherever possible. Consult with local Aboriginal
communities and education consultants for such examples. Refer also
to background information in Three-Dimensional Space 1. Language
Students should be able to communicate using the following
language: object, two-dimensional shape (2D shape),
three-dimensional object (3D object), cone, cube, cylinder, prism,
pyramid, sphere, top view, front view, side view, isometric grid
paper, isometric drawing, depth. Refer also to language in
Three-Dimensional Space 1.
Investigate and represent three-dimensional objects using
drawings identify prisms (including cubes), pyramids, cylinders,
cones and spheres in the environment and from drawings, photographs
and descriptions investigate types of three-dimensional objects
used in
commercial packaging and give reasons for some being more
commonly used (Communicating, Reasoning)
sketch prisms (including cubes), pyramids, cylinders and cones,
attempting to show depth compare their own drawings of
three-dimensional
objects with other drawings and photographs of three-dimensional
objects (Reasoning)
draw three-dimensional objects using a computer drawing tool,
attempting to show depth (Communicating)
sketch three-dimensional objects from different views, including
top, front and side views investigate different two-dimensional
representations of
three-dimensional objects in the environment, eg in Aboriginal
art (Communicating)
draw different views of an object constructed from connecting
cubes on isometric grid paper interpret given isometric drawings to
make models of three-dimensional objects using connecting cubes
Learning Across The Curriculum
Cross-curriculum priorities Aboriginal &Torres Strait
Islander histories & cultures Asia & Australias engagement
with Asia Sustainability General capabilities
Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
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Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Investigate and represent three-dimensional objects using
drawings
2
Identify Prisms Identify and name 3D objects, using concrete
models, including; cubes, pyramids, cylinders, cones and spheres.
Discuss the properties of each shape. Using the school play
equipment as an example of 3D shapes in the environment, have the
students make a detailed sketch of the equipment. Students should
then be encouraged to identify the different 3D shapes used in the
make-up of the school play equipment and label these on their
diagram. Discuss the different 3D shapes used in the play
equipment. Possible questions: - What 3D shapes did you identify
within the play equipment? - What was the predominant shape used?
Why do you think this was the case? - What purpose did each shape
play in the functionality of the play equipment? - Is there any
piece of the equipment that you think could have better utilised a
different 3D shape? Why?
Support: individual assistance as required
3D models, paper and pencils
3
3D Shapes in Construction Provide students with a variety of
pictures of buildings from around the world. Have them identify and
draw the 3D shapes that make up the buildings construction. Explain
to students that a photograph is a 2D representation of a 3D object
and so we are unable to see the entire object. Keeping this in
mind, have the students predict, how many of the 3D shapes they
believe went into the construction of the building (obviously this
does not involve predicting the number of bricks, for example, but
rather columns etc)
Students should be encouraged to compare their drawings to exact
drawings of 3D shapes to check for accuracy and to determine how to
improve on their attempts.
Support: individual assistance as required
3D models, pictures of buildings and/or landmarks, paper and
pencils
4
Different Views Provide students with concrete examples of 3D
shapes. Students work in small groups to share a selection of
shapes. Students need to select one shape at a time and draw and
label the shape. They then need to draw the shape from different
views including; top, front and side views. Students should present
their sketches in a table using the headings; 3D Shape, Top View,
Front View and Side View at the top of each column.
Support: individual assistance as required, peer tutor grouping
strategies, provide ready-made tables
3D models, paper and pencils
5
Drawing Shapes on Isometric Paper Teach the students how to draw
a cube on isometric paper:
Extension: make and draw shapes, such as, the following:
Isometric paper, pencils. Whiteboard, IWB isometric paper
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Sharon Tooney
Provide students with examples of isometric drawings of
interlocking cubes. Have the students use the images to create the
shapes themselves using interlocking cubes and then draw the shapes
on isometric paper using the diagrams as a guide. Possible
examples:
6
Creating and Drawing Shapes from Interlocking Cubes Have
students create a series of shapes using 3,4, 5............
