Basic Maths Booklet VB2: Mathematics behind Whole-Number ... · DEADLY MATHS VET Basic Mathematics...

YuMi Deadly Maths Past Project Resource

Basic Mathematics

Mathematics behind Whole-Number Numeration and Operations

Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure

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

Basic Mathematics

MATHEMATICS BEHIND WHOLE NUMBER NUMERATION AND OPERATIONS

BOOKLET VB2

USING 99 BOARDS, NUMBER LINES, ARRAYS,

AND MULTIPLICATIVE STRUCTURE

VERSION 1: 08/05/09

Research Team:

Tom J Cooper

Annette R Baturo

Chris J Matthews

with

Kaitlin M Moore

Elizabeth Duus

Fiona Hobbs

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 VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009

THIS BOOKLET

This booklet VB2 is Version One of the second booklet produced as material to support

Indigenous students completing a variety of vocational certificates at Shalom Christian

College and Wadja Wadja 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 VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009

CONTENTS

Page

OVERVIEW .......................................................................................................... 1

1. VIRTUAL MATERIALS AND MATHEMATICS LEARNING ...................................... 3

1.1 ROLE OF VIRTUAL MATERIALS ................................................................ 3

1.2 EXAMPLES OF VIRTUAL MATERIALS ........................................................ 4

1.3 EXAMPLES OF VIRTUAL MATERIAL LESSONS ........................................... 6

2. WHOLE NUMBER NUMERATION ...................................................................... 8

2.1 MEANINGS, MODELS, APPROACH AND MATERIALS .................................. 8

2.2 99 BOARDs ............................................................................................ 9

2.3 99 BOARD MATERIALS AND ACTIVITIES .................................................11

2.4 NUMBER LINE .......................................................................................16

2.5 COMPARING AND ORDERING GAMES AND GAME BOARDS ......................17

2.6 DIGIT CARDS AND PLACE VALUE CHART ................................................18

2.7 PLACE VALUE AND COMPARING GAMES .................................................21

3. WHOLE NUMBER ADDITION AND SUBTRACTION ............................................23

3.1 METHODS FOR ADDING AND SUBTRACTING ..........................................23

3.2 99 BOARD .............................................................................................24

3.3 NUMBER LINE .......................................................................................26

4. WHOLE NUMBER MULTIPLICATION AND DIVISION .........................................30

4.1 METHODS FOR MULTIPLYING AND DIVIDING .........................................30

4.2 ARRAYS AND MULTIPLICATION..............................................................31

4.3 Arrays and division ................................................................................34


Page 1 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009

OVERVIEW

The ASISTM VET project funded in 2008 by the Australian Schools Innovations in Science,

Technology and Mathematics (ASISTM) scheme had 6 sites: Wadja Wadja High School at

Woorabinda, Shalom Christian College in Townsville, Palm Island Post Year 10 Campus (run

by Kirwan SHS and Barrier Reef TAFE), Tagai College Secondary Campus at Thursday Island,

Tropical north Queensland TAFE Campus, and Northern Peninsula Area College at Bamaga.

All these sites have only Indigenous students and the project focused on developing

instruments and materials to assist the teaching of mathematics needed for certification for

Indigenous VET students with little previous success in school.

Discussions with Wadja Wadja High School and Shalom Christian College VET and

Mathematics staff decided that the project should develop basic mathematics material to

assist low achieving VET students learn prerequisites for VET courses such as understanding

of number, operations, time and simple shape and measurement.

Booklets

The first booklet, VB1, looked at 2 and 3 digit whole numbers in terms of numeration,

addition and subtraction, and multiplication and division. It focused on concepts and

strategies associated with viewing number using set model. This model represents number

in separated place values. Its most common materials are bundling sticks or MAB on place

value charts, and Money on place value charts.

