© Dianne Siemon 1

Nailing the Big Ideas in Number

Professor Dianne Siemon

National Partnerships Schools Forum

Melbourne Convention and Exhibition Centre, Room 204

7 March 2011

© Dianne Siemon 2

Overview

• What we’ve learnt from Middle Years‟

research?

• Effective teachers of mathematics

• The relevance of the E5 Instructional Model

to the teaching and learning of mathematics

• E5- Assessment FOR learning

• E5- Strengthening connections – The case of

fractions and decimals

• E5 - Facilitating substantive conversations –

Challenging but accessible tasks

© Dianne Siemon 3

0. No response or „yes‟ or „no‟ without

a reason.

1. Reasoning based on numbers

alone, no recognition that „big‟ is

relative.

2. Reasoning shows some

recognition that „big‟ is relative to

total sales, but unsupported

conclusion, little/no explanation,

eg, “it depends ...”.

3. Reasoning concludes that increase

is not „big‟ relative to total sales,

some attempt to relate this to

proportion, eg, “15 out of 725 is not

very big”.

4. Correct conclusion, “not big”, %,

fractions, ratio used correctly to

support well-reasoned explanation.

MYNRP Report (Siemon, Virgona & Corneille, 2001)

Rich tasks and scoring rubrics.

What we’ve learnt from research in MYS

© Dianne Siemon 4

The Middle Years Numeracy Research Project

(MYNRP, 1999-2001):

• there is a significant „dip‟ in Year 7 and 8 performance

relative to Years 6 and 9

9

9.2

9.4

9.6

9.8

10

10.2

10.4

10.6

10.8

Year 5 Year 6 Year 7 Year 8 Year 9

Mean Adjusted Logit Scores by Year Level, November 1999 (N

= 6859)

Differences

between all

year levels

significant

except for

Year 6/Year 9

comparison

© Dianne Siemon 5

An interesting observation:

Mean Adjusted Logit Scores by location, November 1999

(Nmetro = 4303, Nrural = 2556)

8.5

9

9.5

10

10.5

11

11.5

Year 5 Year 6 Year 7 Year 8 Year 9

Rural

Metro

© Dianne Siemon 6

• there is as much difference within Year levels as

between Year levels (spread);

• there is considerable within school variation

(suggesting individual teachers make a significant

difference to student learning);

• the needs of many students, but particularly those „at

risk‟ or „left behind‟, are not being met; and

• differences in performance were largely due to an

inadequate understanding of fractions, decimals, and

proportion, and a reluctance/inability to explain/justify

solutions.

Siemon, D., Virgona, J. & Corneille, K. (2001) Final Report of Middle Years Numeracy

Research Project 1999-2001, RMIT University: Melbourne

Further observations:

© Dianne Siemon 7

“Change the way it’s explained, they need to think about how

you understand, not how they explain” (Vincent, Year 9)

The most critical element in their learning from the students‟

perspective is the quality of teacher explanations, in

particular, the capacity of teachers to connect with their level

of understanding and communicate effectively.

In their own words (MYNRP, 1999-2001):

“Don’t understand how it is set out, don’t like to write it down

if I don’t understand … idea there, but how to write it, what to

do with it ” (Carl, Year 9)

Student engagement is related to the capacity to read, write,

speak and listen to mathematical texts (communicative

competence), that is, access to the forms of communication

used in mathematics.

© Dianne Siemon 8

[Last enjoy maths?] “in class recently, doing fractions,

changing fractions to decimals, it was good because I

actually understood it and I felt better” (Matt, Year 6)

Success is crucial to engagement.

Relevance is about connectedness, not necessarily about

immediately applicable, „real-world‟ tasks, but about being

able to access what is seen to translate to further

opportunities to study „real maths‟ and access to „good‟

jobs.

Self-esteem - students believe that mathematics is

important and relevant, they generally want to learn and

be able to apply mathematics. Mathematics is not

perceived to be as „boring‟ or irrelevant as is often

assumed.

© Dianne Siemon 9

• Plan to provide access and success for all

• Accurate and reliable assessment is essential to identify

where to start teaching.

