
What Are The Stages of the The Number Framework NZ

What stage is Strategy Stage: Emergent?
 Stage ZERO:
 unable to consistently count a given number of objects
 lack knowledge of counting sequences
 no ability to match things in onetoone correspondence.

What stage is Strategy Stage: One to One Counting?
 Stage ONE:
 can count and form a set of objects up to ten
 cannot solve simple problems that involve joining and separating sets, like 4 + 3.

What stage is Strategy Stage:Counting from One on Materials?
 Stage TWO:
 rely on counting physical materials/fingers.
 They count all the objects in both sets
 i.e. in “Five lollies and three more lollies.
 How many lollies is that altogether?”

What stage is Strategy Stage:Counting from One by Imaging?
 Stage Three:
 counting all of the objects
 simple joining and separating
 image visual patterns of objects in their mind and count them.

What stage is Strategy Stage: Advanced Counting (Counting On)?
 Stage Four:
 understand that the end number ina counting sequence measures the whole set
 can relate the addition or subtraction of objects to the forward and backward number sequences by ones, tens, etc.
 For example, instead of counting all objects to solve 6 + 5, the student recognises that “6” represents all six objects and counts on from there... “7 , 8 , 9, 10, 11.”
 have the ability to coordinate equivalent counts, such as “10, 20, 30, 40, 50,” to get $50 in $10 notes.
 This is the beginning of grouping to solve multiplication and division problems.

What stage is Strategy Stage: Early Additive PartWhole?
 Stage Five:
 recognise that numbers are abstract units that can be treated simultaneously as wholes or can be partitioned and recombined.
 This is called partwhole thinking.
 A characteristic of this stage is the derivation of results from related known facts, such as finding addition answers by using doubles or teen numbers.
 The strategies that these students commonly use can be represented in various ways, such as empty number lines, number strips, arrays, or ratio tables.

What stage is Strategy Stage: Advanced Additive–Early Multiplicative PartWhole?
 Stage SIX:
 Can choose from a repertoire of partwhole strategies addition and Subtraction
 see numbers as whole units
 Can understand that “nested” within these units is a range of possibilities for subdivision and recombining.
 Can derive multiplication answers from known facts.
 can also solve fraction problems using a combination of multiplication and additionbased reasoning.

What stage is Strategy Stage: Advanced Multiplicative–Early Proportional PartWhole?
 Stage Seven:
 Can choose appropriately from a range of partwhole strategies  for multiplication and division.
 These strategies require one or more of the numbers involved in a multiplication or division to be partitioned, manipulated, then recombined.
 reversibility, in particular, solving division problems using multiplication.
 able to solve problems with fractions using multiplication and division.
 Can understand the multiplicative relationship between the numerators and denominators of equivalent fractions, e.g. 3/4 = 75/100

Knowledge is...(Don't say POWER!)
The key items of knowledge that students need to learn.
..What do you NEED to solve a problem?
Example:
Understand whole numbers and multiplication to multiply 5 x 5.

Strategy is...
The mental processes students use to estimate answers and solve operational problems with numbers.
 ...HOW to solve a problem?
 ...HOW did you do it?
 Example:
 Use the knowledge of whole numbers and nested numbers to use part whole strategy.

What is PartWholing?
What stage does it start at?
Numbers are abstract units that can be treated simultaneously as wholes or can be partitioned and recombined.
Example: Compensation from known facts like
DOUBLING
7 + 8 as 7 + 7 is 14, so 7 + 8 is 15.
STANDARD PLACE VALUE PARTITIONING
43 + 35 is (40 + 30) + (3 + 5) = 70 + 8.
Stage Five: Early Additive PartWhole

What is DOUBLING?
What stage does it start at?
7 + 8 as 7 + 7 is 14, so 7 + 8 is 15
3 x 8 = 24, so 6 x 8 = 24 + 24 = 48
Stage Five: Early Additive PartWhole

What is STANDARD PLACE VALUE PARTITIONING or PVP?
What stage does it start at?
43 + 35 is (40 + 30) + (3 + 5) = 70 + 8.
Standard place value with TIDY NUMBERS and COMPENSATION
63 – 29 = □ as 63 – 30 + 1 = □.
Stage Five: Early Additive PartWhole

What is REVERSIBILITY?
What stage does is start?
 53 – 26 = □ as 26 + □ = 53.
 26 + (4 + 20 + 3) = 53, so 53 – 26 = 27.
Stage Six: Advanced Additive–Early Multiplicative PartWhole

