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YuMi Deadly Maths Past Project Resource
Retail
Mathematics behind Handling Money Booklet VR1: Whole-Number and Decimal Numeration,
Operations and Computation
YUMI DEADLY CENTRE School of Curriculum
Enquiries: +61 7 3138 0035 Email: [email protected]
http://ydc.qut.edu.au
Acknowledgement
We acknowledge the traditional owners and custodians of the lands in which the mathematics ideas for this resource were developed, refined and presented in professional development sessions.
YuMi Deadly Centre
The YuMi Deadly Centre is a Research Centre within the Faculty of Education at Queensland University of Technology which aims to improve the mathematics learning, employment and life chances of Aboriginal and Torres Strait Islander and low socio-economic status students at early childhood, primary and secondary levels, in vocational education and training courses, and through a focus on community within schools and neighbourhoods. It grew out of a group that, at the time of this booklet, was called “Deadly Maths”.
“YuMi” is a Torres Strait Islander word meaning “you and me” but is used here with permission from the Torres Strait Islanders’ Regional Education Council to mean working together as a community for the betterment of education for all. “Deadly” is an Aboriginal word used widely across Australia to mean smart in terms of being the best one can be in learning and life.
YuMi Deadly Centre’s motif was developed by Blacklines to depict learning, empowerment, and growth within country/community. The three key elements are the individual (represented by the inner seed), the community (represented by the leaf), and the journey/pathway of learning (represented by the curved line which winds around and up through the leaf). As such, the motif illustrates the YuMi Deadly Centre’s vision: Growing community through education.
More information about the YuMi Deadly Centre can be found at http://ydc.qut.edu.au and staff can be contacted at [email protected].
Restricted waiver of copyright
This work is subject to a restricted waiver of copyright to allow copies to be made for educational purposes only, subject to the following conditions:
1. All copies shall be made without alteration or abridgement and must retain acknowledgement of the copyright.
2. The work must not be copied for the purposes of sale or hire or otherwise be used to derive revenue.
3. The restricted waiver of copyright is not transferable and may be withdrawn if any of these conditions are breached.
© QUT YuMi Deadly Centre 2008 Electronic edition 2011
School of Curriculum QUT Faculty of Education
S Block, Room S404, Victoria Park Road Kelvin Grove Qld 4059
Phone: +61 7 3138 0035 Fax: + 61 7 3138 3985
Email: [email protected] Website: http://ydc.qut.edu.au
CRICOS No. 00213J
This material has been developed as a part of the Australian School Innovation in Science, Technology and Mathematics Project entitled Enhancing Mathematics for Indigenous Vocational Education-Training Students, funded by the Australian Government Department of Education, Employment and Workplace Training as a part of the Boosting Innovation in Science, Technology and Mathematics Teaching (BISTMT) Programme.
Queensland University of Technology
DEADLY MATHS VET
Retail
MATHEMATICS BEHIND HANDLING MONEY
BOOKLET VR1
WHOLE-NUMBER & DECIMAL NUMERATION, OPERATIONS, AND COMPUTATION
08/05/09
Research Team:
Tom J Cooper
Annette R Baturo
Chris J Matthews
with
Kaitlin Moore
Elizabeth Duus
Deadly Maths Group
School of Mathematics, Science and Technology Education, Faculty of Education, QUT
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page ii ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
THIS BOOKLET
This booklet (VR1) was produced as material to support Indigenous students completing
certificates associated with Retail at the TAFE and post senior campuses on Palm Island run
by Barrier Reef Institute of TAFE in conjunction with Kirwan State High School. It has been
developed for teachers and students as part of the ASISTM project, Enhancing Mathematics
for Indigenous Vocational Education-Training Students. The project has been studying better
ways to teach mathematics to Indigenous VET students at Tagai College (Thursday Island
campus), Tropical North Queensland Institute of TAFE (Thursday Island Campus), Northern
Peninsula Area College (Bamaga campus), Barrier Reef Institute of TAFE/Kirwan SHS (Palm
Island campus), Shalom Christian College (Townsville), and Wadja Wadja High School
(Woorabinda).
At the date of this publication, the Deadly Maths VET books produced are:
VB1: Mathematics behind whole-number place value and operations
Booklet 1: Using bundling sticks, MAB and money
VB2: Mathematics behind whole-number numeration and operations
Booklet 2: Using 99 boards, number lines, arrays, and multiplicative structure
VC1: Mathematics behind dome constructions using Earthbags
Booklet 1: Circles, area, volume and domes
VC2: Mathematics behind dome constructions using Earthbags
Booklet 2: Rate, ratio, speed and mixes
VC3: Mathematics behind construction in Horticulture
Booklet 3: Angle, area, shape and optimisation
VE1: Mathematics behind small engine repair and maintenance
Booklet 1: Number systems, metric and Imperial units, and formulae
VE2: Mathematics behind small engine repair and maintenance
Booklet 2: Rate, ratio, time, fuel, gearing and compression
VE3: Mathematics behind metal fabrication
Booklet 3: Division, angle, shape, formulae and optimisation
VM1: Mathematics behind handling small boats/ships
Booklet 1: Angle, distance, direction and navigation
VM2: Mathematics behind handling small boats/ships
Booklet 2: Rate, ratio, speed, fuel and tides
VM3: Mathematics behind modelling marine environments
Booklet 3: Percentage, coverage and box models
VR1: Mathematics behind handling money
Booklet 1: Whole-number and decimal numeration, operations and computation
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page iii ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
CONTENTS
Page
1. MEANINGS FOR NUMBER AND OPERATIONS ................................................... 1
1.1 Whole numbers ...................................................................................... 1
1.2 Decimal numbers .................................................................................... 2
1.3 Whole numbers and decimal processes .................................................... 4
1.4 Operation concepts ................................................................................. 5
1.5 Computation strategies ........................................................................... 6
1.6 Pedagogy ............................................................................................... 7
1.7 Physical and virtual activities ................................................................... 7
2. MONEY AS WHOLE NUMBER ........................................................................... 9
2.1 Materials and sequences ......................................................................... 9
2.2 Activities ................................................................................................ 9
2.3 Materials and games ..............................................................................10
BINGO CARDS AND BOARDS ................................................................................19
3. MONEY AS DECIMAL NUMBERS......................................................................23
3.1 Decimal place values .............................................................................23
3.2 Activities, sequences and games .............................................................23
4. ADDITION AND SUBTRACTION OF MONEY .....................................................31
4.1 Determining which operation to use ........................................................31
4.2 Computation .........................................................................................32
4.3 Sequences, activities and games ............................................................33
5. MULTIPLICATION AND DIVISION OF MONEY ..................................................37
5.1 Determining which operation to use ........................................................37
5.2 Computation .........................................................................................38
5.3 Sequences, activities and games ............................................................40
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 1 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
1. MEANINGS FOR NUMBER AND OPERATIONS
1.1 Whole numbers
There are 2 major models for number:
(1) Set: In this form of modelling, numbers are represented in terms of each place value
position. For example, on a simple abacus 423 is represented by:
To teach this for the first time, it is best to use material which changes at each place
value to show that the position on the left is 10 times the position on the right. For
example, 423 is represented by MAB as:
The MAB in the hundreds position is 10 times larger than the material in the tens
position, which in turn is 10 times larger than the material in the ones position.
