THE NUMERACY DEVELOPMENT PROJECTS - nzmaths · the majority of teachers of year 7 and 8 ... (2) The...
Transcript of THE NUMERACY DEVELOPMENT PROJECTS - nzmaths · the majority of teachers of year 7 and 8 ... (2) The...
THE NUMERACY DEVELOPMENT PROJECTS
Teachers are key figures in changing the way in whichmathematics and statistics is taught and learned inschools. Their subject matter and pedagogicalknowledge are critical factors in the teaching ofmathematics and statistics for understanding. Theeffective teacher of mathematics and statistics has athorough and deep understanding of the subject matterto be taught, how students are likely to learn it, and thedifficulties and misunderstandings they are likely toencounter.
The focus of the Numeracy Development Projects is toimprove student performance in mathematics throughimproving the professional capability of teachers. Todate, almost every teacher of year 1 to 6 children andthe majority of teachers of year 7 and 8 children havehad the opportunity to participate.
A key feature of the projects is their dynamic andevolutionary approach to implementation. This ensuresthat the projects can be informed by developingunderstanding about mathematics learning and effectiveprofessional development and that flexibility in approachand sector involvement is maximised.
Numeracy Professional Development Projects 2008Published by the Ministry of Education.PO Box 1666, Wellington, New Zealand.
Copyright © Crown 2008. All rights reserved.Enquiries should be made to the publisher.
ISBN 0 478 13381 2Dewey number 372.7Topic Dewey number 510
Item number 13200
Note: Teachers may copy these notes for educational
purposes.
This book is also available on the New Zealand Maths website,
at www.nzmaths.co.nz/Numeracy/2008numPDFs/pdfs.aspx
The projects continue to build on the findings andexperience associated with the numeracy professionaldevelopment projects that operated in 2002–2007.These projects made an important contribution to whatwe know about:• children’s leaming and thinking strategies in early
mathematics;• effective identification of, and response to,
children’s learning needs;• the characteristics of professional development
programmes that change teaching practice; and• effective facilitation.
Such findings continue to inform the modification andfurther development of the projects. National co-ordinators and facilitators from each region provideongoing feedback about aspects of the projects.
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Numeracy Project Assessment (NumPA)
IntroductionThe Numeracy Project Assessment (NumPA) is a diagnostic tool that is designed to give qualityinformation about the knowledge and mental strategies of the students you work with. Thisinformation is aligned to The Number Framework (see Book 1: The Number Framework).
NumPA takes the form of an individual interview with students. This is necessary for tworeasons:
(1) Uncovering students’ mental strategies involvesfinding out how they solve number problems.Pencil and paper assessment tasks show whetheror not a student is getting correct answers.However, it is often difficult to find out thestrategies that the student uses with pencil andpaper assessment.
(2) The interview process is invaluable for your ownprofessional development.
To decide which form of NumPA to use, you will needto use the strategy window questions discussed onpages 4 and 5 or the flowchart on page 7. Use yourprior knowledge of the student to choose whichquestion to begin with.
To develop the confidence of the student, it is bestto start with a question he or she will find easy.
Use the sequence of questions on pages 4 and 5 toclassify the student’s responses. From that, use thetable on page 6 or the flowchart on page 7 to choosewhich of the assessment forms, A, B, or C, to use.
The interview consists of two main parts; strategyquestions and knowledge questions.
When you first use the assessment, it is importantto follow the script provided. This script directsyou to skip questions where it is clear that they willbe too difficult for the student. This saves time andpotential anxiety.
With the strategy questions, your primary focus ison how the student solved the problem. It issuggested that you ask the student, “How did youwork that out?” for each problem. Sometimes this will not provide adequate information aboutthe strategies the student used, and you will need to ask other more directed questions. It is wisepractice where the answer is correct to reassure the student by saying, “Yes, that’s right” beforeasking them about the strategy they used.
With the knowledge questions, you are looking for fluent responses. Students who take sometime to answer a knowledge question are usually strategising, which means that they do notautomatically know the item of knowledge concerned and are working out a solution.
As you become more familiar with the items and how to evaluate students’ responses, you willbecome much quicker at administering the NumPA. You will get better at assigning stages foreach area of strategy and knowledge from the assessment using the least possible number ofquestions.
Oh, you want to knowhow I worked it out inmy head.
It’s brilliant professionaldevelopment for me too.
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Materials Needed for NumPAThis section provides instructions for assembling the NumPA. Copymasters are provided for allof the materials needed. Since the assessment materials are used frequently, it is advisable tolaminate the card items. For each form of the test use the colour of card suggested. Keep all thematerials in a see-through plastic bag.
Hardware Cardware• Twelve counters of one colour
• Ten counters of another colour
• Two masking cards (A5 size)
Strategy Windows: None
NumPA Form A: Blue
Make up the sets of cards and strips,from pages 14 and 15.
NumPA Form B: Yellow
Make up the sets of cards frompages 24 to 25.
NumPA Form C: Green
Make up the sets of cards frompage 39.
Test Booklets: Spiral Bind Test Scripts: Spiral BindStrategy Windows: White
Photocopy pages 8 and 9.
NumPA Form A: Blue
Photocopy page 16.
NumPA Form B: Yellow
Photocopy pages 26–30.
NumPA Form C: Green
Photocopy pages 39–47.
Strategy Windows: White
Photocopy pages 4–7.
NumPA Form A: Blue
Photocopy pages 10–13.
NumPA Form B: Yellow
Photocopy pages 17–23.
NumPA Form C: Green
Photocopy pages 31–38.
Individual Assessment SheetsOn pages 48 to 53, you will find individual assessment sheets for each form of NumPA. Thesesheets make up a three-page profile where you can enter the initial and follow-up NumPA datafor a student. Such an individual assessment sheet is a significant document in a student’s recordof achievement. Note that page 54 provides answers for the Form C questions.
Before beginning the interview, write the student’s personal information in the top section of theappropriate assessment sheet. You need to make a judgement about the student’s developmentalstage after asking the appropriate questions within each section of the assessment.
The strategy windows questions provided in the first box under the student identification detailswill enable you to determine the student’s strategy stage for addition and subtraction. Theinterview questions are given in the left-hand column of each box. In time, you will learn tointerview students directly from the assessment sheets without referring to the detail included inthe test scripts.
The panels for entering each judgement are in the columns to the right of the questions beingasked. There is a space within each stage to enter either an initial interview result (clear box) or afollow-up result (shaded box). The entry should include the date in abbreviated form, for
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example, 4/5. Any comments that you would like to make, including details on how the studentsolved the strategy windows questions, can be written in the comments space underneath. It isoften helpful to write the student’s responses for later reference. Where possible, record thestudent’s responses as equations or empty number lines. In deciding which cell to make theentry in, you should use the following criteria:
Strategy Questions: Enter the highest stage the student demonstrates within each operationaldomain.
Knowledge Questions: Enter the highest stage at which the student answers every itemcorrectly with fluency and certainty.
From these individual assessment sheets, you can transfer the results onto a Class GroupingSheet for NumPA for your class (see Book 3: Getting Started).
The stages that a student achieveswithin the Knowledge sections ofthe interview are often differentfrom the stages that they achieve inthe Strategy sections. Knowledgecan be learned independently of anymeaningful ability to apply it.Similarly, students can inventadvanced strategies withoutsufficient knowledge to apply themto a broad range of problems andnumbers.
Since the original numeracy development projects in 2001–2002, the assessment tool has beenrevised annually. In 2003, the operational strategy windows were introduced to refer teachers toeach form of NumPA. This allowed the interviews to cover a wide range in studentunderstanding.
In 2004, minor changes were made to separate the place value and basic facts knowledgedomains. This followed weak performances in this area and a recommendation from the TePoutama Tau project evaluation and the Numeracy Reference Group.
The changes for 2005 included extending the addition, subtraction, multiplication and divisionstrategy domains to include fraction and decimal items. The place value knowledge questionswere extended to include items related to students’ ability with tenths. These items were also theresult of recommendations from the project evaluations, 2003.
The 2006 changes included refining the early stages for place value and basic facts to put moreemphasis on five as a sub-base and a simplifying of a place value task with large whole numbers.These were the result of feedback from facilitators and a Rasch analysis carried out on theknowledge items.
Interactive review of the project materials is a strength of the numeracy development projects,and will continue.
Sometimes myknowledge andstrategy stages
are different.
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Operational Strategy WindowsAddition and Subtraction
The answers to these tasks determine which form of NumPA to use. Keep a note of the student’sresponse to each question.
Task (1): Count eight objects.
Actions: Provide the student with access to a pile of counters of the same colour.
Say: Please get 8 counters for me.
Decision: If the student did not count eight items, rate him/her as stage 0 on operationalstrategies. Proceed to form A. Otherwise proceed to task (2).
Task (2): Work out 4 + 3 on materials.
Actions: Place four counters in the student’s hand. Place three counters in the student’sother hand. Ask the student to close their hands. (Open later if necessary).
Say: Please hold out your hands for me. Here are 4 counters. Here are another 3counters. How many counters have you got altogether?
Decision: If the student was unable to solve 4 + 3 correctly, rate them at stage 1. Proceed toform A. If the student solved 4 + 3 by counting the materials, rate her/him atstage 2. Proceed to form A. Otherwise proceed to task (3).
Task (3): Find 8 + 5 = □.
Actions: Place 8 counters of one colour under a card and 5 counters of another colourunder another card. Reveal the collections to the student then cover them, one at atime. Show the problem card to the student.
Say: There are 8 counters under this card and 5 counters under this card. How manycounters are there altogether?
Decision: If the student solved task (2) by imaging but did not solve task (3) by counting on,rate her/him at stage 3. Proceed to form A. Otherwise proceed to task (4). If thestudent solved task (3) by counting on, still give her/him task (4) as this mayprovoke part-whole thinking.
Task (4): Find 9 + 8 = □.
Actions: Place 9 counters of one colour under a card and 8 counters of another colourunder another card. Reveal the collections to the student then cover them, one at atime. Show the problem card to the student.
Say: There are 9 counters under this card and 8 counters under this card. How manycounters are there altogether?
Decision: If the student solved both tasks (3) and (4) by counting on, rate her/him at stage4. Proceed to form B. For students who used any part-whole strategy for either orboth tasks (3) and (4), continue to task (5).
Task (5): Find 37 – 9.
Actions: Show the card with the lolly problem on it to the student.
Say: You have 37 lollies, and you eat 9 of them. How many lollies have you got left?
Decision: If the student used any part-whole strategies on tasks (3) and (4) but counted backto solve task (5), rate her/him at stage 5. Proceed to form B. If the student used apart-whole strategy on task (5), proceed to task (6).
White
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Task (6): Find 53 – 26.
Actions: Show the card with the bus problem written on it.