What is PARTITIONING FACTORS ADDITIVELY?
What stage does it Start?
 (Using DOUBLING)
 3x8 = 24, so 6x8 = 24 + 24 = 48.
 (Using Compensation)
 5x3 = 15, so 6x3 = 18 (three more; compensation using addition)
10x3 = 30, so 9x3 = 27 (three less; compensation using subtraction)
Stage Six: Advanced Additive–Early Multiplicative PartWhole

What is DOUBLING AND HALVING OR TREBLING AND DIVIDING BY THREE (THRIVING)
What stage does it start?
 4x16 as 8x8 = 64 (doubling and halving)
 72 ÷ 4 as 72 ÷ 2 = 36, 36 ÷ 2 = 18 (Halving and halving)
 3 x 24 as 9 x 8
Stage Seven: Advanced Multiplicative–Early Proportional PartWhole

What stage is this?
To solve 27x6...
27 might be split into 20 + 7 and these parts multiplied then recombined, as in
20x6 = 120, 7x6 = 42, 120 + 42 = 162,
or 2x27 = 54, 3x54 = 162.
The first strategy partitions 27 additively, the second strategy partitions 6 multiplicatively.
Stage Seven: Advanced Multiplicative–Early Proportional PartWhole

What is REVERSIBILITY AND PLACE VALUE PARTITIONING?
What stage does it start?
72 ÷ 4 as 10x4 = 40, 72 – 40 = 32, 8x4 = 32, 10 + 8 = 18, so 18x4 = 72
Stage Seven: Advanced Multiplicative–Early Proportional PartWhole

What is MULTIPLYING WITHIN?
What stage does it start?
Every packet in a jar has 8 nuts in it. Three of the 8 nuts are peanuts. The jar contains 40 nuts altogether, all in packets. How many of the nuts are peanuts?(3:8 as □:40, 5x8 = 40 so 5x3 = □)Stage Seven: Advanced Multiplicative–Early Proportional PartWhole

What is THE COMMUTATIVE PROPERTY
?
5 x 6 is the SAME as 6 x 5

What is USING UNIT FRACTIONS AND CONVERSION FROM PERCENTAGES
?
What stage does it start?
 The tracksuit is usually $56, but the shop has a 25% off sale. How much does Mere pay?
 25% is 1/4 , so Mere pays 3/4 of $56. 1/4 of 56 is 56 ÷ 4 = 14.
 3/4 = 3x1/4 , 3x14 = 42
Stage Seven: Advanced Multiplicative–Early Proportional PartWhole

What is THE DISTRIBUTIVE PROPERTY
?
5 x 6 is the same as...
 (4 x 6) + (1 x 6) OR
 (3 x 6) + (2 x 6)
 Albert has 32 matchbox toys. Fiveeighths of them are sports cars. How many of the matchbox toys are sports cars?
 Method 1: 5/8 of 32 as 1/2 of 32 is 16 and 1/8 of 32 is 4, so 5/8 of 32 is 16 + 4 = 20
 Method 2: 1/8 of 32 is 4, so 5/8 of 32 is 5x4 = 20

Strategy Stage: Advanced Proportional PartWhole
 Stage Eight:
 Can select from a repertoire of partwhole strategies for fractions, proportions, and ratios.
 Can find common factors and include strategies for the multiplication of decimals and the calculation of percentages.
 Can find the multiplicative relationship between quantities of two different measures. This can be thought of as a mapping. For example, consider this problem  “You can make 21 glasses of lemonade from 28 lemons. How many glasses can you make using 8 lemons?”
 To solve the problem, students may need to find a relationship between the number of lemons and the number of glasses.
 This involves the creation of a new measure, glasses per lemon. The relationship is that the number of glasses is threequarters the number of lemons.
 This could be recorded as 21:28 is equivalent to □ :8? 21 is 3/4 of 28 ⇒ 6 is 3/4 of 8.

How does DOUBLING AND HALVING work for solving MULTIPLICATION problems? (MULTIPLICATIVE CONNECTIONS)
You can see that 4 x 16 (64) is the same as 8 x 8 or 2 x 32. Therefore you will start to see the relationships between these numbers and their common factors also. AND...
 8 x 64 = 512
 16 x 64 = DOUBLE 1024
 16 x 32 = HALF 512
 8 x 32 = HALF 256
 80 x 32 = x10 2560
 0.8 x 32 = ÷100 25.60
 0.08 x 32 = ÷10 2.56
 0.08 x 3.2 = ÷10 0.256

What is PARTITIVE division?
 Partitive division is about how you SHARE:
 1 by 1
Jenny baked 12 cookies.
 She SHARED them equally into 3 packets.
 How many cookies were in each packet?

