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Early Years Number
Teach for Understanding
Early Years Number
MAV, 2011
This resource is organised around the well-known and thoroughly researched Early Years Growth Points for
Number. The four topics covered are:
Counting,
Place value,
Strategies for addition & subtraction, and
Strategies for multiplication & division.
This has been chosen as the structure of this document as the growth points are used to identify important
stages of learning experienced by most students. The classroom teacher can then use these different stages
of understanding to plan teaching and learning experiences for their early years classroom, and scaffold
learning across the different levels of student understanding.
Structure within each growth point
Targets for learning (almost, but not quite, outcomes) are listed; these are the next learning goals in the
normal trajectory of success for most children.
Evidence: There is no simple match between these growth points and the Early Years Interview, but, as an
indicator of the point the child is at, we have listed the last questions a child might have successfully
answered. This indicates that the child is just beginning to work on this particular growth point.
Activities: Some suggested activities are described, and some of the key resources are listed to help a
teacher of a child at this level to move forward. Inevitably some great resources are omitted, and more will
be written,
Resources: You should feel free to add your own favourite resources to each resource list. Those listed are
either free through the internet or your sector office, or MAV resources from <www.mav.vic.edu.au/shop>.
The Student Resources listed are either hands-on tasks or for use at a computer.
Reporting: Finally there is no simple match between the VELS progression points and the growth points.
However summary statements from VELS are listed below each one suggesting the reporting criteria that
might be chosen to describe a child at that level. It is not being suggested that the VELS Standards or
progression points should form the basis of the curriculum; they were written as a reporting guide and should
only be used as such.
The approximate relationship between the growth points in the four sections is set out below.
Counting
Place value
Adding & subtracting
Multiplying & dividing
1 Rote count to 20
2 Count collections to 20
1 PV: single digits only
3 Count by 1s from any
starting number
4 Count by 2s, 5s, 10s
from 0
1 Count all to add
1 Count all in groups
2 Count on from any
start
2 PV: two-digit numbers
– tens and ones
3 Count to subtract
2 Model multiplication &
division (hands-on)
5 Count by 1s, 2s, 5s,
10s up or down from any
starting number
4 Basic adding &
subtracting strategies
6 Skip counting
5 Derived addition &
subtraction strategies
3 Abstract multiplication
& division
6 Extend addition &
subtraction
4 Basic & derived
strategies for
multiplication
3 PV three-digit numbers
– hundreds, tens and
ones
4 PV four-digit numbers thousands
5 Basic & derived
strategies for division
5 Extend to millions
6 Extend
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Early Years Number
Contents
Counting
towards VELS Interview
0 Beginning counting
[0.5] Q1 – 10
3
1 Rote count to 20, but not a collection
[1.0]
Q11
5
2 Count collections to 20
[1.5]
Q12
7
3 Count by 1s from any starting number
[2.0]
Q18
9
4 Count by 2s, 5s, 10s from 0
[2.5] Q22, 23
11
5 Count by 1s, 2s, 5s, 10s up or down from any starting number
[2.5]
Q28
13
6 Skip counting
[3.0] Q33, 34
14
1 Reading, writing interpreting and ordering single digits only
[1.0]
Q13
16
2 Reading, writing interpreting and ordering two-digit numbers
[1.5]
Q20
17
3 Reading, writing interpreting and ordering three-digit numbers
[2.0]
Q24
19
4 Reading, writing interpreting and ordering numbers beyond 1000
[2.5]
Q29
20
5 Extend and apply place value knowledge
[3.0]
Q35
21
1 Count all to add two collections
[1.0]
Q14
23
2 Count on to add from any start
[1.5]
Q16
25
3 Count back / count down to / count up from … to subtract
[2.0]
Q17
28
4 Basic strategies for adding and subtracting
[2.5] Q21, 25
31
5 Derived strategies for adding and subtracting
[2.5]
Q30
35
6 Extending and applying addition & subtraction
[3.0]
Q36
40
1 Count all in groups
[2.0]
Q15
42
2 Model multiplication & division (hands-on)
[2.5]
Q26
44
3 Abstract multiplication & division
[3.0]
Q27
47
4 Basic, derived & intuitive strategies for multiplication
[3.0]
Q31
50
5 Basic, derived & intuitive strategies for division
[3.0]
Q32
54
6 Extending and applying multiplication & division
[3.0]
Q37
57
Place value
Strategies for Addition and Subtraction
Strategies for Multiplication and Division
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A.
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Early Years Number
Counting
0. Beginning counting – numbers to 10, ordinals to five [VELS 0.5]
Targets
Strengthen one-one correspondence and matching a numeral symbol to a quantity. Note that with a
set of objects, the order of counting does not matter, but what is important is that each object gets
counted once – there is a 1-1 correspondence between the number name and the object. The last
name we come to represents the number of objects (the ‘cardinal’ number).
Evidence
Mathematics Online Interview
Questions 1-10
Activities (from Guidelines in Number)
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Resources
Teacher references for activities
Continuum
0.5 One-to-one correspondence
Activity 1 Counting the digits 1-10
Activity 2 Counting objects
Activity 3 Matching written numbers with the collection
Activity 4 Variations on counting
0.5 Ordinal number
Activity 1 Incidental Classroom Activities
Activity 2 Toy Line Up
Activity 3 Races
Activity 4 Visualising order
Assessment for common misunderstandings
Level 1: Trusting the count
Subitising tool – with cards sets 1 to 5
Mental objects tool - with mental objects card
MAV Resources
People count 2 Counting
Guidelines in Number p1-6
Practical teaching strategies for children with learning difficulties
Book 1 Prenumber
Cornerstones in Number - Counting Ages 5-9
Up to 10 (p48-63) – 7 activities
Hands-on Maths: Developing mathematics with UNIFIX
Pattern (p10-35) – 8 activities
Early counting
Illuminations: Let’s count to 5 U74 (L57, 90-94)
Digits 1 to 5 http://www.literacycenter.net/numbers_en/numbers_en.php
Illuminations: Let’s count to 10 U147 (L501-508), L870, 869
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Counting beetles, Count and compare, Number trains
Count Us In – game 4 Ordinal Number
http://www.abc.net.au/countusin/games/game4.htm
MAV Resources (hands-on or computer)
Spreadsheets from Interactive Learning Subitising
Hands-on Maths: Developing mathematics with UNIFIX
Three blocks on your fingers (p38-39)
Cornerstones in Number: Place Value Ages 6-9
Tens Frames (p10)
Early Years Number
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Early Years Number
1. Rote counting to 20, but not a collection [towards VELS 1.0]
Targets
Strengthen confidence with the verbal sequence. Move onto counting collections and finding more
efficient counting strategies. The way the objects are arranged does not matter – to some children,
more widely spaced looks bigger, but they must learn to ‘trust their counting’.
Evidence
Mathematics Online Interview Question 11
Activities (from Guidelines in Number)
Resources
Teacher references for activities
Continuum
1.0 Counting groups of up to 20 objects
Activity 1 Verbal counting sequence
Activity 2 Efficient counting strategies
Activity 3 Hidden collections
Assessment for common misunderstandings
Level 1: Trusting the count
Subitising tool – with cards sets 1 to 5
Mental objects tool - with mental objects card
MAV Resources
People count 2 Counting
Guidelines in Number p7-9
Practical teaching strategies for children with learning difficulties
Book 1 Prenumber
Working Mathematically: Investigations
Unit 1: Early exploration of number, estimation and graphing
Cornerstones in Number: Counting Ages 5-9
Learning names and symbols for numbers 1 to 10 (p48)
Hands-on Maths: Developing mathematics with UNIFIX
Counting, place value and operations (p36-47) – 5 activities
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Early Years Number
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Counting beetles, Count and compare, Number trains, Count Us In (NW6C4U)
Teen numbers http://www.nzmaths.co.nz/resource/teen-numbers?parent_node=
Tens frames for teen numbers http://nzmaths.co.nz/resource/using-tens-frames-teen-numbers
Teens and fingers http://nzmaths.co.nz/resource/teens-and-fingers
Count Us In – Games for basic number concepts http://www.abc.net.au/countusin/default.htm
Count Us In – Game 11 the next highest number http://www.abc.net.au/countusin/games/game11.htm
Hang out the shirts on the washing line http://www.crickweb.co.uk/ks1numeracy.html#washindex
MAV Resources
Spreadsheets from Interactive Learning Subitising, Twenty frame
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Early Years Number
2. Counting collections to 20 [towards VELS 1.5]
Targets
Strengthen counting on, from the larger number. See Strategies for addition and subtraction #2.