• The teaching focus needs to be carefully targeted on

scaffolding student‟s learning needs

• Extensive professional development is needed to equip

teachers of mathematics with knowledge and skills to

probe students understanding, support conversations

about the ways in which mathematics is represented

and used, and scaffold mathematical thinking.

• „Traditional‟ text-only based approaches are seen as a

major impediment to engagement and successful

learning.

Implications for teaching and learning:

(MYNRP, Final Report, 2001)

© Dianne Siemon 10

0

2

4

6

8

10

12

14

Year 4 Year 5 Year 6 Year 7 Year 8

Overall

Boys

Girls

Adjusted Mean Logit Scores by Year Level and Gender for Victorian

Students, Initial Phase, May 2004 (N=2064)

The Scaffolding Numeracy in the Middle Years

Research Project (SNMY, 2003-2006):

Rich tasks designed to assess the development of

multiplicative thinking in Years 4 to 8

© Dianne Siemon 11

• the vast majority of students appear to have little difficulty

using additive thinking strategies to solve problems

involving relatively small whole numbers;

• students can work with sharing division, simple

proportion, and simple Cartesian product problems earlier

than expected;

• while initial ideas for multiplication and division appear

relatively early, students may take many years to develop

a flexible capacity for multiplicative thinking, particularly

as it applies to rational number;

• a significant number of students are performing below

curriculum expectations in relation to multiplicative

thinking - at least 25% of students at each Year level

might be regarded as „learners left behind‟.

Observations from the SNMY (2003-2006):

© Dianne Siemon 12

Learning and

Assessment

Framework for

Multiplicative

Thinking

(SNMY, 2004)

4/5

5

Inferred

relationship

between LAF

Levels

(Zones) and

CSF/VELS

Levels

4

3/4

3

2/3

2

1/2

© Dianne Siemon 13

0%

20%

40%

60%

80%

100%

Year 4 Year 5 Year 6 Year 7 Year 8

Level 8

Level 7

Level 6

Level 5

Level 4

Level 3

Level 2

Level 1

Proportion of Victorian Students at each Level of the LAF by Year Level,

Initial Phase, May 2004 (N=2064)

Distribution of students by LAF level

© Dianne Siemon 14

This suggests that up to 25% of Australian Year 8 and 9

students do not have the foundation knowledge and

skills needed to participate effectively in further

school mathematics, or to access a wide range of post-

compulsory training opportunities (Siemon & Virgona,

2001; Thomson & Fleming, 2004; Siemon et al, 2006).

The personal, social and economic costs of failing to

address this issue are extremely high. It has been

estimated that the cost of early school leaving, a direct

consequence of underachievement in literacy and

numeracy according to McIntyre and Melville (2005), is

$2.6 billion/year!

CHANGE IS NEEDED

© Dianne Siemon 15

Effective teachers of mathematics:

From: Clarke, D. & Clarke, B. (2002) Stories from the classrooms of successful

mathematics teachers: Painting a picture of effective practice. Paper presented

to Early Numeracy Trainers, Melbourne, March 25-27.

• Focus on important mathematical ideas

• Structure purposeful tasks

• Use a range of materials,

representations, contexts, and

language

• Use teachable moments as they occur

and make connections to relevant prior

knowledge/experience

• Use a variety of question types to

probe and challenge children‟s thinking

© Dianne Siemon 16

From: Clarke, D. & Clarke, B. (2002) Stories from the classrooms of successful mathematics

teachers: Painting a picture of effective practice. Paper presented to Early Numeracy Trainers,

Melbourne, March 25-27.

• Have high but realistic expectations of all

children, promote and value effort, persistence

and concentration

• Use a variety of assessment methods to

collect data and modify planning as a result …

• Believe that mathematics learning can and

should be enjoyable

• Are confident in their own knowledge of

mathematics at the level they are teaching

• Show pride and pleasure in individuals‟

success.

Effective teachers of mathematics:

© Dianne Siemon 17

Effective teachers of mathematics pay attention to:

From: Askew, M. (1999) It ain‟t (just) what you do: Effective teachers of numeracy. In I. Thompson (Ed.)

