MATHS
with
SMARTIES
© Northern Territory of Australia 2005
For further information please contact
Teaching, Learning and Standards Division
Northern Territory Department of Employment, Education and Training
GPO Box 4821, Darwin NT 0801
Telephone (08) 8999 3707
Apart from any use permitted by the Copyright Act 1968, the Department of Employment,
Education and Training grants a licence to download, print and otherwise reproduce this material for educational (within the meaning of the Act) and non-commercial purposes.
This resource was originally produced by the Implementing the Common Curriculum in Aboriginal Schools Program, Darwin, 1996 – 1998. It was revised in 2006.
Original Project Team:
Kate Le Rossignol
Warwick Pascoe
Education Officer
Graphic Artist
Developed from ideas by:
Les Holmes
Warruwi School
MArk Wilson
Non-contact Teacher - Mt Allan Cluster
Elizabeth Thorne
Shepherdson College CEC
Cheryl-anne Courtney
Katherine School of the Air
Kate Tyrell
Nightcliff Primary School
Debbie Scott
Numeracy in Schools Project Officer
Cheryl Morgan
Umbakumba School
Peter Achurch
Utopia School
Josie Roberts
Education Officer Assessment Research
i
Maths with Smarties
Contents
Introduction
Page
Do-Talk-Record................................................................................................iii
Explicit Teaching..............................................................................................iv
Resources / Assessment................................................................................v
Possible Assessment Structure....................................................................vi
Learning Experience 1.Length/Area..................................................................1
How to pack Smarties................................................................................3
Learning Experience 2.Fraction/Percentage................,,,,,,,,............................4
Are there more blue than red Smarties?....................................................5
Learning Experience 3. Probability................................................................. ..7
What chance do you have of drawing a red Smartie?...............................8
Learning Experience 4. Average.....................................................................12
What size is an elephant’s Smartie?.......................................................13
Learning Experience 5. Volume/Area..............................................................15
How does volume affect the shape of a container? ................................16
Learning Experience 6.Equal fractions...........................................................19
Which fraction is biggest?........................................................................21
Learning Experience 7. Mid -page origin........................................................23
Hit the Smartie..........................................................................................25
Learning Experience 8. Number.....................................................................30
What shape number is that?....................................................................31
Learning Experience 9. Area...........................................................................34
How many triangles in a square?.............................................................35
Extension Ideas...............................................................................................38
ii
Maths with Smarties
Do - Talk - Record
Do-Talk-Record is a teaching/learning model that many Northern Territory teachers have
used with success with ESL learners. The model requires learners to thoroughly discuss and do
activities until they have fully grasped the concept before recording any results.
Learners need the opportunity to flow between talking and doing as often as required, before
recording. They may also need to repeat an activity or dicuss it further before continuing
recording.
Do an activity or experience to create a context and meaning. It should:
•
•
•
•
•
be shared by the teacher and students
provide the language for real situation and purpose
provide lots of meaning and help learners to understand new language (it builds the
context for the language)
have the teacher playing an important role in planning and initiating the language
used, providing a language model, sharing cultural information and extending the use
of appropriate language
possible for unplanned results to occur
weigh
compare/order
calculate/count
draw/games
sing/dance
- cooperative
DO
learning
- share with
partner
Describe
- effect
- sequence
- process
- cause
Retell
- observations
- results
TALK
Talk
IDEAS
write what
happened
- negotiated
- individual
nets
models
games
bargraphs
lists
displays
grids
tables
video
piegraphs
RECORD
about the activity with the learners and repeat and practise the language:
•
•
•
•
•
comment on what happened and what was said
recall events
recall the language
practise the language
use the language for the basis for other language development
Record the language used from the doing and the talking to show what they can do and what
they have learned. Reflect on the activity, the learning, progress, possible changes that can be
made etc.
iii
Maths with Smarties
Explicit Teaching
Explicit teaching refers to active instruction. It is where new learning is to be developed, deeply
and strongly. The explicit teaching stage will vary in length depending on the complexity of what
is to be learned and on the learners themselves. Active instruction involves modelling, active
teaching and learners being engaged in dialogue. Learners are to be provided with a variety of
learning tasks with the teacher guiding them to complete and learn from the tasks. Explicit
teaching is used for teaching new concepts and skills.
Because a child is at school it does not mean that they know how school works or how to make
the most of posssible learning opportunities. Explicit teaching means that what is to be learned
is not implied and includes the concept of learning being ‘taught, not caught’.
Explicit teaching of numeracy means
•
assessing where learners are at and starting at needs level
•
particular needs of learners being catered for eg language needs of ESL/ESD
learners
•
conscious planning towards clear outcomes
•
all stakeholders being informed of learning outcomes
teachers knowing what, how and why they are teaching and learners knowing what,
how and why they are learning ie learners are not left guesing about what is
happening and why
•
negotiating the curriculum, cleary defining what learners need to know and want to
learn
•
teaching the skills and processes to reach the outcomes
•
enabling skills, processes and concepts to be understood and applied
•
using a range of materials/representations/contexts for the same concept
•
mathematical language (vocabulary) is an integral componenet of mathematical
•
literacy, leading to numeracy
•
modelling oral language
•
presenting new vocabulary through shared experiences, using concrete objects etc
and providing opportunities for learners to explore and practise the new language
•
using appropriate mathematical equipment to undertake mathematical processes
and solve mathematical problems
•
modelling of numeracy strategies and/or processes eg calculation strategies
•
planning and programming that caters for multi-levels, cooperative learning groups
and catering for different learning styles and multiple intelligences
•
teaching and learning being embedded within a context that makes sense to the
learners and is linked to previous lessons and real life experiences
•
clear, high but realistic expectations - learners knowing what these are
•
learners being provided with time and opportunities to engage, practise, internalise
and reflect on learning
•
learners and teachers engaging in metacognitive processes including
self-evaluation and goal setting
•
finding a medium that is culturally appropriate
•
drawing out key mathematical ideas during and/or towards the end of the lesson
iv
Maths with Smarties
Resources
•
Smarties
•
rulers - 30 cm and 1 metre
•
stop watch
•
calculators
•
colour pencils or textas
•
video camera
•
card
•
grid paper
•
tape /glue
•
scissors
•
beads
•
string
•
bag/box
•
scales
•
newspaper
•
paint
•
glue
•
MAB cubes
Assessment
The term ‘assessment’ in Mathematics refers to the identification and appraisal of students’
knowledge, insight, understanding, skills, achievement, performance, and capability in Mathematics. (Niss 1998)
Assessment is the purposeful, systematic and ongoing collection of information for use in
making judgements about learners’ demonstrations of outcomes. It is an integral part of the
teaching/learning process. As teachers plan learning experiences, they also need to plan how
they will collect and monitor learners’ evidence of learning.
A possible assessment structure for this module has been provided. The outcomes chosen by
an individual teacher will depend on the emphasis taken when using the module and should
reflect only the outcomes that will be directly monitored and for which evidence of learning will
be gathered.
Niss, M. 1998. Assessment in Geometry, In C. Mammana and V. Villani (eds). Perspectives on the
Teaching of Geometry for the 21st Century, (pp. 263 – 274), The Netherlands: Kluwer Academic
Publishers.
v
Possible Assessment Structure
Length / Area: How to pack Smarties?
Indicators – Can the student:
(Relate measurement of length and area to other
measures and record on graph).
Outcome
MDS 3.1
MDS 3.2
Construct, present, read and interpret data/graphs
MDS 3.5
MDS 3.1
MDS 3.2
Evidence
Work Sample
(Record – length of tubes,
area…
their pattern /relationship
eg 2 smarties 1cm long, 3
smarties … cm, 4
smarties… then graph it)
Work Sample
On graph paper draw
different sizes and calculate
area
NS 3.3
Work Sample
(Calculating and applying
number in the collected
samples)
pg 324.
