Teaching Number/Algebra to
Students in NE-Y8
From: Nicky Knight: The Education Group Ltd
“The world-wide focus on numeracy has highlighted the importance of highquality mathematics programmes, which emphasis both numerical knowledge
and advanced mental strategies.” (Howard Fancy, New Zealand Secretary for
Education, Curriculum Update 45: The Numeracy Story, 2001, revisited in Update 55, 2005).
Numeracy teaching and learning is focused on developing student's understanding of
numbers, and their ability to use numbers to solve problems. Students may solve number
problems by counting, adding, subtracting, multiplying, dividing, or combinations of these.
During mathematics classes students should be learning to:
o
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o
o
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enjoy working with numbers
make sense of numbers - how big they are, how they relate to other numbers, and how
they behave
solve mathematical problems - whether real life or imaginary
calculate in their heads whenever possible, rather than using a calculator or pen and
paper
show that they understand mathematics, using equipment, diagrams and pictures
explain and record the methods they use to work out problems
accept challenges and work at levels that stretch them
work with others and by themselves
discuss how they tackle mathematical problems - with other students, their teacher and
you!
Students are encouraged to learn a range of different ways to solve problems and to choose
the most appropriate one for each problem. You may be familiar with certain 'rules' for doing
mathematics. While these will still work, your child may learn different ways to solve
problems. Often these methods involve mental strategies, or working things out in your head,
rather than written methods.
WHY?
This change in approach to mathematics education reflects changes in the world that impact
on the mathematics that people need to know. Employers are increasingly looking for staff
who have problem solving skills and an understanding of concepts, rather than just the ability
to follow rules for calculating. The increasing use of technology has also meant that a
calculator or computer is almost always available in the workplace for larger calculations.
1
THE NUMBER FRAMEWORK
The Number Framework is intended to help teachers, parents and students understand the
stages of learning of number knowledge and strategies.
There are two sections to the Number Framework. The Strategy section describes the
processes students use to solve problems involving numbers - how they work things out. The
Knowledge section describes the key items about number that children know and can recall
quickly. The two sections are linked, with children requiring knowledge to improve their
strategies, and using strategies to develop new knowledge.
NATIONAL STANDARDS
National standards in Mathematics for primary school students outline standards that students
are expected to reach by the end of a child’s first, second and third year at school and then
subsequently by the end of years four, five, six, seven and eight. The table below outlines the
standards.
Ministry of Education: National Standards in Mathematics
Age of student
Numeracy Stage
Curriculum level and sublevel equivalent
After one year at school
Stage 2/3
Level 1 Basic
After two years at school
Stage 4
Level 1 Proficient or
Advanced
After three years at school
Early stage 5
Level 2 Basic
Year 4
Stage 5
Level 2 Proficient or
Advanced
Year 5
Early stage 6
Level 3 Basic
Year 6
Stage 6
Level 3 Proficient or
Advanced
Year 7
Early stage 7
Level 4 Basic
Year 8
Stage 7
Level 4 Proficient or
Advanced
2
The Number Framework Strategy Stages
NE to Y8
Numeracy Strategy Stage and Behavioural Indicators
Emergent
0
Addition/Subtraction: The student has no reliable strategy to count an
unstructured collection of items.
1
Addition/Subtraction: The student has a reliable strategy to count an
unstructured collection of items.
One to One Counting
Counting from One on Materials
2
Addition/Subtraction: The student’s most advanced strategy is counting
from one on materials to solve addition problems.
Multiplication/Division: The student’s most advanced strategy is counting
from one on materials to solve addition problems.
Fractions: The student’s is able to divide a region or set into equal parts
using materials.
Counting from One by Imaging
3
Addition/Subtraction: The student’s most advanced strategy is counting
from one without materials to solve addition problems.
Multiplication/Division: The student’s most advanced strategy is counting
from one without materials to solve addition problems.
Fractions: The student’s is able to divide a region or set into equal parts
using imaging.
Advanced Counting
4
Addition/Subtraction: The student’s the most advanced strategy is
counting-on/counting-back to solve addition or subtraction tasks.