Evidence
Mathematics Online Interview Question 12
Activities (from Guidelines in Number)
Resources
Teacher references for activities
Continuum
1.0 Counting groups of up to 20 objects
Activity 1 Verbal counting sequence
Activity 2 Efficient counting strategies
Activity 3 Hidden collections
1.0 Making better estimates
Activity 1 How many marbles?
Activity 2 Sharing strategies for checking estimates
Activity 3 Estimating calculations
Assessment for common misunderstandings
Level 1: Trusting the count
Subitising tool – with cards sets 1 to 5
Mental objects tool - with mental objects card
MAV Resources
People count 2 Counting
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Early Years Number
Guidelines in Number p7-9
Cornerstones in Number: Counting Ages 5-9
Teens numbers (p74-83) – 7 activities
Practical teaching strategies for children with learning difficulties
Books 2 (numbers to 5), 3 numbers to ten) and 4 (numbers to twenty)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Counting beetles, Count and compare, Number trains
Problem solving from Mathematics Task Centre
Cookie count, Matching cards, Farmyard, Button sort, Squound, Animal farm,
Flowers in the field, Pack up your bears
Teen numbers http://www.nzmaths.co.nz/resource/teen-numbers?parent_node=
Using Tens Frames for “Teen” Numbers http://nzmaths.co.nz/resource/using-tens-frames-teen-numbers
Teens and fingers http://nzmaths.co.nz/resource/teens-and-fingers
Count Us In – Games for basic number concepts http://www.abc.net.au/countusin/default.htm
Count Us In – Game 11 the next highest number http://www.abc.net.au/countusin/games/game11.htm
Hang out the shirts on the washing line http://www.crickweb.co.uk/ks1numeracy.html#washindex
MAV Resources
Cornerstones in Number: Counting Ages 5-9
Teens numbers (p74-83)
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Early Years Number
3. Counting by 1s (forward/backward, including variable starting
points from 1 to 100; names numbers before or after)
[towards VELS 2.0]
Targets
Find more efficient counting strategies, and improve fluency and confidence.
Evidence
Mathematics Online Interview Question 18
Activities
Ten-frames – see Addition and subtraction 1. Twenty-frames – see Addition and subtraction 2.
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Early Years Number
Resources
Teacher references for activities
Continuum
1.25 Counting with Two Digit Numbers
Activity 1 Bridging the decades
Activity 2 Counting from any number
Activity 3 Efficient strategies for counting
Activity 4 Number Rolls
MAV Resources
People count 2 Counting
Guidelines in Number p9-17
Paper abacus
Snake lines and sewing tapes
MAV-money
Cornerstones in Number: Counting Ages 5-9
Hundred by ones (p84-92) – 8 activities
Practical teaching strategies for children with learning difficulties
Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Counting beetles, Count and compare, Number trains
Know the forward and backward counting sequences of whole number to 100
http://www.nzmaths.co.nz/ao/ao1-know-forward-and-backward-counting-sequences-whole-numbers-100
Make 100
http://www.nzmaths.co.nz/resource/make-100
100th Day of School
http://enchantedlearning.com/themes/hundred.shtml
Printable book
http://enchantedlearning.com/books/howmany/100days/
MAV Resources
Tuning In (Lower) Counting to 100 p43, Tens p69, The five bike family p50, Counting patterns p59
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Early Years Number
4. Counting from 0 by 2s, 5s, and 10s [towards VELS 2.5]
Targets
Develop fluency with numbers up to 100, particularly multiples and patterns with 2s, 5s and 10s both
upwards and downwards.
Evidence
Mathematics Online Interview Questions 22, 23
Activities (from Guidelines in Number)
Resources
Teacher references for activities
Continuum
1.5 Using a 100s chart for mental calculation
Activity 1 Guess my number
Activity 2 Number neighbours
Activity 3 Missing numbers
Activity 4 Challenges
MAV Resources
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 3 Basic facts (Multiplication, Division)
Module 4 Two-digit whole numbers
People count 2 Counting
Guidelines in Number p15, 31
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Early Years Number
Paper abacus
Snake lines and sewing tapes
Tuning In (Lower) What if? p16, Guess what pattern p66, Skip count pattern p65, Odds p67, How many p69,
Twenty-nine p68, Twenty-three p70
Cornerstones in Number: Counting Ages 5-9
Twos, fives and tens (p93-102) – 9 activities
Student Resources (hands-on or computer)
FUSE Counting beetles, Count and compare, Number trains
Counting Caterpillar http://www.ictgames.com/caterpillar_slider.html
Count Us In – Game 6 Matching words and numbers http://www.abc.net.au/countusin/games/game6.htm
MAV Resources
Spreadsheets from Interactive Learning Skip count to 140, Skip count back from 140, Odometer
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Early Years Number
5. Counting by 2s, 5s, and 10s up or down from any number
[VELS 2.5]
Targets
Develop familiarity with patterns of ones and tens digits while counting up and down by 2s, 5s and
10s from any number
Evidence
Mathematics Online Interview Question 28
Activities
On a 100s chart, start at 7 and go up in 5s. Describe and explain the pattern.
1
2
3
4
5
6
7
8
9
10
11
12
13 14
15
16
17
18
19
20
21
22
23 24
25
26
27
28
29
30
31
32
33 34
35
36
37
38
39
40
41
42
43 44
45
46
47
48
49
50
Resources
Teacher references for activities
Continuum
2.0 Skip counting
Activity 1 Counting games
Activity 2 Whisper count
Activity 3 How far can you go?
Activity 4 Using the hundreds grid for counting
MAV Resources
People count 2 Counting
Guidelines in Number p15-31
Paper abacus
Snake lines and sewing tapes
Cornerstones in Number: Counting Ages 5-9
Twos, fives and tens (p93-102)
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Problem solving from Mathematics Task Centre
Cookie count, Matching cards, Farmyard, Button sort, Squound, Animal farm, Flowers in the
field, Pack up your bears
MAV Resources
Spreadsheets from Interactive Learning Skip count to 140, Skip count back from 140
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Early Years Number
6. Skip counting and applying counting skills [VELS 3.0]
Targets
Skip counting not only familiarises the child with the numbers, but forms the basis for adding,
subtracting, multiplying and dividing – where place value is not involved. It should be related to
number lines.
Evidence
Mathematics Online Interview Questions 33, 34
Activities
 Skip counting
Children should be able to skip count, starting from any number and counting in steps of any single digit
number. (Example: skip up by 90s – 3, 12, 21, 30, 39, 48,...)
On a 100s chart, start at 41 and go down in 8s. Describe and explain the pattern.
1
2
3
4
5
6
7
8
9
10
11
12
13 14
15
16
17
18
19
20
21
22
23 24
25
26
27
28
29
30
31
32
33 34
35
36
37
38
39
40
41
42
43 44
45
46
47
48
49
50
This should be done with a 100-chart. If children put markers on the numbers they reach, they will see the
patterns: numbers are in lines. It will assist with addition and subtraction as it reinforces the regrouping that
occurs at each ‘ten’.
NOTE: Multiplication tables derive from skip counting from 0 (e.g. skip count up by 7s to get 6 x 7: 7, 14, 21,
28, 35, 42.). However this is not the same as recalling multiplication facts; 6 x 7 = 42 instantly called requires
memory training.