(Carry out area calculations for squares and other
rectangles – use mathematical language to
describe).
Describe, measure, compare, order areas Find/use different ways of measuring area pg 324.
(Demonstrate a capacity to solve problems)
Fluently and flexibly use addition, multiplication and
related subtraction, division facts, use the inverse
relationship between + - and x ÷ for solving
problems and estimating answers in familiar
contexts pg 333.
(Relate measurement of volume and capacity to
other measures)
Volume and capacity – describe measure,
compare, order pg 324.
(Be systematic in thinking about key features of a
problem, organising information and checking
answers)
Implement and evaluate methods and results and
verify against own and established criteria pg 50.
MDS 3.1
MDS 3.2
Con 2
Band 3
Work Sample
Discussion
(Working the data – 2
different sized tubes, same
number of smarties)
Discussion
(Conclusion – most
practical tube).
Fraction / Percentage: Are there more blue than red smarties?
Indicators – Can the student:
Outcome Evidence
NS 3.1
Work Sample
Read, name, record, compare equivalent fractions
NS 3.2
(Exploring the problem –
and determine relationships pg 333.
Fraction of each coloured
(Work with mixed numbers with like denominators,
smartie)
extend to unlike denominators, equivalent fractions
and ordering fractions with like denominators)
Use addition and related subtraction pg 333
NS 3.3
Understand and represent the relationship
between decimal-fraction-percentage and use
common equivalences in relation to comparing
quantities… convert numbers to assist in efficient
computation, eg fractions-decimals pg 334.
Collect, present, read, interpret data in variety of
ways pg 324.
N 4.1
N 4.3
MDS 3.5
Check
Check
Work Sample
(Working the Data)
Work Sample
(Working the Data –
Construct own Table from
circle of smarties)
Work Sample
(Record-Pie Chart)
Text that is in ( ) is – directly/indirectly - taken from this Maths with Smarties Module document. Text that has been taken – indirectly/directly – from the NTCF has the appropriate page
numbers listed with that text. This assessment section has been completed/compiled by Sharon Gierus October 2006.
vi
Fraction / Percentage: Are there more blue than red smarties?
(Be systematic in thinking about key features of a
Con 2
Discussion
(Working the Data and
problem…organising the information)
Conclusion)
Arrange information independently into a useable
Band 3
form…table….pie chart
Band 4
Recognise data…find patterns…answer questions
pg50.
Probability: What chance do you have of drawing a red smartie?
Indicators – Can the student:
Outcome Evidence
(Work with equivalent fractions)
Coloured Smartie sorting
Use common equivalences…when comparing
NS 4.1
(Do)
quantities pg 334.
Work Sample (RecordInterpret and evaluate data pg 325.
CD 4.4
fractions in the tables)
Before -Discuss outlook of
(Demonstrate a capacity to solve problems)
probability of specific
CD 4.2
List possible outcomes, use results and predict
situation After – Discuss
outcomes of repetition pg 325.
how it turned out
CD 4.2
Work Sample
List all possible outcomes…simple experiment pg
325
(completed table–Working
(Numerical probability assigned to a single event)
with Data)
…use fractions to assign probabilities pg 325.
Implement and interpret experiment…use fractions
CD 4.2
Work Sample
to assign probability pg 325.
(completed table (Numerical probability assigned to consecutive
consecutive events)
events)
CD 4.3
Work Sample
Organise data from a table into a graph pg 325
CD 4.4
(From Record and
(Record and interpret data using standard tally)
Exploring the Problem)
Extract and interpret information from graphs and
tables pg 325.
Con 2
Discussion
(Be systematic in thinking about key features of a
problem…organising information)
Band 4
(Conclusion)
Recognise data…find patterns…further
questioning pg 50.
Check
Text that is in ( ) is – directly/indirectly - taken from this Maths with Smarties Module document. Text that has been taken – indirectly/directly – from the NTCF has the appropriate page
numbers listed with that text. This assessment section has been completed/compiled by Sharon Gierus October 2006.
vii
Average: What size is an elephant’s Smartie?
Indicators – Can the student:
Read and measure mass using appropriate and
standard units pg 324. (Kilograms/grams in practical
situations using required measuring apparatus)
Apply ratio concepts to different situations pg 334
Fluently and flexibly use + x and related – and /
facts …choose different methods to solve
problems with whole numbers and
decimals…solve problems that involve simple
ratios of whole numbers and money pg 333
Divide decimals by one-digit numbers and can
interpret the reminders including recurring
decimal…pg 334.
Determine and use the mean (average) of
scores…when summarising data pg 324.
(Demonstrate a capacity to solve and extend
problems by posing their own questions)
Implement…evaluate methods/results…verify
against own and established criteria pg 50.
Apply whole action learning cycle to a task
…collect…analyse information…plan…reflect pg 50.
Outcome
MDS 3.1
N 4.1
NS 3.3
N 4.3
MDS 3.5
Con2
Band 3
Evidence
Work Sample
(Do-student/ smartie
weights)
Work Sample (Do)
Discuss (Talk)
Demonstrated throughout
(And find mass of two
different things/objects and
determine ration
Work Sample
(Exploring the Problem)
Work Sample
(Conclusion – Find an
appropriate sized Smartie
for an animal)
Band 4
Volume / Area: How does shape effect the volume of a container?
Indicators – Can the student:
Outcome Evidence
Work Samples
(Make models and drawings of 3D objects
(Construction of Net/modelincluding cylinder, cone, prism, pyramid and
SS 3.1
3D
octahedron)
drawn object
Design nets to construct a range of 3D
Attention to labels objects…draw recognisable 3D objects…including
vertices/faces/edges)
rectangular prisms…recognise and use
appropriate geometrical language pg 315.
Describe measure, compare and order areas
MDS 3.1
Work Sample
(Measure/represent areas
Make a variety of closed shapes…same
MDS 3.2
with Smarties –
perimeter…compare area…find and use different
Then show largest area to
ways to measure area of 2D shapes pg 324.
smallest on graph)
Use appropriate and familiar measures for
MDS 3.1
Discussion and explore
describing, comparing and ordering objects in
possibilities
relation to capacity of containers pg 324.
(Drawing Conclusions)
(Relate measurement of volume and capacity to
other measures) - Describe, measure, and
compare areas and volumes…of made
objects/models pg 324.
(Use several problem solving strategies and, with
some reminding, check their solutions)
Implement and evaluate methods and
results…verify against own…established data pg 50.
Check
MDS 3.1
Work Sample
(Exploring the Problem table)
Con 2
Band 3
Discussion
(Drawing Conclusions)
Check
Text that is in ( ) is – directly/indirectly - taken from this Maths with Smarties Module document. Text that has been taken – indirectly/directly – from the NTCF has the appropriate page
numbers listed with that text. This assessment section has been completed/compiled by Sharon Gierus October 2006.
viii
Equal Fractions: Which fraction is bigger?
Indicators – Can the student:
Read, name, record, and compare fractions and
recognise equivalences pg 333.
(Equivalent fraction…finding common
denominators …order fraction with like
denominators. Use mathematical language to
describe objects/ relationships) eg Say 3/8 is less
than ½ as ½ is equal to 4/8 pg 333.
(Use common fractions, investigate number
patterns and solve simple number puzzles)
Create, continue, and describe number
patterns…explain and justify their conclusions
about sequenced items…by referring to previous
element pg 333.
Identify independently an issue…problem within
broader area of discussion/experience pg 50 in relation to needing to compare fractions
Mid-page Origin: Hit the Smartie.
Indicators – Can the student:
Use positive and negative numbers…model,
compare, order measurements pg 334.
(Movement and position)
Identify and describe locations using simple
coordinates pg 315.
Outcome
NS 3.1
NS 3.2
Con 2
Band 3
Outcome
N 4.1
SS 3.3
CD 4.4
Analyse and evaluate…appropriateness of
graphical representations pg 325.