Issues in teaching numeracy in primary schools (91-102). Buckingham, UK: Oxford University Press

• connections between different aspects of

mathematics, for example, addition and

subtraction or fractions, decimals and

percentages

• connections between different

representations of mathematics: moving

between symbols, words, diagrams and objects

• connections with children’s methods –

valuing these and being interested in children‟s

thinking but also sharing their methods

© Dianne Siemon 18

Activity 1.

Calculate: 35 x 25

Student A

35

x 25

125

+ 75

875

Student B

35

x 25

175

700

875

Student C

35

x 25

25

150

100

+ 600

875

Which of these students is using a method that could be

used to multiply any two whole numbers?1

1 Copyright © 2006 The Regents of the University of Michigan. For information, questions, or permission requests please contact Merrie Blunk,

Learning Mathematics for Teaching, 734-615-7632. not for reproduction or use without written consent of LMT. Measures development

supported by NSF grants REC-9979873, REC-0207649, EHR-0233456 & EHR-0335411, and by subcontract to OPRE on Department of

Education (DOE), Office of Educational Research and Improvement (OERI) award #R308A960003.

© Dianne Siemon 19

Prompts inquiry,

structures inquiry,

maintains

momentum.

Assesses performance

against standards,

facilitates student self

assessment

Develops shared norms,

determines readiness for

learning, establishes

learning goals, develops

metacognitive capacity.

The E5 model and mathematics

ENGAGE

EXPLORE

EXPLAIN

ELABORATE

EVALUATE

Presents new content,

develops language and

literacy, strengthens

connectionsTeacher Capabilities

Facilitates

substantive

conversation,

cultivates higher order

thinking, monitors

progress

© Dianne Siemon 20

A ‘really big’ idea …

Is an idea, strategy, or way of thinking about some

key aspect of mathematics, without which students‟

progress in mathematics will be seriously impacted

Encompasses and connects many other ideas and

strategies

Provides an organising structure or a frame of

reference that supports further learning and

generalisations

Cannot be clearly defined but can be observed in

activity …(Siemon, 2006)

Working with the Big Ideas in Number:

© Dianne Siemon 21

Why is this important?

Too many students are being „left behind‟ in the middle

years;

Overcrowded, undifferentiated, often obscure

mathematics curriculum;

Need to support the growing number of out-of-field

teachers of mathematics in more appropriate ways; and

Need to challenge the pedagogical assumptions

inherent in the represented or „packaged‟ curriculum.

© Dianne Siemon 22

For example,

Place-Value

Counts using number

naming sequence to

determine how many

Counts collections

by 2s, 5s and 10s

Composite units

Subitising

Matches number

words and symbols to

collections

Recognises

numbers 0 to 5

without counting

Ten of these is

one of those

Identifies 1 more

than/1 less than a

given number

Demonstrates a

knowledge of

numbers to 10 in

terms of their parts

Trusts the count

A thousand of

these is one of

those

Demonstrates a

sense of numbers

beyond 10

One tenth of

these is one of

those

Base ten

structure

© Dianne Siemon 23

E5 - Assessment FOR learning:

Teaching informed by quality assessment data has long

been recognised as an effective means of improving

learning outcomes (eg, Ball, 1993; Black and Wiliam, 1998;

Callingham & Griffin, 2000; Clark, 2001).

Typically, the tasks used:

• focus on what the student understands and can do

(Darling-Hammond et al, 1995);

• allow all learners to make a start,

• accommodate multiple solution strategies; and

• relate to the kinds of activities used in teaching and

learning (Clarke & Clarke,1999; Callingham & Griffin,

2000).

© Dianne Siemon 24

Tools to Determine Readiness …

Level 1 – Trusting the Count

Level 2 – Place-Value

Level 3 – Multiplicative Thinking

Level 4 – Partitioning

Level 5 – Proportional Reasoning

Level 6 - Generalising

Available on the DEECD website

under Assessment on the

Mathematics Domain page

• Early Numeracy and Fraction Online Interviews

• Scaffolding Numeracy in the Middle Years (SNMY) -

Assessment Options linked to the Learning and

Assessment Framework for Multiplicative Thinking

(LAF). Eight developmental zones with advice on what

to establish and consolidate and what to introduce and

develop at each level of understanding

• Assessment for Common Misunderstandings and

teaching advice related to the „big ideas‟ at each level

of the Number strand:

ENGAGE

© Dianne Siemon 25

Advice to identify learning goals

Scaffolding student learning is the primary task of teachers

of mathematics.