Say: There are 53 people on the bus. 26 people get off. How many people are left onthe bus?
Decision: If the student imagined using a standard written method, read the notes at theend of task (8). If the student failed to solve this problem correctly, rate her/him atstage 5 and proceed to form B.
Task (7): Find 394 + 79.
Actions: Show the card with the stamp problem written on it.
Say: Sandra has 394 stamps. She gets another 79 stamps from her brother. Howmany stamps does she have then?
Decision: If the student gets both of tasks (6) and (7) correct using part-whole strategies,proceed to task 8. Otherwise rate the student at stage 5 and proceed to form B.
Task (8): Find 5.3 – 2.89 metres.
Actions: Show the card with the sewing problem written on it.
Say: Marija has a 5.3 metre length of fabric. She uses 2.89 metres of it to make atracksuit. How much fabric has she got left?
Decision: Regardless of the student’s answer, proceed to task 9.
Task (9): Find 2 – ( + ) pizza.
Actions: Show the card with the pizza problem written on it.
Say: Harry and Sally buy two pizzas. Harry eats of a pizza while his friend Sallyeats of a pizza. How much pizza is left over?
Decision: If the student gets both Tasks (8) and (9) correct using part-whole strategies, rateher/him at stage 7 for addition and subtraction. Otherwise rate the student atstage 6. Proceed to form C.
Note about Standard Written FormsIf the student’s method on any problem was to perform standard written form in their head,then no conclusions should be drawn about their strategy stage. Further questioning is needed toestablish their strategy stage. For example:
• “Can you do that in a different way?”
• “Explain how the written method works.” Look for evidence of part-whole reasoning; forexample, for 53 – 26, the student clearly understands that 53 is the same as four tens and 13ones.
White
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Stage & Behavioural Indicator Use NumPAform:
0 Emergent AThe student has no reliable strategy for counting an unstructuredcollection of items.
1 One-to-one Counting AThe student has a reliable strategy for counting an unstructuredcollection of items.
2 Counting from One on Materials AThe student’s most advanced strategy is counting from one on materialsto solve addition problems.
3 Counting from One by Imaging AThe student’s most advanced strategy is counting from one without theuse of materials to solve addition problems.
4 Advanced Counting BThe student’s most advanced strategy is counting on or countingback to solve addition or subtraction tasks.
5 Early Additive Part-Whole BThe student uses any part-whole strategy to solve addition orsubtraction problems mentally by reasoning the answer from basicfacts and/or place value knowledge.
6 Advanced Additive–Early Multiplicative Part-Whole CThe student is able to use at least two different mental strategies to solveaddition or subtraction problems with multi-digit numbers.
7 Advanced Multiplicative–Early Proportional Part-Whole CThe student is able to use at least two different mental strategies to solveaddition or subtraction problems with decimals and fractions withrelated denominators.
White
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Operational Strategy Window Flowchart
Task Form of NumPA to use
Task (1)Count eight
objects
Task (2)Work out 4 + 3on materials
Task (3)Find 8 + 5 = □
Task (4)Find 9 + 8 = □
Task (5)Find 37 – 9
Task (6)Find 53 – 26
Task (7)Find 394 + 79
Task (8)Find 5.3 – 2.89
Form A
Form B
Form C
Unable to count. Rate as stage 0.
Unable to join sets. Rate as stage 1.
Yes, using materials. Rate as stage 2.
No. Rate as stage 3.
Yes, by counting on. Rate as stage 4.
No, or Yes, by counting back. Rate as stage 5.
No, or Yes, using one part-wholestrategy. Rate as stage 5.
Yes, using at least two differentpart-whole strategies. Rate as stage 6.
Yes
Yes, using imaging
Yes, by counting on or part-whole strategy.
Yes, by part-whole strategy.
White
Task (9)Find 2 – ( + )7
834
Yes, using at least two differentpart-whole strategies. Rate as stage 7.
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Photocopiable MaterialTasks (3) to (8)
Task (3)
8 + 5 =Task (4)
9 + 8 =Task (5)
You have 37 lollies, and you eat 9 of them.How many lollies have you got left?Task (6)
There are 53 people on the bus. 26people get off. How many people areleft on the bus?
White
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Task (7)
Sandra has 394stamps. She getsanother 79 stampsfrom her brother.How many stampsdoes she have then?Task (8)
Marija has a 5.3metre length offabric. She uses2.89 metres ofit to make atracksuit.How much fabric has she got left?
Task (9)
Harry and Sally buytwo pizzas. Harryeats of a pizza whilehis friend Sally eatsof a pizza. How muchpizza is left over?
White
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Numeracy Project Assessment (NumPA) Form ATransfer the notes from the Strategy Windows tasks (pages 4 and 5) to the addition andsubtraction stage boxes on the Individual Assessment Sheet.
Knowledge Questions
Forwards Number Word Sequence (FNWS)Things the interviewer says are in bold. Comments for the interviewer appear in plain type.
(1) Start counting from 1. I will tell you when to stop. Stop at 32.
What’s the next number after...? If the student does not understand the meaning of thequestion, say: The next number after 2 is 3. So if I say 2, you say 3.What is the next number after …? (2) 5 (3) 9
For questions (4) to (7), listen carefully for confusion between “teen” and “ty”. If the student hasthis confusion, they are assessed at no higher than stage 2.
What’s the next number after ...?(4) 13 (5) 19 (6) 12 (7) 15If the student confuses “teen” and “ty” in questions (4) to (7), it is still worthwhile to askquestions (8) to (11) to see if the confusion is only with the “teen” numbers.
What’s the next number after ...?(8) 29 (9) 46 (10) 69 (11) 80 (12) 139 (13) 899
Stage & Behavioural Indicator
0 Emergent FNWSThe student cannot produce the FNWS from 1 to 10.
1 Initial FNWS up to 10The student can produce the FNWS from 1 to 10 but cannot produce the number justafter a given number in the range 1 to 10.
2 FNWS up to 10The student can produce the number just after a given number in the range 1 to 10without dropping back.
3 FNWS up to 20The student can produce the number just after a given number in the range 1 to 20without dropping back.
4 FNWS up to 100The student can produce the number just after a given number in the range 1 to 100without dropping back.
5 FNWS up to 1 000The student can produce the number just after a given number in the range 1 to1 000 without dropping back.
Blue
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For the forwards and backwards number word sequences, dropping back means that the studentsays or mentally counts several numbers up to the given number. He/she then says the numberbefore or after that number. For example, the student finds the number after five by saying one,two, three, four, five, six. A student finds the number before 14 by saying 10, 11, 12, 13, 14, thensays 13.
Backwards Number Word Sequence (BNWS)(14) Count backwards from 10. I will tell you when to stop. Stop at 0 or 1.
(15) Count backwards from 24. I will tell you when to stop. Stop at 11.
What number comes before ...?If the student does not understand the meaning of the question, say: The number that comesbefore 2 is 1. So if I say 2, you say 1.What number comes before …?(16) 3 (17) 9 (18) 5 (19) 8
For questions (20) to (23), listen carefully for confusion between “teen and “ty”. If the student hasthis confusion, they are assessed at no higher than stage 2.
What number comes before ...?(20) 16 (21) 20 (22) 17 (23) 11 (24) 13
If the student confuses “teen” and “ty” in questions (20) to (24), it is still worthwhile to askquestions (25) to (26) to see if the confusion is only with the teen numbers.
What number comes before ...?(25) 31 (26) 47 (27) 70 (28) 236 (29) 600
Stage & Behavioural Indicator
0 Emergent BNWSThe student cannot produce the BNWS from 10 to 0.
1 Initial BNWS back from 10The student can produce the BNWS from 10 to 0 but cannot produce the number justbefore a given number in the range 0 to 10.
2 BNWS back from 10The student can produce the number just before a given number in the range 0 to 10without dropping back.
3 BNWS back from 20The student can produce the BNWS from 20 to 0, and the number just before a givennumber in the range 0 to 20 without dropping back.
4 BNWS back from 100The student can produce the BNWS from 100 to 0, and the number just before a givennumber in the range 0 to 100 without dropping back.
5 BNWS back from 1 000The student can produce the BNWS from 1 000 to 0, and the number just before a givennumber in the range 0 to 1 000 without dropping back.
Blue
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Numeral Identification
What is this number? Show cards with the number written on them.
(30) 3 (31) 9 (32) 5 (33) 1 (34) 8(35) 6 (36) 0 (37) 4 (38) 2 (39) 7 (40) 10For questions (41) to (45), listen carefully for confusion between “teen” and “ty”. If the studenthas this confusion, they are assessed at stage 1.
What is this number?(41) 13 (42) 19 (43) 11 (44) 16 (45) 12If the student confuses “teen” and “ty” in questions (41) to (45), it is still worthwhile to askquestions (46) to (49) to see if the confusion is only with the “teen” numbers.
What is this number?(46) 66 (47) 43 (48) 80 (49) 38 (50) 137 (51) 702
Stage & Behavioural Indicator
0 Emergent Numeral IdentificationThe student cannot identify most of the numerals in the range 0 to 10.
1 Numerals to 10The student can identify the numerals in the range 0 to 10.
2 Numerals to 20The student can identify the numerals in the range 0 to 20.
3 Numerals to 100The student can identify one- and two-digit numbers.
4 Numerals to 1 000The student can identify two- and three-digit numbers.
Place Value
(52) Place a four-strip horizontally on the table. Now place a ten-strip beneath the four-strip.Say: Here are four dots. Here are ten more dots. How many dots are there now?Continue adding ten-strips to show 24, 34, 44, 54, 64, 74, asking the student to name thetotal number of dots each time.
If the student cannot count the total number of dots for four and ten, rate him/her at stage 0–1. Ifthe student counts all the dots to find totals of 14 and 24, rate him/her at stage 2. If the studentcounts in fives and ones to find totals of 14 and 24, rate him/her at stage 3. For students who addten each time to their previous answer when a new strip is added, 14, 24, 34, 44, ..., tentativelyrate them at stage 4. You may wish to further assess their place value understanding usingquestions 27 and 28 of Form B.
Blue
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Stage & Behavioural Indicator
0–1 EmergentThe student cannot count the number of objects in combined collections.
2 One as a UnitThe student finds the total number of objects in collections by counting all of theobjects by ones. He/she does not use ten as a counting unit.
3 Five as a Counting UnitThe student uses five as a counting unit, for example, 5, 10, 11, 12, 13, 14.
4 Ten as a Counting UnitThe student uses ten as a counting unit, for example, 10, 20, 30, 40, 41, 42, 43, 44.
Basic Facts
For questions (53) to (59), show the equation from the test booklet and read it out aloud.Instant recall of the answers is required rather than counting methods.