Resources
Teacher references for activities
Continuum
2.0 Skip counting
Activity 1 Counting games
Activity 2 Whisper count
Activity 3 How far can you go?
2.5 Advanced skip counting
Activity 1 Using the hundreds grid for counting
Activity 2 Advanced whisper count
Activity 3 How far can you go?
Activity 4 Using counting grids
Activity 5 Counting games
MAV Resources
People count 2 Counting
Guidelines in Number p31
Paper abacus
Snake lines and sewing tapes
Cornerstones in Number: Counting Ages 5-9
Threes and fours (p113-118), Nines (p119-122), Odd and even (p123-125)
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
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Student Resources (hands-on or computer)
Spreadsheets from Interactive Learning
Skip count to 9999, Skip count back from 9999, Odometer
Early Years Number
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B.
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Early Years Number
Place Value
1. Reading, writing, interpreting, and ordering single digit numbers
[VELS 1.0]
Targets
At this level place value is just an understanding tens and ones as the basis of two-digit numbers.
However this requires fluency with counting and awareness of patterns in the numbers up to 99.
Evidence
Mathematics Online Interview Question 13
Activities
See Counting.
Resources
Teacher references for activities
Continuum
1.0 Counting groups of up to 20 objects
Activity 1 Verbal counting sequence
Activity 2 Efficient counting strategies
Activity 3 Hidden collections
Assessment for common misunderstandings
Level 1: Trusting the count
MAV Resources
People count 3 Place value
Guidelines in Number p17-18
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Wishball
MAV Resources
Spreadsheets from Interactive Learning Skip count back (140), Skip count from 9999
Hands-on Maths: Developing mathematics with BASE TEN
Before Base Ten (p2-13)
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Early Years Number
2. Reading, writing, interpreting, and ordering two-digit numbers
[VELS 1.5]
Targets
At this level place value is just an understanding tens and ones as the basis of two-digit numbers.
However this requires fluency with counting and awareness of patterns in the numbers up to 99.
Evidence
Mathematics Online Interview Question 20
Activities (from Guidelines in Number)
Base ten blocks
Use longs and minis, and play trading games, such as “First to 100” with one or two dice.
Resources
Teacher references for activities
Continuum
1.25 Counting with Two Digit Numbers
Activity 1 Bridging the decades
Activity 2 Counting from any number
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Early Years Number
Activity 3 Efficient strategies for counting
Activity 4 Number Rolls
1.5 Using a Hundreds Chart
Activity 1 Guess my number
Activity 2 Number neighbours
Activity 3 Missing numbers
Activity 4 Developing mental strategies
Assessment for common misunderstandings
Level 2: Place value
2.1 Number naming, 2.2 Efficient counting, 2.3 Sequencing, 2.4 Renaming
Maths300
Nine and over, Highest number, Human computer, Counting machines
MAV Resources
Teach for Understanding (Whole Numbers)
Place value p19
People count 3 Place value
Guidelines in Number p18, 36
Cornerstones in Number: Place value ages 6-9
Understanding the nature of place value (p38-43)
Modelling numbers (p44-61)
Extending numbers (p62-79)
Renaming numbers (p80-101)
Hands-on Maths: Developing mathematics with UNIFIX
Trading games and beyond (p48-76)
Hands-on Maths: Developing mathematics with BASE TEN
Trading games p16-32
The concept of place value
Paper abacus
Snake lines and sewing tapes
Cornerstones in Number: Place value ages 6-9 Modelling numbers (p44-61)
Hands-on Maths: Developing mathematics with BASE TEN Trading games (p14-33)
Hands-on Maths: Developing mathematics with UNIFIX Trading games and beyond (p48-76)
Practical teaching strategies for children with learning difficulties Book 5 Numbers to 100
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Wishball, Count us in [NW6C4U]
Count Us In – Games for basic number concepts http://www.abc.net.au/countusin/default.htm
Count Us IN – Game 11 the next highest number http://www.abc.net.au/countusin/games/game11.htm
Hang out the shirts on the washing line http://www.crickweb.co.uk/ks1numeracy.html#washindex
Learning about Number Relationships and Properties of Numbers Using Calculators and Hundred Boards
http://standards.nctm.org/document/eexamples/chap4/4.5/index.htm
Create a House Number http://illuminations.nctm.org/LessonDetail.aspx?id=L225
Counting Caterpillar http://www.ictgames.com/caterpillar_slider.html
Count Us In – Game 6 Matching words and numbers http://www.abc.net.au/countusin/games/game6.htm
MAV Resources
Spreadsheets from Interactive Learning Twenty-frames, Montessori cards, Odometer
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Early Years Number
3. Reading, writing, interpreting, and ordering three-digit numbers
[VELS 2.0]
Targets
Develop a sound understanding of place value up to 999. ‘Interpreting’ requires an awareness that
ten = 10 ones and that one hundred = 10 tens.
Evidence
Mathematics Online Interview Question 24
Activities (from Guidelines in Number)
Base ten blocks (flats, longs and minis)
Resources
Teacher references for activities
Continuum
2.25 Renaming three-digit whole numbers
Activity 1 Using pop sticks to rename
Activity 2 Using MAB to rename
Activity 3 Using a number expander to rename
Maths300
Highest number, Making 1000, Human computer, Counting machines
MAV Resources
People count 3 Place value
Guidelines in Number p31, 36, 53-55, 57-58, 79-82
The concept of place value
Paper abacus
Hands-on Maths: Developing mathematics with BASE TEN
Hundreds and thousands p33
Practical teaching strategies for children with learning difficulties
Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Place value to 1000, Wishball
Spreadsheets from Interactive Learning
Skip count to 9999, Skip count from 9999, Montessori cards, Odometer
Problem solving from Mathematics Task Centre
standards.nctm.org/document/eexamples/index.htm 4.5 and 4.6
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Early Years Number
4. Reading, writing, interpreting, and ordering numbers beyond
1000 [VELS 2.5]
Targets
Interpreting means understanding the meaning of these numbers. Montessori cards, showing the
actual value of each digit, may be very useful.
Evidence
Mathematics Online Interview Question 29
Activities
Resources
Teacher references for activities
Maths300
Making one million
MAV Resources
The concept of place value
Paper abacus
Snake lines and sewing tapes
Hands-on Maths: Developing mathematics with UNIFIX
Trading games and beyond (p48-76)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Wishball
MAV Resources
Spreadsheets from Interactive Learning
Skip count to 9999, Skip count from 9999, Odometer
Blue = free hyperlink, black = book, no hyperlink
21
Red = MAV product hyperlink
Early Years Number
5. Extending and applying place value knowledge [VELS 3.0]
Targets
Place value can be applied in many ways. One common application is counting money. The ideas of
round also depend on understanding place value; for example rounding two digit numbers to the
nearest ten, or rounding four digit numbers to the nearest thousand, hundred or ten. Estimation
activities depend on the ability to round numbers, and this applies also to the use of number lines.
Evidence
Mathematics Online Interview Question 35
Activities (from Guidelines in Number)
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Resources
Teacher references for activities
Maths300
Human computer
MAV Resources
People count 3 Place value
Guidelines in Number p153-156
The concept of place value
Paper abacus
Snake lines and sewing tapes
Student Resources (hands-on or computer)
Learning Objects (FUSE) Wishball
Spreadsheets from Interactive Learning
Skip count to 9999, Skip count from 9999, Montessori cads, Odometer
Early Years Number
Blue = free hyperlink, black = book, no hyperlink
C.
23
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Early Years Number
Strategies for Addition and Subtraction
1. Count all to add (two collections) [towards VELS 1.0]
Targets
Children usually start to add by recounting both numbers of objects as an entirely new set to be
counted. The next step is to build confidence and develop ‘counting on’. To add two numbers, the
quickest strategy is to start at the larger (requiring an understanding of order) and then count forward
the correct number of counts.