(2) Line: In this form of modelling numbers are represented by their position on a line in
relation to numbers assigned to the ends of the line. For example, 83 on 0–100 and on
80–100 lines would look like this:
As described in booklet VB1, there are 4 meanings associated with understanding whole
numbers, including 2 and 3 digit numbers.
(1) Place value/Separation: The understanding that 237 is 2 hundreds, 3 tens and 7 ones
and is said/written “two hundred and thirty-seven”. It is a positional order of place
value positions around a base of ten – ten ones forms 1 ten and ten tens forms 1
hundred, while 1 tenth of a hundred is 1 ten and 1 tenth of a ten is 1 one. Materials
Hundreds Tens Ones
Hundreds Tens Ones
0 100
83
80 100
83
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 2 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
used are set model – bundling sticks, MAB and money to show value or base, and a
Place Value Chart (PVC) to show position.
(2) Counting (odometer): The understanding that each PV position counts forward and
backward, and that this counting follows rules – forward is 1, 2, 3, 4, 5, 6, 7, 8, 9 and
then the position becomes a zero and the position on the left goes up by 1; and
backward is 8, 7, 6, 5, 4, 3, 2, 1, 0 and then the position becomes a 9 and the position
on the left goes down by 1. Materials used are calculators, flip cards, MAB and, if
available, a car odometer.
(3) Number line (rank): The understanding that no matter how many ones, tens and
hundreds there are, or whether there are 2 or 3 or more digits, each number is one
point on a number line with the distance it is from 1 determining how it compares with
other numbers (e.g. ranked with other numbers). Materials used are predominantly
line number model – pegs, rope and cords, number lines, and 99 boards.
(4) Multiplicative structure: The understanding that adjacent place values are related left
to right by ÷10 and right to left by ×10 and that this is continuous across the place
value positions.
Materials used are predominantly set model – digit cards, PVCs, and slide rules.
1.2 Decimal numbers
Decimal numbers are an extension of whole numbers, using fractions to extend place value
positions to tenths, hundredths, and so on. The meaning of fraction has 5 components.
(1) Part of a whole: A whole is taken and partitioned/divided into, say, 7 parts, all equal.
Five of these parts are considered. Then the fraction is 5/7. The steps are:
consider a whole, e.g.,
break it into 7 equal parts, e.g.,
each part is named by the number of parts,
here sevenths
five of these are considered
the fraction is named in terms of the number of parts
considered and the total number of equal parts in the
whole, here five sevenths.
The secret is to always know what the whole is.
Hundreds Tens Ones
Seventh or 1/7
5 sevenths or 5/7
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Page 3 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
(2) Part of a set: A set, say of 12 things, is divided/partitioned into equal groups. The
5 steps are the same as (1):
make the set a whole, e.g.,
break into 4 equal groups, e.g.,
each group is named by the number of
groups, here, fourths
consider three of these groups
name the fraction – three fourths or
three quarters – ¾
The secret is always to know the one – to ensure students know it is not 12 things but
one thing.
(3) Number line: The whole is the distance from 0 to 1 on a line and the fraction is a
position on the line which signifies its value, for example:
Take the one
Partition into 5
equal lengths
Name one length - here
one fifth or 1/5
Consider 3 of these lengths
Name the fraction three-fifths, 3-fifths or 3/5
(4) Division: A fraction, say ¾, is the same as the numerator divided by denominator,
that is, 3÷4. Consider 3 cakes to be shared amongst 4 people – each person gets ¾.
(5) Multiplier: A fraction, say ¾, is that which multiplies by the numerator and divides by
the denominator, that is, ¾ is the same as x 3 and ÷ 4
Whole numbers involve digits in position to the left of the ones, increasing by 10 times as
move to left. (and ÷ 10 as move to right). The value of each digit is determined by the ones
digit which is the right-most digit for whole numbers.
The extension of whole numbers to decimals is straight forward. New positions (which are
fractions because ÷ 10) are added to the left in a symmetry about the ones, e.g.
0 1
0 1
1/5
0 1
Three
¼ from each caketo each person
Thousands Hundreds Tens Ones tenths hundredths thousandths
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 4 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
The decimals are the same as whole numbers in that:
(1) x 10 so move to left ÷ 10 so move to right; and
(2) value of digit determined by relationship to ones position.
The decimals are different to whole number in that the ones position is determined by a dot
after it. So, 367.48 means the 7 is ones, 6 is tens, 3 is hundreds and, in the other direction,
4 is tenths and 8 is hundredths. Similar to whole numbers, the decimal numbers have 4
meanings:
(1) place value/separation – 0.07 means 7 hundredths as the 0 is in the ones position;
(2) counting – the tenths and hundredths and the rest of the fraction positions count up
and down the same as whole-number positions;
(3) rank – decimal numbers have positions on the number line (showing order); and
(4) multiplicative structure – x 10 to left and ÷ 10 to right remain for all positions.
1.3 Whole numbers and decimal processes
Whole-numbers and decimal numbers have the same processes that students should
understand. These are:
(1) Reading and Writing: Students need to be able to read and write numbers. Note that
reading decimals (and whole numbers) is not the same as spelling the numbers:
Eg. 327 is three hundred and twenty seven (not three two seven)
403 is four hundred and three
613 is six hundred and thirteen
6.85 is six and eight five hundredth
0.026 is 26 thousandths
(2) Seriation: Students need to be able to work out 1 more and 1 less, 10 more and 10
less, one tenth more and less, 100 more and less, and so on. For example:
What is 10 more than 290?
What is one tenth less than 30?
(3) Order: Students need to be able to compare/order numbers. This means comparing
place value to place value position not looking at length of number. For
example: 3.8 is larger than 3.654 because
H T O t h th
33
86 5 4
Same Less
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 5 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
(4) Renaming: Students need to be able to think of numbers in more than one
way(particularly for money). For example:
3.6 = 3 ones 6 tenths (3 $1, 6 10 c)
= 2 ones 16 tenths (2 $1, 16 10 c)
= 36 tenths (36 10 c)
42.5 = 4 tens 2 ones 5 tenths (4 $10, 2 $1, 5 10c)
= 3 tens 11 ones 15 tenths (3 $10, 11 $1, 15 10c)
(5) Rounding/Estimating: Students need to be able to calculate how to round to a given
place value. For example:
27.4 is 27 to nearest one and 30 to the nearest 10
356.8 is 360 to the nearest 10 and 400 to nearest 100
1.4 Operation concepts
There are 4 models with which to think of operations:
(1) Sets or groups: Look at operations in terms of sets of objects: 2+3 is 2 objects joining
3 objects, 6–2 is 2 objects leaving 6 objects, 3x4 is 3 groups of 4 objects while 15÷5 is
15 objects partitioned into 5 groups or 15 objects partitioned in groups of 5.
(2) Number lines: Look at operations in terms of number tracks or number lines: 2+3 is a
jump of 2 followed by a jump of 3; 6–2 is a jump of 6 and a jump back of 2; 3x4 is 3
jumps of 4; 15÷5 is finding 5 equal jumps to make 15 or how many jumps of 5 make
15.
(3) Array/Area: Multiplication and division can be done
by arrays or area: 3x4 is 3 rows of 4 or 3 by 4; and
15÷5 is 15 in rows of 5 (how many rows) or 15 in 5
rows (how many in each row) as on right.