The second booklet, which is this booklet, again looks at 2 and 3 digit whole numbers in

terms of numeration, addition and subtraction, and multiplication and division. However, it

focuses on concepts and strategies associated with a variety of models and materials (99

board, number line, arrays and digit cards on place value charts) that were not included in

VB1 – seeing number in a number line/rank way (99 board, number line) as well as place

value/separated way (digit cards on place value charts). As well, number is also seen in a

multiplicative way (arrays and digit cards).

This booklet covers:

(1) the nature of virtual materials in mathematics teaching as perceived by this booklet;

(2) numeration physical material activities with 99 board, number line and digit cards that

should precede the virtual material work, with 99 board and number line focusing on

seriation (e.g., adding and subtraction 10) and order (rank), and the digit cards

focusing on the multiplicative structure in the number system (e.g., x10 moves digits

one place to the left);

(3) addition and subtraction physical material activities with 99 board and number line that

use the mental computation strategies of sequencing (e.g., 34+47 = 74+7 = 81) and

compensation (e.g., 65-28 = 65-30+2 = 37), including additive subtraction (e.g., 65-

28 is the same as 28+? = 65, since 28+2 = 30, 30+30 = 60, and 60+5 = 65, the

answer is 2+30+5 = 37); and

(4) multiplication and division physical material activities using arrays (and the area model

of multiplication) which show the distributive law in action in more than one way (e.g.,

38x7 = 30x7+8x7, 38x7 = 38x1+38x2+38x4, and 38x7 = 40x7-2x7).



Files of virtual materials

This booklet is supported by virtual online materials which provide practice on computers to

support basic mathematics. They are designed to be used by students to practice activities

initially developed with physical materials. The virtual-material files for VB1 consists of three

folders, one for bundling sticks, one for MAB and one for money. They reinforce how these

materials are used to teach number, and the four operations. The virtual-material files for

VB2 consists of four folders one each covering virtual copies of 99 boards, number lines,

arrays and digit cards. The online materials for VB2 can be found in the virtual resource

folder at http://ydc.qut.edu.au/yumi-deadly-resources/past-projects.html; under BASIC

MATHS. Here you will find links to the four previously stated folders. These materials are

used to:

(1) represent 2 and 3 digit numbers written as names and written as symbols with virtual

materials;

(2) write 2 and 3 digit number-names and symbols for virtual-material representations of

these numbers;

(3) show seriation and order of numbers and the multiplicative structure of the number

system;

(4) sequencing and compensation strategy for the addition and subtraction algorithms;

and

(5) array/area approaches for multiplication and division algorithms.

Reason for this second booklet

Historically it was common to look at number in terms of place value. That is, 237 is 2

hundreds, 3 tens and 7 ones. This is a very important and crucial way to look at number

because it is the way we write and say numbers. It leads to a separation strategy for

operations – that is to divide numbers into their place values, operate separately and then

recombine. These methods are effective but difficult to do mentally and tend not to relate

well to estimation methods.

In the real world, the place value approach to number is most commonly applied in number.

However, it is not quite as effective for measurement.

A common way to use number in measurement is in measuring instruments where number is

presented on a line (in some cases, the line is curved). This has led to a growth in looking

at number as position on a line. This has been found useful in ordering numbers and also in

methods for operations that are more in harmony with estimation methods and work better

mentally (what are now called mental computation strategies – namely, sequencing and

compensation).

Thus, this booklet is an important adjunct to VB2 in providing this alternative, but important,

way of looking at number. Thus, the initial focus in VB2 on 99 board and number line, and

the later focus on the array and then area model with number lines for length and width.

Finally, the booklet includes materials that should have been put in VB1, digit cards and

place value charts. These are used to show the x10 and /10 relationship between adjacent

place value positions.

http://ydc.qut.edu.au/yumi-deadly-resources/past-projects.html



1. VIRTUAL MATERIALS AND MATHEMATICS LEARNING

1.1 ROLE OF VIRTUAL MATERIALS

Current pedagogical beliefs emphasise that the abstraction of mathematical concepts and

processes is best served by a combination of work with appropriate manipulatives and

reflection with peers and teacher. Manipulatives are most obviously physical but mental

manipulation can also be undertaken with pictures and diagrams. Reflection is with

language and symbols.

