Number Strategies (ACMN076) MAG 4.4.1
Draft-This is a work in progress. MAG writing project 2013
Australian Curriculum YR 4
ACMN076 Develop efficient mental and written strategies and
use appropriate digital technologies for multiplication and for
division where there is no remainder
Key IdeaMultiplication and division are the inverse operation of one
another.
5 X 10 = 50
10 X 5 = 50
50 ÷ 2 = 25
multiplication facts ( see document in resources) Revise what a
turnaround is and ask students to state the turnarounds for a
set of multiplication facts. Ask learners to prove the statement:
if 20 x 5 is the same as 100
5 x 20 is the same as 100
2. Using concrete materials, ask the students to show two
different arrays for a given multiplication fact and its
turnaround. Use MAG 3.4.1 Think board activities. Remind
learners that they need to estimate products before computing
the actual answers, to give them a sense of size of the answer.
50 ÷ 5 = 50
Multiplication is the most effective way to solve problems
where the same number is repeatedly added.
http://www.brainpopjr.com/math/additionandsubtraction/rep
eatedaddition/grownups.weml
Division is about sharing and making equal groups.
http://www.brainpop.com/educators/community/lessonplan/making-equal-groups-background-information-forteachers-and-parents/?bp-jr-topic=making-equal-groups
Its important to develop strategies for number facts to
strengthen the learners understanding of the relationship
between numbers and patterns in mathematics
Resources
● FISH problem solving kit
● Concrete material eg unifix cubes, counters.
● Multiplication facts charts
● 10 sided dice
● Charts
Introductory Activity Process
1.Complete a series of revision activities in relation to basic
40 x 50 = 4 x 5 = 9 x 100 = 900
Give students a series of ‘Place value’ questions to practise
with and check for understanding
10 (b) Division Strategy:
Introduce the fourth mental strategy for division Place Value
Look for division facts then use place value
eg. 280 / 7 = 28 / 7 = 4 tens= 40
210 / 3 = 21 / 3 = 7 tens = 70
Give students a series of ‘Place Value’’ questions to practise
with and check for understanding.
3. Introduce or revise the inverse operation multiplication facts
chart, investigate the patterning between each set of facts
(repeated addition) and also the inverse to multiplication
(Division). Using the two arrays, instruct students to model the
inverse facts, is 10 divided by 2 the same as 5 lots of 2?
4. Ask learners to use the relationship between sets of facts 5 x facts are
half of ?, facts 4 x facts and double ?, facts, 9 times facts are 1 less than ?,
facts etc to make statements about sets of numbers. Eg 10 times table is
half of? is 1 less than, is double ?
5. Explain the importance of (algorithmic calculations) working out
mentally and brainstorm activities in our daily lives where we would need
to use mental strategies to calculate problems.
Inverse operation strategies
Highlight and focus on the relationship between the two operations.
6 (a) Multiplication Strategy:
Introduce the first mental strategy for multiplication - ‘Doubles’ You
double the given number by 2.
eg. 63 x 2 = double 60 and double 3 = 66
Give students a series of ‘ doubling’ questions to practise with and check
for understanding (you can increase to 3 digit numbers for those who
require extension).
6 (b) Division Strategy:
Introduce the first mental strategy for division Halving
One half of the number is the same as / by 2
eg. 18 / 2 = Half of 18 = 9
400 / 2 = Half of 400 = 200
Give students a series of ‘Halving’ questions to practise with and check for
understanding.
7 (a).Multiplication Strategy:
Introduce the second mental strategy for multiplication - ‘Double doubles’
Double the number and double again for 4 x tables.
eg. 31 x 4 = double 31 = 62 then double again = 124
Give student a series of ‘double doubles’ multiplication
Give students a series of ‘Halving, halving’ questions to
Sing the facts. Use a the following suggested websites.
practise with and check for understanding.
7 (b) Division Strategy:
Introduce the second mental strategy for division -
1
Halving, halving. One quarter of a number is half and half again,
the same as / 4
eg. 68 / 4 = 68 / 2 = 34 / 2 = 17
330 / 4 = 440 / 2 = 220 / 2 =110
8 (a). Multiplication Strategy:
Introduce the third mental strategy for multiplication - ‘Ten and
halve’
Multiply by 10 then halve the answer.
eg. 24 x 5 = 24 x 10 = 240 then halve = 120
Give students a series of ‘Ten and halve’ multiplication
questions to practise with and check for understanding
8 (b) Division Strategy:
Introduce the third mental strategy for division Split and Divide
Split the dividend and divide the parts
eg. 36 / 3 = 30 / 3 and 6 / 3 = 12
525 / 5 = 500 / 5 and 20 / 5 and 5 / 5 = 105
Give students a series of ‘Split and Divide’ questions to practise
with and check for understanding.
9 (a). Multiplication Strategy:
Introduce the fourth mental strategy for multiplication ‘Friendly and fix’
Rounding or making the number easier to work with then fix
the change.
eg. 16 x 9 = 16 x 10 = 160 - 16 = 144
7 x 19 = 7 x 20 = 140 - 7 = 133
99 x 7 - 100 x 7 = 700 - 7 = 693
Give students a series of ‘Friendly and fix’ questions to practise
with and check for understanding.