Tell me the answer to ...(53) 2 + 3 (54) 5 + 4 (55) 6 and what makes 10?(56) 6 + 6 (57) 9 + 9 (58) 10 + 4 (59) 7 + 10
Stage & Behavioural Indicator
0–1 EmergentThe student is unable to recall instantly facts to five, for example, 2 + 3.
2 Addition Facts to FiveThe student instantly recalls facts to five, for example, 2 + 3.
3 Addition Facts to TenThe student instantly recalls facts to ten, for example, 5 + 4, 6 + □ = 10.
4 Addition Facts with Tens and DoublesThe student recalls the doubles to 20, and teen facts, for example, 14 = 10 + 4.
Based on: Wright, R. J., Martland, J., & Stafford, A. (2000). Early Numeracy: Assessment for Teaching andIntervention. London: Paul Chapman Publications/Sage. Acknowledgement is made that some of the ideasfor questions were originally sourced from the New South Wales’s Count Me In Too Professional DevelopmentPackage (1999). Department of Education and Training Publishers. New South Wales, Australia.
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Photocopiable MaterialTasks (30) to (51)
0 1 2
3 4 5
6 7 8
9 10 11
12 13 16
19 38 43Blue
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Question 53 Question 54
2 + 3 5 + 4Question 55 Question 56
6 + = 10 6 + 6Question 57 Question 58
9 + 9 10 + 4Question 59
7 + 10
Blue
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Numeracy Project Assessment (NumPA) Form BTransfer the notes from the Strategy Windows tasks (pages 4 and 5) to the addition andsubtraction stage boxes on the Individual Assessment Sheet.
Operational Strategy Questions
Multiplication and DivisionThings the interviewer says are in bold. Comments for the interviewer appear in plain type.
(1) Here is a forest of trees. There are 5 trees in each row, and there are 8 rows. Usehorizontal and vertical sweeps with the index finger. Mask all but one horizontal and onevertical edge of the array. How many trees are there in the forest altogether? If thestudent is unable to give an answer, uncover the rest of the sheet. If I planted 15 moretrees, how many rows of 5 would I have then altogether?
If the students solves question (1) using one-to-one counting and/or skip-counting, omitquestions (2) and (3). Rate him/her at stages 2–3 or 4, as appropriate. Stop the multiplicationquestions and proceed to the questions on proportions and ratios.
For questions (2) and (3), screen the answer then uncover it if the student responds correctly. Ifthe student gives no response or an incorrect one, go to the proportions and ratios section.
(2) What is 3 � 20?If 3 � 20 = 60, what does 3 � 18 equal?Does the student derive 3 � 18 by 60 – 6 = 54?
(3) What is 5 � 8? If 5 � 8 = 40, what does 5 � 16 equal ?Does the student derive 5 � 16 = 80 by doubling 40?
Rate the student according to the most advanced strategy he/she uses on questions (2) and (3).Note that the student can know how to derive multiplication facts from other known facts, thatis, he/she could be at stage 6 but use counting on or back.
Stage & Behavioural Indicator
2–3 Counting from OneThe student solves multiplication problems by counting all of the objects.
4 Advanced CountingThe student solves multiplication problems by skip-counting, where he/she has aknown sequence or by using a combination of skip-counting and counting in ones, forexample, 5, 10, 15, 20.
5 Early Additive Part-WholeThe student solves multiplication problems by forming the factors where they have aknown multiplication fact or by using repeated addition, for example, for 5 � 8:5 + 5 = 10, 10 + 10 + 10 + 10 = 40
6 Advanced Additive–Early Multiplicative Part-WholeThe student solves multiplication problems by deriving from known multiplicationfacts, for example, 3 � 20 = 60 so 3 � 18 = 60 – (3 � 2) = 54.
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Proportions and Ratios
(4) Show the student the fraction circle sheet. Which of these cakes has been cut intothirds? If the student responds incorrectly, point to the thirds. Here are 12 jelly beans tospread out evenly on top of the cake. You eat one-third of the cake. How many jellybeans do you get? If the student cannot answer the question, allow them to manipulatethe beans or counters to solve it. If the student needs to manipulate the materials to solvequestion (4), rate them at stage 1 or 2–4, as appropriate, and proceed to the knowledgequestions.
(5) What is of 28? Does the student use a part-whole strategy based on addition and/ormultiplication?
Stage & Behavioural Indicator
1 Unequal SharingThe student is unable to find a fraction of a number by sharing the objects into equalsubsets.
2–4 Equal SharingThe student finds a fraction of a number by sharing the objects into equal subsets,physically or by imaging.
5 Early Additive Part–WholeThe student finds a unit fraction of a number mentally, using trial and improvementwith addition facts, for example, of 12 as 4 + 4 + 4 = 12.
6 Advanced Additive–Early Multiplicative Part–WholeThe student finds a fraction of a number mentally, using a combination of additionfacts and multiplication, for example, of 28 as: of 20 = 5 so of 24 = 6 so of 28 = 7, 3 � 7 = 21; or of 28 is 14, of 14 is 7,
14 + 7 = 21.
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14
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Knowledge Questions
Forwards Number Word Sequence (FNWS)
Ask question (6) only if the student is at the advanced counting stage. For other students proceedto question (7).
(6) Start counting from 10. I will tell you when to stop. Stop at 32. If the student hasproblems counting up through the teens, rate him/her at stage 2 and proceed to the BNWSquestions.
For each number I show you, read the number then tell me the number that comes justafter it, the number that is one more. For example, if I show you 4, you say 5. Show theFNWS cards. Stop at the point at which the student encounters difficulty and proceed to theBNWS questions.
(7) 12 (8) 17 (9) 29 (10) 99 (11) 209(12) 999 (13) 3 049 (14) 989 999
For the forwards and backwards number word sequences, dropping back means that the studentsays or mentally counts several numbers up to the given number. He/she then says the numberbefore or after that number. For example, the student finds the number after 25 by saying 21, 22,23, 24, 25, 26. A student finds the number before 14 by saying 10, 11, 12, 13, 14, then says 13.
Rate the student at the highest stage in which they get all relevant questions (7) to (14) correct.
Stage & Behavioural Indicator
2 FNWS up to 10The student can read and give the number just after a given number in the range 1 to 10without dropping back.
3 FNWS up to 20The student can read and produce the number just after a given number in the range 1 to20 without dropping back.
4 FNWS up to 100The student can read and produce the number just after a given number in the range 1 to100 without dropping back.
5 FNWS up to 1 000The student can read and produce the number just after a given number in the range 1 to1 000.
6 FNWS up to 1 000 000The student can read and produce the number just after a given number in the range 1 to1 000 000.
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Backwards Number Word Sequence (BNWS)
Ask question (15) only if the student is at the advanced counting stage. For other students,proceed to question (16).
(15) Start counting backwards from 23. I will tell you when to stop. Stop at 10. If thestudent has problems counting back through the teens, rate him/her at stage 2 andproceed to the fractional numbers questions.
For each number I show you, read the number then tell me the number that comes justbefore it, that is, the number that is one less. For example, if I show you 4, you say 3.Show the BNWS cards. Stop at the point the student encounters difficulty and proceed to thefractional number questions.
(16) 13 (17) 19 (18) 30 (19) 100 (20) 680(21) 900 (22) 2 400 (23) 603 000
Rate the student at the highest stage in which they get all relevant questions (16) to (23) correct.
Stage & Behavioural Indicator
2 BNWS back from 10The student can read and give the number just before a given number in the range 1 to 10without dropping back.
3 BNWS back from 20The student can read and produce the number just before a given number in the range 1to 20 without dropping back.
4 BNWS back from 100The student can read and produce the number just before a given number in the range 1to 100 without dropping back.
5 BNWS back from 1 000The student can read and produce the number just before a given number in the range 1to 1 000.
6 BNWS back from 1 000 000The student can read and produce the number just before a given number in the range 1to 1 000 000.
Yellow
The Diagnostic Interview
21
Fractional Numbers
(24) Here are some fractions. Say each fraction as I show it. Give the student the symbolcards for , , , , .
(25) Give the student the unit fraction cards from question (24). Put these fractions in orderfrom smallest over here, indicating left, to largest over here, indicating right. If correctask, Why do you think one-quarter is less than one-third? Does the student explain theeffect of increasing the bottom number (denominator) as decreasing the value of thefraction?
(26) Show the student the test booklet page with , 1 , 1 , 1, on it. Point to the fraction .
Which of these numbers are the same as ? If correct, check that the answer is not a
guess by asking Explain how you know this.
If the student orders unit fractions but cannot recognise that is equivalent to 1 or 1 , rate him/her at stage 5.
Rate the student at the highest stage in which he/she gets all relevant questions (24) to (26)correct.
Stage & Behavioural Indicator
2–3 Unit Fractions Not RecognisedThe student cannot identify symbols for unit fractions.
4 Unit Fractions RecognisedThe student can read unit fraction symbols, for example, the student can read asone-third, as one-quarter.
5 Ordered Unit FractionsThe student can compare unit fractions, for example, >
6 Co-ordinated Numerators and DenominatorsThe student describes the size of fractions with reference to both the numerator anddenominator, for example, is one whole and two-sixths or one whole and one-third.
Yellow
13
13
14
15
12
16
68
26
13
86
86
26
13
86
14
13
14
86
214
The Diagnostic Interview
22
Place Value
For the following questions, students should be rated by their fluent recall. Prolonged use ofstrategising suggests the student does not know the answer.
For each question (27) to (33), show the equation and read it aloud, or use the cards provided[Question (33)].
The student must correctly answer all of questions ...
(27), without counting, to be rated at stage 4, otherwise rate them at stage 3
(28) and (29) to be rated at stage 5
(30) and (31) to be rated at stage 6
(32) and (33) to be rated at stage 7.
Where the student shows knowledge gaps, rate him/her at the previous stage, and move to theBasic Facts section.
Tell me the answer to ...(27) A toy costs $80. How many $10 notes do you need to pay for it?(28) A radio costs $230. How many $10 notes do you need to pay for it?(29) What number is the arrow pointing to? How do you know?
Both 6.8 and 6 and 8 tenths are acceptable answers.
(30) You have $26,700 in $100 notes. How many notes do you have?(31) What number is three tenths more than 4.8? How do you know?(32) How many tenths are in all of this number? 4.67 Circle 4.67 with index finger. While 46
tenths is the expected answer, 46.7 tenths is also acceptable.
(33) Put these decimals (0.39, 0.478, 0.8) in order from smallest over here, indicating left, tolargest over here, indicating right.
Stage & Behavioural Indicator
4 Ten as a Counting UnitThe student uses ten as a counting unit, for example, 10, 20, 30, 40, 50, 60, to find thenumber of tens in 60.
5 Tens in numbers to 1 000, Tenth as a Counting UnitThe student knows how many tens are in whole numbers to 1 000 and recognisestenths among whole numbers.