Evidence
Mathematics Online Interview Question 14
Activities
 Numbers to 10 using ‘10-frames’
A basic skill involves great familiarity with combinations that total 10 or less. The best tool for this is the 10frame. Students can explore the many combinations possible and learn about odd and even numbers.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
Blue = free hyperlink, black = book, no hyperlink
24
Red = MAV product hyperlink
Early Years Number
Resources
Teacher references for activities
Continuum
1.0 Counting groups of up to 20 objects
Activity 1 Verbal counting sequence
Activity 2 Efficient counting strategies
Activity 3 Hidden collections
1.0 Making better estimates
Activity 1 How many marbles?
Activity 2 Sharing strategies for checking estimates
Activity 3 Estimating calculations
Assessment for common misunderstandings
Level 1: Trusting the count
Subitising tool – with cards sets 1 to 5
Mental objects tool - with mental objects card
Mental computation: a strategies approach: Module 2 Basic facts (Addition, Subtraction)
Developing computation p14,15
MAV Resources
Teach for Understanding (Whole Numbers)
1 Add single digits, mentally
People count 4 Basic addition and subtraction
Guidelines in Number p11, 12
Practical teaching strategies for children with learning difficulties Book 4 (numbers to 20)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Number partner
Using Tens Frames to Build the Addition and Subtractions Facts to Ten
http://www.nzmaths.co.nz/resource/using-tens-frames-build-addition-and-subtraction-factsten?parent_node=
Lollies http://www.nzmaths.co.nz/resource/lollies?parent_node=??
Do it with dominoes 1 Counting to Find Sums http://illuminations.nctm.org/LessonDetail.aspx?id=U47
Do it with dominoes 2 Hopping on the Number Line http://illuminations.nctm.org/LessonDetail.aspx?ID=L53
Do it with dominoes 3 Exploring Adding with Sets http://illuminations.nctm.org/LessonDetail.aspx?ID=L54
Using Tens Frames to Build the Addition and Subtraction Facts to Ten
http://www.nzmaths.co.nz/resource/using-tens-frames-build-addition-and-subtraction-factsten?parent_node=
MAV Resources
Early Years Number Learn differences from 10 p28
Spreadsheets from Interactive Learning Subitising, Twenty-frames, Add to 20
Blue = free hyperlink, black = book, no hyperlink
25
Red = MAV product hyperlink
Early Years Number
2. Count on to add [towards VELS 1.5]
Targets
Develop improved strategies, such as ’Complements to ten’ (counting up to the next ten, and then
using the rest of the number being added as the ones digit), for example 24 + 8 is 24 + 6 (30) + 2
more to make 32.
Evidence
Mathematics Online Interview Question 16
Activities
 Numbers to 20, with ‘20-frames’
As a logical extension of ‘10-frames’, make use of ‘20-frames’. Use them for finding and understanding the
combinations of single digits that go over 10. These can be understood as ‘making up to 10’ and then ‘some
over’.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
 Tens and ones
Try to stress the place value idea that a number can be broken into tens and ones. It is the start of many
similar approaches with larger numbers, and is also the basis of subtraction when the process is reversed.
Here is an example: 8 + 7.
Find the first number, and count on. Note that you can think of 7 as 2 (to make 10 with the 8) and 5
Blue = free hyperlink, black = book, no hyperlink
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Red = MAV product hyperlink
Early Years Number
Start here
Although children can find the answers by counting, they should get used to ‘regrouping’ into ten and what is
left. Children can answer by colouring over the first number in one colour and the part added in another.
 Flashcards
Use flashcards if you have them. Confident automatic response to these additions is vital.
 Double any single digit.
Treat this as adding. Students should learn to recognise that doubling 5, 6, 7, 8 or 9 will produce a tens digit.
Instant recall of the pairs of numbers that add to 9 and to 10 is particularly useful later.
Resources
Teacher references for activities
Continuum
1.0 Counting groups of up to 20 objects,
Activity 1 Verbal counting sequence
Activity 2 Efficient counting strategies
Activity 3 Hidden collections
1.5 Counting on
Activity 1 Teddies in the buses
Activity 2 Counters in the bag and in the hand
Counting on- More About
http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/N
15002G.htm
1.5 Complements to Ten
Activity 1 Make ten
Activity 2 Using tens frames
Activity 3 Hiding dots
Activity 4 Find your partner
Activity 5 Bead frame patterns
Assessment for common misunderstandings
Level 1: Trusting the count
Subitising tool – with cards sets 1 to 5
Mental objects tool - with mental objects card
Mental computation: a strategies approach: Developing computation p14,15, 16
Module 2 Basic facts (Addition, Subtraction)
Do It with Dominoes
Lesson 1 Counting to Find Sums http://illuminations.nctm.org/LessonDetail.aspx?id=U47
Lesson 2 Hopping on the Number Line http://illuminations.nctm.org/LessonDetail.aspx?ID=L53
Lesson 3 Exploring Adding with Sets http://illuminations.nctm.org/LessonDetail.aspx?ID=L54
MAV Resources
Teach for Understanding (Whole Numbers)
1 Add single digits, mentally
People count 5 Place value, addition and subtraction
Guidelines in Number p11-12, 20-21
Practical teaching strategies for children with learning difficulties Book 4 (numbers to 20)
Blue = free hyperlink, black = book, no hyperlink
27
Red = MAV product hyperlink
Early Years Number
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Number trains [K5W9Q2], Number partner
Frogs on a Log: Finding One More than a Number
http://illuminations.nctm.org/LessonDetail.aspx?id=L867
A range of Number Activities for students:
http://www.copacabana-p.schools.nsw.edu.au/Get_Smart_Pages/Get_Smart_Maths_s1_Number.html
Test the toad – basic addition
http://www.bbc.co.uk/schools/numbertime/games/test.shtml
Lesson 1 – Finding Addition Patterns
http://illuminations.nctm.org/LessonDetail.aspx?ID=L97
Finding Sums to Six
http://illuminations.nctm.org/LessonDetail.aspx?ID=L98
Balancing Act- Preparation for writing equations
http://illuminations.nctm.org/LessonDetail.aspx?id=L166
Sum Search
http://illuminations.nctm.org/LessonDetail.aspx?id=L58
Using Tens Frames to Build the Addition and Subtraction Facts to Ten
http://www.nzmaths.co.nz/resource/using-tens-frames-build-addition-and-subtraction-factsten?parent_node=
That’s Odd http://www.nzmaths.co.nz/resource/thats-odd
Beetle Wheels – skip counting http://www.nzmaths.co.nz/resource/beetle-wheels
MAV Resources
Spreadsheets from Interactive Learning Adding to 20, Addition facts
Blue = free hyperlink, black = book, no hyperlink
28
Red = MAV product hyperlink
Early Years Number
3. Count back / count down to / count up from … to subtract
[towards VELS 2.0]
Targets
This is exactly the reverse of the process of adding two single digit numbers. This method stresses the idea
that a number in tens and ones can be broken into separate parts. It is the start of many similar approaches
with larger numbers, and is also the basis of subtraction when the process is reversed.
Evidence
Mathematics Online Interview Question 17
Activities
 Subtract using only numbers to 10 using ‘10-frames’
A basic skill involves great familiarity with combinations that total 10 or less. The best tool for this is the 10frame. Students can learn to ‘see’ many combinations and represent them as additions or subtractions.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
This shows 9 – 4 = 5
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
This shows 9 – 5 = 4
 Find differences by counting backwards
Here is an example: 16 – 9. Start at the first number. Count backwards the amount of the second number.
Note that you can count back to 10 by taking off the amount of the ones digit, then take off the rest.
Start here
 Learn differences from 10
Because taking single digits from the ‘teens’ can be done by subtracting back to 10 then from 10 (see 4
below), memorising differences from 10 is an essential skill.