interlocking cubes. Have students draw their shapes on isometric
paper or alternatively, have the students work in pairs and draw
their partners shapes. Provide students with a series of
interlocking cube shape diagrams and have them determine the number
of cubes within each one. Possible examples:
Support: Allow students who are unable to visualise the number
of cubes used in the 2nd part of the lesson, make a model using
interlocking cubes Extension: Use more complex diagrams, such as,
the following:
Isometric paper, pencils. Whiteboard, IWB isometric paper
7
Drawing 3D Shapes With Computers Have students experiment with
drawing individual 3D shapes using computer software and/or online
tools. When students have become proficient in drawing individual
shapes, see if they can create a 3D image of a building, for
example. Possible online tools/downloadable programs include:
http://www.sketchup.com/ Alternatively 3D shapes can be created
using Word: Step 1: Launch Microsoft Word, and click the Insert tab
at the top of the screen, then click the Shapes button. Step 2:
Click one of the shapes, such as a circle, from the drop-down
selection menu. None of the shapes are 3D; youll add that look in a
later step. The cursor turns into a plus sign. Step 3: Drag the
cursor on the Word page to form the shape. Click the shape to open
the new orange Drawing Tools tab at the top of the screen and the
related ribbon below the
Support: individual support as needed Extension: encourage
students that are capable of creating shapes without assistance to
incorporate shapes into 3D designs and or constructions
Computers, paper and pencils
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Sharon Tooney
tab. Step 4: Click the 3-D Effects button on the ribbon. Without
clicking, hover the cursor over the options available, moving from
button to button for options such as turning a flat circle into a
3D cone. Step 5: Experiment with hovering over the options in the
drop-downs fly-out menus as well, with 3D shape lighting and
direction choices. Step 6: Click an actual 3D effect to see it
instantly take shape on the Word page.
10
Revision and Assessment
ASSESSMENT OVERVIEW
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Sharon Tooney
MATHEMATICS PROGRAM PROFORMA
STAGE: Year 4 ES1 S1 S2 S3
STRAND: STATISTICS AND PROBABILITY
TERM: 1 2 3 3
WEEK: 1 2 3 4 5 6 7 8 9 10
SUBSTRAND: Chance 2 KEY CONSIDERATIONS OVERVIEW OUTCOMES A
student: uses appropriate terminology to describe, and symbols to
represent, mathematical ideas MA2-1WM describes and compares chance
events in social and experimental contexts MA2-19SP
Background Information Theoretically, when a fair coin is
tossed, there is an equal chance of obtaining a head or a tail. If
the coin is tossed and five heads in a row are obtained, there is
still an equal chance of a head or a tail on the next toss, since
each toss is an independent event. Language Students should be able
to communicate using the following language: chance, event,
possible, impossible, likely, unlikely, less likely, more likely,
most likely, least likely, equally likely, experiment, outcome.
Describe possible everyday events and order their chances of
occurring (ACMSP092) use the terms 'equally likely', 'likely' and
'unlikely' to describe the chance of everyday events occurring, eg
'It is equally likely that you will get an odd or an even number
when you roll a die' compare the chance of familiar events
occurring and describe the events as being 'more likely' or 'less
likely' to occur than each other order events from least likely to
most likely to occur, eg 'Having 10 children away sick on the same
day is less likely than having one or two away' compare the
likelihood of obtaining particular outcomes in a simple chance
experiment, eg for a collection of 7 red, 13 blue and 10 yellow
marbles, name blue as being the colour most likely to be drawn out
and recognise that it is impossible to draw out a green marble
Identify everyday events where one occurring cannot happen if the
other happens (ACMSP093) identify and discuss everyday events
occurring that cannot occur at the same time, eg the sun rising and
the sun setting Identify events where the chance of one occurring
will not be affected by the occurrence of the other (ACMSP094)
identify and discuss events where the chance of one event occurring
will not be affected by the occurrence of the other, eg obtaining a
'head' when tossing a coin does not affect the chance of obtaining
a 'head' on the next toss explain why the chance of each of the
outcomes of a second
toss of a coin occurring does not depend on the result of the
first toss, whereas drawing a card from a pack of playing cards and
not returning it to the pack changes the chance of obtaining a
particular card or cards in future draws
compare events where the chance of one event occurring is not
affected by the occurrence of the other, with events where the
chance of one event occurring is affected by the occurrence of the
other, eg decide whether taking five red lollies out of a packet
containing 10 red and 10 green lollies affects the chance of the