6 Hundreds in Whole Numbers, Connected Tenths and OnesThe student knows how many hundreds are in any whole number to 100 000 andrecognises that ten tenths make one.
7 Tenths in Decimals/Ordered DecimalsThe student knows how many tenths are in numbers with two decimal places, forexample, 7.56 has 75 or 75.6 tenths, and orders decimals to three places, for example,0.539, 0.6, 0.72.
Yellow
The Diagnostic Interview
23
Basic Facts
For the following questions, students should be rated by their fluent recall. Prolonged use ofstrategising suggests the student does not know the answer, and must work it out. For eachquestion (34) to (48), show the equation in the test booklet and read it aloud. Cease the interviewat the line of questions at which the student has knowledge gaps and rate them using theindicators below.
What is the answer to ...(34) 2 + 3 (35) 5 + 4 (36) 6 and what makes 10?(37) 6 + 6 (38) 9 + 9 (39) 10 + 4 (40) 7 + 10(41) 8 + 6 (42) 6 + 9 (43) 8 � 5 (44) 5 � 7(45) 17 – 9 (46) 15 – 6 (47) 6 � 7 (48) 8 � 4
Stage & Behavioural Indicator
2 Addition Facts to FiveThe student instantly recalls facts to five, for example, 2 + 3.
3 Addition Facts to TenThe student instantly recalls facts to ten, for example, 5 + 4, 6 + □ = 10.
4 Addition Facts with Tens and DoublesThe student recalls the doubles to 20, and teen facts, for example, 14 = 10 + 4.
5 Addition FactsThe student recalls the basic addition facts, and the multiplication facts for 2, 5, and10.
6 Subtraction and Multiplication FactsThe student recalls the basic subtraction and multiplication facts.
Yellow
The Diagnostic Interview
24
Photocopiable MaterialQuestions (7)-(14)
12 17 29 99
209 999 3 049 989 999
Questions (16)-(23)
13 19 30 100
680 900 2 400 603 000
Yellow
The Diagnostic Interview
25
Questions (24) and (25)
1 1 1 1 13 4 5 2 6
Question 33
0.8 0.39 0.478
Yellow
The Diagnostic Interview
26 Yellow
Question 1
Here is a forest of trees. There are 5 trees in each row, and there are 8 rows.How many trees are there in the forest altogether?
If I planted 15 more trees, how many rows of 5 would I have then altogether?
The Diagnostic Interview
28 Yellow
34
Question 4
Which of these cakes has been cut intothirds?
Question 5
What is of 28?
The Diagnostic Interview
29
Question 26
Which of these numbers are thesame as ?
1 1 1Question 27
A toy costs $80. How many $10 notesdo you need to pay for it?Question 28
A radio costs $230. How many $10notes do you need to pay for it?Question 29
What number is the arrow pointing to?How do you know?
Question 30
You have $26,700 in $100 notes.How many notes do you have?Question 31
What number is three tenths more than4.8? How do you know?
Yellow
86
26
68
13
2 14
The Diagnostic Interview
30
Question 32
How many tenths are in all of thisnumber? 4.67Question 34 Question 35
2 + 3 5 + 4Question 36 Question 37
6 + = 10 6 + 6Question 38 Question 39
9 + 9 10 + 4Question 40 Question 41
7 + 10 8 + 6Question 42 Question 43
6 + 9 8 � 5Question 44 Question 45
5 � 7 17 – 9Question 46 Question 47
15 – 6 6 � 7Question 48
8 � 4
Yellow
The Diagnostic Interview
31
Numeracy Project Assessment (NumPA) Form CTransfer the notes from the Strategy Windows tasks (pages 4 and 5) to the addition andsubtraction stage boxes on the Individual Assessment Sheet.
Operational Strategy Questions
Multiplication and Division
Things the interviewer says are in bold. Comments for the interviewer appear in plain type.
(1) Here is a forest of trees. There are 5 trees in each row, and there are 8 rows.Use horizontal and vertical sweeps with index finger. Mask all but one horizontal and onevertical edge of the array. How many trees are there in the forest altogether? If thestudent is unable to give an answer, uncover the rest of the sheet.
If I planted 15 more trees, how many rows of 5 would I have then altogether?
For questions (2) and (3), screen the answer then uncover it if the student responds correctly. Ifthe student gives no response or an incorrect response, rate him/her at no higher than stage 5 onmultiplication and division and proceed to the proportions and ratios section.
(2) What is 3 � 20? If 3 � 20 = 60, what does 3 � 18 equal?Does the student derive 3 � 18 by 60 – 6 = 54?
(3) What is 5 � 8? If 5 � 8 = 40, what does 5 � 16 equal?Does the student derive 5 � 16 = 80 by doubling 40?
If the student does not derive the answers to questions (2) and (3), rate him/her at either stage 4or 5 and proceed to the questions on proportions and ratios.
(4) There are 24 muffins in each basket. How many muffins are there altogether?Does the student use a part-whole strategy such asplace value partitioning, for example, 6 � 20 = 120, 6 � 4 = 24, 120 + 24 = 144;tidy numbers, for example, 6 � 25 = 150, 150 – 6 = 144;or proportional reasoning, for example, 6 � 24 = 12 � 12 = 144 (doubling and halving)?
(5) At the car factory, they need 4 wheels to make each car. How many cars could theymake with 72 wheels?Does the student use a part-whole strategy such as standard place value partitioning,for example, 40 ÷ 4 = 10, 72 – 40 = 32, 32 ÷ 4 = 8, 10 + 8 = 18;tidy numbers, for example, 80 ÷ 4 = 20 so 72 ÷ 4 = 20 – (8 ÷ 4) = 18;reversing, for example, 10 � 4 = 40, 8 � 4 = 32 so 18 � 4 = 72 (multiplying to solve adivision problem);proportional reasoning and reversing, for example, 9 � 8 = 72 so 18 � 4 = 72(doubling and halving) so 72 ÷ 4 = 18 (reversing)?
If the student solves any of questions (4) or (5) successfully using at least two different advancedstrategies, rate him/her at stage 7 for multiplication and division and proceed to questions (6)and (7). Otherwise rate the student at stage 6.
(6) Ivan has 2.4 kilograms of mince. Each pattie takes 0.15 kilograms of mince. How manypatties can Ivan make?Does the student use a mental part-whole strategy such asdoubling, 2 � 0.15 = 0.3 so 4 � 0.15 = 0.6 so 16 � 0.15 = 2.4;reversing with rounding and place value, 10 � 0.15 = 1.5so 20 � 0.15 = 3.0 so 16 � 0.15 = 2.4?
Green
The Diagnostic Interview
32
(7) Each day on the life raft, 22 litres of water are shared equally among the 8 survivors.How much water, in litres, does each person get each day?Does the student use mental part-whole strategies such as:standard place value, 16 ÷ 8 = 2, 6 ÷ 8 = 0.75, so 22 ÷ 8 = 2.75;converting equivalent fractions to decimals, 22 ÷ 8 = 2 = 2 = 2.75?
If the student solves both questions (6) and (7) using two different advanced strategies, ratehim/her at stage 8 for multiplication and division. Otherwise rate the student at stage 7.
Where the student images a written algorithm, no assumption can be made about their stage.Question the student about their understanding of the processes involved in the algorithm andwhat other strategies they could use to solve the given problem.
Stage & Behavioural Indicator
4 Advanced CountingThe student solves multiplication problems by skip-counting where he/she has a knownsequence or by using a combination of skip-counting and counting in ones, for example,5, 10, 15, 20.
5 Early Additive Part-WholeThe student solves multiplication problems by forming the factors where they have aknown multiplication fact or using repeated addition, for example,for 6 � 4: 4 + 4 = 8, 8 + 4 = 12, 12 + 12 = 24.
6 Advanced Additive–Early Multiplicative Part-WholeThe student solves multiplication problems by deriving from known multiplication facts,for example, 3 � 20 = 60 so 3 � 18 = 60 – (3 � 2) = 54.
7 Advanced Multiplicative–Early Proportional Part-WholeThe student is able to use at least two differ ent advanced mental strategies to solvemultiplication and division problems with whole numbers.
8 Advanced Proportional Part-WholeThe student is able to use at least two differ ent advanced mental strategies to solvemultiplication and division problems with decimals and fractions with relateddenominators.
Green
68
34
The Diagnostic Interview
33
Proportions and Ratios
(8) Show the student the fraction circle sheet in the test booklet (page 43). Which of thesecakes has been cut into thirds? If the student responds incorrectly, identify (point to) thethirds. Here are 12 jelly beans to spread out evenly on top of the cake. You eat one-third of the cake. How many jelly beans do you get? If the student cannot answer thequestion, allow them to manipulate the beans or counters to find it.
For students who need to equally share the beans/counters, either with materials or by imagingthe movement of the materials, rate them at stages 2–4 for proportions and ratios and proceed tothe knowledge questions (page 36).
(9) What is of 28? Does the student use a part-whole strategy based on addition and/ormultiplication? If the student uses part-whole strategies based on addition and/or halving,continue on to question (10) as this item may lead them to using division.
(10) 12 is of a number. What is the number?
For students who are unsuccessful at question (10), rate them at either stage 5 or 6, whatever isappropriate from their response to question (9), and proceed to the knowledge questions.Answering question (10) using multiplication and division places the students at stage 7 at least.If the student uses part-whole strategies successfully for both questions (11) and (12), rate them atstage 8.
(11) It takes 10 balls of wool to make 15 beanies. How many balls of wool does it take tomake 6 beanies? Show the question in the test booklet (page 44). Does the student use apart-whole strategy based on equivalent fractions such as finding relationships betweendifferent units, for example, 10 ➝ 15 so 1➝ 1.5 so 4 ➝ 6 (unit fractions), or 6 � 2 = 15 so□ � 2 = 10; or finding relationships within the same units, for example, 10 ➝ 15 so20➝ 30 so 4 ➝ 6?
(12) There are 21 boys and 14 girls in Ana’s class. What percentage of Ana’s class areboys? Show the question in the test booklet (page 44). Does the student use a part-wholestrategy based on equivalent fractions such as finding relationships between differentunits, for example, 21 ÷ 7 = 3, 14 ÷ 7 = 2 (common factor); 21 ➝ 35 so 3 ➝ 5 so 60 ➝ 100,so = 60% are boys; or finding relationships within the same units, for example,35 � 3 = 105 so 21 � 3 = 63 and adjust down to 60%.
34
23
35
12
12
Green
The Diagnostic Interview
34
Stage & Behavioural Indicator
2–4 Equal SharingThe student finds a fraction of a number by sharing the objects into equal subsetsphysically or by imaging.