Blue = free hyperlink, black = book, no hyperlink
29
Red = MAV product hyperlink
Early Years Number
 Subtract from numbers to 20, with ‘20-frames’
As a logical extension of ‘10-frames’, make use of ‘20-frames’. Use them for finding and understanding the
subtractions that go under 10 when taking single digits from the ‘teens’. This is done by breaking the number
being subtracted into two parts.
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.
 Use flashcards if you have them.
Resources
Teacher references for activities
Continuum
1.75 Fact families - addition & subtraction
Activity 1 Fact families using materials
Activity 2 Domino fact families
Activity 3 Dice fact families
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 2 Basic facts (Addition, Subtraction)
MAV Resources
Teach for Understanding (Whole Numbers)
2 Subtract a single digit from a number between 10 and 18, mentally
People count 5 Place value, addition and subtraction
Guidelines in Number p12, 22-23
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Wishball, Number partner
Problem solving from the Mathematics Task Centre
4 Window Frames, 7 Consecutive Sums, 9 Row Points, 15 Domino Trails, 17 Truth Tiles 2,
30 Truth Tiles, 35 Crosses, 39 Criss-Cross Numbers, 45 Eric The Sheep, 52 Which Floor,
56 Challenge, 75 What's It Worth? 99 How Many Beans? 120 Nim, 127 Highest Number 1,
211 Soft Drink Crates, 218 Guessing Colours Game
MAV Resources
Spreadsheets from Interactive Learning
Subtract from 20, Subtraction facts, Backwards adding
Take Away
http://illuminations.nctm.org/LessonDetail.aspx?id=L192
Comparing Connecting Cubes
http://illuminations.nctm.org/LessonDetail.aspx?id=U41
Lesson 1 Counting Back and Counting on
http://illuminations.nctm.org/LessonDetail.aspx?ID=L35
Lesson 2 Comparing Sets
http://illuminations.nctm.org/LessonDetail.aspx?ID=L36
Blue = free hyperlink, black = book, no hyperlink
30
Red = MAV product hyperlink
Lesson 3 Using the Number Line to Compare
http://illuminations.nctm.org/LessonDetail.aspx?ID=L37
Lesson 4 Balancing
http://illuminations.nctm.org/LessonDetail.aspx?ID=L40
Lesson 5 Fact Families
http://illuminations.nctm.org/LessonDetail.aspx?ID=L41
Lesson 6 Comparing Connecting Cubes
http://illuminations.nctm.org/LessonDetail.aspx?ID=L43
Subtraction Game
http://www.nzmaths.co.nz/resource/subtraction-game-0?parent_node=
Subtraction Facts
http://www.nzmaths.co.nz/taxonomy/term/221
Five Models of Subtraction
Lesson 1 Counting Back and Counting On
http://illuminations.nctm.org/LessonDetail.aspx?ID=L103
Lesson 2 Taking Away Sets
http://illuminations.nctm.org/LessonDetail.aspx?ID=L104
Lesson 3 Hopping Backward on the Number Line
http://illuminations.nctm.org/LessonDetail.aspx?ID=L105
Finding the Balance
http://illuminations.nctm.org/LessonDetail.aspx?ID=L106
Finding Fact Families
http://illuminations.nctm.org/LessonDetail.aspx?ID=L107
Practice Makes Perfect
http://illuminations.nctm.org/LessonDetail.aspx?ID=L108
Looking Back and Moving Forward
http://illuminations.nctm.org/LessonDetail.aspx?ID=L109
Early Years Number
Blue = free hyperlink, black = book, no hyperlink
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Red = MAV product hyperlink
Early Years Number
4. Basic strategies for adding or subtracting (doubles,
commutativity, adding 10, tens facts, other known facts)
[towards VELS 2.5]
Targets
Build confidence with mental computation, by the memorisation of a basic set of facts (already
understood) and their extensions.
‘Commutativity’ means that adding can be done in either order, and this is the essence of the
flexibility for adding. For example 36 + 58: tens first (80 then add 14); ones first (14 then add 80);
mixed order (e.g. 36 + 8 + 50); shifting numbers about (e.g. add 34 + 60, or 40 + 54).
Evidence
Mathematics Online Interview Questions 21, 25
Activities
Adding
Blue = free hyperlink, black = book, no hyperlink
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Red = MAV product hyperlink
Early Years Number
 Use a number chart
Here is an example: 25 + 8. Start at the first number. Count the second number from the end of the first.
Start here
 Lead towards an appreciate of the ‘extensions’ of each basic addition fact
The idea is that children will recognise patterns like these:
5 + 6 = 11
6 + 8 = 14
7 + 6 = 13
8 + 9 = 17
15 + 6 = 21
16 + 8 = 24
17 + 6 = 23
18 + 9 = 27
25 + 6 = 31
26 + 8 = 34
27 + 6 = 33
28 + 9 = 37
35 + 6 = 41
36 + 8 = 44
37 + 6 = 43
38 + 9 = 47
and so on. If you know one, you also know the others.
 Use flashcards if you have them.
Subtracting

Bundles of sticks and Base 10 blocks
We are subtracting single digit numbers from double-digit numbers. There are two types of problem:
simple ones without any regrouping, and those involving regrouping of bundles of tens (or longs).
We are also subtracting double-digit numbers from double-digit numbers without any regrouping,
Blue = free hyperlink, black = book, no hyperlink

33
Red = MAV product hyperlink
Early Years Number
Mental computation
This time the process is similar, but there are some extra tens involved. There are two cases, one when
there is no ‘carrying’ (no ten needs to be regrouped) and one when we regroup a ten so we can take away
the ones. Here is an example: 35 – 8.
Start with the first number and count backwards.
Note that the 8 can be split into 5 (to take from 35) and 3 (to take from 30 leaving 27).
Start here
 Lead towards an appreciation of the ‘extensions’ of each basic subtraction fact
The idea is that children will recognise patterns like these:
15 – 6 = 9
16 – 8 = 8
17 – 9 =
8
18 – 9 = 7
25 – 6 = 19
26 – 8 = 18
27 – 9 = 18
28 – 9 = 17
35 – 6 = 29
36 – 8 = 28
37 – 9 = 28
38 – 9 = 27
45 – 6 = 39
46 – 8 = 38
47 – 9 = 38
48 – 9 = 37
and so on. If you know one, you also know the others.
Use flashcards if you have them. Quick mental response is very valuable.
Resources
Teacher references for activities
Continuum
1.5 Counting on
Activity 1 Teddies in the buses
Activity 2 Counters in the bag and in the hand
1.5 Complements to Ten part 2
Activity 1 Make ten
Activity 2 How many more to make ten
Activity 3 Hidden dots
Activity 4 Find your partner
Activity 5 Bead frame patterns
1.5 Using a hundreds chart for mental calculation
Activity 1 Guess my number
Activity 2 Number neighbours
Activity 3 Missing numbers
Activity 4 Challenges
1.75 Fact families (Addition and subtraction)
Activity 1 Fact families using materials
Activity 2 Domino fact families
Activity 3 Dice fact families
2.0 Skip counting
Activity 1 Counting games
Activity 2 Whisper count
Activity 3 How far can you go?
Activity 4 Using the hundreds grid for counting
Maths300
Take away of the day
Blue = free hyperlink, black = book, no hyperlink
34
Red = MAV product hyperlink
Early Years Number
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 2 Basic facts (Addition, Subtraction)
Module 4 Two-digit whole numbers
MAV Resources
Teach for Understanding (Whole Numbers)
9 Add a one-digit number to a two-digit number, mentally
10 Subtract a one-digit number from a two-digit number, mentally
People count 5 Place value, addition and subtraction
Guidelines in Number p15-18, 35-36, 53-55, 57-58
Add and subtract whole numbers with MAB
Snake line and sewing tape
MAV-money
Paper abacus
Hands-on Maths: Developing mathematics with BASE TEN
Addition (p48-53)
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Wishball, The number partner
Spreadsheets from Interactive Learning
Adding single digits, Skip count to 140, Skip count to 9999, Grid patterns,
Add with diagrams, Subtract with diagrams
Blue = free hyperlink, black = book, no hyperlink
35
Red = MAV product hyperlink
Early Years Number
5. Derived strategies for adding or subtracting (near doubles,
adding 9, build to next ten, fact families, intuitive strategies)
[VELS 2.5]
Targets
Build awareness of fact families, patterns and other strategies for adding and subtracting.