(4) Tree diagrams: Multiplication and division can be
done by combinations (e.g. tree diagrams):
4
3
4
3
15
15
3x4 is 3 shirts and 4 pants, so
how many outfits (as on right);
and
15÷5 is 15 outfits and 5 pants,
so how many shirts.
Start
Start
Shirt 1
Shirt 2
Shirt 3
Pant 1
Pant 2
Pant 3
Pant 4
Pant 5
4 Pants
4 Pants
4 Pants
15 Outfits
12 Outfits
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 6 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
There are 4 meanings for operations.
(1) Traditional or forward: Addition is joining; subtraction is take-away; multiplication is
lots of objects, rows or jumps; and division is sharing (putting into ? groups) or
grouping (putting in groups of ?)
(2) Inverse as backward: Addition is inverse of take away (2 boys ran away, left 3 how
many at the start?); subtraction is inverse of joining (2 boys joined the others to make
6, how many others were there?); multiplication is inverse of division (apples were put
into 3 groups of 5 or 5 groups of 3, how many apples at the start?); and division is
inverse of multiplication (3 groups makes 12 or groups of 4 make 12, how many in
each group are how many groups?)
(3) Comparison: Addition is more (I have $2 you have $3 more than me, how much do
you have?); subtraction is inverse of more or difference (you have $2 more than me,
you have $6, how much do I have; I have $2, you have $6, how many times is the
difference?); multiplication is multiplier (I have $3, you have 4 times what I have, how
much do you have?); and division is inverse of multiplier (I have $3 you have $15, how
many times more do you have than me?; you have $15 which is 5 times what I have,
how much do I have?)
(4) Combinations: Multiplication and Division is bringing things together that are not
related so you can count the number of combinations; multiplication is 3 shirts and 4
pants to make 12 outfits; division is 15 outfits and 5 shirts, how many pants? It is
based on tree diagrams and tables as follows.
1.5 Computation strategies
There are 3 strategies for computation (as also described in VET booklet VB1)
(1) Separation: The understanding that numbers can be added, subtracted, multiplied, and
divided by separating the numbers into their place value parts, operating on the parts
separately then combining the parts to get the answer. For example: 48+25 =
(40+20) + (8+5) = 60+13 = 73; and 38x7 = (30x7) + (8x7) = 210+56 = 266.
(2) Sequencing: The understanding that number can be added, subtracted, multiplied and
divided by taking one number, leaving it as it, and then breaking the second number
into parts and operating the parts of the second number with the first whole number in
a sequence so that the answer emerges. For example: 48 + 25 = (48 + 20) + 5 = 68
+ 5 = 73; and 38 x 7 = (38 x 5) + (38 x 2) = 190 + 76 = 266.
(3) Compensation: The understanding that number can be added, subtracted, multiplied
and divided by finding an easy operation and then compensating for the change to the
Start
4 Pants
4 Pants
4 Pants
3 Shirts
Pants 1 Pants 2 Pants 3 Pants 4
Shirt 1
Shirt 2
Shirt 3
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 7 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
easy operation. For example: 48 + 25; 50 + 25 = 75’ subtract to compensation 22 =
73; and 38 x 7; 40 x 7 = 280, subtract 2 x 7 = 14, = 280 – 14 = 266.
1.6 Pedagogy
Similar to VB 1, the pedagogy of VR1 is built around the Payne
Rathmell triangle. Real world instances are the starting point,
then numbers and operations are represented with models,
then language is developed, then symbols, and finally all parts
are interconnected.
The models and general approach being used are:
As can be seen, virtual models come after physical and precede pictorial and patterns. To
achieve this, the teaching will also be a cycle of introduction, consolidation and application.
Teaching should also ensure that there are instructional activities that:
(1) Generalise – show rules that hold for all numbers; eg., 48+25 = 25+48;
¾+7/11 = 7/11+¾ and so on;
(2) Reverse – go both ways, e.g., ask students to add 48+25, then give answer of 73 and
ask for a question; and
(3) Encourage flexibility – show a variety of methods, e.g., 48+25 = 48+20+5,
or = 48+20+2+3, or = 48+10+10+2+3 and so on.
1.7 Physical and virtual activities
This booklet is to be used with a set of virtual materials. Booklet VB1 and VB2 discuss the
roles of virtual materials. The virtual materials we built around money and though there are
money 99 boards and money number lines which are useful, the predominant method used
is set model – based on $100 and $10 notes and $1 coins for whole number, adding 10 c
and 5 c coins for decimal numbers.
Body Active (whole body)
Hand Physical, virtual, pictorial
Mind Mental model
Real World
Physical
Virtual
Pictorial
Patterns
Real World
Models
SymbolsLanguage
Problem Solving
Application
Extensions
Introduction
Informal
Formal
Practice
Consolidation
Connections
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 8 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
However, for realism the virtual materials also use $5, $20 and $50 notes and $2, 50c and
20c coins.
The idea for this booklet is that there needs to be activity with the body and with materials
before virtual activity (see diagram below). There also needs to be activity with the mind
during and after virtual activities to ensure the virtual models are stored as mental models in
the mind. Thus, this booklet provides the necessary first steps before the virtual materials
are used.
BODY HAND MIND
Kinaesthetic Physical Virtual Mental
activity material material
models
activity activity
Lastly, although a reasonable number of virtual material activities are supplied, the idea of
the materials is that they copy activity with real materials and so can be modified and added
to by teachers. The idea is to replicate what happens with real physical materials with the
virtual materials - to reproduce the good physical-material activities with the virtual-materials
activities.
The virtual materials are provided either:
in a CD stuck to the back cover of this booklet, or
as the next folder in the CD/website, if this booklet is on a CD or being accessed via
a website.
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 9 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
2. MONEY AS WHOLE NUMBER
The most basic thing to do with money is to count it and see if one has enough for a
purchase. This means that you understand money in terms of whole numbers and, with
cents, decimal numbers.
2.1 Materials and sequences
The physical and pictorial materials that should be used are:
(1) $100 and $10 notes and $1 coins;
(2) $100, $50, $20, $10 and $5 notes and $2, $1, 50c, 20c, 10c and 5c coins;
(3) $0 to $99 board;
(4) $0 to $100 and $0 to $1000 number lines with amounts marked; and
(5) the same number lines with only the end points marked.
The $0 to $99 board and some number lines are attached. To enrich the activity with the
physical and pictorial materials, activities with virtual materials are provided for whole
numbers ($100, $50, $20, $10 and $5 notes and $2, $1 coins), and for decimal numbers
($100, $50, $20, $10 and $5 notes and $2, $1, 50c, 20c, 10c and 5c coins)
The sequencing of materials should be:
(1) physical materials to virtual
materials to pictorial materials;
(2) one material for each place-value
position and a place value chart
(PVC) before all materials on a PVC,
as on right;
(3) play money and PVCs before 99
boards and number lines;
(4) 2 digit numbers before 3 digit
numbers; and
(5) straight-forward numbers (e.g., 46,
234) before numbers with zeros
(e.g., 40, 204, 360) before teen
numbers (e.g., 14, 317)
2.2 Activities
The sequence of activities should be built around the processes. The initial activities should
focus on place value and reading and writing amounts of money: (a) give money say/write
how much; and (b) say/write how much put out money. The materials here are set model,
e.g., money and PVC.