5 Early Additive Part-WholeThe student finds a unit fraction of a number mentally using trial and improvementwith addition facts, for example, of 12 as 4 + 4 + 4 = 12.
6 Advanced Additive–Early Multiplicative Part-WholeThe student finds a fraction of a number mentally using a combination of additionfacts and multiplication, for example, of 28 as: of 20 = 5 so of 24 = 6 so of 28 = 7, 3 � 7 = 21;
or of 28 is 14, of 14 is 7, 14 + 7 = 21.
7 Advanced Multiplicative–Early Proportional Part-WholeThe student finds a fraction of a number using division and multiplication, forexample, � □ = 12 so � □ = 6 so □ = 6 � 3 = 18, or 1 � 12 = □ so □ = 18.
8 Advanced Proportional Part-WholeThe student uses at least two different strategies to solve problems that involveequivalence with and between fractions, ratios, and proportions, for example,75% of 36 as of 36; or 12 ➝ 8 as □ ➝ 18: 12 ➝ 8 so 3 ➝ 2 (dividing by four) so27 ➝ 18 (multiplying by nine).
13
34
14
14
14
12
12
23
13
12
34
Green
The Diagnostic Interview
35
Knowledge Questions
Forwards and Backwards Number Word SequenceShow the number sequence cards. Stop at the point the student encounters difficulty andproceed to the fractions questions. For each number I show you, tell me the number thatcomes just after it, the number that is one more. Also tell me the number that comes justbefore it, the number that is one less.(13) 2 400 (14) 3 049 (15) 603 000 (16) 989 999
Stage & Behavioural Indicator
5 FNWS and BNWS within 1 000The student can produce the number before and after a given number in the range1 to 1 000.
6 FNWS and BNWS within 1 000 000The student can produce the number before and after a given number in the range1 to 1 000 000.
Fractional Numbers
(17) Here are some fractions. Say each fraction as I show it. Show the student the symbolcards for , , , , one at a time. Lay the cards on the table as they are shown.
If the student is unable to recognise the fractions, rate him/her at stage 2–3.
(18) Referring to the fraction cards from question (17) … Put these fractions in order fromsmallest over here, indicating left, to largest over here, indicating right. If correct, ask,Why do you think one-quarter is less than one-third? Does the student explain the effectof increasing the bottom number (denominator) as decreasing the value of the fraction?
If the student can recognise the unit fractions but cannot order them, rate him/her at stage 4.
(19) Show the student the test booklet page (page 45) with , 1 , 1 , 1, and on it. Point to the
fraction . Which of these numbers are the same as ? If the student is correct, check
that the answer is not a guess by asking Explain how you know this.
If the student orders unit fractions but cannot recognise that is equivalent to 1 or 1 , ratehim/her at stage 5.
(20) Give the student the set of mixed fractions, , , , , , . Here are some fractions. Putthem in order from smallest over here, indicating left, to largest over here, indicatingright. Does the student recognise the equivalence of two-thirds and six-ninths? If so, ratethem at stage 7 at least, if not rate them at stage 6. Rate a student who successfully ordersall the fractions in question (20) at stage 8.
86
13
14
15
12
16
68
26
13
86
25
12
23
69
86
26
13
34
Green
214
716
The Diagnostic Interview
36
Stage & Behavioural Indicator
2-3 Unit Fractions Not RecognisedThe student cannot identify symbols for unit fractions.
4 Unit Fractions RecognisedThe student can read unit fraction symbols, for example, the student can read asone-third, as one-quarter.
5 Ordered Unit FractionsThe student can compare unit fractions, for example, > .
6 Co-ordinated Numerators and DenominatorsThe student describes the size of fractions with reference to both the numerator anddenominator, for example, is one whole and two-sixths.
7 Equivalent FractionsThe student names equivalent fractions from a set of fractions with differentdenominators, for example, = , = .
8 Ordered FractionsThe student orders fractions with unlike denominators and numerators, for example, < .
13
14
13
14
86
23
34
68
25
Green
812
716
The Diagnostic Interview
37
Place ValueFor the following questions, students should be rated by their fluent recall. Prolonged use ofstrategising suggests the student does not know the answer, and must work it out.
The student must correctly answer all of questions...(21) and (22) to be rated at stage 5, otherwise rate them at stage 4.(23) and (24) to be rated at stage 6,(25) and (26) to be rated at stage 7,(27) to (30) to be rated at stage 8.
Where the student shows knowledge gaps, rate him/her at the previous stage.
(21) A radio costs $230. How many $10 notes do you need to pay for it?(22) What number is the arrow pointing to? How do you know?
Both 6.8 and 6 and 8 tenths are acceptable answers.
(23) You have $26,700 in $100 notes. How many notes do you have?(24) What number is three tenths less than 2? How do you know?(25) How many tenths are in all of this number? 4.67 Circle 4.67 with index finger.
While 46 tenths is the expected answer, 46.7 tenths is also acceptable.
(26) Put these decimals (0.39, 0.478, 0.8) in order from smallest over here, indicating left,to largest over here, indicating right.
(27) How many hundredths are in all of this number? 2.097 Circle 2.097 with index finger.While 209 hundredths is the expected answer, 209.7 hundredths is acceptable.
(28) Round 7.649 to the nearest tenth.(29) Give three numbers that are between 7.59 and 7.6. If you had time, how many
numbers could you find?(30) Name 137.5% as a decimal.
Stage & Behavioural Indicator
5 Tens in numbers to 1 000, Tenth as a Counting UnitThe student knows how many tens are in numbers to 1 000, and recognises tenths amongwhole numbers.
6 Hundreds in Whole Numbers, Connected Tenths and OnesThe student knows how many hundreds are in any whole number to 100 000, andrecognises that ten tenths make one.
7 Tenths in Decimals/Ordered DecimalsThe student knows how many tenths are in numbers with two decimal places, forexample, 7.56 has 75 or 75.6 tenths, and orders decimals to three places, for example,0.539, 0.6, 0.72.
8 Decimal ConversionsThe student knows how many hundredths are in decimals, and rounds numbers to thenearest tenth, for example, 7.649 ➝ 7.6 to the nearest tenth, not 7.7. The student canidentify decimals between others and name a percentage as a decimal and vice versa,for example, 137.5% as 1.375.
Green
The Diagnostic Interview
38
Basic Facts
For the following questions, students should be rated by their fluent recall. Prolonged use ofstrategising suggests the student does not know the answer, and must work it out. For eachquestion (31) to (43), show the equation in the test booklet (page 48) and read it aloud. Cease theinterview at the line of questions at which the student has knowledge gaps and rate them usingthe indicators below.
What is the answer to ...(31) 8 + 6 (32) 6 + 9 (33) 8 � 5 (34) 5 � 7(35) 17 – 9 (36) 15 – 6 (37) 6 � 7 (38) 8 � 4(39) 56 ÷ 7 (40) 63 ÷ 9For questions (41) to (43) explain the meaning of the terms, factor, common factor, and leastcommon multiple, if necessary.
(41) Name all the factors of 81.(42) What is the highest common factor of 72 and 81?(43) What is the least common multiple of 8 and 12?
Stage & Behavioural Indicator
4 Addition Facts with Tens and DoublesThe student recalls the doubles to 20, and “teen” facts, for example, 14 = 10 + 4.
5 Addition FactsThe student recalls the basic addition facts, and the multiplication facts for 2, 5, and 10.
6 Subtraction and Multiplication FactsThe student recalls the basic subtraction and multiplication facts.
7 Division FactsThe student recalls the basic division facts and names all the factors of numbers to 100.
8 Common Factors and MultiplesThe student names all the common factors of two numbers to 100, and the least commonmultiple of numbers to 10.
Green
The Diagnostic Interview
39
Question (20)
2 3 2 6 7 13 4 5 9 16 2
Photocopiable Material
Questions (13)–(16)
2 400 3 049 603 000 989 999
Questions (17) and (18)
1 1 1 1 13 4 5 2 6
Green
Question (26)
0.8 0.39 0.478
The Diagnostic Interview
40
Question 1
Here is a forest of trees. There are 5 trees in each row, and there are 8 rows.How many trees are there in the forest altogether?
If I planted 15 more trees, how many rows of 5 would I have then altogether?
Green
The Diagnostic Interview
41
Question 2
3 � 20 = 603 � 18 =
Question 3
5 � 8 = 405 � 16 =
Question 4
There are 24 muffins in each basket. Howmany muffins are there altogether?
Green
The Diagnostic Interview
42
Question 5
At the car factory,they need 4 wheelsto make each car.How many cars could they make with 72wheels?
Question 6
Ivan has 2.4 kilogramsof mince. Each pattietakes 0.15 kilogramsof mince.How many pattiescan Ivan make?
Question 7 Each day on thelife raft, 22 litresof water is sharedequally betweenthe 8 survivors.How much water,in litres, does eachperson get eachday?
Green
The Diagnostic Interview
43
Question 8
Which of these cakes has been cut intothirds?
Question 9
What is of 28?34
Green
The Diagnostic Interview
44
Question 10
12 is of a number.What is the number?
Question 11
It takes 10 balls of wool to make15 beanies. How many balls of wool doesit take to make 6 beanies?
Question 12
There are 21 boys and 14 girls in Ana’sclass.What percentage of Ana’s class are boys?
23
Green
The Diagnostic Interview
45
Question 19
Which of these numbers are thesame as ?
1 1 1Question 21
A radio costs $230.How many $10 notes do you need topay for it?
Question 22
What number is the arrow pointing to?How do you know?
Question 23
You have $26,700 in $100 notes.How many notes do you have?
86
26
68
13
2 14
Green
The Diagnostic Interview
46
Question 24
What number is three tenths less than2? How do you know?
Question 25
How many tenths are in all of thisnumber? 4.67
Question 27
How many hundredths are in all of thisnumber? 2.097
Question 28
Round 7.649 to the nearest tenth.
Question 29
Give three numbers that are between7.59 and 7.6. If you had time, howmany numbers could you find?
Question 30
Name 137.5% as a decimal.
The Diagnostic Interview
47
Question 31 Question 32
8 + 6 6 + 9Question 33 Question 34
8 � 5 5 � 7Question 35 Question 36
17 – 9 15 – 6Question 37 Question 38
6 � 7 8 � 4Question 39 Question 40
56 ÷ 7 63 ÷ 9Question 41
Name all the factors of 81.Question 42
What is the highest common factor of72 and 81?Question 43
What is the least common multiple of8 and 12?
Green
48
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stio
ns
Forw
ards
Num
ber
Wor
d Se
quen
ce (F
NW
S)(1
)St
art
coun
ting
fro
m 1
. I
will
tell
you
when
to
stop
. (St
op a
t 32.
)W
hat’s
the
nex
t nu
mbe
r af
ter
...?Th
e ne
xt n
umbe
r af
ter
2 is
3.
