Post on 05-Aug-2018
© 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 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 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)