Virtual materials provide another collection of manipulations to add to the physical, and

pictorial and diagrammatic. Therefore, teaching mathematics can be seen to involve the use

of manipulation (physical, virtual, pictorial and diagrammatic materials, written symbols,

spoken words) in order to facilitate student development of mental models (internal

representations). The kinaesthetic actions associated with physical and virtual manipulatives

(physical and mouse movements) assist abstraction by providing mental images to scaffold

the symbolism.

Most activity with physical materials involves sliding, joining, separating, grouping,

ungrouping, partitioning, turning and flipping actions. All of these actions are available on

computer through mouse movements and images of the concrete materials (virtual

materials) using computers with commonly available generic software. Thus, computer

activities with virtual materials reflect activities with concrete and pictorial materials and their

efficacy is bound up with the effectiveness of physical materials.

Parts of what learners construct from interaction with materials are built into the medium of

the materials. Students can mentally replicate (in their schemas) the relations and

transformations represented by the concrete material, and abstract this mental replication to

symbols and mental models; however, there is a gap between action and expression that is

difficult to bridge. Physical materials are often very multi-sensory (e.g,, they involve colour

or have interesting textures or shapes) which can hinder the abstraction process. Pictorial

materials are more abstract than concrete materials as the child is expected to imagine any

manipulation that may have been required to transform. Physical and virtual materials are

also inflexible and can be messy.

Thus, virtual materials and actions can be effective in teaching mathematics because they:

(1) are a bridge between physical and pictorial being not as overt as physical

representations nor as covert as pictorial representations;

(2) can be “debugged, reconstructed, transformed, separated and combined together” and

saved for later reuse with the same or other students;

(3) provide teachers with unique knowledge of all students’ proficiency with all

components of the manipulations as the manipulations can to be saved and stored for

later assessment;

(4) can integrate with physical materials in a way that enhances mathematics learning

(because of the way they reinforce physical materials); and

(5) have capacities for actions, activities and representations not easily available with

concrete materials; for example, shapes can be enlarged by specific amounts, turned

by specific degrees.



In this way, virtual materials use the visual, symbolic and operational power of the

technological media and provide another pedagogical and didactical tool for the media.

Note: overall, the virtual materials developed for this booklet are a very different use of

computers in mathematics education than that commonly seen in schools, at least in

Queensland. As computer activities, they are relatively simple – students manipulate

computer drawn copies of real materials using PowerPoint. As such, virtual materials have

comforting similarities to concrete activities and this aspect seems to make it easier for

teachers to translate their mathematics teaching to virtual materials and, thus, to computers.

The best way to use virtual materials is to integrate it with other representations:

(1) virtual materials should follow work with the physical materials that they copy;

(2) virtual materials also work well if integrated with physical materials (interchanging

from physical to virtual);

(3) virtual materials require student expertise and familiarity with the PowerPoint actions

that they use (e.g. copy, paste, click and move)

An effective way to teach mathematics is to use the Payne-Rathmell model as below. This

means starting with real world problems, modelling with physical, virtual and pictorial

materials, and then introducing language and symbols. Then, this should be followed with

activities to connect all these 5 forms in all directions (reversing). The models used should

follow the order on right, starting with real world moving to physical, virtual and pictorial and

finishing with patterning activities (as below).

Payne & Rathmell triangle Models

Real world situations Sequence Physical

Virtual

Pictorial

Patterns

1.2 EXAMPLES OF VIRTUAL MATERIALS

(1) Place value for two-digit whole numbers can be effectively developed by activities with

bundling sticks (singly and in bundles of ten) and a tens/ones place-value chart as

follows.

The strength of this learning approach is that it is multi-representational (providing

visuals, language & symbols) and dynamic (showing transformations and changes as

well as relations). It also leads onto pictorial representations. Concrete to pictorial

Models Language Symbols

Tens Ones

Child constructs or sees

forty-three

Child says

T Ones

4 3

Child writes



material represents a sequence of abstraction because concrete materials can be

physically manipulated whereas pictures cannot the child is expected to imagine any

manipulation.