So if
I s
ay 2
, you
say
3. W
hat
is t
he n
ext
num
ber
afte
r ...?
(2)
5(3
)9
(4)
13(5
)19
(6)
12(7
)15
(8)
29(9
)46
(10)
69(1
1)80
(12)
139
(13)
899
Stag
e 0
Emer
gent
FNW
SD
oes
not s
ayFN
WS
up to
10 Com
men
ts
Blue
49
Bac
kwar
ds N
umbe
r W
ord
Sequ
ence
(BN
WS)
(14)
Coun
t ba
ckwa
rds
from
10.
I w
ill t
ell y
ou w
hen
to s
top.
(Sto
p at
0 o
r 1.
)(1
5)Co
unt
back
ward
s fr
om 2
4. I
will
tel
l you
whe
n to
sto
p. (S
top
at 1
1.)
Wha
t nu
mbe
r co
mes
bef
ore
...? T
he n
umbe
r th
at c
omes
bef
ore
2 is
1. S
o if
I s
ay 2
,yo
u sa
y 1.
Wha
t nu
mbe
r co
mes
bef
ore
...?(1
6)3
(17)
9(1
8)5
(19)
8(2
0)16
(21)
20(2
2)17
(23)
11(2
4)13
(25)
31(2
6)47
(27)
70(2
8)23
6(2
9)60
0C
omm
ents
Stag
e 1
Initi
alBN
WS
back
from
10
Says
BN
WS
back
from
10
Stag
e 2
BNW
S ba
ckfr
om 1
0Sa
ys B
NW
Sba
ck fr
om 1
0an
d n
umbe
rbe
fore
Stag
e 3
BNW
Sfr
om 2
0Sa
ys B
NW
Sba
ck fr
om 2
0an
d n
umbe
rbe
fore
Stag
e 4
BNW
Sba
ck fr
om10
0Sa
ys B
NW
Sba
ck fr
om 1
00an
d n
umbe
rbe
fore
Stag
e 5
BNW
S ba
ckfr
om 1
000
Says
BN
WS
back
from
1 0
00an
d n
umbe
rbe
fore
Stag
e 0
Emer
gent
BNW
SD
oes
not s
ayB
NW
S fr
om10 St
age
0Em
erge
ntN
umer
alId
entif
icat
ion
Stag
e 1
Iden
tifie
snu
mer
als
to10
Stag
e 2
Iden
tifie
snu
mer
als
to20
Stag
e 3
Iden
tifie
snu
mer
als
to10
0
Stag
e 4
Iden
tifie
snu
mer
als
to1
000
Num
eral
Iden
tifi
cati
on*
Wha
t is
thi
s nu
mbe
r?(3
0)3
(31)
9(3
2)5
(33)
1(3
4)8
(35)
6(3
6)0
(37)
4(3
8)2
(39)
7(4
0)10
(41)
13(4
2)19
(43)
11(4
4)16
(45)
12(4
6)66
(47)
43(4
8)80
(49)
38(5
0)13
7(5
1)70
2C
omm
ents
Plac
e V
alue
*(5
2)H
ere
are
four
dot
s. H
ere
are
ten
mor
e do
ts. H
ow m
any
dots
are
the
re n
ow?
14, 2
4, 3
4, 4
4, 5
4, 6
4, 7
4.
Bas
ic F
acts
#Te
ll m
e th
e an
swer
to
...(5
3)2
+ 3
(54)
5 +
4(5
5)6
and
wha
t mak
es 1
0?(5
6)6
+ 6
(57)
9 +
9(5
8)10
+ 4
(59)
7 +
10
Blue
Stag
e 0–
1C
anno
t cou
nt th
enu
mbe
r of
obj
ects
Stag
e 2
Cou
nts
in o
nes
Stag
e 3
Cou
nts
in fi
ves
and
ones
Stag
e 4
Cou
nts
in te
ns,
know
s nu
mbe
r of
tens
, kno
ws
plac
eva
lue
of te
ns d
igit
Com
men
ts
Stag
e 0–
1U
nabl
e to
rec
all
addi
tion
fact
s to
five
Stag
e 2
Inst
antly
reca
llsfa
cts
to fi
ve
Stag
e 3
Inst
antly
reca
llsfa
cts
to te
n
Stag
e 4
Rec
alls
dou
bles
and
teen
fact
s
Com
men
ts
50
Yello
w
Num
PA F
orm B
Ind
ividua
l Ass
essm
ent Sh
eet
* de
note
s ca
rds
need
ed#
test
boo
klet
nee
ded
Chi
ld’s
Nam
e:D
oB:
Year
:Et
hnic
ity: E
M P
A O
Gen
der:
M F
Dat
e:
Stag
e 4
Adv
ance
d C
ount
ing
Cou
nts
on
Stag
e 5
Earl
y A
dditi
ve P
art-
Who
le
Der
ives
ad
dit
ion
and
sub
trac
tion
fact
s
Ope
rati
onal
Str
ateg
y Q
uest
ions
Add
itio
n an
d Su
btra
ctio
n (S
trat
egy
Win
dow
s)#
(3)
Ther
e ar
e 8
coun
ters
und
er t
his
card
and
5 c
ount
ers
unde
r th
is c
ard.
How
man
yco
unte
rs a
re t
here
alt
oget
her?
(4)
Ther
e ar
e 9
coun
ters
und
er t
his
card
and
8 c
ount
ers
unde
r th
is c
ard.
How
man
yco
unte
rs a
re t
here
alt
oget
her?
(5)
You
have
37
lolli
es, a
nd y
ou e
at 9
of
them
. How
man
y lo
llies
hav
e yo
u go
t le
ft?
(6)
Ther
e ar
e 53
peo
ple
on th
e bu
s. 26
peo
ple
get o
ff. H
ow m
any
peop
le a
re le
ft o
n th
e bu
s?
Com
men
ts
Mul
tipl
icat
ion
and
Div
isio
n#
(1)
Her
e is
a f
ores
t of
tre
es. T
here
are
5 t
rees
in e
ach
row,
and
the
re a
re 8
row
s. H
owm
any
tree
s ar
e th
ere
in t
he f
ores
t al
toge
ther
?If
I p
lant
ed 1
5 m
ore
tree
s, h
ow m
any
rows
of
5 wo
uld
I ha
ve t
hen
alto
geth
er?
(2)
Wha
t is
3 �
20?
If 3
� 2
0 =
60, w
hat
does
3 �
18
equa
l?(3
)W
hat
is 5
� 8
?If
5 �
8 =
40,
wha
t do
es 5
� 1
6 eq
ual?
Stag
e 2–
3C
ount
from
One
Cou
nts
all t
he o
bjec
ts
Stag
e 4
Adv
ance
dC
ount
ing
Use
s sk
ip-c
ount
ing
Stag
e 5
Earl
y A
dditi
vePa
rt–W
hole
Use
s re
peat
ed a
dd
itio
nan
d/o
r us
es k
now
nm
ulti
plic
atio
n fa
cts
Com
men
ts
Stag
e 6
Adv
ance
d A
dditi
ve–
Earl
y M
ultip
licat
ive
Part
–Who
leD
eriv
es m
ulti
plic
atio
nfa
cts
Prop
orti
ons a
nd R
atio
s#
(4)
Whi
ch o
f th
ese
cake
s ha
s be
en c
ut in
to t
hird
s? H
ere
are
12 je
lly b
eans
to
spre
adou
t ev
enly
on
top
of t
he c
ake.
You
eat
one
-thi
rd o
f th
e ca
ke. H
ow m
any
jelly
bea
nsdo
you
get
?(5
)W
hat
is
of
28?
Stag
e 1
Une
qual
Sha
ring
Une
qual
ly s
hare
s ob
ject
s
Stag
e 2–
4Eq
ual S
hari
ngSh
ares
obj
ects
phy
sica
llyor
by
imag
ing
Stag
e 5
Earl
y A
dditi
vePa
rt–W
hole
Use
s ad
dit
ion
fact
s
Com
men
ts
Sta
ge 6
Adv
ance
d A
dditi
ve–
Earl
y M
ultip
licat
ive
Part
–Who
leU
ses
addi
tion
and
mul
tiplic
atio
n fa
cts
3 4
Kno
wle
dge
Que
stio
ns
Forw
ards
Num
ber W
ord
Sequ
ence
(FN
WS)
*(6
)St
art
coun
ting
fro
m 1
0. I
will
tel
l you
whe
n to
sto
p. (S
top
at 3
2.)
For
each
num
ber
I sh
ow y
ou, r
ead
the
num
ber
then
tel
l me
the
num
ber
that
com
esju
st a
fter
it, t
he n
umbe
r th
at is
one
mor
e. F
or e
xam
ple,
if I
sho
w yo
u 4,
you
say
5.
(7)
12(8
)17
(9)
29(1
0)99
(11)
209
(12)
999
(13)
3 04
9(1
4)98
9 99
9
Stag
e 2
Says
FN
WS
up to
10
Stag
e 3
Says
FN
WS
up to
20
Stag
e 4
Says
num
ber
afte
r up
to 1
00
Stag
e 5
Says
num
ber
afte
r up
to 1
000
Stag
e 6
Says
num
ber
afte
r up
to1
000
000
Com
men
ts
51
Yello
w
Stag
e 2
Says
BN
WS
back
from
10
Stag
e 3
Says
BN
WS
back
from
20
and
num
ber b
efor
e
Stag
e 4
Says
num
ber
befo
re u
p to
100
Stag
e 5
Says
num
ber
befo
re u
p to
1�00
0
Stag
e 6
Says
num
ber
befo
re u
p to
1�00
0�00
0
Bac
kwar
ds N
umbe
r Wor
d Se
quen
ce (B
NW
S)*
(15)
Star
t co
unti
ng b
ackw
ards
fro
m 2
3. I
will
tel
l you
whe
n to
sto
p. (S
top
at 1
0.)
For
each
num
ber
I sh
ow y
ou, r
ead
the
num
ber
then
tel
l me
the
num
ber
that
com
es ju
stbe
fore
it, t
hat
is, t
he n
umbe
r th
at is
one
less
. For
exa
mpl
e, if
I s
how
you
4, y
ou s
ay 3
.(1
6)13
(17)
19(1
8)30
(19)
10
0(2
0)68
0(2
1)90
0(2
2)
2 40
0(2
3)60
3 00
0C
omm
ents
Stag
e 2–
3D
oes
not r
ecog
nise
unit
frac
tions
Stag
e 4
Rec
ogni
ses
unit
frac
tions
Stag
e 5
Ord
ers
unit
frac
tions
Stag
e 6
Co-
ordi
nate
snu
mer
ator
s an
dde
nom
inat
ors
Com
men
ts
Frac
tion
al N
umbe
rs#
*(2
4)H
ere
are
som
e fr
acti
ons.