(In relation to investigations of chance outcomes
and selecting most likely occurrences)
Design, implement, interpret experiments …List all
possible outcomes…use results, giving reasons, to
predict outcomes of repetition of the experiment
pg 325. Which coloured Smartie is more likely to be
selected?
(Contribute to discussions on different ways of
using mathematics for solving problems)
Evaluate statements and assertions about
situations represented in displayed data pg 325.
(Demonstrate a capacity to solve mathematical
problems - conjecture/problems solving strategies)
Independently identify…issue/problem to
investigate pg 50. - with using coordinates in your
daily life.
CD 4.2
CD 4.4
Con 2
Band 4
Evidence
Work Sample
(Record – The grid)
(Exploring the Problem –
the fraction of each colour
smartie and order them)
Check
Explore by yourself
Work Sample
(Show colour and number
of smarties as a fractionrepeat with half/double
Smartie amount)
Discussion
(Drawing Conclusions)
Evidence
Reference to and use of
coordinates
(Drop the Smartie - Hit the
Smartie)
Work Sample
(Hit the Smartie-keep
record sheet of hits/misses)
(Drop the Smarties –
representation of where
smarties have dropped by
coloured circles, together
with coordinate plots)
Discussion
(Four Corners –any
relationship
between amount and
colour of Smarties selected
from container?)
Check
Discussion/Notes
( Exploring the Problem –
Drop the Smartie –
estimate drops after first
game)
Discussion and exploring
possibilities
(Drawing Conclusions)
Text that is in ( ) is – directly/indirectly - taken from this Maths with Smarties Module document. Text that has been taken – indirectly/directly – from the NTCF has the appropriate page
numbers listed with that text. This assessment section has been completed/compiled by Sharon Gierus October 2006.
ix
Number: What shape is that?
Indicators – Can the student:
Create, continue, and describe number
patterns…and generate the rule for the pattern
Explain and justify their conclusions on sequenced
items by referring to the previous element and
Continue a growth pattern based on shape and
identify/describe resulting number pattern pg 333.
Choose different methods when solving problems
when solving + - x ÷ problems in familiar contexts
this includes using including whole numbers,
decimals, unit fractions pg 333.
(Demonstrate a capacity to solve problems)
Arrange information independently into a useable
form…in order pg 50.
Area: How many triangles in a square?
Indicators – Can the student:
Describe, measure, compare, order areas pg 324.
(use informal measurement of various triangular
regions and surfaces)
(Investigate the relationship between areas of
triangles and areas of rectangles)
Find and use different ways of measuring the area
of 2D shapes, eg. Cutting a triangle…arranging it
into a rectangle helping to determine its area pg 324.
Use appropriate geometrical language (eg.
diagonal, right angle, parallel, and perimeter) to
describe shapes and representations pg 315.
(Generate problem…general solution)
Identify independently…issue or problem within
broader area of discussion/experience pg 50.
Outcome
NS 3.2
Evidence
Work Sample
(Exploring the Problem-Use
Smarties for Triangular,
Square, Cubic number
patterns then draw them)
NS 3.3
Work Sample
(Working the Data)
Con 2
Band 3
Take notice of this in the
above, and Discuss
(Drawing Conclusions)
Outcome
MDS 3.1
Evidence
Work Sample
(Table of Shape and
Number of Smarties)
Involvement in class chart
or graph representing this
(Record)
MDS 3.2
SS 3.1
Con 2
Band 3
Check
Check
Mathematical Learning
Journal
(Working the Data Describe objects and
specific relationships –
triangle to rectangle)
Discuss/Explore
Work Sample
(Drawing Conclusions –
and Discover and draw to
represent – How many
pairs of congruent triangles
can you find in a
rectangle?)
Text that is in ( ) is – directly/indirectly - taken from this Maths with Smarties Module document. Text that has been taken – indirectly/directly – from the NTCF has the appropriate page
numbers listed with that text. This assessment section has been completed/compiled by Sharon Gierus October 2006.
x
Extension Ideas.
Indicators – Can the student:
(Estimate number of smarties in jar/container and
ratio of different coloured ones in that container)
Outcome
Evidence
Work Sample
(Processes of problem
solving)
Check
Solve problems involving simple ratios pg 333.
NS 3.3
Work Sample
(Processes of problem
solving)
(Estimate the length of the smarties in a container
if placed end to end)
Length - use appropriate familiar measures to
describe, compare, and order objects…make
estimations and measure pg 324.
(Estimate the weight of the smarties and find
different objects of a similar weight)
MDS 3.1
Mass – use appropriate familiar measures to
describe, compare, and order objects…make
estimations and measure pg 324.
(Hunt for smarties using grid reference)
MDS 3.1
Identify and describe locations by using coordinate
maps - simple ones pg 315.
Implement and evaluate methods/results….verify
against own and established criteria
SS 3.3
Reflect on own learning experiences…from
throughout this module…through journal writing
Pg 50.
Work Sample
(Processes of problem
solving)
Did the student find the
smarties?
Con 2
Band 3
Mathematical Learning
Journal
(What did I like learning
about the most? How did I
overcome a specific
problem? How different
were my ideas to my
peers? What relationship
did I discover? – numbers,
patterns, shapes,
measuring)
Conclusion and Summary
Text that is in ( ) is – directly/indirectly - taken from this Maths with Smarties Module document. Text that has been taken – indirectly/directly – from the NTCF has the appropriate page
numbers listed with that text. This assessment section has been completed/compiled by Sharon Gierus October 2006.
xi
Smarties Learning Experience 1
Length / Area
How to pack Smarties
EsseNTial Learnings
•
Con 2
Mathematics Learning Outcomes
•
MDS 3.1
•
MDS 3.2
•
MDS 3.5
•
NS 3.3
This learning experience provides opportunities for learners to:
•
•
•
•
•
•
•
•
Relate measurement of length to other measures
Carry out area calculations for squares and other rectangles
Relate measurement of area to other measures
Relate measurement of volume and capacity to other measures
Construct and read graphs and interpret graphical information
Demonstrate a capacity to solve mathematical problems using conjecture and
problem solving strategies
Use mathematical language to describe objects and relationships
Be systematic in thinking about key features of a problem, organising the information
to make it easier to deal with and checking that their answers fit the specifications.
Language focus
Exchanging information (asking questions, making statements and reacting)
•
Comparing
•
Enquiring about or expressing opinions/knowledge
Characteristics
•
Shape
•
Colour
Relationship between units of meaning
•
Comparison
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
length
volume
area
column
compare
1
Smarties Learning Experience 1
Do - Talk - Record
Do
The aim of this learning experience is to find a relationship between length and area.
Use the nets to construct tubes being careful to match the edge of the paper to the
dotted line, then tape them to a sturdy base. The students then sort the Smarties into
the tubes and cut the tubes off to the level of the Smarties.
Measure the length of each tube and count the Smarties to see if there is a
relationship. The students flatten the tubes and mount them on card to make a
graph. They then calculate the area of each flattened tube to find if there is a
relationship between the number of Smarties and the length and area of the tubes.
Next have them construct a tube to fit all their Smarties, with the diameter of one
Smartie. They then compare the length and area of the single tube to the other tubes
added together. Have the students finish by constructing two other tubes of different
diametres which will hold all the Smarties, to decide if there is a difference in the
area they take up and which would be the best packaging.
Talk
The concepts in this learning experience should be familiar to the students. The
activity highlights the relationship between measures and the way one will effect the
other.
‘If a tube with 2 Smarties is 1 cm long, how long would a tube with 4 Smarties be?’
‘What length of the tube does 1 Smartie take?’
‘How much bigger is the area for 4 Smarties compared to the area for 2 Smarties?’
‘Can you see how the length affects the area?’