The next activities should focus first on seriation, e.g., what is $10 more then $42, what is
$1 less than $190, what is $10 less than $205? The materials here are $0 - $99 board and
money/PVC.
H T O
$100$100
$100
$10
$10
$10
$10
$10
$10
$10
$1
$1
$1
$1
$1
$1
$1
$1
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 10 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
After this we should move onto order, working out what amount is larger (and whether we
have enough money to buy something). The materials here are $0 - $99 board and number
lines ($0 to $100 and $0 to $1000).
Then, there is renaming, e.g., looking at different ways to make numbers. With the obvious
importance of this process with respect to money, this activity should be taught early.
Examples of activities are:
(1) working out how much money there is in a pile of notes and coins; and
(2) making up all different ways to make a cert amount (eg. $50 = $20 + $20 + $10 and
so on).
Finally, there is rounding and estimation, e.g., $47 is $50 to the nearest $10, but is $45 to
nearest $5.
2.3 Materials and games
A lot of learning can be had from using materials and playing games. Here are a few:
(1) A $0-$99 board and some activities that extend it (use to show $1 and $10 more and
less);
(2) $0-$100 and $0-$1000 lines (used to place numbers to determine larger);
(3) Mix and match three digit money cards (print on cardboard and cut up, and then
students reform as “jigsaws” representing the same number); and
(4) Bingo cards and boards (boards printed on different colours and cards cut out,
someone calls out cards, others cover any number on their board that is the same as
that called, first to get three-in-a-row wins (horizontal, vertical and diagonal).
Note: The rules to Mix & Match and to Bingo are given in Section 3.2.
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Page 11 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
$0-$99 BOARD
0 1 2 3 4 5 6 7 8 9
10 11 12 13 14 15 16 17 18 19
20 21 22 23 24 25 26 27 28 29
30 31 32 33 34 35 36 37 38 39
40 41 42 43 44 45 46 47 48 49
50 51 52 53 54 55 56 57 58 59
60 61 62 63 64 65 66 67 68 69
70 71 72 73 74 75 76 77 78 79
80 81 82 83 84 85 86 87 88 89
90 91 92 93 94 95 96 97 98 99
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Page 12 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
$99 BOARD EXTENSION ACTIVITIES
(a) (b)
$614 $837
(c) (d)
$287 $355
(e)
(f)
$964
$499
YuMi Deadly Maths Past Project Resource © 2008, 2011 QUT YuMi Deadly Centre
Page 13 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
$0-$100 AND $0-$1000 NUMBER LINES
$0 ___________________________________________________________________$100
$0 ___________________________________________________________________$100
$0 ___________________________________________________________________$100
$0 ___________________________________________________________________$100
$0 ___________________________________________________________________$100
$0 ___________________________________________________________________$100
$0 ___________________________________________________________________$1000
$0 ___________________________________________________________________$1000
$0 ___________________________________________________________________$1000
$0 ___________________________________________________________________$1000
$0 ___________________________________________________________________$1000
$0 ___________________________________________________________________$1000
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Page 14 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
MIX AND MATCH CARDS
Seven $100s, one $10 and two $1s Seven
hundred and twelve dollars
$712
$1
100 10 1
$10
$100
Two $100s, one $10 and six $1s Two
hundred and sixteen dollars
$216
$1
100 10 1
$10
$100
$100
$100
$100
$1
$1
$1
$1
$1
$1
$100
$100
$100
$100
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Page 15 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
Four $100s, four $10s and four $1s Four
hundred and forty-four
dollars
$444
$1
100 10 1
$10
$100
Two $100s, six $10s and one $1 Two
hundred and sixty one dollars
$261
$1
100 10 1
$100
$100
$100
$10
$10 $1
$1
$100
$10
$10
$10
$10
$10
$10
$100
$10 $1
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Page 16 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
Five $100s, two $10s and five $1s Five
hundred and twenty-five
dollars
$525
$1
100 10 1
$10
$100
One $100, eight $10s and three $1s One
hundred and eighty-three
dollars
$183
$1
100 10 1
$10
$100
$100
$100
$10
$10 $1
$1
$10
$10
$10
$10
$1
$10
$1
$10
$100
$100
$1
$1
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Three $100s and three $10s The hundred
and thirty dollars
$330 100 10 1
$10
$100
Four $100s and seven $1s Four
hundred and seven dollars
$407
$1
100 10 1
$100
$100
$10
$10
$100
$100
$1
$1
$100
$1
$1
$1
$1
$100
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One $100s, three
$10s and four $1s One hundred
and thirty-
four dollars
$134
$1
100 10 1
$10
$100
$10
$10 $1 $1
$1
Three $100s, six $10s
and two $1s Three
hundred and
sixty-two
dollars
$362
$1
100 10 1
$10
$100
$100
$100 $1
$10
$10
$10
$10
$10
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BINGO CARDS AND BOARDS
BINGO CARDS
$934 $629 $450
$502 $317 $487
$161 $684 $721
$211 $473 $371
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Page 20 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
BINGO BOARDS
Nine $100s, three $10s
and four $1s
Six hundred and twenty-nine dollars
Three $100s, one $10 and seven $1s
Four hundred and eighty-
seven
One hundred and sixty-one
dollars
Six hundred and eight-four dollars
Three $100s, seven $10s and one $1
100 10 1
$100
$100
$100
$100
$100
$100
$100
$100
$100
$10
$10
$10
$1 $1
$1 $1
100 10 1
$1 $1
$1 $1
$100
$100
$100
$100
$100
$100
$10
$10
$10
$10
$10
$10
$10
$10
100 10 1
$100
$100
$100
$100
$100
$100
$10
$10 $1 $1
$1 $1
$1 $1
$1 $1
$1
100 10 1
$1
$10
$10
$100
$100
$100
$100
$100
$100
$100
100 10 1
$1 $1
$1
$100
$100
$100
$100
$10
$10
$10
$10
$10
$10
$10
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Nine hundred and thirty-four dollars
Four $100s and five $10s
Five hundred and two dollars
Four $100s, eight $10s and seven
$1s
One $100, six $10s and
one $1
Seven $100s, two $10s
and one $1
Two hundred and eleven
dollars
Four hundred and seventy-
three
Four $100s, seven $10s and three
$1s
100 10 1
$10
$10
$10
$10
$10
$100
$100
$100
$100
100 10 1
$1 $1
$100
$100
$100
$100
$100
100 10 1
$1
$100
$100
$10
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Six $100s, two $10s and
nine $1s
Four hundred and fifty dollars
Five $100s and two $1s
Three hundred and seventeen
dollars
Six $100s, eight $10s
and four $1s
Seven hundred and twenty-one
dollars
Two $100s, one $10 and
one $1
Three
hundred and seventy-one
dollars
100 10 1
$10 $1 $1
$1 $1
$1 $1
$1
$100
$100
$100
100 10 1
$1 $1
$1 $1
$1 $1
$1
$100
$100
$100
$100
$10
$10
$10
$10
$10
$10
$10
$10
100 10 1
$1
$100
$10
$10
$10
$10
$10
$10
100 10 1
$1
$100
$100
$100
$10
$10
$10
$10
$10
$10
$10
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Page 23 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
3. MONEY AS DECIMAL NUMBERS
When add cents, money can be seen as a decimal number to hundredths. However, this is
not straightforward as with the removal of the 1c and 2c coins, there are only 5c in money
and so the money representation of hundredths is incomplete.