(2) Virtual base-10 blocks on a computer can be manipulated (“clicked and dragged”)

similar to the real blocks and provides another way in which numeration can be taught.

As follows, such virtual materials should provide a conceptual bridge from concrete to

pictorial representations.

(3) Space and shape activities can also be taught by sliding, flipping and turning, as

follows. In particular, students can easily undertake tessellations with virtual materials.

Assembling a class set of real materials is time-consuming.

(4) Tessellations can be easily completed virtually. Sliding, flipping and turning virtual

shapes requires only one template, which can be downloaded for individual student’s

use. The students themselves can then quickly copy the shapes required and, with

respect to tessellations, have access to a variety of colours to enhance the final

product (as follows).

Sliding the shape from one

position to another

Flipping the shape from one

position to another

Rotating the shape from one

position to another

Start with ... … copy and tessellate. Start with ... … copy and tessellate.

Concrete materials to represent place value and size relationships between places

Place value chart used in conjunction with

concrete materials or digit cards to represent

position and order of the places

Pictorial representation

TensHundredsThousands Ones

1 3

Bundling sticksUnifix cubes Base-10 blocks

Tens Ones



1.3 EXAMPLES OF VIRTUAL MATERIAL LESSONS

(1) 2-digit numeration

Two-digit numeration activity uses base-10 blocks (as follows) to represent numbers.

The mouse actions involved in picking up and placing the blocks on the virtual place

value chart are very similar to the hand movements that pick up and concrete place

blocks on a real place value chart.

(2) Polygons

Students “click and drag” shapes to assess the extent to which they understand the

polygon concept.

(3) Telling the time

Students “told time” on a virtual clock (as follows). This was after they had to stand

up and physically rotate their bodies using their extended arms to point to a given

position (e.g., 4 o’clock) on a large circle (clock face) drawn on the floor.

Polygons Not polygons

Drag the shapes to the correct box.

Polygons are .

Use the blocks to show these numbers: 23; 35; 41

Tens Ones

Make the clock show

4 o’clock



(4) Flips, slides, and turns

Flips, slides and turns can be introduced through tangrams (as follows).

(5) Patterning

Assist recall of basic multiplication facts by exploring patterns by colouring squares on

virtual hundred boards to show multiples of 5, 3 or 9 and so on. The action of the

mouse to colour the squares is very different to the action of a coloured pen on a

paper 100s board. However, the ability of the repeat button to quickly colour squares

and the PowerPoint program to edit errors made the virtual boards very attractive to

the students.

(6) Scales

This involves placing number on number lines to reinforce teaching of scales (as

below).

Identify the number at A. Show 330 on the scale.

0 A 500

The virtual scales developed to practice this identification and finding of numbers were

based on authentic scales, for example, tachometers, measuring cylinders and

altimeters. The students could move arrows or empty and fill cylinders to identify the

numbers. The students found this motivating and the virtual activities were well

received.

USE THE TWO SMALL TRIANGLES ...

…TO MAKE THE SQUARE AND THE RHOMBUS.

USE ALL THE TANGRAM PIECESTO MAKE THE DOG!



2. WHOLE NUMBER NUMERATION

2.1 MEANINGS, MODELS, APPROACH AND MATERIALS

Whole-number numeration has 4 components/meanings.

(1) Place Value

Students should know that values of numbers are determined by the position of their

digits in relation to the 1’s position (e.g. 257 is 2 hundreds, 5 tens and 7 ones). When

students see a number, they should realise that what the number means is determined

by position and the value of that position (where the value is based on multiples of 10)

(2) Counting

Students should know that numbers follow the same counting pattern in each place-

value position, e.g. counting forward – 472, 482, 492, 502, 512 (go up to 9, then back

to 0, number on left increases by 1); counting backward – 5324, 5224, 5124, 5024,

4924 (go down to 0, then back to 9, number on left decreases by 1)

(3) Rank

Students should know that, regardless of how many different digits are in it, the

number represents one position on a number line and that its overall value is

determined by how far down the line it is.