Record
The graph can be done in white or coloured card. If you wish to use it for display
purposes the colour may be more effective.
How to pack Smarties
Do
What you need
What you do
50g box of Smarties
scissors
tape
coloured or white card
ruler
Make 8 tubes from the nets.
Tape the tubes to a base.
Sort a box of Smarties by colour into the tubes.
Cut off the tube down to the level of the Smarties.
For each colour, measure the column and count the
Smarties.
Flatten each tube and make into a graph.
Calculate the area of each colour of flattened tube.
Make one long skinny tube for all the Smarties, measure the
area when flattened.
2
Smarties Learning Experience 1
Getting Started
•
How many different ways are Smarties packaged?
Record
Exploring the Problem
•
Using nets of tubes provided, construct tubes from card. Be careful to match up to
the dotted line.
•
Tape the tubes upright in a row onto a piece of card.
•
Sort the Smarties into the tubes according to colour. Cut off the tubes at the top
of each Smartie column. Measure the length of each column.
•
How many Smarties of each colour?
•
Is there a pattern between the number of Smarties
and the length of the tubes?
Working the Data
•
Empty the tubes, flatten and mount them on card to make a 2D graph.
•
Calculate the area of each flattened tube.
•
What is the relationship between the number of Smarties and the area of the
flattened tube?
•
Make a tube with the diameter of one Smartie to fit the whole packet.
•
What is the length and area of the tube?
•
Is the length and area of the tube the same as all the other tubes added together?
•
What would be the area of it when flattened?
•
Make two other tubes of different diameters to fit the same number of Smarties.
•
What are the lengths of your shorter, fatter tubes?
•
What are the areas of the shorter, fatter tubes?
•
Is there a difference in the area they used?
Drawing Conclusions
•
What is the relationship between length, area and the number of Smarties?
•
Explain which would be the most practical tube to package the Smarties.
3
Smarties Learning Experience 2
Fraction / Percentage
Are there more blue than red Smarties?
EsseNTial Learnings
•
Con 2
Mathematics Learning Outcomes
•
NS 3.1
•
N 4.1
•
NS 3.2
•
NS 3.3
•
N 4.3
•
MDS 3.5
This activity provides opportunities for learners to:
•
Add and subtract fractions and mixed numbers with like denominators. Extension
may include unlike denominators.
•
Carry out activities which demonstrate the subset/set nature of fractions.
•
Carry out activities which give experience with equivalent fractions.
•
Carry out activities involving the ordering of fractions with like denominators.
•
Carry out activities involving relations between decimal, fraction and percentage
notation.
•
Record data in and interpret tables.
•
Be systematic in thinking about key features of a problem, organising the information
to make it easier to deal with.
Language Focus
Exchanging information (asking questions, making statements and reacting)
•
Identifying
People, places, things, events, qualities and ideas
•
Number
Characteristics
•
Physical appearance
•
Colour
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
percentage
fraction
denominator
4
Smarties Learning Experience 2
Do - Talk - Record
Do
You will be able to carry the information from Learning Experience 1 and have a pair
or small group to each pack of Smarties. Generally there are 46 - 47 Smarties in a
packet, so you will need some extras to round the students, packs off to 50 (an easy
number to relate to 100). Have the students sort the Smarties and then write the
colours as a fraction. Place the Smarties in a circle, but keep the colours together.
String up 100 beads and put the strung beads around the Smarties. Tie eight pieces
of string onto a pencil, put a knob of Blu tac in the centre of the Smartie circle with
the pencil resting in it. Then use the string to divide the Smartie circle to show the
fractions.
Rather than completing a table have the students construct their own. It needs to
show the number of Smarties, the fraction, an equivalent fraction out of 100 and the
fraction converted to a percentage. Students finish by answering questions taken
from the table, sometimes involving the adding of pieces of information.
Talk
Setting up the pie graph could be difficult and needs to be talked about in detail.
‘Keep the Smarties in their colours in the circle, can you still see the fractions?’
‘Can you see the percentage for each colour?’
‘Is percentage normally based on 50?’
‘Then what number is it normally based on?’
‘With the string of beads around the circle it makes it possible to find the percentage
of each colour.’
‘Why is 50 such a good number to be using when trying to find a percentage?’
Record
You may want your students to do some sort of drawing to represent the pie graph
they made with the Smarties and beads.
Are there more blue than red Smarties?
Do
What you need
What you do
a packet of Smarties
a few extra Smarties
100 beads
string
Add to your Smarties to make 50.
Sort the Smarties into colours and write the fraction for each.
Put the Smarties in a circle.
Make a string of a hundred beads and put them around the
Smarties.
Tie 8 pieces of string to a pencil and put it in the centre of the
circle.
Use the string to divide the Smarties into a pie graph.
Make a table showing the fractions and percentages of your
Smarties.
5
Smarties Learning Experience 2
Getting Started
Record
•
•
•
Exploring
•
•
•
•
•
•
•
•
Count how many Smarties are in your packet.
How many more do you need to make 50?
What will be the denominator when writing a fraction about your Smarties?
the Problem
Sort the Smarties into colours.
Write the fraction for each colour.
Keep the colours together and make a circle with the Smarties.
Make a string of 100 beads.
Put the beads around the circle of Smarties.
Tie 8 lengths of string to a pencil.
Put a knob of Blu tac in the centre of the circle and place the pencil in it.
Use the string to divide the circle into a pie graph.
Working the Data
•
Construct a table showing the number of Smarties of each colour.
•
Record them in order of increasing number.
•
Show the number of Smarties, as a fraction, as an equivalent fractions out of 100,
and as a percentage.
•
Are any colours equal to a 1/4?
•
What colours can you add together to make a 1/2?
•
How many more Smarties would you need for 50% to be red?
•
Is any colour exactly 10%?
•
How many Smarties make 10%?
Drawing Conclusions
•
What have you learnt about percentages and fractions with a denominator of 50?
•
What have you learnt about percentages and fractions with other denominators?
6
Smarties Learning Experience 3
Probability
What chance do you have of drawing a red Smartie?
EsseNTial Learnings
•
Con 2
Mathematics Learning Outcomes
•
NS 4.1
•
CD 4.2
•
CD 4.3
•
CD 4.4
This learning experience provides opportunities for learners to:
•
•
•
•
•
Carry out activities which give experience with equivalent fractions.
Record and interpret data using standard tally.
Carry out activities in which a numerical probability is assigned to:
(a) a single event
(b) consecutive events.
Demonstrate a capacity to solve mathematical problems using conjecture and
problem solving strategies.
Be systematic in thinking about key features of a problem, organising the information
to make it easier to deal with.
Language Focus
Exchanging information (asking questions, making statements and reacting)
•
Identifying
•
Asking for and giving information
•
Comparing
People, places, things, events, qualities and ideas
•
Actions / events
Characteristics
•
Colour
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
probability
chance
event
consecutive
tally
7
Smarties Learning Experience 3
Do - Talk - Record
Do
Have the students work in groups and then each group share their results. If the
students have not already done Activity 2 they can just sort a packet of Smarties into
colours and count them. This activity is about two types of probability, single event
and consecutive events. When the students draw a Smartie and replace it before
drawing another, they are creating a single event. When they draw a Smartie and do
not replace it before drawing the next they are creating consecutive events.
The students are to discuss and share their results and try to identify similaritie/
differences in the data they have collected before they progress to the tables
showing the theory attached to probability. After that they can decide if the theory
matches their data.
Talk
Probability can be a very difficult concept to grasp so it is essential the necessary
terms are clearly defined and the students have ample opportunity to discuss what is
happening.
‘How many sides on a dice?’
‘Yes, six sides.’ (All Possible Outcomes)
‘How many threes on a dice?’
‘Only one three.’ (Possible Outcome)
‘So that means there is only one chance in six of rolling a 3’ (Probability)
‘What is the chance of rolling a one?’