Multiplication/Division: On multiplication tasks the student uses skip
counting.
Fractions: The student is able to use symmetry to create halves, quarters,
eighths etc. of a set.
Early Additive Part-Whole Thinking
5
Addition/Subtraction: The student shows any Part-Whole strategy
to solve addition or subtraction problems mentally by reasoning the
answer from known basic facts/place value.
Multiplication/Division: On multiplication tasks the student uses a
combination known multiplication facts and repeated addition e.g.
4 x 6 = (6 + 6) + (6 + 6) = 12 + 12 = 24.
Fractions: The student finds a unit fraction of a number by trial
and improvement e.g.
, 12 ÷ 3 ≠ 3
because 3 + 3 + 3 = 9, but
6
because 4 + 4 + 4 = 12
Advanced Additive Part-Whole Thinking
Addition/Subtraction: The student is able to use at least two
3
different mental strategies to solve addition or subtraction problems
with multi-digit numbers.
Multiplication/Division: The student uses known facts to derive
answers to multiplication and division problems
E.g. 4 x 6 = 2 x 12 = 24. E.g. 9 x 6 = 10 x 6 - 6 = 54
Fractions: The student uses known multiplication facts to find
fractions of a set. e.g.
Advanced Multiplicative Part-Whole
7
Addition/Subtraction: The student is able solve addition and subtraction
problems for integers and decimals.
Multiplication/Division: The student is able to use at least two different
mental strategies to solve multiplication and division problems with whole
numbers.
Fractions/Ratios: The student uses known multiplication facts to solve
problems with fractions, proportions and rations.
E.g. 3: 5 = � : 40, � = 8 x 3 = 24. e.g.
.
Advanced Proportional Part-Whole
8
Addition/Subtraction: The student can add and subtract mixed fractions
with unlike denominators. e.g.
.
Multiplication/Division: The student uses at least two different strategies to
solve problems that involve equivalence with and between fractions, ratios
and proportions.
e.g. 7.2 ÷ 0.4: 7.2 ÷ 0.8 = 9, so 2 x 9 = 18 is the answer.
E.g. 3.6 x 0.75 =
Fractions/Ratios: The student uses a broad range of strategies to solve
problems with fractions, proportions and ratios.
E.g. 65% of
24: 50% of 24 = 12, 10% of 24 = 2.4, so 5% of 24 = 1.2, answer = 12 + 2.4
+ 1/2 = 15.6.
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How can I support my daughter’s
Mathematical learning?
You can help in many ways to support your child’s number development even if the
approaches are different to how you learned mathematics in school.
Talk to your child’s teacher and use the following support tools to inform and assist
you to help your child in their number learning:
•
See the Ministry of Education brochure: “Help your child to develop
Numeracy… What you do counts!”
•
Visit the Ministry of Education mathematics web site:
www.nzmaths.co.nz
Visit in particular the Numeracy
Project section for materials, the
Learning Framework for
Number and Evaluation reports
about the effectiveness of the
Numeracy Project.
•
Talk to your child’s teacher and ask what basic Number knowledge facts
would be useful for you to assist your child with practising at home.
•
Support your child to see and use numbers in real contexts in their daily lives
wherever you can.
Numeracy: “… to be numerate is to have the ability and inclination to use
mathematics effectively – at home, at work and in the community.” (Howard
Fancy, Update 4)
Quick Games
 Play games using decks of cards for speedy adding/ subtraction e.g. Two
players. One is the dealer. Dealer turns two cards over and places them in one
pile. The other player adds the numbers on them (or subtracts). The dealer
adds another card to the pile. Other player instantly adds the numbers on that
card and the last face-up card. Or adds new card to the existing total. Who can
add all the deck of cards in the fastest time?