This cannot be achieved without accurate information

about what each student knows already and what might

be within the student’s grasp with some support from the

teacher and/or peers.

This requires (i) assessment techniques that expose

students thinking.

But it also requires (ii) an interpretation of what different

student responses might mean, and (iii)some practical

ideas to address the particular learning needs identified.

© Dianne Siemon 26

Adapted from „Butterflies and Caterpillars‟ (Kenney, Lindquist &

Heffernan, 2002) for the SNMY Project (2003-2006)

The SNMY Materials This task had 9 items

altogether including:

Items of increasing

complexity, eg, “How many

complete model butterflies

could you make with 29

wings, 8 bodies and 13

feelers?”

Items involving simple

proportion and rate, eg,

“To feed 2 butterflies, the

zoo needs 5 drops of nectar

per day. How many drops

would be needed per day to

feed 12 butterflies?” …and

Items involving the

Cartesian product, eg,

given 3 different body

colours, 2 types of feelers

and 3 different wing colours,

“How many different model

butterfles could be made?”

© Dianne Siemon 27

Reading and

interpreting

quantitative data

relative to context

Open-

ended

question

Recognising

relevance of

proportion

Mathematics used,

eg, percent,

fractions, ratio

Solution

strategy

unclear,

problem

solving

SNMY Project (2003-2006)

A Short

Task

© Dianne Siemon 28

ADVENTURE CAMP …

TASK: RESPONSE: SCORE

a. No response or incorrect or irrelevant statement 0

One or two relatively simple observations based on

numbers alone, eg, “Archery was the most popular

activity for both Year 5 and Year 7 students”, “More

Year 7 students liked the rock wall than Year 5 students”

1

At least one observation which recognises the

difference in total numbers, eg, “Although more Year 7s

actually chose the ropes course than Year 5, there were

less Year 5 students, so it is hard to say”

2

b. No response 0

Incorrect (No), argument based on numbers alone, eg,

“There were 21 Year 7s and only 18 Year 5s”

1

Correct (Yes), but little/no working or explanation to

support conclusion

2

Correct (Yes), working and/or explanation indicates that

numbers need to be considered in relation to respective

totals, eg, “18 out of 75 is more than 21 out of 100”, but

no formal use of fractions or percent or further argument

to justify conclusion

3

Correct (Yes), working and/or explanation uses

comparable fractions or percents to justify conclusion,

eg, “For Year 7 it is 21%. For Year 5s, it is 24% because

18/75 = 6/25 = 24/100 = 24%”

4

A Year 6 Student Response to Adventure Camp Short Task (SNMY, May 2004)

© Dianne Siemon 29

Allow all learners to make a start

Two Year 4 Student Responses to Missing Numbers Short Task (SNMY, May 2004)

© Dianne Siemon 30

Learning and

Assessment

Framework

Eight differentiated

levels of multiplicative

thinking determined

on the basis of item

analysis*

Teaching advice

provided for each

level (or zone)

* From the Scaffolding

Numeracy in the

Middle Years

Research Project

(SNMY, 2006)

© Dianne Siemon 31

Assessment for learning – class

administered materials:

From the Assessment

Materials for Multiplicative

Thinking (SNMY, 2006)

© Dianne Siemon 32

1. Work in pairs – use the SNMY Option 1 Scoring

Rubrics to mark one of the student work samples.

Record agreed scores on a Student Assessment

Report (SAR) Form.

2. Complete item 2 of the SAR Form on the basis of the

student‟s overall response using the Option 1 Raw

Score Translator as a guide - be as specific as you

can.

3. Refer to the Learning Assessment Framework for

Multiplicative Thinking and your own experience to

complete item 3 of the SAR Form – give examples

where appropriate.