Record
There are the tables the students will have completed, but you may also want them
to convert the information they have gathered into some form of graph.
What chance do I have of drawing a red Smartie?
What you need
information from activity 2 OR
Fun packet of Smarties
bag or box to put the Smarties in
What you do
Put the Smarties into a bag.
Draw 10 Smarties, put them back after each draw and tally
them.
Share your results with the class and tally again.
Draw another 10 Smarties and DO NOT put them back after
each draw.
Tally the results, share them and tally again.
Complete the two tables about the probability of an event
happening.
8
Smarties Learning Experience 3
Getting Started
Record
•
•
Exploring
•
•
•
From the results of Learning Experience 2, which colour Smartie do you think would
be drawn most?
How many Smarties are in your packet?
the Problem
Put the Smarties into a bag or a box.
Students choose 10 Smarties from the bag and tally the results.
They take out one Smartie at a time and put it back before choosing the next one.
eg blue llll
red lll
green l
purple 1
TRIAL 1A
blue_________________________________
yellow_______________________________
purple_______________________________
brown_______________________________
red_______________________________
green_____________________________
pink______________________________
orange____________________________
TRIAL 1B
•
Collect the whole class data and tally.
blue_________________________________
yellow_______________________________
purple_______________________________
brown_______________________________
•
•
•
red_______________________________
green_____________________________
pink______________________________
orange____________________________
Is there anything similar about the two results?
Students choose another 10 Smarties and tally the results.
DO NOT return them to the bag after each draw.
TRIAL 2A
blue_________________________________
yellow_______________________________
purple_______________________________
brown_______________________________
red_______________________________
green_____________________________
pink______________________________
orange____________________________
TRIAL 2B
blue_________________________________
yellow_______________________________
purple_______________________________
brown_______________________________
•
red_______________________________
green_____________________________
pink______________________________
orange____________________________
Is there anything similar about the above results?
9
Smarties Learning Experience 3
Working the Data
•
•
•
•
•
•
Make a bar graph of each set of results.
Discuss your results with other students.
Is there a difference between the types of results from Trials 1 and Trials 2?
The results from Trials 1 A and B are about a single event.
Look at the table below to see how to find the probability of a single event.
Use the results from Trial 1B to complete the table.
EVENT
ALL POSSIBLE
OUTCOMES
POSSIBLE
OUTCOMES
1
The chance of getting a tail
with one toss of a coin.
H,T
1 (T)
The chance of getting one head and
one tail by tossing 2 coins together.
HH, HT, TH, TT
2 ( HT, TH )
The chance of drawing a court card
from a pack of 52 cards.
The chance of drawing a red
Smartie first time ( 3 red in pkt).
52 CARDS
10 SMARTIES
PROBABILITY
16 (J-A in 4 suits)
3
/2
2
/4
16
/52 or 4/13
3
/10
The chance of drawing a blue
Smartie first time.
The chance of drawing a yellow
Smartie first time.
The chance of drawing a colour
beginning with the letter “P”.
10
Smarties Learning Experience 3
The next table shows how the probability changes for consecutive events.
(Use the information in the table on page 10)
POSSIBLE OUTCOMES
POSSIBLE SUCCESSES
PROBABILITY
This is how the results change for the next draw, if you drew a court card, but did not put it back.
51 CARDS
15
15
/51 or 5/
If the first card had not been a court card, this is what the probability would change to:
51 CARDS
16
16
/51
This is how the results change for red Smarties in the next draw if you drew one,but did not put it
10 SMARTIES
2
2
/10 or 1/5
If the first Smartie had not been red, what would the probability change to for red Smarties?
How do the results change for yellow Smarties in the next draw if you drew one,but did not put it
If the first Smartie had not been yellow, what would the probability change to for yellow Smarties?
How do the results change for Smarties beginning with the letter “P” in the next draw if you drew
If the first Smartie had not been pink or purple, what would the probability change to for Smarties
Drawing Conclusions
•
Does the number of items that can be drawn change the probability?
•
Does the information in the tables have results much like the draws you made?
11
Smarties Learning Experience 4
Average
What size is an elephant’s Smartie?
EsseNTial Learnings:
•
Con 2
Mathematics Learning Outcomes
•
MDS 3.1
•
MDS 3.5
•
N 4.1
•
N 4.3
•
NS 3.3
This learning experience provides opportunities for learners to:
•
Measure mass using kilograms and grams in practical situations. Select the
appropriate units of measure and the required measuring apparatus.
•
Divide whole numbers and decimals by a counting number to 9. Include examples
where the quotient is a recurring decimal.
•
Multiply and divide whole numbers and money by whole numbers with no
remainders.
•
Carry out activities to determine mean of scores.
•
Demonstrate a capacity to solve mathematical problems using conjecture and
problem solving strategies.
•
Extend problems by posing their own questions.
Language Focus
•
During the activity it is anticipated that the students will use the appropriate language
in the following ways.
Exchanging information (asking questions, making statements and reacting)
•
Enquiring about or stating facts
•
Explaining
People, places, things, events, qualities and ideas
•
Number
Characteristics
•
Physical appearance
•
Colour
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
mean
average
ratio
12
Smarties Learning Experience 4
Do - Talk - Record
Do
Talk
You will need a number of packets of Smarties so the students get a reasonable
variation of numbers to do their calculations. For this activity they all need to be the
same weight, e.g. 85g. Even though the packets have the same weight printed on
them, the weights will vary, but the students will need accurate scales. They then do
the calculations to find the average (mean) of each packet, and decide how accurate
the weight printed on the packet is. The number of Smarties per packet is recorded
and averaged, and the average number of each colour is calculated.
From the earlier calculations they find the weight of one Smartie. Students then
calculate the average weight of a student in the classroom and use the two figures to
get the ratio. To help understand the concept of a ratio, the students can design a
Smartie for an elephant. You will need books for them to find the size of an elephant
and then they can calculate the necessary size of the Smartie. You may want to
have the students to produce a few other samples on ratio.
How to calculate the ratio of a Smartie for an elephant with the same ratio as a
person to a Smartie, may take some discussion.
‘If the Smartie weighed 10g and a person weighed 100g, what would be the ratio?’
‘Yes, 1:10.’
‘So how many times bigger is the Smartie than the person?’
‘10 times.’
‘If the elephant weighed10 tonnes, then how big would the Smartie need to be?’
Record
Quite a fun display could be made of all sorts of different sized Smarties to suit lots
of different animals.
What size is an elephant’s Smartie?
What you need
What you do
scales
packets of Smarties
newspaper
tape
paint
Weigh all the Smartie packets and average the mass.
Count the Smarties in each packet and average.
Calculate the average number of each colour of Smartie.
Find the average mass for one Smartie.
Find the average massof the students in your class and
calculate the ratio of student to Smartie.
Calculate what size Smartie which would suit an elephant
and make it.
13
Smarties Learning Experience 4
Getting Started
Record
•
Exploring
•
•
•
•
•
•
•
Do you think all Smartie packets weigh the same?
the Problem
Weigh and record each of the Smartie packets.
Calculate the average mass of Smarties packets.
Count the Smarties in each packet. Record.
Calculate the average number of Smarties per packet.
Calculate the average number of each colour per packet.
Calculate the mass of one Smartie.
Calculate the average mass of students in your class.
Working the Data
•
What is the ratio of the average mass of a Smartie to the average mass of the
students in your class.
•
Find out how much an elephant weighs.
•
Calculate the mass of a Smartie to suit the size of an elephant.
•
Make an actual size Smartie for an elephant using newspaper and paint it.
Drawing Conclusions
•
Would a Smartie made for an elephant suit us?
•
What would be the ratio of the elephant’s Smartie to you?
•
What would be the ratio of a normal Smartie to an elephant?
•
Design another Smartie for some other animal and explain to the class why the size
and ratio are right.