 Children love “Maths Bingo”. It’s just a grid of 9 squares, where they put in
any 9 numbers e.g. numbers between 1-20, or multiples of 3 or 6 etc
depending on their needs. The ‘bingo caller’ gives maths problems and if they
have the answer in their grid they can cross it out. Works well when two
people do it so that there is a ‘winner’
5

“Guess my number” is a useful game for playing on a journey. As your child
plays the game they will practise thinking about the order of numbers. Start
the game by saying to your child ' I am thinking of a number between 1 and
10'. Explain that the aim of the game is to guess the mystery number by asking
questions and that you will only answer 'yes' or 'no'. Children soon learn that it
is more useful to ask "Is the number bigger than 5?" than to ask 'Is it 7?" Older
children can progress to guessing mystery numbers up to 100, and by Year 2
they should be able to use questions such as:
'Is it an odd number?'
'Is the number a multiple of 10?' (e.g. 20, 30, 40)
Numbers Around Us.
• Use shopping opportunities to estimate, count and compare quantities.
• Talk about fractions or dividing when items are being cut or shared.
• Relate decimal numbers to money. Look for numbers in the environment and
practice reading them and adding or taking away 100s or 1000s etc to them.
 Use numbers on letterboxes/ number plates – compare/ order/ subtract/ add/
say the one before/ after/ what is the number of the house that’s 2 (20, 200,
2000) houses further on.
STAGE 0 AND 1-3: NUMBER KNOWLEDGE
To have mastery of knowledge, students will identify or state what they know
instantly after being asked.
Columns to  learning
Stage 0 (Emergent)
Knowing about the numbers 0-10
I can count forwards and backwards to and from 10
I can read, say and order the numbers 1- 10 (1, 2, 3 etc)
I can say what number comes before, after and in between any number
up to 10. e.g. ___ 5,6
6,7 ____ 8 ___ 10
I can recognise the patterns for numbers up to 5. e.g. patterns for
numbers up to 5 on dice, playing cards, dominoes, fingers etc.
Stages 1, 2 and 3 (Counting from one)
Knowing about the numbers 0-20
I can count forwards and backwards to and from 20
I can say and put the numbers 1- 20 in order (1, 2, 3 …)
I can say what number comes before, after and in between any number
up to 20.
I can write the numbers 0 – 20 (0, 7, 15).
I can read the numbers 0 – 20 (18 eighteen).
I can show amounts between 0 and 20 with pictures, objects, symbols,
diagrams etc.
I can skip-count by twos and by fives to 20 e.g. 2, 4, 6…
5, 10, 15…
6
I can say the number groups within 5 e.g. 0 and 5; 2 and 3; etc.
I can say the number groups with 5 (up to 10) e.g. 5 and 0; 5 and 1; 5
and 2; 5 and 3; 5 and 4; 5 and 5.
I can say the number groups within 10 e.g. 1 and 9; 2 and 8; etc.
I can recognise the patterns for numbers up to 10 e.g. (dice, playing
cards, fingers etc.)
I can say the doubles within 10 (2 + 2; 3 + 3; 4 + 4; 5 + 5).
I can instantly recall all the + - basic facts to 5 in any order
1 + 0; 1 + 1; 1 + 2 ; 1 + 3; 1 + 4; 2 +1; 2 + 2 etc.
STAGE 4: NUMBER KNOWLEDGE
To have mastery of knowledge, students will identify or state what they know
instantly after being asked.
Columns to  learning
Stage 4 (Advanced counting).
Knowing about the numbers 0-100, simple fractions
I can count forwards and backwards within the 0 -100 range e.g. 20, 21,
22 … 50; or 80, 79, 78…. 30
I can say and put the numbers 1- 100 in order
I can say what number comes before, after and between any numbers
within the 0 -100 range
I can read and write the numbers within the 0 -100 range (34 thirtyfour).
I can skip-count forwards and backwards by twos, fives and tens within
the
0 - 100 range e.g. 32, 34, 36…. 60 20, 25, 30… 10, 20, 30…
I can say the number groupings with 10 e.g. 10 and 1; 10 and 2, 10 and
3… and the pattern of the ‘teen’ numbers
(13 = ten and 3...)
I can say the number groupings within 20 e.g. 11 and 9; 12 and 8; etc.
I can say the number of tens in decades within the 0 - 100 range e.g.
‘how many tens in 40? 60? 100?