Activity 2. Interpreting data exercise

© Dianne Siemon 33

The Assessment for Common Misunderstanding Tools

comprise a number of easy to administer,

performance-based assessment tasks designed to

address a key area of Number at each Level of the

Victorian Essential Learning Standards (VELS).

A hierarchy of student responses is identified for each

task.

For each response, an interpretation of what the

response might mean is provided together with some

targeted teaching suggestions.

* From the Assessment for Common Misunderstandings (prepared by Dianne

Siemon, 2006, available on DEECD website)

Assessment for learning – individually

administered materials:

© Dianne Siemon 34

The Big Ideas:

LEVEL 1 – Trusting the count, developing flexible mental objects for the

numbers 0 to 10

LEVEL 2 – Place-value, the importance of moving beyond counting by

ones, the structure of the base ten numeration system

LEVEL 3 – Multiplicative thinking, the key to understanding rational

number and developing efficient mental and written computation strategies

in later years

LEVEL 4 – Partitioning, the missing link in building common fraciton and

decimal knowledge and confidence

LEVEL 5 – Proportional reasoning, extending what is known about

multiplication and division beyond rule-based procedures to solve

problems involving fracitons, decimals, percent, ratio, rate and proportion

LEVEL 6 – Generalising, skills and strategies to support equivalence,

recognition of number properties and pattersn, and the use of algebraic

text without which it is impossible to engage in broader curricula

expectations at this level

© Dianne Siemon 35

Resources

Diagnostic Tool

Assessment for Common Misunderstanding Tools:

© Dianne Siemon 36

* From the Assessment for Common

Misunderstandings (prepared by Dianne

Siemon for DE&T, October 2006)

(Adapted from Clarkson, 1989)

Diagnostic Tool

Advice

© Dianne Siemon 37

Level 1 – Trusting the count

Student: Date:

Card Set: Pile A Pile B

1. Single Digit (2, 4, 5, 8, 10)

2. Ten-Frame Doubles (1, 3, 4, 6, 9)

3. Ten-Frame To Five (3, 6, 8, 0)

4. Ten-Frames Random (2, 4, 5, 7, 10)

5. Two Ten-Frames (12, 14, 17, 19)

1.1 Subitising

Set 1

Set 2

Set 4

Set 5

Dianne Siemon, RMIT University

© Dianne Siemon 38

Level 1 – Trusting the count

1.2 Mental Objects Tool

There are 5 here and 4 under the

container … How many altogether?

Three levels of response …

• perceptual,

• figural, and

• abstract

Show the nine dots briefly

then cover with the flap

This tool assesses students

knowledge of part-part-whole

for the numbers to ten and

beyond

© Dianne Siemon 39

Level 2 – Place-Value

2.1 Number Naming Tool

Ask student to

count and

record how

many …

Make 34 …

Observe

response

26

Circle the 6 and

ask: What has this

got to do with

what you‟ve got

there?

Then circle the 2 and ask:

What has this got to do with

what you‟ve got there?

Then ask student to count counters,

record and make groups of four …

Repeat earlier questions …

If appropriate, ask students to count

forwards and backwards using a 0-99

Number Chart and mask provided

© Dianne Siemon 40

Level 2 – Place-Value

2.2 Efficient Counting ToolAssesses student‟s capacity to

recognise small numbers as

countable units … see

materials

2.3 Sequencing Tool

2.4 Renaming and Counting Tool

Assesses student‟s capacity to make, name, record, and

rename 3 digit numbers … see materials

0 100

48

Ask student to peg each number on the rope, starting with

48 (if too difficult change 100 card to 20 and proceed as

advised … Observe student’s strategy

© Dianne Siemon 41

Level 3 – Multiplicative Thinking

3.2 Additive Strategies Tool3 2

4 9

Assesses student‟s capacity to add and

subtract mentally

6 8

9

Do you agree that the sum of these numbers is 9? Sum of

3, 2

and 4

5

7 24

34 72

58

18

22 87

Explore thinking involved and identify strategies.

Stop as soon as student experiences difficulty.