3.1 Decimal place values
The $100 note is worth ten $10 notes and the $10 note is worth ten $1 coins. So, in terms
of whole numbers, the dollars are a good example as follows:
Now $1 coin is worth ten 10c pieces, so 10 c is 1/10 of $1. A 10c coin is 10 1c pieces, if
they existed, so a cent is 1/10 of a 10c coin and 1/100 of $1. This means that decimal place
values to hundredths are as follows
This also means that rounding is important because money is in 5 cent pieces, e.g., 3.27 is
$3.25 and 3.28 is $3.30 for the paying customer.
Finally, there are ways other than decimal points that prices show hundreds, tens, ones,
tenths and hundredths. For example, three $10 notes, four $1 coins, seven 10 cent pieces
and a 5 cent piece is 34.75. However, some publications/prices write this as: 34 75 or 34 - 75.
3.2 Activities, sequences and games
Decimal numbers are taught using the same sequencing as whole number (so re-read
section 2.2) Once again, money and number lines are useful in teaching and there are
games that are also useful, e.g., as follows.
(1) Mix and Match – cards contain pictures of different representations of the same
amount of money all in the same jigsaw form – all the cards are on the same colour
cardboard – all are cut up into pieces and mixed together - the students have to
H T O
4 3 6
$100
$100$100
$10
$10
$10 $1
$1
$1
$1
$1
$1$100
H T O t h
3 2 6 5
$10
$10
$10
$1
$1
50c
10c
5c
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Page 24 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
reform the cards by putting the representations of the same amounts of money
together.
(2) Cover the Board – cardboard A4 sheets of different colours are divided into squares.
Each sheet shows a different way to represent the value with the squares on each
sheet each having different amounts of money but represented in the same way (e.g.,
one sheet has the money amounts all in symbols, the next has it in words, the third
pictures of notes and coins and so on) – one sheet is left whole (this becomes the
board) – the rest are cut into card decks of different colour – players take different
decks and, in turn, place one of their cards on the board, or an opponent’s card on the
board, showing the same amount of money until all players have played all their cards
– the player with the most of their coloured cards on top at the end wins.
(3) Bingo (or cover the monster) – flash cards are made with many examples of amounts
of money but all in the same representations – boards are made with different
representations of these numbers, or some of these numbers, randomly across a grid
or placed within monster shapes – one player shows the flash cards one at a time –
the other players cover (with a counter) any amount on their board that is the same as
the flash card - the first player with 3 in a row, column or diagonal covered, or with all
amounts on a monster covered, wins.
Three Cover the Board sheets and three Bingo boards follow in the next 6 pages.
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COVER THE BOARD CARDS [1]
Instructions: Photocopy all 3 pages on different coloured cardboard – do not cut up symbol sheet
$934.25 $146.50 $604.15
$640.30 $561.50 $943.25
$516.05 $704.40 $146.15
$229.05 $560.05 $392.15
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COVER THE BOARD CARDS [2]
Nine hundred and thirty-four dollars and twenty-five
cents
One hundred and forty-six dollars and fifty cents
Six hundred and four dollars and
fifteen cents
Six hundred and forty dollars and
thirty cents
Five hundred and sixty-one dollars and fifty cents
Nine hundred and forty-three dollars and twenty-five
cents
Five hundred and sixteen dollars and five cents
Seven hundred and four dollars and forty cents
One hundred and forty-six dollars and fifteen cents
Two hundred and twenty-nine
dollars and five cents
Five hundred and sixty dollars and
five cents
Three hundred and ninety-two
dollars and fifteen cents
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Page 27 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
5c $1 $1
$1 $1
10c
COVER THE BOARD CARDS [3]
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$100
$100
$100
$10
$10
$10
$1 $1
$1 $1
10c
10c 5c
100 10 1 1/10 1/100
$100
$10
$10
$10
$10
$1 $1
$1 $1
$1 $1
10c
10c
10c
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$10
$10
$10
$10
$10
$10
$1
10c
10c
10c
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
5c
$10 $1 $1
$1 $1
$1 $1
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$100
$1 $1
$1 $1
10c
10c
10c
10c
100 10 1 1/10 1/100
$100
$100 5c
$10
$10 $1 $1
$1 $1
$1 $1
$1 $1
$1
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$100
$100
$100
5c
$10
$10
$10
$10
$1 $1
$1
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$10
$10
$10
$10
10c
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
5c
$10
$10
$10
$10
$10
$10
100 10 1 1/10 1/100
$100 5c
$10
$10
$10
$10
$1 $1
$1 $1
$1 $1
10c
100 10 1 1/10 1/100
$100
$100
$100
5c
$10
$10
$10
$10
$10
$10
$10
$10
$10
$1 $1
10c
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Page 28 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
BINGO BOARDS [1]
Instructions: Photocopy on different colour cardboard – Use first Cover the Board cards as flash cards
Nine hundred and thirty-four dollars and twenty-five
cents
One hundred and forty-six dollars and fifty cents
Six hundred and forty dollars and
thirty cents
Five hundred and sixty-one dollars and fifty cents
Seven hundred and four dollars and forty cents
Five hundred and sixty dollars and
five cents
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
5c $1 $1
$1 $1
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$100
$100
$100
5c
$10
$10
$10
$10
$1 $1
$1
10c
10c
100 10 1 1/10 1/100
$100 5c
$10
$10
$10
$10
$1 $1
$1 $1
$1 $1
10c
100 10 1 1/10 1/100
$100
$100
$100
5c
$10
$10
$10
$10
$10
$10
$10
$10
$10
$1 $1
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
5c
$10 $1 $1
$1 $1
$1 $1
100 10 1 1/10 1/100
$100
$100 5c
$10
$10 $1 $1
$1 $1
$1 $1
$1 $1
$1
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Page 29 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
BINGO BOARDS [2]
Nine hundred and thirty-four dollars and twenty-five
cents
Seven hundred and four dollars and forty cents
One hundred and forty-six dollars and fifty cents
Three hundred and ninety-two dollars and fifteen cents
Two hundred and twenty-nine dollars
and five cents
Five hundred and sixty-one dollars and fifty cents
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$100
$100
$100
5c
$10
$10
$10
$10
$1 $1
$1
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
5c
$10 $1 $1
$1 $1
$1 $1
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$10
$10
$10
$10
10c
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
5c $1 $1
$1 $1
10c
100 10 1 1/10 1/100
$100 5c
$10
$10
$10
$10
$1 $1
$1 $1
$1 $1
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
5c
$10
$10
$10
$10
$10
$10
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Page 30 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
BINGO BOARDS [3]
Nine hundred and forty-three dollars and twenty-five
cents
Five hundred and sixty dollars and
five cents
One hundred and forty-six dollars and fifteen cents
Five hundred and sixteen dollars and
five cents
Three hundred and ninety-two dollars and fifteen cents
Two hundred and twenty-nine dollars
and five cents
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
5c $1 $1
$1 $1
10c
100 10 1 1/10 1/100
$100
$10
$10
$10
$10
$1 $1
$1 $1
$1 $1
10c
10c
10c
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$10
$10
$10
$10
$10
$10
$1
10c
10c
10c
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$100
$1 $1
$1 $1
10c
10c
10c
10c
100 10 1 1/10 1/100
$100
$100
$100
$100
$100
$100
$100
$100
$100
5c
$10
$10
$10
$10
$1 $1
$1
10c
10c
100 10 1 1/10 1/100
$100 5c
$10
$10
$10
$10
$1 $1
$1 $1
$1 $1
10c
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Page 31 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
4. ADDITION AND SUBTRACTION OF MONEY
The basis of handling money is to work out if you have enough for want you want to buy
and, if you have, how much change you will get. To know these things requires the ability
to add and subtract. When working in retail, the rules for addition and subtraction are more
complex, e.g., being able to balance books, being able to equate money in cash register with
initial starting money, sales take in and change given.