I can instantly recall all the + - basic facts to 10 e.g. 4 + 3 = 7; 8 - 3 =
5; 10 – 2 = 8 etc.
I can instantly recall all the doubles and corresponding halves to 20 in
any order e.g. 6 + 6, 7 + 7 ½ of 2 ½ of 14
I can say the “ten and…” facts to 20 e.g. 10 + 4, 10 + 8
I can say the compatible ‘ten’ numbers that add to 100 e.g. 40 + 60, 70 +
30.
I can recognise and read the symbols for halves, quarters, thirds and
fifths e.g.
1
2
1
4
1
3
1
5
7
STAGE 5: NUMBER KNOWLEDGE
To have mastery of knowledge, students will identify or state what they know
instantly after being asked.
Columns to learning
Stage 5 (Early Additive)
Knowing about the numbers 0-1000, most common fractions
I can count in number sequences forwards and backwards, by ones, tens
and hundreds within the 0 -1000 range
I can say what number comes 1, 10, 100 before and after any number
within the 0 -1000 range
I can read, write and order the numbers within the 0 -1000 range
I can skip-count forwards and backwards by twos, threes, fives and tens
within the
0 – 1000 e.g. 3, 6, 9, 12, etc.
I can order fractions with the same denominator e.g.
1 2
5 5
3
5
4 5
5 5
I can say the number groups within 100 e.g. 49 and 51 (particularly
multiples of 5 e.g. 25 and 75).
I can say how many twos in numbers up to 20 e.g. 2 groups of 8 in 17 (at
this level children don’t need to state the remainders).
I can say how many fives in numbers up to 50 e.g. 9 groups of 5 in 47.
I can say the grouping of tens that can be made from a three-digit number
e.g. tens in 763 is 76.
I can say the number of hundreds in centuries and thousands e.g. hundreds
in 800 is 8 and in 4000 is 40.
I can round three-digit whole numbers to the nearest 10 or 100 e.g. 561
rounded to the nearest 10 is 560 and to the nearest 100 is 600.
I can instantly recall all the + basic facts to 20 e.g. 7 + 2 = 9; 11 + 3 = 14
I can instantly recall the 2 times, 5 times and 10 times tables and know
their corresponding division facts e.g. 4 x 5 + 20
20 ÷ 5 = 4 .
I can say the multiples of 100 that add to 1000 e.g. 400 + 600, 700 + 300.
I can recognise and read the symbols for most common fractions including
fifths and tenths and improper fractions (numerator bigger than
denominator) e.g
7
3
STAGE 6: NUMBER KNOWLEDGE
To have mastery of knowledge, students will identify or state what they know
instantly after being asked.
Columns to learning
Stage 6 (Advanced Additive)
Knowing about the numbers 0-1 000 000, decimals to three places,
most fractions.
I can count in whole-number sequences forwards and backwards, by ones,
tens, hundreds and thousands within the 0 - 1 000 000 range.
I can say what number comes 1, 10, 100, 1000 before and after any
number within the 0 - 1 000 000 range
I can say the forwards and backwards word sequences for halves, quarters,
thirds, fifths and tenths e.g.
1
3,
2
3,
1,
4
3,
5
3,
2,
7
3 etc
I can count forwards and backwards in tenths and hundredths (between
8
given amounts) e.g.
From 0.00 to 3.00 by tenths. 0.1, 0.02, 0.03… 0.09, 1, 1.01, 1.02… 1.09,
2.00 etc
From 0.00 to 1.00 by hundredths 0.01, 0.02…0.09, 0.10, 0.11, 0.12…0.20,
0.21…0.29, 0.30…, 0.99…, 1.00
I can order whole numbers within the 0 - 1 000 000 range e.g. 997, 307
979 703 309 709
I can order unit fractions for halves, thirds, quarters, fifths and tenths e.g.
1 1 1 1
3 2 5 4
I can say the number groupings within 1000 e.g. 240 and 760.
I can say how many twos, threes, fives and tens in numbers up to 100 and
state any remainders.
I can say the grouping of tens and a hundreds that can be made from a
four-digit number e.g. tens in 4763 = 476 remainder 3, hundreds = 47
remainder 63.