14
Smarties Learning Experience 5
Volume/Area
How does shape effect the volume of a container?
EsseNTial Learnings
•
Con 2
Mathematics Learning Outcomes
•
MDS 3.1
•
MDS 3.2
•
SS 3.1
This learning experience provides opportunities for learners to:
•
•
•
•
•
•
Make models and drawings of a variety of three-dimensional objects including
cylinder, cone, prism, pyramid and octahedron.
Carry out activities involving the informal measurement of areas of various regions
and surfaces, including parallelograms and other polygons and circles as well as
surface areas of three- dimensional shapes.
Relate measurement to other measures.
Use cubes of known volume to:
(a) construct a variety of three-dimensional shapes including rectangular prisms
(b) measure the volume of constructed models
(c) measure the capacity of containers.
Relate measurement of volume and capacity to other measures.
Use several problem solving strategies and, with some reminding, check their
solutions.
Language Focus
Exchanging information (asking questions, making statements and reacting)
•
Comparing
•
Enquiring about or stating facts
People, places, things, events, qualities and ideas
•
Qualities
•
Number (counting) etc.
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
volume
area
dodecahedron
prism
pyramid
octahedron
cylinder
cone
icosahedron
15
Smarties Learning Experience 5
Do - Talk - Record
Do
By now students should be able to recognise and name many three-dimensional
shapes. At Band 3 it is required they also make and draw them. Have them name
and estimate the capacity of each shape using Smarties. The students can then
sequence the shapes from the one which will hold the least to the most. Have the
students choose three shapes to work with. They could work in groups of two or
three so they can complete the table of shapes. They then measure the area of the
nets using Smarties and then find the volume of the shapes by also using Smarties.
The students answer questions about any similarities between the area and volume
of a shape and finish off drawing the shapes they have made.
Talk
To find if there are any patterns or relationships between the area of a net and the
volume of its shape will require considerable discussion, initially on a classroom level
led by you and then in small groups.
‘Was there any shape which had the same number of Smarties to cover the surface
of the net and fill the volume?’
‘Can you give a reason why?’
‘Were there certain shapes which held more Smarties?’
‘What did they have in common?’
‘Can you notice any pattern between the area and the volume of the shapes?’
Record
The made up shapes will make an excellent display. The students could make
various ones of different colours and sizes to hang around the room.
How does shape effect the area of a container?
What you need
Smarties
scissors
card
tape/glue
pencil
ruler
What you do
Name three-dimensional shapes and esimate their volume
using Smarties.
Measure the area of three nets.
Make the nets.
Find the volume of each net.
Look for any patterns between the number of Smarties
needed for the area compared to the number of Smarties
needed for the volume.
16
Smarties Learning Experience 5
Getting Started
Record
•
Exploring
•
•
•
•
•
•
•
Name these three-dimensional shapes.
the problem
Estimate how many Smarties each shape will hold.
Then place them in order from the shape which will hold the least to the most.
Measure the area of three nets using Smarties. Include a dodecahedron or
icosahedron.
Record the number of Smarties.
Ask others for their information to help complete the table.
Fold the nets of the shapes you have chosen. (Make sure you leave an opening to
put the Smarties in.)
Count Smarties into each shape. Record.
Shape
No. of Smarties
Area
No. of Smarties
Volume
17
Smarties Learning Experience 5
Working the Data
•
•
•
•
•
•
•
•
Were your estimates of which shape would hold the most Smarties to which shape
would hold the least Smarties accurate?
Which shape held the most Smarties?
Which shape used the most Smarties to cover the net?
Name any shapes which used an equal number of Smarties to cover the area of the
net as to fill the made up solid.
Did any shapes use an equal number of Smarties to cover the nets, but different
amounts for the volume?
Was there a pattern in the number of Smarties needed to cover the area of the net
compared to the number of Smarties needed to fill the made upsolids?
Draw your three shapes below.
Count their faces, edges and vertices
faces_______________________
edges______________________
vertices_____________________
faces_______________________
edges______________________
vertices_____________________
faces_______________________
edges______________________
vertices_____________________
Drawing Conclusions
•
Which shapes had a large area, but held a small number of Smarties. Why?
•
If you used a liquid instead of Smarties, do you think it would make a difference to
how much you could fit into a container? Why?
18
Smarties Learning Experience 6
Equal Fractions
Which fraction is biggest?
EsseNTial Learnings
•
Con 2
Mathematics Learning Outcomes
•
NS 3.1
•
NS 3.2
This learning experience provides opportunities for learners to:
•
•
•
•
Carry out activities which give experience with equivalent fractions by:
finding a common denominator for two or more fractions
Carry out activities involving the ordering of fractions with like denominators.
Use mathematical language to describe objects and relationships.
Use common fractions, investigate number patterns and solve simple number
puzzles.
Language Focus
Exchanging information (asking questions, making statements and reacting)
•
Identifying
•
Comparing
People, places, things, events, qualities and ideas
•
Number
Characteristics
•
Colour
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
fraction
equal
equivalent
19
Smarties Learning Experience 6
Do - Talk - Record
Do
This activity needs to be done in pairs. Each student begins with a Fun Pack of
Smarties and later one of each pair is given a second pack. It is assumed there are
10 Smarties in each pack, but sometimes extras end up in the packs, so check and
make sure there are definitely 10 per pack.You will also need to cut out the fraction
cards ready to be passed around. The aim of this activity is to get students to realise
the need for a common denominator before comparing fractions. It is also intended
they recognise the biggest fraction may not be the biggest number of items when
dealing with fractions of different denominators.
Have the students sort their Smarties by colour and then order them from smallest to
largest fraction. As they have the same denominator the students can compare their
fractions with their partners. Next give one person in each pair a second pack of Fun
Smarties, ensuring it contains 10 Smarties. Then have the students colour the strip
of paper which matches their number of Smarties, cut it out and overlay their
partner’s. Students should be given a chance to discuss and recognise how different
denominators affect the result.
Hand out the fraction cards to each student and ask them to make the number of
Smarties match the denominator. Each pair of students then selects a colour they
both have in common and try to decide who has the biggest fraction. Next have them
make a grid or strip to help find the common denominator of the colour. They then
compare the fractions again. This can be repeated with pairs of students swapping
cards until they are confident about common denominators.
Talk
Students need time to discuss which fraction is the largest when the denominator is
the same or different.
‘Do the fractions have the same denominator?’
‘Yes, so is it easy to see who has the biggest fraction?’
‘Do the fractions have the same denominator?’
‘No, so is it easy to see who has the biggest fraction?’ ‘Why?’
‘Does the person with the most Smarties of one colour always have the biggest
fraction?’
Record
The grid the students make will show how well they understand finding a common
denominator.
20
Smarties Learning Experience 6
Which fraction is biggest?
Do
What you need
Smarties Fun Pack
coloured pencils or textas
scissors
What you do
Order your Smartie fractions from biggest to smallest.
Compare information with your partner.
Colour the strips to match your Smarties.
Cut out and compare to your partner’s.
Match your group of Smarties to the card from the teacher.
Write the fraction for one colour.
Draw a grid to compare the fractions.
Getting Started
Record
•
Which would you take 1/5 of a packet of Smarties or 1/10?
Exploring the Problem
•
Write the fraction for each of your Smartie colours and order them smallest to
biggest.
•
Who has the biggest fraction of yellow Smarties?
•
Now keep one packet and your partner needs two.
•
Estimate who has the biggest fraction of yellow Smarties?
•
Colour one of the strips to match the number and colour of Smarties you have.
Working the Data
•
Cut out the strip and put on top of your partner’s strip.
•
Who has the biggest fraction of red Smarties?
•
Who has the smallest fraction of green Smarties?
•
Who has the most green Smarties?
•
Now change the number of Smarties you have to match the card from your teacher.