These real vocational requirements should be the basis of what is taught. However, this
booklet focuses on the underlying mathematics, which is the meanings of addition and
subtraction, and addition and subtraction computation. Since calculators can, and probably
must, be used to calculate answers, this book focuses on:
(1) being able to determine which operation to use;
(2) knowing strategies for computing; and
(3) being able to use a calculator
4.1 Determining which operation to use
The secret behind addition and subtraction is to realise that addition is 2 or more parts
joining, where the total is unknown and subtraction is a total splitting into 2 or more parts
where one part is unknown. Thus, each situation has to be thought of as one part, a second
part and a total. In this way of thinking (called the part-part-total approach), the following
holds:
(a) addition is when know both parts and want to know the total; and
(b) subtraction is when do know the total and one part and want to find the other part.
Let us think of some situations:
(1) A shopper pulls out $12.50 for $27.35 bill, how much more money does she have to
find?
Total = $27.35 One part = $12.50 Other part = Unknown?
Thus, this is subtraction and we use our calculator to find $27.35 - $12.50
(2) A shopper gives you $32.65 and says they have another $28.50 to spend. How much
did they have to spend?
Total = unknown? One part = $32.65 Other part = $28.50
Thus this is addition and we use one calculator to find $32.65 + $28.50.
(3) One bag was $157/60 and the other $89.85. What is the difference in price?
Total = $157.60. One part = $89.85. Other part = difference.
Thus, this is subtraction and we use our calculator to find $57.60 - $89.85.
(4) The dress was $55.60 less than the pants and the pants cost $86.80.How much was
the dress?
Total = the dress One part = $55.60 Other part = $86.80.
Thus, this is addition and we use our calculator to find $55.60 + $86.60.
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4.2 Computation
When using a calculator, 3 things are essential:
(1) the user knows the operation and the numbers to which it applies:
(2) the user has proficiency with the calculator and pushes the right buttons; and
(3) the user can spot an obvious nonsensical answer.
Thus in training, need to:
(1) ensure knowledge of part-part-total approach (as in Section 4.1)
(2) ensure cover calculator proficiency; and
(3) develop ideas that help see sense (e.g. rules for addition and subtraction such as
addition makes things larger; mental models to enable understanding; techniques for
ease of computation).
Calculator proficiency can be achieved simply by practice: (a) give many examples to use on
the calculator; (b) explore the calculator looking at all the hidden components (look up
power calculation, constant calculation, truncation error, and so on); and (c) try “talking
calculator” problems (Google this).
Ideas that can help see the sense are as follows:
(1) Rules for addition and subtraction: Look at addition and subtraction and explore their
rules. For example, in 23 + 14 = 37
(i) What is the relationship of 37 to the other numbers? Does this always hold?
(ii) What happens if 14 increases? What if we want 37 to stay the same?
(iii) What happens if we turn it around, e.g., 37 = 23 + 14? What happens if we turn
around the left hand side, e.g., what does 14 + 23 equal?
(iv) Repeat this for 56 - 35 = 21. What happens if 56 increases? What happens if 35
increases? What if we want 21 not to change? What about 21 = 56 – 35 or
35 – 56 ?
(2) Mental models:
These are ways of thinking about addition and subtraction:
(i) addition and subtraction as separation – In this way of thinking, numbers are
separated into ones, twos, etc, then operated on and recombined. It works well
for measures (3 m 12 cm + 4 m 41 cm = 7 m 53 cm), fractions (3 3/5 + 4 1/5 = 7
4/5) and algebra (2x + 3y + 4x + 6y = 6x + 9y). It is good for estimation, e.g.,
$375 + $414 is bigger than $300 + $400.
(ii) addition and subtraction as change – In this way of thinking 23 + 32 = 55 is 23 being
changed to 55 by the act of adding 32
so it looks like as on right:
This leads to inverse – how do we get back from 55 to 23 – we subtract 32. It
helps understand inverse meanings of operations, e.g., I have $24, the dress is
$47. How much more money do I need?
This is thought of as in the diagram as on
right, and so ? = $47 - $24.
+ 325523
+?$47$24
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(iii) addition and subtraction is movement on a line. In this way of thinking addition is
moving forward and subtraction backwards on a line.
(iv) For example:
23 + 32 is:
64 – 43 is:
(3) Techniques: Sometimes it really helps to have a variety of techniques. For instance,
for 25 + 48, we can think of this as 20 + 40 + 5 + 8 = 60 + 13 = 73, or 25 + 40 + 8
= 65 + 48 = 73, or 25 + 50 – 2 = 75 – 2 = 73.
One of the most useful techniques is additive subtraction or, in terms of money, the
shopkeeper’s algorithm. In this technique, 6 – 2 = ? is thought of as 2 + ? = 6 and we
build from smaller to larger numbers.
For example:
(a) 2 + 4 = 6 6 – 2 = 4
(b) 64 – 43 is
which is 10 + 10 + 1 = 21
(c) 404 – 186 is
4.3 Sequences, activities and games
Sequencing addition and subtraction should take account of:
(i) no carrying before/carrying before 2 or more carrying;
(ii) 2 digits before 3 digits; and
(iii) addition before subtraction
Because calculators are being used, addition and subtraction activities and games should
focus on meanings and not answers. However, if activities do focus on answers, it is good to
motivate them in some way. E.g., (puzzles, tricks). Some of these worksheet ideas and
some activities and games are presented in the following pages.
+30 +2
23 53 55
-3 -40
21 24 64
+10 +10
43 53 64
+1
186
+14
200 400
+200 +4
404
218
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WORKSHEET [1]: WHAT DID THE SEA SAY TO THE SAND?
Name: ______________________ Year: _____________ School: __________________________
1. Jack gave Bill $213 and Joe $362. How much did he give away? (Use
separation) __________ = H
2. Jill bought a sound system for $322 and a fridge for $547. How much did
she pay? (Use sequencing) __________ = G
3. Frank paid $476 rent and $213 electricity. How much did he pay? (Use compensation) __________ = N
4. Sue bought dresses for $156 and shoes for $218. How much did these cost? (Use separation) __________ = V
5. Jake received $473 from CDEP and $228 from a friend. How much money
did he get? (Use sequencing) __________ = T
6. Mel got her payment of $362 and her allowance of $285. How much did she
receive? (Use compensation) __________ = J
7. John bought an MP3 player for $168 and paid the power bill for $374. How
much did he pay? (Use separation) __________ = W
8. Sue paid both her credit card amounts, $467 and $339. How much did she
pay? (Use compensation) __________ = D
9. Arthur went to town and paid $385 for his trailer to be repaired and $397 for his car repairs. How much did he pay? (Use compensation) __________ = I
10. Joe paid me $652 and Frank paid me $269. How much did I get? (Use sequencing) __________ = S
What did the sea say to the sand?