Say
eac
h fr
acti
on a
s I
show
it. (
, ,
, ,
)(2
5)Pu
t th
ese
frac
tion
s (fr
om q
uest
ion
24) i
n or
der
from
sm
alle
st o
ver
here
to
larg
est
over
here
. (If
corr
ect a
sk ..
.) W
hy d
o yo
u th
ink
one-
quar
ter
is le
ss t
han
one-
thir
d?(2
6)W
hich
of
thes
e nu
mbe
rs is
the
sam
e as
(p
oint
ing
to
)? (S
how
the n
umbe
rs,
, 1, 1
, 1,
,in
the t
est b
ookl
et) E
xpla
in h
ow y
ou k
now
this
.
Plac
e V
alue
*#
Tell
me
the
answ
er t
o ...
(27)
A t
oy c
osts
$80
. How
man
y $1
0 no
tes
do y
ou n
eed
to p
ay f
or it
?(2
8)A
rad
io c
osts
$23
0. H
ow m
any
$10
note
s do
you
nee
d to
pay
for
it?
(29)
Wha
t nu
mbe
r is
the
arr
ow p
oint
ing
to?
How
do
you
know
?(3
0)Yo
u ha
ve $
26,7
00 in
$10
0 no
tes.
How
man
y no
tes
do y
ou h
ave?
(31)
Wha
t nu
mbe
r is
thr
ee t
enth
s m
ore
than
4.8
? H
ow d
o yo
u kn
ow?
(32)
How
man
y te
nths
are
in a
ll of
thi
s nu
mbe
r? 4
.67
(33)
Put t
hese
dec
imal
s (0
.39,
0.4
78, 0
.8) i
n or
der
from
sm
alle
st o
ver
here
to la
rges
t ove
r he
re.
Bas
ic F
acts
#W
hat
is t
he a
nswe
r to
...?(3
4)2
+ 3
(35)
5 +
4(3
6)6
and
wha
t mak
es 1
0?(3
7)6
+ 6
(38)
9 +
9(3
9)10
+ 4
(40)
7 +
10(4
1)8
+ 6
(42)
6 +
9(4
3)8
� 5
(44)
5 �
7(4
5)17
– 9
(46)
15 –
6(4
7)6
� 7
(48)
8 �
4
1 31 4
1 51 2
1 6
6 82 6
1 38 6
8 6
Stag
e 4
Cou
nts
in te
ns
Stag
e 5
Kno
ws
tens
innu
mbe
rs to
100
0,te
nths
am
ong
who
le n
umbe
rs
Stag
e 6
Kno
ws
hund
reds
in w
hole
num
bers
,co
nnec
ts te
nths
and
ones
Stag
e 7
Kno
ws
num
ber
ofte
nths
in d
ecim
als,
orde
rs d
ecim
als
Stag
e 2
Inst
antly
reca
lls fa
cts
tofiv
e
Stag
e 3
Inst
antly
reca
lls fa
cts
for
ten
Stag
e 4
Rec
alls
doub
les
and
teen
fact
s
Stag
e 5
Add
ition
fact
san
dm
ultip
licat
ion
fact
s fo
r 2,
5, 1
0
Stag
e 6
Subt
ract
ion
and
mul
tiplic
atio
nfa
cts
Com
men
ts
Com
men
ts
2 14
52
Gree
n
Num
PA F
orm C
Ind
ividua
l Ass
essm
ent Sh
eet
* de
note
s ca
rds
need
ed#
test
boo
klet
nee
ded
Chi
ld’s
Nam
e:D
oB:
Year
:Et
hnic
ity: E
M P
A O
Gen
der:
M F
Dat
e:
Stag
e 6
Adv
ance
d A
dditi
ve–E
arly
Mul
tiplic
ativ
ePa
rt–W
hole
Use
s at
leas
t tw
o d
iffer
ent a
dva
nced
men
tal p
art-
who
le s
trat
egie
s
Com
men
ts
Ope
rati
onal
Str
ateg
y Q
uest
ions
Add
itio
n an
d Su
btra
ctio
n (S
trat
egy
Win
dow
s)#
(6)
Ther
e ar
e 53
peo
ple
on th
e bu
s. 26
peo
ple
get o
ff. H
ow m
any
peop
le a
re le
ft o
n th
e bu
s?(7
)Sa
ndra
has
394
sta
mps
. She
get
s an
othe
r 79
sta
mps
fro
m h
er b
roth
er. H
ow m
any
stam
ps d
oes
she
have
the
n?(8
)M
arija
has
a 5
.3 m
etre
leng
th o
f fa
bric
. She
use
s 2.
89 m
etre
s of
it t
o m
ake
atr
acks
uit.
How
muc
h fa
bric
has
she
got
left
?(9
)H
arry
and
Sal
ly b
uy t
wo p
izza
s. H
arry
eat
sof
a p
izza
whi
le h
is f
rien
d Sa
llyea
tsof
a p
izza
. How
muc
h pi
zza
is le
ft o
ver?
Mul
tipl
icat
ion
and
Div
isio
n#
(1)
Her
e is
a f
ores
t of
tre
es. T
here
are
5 t
rees
in e
ach
row,
and
the
re a
re 8
row
s. H
owm
any
tree
s ar
e th
ere
in t
he f
ores
t al
toge
ther
?If
I p
lant
ed 1
5 m
ore
tree
s, h
ow m
any
rows
of
5 wo
uld
I ha
ve t
hen
alto
geth
er?
(2)
Wha
t is
3 �
20?
If 3
� 2
0 =
60, w
hat
does
3 �
18
equa
l?(3
)W
hat
is 5
� 8
?If
5 �
8 =
40,
wha
t do
es 5
� 1
6 eq
ual?
(4)
Ther
e ar
e 24
muf
fins
in e
ach
bask
et. H
ow m
any
muf
fins
are
the
re a
ltog
ethe
r?(5
)A
t th
e ca
r fa
ctor
y, t
hey
need
4 w
heel
s to
mak
e ea
ch c
ar. H
ow m
any
cars
cou
ld t
hey
mak
e wi
th 7
2 wh
eels
?(6
)Iv
an h
as 2
.4 k
ilogr
ams
of m
ince
. Eac
h pa
ttie
tak
es 0
.15 k
ilogr
ams
of m
ince
.H
ow m
any
patt
ies
can
Ivan
mak
e?(7
)Ea
ch d
ay o
n th
e lif
e ra
ft, 2
2 lit
res
of w
ater
is s
hare
d eq
ually
am
ong
the
8 su
rviv
ors.
How
muc
h wa
ter,
in li
tres
, doe
s ea
ch p
erso
n ge
t ea
ch d
ay?
Prop
orti
ons a
nd R
atio
s#
(8)
Whi
ch o
f th
ese
cake
s has
bee
n cu
t int
o th
irds?
Her
e ar
e 12
jelly
bea
ns to
spre
ad o
utev
enly
on
top
of th
e ca
ke. Y
ou e
at o
ne-t
hird
of
the
cake
. How
man
y je
lly b
eans
do
you
get?
(9)
Wha
t is
of 2
8?(1
0)12
isof
a n
umbe
r. W
hat
is t
he n
umbe
r?(1
1)It
tak
es 1
0 ba
lls o
f wo
ol t
o m
ake
15 b
eani
es. H
ow m
any
balls
of
wool
doe
s it
tak
e to
mak
e 6
bean
ies?
(12)
Ther
e ar
e 21
boy
s an
d 14
girl
s in
Ana
’s cl
ass.
Wha
t per
cent
age
of A
na’s
clas
s ar
e bo
ys?
3 42 3
3 47 8
Stag
e 4
Adv
ance
dC
ount
ing
Use
s sk
ip-c
ount
ing
Stag
e 5
Earl
yA
dditi
vePa
rt–W
hole
Use
s re
peat
edad
dit
ion
and
/or
uses
kno
wn
mul
tipl
icat
ion
fact
s
Stag
e 6
Adv
ance
dA
dditi
ve–
Earl
yM
ultip
licat
ive
Part
–Who
leD
eriv
esm
ulti
plic
atio
n fa
cts
Stag
e 7
Adv
ance
dM
ultip
licat
ive
–Ear
lyPr
opor
tiona
lPa
rt–W
hole
Use
s at
leas
t tw
od
iffer
ent a
dva
nced
men
tal s
trat
egie
s
Stag
e 2–
4Eq
ual s
hari
ngof
obj
ects
phys
ical
ly o
rby
imag
ing
Stag
e 5
Earl
yA
dditi
vePa
rt–W
hole
Use
s ad
dit
ion
fact
s
Stag
e 6
Adv
ance
dA
dditi
ve–
Earl
yM
ultip
licat
ive
Part
–Who
leU
ses
add
itio
n w
ith
mul
tipl
icat
ion
and
div
isio
n fa
cts
Stag
e 7
Adv
ance
dM
ultip
licat
ive–
Earl
yPr
opor
tiona
lPa
rt–W
hole
Find
s fr
acti
ons
ofnu
mbe
rs u
sing
mul
tipl
icat
ion
and
div
isio
n
Stag
e 8
Adv
ance
dPr
opor
tiona
lPa
rt–W
hole
Use
s at
leas
t tw
od
iffer
ent a
dva
nced
men
tal s
trat
egie
s
Com
men
ts
Stag
e 8
Adv
ance
dPr
opor
tiona
lPa
rt–W
hole
Use
s tw
o d
iffer
ent
adva
nced
stra
tegi
es fo
rd
ecim
als
and
frac
iton
s
Stag
e 7
Adv
ance
d M
ultip
licat
ive–
Earl
yPr
opor
tiona
l Par
t–W
hole
Use
s at
leas
t tw
o d
iffer
ent s
trat
egie
s w
ith
dec
imal
san
d fr
acti
ons
Com
men
ts
53
Gree
n
Bas
ic F
acts
#W
hat
is t
he a
nswe
r to
...?(3
1)8
+ 6
(32)
6 +
9(3
3)8
� 5
(34)
5 �
7(3
5)17
– 9
(36)
15 –
6(3
7)6
� 7
(38)
8 �
4(3
9)56
÷ 7
(40)
63 ÷
9(4
1)N
ame
all t
he f
acto
rs o
f 81
.(4
2)W
hat
is t
he h
ighe
st c
omm
on f
acto
r of
72
and
81?
(43)
Wha
t is
the
leas
t co
mm
on m
ulti
ple
of 8
and
12?
Kno
wle
dge
Que
stio
ns
Forw
ards
and
Bac
kwar
ds N
umbe
r Wor
d Se
quen
ce*
For
each
num
ber
I sh
ow y
ou, t
ell m
e th
e nu
mbe
r th
at c
omes
just
aft
er it
, the
num
ber
that
is o
ne m
ore.