•
Choose a colour that both you and your partner have.
•
Write the fraction for both of them. e.g. 1/8 red or 3/7 green.
•
Draw a grid or strip to help you compare the two fractions.
•
Who has the biggest fraction?
Drawing Conclusions
•
What do you need to compare fractions?
21
Smarties Learning Experience 6
1 PACKET
2 PACKETS
22
Smarties Learning Experience 7
Mid-page origin
Hit the Smartie
EsseNTial Learnings
•
Con 2
Mathematics Learning Outcomes
•
CD 4.2
•
CD 4.4
•
SS 3.3
•
N 4.1
This activity provides opportunities for learners to:
•
•
•
•
Carry out activities based on movement and position by locating and plotting points
on a grid with mid-page origin.
Investigate everyday events, games and activities that have a chance outcome by
choosing the situation in which the event is more likely to occur..
Contribute to discussions on different ways of using mathematics for solving
problems.
Demonstrate a capacity to solve mathematical problems using conjecture and
problem solving strategies.
Language Focus
Exchanging information (asking questions, making statements and reacting)
•
Asking for and giving information
People, places, things, events, qualities and ideas
•
Actions /events
Space
•
Location
Cognitive Processing Skills
•
Make hypotheses from specific data, test these out, and reformulate them if
necessary.
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
predict
cylinder
co-ordinates
quadrant
axis
positive
negative
23
Smarties Learning Experience 7
Do - Talk - Record
Do
Introduce this activity with a game of Four Corners to develop the concept of
quadrants. Mark or label the corners of the room four different colours, you also need
four Smarties the same colour in a container. The students move into the corner of
their choice and you select a Smartie from the container. Any students in the corner
with the same colour as the Smartie is out. The Smarties goes back in the container
and the students then choose another corner (or they can stay where they are).
Another Smartie is drawn. and any students in the corner with the same colour as
the Smartie is out. Keep following the same procedure until only one student is left.
When you ask students to move try to use the term quadrant rather than corner to
get them familiar with the word.
To do the drops on the XY axes with the Smarties, the students will need to make a
cylinder about 5 cm in diameter. The Smarties go in the cylinder with their palm
underneath and held directly above the centre of the axes. It is important the
Smarties are released quickly to get an even spread so the students may need to do
a few practices before they begin recording. After they have made 5 drops and
recorded the results they use the information to predict where the Smarties will land
on the next two drops and tick their successful answers. They drop the Smarties a
further three times, but this time record the co-ordinates of those which land exactly
on a set of co-ordinates. For this game make them aware the Smarties in spaces
have no co-ordinates which will create fewer problems in the game which is to
follow. Finish with a game of Hit the Smartie, which is the same as Battleships and
Cruisers, and the rules of play are on the student activity sheet.
Talk
When the class is playing Four Corners keep asking the students which quadrant
they are in have them reply using the term.
‘I am in the red quadrant.’
The next section of the activity familiarizes the students with the terms relating to
each quadrant - positive, positive; negative, positive; etc.
‘How many Smarties landed in the negative, negative quadrant ’
Record
Have the students colour where their Smarties will be on the grid of Hit the Smartie
and then there will be a record of how accurate they are at marking co-ordinates.
24
Smarties Learning Experience 7
Hit the Smartie
Do
What you need
Fun pack of Smarties
piece of card
tape
pen
What you do
Play a game of 4 corners.
Make a 5 cm diameter cylinder.
Put the Smarties in the cylinder and drop over the axes 5
times.
From the results predict how the Smarties will land next time.
Plot the co-ordinates of the Smarties for the next three
throws.
Play a game of Hit the Smartie.
25
Smarties Learning Experience 7
Getting Started
Record
•
Play a game of 4 corners.
Exploring
•
•
•
•
•
the Problem
Make a cylinder from card with a diameter of about 5 cm.
Place 10 Smarties in it with your hand underneath.
Hold the cylinder 10 cm above the X Y axes and drop the Smarties (quickly).
Do this 5 times and record how many Smarties land in each quandrant.
Record if they land off the page or exactly on a line.
( -,+ )
•
•
( +,+ )
( -,- )
( +,- )
off
line
From your 5 drops estimate how many Smarties will land in each quadrant.
Record your partners and your estimates, and then test. Tick the ones you get right.
( -,+ )
( +,+ )
( -,- )
( +,- )
off
line
( +,- )
off
line
1
1
1
•
( -,+ )
Estimate again, record and test.
( +,+ )
( -,- )
2
2
2
26
Smarties Learning Experience 7
•
•
•
•
How many estimates did you get right?
How did it help to know where the Smarties had landed on other throws?
Drop the Smarties again and plot the co-ordinates for the ones which land exactly on
two lines.
Repeat another two times.
1.
2.
3.
Working the Data
•
Play a game of Hit the Smartie.
Rules for Hit the Smartie
•
•
•
•
•
•
•
•
•
•
•
•
Place Smarties on a grid or colour the squares where you would like them to be.
Make sure they are exactly on a pair of co-ordinates.
Keep each colour together in a line.
Keep them hidden from your partner, (a book could be stood between you).
The second grid is to record your tries to hit your partners Smarties.
Give a pair of co-ordinates to your partner who tells you if you hit one of their
Smarties.
Record the turn on the second grid with a “X” for a miss and a “H” for a hit.
Your partner then has their turn to try and hit your Smarties.
You must tell them if they hit one.
You must also say when all the Smarties in the line have been hit.
If there is only one Smartie in the line, you still tell your partner the whole line is
gone.
The first to hit all the Smarties is the winner.
Drawing Conclusions
•
When would you use co-ordinates in your daily life?
27
( +,+ )
( -,+ )
( +,- )
( -,- )
Smarties Learning Experience 7
28
Smarties Learning Experience 7
y
5
4
3
2
1
-5
-4
-3
-2
-1
0
1
2
3
4
5
x
1
2
3
4
5
x
-1
-2
-3
-4
-5
y
5
4
3
2
1
-5
-4
-3
-2
-1
0
-1
-2
-3
-4
-5
29
Smarties Learning Experience 8
Number
What shape number is that?
EsseNTial Learnings
•
Con 2
MathematicsLearning Outcomes
•
NS 3.2
•
NS 3.3
This learning experience provides opportunities for learners to:
•
•
•
•
Carry out activities with particular sets of numbers:
odd, even, prime and composite, square, triangular and cubic numbers.
Carry out pattern making and pattern searching activities in which the rule followed to
generate a sequence of numbers is discovered or used.
Analyse, discuss and practise methods which make mental calculations easier by
reordering factors.
Demonstrate a capacity to solve mathematical problems using conjecture and
problem solving strategies. Students also use mathematical language to describe
objects and relationships.
Language Focus
Exchanging information (asking questions, making statements, and reacting)
•
Identifying
•
Comparing
People, places, things, events, qualities and ideas
•
Number (counting) etc.
Cognitive Processing Skills
•
Draw conclusions, using given information.
Vocabulary
In order to do this activity, students should be familiar with these terms and be able to use them
appropriately in mathematical situations.
prime
composite
triangular
square
cubic
arrange
factors
30
Smarties Learning Experience8
Do - Talk - Record
Do
The activity begins by showing students how a number can be represented by
different arrangements. You may find it necessary for your class to make the
arrangements for extra numbers before you feel they are confident enough to
recognise the difference between a composite and a prime number - one gives a
multiple of arrangements, the latter only a line. The activity then looks at triangular,
square and cubic numbers and the students need to identify the pattern and continue
with it using the Smarties to assist where necessary. Using their pack of Smarties
and the information they have gathered from the activity, they classify the numbers
(of Smarties) into prime, composite, triangular, square and cubic. Some numbers
will belong in more than one category.
The students should have done all of this before except for cubic numbers.