O , $689 $701 $575 $782 $689 $869 .
U
$782 $701 $647 $921 $701
.
A E
$542 $374 $806
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WORKSHEET [2]: WHAT DID THE CHEWING GUM SAY TO THE SHOE?
Name: ______________________ Year: _____________ School: __________________________
1. Frank had $500 to pay the $329 power bill. How much would he have left
to spend on other things? __________ = U
2. Eloise had to pay her car repair bill of $378. She had $450. How much
would Eloise have left? __________ = O
3. Larissa earned $640 and had to pay $250 in groceries. How much does Larissa have left? __________ = S
4. Katy saw a laptop advertised for $999 with $35 off if she paid cash. How much would she pay for the laptop if she paid with cash? __________ = N
5. Arnold received $236 for his birthday. He decided to spend $128 on clothes
and a CD and put the rest in the bank. How much money would Arnold put in the bank? __________ = K
6. The tickets cost $812. Mark said he would pay $406 towards the total cost. How much does Jeremy have to pay to buy the tickets? __________ = T
7. Emily paid $328 for food for the party. She started with $517. How much
would she have left to spend on decorations? __________ = C
8. Max had a loan of $430. He paid $159 towards the total. How much more
money does max owe? __________ = Y
9. The rent was $365. Ruby paid $167 of it. How much does her flatmate Nicole have to pay? __________ = I
10. Piper found a fridge advertised as on sale from $786 down to $592. How much money would she save if she bought the fridge? __________ = M
What did the chewing gum say to the shoe?
‘ $198 $194 $390 $406 $171 $189 $108
$72 $964 $271 $72 $171
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WORKSHEET [3]: WHAT DO LAZY DOGS DO FOR FUN?
Name: ______________________ Year: _____________ School: __________________________
1. Maria went shopping. She bought a pair of jeans for $57.45 and a belt for $16.93. How much did she spend? __________ = D
2. Angus paid $186.32 for the phone bill and $82.84 for the gas bill. How
much did he spend on bills? __________ = A
3. Julia paid $56.23 at the fruit shop and Richard paid $82.80 at the butcher’s
shop. How much did they spend on food altogether? ___________ = S
4. For theme park tickets, Ellie paid $45.50 for herself and $28.95 for her son.
How much did it cost for theme park tickets altogether? ___________ = P
5. It cost Marcus $35.82 for boat hire and $24.61 for a fishing rod and bait. How much did it cost Marcus to go fishing? __________ = C
6. At the supermarket, Matthew paid $100 for groceries worth $76.84. What was his change? __________ = K
7. Nicholas bought some art supplies for $34.71. What was his change when he paid the shopkeeper $50? ___________ =H
8. Raelene bought some goldfish and a fish bowl. The cost was $72.79. How
much money did she have left for fish food if she paid $90? ___________ = E
9. Tessa bought pizzas for dinner for her family. The cost was $39.82. What
was her change from $50? __________ = R
10. Phil bought a new pair of football boots with $150 cash. What was his
change if they cost $138.47? ___________ = !
What do lazy dogs do for fun?
$60.43 $15.29 $269.16 $139.03 $17.21
$74.45 $269.16 $10.18 $23.16 $17.21 $74.38 $60.43 $269.16 $10.18 $139.03 $11.53
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Page 37 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
5. MULTIPLICATION AND DIVISION OF MONEY
Although addition and subtraction are everyday in money situations, it is also common for
purchases of more than one object or length of material or etc. For example, 13 bottles of
Coke at $2.65 or 26 m of wood at $13.45. In these situations, multiplication is necessary.
Sometimes amounts are per pack, that is, 12 cans for $8.40. To find the cost of one can is
division (e.g., $8.40 ÷ 12)
As for addition and subtraction, calculators are used for calculation. Therefore, the
important thing is to know what buttons to press. The two foci of addition and subtraction
also hold here. Thus, multiplication and division for this booklet focus on:
(a) problem solving – being able to determine which operation to use; and
(b) computation – knowing strategies for computing and being able to use a
calculator.
5.1 Determining which operation to use
Multiplication and division are different to addition and subtraction in that
(1) for addition and subtraction, all the numbers refer to the same thing (e.g. balls,
people, money); and
(2) for multiplication and division, all the numbers do not refer to the same thing – one
number refers to the number of groups, sets, lengths of that one thing (e.g. 3 packs
of balls, 5 groups of children, 3 bottles at $3.50)
Once it has been determined that the operation is either multiplication or division, then have
to choose which of these two it is. The way to do this is to use the factor-factor-product
approach which states:
(1) the operation is multiplication when know the factors and want the product; and
(2) the operation is division when know the product and one factor and want the other
factor.
Multiplication is a situation where there is a number of groups (one factor) each with the
same number in them (the other factor) which when combined give a product. For example,
3 sets of 4 apples is 12 apples. Division is when you start with a number and partition it into
a number of groups (one factor) with the same number in each group (other factor). In this
situation, the starting number is a product. For example, 15 apples can be partitioned into 5
sets of 3 apples.
The various situations that are possible are described in Section 1.4 some of the more
complex of these situations will show how the factor-factor-product approach works.
(a) The packages of chocolates each contain 8 chocolates. The cost is $56.16. How
many packages. Product is $56.16, one factor giving number of groups is
unknown?, the other factor giving the number in each group is 8. Thus, the
operation is division as a factor is unknown. Therefore, we use our calculator to
find $56.16 ÷ 8.
(b) The money was divided amongst the 6 children. Each got $24.25. How much
money was there? Product is unknown (?), one factor is number of groups (6),
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Page 38 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
the other factor is the number in each group ($24.25) Thus the operation is
multiplication and the calculator is used to find 6 x $24.25.
(c) Jane spent 3 times as much as Bill. Jane spent $85.23. How much did Bill
spend? Product is $85.25, one factor is 3, other factor is unknown. Thus the
operation is division and the calculator is used to find $85.25 ÷ 3.
5.2 Computation
Similar to addition and subtraction, 3 things are essential:
(1) the user knows the operation and the numbers to which it applies;
(2) the user is proficient with the calculator and pushes the right buttons; and
(3) the user can spot an obvious nonsensical answer.
This requires training to:
(1) ensure knowledge of factor-factor or product approach (as in section 5.1);
(2) ensure cover calculator proficiency; and
(3) develop ideas that help see sense (eg., rules, mental models & techniques).
Similar to addition and subtraction, calculator proficiency comes from practice, exploration
and “talking calculator” activities.
Ideas that can help sense-making use as follows:
(1) Rules for multiplication and division:
Look at a multiplication like 3 x 8 = 24:
What is relation of 24 to other number?
What happens if 3 increases? What about if 24 has to stay the same?
What happens if we turn around the equation and the LH expression (e.g. 8 x
3 = ?, 24 = 3 x 8)?
Do these “rules” differ for division (e.g. what happens when the 3 in
24 ÷ 3 = 8 increases?)?