© Dianne Siemon 42

Level 3 – Multiplicative Thinking

3.3 Sharing ToolAssesses

student‟s

capacity to share

equally,

recognise

commutativity,

and work with

the language of

multiplication

Can you share these among 6?

Same or

different?

Imagine you have 2 lollies and your sister has 3 times

as many … How many lollies does your sister have?

3.4 Array and Regions ToolHow many dots altogether?

How many name-tags like this could be

made from a sheet of paper this size?

Assesses extent to which student‟s can work with arrays and regions

© Dianne Siemon 43

Level 4 – Partitioning

4.3 Fraction Making ToolAssesses

student‟s

capacity to

generate fraction

models (a) Can you give me half?

… 1 third? … 5 eighths?

(b) Cut to make 8 equal pieces … If 3 quarters of

the pizza was eaten, show how much was eaten.

(c) Can you use the

ball of plasticine to

show 5 thirds?

(e) Divide this rectangle into 3 equal parts … name of

each part?

plasticine

(d) Can you use

these to show 2 and

5 sixths?

(f) Can you divide this line into 5 equal

parts? Name of each part?(g) If this

is 2 thirds,

what is 1? 0 6 fifths(h) Where is 1?

© Dianne Siemon 44

E5 - Structuring Inquiry

Trudy Sady: Year 1/2, Lakes

Entrance PS (PNRP, June 2002)

Starting Point: children

can count by ones, twos,

fives and tens using a 0-

99 number chart

Teacher’s Intent: to

develop more efficient

counting strategies for

larger collections

Tasks: skip counting

using a number chart,

„Chicken Scramble‟, and

„Dog-Food Pie and

Pasta Salad‟.

© Dianne Siemon 45

Chicken Scramble:

Children collect a

large number of

counters

Trudy draws

attention to different

patterns and

counting strategies

© Dianne Siemon 46

Dog

Food

Pie and

Pasta

Salad

© Dianne Siemon 47

What we learnt from the PNRP:

A number of factors seem to be critical in maximising

student learning. These include:

• Matching the learning experience to student‟s learning needs

• Remaining focussed on the key mathematical ideas/strategies

• Encouraging and using student discussion.

Explaining, conjecturing, risk-taking cannot

be implemented „out-of-the-blue‟, it is

achieved as a consequence of a carefully

negotiated classroom culture

Teaching appears to be more effective

where teachers

• are VERY clear about the mathematics they want the children to learn;

• know where the mathematics is going; and

• take advantage of opportunities to make connections.

© Dianne Siemon 48

Activity 3. 35 feral cats were estimated to live in

a 146 hectare nature reserve.

27 cats were estimated to live in

another nature reserve of 103

hectares.

Which reserve had the biggest feral

cat problem?

What do you need to know to do this?

At what Year level might you expect

students do be able to solve this

problem?

E5 - Strengthening Connections …

© Dianne Siemon 49

Proportionality

„for each‟ idea

„ratio‟ idea

represent

calculate

estimate

„factor‟ idea

„array‟ & „region‟

ideas

PartitioningMultiplicative

Thinking

© Dianne Siemon 50

The case of Fractions and Decimals

While students come to school with well developed

notions of a fair share, some capacity to share

equally, and a common sense understanding of

some fraction names such as half and quarter, they

need considerable practical experience parts and

whole to develop key generalisations.

BUT before they can be expected to work

effectively and meaningfully with fraction diagrams

students need to be exposed to a broader range

of ideas and representations for multiplication

See Siemon, D. (2004). Partitioning the Missing Link available on DEECD

website

© Dianne Siemon 51

What do you see?

1 and a half?

5 thirds?

2

3

3

5

3

2

2

5

See Siemon, D. (2004). Partitioning

the Missing Link available on

DEECD website

of ? of ?

1 divided by ?

© Dianne Siemon 52

• notice key generalisations;

• create fraction diagrams and

number line models;

• make connections to the

region, for each, and factor

ideas for multiplication; and

• make, name, compare, and

rename mixed and proper

fractions.

Partitioning - physically dividing region models or line

segments into equal parts - is the key to formalising and

extending fraction ideas.