Evidence
Mathematics Online Interview Question 30
Activities
Adding
 Use bundles of sticks
 Use Base 10 blocks
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36
Red = MAV product hyperlink
Early Years Number
 Mental computation
Here is an example: 25 + 38.
a Add the tens first (20 + 30 = 50).
b Then add the two ones digits, which should be a quick mental calculation at this stage (5 + 8 = 13).
c These will be in addition to the 50 we have already, so that makes 63.
 Doubling
Doubling any single tens digit.
Students should learn to recognise that doubling 50, 60, 70, 80, 90 will produce a hundreds digit.
Doubling small two-digit numbers,
e.g. double 13 (for 20 + 6 = 26), 16 (for 20 + 12 = 32)
Students should learn to recognise that doubling numbers ending in 5 or more etc. will produce an extra tens
digit. Extend this to doubling any two digit number.
 Use flashcards if you have them. Aim for quick mental response.
Blue = free hyperlink, black = book, no hyperlink
Subtracting
• Use bundles of sticks.
 Use Base 10 blocks
37
Red = MAV product hyperlink
Early Years Number
Blue = free hyperlink, black = book, no hyperlink

38
Red = MAV product hyperlink
Early Years Number
Mental computation
Mostly this is very similar to the situation above. As well as subtracting the ones, we also subtract tens. It is
important to recognise that when operating mentally with a ‘regrouping question’ (such as 45 – 28) very few
people actually subtract the ones first. Many people use alternatives that feel comfortable to them. For
example, here are three legitimate strategies with 45 – 28:
a subtract 20 first to get 25, then take 8 to get 17.
b subtract the 8 first to get 37, then subtract 20.
c add 2 to both numbers to change the question to 47 – 30 = 17. (This strategy makes the subtracted
number up to a multiple of ten. In so doing the difficulty with the ones disappears.)
 Here is an example for strategy (a): 45 – 28.
a Start with 45. First take away 20 to get 25.
b Split the 45 into 30 and 15, in your head.
c Take the 8 from the 25, by splitting 8 into two parts (5 and 3)
so 25 – 8 = 25 – 5 – 3 = 20 – 3 = 17.
 Use flashcards if you have them.
Resources
Teacher references for activities
Continuum
1.5 Counting on
Activity 1 Teddies in the buses
Activity 2 Counters in the bag and in the hand
1.5 Complements to Ten part 2
Activity 1 Make ten
Activity 2 How many more to make ten
Activity 3 Hidden dots
Activity 4 Find your partner
Activity 5 Bead frame patterns
1.5 Using a hundreds chart for mental calculation
Activity 1 Guess my number
Activity 2 Number neighbours
Activity 3 Missing numbers
Activity 4 Challenges
1.75 Fact families (Addition and subtraction)
Activity 1 Fact families using materials
Activity 2 Domino fact families
Activity 3 Dice fact families
Blue = free hyperlink, black = book, no hyperlink
39
Red = MAV product hyperlink
Early Years Number
2.0 Skip counting
Activity 1 Counting games
Activity 2 Whisper count
Activity 3 How far can you go?
Activity 4 Using the hundreds grid for counting
Maths300
Domino trails
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 2 Basic facts (Addition, Subtraction)
Module 4 Two-digit whole numbers
MAV Resources
Teach for Understanding (Whole Numbers)
11 Add two-digit numbers, mentally
12 Subtract two-digit numbers, mentally
People count 5 Place value, addition and subtraction
Guidelines in Number p22-23, 42-43
Add and subtract whole numbers with MAB
Snake line and sewing tape
MAV-money
Paper abacus
Cornerstones in Number: Place value ages 6-9
Adding and subtracting tens and hundreds (p102-112)
Hands-on Maths: Developing mathematics with BASE TEN
Addition and subtraction (p48-58)
Practical teaching strategies for children with learning difficulties
Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Number partner, Take away bars, Difference bars
MAV Resources
Spreadsheets from Interactive Learning
Add with diagrams, Subtract with diagrams, Subtract from 100, Skip count back (140),
Skip count from 9999, Odometer, Change
Blue = free hyperlink, black = book, no hyperlink
40
Red = MAV product hyperlink
Early Years Number
6. Extending and applying addition and subtraction using basic,
derived and intuitive strategies mentally [VELS 3.0]
Targets
Using the mental image of the 100 chart, developed from its frequent use in the classroom, students
should be able to add and subtract numbers fluently.
Evidence
Mathematics Online Interview Question 36
Activities
 Teach the extensions
The idea is that children will recognise patterns like these: if the ones digits go over 10, then add an extra 10.
Add any tens:
5 + 6 = 11
6 + 8 = 14
7 + 6 = 13
8 + 9 = 17
15 + 26 = 41
16 + 48 = 64
67 + 26 = 93
18 + 79 = 97
35 + 56 = 91
56 + 28 = 84
57 + 26 = 83
28 + 69 = 97
45 + 36 = 81
36 + 48 = 84
47 + 36 = 83
38 + 59 = 97
and so on. If you know one, you also know the others.
 Play the dice game.
Each child rolls a die (6 sided or 10 sided) and adds that number of ones to their current total.
This means that single digit numbers are being added, mentally. If any child needs help, use the number
chart or the material.
The first child to reach 100 or more wins the game.
Subtracting
 Teach the extensions
The idea is that children will recognise patterns like these:
15 – 6 = 9
16 – 8 = 8
17 – 9 = 8
18 – 9 = 7
55 – 16 = 19
96 – 18 = 78
87 – 59 = 18
78 – 59 = 17
65 – 26 = 29
86 – 28 = 58
67 – 49 = 18
58 – 19 = 27
75 – 36 = 39
76 – 38 = 38
47 – 39 = 18
38 – 29 = 7
and so on. If you know one, you also know the others.
 Play the dice game, backwards.
Each child starts with 100. On their turn they roll a die (6 sided or 10 sided) and take that number of ones
from their current total. This means that single digit numbers are being subtracted, mentally. If any child
needs help, use the number chart. The first child to reach 0 or less wins the game.
 Skip counting backwards
Children should be able to skip count, starting from any number and counting backwards in steps of any
single digit number. (Example 55, 48, 41, 34, 27,...)
This should be done with a number chart (to check answers). It will assist with addition and subtraction as it
reinforces the regrouping that occurs at each ‘ten’.
The tough part of skip counting backwards is going back past the previous ten.
Suppose we are counting backwards by 3s. So 45 – 3 = 42 is easy, but 42 – 3 = 39 is much harder.
Each case should be seen as an extension of the idea between 0 and 20.
For example, 42 – 3 is just an extension of 12 – 3 = 9, so 42 – 3 = 39.
Blue = free hyperlink, black = book, no hyperlink
41
Red = MAV product hyperlink
Early Years Number
This idea is usefully shown on a number line.