(2) Mental Models
(i) Multiplication as lots of and separation – In this way of thinking 24 x 3 is 3 lots of
24, which is 3 lots of 20 plus 3 lots of 4. Works well for measures
(e.g., 5 x (6 m 14 cm) = 30 m 70 cm), fractions (e.g., 7 x 3 ½ = (7x3) + (7x½) =
21 7/2 = 24 1/2 ), and algebra (e.g., 4 x (2x + 3) = 8x +12).
(ii) Multiplication as change – In this way of
thinking 3 x 4 = 12
This leads to inverse
(iii) Multiplication and division as area – In this way of thinking 5 x 7 is the area of a 5 x 7
rectangle or 5 rows of 7 square units. This is so powerful:
4
123
x4123
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(3) Techniques: It is useful to have a variety of techniques (see Section 1.3) For example,
5 x 37 can be thought of as:
One technique that can be useful is the grouping method for division. Most schools
teach the sharing method only, e.g. $156 shared amongst 4 people. However,
the grouping method thinks of this as how many $4’s in $156.
8 x 37 is rectangle 8 x 37
This shows that
8 x 37 = (8 x 30) + (8 x 7)
4 x 3 2/9
= (4 x 3) + (4 x 2/9)
= 12 8/9
X (X + 2)
= (X x X) + (X x 2)
X 2
X x 2X x XX
3 2/9
4 x 2/94 x 34
30 7
8 x 78 x 308
37
X 5
35 7 x 5
150 30 x 5
185
37
X 5
37 37 x 1
74 37 x 2
148 37 x 4
185
37
X 5
200 40 x 5
-15 3 x 5
185
The algorithm is similar to the traditional
showing algorithm but you do not have to be
accurate in determining “how many”.
Try removing 10 lots of 4, not enough, try 20
lots of 4, keep going until there is not enough
left for another 10. Then start taking less than
10 lots of 4. Add all the lots removed and have
answer (39)
156
- 40
116
- 80
36
- 20
16
- 16
0
10 lots of 4
20 lots of 4
5 lots of 4
4 lots of 4
39
4
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5.3 Sequences, activities and games
Sequencing should take account of
(1) no carrying multiplication before carrying one place value before carrying more them 2
place values, and so on;
(2) 2 digits x 1 digit, before 3 digits x 1 digit before 2 digits x 2 digits and so on;
(3) multiplication before division; and
(4) no carrying division before carrying before medial zero (e.g. 816 ÷ 4 which is 204 not
24).
Once again activities and games should focus on meaning and not correct answers.
However, if answers were required, try to motivate them with a puzzle or trick.
Some activities and games are presented on following pages.
How about a second example: $1561/$7?
Here you start with 100 lots of 7.
Note: this method is good when estimating,
e.g., 888 ÷ 24.
Think – how many 24’s is 888 – is there 10
lots of 24 (240), what about 20 lots of 24
(480), 30 lots of 24 (720), 40 lots of 24
(960). Therefore, the estimate is
somewhere between 30 and 40 – nearer to
40 as 888 is closer to 960 than 720. So
about 36 or 37 is a good estimate.
1561
- 700
861
- 700
161
- 70
91
- 70
21
- 21
0
100 lots of 7
100 lots of 7
10 lots of 7
10 lots of 7
3 lots of 7
223
7
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WORKSHEET [1]: WHAT DID ZERO SAY TO EIGHT?
Name: ______________________ Year: _____________ School: __________________________
Complete the following to solve the puzzle
1. 8 t-shirts cost $128. What was the cost of 1 t-shirt? Use separation. ________ = U
2. Andrew paid $96 for 16 hamburgers for the team. How much did
each hamburger cost? Use sequencing. ________ = I
3. Michael 18 exercise books for school. He paid $54. How much was
each exercise book? Use compensation. ________ = R
4. It cost Jordan $980 to rent his house for 4 weeks. How much rent did
he pay each week? Use separation. ________ = K
5. The team bought footballs to sell at their games to raise money. How much did each football cost if they paid $855 for 45 footballs? Use
sequencing. ________ = L
6. Amy bought 16 people a big box of chocolates each. She paid $288.
How much did each box of chocolates cost? Use compensation. ________ = O
7. If 15 PlayStation games cost $630, how much does one PlayStation game cost? Use separation. ________ = E
8. The teacher bought dictionaries for her class. How much did each dictionary cost if she bought 25 for $800? Use sequencing. ________ = Y
9. Tamara worked out that she spent $572 on lunches over 52 weeks. How much did she spend per week on lunches? Use compensation. ________ = B
What did Zero say to Eight?
$6 $19 $6 $245 $42
T
$32 $18 $16 $3 $11 $42 $19
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Page 42 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
FIRST TO 15
Method
Select a number (cross it out)
Use a calculator to multiply this number by 8.
Check where this number is in the framework.
First to 15 points wins.
Note – try to finish in 5 selections.
Numbers
$68-70 $56-85
$88-40 $78-55 $47-81
$49-75 $62-15 $63-45
$68-84 $74-85 $73-35
$88-85 $73-05 $65-15
$26-30 $67-20 $81-65
$79-55 $59-75 $90-70
Framework
1 POINT 2 POINTS 3 POINTS 2 POINTS 1 POINT
300 400 500 600 700 800
TALLY
Trial Points
TOTAL
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Page 43 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
MULTIPLICATION NOUGHTS AND CROSSES
1.
2.
3.
$64 $37 $48
X
6 8 7
$343 $336 $296
$288 $384 $512
$448 $259 $384
$78 $55 $63
X
7 9 5
$441 $275 $702
$546 $385 $495
$315 $390 $567
$49 $81 $68
X
8 7 9
X
$648 $544 $343
$441 $392 $729
$476 $567 $612
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Page 44 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
MIX & MATCH CARDS
Instructions: Photocopy all pages onto same colour card
John bought 6 cameras at $92 each.
How much did he spend?
6 lots of $92
$ 92 x 6
6 x $90 and 6 x $2
Freda bought 11 meals for $28 each.
How much for the food?
11 lots of $28
$ 28 x 11
11 x $20 and 6 x $8
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Page 45 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
MIX & MATCH CARDS
The pants were $46, I bought 6 pairs.
How much did I spend? 6 groups of $46
$ 46 x 6
6 x $40 and 6 x $6
Fred gave his 9 friends $62 each. How much did he give away?
9 lots of $62
$ 62 x 9
9 x $60 and 9 x $2
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Page 46 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
MIX & MATCH CARDS
Alan paid 6 weeks of rent at $134 a week.
How much did he pay?
6 lots of $134
$134 x 6 6 x $100, 6 x $30
and 6 x $4
Sue bought 14 DVDs at $28 each. How much did she pay?
14 lots of $28
$ 28 x 14
14 x $20 and 14 x $8
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Page 47 ASISTEMVET08 Booklet VR1: Whole-Number & Decimal Numerations, Operations and Computations, 08/05/2009
MIX & MATCH CARDS
Joe paid the power bills. For 8 months it cost $115 per month.
How much did he pay? 8 lots
of $115
$115 x 8 8 x $100, 8 x $10
and 8 x $5
Jacquie bought 13 uniforms at $57 each.
How much did she pay?
13 lots of $57
$ 57 x 13
13 x $50 and 13 x $7