I can say the number tenths and hundredths in decimals to two places e.g.
tenths in 7.2 is 72, hundredths in 2.84 is 284.
I can round whole numbers to the nearest 10, 100 or 1000.
I can round decimals with up to two decimal places, to the nearest whole
number e.g. 6.49 rounds to 6
and 19.91 rounds to 20.
I can instantly recall all the + - basic facts to 20
e.g.14 + 1 = 15; 17 - 3 = 14; 16 – 12 = 4 etc.
I can instantly recall all the multiplication and division facts
(up to 10 x 10).
I can recognise and read the symbols for any fractions.
STAGE 7: NUMBER KNOWLEDGE
To have mastery of knowledge, students will identify or state what they know
instantly after being asked.
Columns to  learning
Stage 7 (Advanced Multiplicative)
Extending knowledge about numbers 0-1 000 000, decimals to three places,
fractions.
I can count forward and backwards by 0.001s, 0.01s, 0.10s, ones, tens (between
given amounts).
I can say the number one-thousandth, one-hundredth, one-tenth, one, ten etc
before and after any given whole number.
I can order decimals to three places e.g. 1.01, 1.10, 1.00, 1.11.
I can order fractions e.g.
2 1 3 3 8
3 2 5 4 10
I know the groupings of numbers less than 10 that are in numbers to 100 and
can state the remainders e.g. sixes in 38 = 6 with 2 remainder.
I know the groupings of 10, 100, 1000 in 7 digit numbers e.g. 1000s in 47 562
I know equivalent fractions for ½ 1/3 ¼ 1/5 1/10 with other fractions with
denominators up to 100 and 1000 e.g. ½ is equivalent to 50/100 and 500/1000.
I can round whole numbers and two-place decimals to the nearest whole number
or tenth e.g. 6.49 rounds to 6.5
19.99 rounds to 20.
I can instantly recall the fraction to decimal to percentage conversions for
halves, thirds, quarters, fifths and tenths e.g. ½ = .50 = 50%
I know the divisibility rules for 2,3,5,9
I know the square numbers to 100 and the corresponding square roots e.g.
72 = 49, √ 49 = 7
9
I can identify the factors of numbers to 100 including prime numbers e.g. factors
of 15 = {1,3,5,15}.
I can identify common multiples of numbers to 10 e.g. common multiples of 2
and 9 = {18, 36, 54…}.
I can use columns to write and solve + and – of whole numbers e.g.
24
+36
60
I can do short x and ÷ of a 3- digit x 1-digit whole numbers
STAGE 8: NUMBER KNOWLEDGE
To have mastery of knowledge, students will identify or state what they know
instantly after being asked.
Columns to  learning
Stage 8 (Advanced Proportional)
I can count forward and backwards by 0.001s, 0.01s, 0.10s, ones, tens
(starting at any number).
I can say the number one-thousandth, one-hundredth, one-tenth, one, ten
etc before and after any given decimal number.
I can order fractions, decimals and percentages
I know the number of tenths, one-hundredths, one-thousandths in
numbers up to 3 decimal places e.g. hundredths in 6.57 = 657.
I know what happens when a whole number is x or ÷ a power of 10
e.g. 4.57 x 10 = 45.7
I can round decimals to the nearest 100, 10, 1 1/10 or 1/100. e.g. 6.437
rounds to 6.440 when rounded to nearest 1/100.
I can instantly recall the fraction to decimal to percentage conversions
for given fractions or decimals. e.g. 9/8 = 1.125 = 112.5%
I know the divisibility rules for 2,3,4,5,6,8 and10
I know the simple powers of numbers e.g. 24 = 16
I can identify the highest common factors of numbers to 100 e.g. highest
common factor of 48 and 36 is 12
I can identify the lowest common multiples of numbers to 10 e.g. lowest
common multiples of 2 and 9 = 18.
I can do short x and ÷ of whole numbers and decimals by single- digit
numbers.
I can do x of 3 and 4-digit whole numbers by two- digit whole numbers
e.g.
472
x 23
1416
9440
10856
10