Talk
(Extension - students need to be familar with triangular numbers first to see this
pattern). Being able to physically build the number s into shapes will help to identify
the development of patterns, as will discussion about what is happening.
‘How many layers of cubes are there?’
‘How many on that layer?’
‘So how many cubes all together?’
‘How much bigger is this arrangement than the one before?’
‘Is there a pattern you can notice going from one arrangement to the next?’
Record
When they do the arrangements ensure the students record the factors as well so
they can be aware of the relationship between the two.
What shape number is that?
Do
What you need
50g pack Smarties
MAB cubes or similar
What you do
Arrange Smarties to represent different numbers and record
the factors.
Find the patterns for triangular, square and cubic numbers
and continue.
Using your Smartie pack decide which numbers are prime,
composite, triangular, square and cubic.
31
Smarties Learning Experience 8
Getting Started
•
•
1x6
•
•
•
•
•
•
•
2x3
1x6
•
•
•
•
•
•
•
•
•
•
Numbers can be represented by the arrangement of objects.
These are the ways to arrange the number 6.
•
•
•
•
•
•
Record
•
3x2
•
•
So the factors for 6 are 1, 2, 3 & 6.
Use your Smarties to show the different ways to arrange the number 8 and its
factors.
Exploring
•
•
•
•
•
the Problem
Arrange your Smarties for the numbers 12, 7 and 3.
Record the factors for each arrangement.
Which numbers gave more than one arrangement?
What shape/s did you arrange the Smarties in?
These numbers are called composite numbers because they have factors which
are other than one and itself. 4 - 1, 2, 4
•
Which numbers gave only one arrangement?
•
What shape did you arrange the Smarties?
•
These numbers are called prime numbers because they can only have two factors,
one and itself. 5 - 1, 5
Triangular numbers
Draw the next two arrangements in the pattern with the matching numbers.
1
3
+2
6
+3
10
+4
+
+
Continue the pattern with just the numbers: _____, _____, _____,_____, _____
32
Smarties Learning Experience 8
Square numbers
•
Use different coloured Smarties where the different shapes are used to identify the
pattern. When you know it continue for another 3 numbers.
2
1 =1
2
2
2
2=4
3=9
4 = 16
4 = 1 +3
9 = 4 +5
16 = 9 +7
2
=
5=
=
Extension Activity
Cubic numbers
•
Can you use Smarties to make a cubic Number?
•
What could you use instead?
•
Continue the pattern when you work it out.
3
1=1
1+(1x6)+1=8
3
2=8
8+(3x6)+1=27
3
3 = 27
27+(6x6)+1=64
3
4 = 64
64+(10x6)+1=
Working the Data
•
Using your pack of Smarties record which numbers are prime, composite, triangular,
square and cubic.
•
Some numbers will belong in more than one group.
prime ________________________________________________________________________
composite ____________________________________________________________________
triangular _____________________________________________________________________
square _______________________________________________________________________
cubic _________________________________________________________________________
Drawing Conclusions
•
If a number is a prime it cannot be a ______________________________________.
•
Why would part of the pattern for cubic numbers have “x 6”?
•
As the numbers get bigger what do you notice about the number of groups they will
fit into?
33
Smarties Learning Experience 9
Area
How many triangles in a square?
EsseNTial Learnings
•
Con 2
Mathematics Learning Outcomes
•
MDS 3.1
•
MDS 3.2
•
SS 3.1
This activity provides opportunities for learners to:
•
•
•
•
Carry out activities involving the informal measurement of areas of various triangular
regions and surfaces.
Investigate the relationship between areas of triangles and areas of rectangles.
Generate some problems of their own and identify a general solution.
Use mathematical language to describe objects and relationships.
Language Focus
Exchanging information (asking questions, making statements and reacting)
•
Identifying
•
Comparing
•
Enquiring about or stating facts
People, places, things, events, qualities and ideas
•
Number (counting)
Characteristics
•
Shape
Cognitive Processing Skills
•
Make hypotheses and generalisations from specific data, test these out, and
reformulate them if necessary.
Vocabulary
In order to do this learning experience, students should be familiar with these terms and be able
to use them appropriately in mathematical situations.
area
triangle
rectangle
34
Smarties Learning Experience 9
Do - Talk - Record
Do
In this activity the students should learn the relationship between the area of a
rectangle and the area of a triangle (when the vertices are taken from each corner of
the rectangle). The lesson develops when the students divide the first rectangle in
half to emphasise the resulting triangles are congruent. As large numbers of
Smarties are needed to cover the areas have the students work in small groups.
Have the students fill the rectangle with Smarties and record the number. Fill each
one in turn and record the results. The same process is repeated, but with the
students own rectangle and again recorded.
Have students draw a diagonal to make two triangles.
For the next part try to use one or two colours of Smartie for each triangle. The
students can fill each of the smaller triangles and hopefully recognise the
congruence of each pair. Finish with the Smarties from triangles B and C being used
to cover triangle A, which should then make it clear that the area of triangle A is
exactly half of the rectangle.
Talk
The students will not be dealing wth any new terminology, but may need reminding
about the meaning of congruence.
‘What could you say about these two triangles?’
eg “They are both right-angled triangles and are the same size.”
‘What can you tell me about this pair of triangles ?’
‘No matter which way you turn them or look at them, they are the same.’
‘So what do we call that type of triangle?’
Record
You may want to make a class chart or graph on the students results from their
individual rectangles and triangles to highlight the pattern.
How many triangles in a square?
Do
What you need
What you do
bulk Smarties
ruler
Count how many Smarties it takes to fill the rectangle.
Draw two triangles in the rectangle and count how many
Smarties in each.
Draw another rectangle and repeat the process.
Use the rectangle with three triangles.
Measure the areas of triangles B and C.
Compare to the halves of triangle A.
35
Smarties Learning Experience 9
Getting Started
Record
•
•
Measure the area of a rectangle by counting with Smarties.
Hint: It helps to keep the Smarties in lines.
•
•
Draw two triangles in the rectangle by drawing a diagonal.
Measure the area of each triangle with Smarties (be as exact as you can).
Record the number of Smarties in a table.
Shape
Number of Smarties
rectangle 1
triangle 1
triangle 2
rectangle 2
triangle 1
triangle 2
•
•
•
•
What do you notice?
Draw another rectangle and fill it with Smarties. Record.
Draw a diagonal and make two triangles. Fill with Smarties. Record.
What do you notice?
36
Smarties Learning Experience 9
;;;
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;;;;;;;;
;;;
;;;;;;;;
;;;
;;;;;;;;
;;;;;;;;
;;;
;;;;;;;;
;;;
;;;;;;;;
;;;;;;;;
;;;
Working the Data
•
•
Draw a line from the top of the triangle to the bottom of the rectangle.
Keep parallel with the sides of the rectangle.
B
C
A
•
•
•
•
•
•
•
•
•
Measure the area of triangle B using one colour Smartie.
Measure the area of the left hand side of triangle A using another colour Smartie.
What do you notice?
Measure the area of triangle C using one colour Smartie.
Measure the area of the right hand side of triangle A using another colour Smartie.
What do you notice?
Remove the Smarties.
Use the Smarties from triangles B and C to try to cover triangle A.
What do you notice?
Drawing Conclusions
•
What is the area of a triangle compared to the area of a rectangle?
•
Describe the types of triangles in the shaded and unshaded sections?
37
Smarties Learning Experience 10
Extension Ideas
Essential Learnings
•
Con 2
Mathematics Learning Outcomes
•
NS 3.3
•
MDS 3.1
•
SS 3.3
•
•
•
•
•
•
estimate the number of Smarties in a jar (fundraiser)
estimate amounts of different colour Smarties in the jar
estimate the length of the Smarties in the jar when placed end-to-end
estimate the weight of the Smarties in the jar
suggest three items that weigh the same as Smarties
a Smartie hunt using grid references
38
SMARTIES