next lolly taken out being red, and compare this to what happens if
the first five lollies taken out are put back in the jar before the
sixth lolly is selected
Learning Across The Curriculum
Cross-curriculum priorities Aboriginal &Torres Strait
Islander histories & cultures Asia & Australias engagement
with Asia Sustainability General capabilities
Critical & creative thinking Ethical understanding
Information & communication technology capability Intercultural
understanding Literacy Numeracy Personal & social capability
Other learning across the curriculum areas Civics & citizenship
Difference & diversity Work & enterprise
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Sharon Tooney
CONTENT WEEK TEACHING, LEARNING and ASSESSMENT
ADJUSTMENTS RESOURCES REG
Describe possible everyday events and order their chances of
occurring Identify everyday events where one occurring cannot
happen if the other happens Identify events where the chance of one
occurring will not be affected by the occurrence of the other
1
Take-away Dice In pairs, students play the following game to
investigate the concept of fairness. In turns, they throw two dice
and subtract the smaller number from the larger number eg if 4 and
6 is thrown, they calculate 6 4 = 2. Player A scores a point if the
answer is 0, 1, or 2. Player B scores a point if the answer is 3,
4, or 5. Students play the game and are asked to comment on whether
the game is fair and why. Students are asked how the rules of the
game could be changed to make the game fairer and how they could be
changed so it is impossible for one student to lose.
Support: concrete materials to solve subtraction problems, peer
tutoring strategies for grouping
Dice, paper and pencils
2
Sample Bags Students place four counters or blocks (eg three
blue and one white) into a bag. The teacher discusses with the
students the chance of drawing out a blue block. Possible questions
include: - would you have a good chance or a poor chance of drawing
out a blue block? Why? - what colour block is most likely to be
drawn out? Why? Students could trial their predictions by drawing a
block out of the bag a number of times, recording the colour and
replacing the block each time. Students discuss their findings. The
colours are then swapped to three white blocks and one blue block.
The teacher discusses with the students the chance of drawing out a
blue block from this new group. Possible questions include: - would
you have a good chance or a poor chance of drawing out a blue
block? Why? - what colour block is most likely to be drawn out?
Why? Students complete a number of trials and discuss the results.
Students are encouraged to make summary statements eg If there are
lots of blue blocks you have a good chance of getting a blue
block.
Support: summary statement that only require insertion of word
explaining likelihood
Counters or blocks, bag, paper and pencils
3
Is It Fair? The class is organised into four teams. Each team is
allocated a colour name: red, blue, green or yellow. The teacher
has a bag of counters composed of 10 red, 5 blue, 4 green and 1
yellow. The students are told that there are twenty counters and
that each colour is represented in the bag. The composition of
counters is not revealed to the students. The teacher draws a
counter from the bag and a point is given to the team with the
corresponding colour. The counter is then returned to the bag and
the process is repeated for twenty draws. Individually, the
students are then asked to predict the composition of coloured
counters in the bag, explain their prediction and state whether the
game is fair. Possible questions include:
Support: peer tutoring strategies for grouping
Bag, counters, paper and pencils
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Sharon Tooney
- what happens if one colour is not included? - have you tried
using a diagram to help you with your predictions? - what are some
possible explanations? - how will you know if your generalisations
are reasonable? Students are then told the composition of colours
in the bag and are asked to name the colours most and least likely
to be drawn out.
4
Musical Chairs Students play the game Musical Chairs removing
one chair each time. The chance of each student getting a chair is
discussed. The game is repeated with three or more chairs removed
at a time and students are asked to comment on whether there is
more or less chance of getting out compared to the original game.
Variation: Other games could be played where an aspect of the game
is changed to affect the chance of various outcomes occurring.
Chairs, music CD player
5
Combination Dressing Students are told that they will be given
three t-shirts and two pairs of trousers and are asked to predict
how many different combinations of clothes they could make from
them. They work out a strategy and follow it to calculate the
number of combinations and compare the results to their
predictions.
Support: provide cardboard cut-outs of clothes to make
combinations with
Paper and pencils
10
Revision and Assessment
ASSESSMENT OVERVIEW
Light Challenges