Resources
Teacher references for activities
Continuum
2.0 Flexible addition and subtraction
Activity 1 Choosing which order to add
Activity 2 Strengthening visual images for mental computation
Activity 3 Games for number fluency
Activity 4 Mixing addition and subtraction
Maths300
14 The Farmer's Puzzle, 17 Eric The Sheep
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 2 Basic facts (Addition, Subtraction)
Module 4 Two-digit whole numbers
MAV Resources
Teach for Understanding (Whole Numbers)
11 Add two-digit numbers, mentally
12 Subtract two-digit numbers, mentally
People count 5 Place value, addition and subtraction
Guidelines in Number p40-43, 62-65
RIME 5&6 Words add up
Add and subtract whole numbers with MAB
Snake line and sewing tape
MAV-money
Paper abacus
Hands-on Maths: Developing mathematics with BASE TEN
Addition and subtraction (p48-58)
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Part-adder, Number partner, Take-away bars, Difference bars
Problem solving from Mathematics Task Centre
4 Window Frames, 7 Consecutive Sums, 9 Row Points, 15 Domino Trails, 17 Truth Tiles 2,
30 Truth Tiles, 35 Crosses, 39 Criss-Cross Numbers, 45 Eric The Sheep, 52 Which Floor,
56 Challenge, 120 Nim, 127 Highest Number 1, 211 Soft Drink Crates, 218 Guessing
Colours Game
MAV Resources
Spreadsheets from Interactive Learning
Adding to 100, Doubling, Add with diagrams, Subtract with diagrams, Subtract any from 100,
Subtract by adding, Subtraction strategies , Three circles puzzle, Checking change
Blue = free hyperlink, black = book, no hyperlink
42
Red = MAV product hyperlink
Early Years Number
D. Strategies for Multiplication and Division
1. Counting grouped items by counting all items [VELS 2.0]
Targets
At this level students are barely aware of the several copies of a single set of objects, e.g. 4 copies
of a set of 3 (see below). It includes equal grouping to make counting easier – counting by twos,
threes, etc.
Evidence
Mathematics Online Interview Question 15
Activities
See Counting. Also
Blue = free hyperlink, black = book, no hyperlink
43
Red = MAV product hyperlink
Early Years Number
Resources
Teacher references for activities
Continuum
1.0 Counting groups of up to 20 objects,
Activity 1 Verbal counting sequence
Activity 2 Efficient counting strategies
Activity 3 Hidden collections
Assessment for Common Misunderstandings
Level 3: Multiplicative thinking
Mental computation: a strategies approach: Developing computation p17-24, 29-38
MAV Resources
Teach for Understanding (Whole Numbers)
3 The meaning of multiplication
People count 6 Basic multiplication
Guidelines in Number p13
Snake line and sewing tape
Practical teaching strategies for children with learning difficulties Book 4 (numbers to 20)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Count and compare, Number trains
Seeing Doubles
http://illuminations.nctm.org/LessonDetail.aspx?id=L875
Smart Doubling
http://www.nzmaths.co.nz/resource/smart-doubling?parent_node=
Marbles
http://www.nzmaths.co.nz/resource/marbles?parent_node=
Dividing Dough
http://www.nzmaths.co.nz/resource/dividing-dough
Baby Weight
http://illuminations.nctm.org/LessonDetail.aspx?id=L170
Count Us In- Game 13 Halves
http://www.abc.net.au/countusin/games/game13.htm
MAV Resources
Spreadsheets from Interactive Learning Doubling, Groups of (small), Dividing, Finding half
Blue = free hyperlink, black = book, no hyperlink
44
Red = MAV product hyperlink
Early Years Number
2. Modelling multiplication and division (all objects perceived)
[VELS 2.5]
Targets
Multiplying means the use of repeated copies of the same set. The first number in a multiplication
means the number of copies, the symbol x means ‘lots of’, and the second number is the size of the
set. So 7 x 3 means 7 lots of 3. It is better to say ‘lots of’ and not ‘times’.
There is a close link between multiplying and dividing. We can think of dividing as finding a missing
number in a multiplying problem. There are two possible ways to do this. For example,
21 objects are shared fairly by 7 people. How much do they get each? 7 x ∆ = 21.
how many lots of 3 makes a total of 21? This is equivalent to ∆ x 3 = 21, since the first number tells us how
many lots.
Evidence
Mathematics Online Interview Question 26
Activities
Focus on the meaning of multiplication
Why do we say that 7 times 3 makes 21? The 7 tells us how many 3s to make; it is called the multiplier. The
things may be arranged randomly, or in piles or in neat rows.
The arrangement in an array makes it clear that 7 lots of 3 is the same as 3 lots of 7, and we need to make
good use of this to help students learn the facts.
However the answer always comes as a number of tens and ones, so place value should always be
stressed.
Focus on the two meanings of division
The two meanings of division
There are two meanings for division: sharing and ‘how many?’.
For example, 12 ÷ 3 can mean
‘12 things shared equally between 3’
or
‘how many 3s are needed to make 12?’.
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Early Years Number
Each corresponds to the same pair of multiplication facts: 3 x 4 or 4 x 3.
This is clearer with a rectangular array. Here the rows divide the number into 3 equal parts, and the columns
show how many 3s there are.
The length and width represent the numbers that can be multiplied. These are given the special
mathematical name of factors. The factors of 12 shown here are 3 and 4, because 3 x 4 = 12. We can also
show that 2 and 6 are factors of 12, and so are 1 and 12.
Resources
Teacher references for activities
Continuum
2.25 Early division ideas
Activity 1 Modelling quotition
Activity 2 A first look at remainders
Activity 3 Number line and repeated subtraction
2.75 Better multiplication strategies
Activity 1 How many chocolates?
Activity 2 Partially covered arrays
2.75 Fact families for multiplication and division
Activity 1 fact families from arrays
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Early Years Number
Activity 2 Recognising different fact families
Activity 3 Fact family fortune
Activity 4 fact family bonanza
Assessment for Common Misunderstandings
Level 3: Multiplicative thinking
Maths300
Spiders and ants,
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 3 Basic facts (Multiplication, Division)
MAV Resources
Teach for Understanding (Whole Numbers)
4 Multiply by single digits
People count
6 Basic multiplication
7 Basic division
Guidelines in Number p13, 24
Snake lines and sewing tape
Multiplication and division with Base ten blocks
Hands-on Maths: Developing mathematics with BASE TEN
Multiplication and division (p59-67)
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
The multiplier, Pebble array, Arrays
Basic Facts
http://www.nzmaths.co.nz/taxonomy/term/196
MAV Resources
Spreadsheets from Interactive Learning Array count, Groups of (small), Groups of
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Early Years Number
3. Abstracting multiplication and division [VELS 2.5]
Targets
In this context ‘abstracting’ means that children are able to recognise that both multiplying and
dividing involve situations with several copies of equal sets. There are several types of situations that
model this, such as arrays and identical sets spread out.
‘Abstracting’ does not mean working with abstract expressions. Instead it means that students
should be learning to choose the correct expression for a given practical computation situation. In
particular they need to be able to recognise when multiple copies of equal sets are used in the
problem, and choose multiplication (to find the total) or division (to find wither the number of sets or
the number in each set).
Evidence
Mathematics Online Interview Question 27
Activities
Multiplying
 Discovering multiplication facts
Here is an example: 6 x 7.
a Arrange 6 lots of 7 in a rectangle. Cuisenaire rods are good for this. This is a concrete representation of
the situation. (See diagram next page.)
b Its value should be found by swapping for tens and ones.
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Early Years Number
c Its value can also be found using the skip counting skills previously developed: starting from 0 this
produces the ‘7 times table’ facts. This is an abstract representation.
 Use chart and line
The same problem (e.g. 6 x 7) should be seen on the number line.
Placing Cuisenaire rods (7s) along a number line links the two representations.