Als
o te
ll m
e th
e nu
mbe
r th
at c
omes
just
bef
ore
it, t
he n
umbe
r th
at is
one
less
.(1
3)2
400
(14)
3 04
9(1
5)60
3 00
0(1
6)98
9 99
9
Frac
tion
al N
umbe
rs#
*(1
7)H
ere
are
som
e fr
acti
ons.
Say
eac
h fr
acti
on a
s I
show
it. (
, ,
, ,
)(1
8)Pu
t th
ese
frac
tion
s (fr
om q
uest
ion
17) i
n or
der
from
sm
alle
st o
ver
here
to
larg
est
over
here
. (If
corr
ect a
sk ..
.) W
hy d
o yo
u th
ink
one-
quar
ter
is le
ss t
han
one-
thir
d?(1
9)W
hich
of
thes
e nu
mbe
rs is
the
sam
e as
(p
oint
ing
to
)? (S
how
the n
umbe
rs,
, 1, 1
, 1,
,in
the t
est b
ookl
et) E
xpla
in h
ow y
ou k
now
this
.(2
0)H
ere
are
som
e fr
acti
ons (
, ,
, ,
, ).
Put
them
in o
rder
fro
m s
mal
lest
ove
r he
re t
ola
rges
t ov
er h
ere.
Plac
e V
alue
#*
Tell
me
the
answ
er t
o?(2
1)A
rad
io c
osts
$23
0. H
ow m
any
$10
note
s do
you
nee
d to
pay
for
it?
(22)
Wha
t nu
mbe
r is
the
arr
ow p
oint
ing
to?
How
do
you
know
? (6
.8 o
r 6
and
8 te
nths
)(2
3)Yo
u ha
ve $
26,7
00 in
$10
0 no
tes.
How
man
y no
tes
do y
ou h
ave?
(24)
Wha
t nu
mbe
r is
thr
ee t
enth
s le
ss t
han
2? H
ow d
o yo
u kn
ow?
(25)
How
man
y te
nths
are
in a
ll of
thi
s nu
mbe
r? 4
.67
(26)
Put t
hese
dec
imal
s (0.
39, 0
.478
, 0.8
) in
orde
r fr
om s
mal
lest
ove
r he
re to
larg
est o
ver
here
.(2
7)H
ow m
any
hund
redt
hs a
re in
all
of t
his
num
ber?
2.0
97(2
8)Ro
und
7.64
9 to
the
nea
rest
ten
th.
(29)
Give
thr
ee n
umbe
rs t
hat
are
betw
een
7.59
and
7.6
? If
you
hav
e ti
me,
how
man
ynu
mbe
rs c
ould
you
fin
d?(3
0)N
ame
137.
5% a
s a
deci
mal
.
Stag
e 5
Says
num
ber
befo
re o
r af
ter
up to
1 0
00
Stag
e 6
Says
num
ber
befo
re o
r af
ter
up to
1 0
00 0
00
Com
men
ts
Stag
e 4
Rec
alls
doub
les
and
teen
fact
s
Stag
e 5
Add
ition
fact
san
dm
ultip
licat
ion
fact
s fo
r 2,
5, 1
0
Stag
e 6
Subt
ract
ion
and
mul
tiplic
atio
nfa
cts
Stag
e 7
Div
isio
n fa
cts
Stag
e 8
Com
mon
fact
ors
and
mul
tiple
s
1 31 4
1 51 2
1 6
6 82 6
1 38 6
8 6
2 33 4
2 51 2
6 9
Stag
e 4
Rec
ogni
ses
unit
frac
tions
Stag
e 5
Ord
ers
unit
frac
tions
Stag
e 6
Co-
ordi
nate
snu
mer
ator
san
dde
nom
inat
ors
Stag
e 7
Rec
ogni
ses
equi
vale
ntfr
actio
ns
Stag
e 8
Ord
ers
frac
tions
with
unlik
ede
nom
inat
ors
and
num
erat
ors
Stag
e2–
3D
oes
not
reco
gnis
eun
itfr
actio
ns
Com
men
ts
Stag
e 4
Cou
nts
in te
ns
Stag
e 5
Kno
ws
tens
innu
mbe
rs to
1000
, ten
ths
amon
g w
hole
num
bers
Stag
e 6
Kno
ws
hund
reds
inw
hole
num
bers
,co
nnec
ts te
nths
and
ones
Stag
e 7
Kno
ws
num
ber
ofte
nths
inde
cim
als,
orde
rsde
cim
als
Stag
e 8
Kno
ws
hund
redt
hs in
deci
mal
s,de
cim
als
betw
een
othe
rs, n
ames
perc
enta
ges
asde
cim
als
and
vice
ver
sa.
Com
men
ts
Com
men
ts
2 14
7 16
The Diagnostic Interview
54
Answers to Form COperational Strategy Questions
Addition and Subtraction(6) 27 people (7) 473 stamps (8) 2.41 metres (8) pizza
Multiplication and Division(1) 40 trees 3 more rows making 11 rows in total
(2) 3 � 18 = 54 (60 – 6) (3) 5 � 16 = 80 (double 40)
(4) 144 muffins (5) 18 cars
(6) 16 patties (7) 2.75 litres or 2 litres
Proportions and Ratios(8) 4 jellybeans (9) of 28 is 21
(10) of 18 is 12 (11) 4 balls of wool makes 6 beanies
(12) 60% of Ana’s class are boys
Knowledge Questions
Forwards and Backwards Number Word Sequence(13) 2 401, 2 399 (14) 3 050, 3 048 (15) 603 001, 602 999 (16) 990 000, 989 998
Fractional Numbers(17) one-third, one-quarter, one-fifth, one-half, one-sixth
(18)
(19) 1 and 1 are the same as
(20)( ) ( = )
Place Value(21) 23 (22) 6.8 or 6 and 8 tenths
(23) 267 $100 notes (24) 1.7 or 1 and 7 tenths
(25) 46 or 46.7 (26) 0.39 0.478 0.8
(27) 209 or 209.7 (28) 7.6
(29) 7.591 – 7.599, etc. (30) 1.375Infinite number ofdecimals are possible
Basic Facts(31) 14 (32) 15 (33) 40 (34) 35
(35) 8 (36) 9 (37) 42 (38) 32
(39) 8 (40) 7 (41) 1, 3, 9, 27, 81 (42) 9
(43) 24
34
23
16
15
14
13
12
26
13
86
25
12
69
34
23
69
23
Green
38
34
716
ACKNOWLEDGMENTSThe Ministry of Education wishes to acknowledge thefollowing people and organisations for their contributiontowards the development of this handbook.
THE PARTICIPANTS:The New Zealand numeracy project personnel –facilitators and principals, teachers, and children frommore than eighteen hundred New Zealand schools whocontributed to this handbook through their participationin the numeracy development projects from 2000–2005.
THE NUMERACYREFERENCE GROUP:Professor Derek Holton, convenor (The University ofOtago), Professor Megan Clark (Victoria University ofWellington), Dr Joanna Higgins (Victoria University ofWellington College of Education), Dr Gill Thomas (MathsTechnology Limited), Associate Professor Jenny Young-Loveridge (The University of Waikato), AssociateProfessor Glenda Anthony (Massey University), TonyTrinick (The University of Auckland Faculty ofEducation), Garry Nathan (The University of Auckland),Paul Vincent (Education Review Office), Dr JoannaWood (New Zealand Association of MathematicsTeachers), Peter Hughes (The University of AucklandFaculty of Education), Vince Wright (The University ofWaikato School Support Services), Geoff Woolford(Parallel Services), Kevin Hannah (Christchurch Collegeof Education), Chris Haines (School Trustees’Association), Linda Woon (NZPF), Jo Jenks (VictoriaUniversity of Wellington College of Education, EarlyChildhood Division), Bill Noble (New ZealandAssociation of Intermediate and Middle Schools), DianeLeggatt of Karori Normal School (NZEI Te Riu Roa),Sului Mamea (Pacific Island Advisory Group, PalmerstonNorth), Dr Sally Peters (The University of WaikatoSchool of Education), Pauline McNeill of ColumbaCollege (PPTA), Dr Ian Christensen (He Kupenga Hao ite Reo), Liz Ely (Education Review Office), Ro Parsons(Ministry of Education), Malcolm Hyland (Ministry ofEducation).
THE ORIGINAL WRITERS,REVIEWERS AND PUBLISHERS:Vince Wright (The University of Waikato School SupportServices), Peter Hughes (The University of AucklandFaculty of Education), Marilyn Holmes (Dunedin Collegeof Education), Sarah Martin (Bayview School), GaynorTerrill (The University of Waikato School of Education),Carla McNeill (The University of Waikato School of
Education), Professor Derek Holton (The University ofOtago), Dr Gill Thomas (Maths Technology Limited),Bruce Moody (mathematics consultant), Lynne Petersen(Dominion Road School), Errolyn Taane (DunedinCollege of Education), Lynn Tozer (Dunedin College ofEducation), MaIcolm Hyland (Ministry of Education), RoParsons (Ministry of Education), Kathy Campbell(mathematics consultant), Tania Cotter, JocelynCranefield, Kirsty Farquharson, Jan Kokason, BronwenWall (Learning Media Limited), Joe Morrison, AndrewTagg (Maths Technology Limited).
REVISED EDITION 2006:Vince Wright (The University of Waikato School SupportServices), Susan Roche (Learning Media Limited),Andrew Tagg (Maths Technology Limited).
In addition, the Ministry of Education wishes toacknowledge Professor Bob Wright (Southem CrossUniversity, Lismore, NSW), Dr Noel Thomas (CharlesSturt University, Bathhurst, NSW), Dr KoenoGravemeijer (Freudenthal Institute, Utrecht,Netherlands), Jim Martland (The University ofManchester, UK), Dr Ian Christensen (MasseyUniversity).
The Ministry of Education also wishes to acknowledgeThe New South Wales Department of Education andTraining for permission to trial Count Me In Too in 2000through a one-year arrangement. The findings from theuse of this pilot project informed the development of thenumeracy policy in New Zealand.
Count Me In Too is the registered Trade Mark of theCrown in the Right of the State of New South Wales(Department of Education and Training). Copyright ofthe Count Me In Too Professional DevelopmentPackage materials (1997–2002), including the LearningFramework in Number and the Schedule for EarlyNumber Assessment, is also held by the Crown in theRight of the State of New South Wales (Department ofEducation and Training) 2002.
The cover design is by Dave Maunder (Leaming MediaLimited) and Base Two Design Ltd. All other illustrationsare by Noel Eley and James Rae.
All illustrations copyright © Crown 2008 except: Digitalimagery of the conch copyright © 2000 PhotoDisc, Inc.