What is QUOTATIVE division?
Quotative division is about counting in GROUPS: 4 by 4
Jenny baked 12 cookies.
 She put 4 cookies in each packet (packets of 4).
 How many packets did she make?

What's the difference in MULTIPLICATION in Stages 4,5,6?
 Take 6 x 5 as an array...

 Stage 4: Skip Counting 5,10,15,20,25,30...
 Stage 5: CLUMPING 10, 20, 30...
 Stage 6: Will know x2, x5, x10...
Stage 6 & 7: BASIC FACTS

What is a PRIME number?
A number that can only be divided by itself or 1.

What is EQUAL ADJUSTMENT?
If you have 70  46...
you add 4 to 46 to make it 50...but ALSO add 4 to 70 to keep the DIFFERENCE THE SAME.
It's always easier to round off the number you are subtracting.

What are the MULTIPLIER and the MULTIPLICAND of 5 x 6?
 The MULTIPLIER is 5: How many packets?
 The MULTIPLICAND is 6: How many in EACH packet?

What is BEDMAS?
 B = Brackets
 E = Exponents
 D = Division
 M = Multiplication
 A = Addition
 S = Subtraction

What is a TABLE MOUNTAIN and how does it work?

How can you use your x2 and x5 tables to work out the 4 x 14?
 4 x 2 = 8
 4 x 5 = 20
 4 x 7 = 28
 4 x 14 = 56 (Double)

What is MULTIPLICATIVE THINKING?
Multiplicative thinking is characterised by the way you think rather than the problems you solve.
Multiplicative thinking involves “constructing and manipulating factors (the numbers being multiplied) in response to a variety of contexts ... [and] deriving [unknown results] from known facts using the properties of multiplication and division [commutative, associative, distributive, inverse].”
(Ministry of Education, Book 6, p. 3).

What is the ROUNDING AND COMPENSATING?
24 x 6 = 144
(25 x 6)  6 = (150  6) = 144

What are the stages of MULTIPLICATION?
Stages of Multiplication
 Onebyone counting (CA) 'Counting All'
 Skip counting (AC) 'Advanced Counting'
 Combination of multiplication facts and repeated addition(EA) 'Early Additive'
 Deriving from known results (AA) 'Advanced Additive'
 Basic Facts known (AM) 'Advanced Multiplicative'

How do PRIME FACTOR TREES work?

How do does the NZC, Number Framework and the NZ maths Standards line up?

What does it mean to be 'NUMERATE?'
To be numerate is to have the ability and inclination to use mathematics effectively in our lives – at home, at work, and in the community.”

What is NUMERACY?
“Numeracy is the ability to process, communicate and interpret numerical information in a variety of contexts.”
...The understanding of maths, to process and manipulate what you know... transferring the mathematics that you learn and being able to apply it...
We want to teach the mathematics and then the NUMERACY so they can use it.

What is MATHEMATICS?
Mathematics is the exploration and use of patterns in quantities, space and time"
...About solving problems...the doing, exploring of the problem...
We want to teach the mathematics and then the NUMERACY so they can use it.

Give me FOUR strategies to work out 4 x 18?
 Algorithm (On Paper) (Booooooo!!!)
 Doubling And Halving 8x9
 Place Value Partitioning (Pvp) 4x10 + 4x8
 Rounding And Compensating (Tidy Numbers) (4x20)(4x2) 808
 Deriving From Known Results = 4 Groups Of 9 + 4 Groups Of 9 Or 2x (4x9) (Both Splitting A Factor/)

What are the DIVISIBILITY RULES for 2 & 3?
Divisible by 2:
All even numbers are divisible by 2
Divisible by 3:
 Add up all the digits in the number.
 If the sum of the digits is divisible by 3, then the number is divisible by 3
e.g. 1425 (1 + 4 + 2 + 5 = 12) 12 is divisible by 3, therefore 1 425 is divisible by 3

What are the DIVISIBILITY RULES for 4 & 5?
Divisible by 4:
Are the last two digits in the number divisible by 4? If so then the number is divisible by 4
 e.g. 36 924 ends in 24 which is divisible by 4, therefore 36 924 is divisible by 4

 Divisible by 5:
All numbers ending in a 5 or a 0 are divisible by 5

What are the DIVISIBILITY RULES for 6?
Divisible by 6:
 1. If the number is even and divisible by 3, it is also divisible by 6
 e.g. 1374 ends in an even number and 1 + 3 + 7 + 4 = 15, which is divisible by 3 therefore the number will be divisible by 6.
 1473 is an odd number and while 1 + 4 + 7 + 3 = 15, which is divisible by 3, the number is NOT divisible by 6
2. Halve the number. If the digits of the halved number add up to 3, the number is divisible by 6.