By developing strategies for thirding and fifthing based on

halving, students can be supported to:

Equal parts needed – link to

sharing division (partition)

As the number of parts

increases, the size of each

part decreases

The number of parts names

the part

If the total number of equal

parts is increased by a

certain factor, the number of

parts required increases by

the same factor

© Dianne Siemon 53

Facilitating Substantive Conversations …

• The teacher engages students in dialogue to

continually extend and refine their understandings

• Students are supported to identify and define

relationships between concepts and to generate

principles or rules

• The teacher selects contexts from familiar to

unfamiliar, which progressively build the students‟

ability to transfer and apply their learning

• In applying their understanding, students are

supported to create and test hypotheses and

to make and justify decisions …

© Dianne Siemon 54

This requires challenging but accessible tasks that:

• involve important mathematics from more than one area

of the mathematics curriculum;

• connect to a real-world context, problem, question or

issue of interest to students;

• generally involve some data collection, analysis,

conjecture, evaluation and result in some sort of product

or outcome;

• provide opportunities for working mathematically and

making problem solving explicit; and

• can be used over a number of Year levels.

Activity 4. Work in groups on

one of the tasks provided. Discuss

in relation to the above criteria

Post and Rail Fences,

Max‟s Matches,

Jan‟s T-Shirts, or

A Matter of Perspective

© Dianne Siemon 55

Jan’s T-Shirts:

How much does each item cost?

Rose

Bowl

Rose

Bowl

Rose

Bowl

$44.00

$30.00

Both tasks from De Lange (2001)

Side elevation

Front elevation

A Matter of Perspective:

Least number of blocks?

Maximum number of blocks?

© Dianne Siemon 56

Max’s Matchsticks: How many matchsticks to make 10

squares.

Max‟s solution

11 x 2 + 9

Di‟s solution

10 x 3 + 1

Sergio‟s solution

4 x 10 - 9Leanne‟s solution

4 + (9 x 3)

Do each of these strategies work? Why?

Explain each person‟s thinking. Use each strategy to work out

how many matchsticks for 5 squares, 12, and 27 squares.

Developed for SNMY (Siemon and Stephens, 2003)

© Dianne Siemon 57

Post and Rail Fences:

Version A:

If fence posts cost $12 and

each each rail costs $9, how

much would it cost to make a

rectangular fence using 48

rails?

Version B:

Imagine you are a fencing contractor specialising in

post-and-rail fences. Prepare a design brief and quote

for a post-and-rail fence using 48 rails. How might your

quote vary for different clients?

E.g., A 4-post, 8-rail fence

might look like this

© Dianne Siemon 58

Rubbish: How much rubbish does this grade

produce in a school year?

Bird-seed: Using 1 sheet of A4 paper only and

some sticky tape, make a container to hold bird-

seed. What sort of container will hold the most

bird-seed when filled to the top? What will hold

the least amount?

Danger Distance: How far away does an on-

coming car need to be before you should cross

the road?

Money Trail: As a fundraiser, let‟s build a money

trail of 20c coins from the classroom to the front

gate. How much would it be worth?From MCTP (Lovitt &

Clarke, 1988)

Investigations

© Dianne Siemon 59

„This Goes With That‟

1. Collect survey data (e.g.,

favourite sport, fast food, type of

television program, …)

2. Discuss how this might be

represented and communicated

3. Make a strip graph by deciding

on a scale (e.g., 3 cm = 1 vote),

join the ends to make a circle

then use a 100-bead circle to ….

4. What possibilities are there?From Maths300, Curriculum Corporation (1995)

An activity that explores data representation,

proportion and percent

© Dianne Siemon 60

Some areas of your

skin have over

2,410,000 microbes

per cm2. How many

would live on 1 m2

of skin?

Droplets of moisture from a sneeze

have been measured travelling at 165

km/hour. How many metres/sec is this?

Given that there are about

160,934,000 metres of

blood vessels in the

average human body and

an extra kilogram of fat

requires 708,000 m of

blood vessels.

What questions might you

ask?

Adapted from the Guiness

Book of Records for HBJ

Mathematics (1987)