10 (a). Multiplication Strategy:
Introduce the fifth mental strategy for multiplication - ‘Place
Value’ Use single digit multiplication facts then multiply by 10 or
100. eg. 8 x 40 = 8 x 4 =32 x 10 = 320
Plus moves from left to right and Minus moves from right to
left. (The students should be encouraged to think about why
that might be.) Take it in turns to throw the two dice and add
the numbers. Move that number of
Extensions and Variations
Ask learners to brainstorm and list the various ways division
facts can be recorded. When students are solving division facts,
remind them to use the multiplication fact to help them (Use of
backtracking) find the answer e.g. for 9 ÷ 3, students think
‘3 x � = 9’
Provide students with a mix up of operations and questions.
Discuss whether answers will be bigger or smaller than the
number students start with . Multiplication makes the answer
larger, division makes an answer smaller (check for students
understanding of this concept)
places in your direction. If the counter reaches 1, Minus has won and so,
of course if the counter reaches 27, Plus has won.
Investigation: Tug of War
One player is called "PLUS"
The other is called "MINUS" so decide who is who.
Think about whether you have to land exactly at 1 or 27 or allowed to end
up beyond those points. I wonder what difference it will make if you are
allowed to go beyond rather than landing exactly on the end numbers?
Once students have got used to the game, they can play a variation, this
time using multiplication or division and or perhaps extending the number
line.
Ask learners to develop a fact with two cards each displayed on
one and a strategy on the second. Mix and match the cards and
ask students to match a fact to a strategy. There is often more
than one strategy.
When the rules are changed students must reflection on whether the
change makes the game better to play.
The teacher needs to ensure that the basic strategies are
present in the samples:
● arrays
● skip counting on a numberline
● area model
● repeated addition
● any order-multiplication
Strategies for multiplication include double, double-double, ten
and halve, friendly and fix , place value.
Useful strategies for division include halving-halving, split then
divide, place value.
Background
To predict the usefulness of strategies for given problems
Place beads in zip-lock bags. Hold up two bags and discuss
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.
Learners come to realise that division is the inverse of multiplication and
visa versa.
Learners should be encouraged to check the answer to a division question
by multiplying their answer by the divisor. To divide learners may recall
division facts or transform the division into a multiplication and use
multiplication facts.
which strategy could be used to add them.
Multiplication and Division Songs and Raps
http://www.songsforteaching.com/math/multiplicationsongs.ht
m
2
Digital Learning
Mulitplication
www.topmarks.co.uk/maths-games/5-7.../multiplication-anddivision
Various games available
Word and Real Life Problems
www.primaryresources.co.uk
Multiplication Interactive Games
https://interactivemaths.wikispaces.com/Multiplication+%26+
Division
Assessment Year 4 Achievement Standard
Vocabulary: Criteria-Learner uses appropriate terminology to describe, and
By the end of Year 4, students choose appropriate strategies for
calculations involving multiplication and division. They recognise
common equivalent fractions in familiar contexts and make
connections between fraction and decimal notations up to two
decimal places. Students solve simple purchasing problems. They
identify unknown quantities in number sentences. They describe
number patterns resulting from multiplication. Students compare
areas of regular and irregular shapes using informal units. They
solve problems involving time duration. They interpret information
contained in maps. Students identify dependent and independent
events. They describe different methods for data collection and
representation, and evaluate their effectiveness. Students use the
properties of odd and even numbers. They recall multiplication
facts to 10 x 10 and related division facts. Students locate familiar
fractions on a number line. They continue number sequences
involving multiples of single digit numbers. Students use scaled
instruments to measure temperatures, lengths, shapes and
objects. They convert between units of time. Students create
symmetrical
Please note this website is a wiki space that you have to
request membership to.
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Division Drag Race
Uses appropriate terminology to describe, and symbols to
represent, mathematical ideas
Selects and uses appropriate mental or written strategies, or
technology, to solve problems
Checks the accuracy of a statement and explains the reasoning
used
Uses mental and informal written strategies for multiplication and
division
Assessment Task
Write as many equations as you can to describe this array
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Turnarounds, arrays, inverse, patterning, inverse, lots of , groups of,
multiplication, division, plus, minus,
The Number Chain Game Cards also have mathematical phrases which
provide an opportunity to work with building mathematical language.
Posssible Assessment Criteria
Division Interactive Games
https://interactivemaths.wikispaces.com/Multiplication+%26+
Division
symbols to represent, mathematical ideas.Use loop card activity to extend
vocabulary understanding.
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Links to other MAGs
The following MAGs can provide differentiation ideas for learners who
need further support
MAG 3.2.4 Multiplication Facts 1
MAG 3.3.1 Multiplication Facts 2
MAG 3.3.4 Multiplication Problems 1
Printable Charts Resources:
http://www.amathsdictionaryforkids.com/mathsCharts.html
Word Wall:
Repeated addition, sharing, equal groups, inverse, patterning,
strategy,
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