What are the DIVISIBILITY RULES for 7?
Divisible by 7:
 Take the last digit in a number
 Double the digit and subtract it from the next two digits (hundreds & tens column).
 If the number you get is divisible by 7 then the whole number is divisible by seven
 e.g.357 – Double 7 to get 14. Subtract 14 from 35 to get 21which is divisible by 7
 Repeat the process for each group of three digits
 e.g.469 357  Double 7 to get 14. Subtract 14 from 35 to get 21 which is divisible by 7
 Double 9 to get 18. Subtract 18 from 46 to get 28 which is divisible by 7

What are the DIVISIBILITY RULES for 8?
 If the last three digits are divisible by 8, the number is
 e.g. 6328 – 328 is divisible by 8 therefore 6 328 is divisible by 8.
 Halve the number. If the last two digits are divisible by 4, the number is divisible by 8
 e.g. 6328 ÷ 2 = 3164. The last two digits of 64 are divisible by 4 therefore the number (6 328) is divisible by 8

What are the DIVISIBILITY RULES for 9?
 Add up all the digits in the number
 If the sum of the digits is divisible by 9, then the number is divisible by 9
 e.g. 3429 (3 + 4 + 2 + 9 = 18) 18 is divisible by 9, therefore 3 429 is divisible by 9

What is th KEY IDEA and the KNOWLEDGE being developed at STAGE 23?
Key Idea:
 Children are counting sets of the same number to
 solve problems that could be solved by
 multiplication.
Knowledge being developed:
 • Skip counting 2, 5 & 10
 • Doubles to 20

What is th KEY IDEA and the KNOWLEDGE being developed at STAGE 4?
Key Idea:
 Children are moving from counting all to learning to
 use addition strategies (skip counting) to solve
 problems that could be solved by multiplication.
Knowledge being developed:
 • Multiplication as repeated addition
 • Division as repeated subtraction
 • Skip counting in threes
 • Multiplication & division fact for 2, 5 & 10
 • Sharing into equal sets
• Grouping, how many sets can be made.

What is th KEY IDEA and the KNOWLEDGE being developed at LOWER STAGE 5?
Using known facts
 • 2 x doubles
 • x 2 equal sets/groups of 2
 • x 10 place value
 • x 5 place value and doubles (halves)

What is th KEY IDEA and the KNOWLEDGE being developed at UPPER STAGE 5?
 Children are learning to derive further multiplication facts using addition and subtraction strategies from multiplication facts they already know and addition facts they already know
 (e.g. trees problem in NumPA solved in groups of 10).
“Teaching the tables” requires teachers to help children make the connections between what they already know and what they are learning.

What is th KEY IDEA and the KNOWLEDGE being developed at STAGE 6?
Students at this stage are developing a range of multiplication & division strategies.
 Students need recall of all multiplication facts to 10 x 10 and the matching division facts
 They are developing Divisibility rules for 2, 3, 5, 9 & 10
 Factors of numbers to 100

What is th KEY IDEA and the KNOWLEDGE being developed at STAGE 7?
Students understanding of multiplication & division progresses at this stage
Knowledge being developed at this stage includes:
 Common Factor (Highest Common Factor),
 Common Multiples (least/lowest common multiple),
 Divisibility rules for 4, 6, & 8.

Show 24 x 36 as an ARRAY?


How do you use CROSS MULTIPLYING to show 24 x 36?

Write a word problem to go with a particular problem structure:
a) Start Unknown
b) Change Unknown
c) Result Unknown
When translating top a word problem make sure you keep the unknown in the same structure.
a) Start Unknown : "Miguel has some of apples. He adds 6 apples so that he now has 10. How many apples did he start with?"
b) Change Unknown: "Miguel has 6 apples. His Gran gives him a few more from her tree. Now he has 10. How many did his Gran give him?"
c) Result Unknown: "Miguel picked 6 apples from his tree and 4 from his Grans. How many apples does he have altogether?"

What is the ASSOCIATIVE property?
 Doubling and halving
 Thirding and trebling, etc.
 e.g. 12 x 25 = 3 x (4 x 25) = 3 x 100
 e.g. 3 x 24 = 9 x 9

What is the INVERSE property?
Reversing, Doing and undoing
 e.g. 42 ÷ 3 = as □ x 3 = 42
 14 x 3 = 42
 so... 42 ÷ 3 = 14