Dividing
Recall that division has two meanings
Resources
Teacher references for activities
Continuum
2.25 Early division ideas
Activity 1 Modelling quotition
Activity 2 A first look at remainders
Activity 3 Number line and repeated subtraction
Assessment for common misunderstandings
Level 3: Multiplicative thinking
www.education.vic.gov.au/studentlearning/teachingResources/maths/common/commisslvl3.htm
Maths300
17 Eric The Sheep, 97 Tackling Times Tables
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Early Years Number
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 3 Basic facts (Multiplication, Division)
MAV Resources
Teach for Understanding (Whole Numbers)
5 Division – reverse multiplication, mentally
People count
6 Basic multiplication,
7 Basic division
Guidelines in Number p13, 25-28, 45-46, 66
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
The multiplier, Pebble array, Arrays
Problem solving from Mathematics Task Centre
4 Window Frames, 7 Consecutive Sums, 9 Row Points, 15 Domino Trails, 17 Truth Tiles 2,
30 Truth Tiles, 35 Crosses, 39 Criss-Cross Numbers, 45 Eric The Sheep, 52 Which Floor,
56 Challenge, 75 What's It Worth? 99 How Many Beans? 120 Nim, 127 Highest Number 1,
211 Soft Drink Crates, 218 Guessing Colours Game
MAV Resources
Spreadsheets from Interactive Learning
Array count, Tables, Multiplication facts, Missing numbers, Multiplying,
Number line multiplying, Multo Random, Multiple patterns
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Early Years Number
4. Basic, derived and intuitive strategies for multiplication
(such as commutativity, skip counting and building up from
known facts) [VELS 3.0]
Targets
Commutativity (such as 4 x 7 = 7 x 4) is quite useful to reduce the memory load for tables; remember
one and you have both!
Skip counting (from 0) deals with sets of multiples (e.g. 6, 12, 18, 24, 35, 42…) and clearly helps with
multiplication understanding and answers.
‘Building up from known facts’ is essentially using the distributive law. For 7 x 6, many students add
6 to the memorised square number 36. Others use 5 x 6 (known from clocks) and add 2 x 6.
Evidence
Mathematics Online Interview Question 31
Activities
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 Combining facts
The value of 6 x 7 can be found by adding known facts. For many students they will be a useful aid to
memory.
 Combining facts to make new facts
Encourage students to use various alternatives to assist in recall. Most will know 1 x, 2 x and 5 x, so learn to
combine them. For example 7 x 8 is 2 x 8 + 5 x 8 = 16 + 40 = 56.
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Early Years Number
Resources
Teacher references for activities
Continuum
2.75 Better multiplication strategies
Activity 1 How many chocolates?
Activity 2 Partially covered arrays
2.75 Fact families for multiplication and division
Activity 1 fact families from arrays
Activity 2 Recognising different fact families
Activity 3 Fact family fortune
Activity 4 fact family bonanza
Assessment for common misunderstandings
Level 3: Multiplicative thinking
Maths300
Cookie count, Tables for 25
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 3 Basic facts (Multiplication, Division)
Module 4 Two-digit whole numbers
MAV Resources
Teach for Understanding (Whole Numbers)
8 Multiply single digits (tables)
People count 6 Basic multiplication,
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Early Years Number
Guidelines in Number p24, 44, 65
Snake lines and sewing tape
Practical teaching strategies for children with learning difficulties Book 5 (numbers to 100)
Student Resources (hands-on or computer)
Learning Objects (FUSE)
The multiplier, Pebble arrays, Arrays
Problem solving from Mathematics Task Centre
4 Window Frames, 7 Consecutive Sums, 9 Row Points, 15 Domino Trails, 17 Truth Tiles 2,
30 Truth Tiles, 35 Crosses, 39 Criss-Cross Numbers, 45 Eric The Sheep, 52 Which Floor,
56 Challenge, 75 What's It Worth? 99 How Many Beans? 120 Nim, 127 Highest Number 1,
211 Soft Drink Crates, 218 Guessing Colours Game
MAV Resources
Spreadsheets from Interactive Learning
Array count, Array Multiply, Tables, Multiplication facts, Missing numbers, Multiplying,
Number line multiplying, Multo Random, Multiple patterns, Multiply by 1 or 2 digits
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Early Years Number
5. Basic, derived and intuitive strategies for division
(such as fact families) [VELS 3.0]
Targets
Build flexibility in use of multiplication facts backwards to achieve division. For example, 56 ÷ 7 might
be done by breaking the 56 into 49 (known as 7 x 7) and another 7, hence 8 lots of 7. Another split is
set out below.
Evidence
Mathematics Online Interview Question 32
Activities
Sharing
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Early Years Number
How many groups?
Resources
Teacher references for activities
Continuum
2.25 Early division ideas
Activity 1 Modelling quotition
Activity 2 A first look at remainders
Activity 3 Number line and repeated subtraction
2.75 Fact families for multiplication and division
Activity 1 fact families from arrays
Activity 2 Recognising different fact families
Activity 3 Fact family fortune
Activity 4 fact family bonanza
Assessment for Common Misunderstandings
Level 3: Multiplicative thinking
Maths300
51 Hunting For Stars
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 3 Basic facts (Multiplication, Division)
Module 4 Two-digit whole numbers
MAV Resources
Teach for Understanding (Whole Numbers)
19 Divide with remainders, mentally
People count 7 Basic division
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Early Years Number
Guidelines in Number p13, 25-28, 45-46, 66
Add links to hands-on activities in number
People count (Teacher reference) 6 Basic multiplication, 7 Basic division
Student Resources (hands-on or computer)
Learning Objects (FUSE)
The divider, Divide it up
Spreadsheets from Interactive Learning
Dividing up to 100, Division facts, 10 or 100 times what? Backwards multiplying,
Common factor, Divide by 1 or 2 digits
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Early Years Number
6. Extending and applying multiplication and division [VELS 3.0]
Targets
Develop awareness of division as the inverse of multiplication, both to find how many groups
(sharing) and to find the size of each group (quotition).
Develop awareness of the relationships between various multiplication (and division) facts; for
example that since 2 x 3 = 6, and 5 x 3 = 15, then 7 x 3 = 6 + 15 = 21.
Evidence
Mathematics Online Interview Question 37
Activities
 Encourage students to use various alternatives to assist in recall.
These are all based on the Distributive law: you can split the number into two parts, divide each of them, and
add the results.
For example 56 ÷ 8 is 40 ÷ 8 + 16 ÷ 8 = 5 + 2 = 7.
 Use arrays to find the pairs of factors for a number.
Good numbers to use are those with many pairs of factors, such as 6, 12, 16, 18, 20, 24, 30.
For example, 12 = 3 x 4, but also 4 x 3, 6 x 2, 2 x 6, 12 x 1 and 1 x 12
 Use flashcards for division facts quick response.
 Find divisors – all the whole numbers that divide exactly into a number.
 Teach about common factors.
A common factor is a number that divides into two different numbers, such as 3 - a common factor of 6 and
9. The number 1 divides into everything, so in the case of 6 and 9, 3 is the highest common factor. This is
shown in the combined array; there are 3 rows in the array for 6 and 3 rows in the array for 9.
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Early Years Number
Resources
Teacher references for activities
Continuum
2.75 Better multiplication strategies
Activity 1 How many chocolates?
Activity 2 Partially covered arrays
2.75 Fact families for multiplication and division
Activity 1 fact families from arrays
Activity 2 Recognising different fact families
Activity 3 Fact family fortune
Activity 4 fact family bonanza
Maths300
Cookie count, Tables for 25
Problem solving from Mathematics Task Centre
Window frames, Consecutive sums, Row points, Domino trails, Truth tiles, Truth tiles 2,
Crosses, Criss-cross numbers, Eric the sheep, What’s it worth? Soft drink crates
Mental computation: a strategies approach: Developing computation p17-24, 29-38
Module 3 Basic facts (Multiplication, Division)
Module 4 Two-digit whole numbers
MAV Resources
Teach for Understanding (Whole Numbers)
19 Divide with remainders, mentally
22 Divide by one digit
People count
6 Basic multiplication
7 Basic division
Guidelines in Number p87-88, 131-137
Multiply and divide with Base ten blocks
Student Resources (hands-on or computer)
Learning Objects (FUSE)
Divide it up, Integer cruncher, Hopper, School canteen
MAV Resources
Spreadsheets from Interactive Learning
Multiply by 1 or 2 digits, Divide by 1 or 2 digits, Number line multiplying, Division in pictures,
Energy use