Retail Booklet VR1: Mathematics behind Handling Money · 2018-04-17 · MATHEMATICS BEHIND HANDLING...

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.

http://www.ydc.qut.edu.au/

mailto:[email protected]

http://ydc.qut.edu.au/

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


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


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



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



(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



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



(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



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



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



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.



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



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.



$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



$99 BOARD EXTENSION ACTIVITIES

(a) (b)

$614 $837

(c) (d)

$287 $355

(e)

(f)

$964

$499



$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



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



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



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



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



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



BINGO CARDS AND BOARDS

BINGO CARDS

$934 $629 $450

$502 $317 $487

$161 $684 $721

$211 $473 $371



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



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



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



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



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.



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




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



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

$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



BINGO BOARDS [1]

Instructions: Photocopy on different colour cardboard – Use first Cover the Board cards as flash cards


cents


Six hundred and forty dollars and

thirty cents




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



BINGO BOARDS [2]


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

$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



BINGO BOARDS [3]

Nine hundred and forty-three dollars and twenty-five

cents


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



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.



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



(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



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



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



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



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),



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



(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



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



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



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



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



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



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



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



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

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Retail Booklet VR1: Mathematics behind Handling Money · 2018-04-17 · MATHEMATICS BEHIND HANDLING...

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