(4) Multiplicative structure

Students should know that adjacent place value positions are related by multiplication

and division by 10 (i.e., move one place to the left is x10, move one place to the right

is ÷10, move two places to the left is x100, and so on).

Whole-number numeration has processes:

reading and writing numbers (words, language and symbols);

place value (e.g., 3 tens, 4 hundreds and 5 ones is 435);

seriation and counting (e.g., 435+10, 435-10 – counting forward and backward in

any place value);

comparing and ordering (e.g., 402 is larger than 295);

renaming (e.g., 435 is 43 tens and 5 ones, is 3 hundreds 11 tens and is 3 hundreds

and 135 ones, and so on); and

rounding and estimating (e.g. 435 is 440 to nearest ten).

These meanings and processes can be demonstrated using various models and materials,

including:

set (bundling sticks, MAB and money) – for place value and counting and for

reading/writing, place value, seriation, and renaming;

number Line (99 board and number tracks and lines) – for rank and for seriation,

order, and rounding; and

digit cards and place value charts – for multiplicative multiplicative structure.



This booklet (VB2) focuses on number line, array and set models and predominantly the rank

and multiplicative structure components.

To do this, VB2 makes use of the following three materials (each in turn is described in this

Section):

(1) 99 Board

99 boards are a board with a 10x10 array of numbers 0 to 99 in rows 0 to 9, 10 to 19,

20 to 29 and so on. Numbers in columns have the same ones place, while numbers in

rows have the same tens place. This means that if we consider a number like 43, the

number on left is 42, on right 44, above 33 and below 53. Tens are given by the

number of jumps down and ones are number of jumps across.

(2) Number Line

Number lines are straight lines divided into ones, tens and hundreds (like a measuring

tape). They can have each number marked, just a few numbers marked and no

numbers marked. Larger numbers are further away from the LHS (usually a zero) than

smaller numbers, so number lines are good for order.

(3) Digit Cards and Place Value Charts

These are cards with digits 0 – 9 printed on them and are used on place value charts.

They can be used to show how numbers are changed when the digits are moved to

the right or the left. If students are given a calculator, they can explore what

multiplications and divisions will give movements of one or more places to the left and

right. Activities can go from movement to operation and operation to movement. (Rule

is that x10 is 1 place to left and /10 is 1 place to right.)

Each of these has virtual materials to support it. The crucial thing is to allow the students

time to familiarise themselves with the real materials before using the virtual materials.

2.2 99 BOARDS

The 99 board represents number in terms of rows down for tens and columns across for

ones. Activities focus on building understanding of position of numbers so that can easily

determine 1 more and less and ten more and less. A sequence of possible activities follows.

(1) Getting to know the patterns of numbers

Have students read columns and rows. For example, 4, 14, 24, ...; and 60, 61, 62, ...., and notice the patterns.

Reading a column:

It can be useful to have students read the column as

“four, onety-four, twoty-four”, and so on. Then the

pattern in the column can be seen – it is that “four”

is said each time with the tens going up, that is, the

ones stay the same and the tens increase.

Reading a row:

The row is read as “sixty, sixty-one, sixty-two”, and

so on. Then the pattern in the row becomes

apparent – it is that the “sixty” is said each time with

the ones increasing, that is the tens stay the same

and the ones increase.



Extension ideas:

a. Cut 99 boards into jigsaw puzzles and get students to reform them. Get

students to make puzzles for each other.

b. Hand out 99 boards with parts missing and students have to complete the

numbers.

(2) Knowing where numbers are – placing numbers by tens and ones

Always start at zero. For position to number - get students to start at zero and move

down and across encouraging students to see pattern, e.g., that 3 down and 7 across

is 37. For number to position – move the tens down and ones across, e.g., 54 is 5

down and 4 across (starting at zero).

Extension ideas: Play “Three in a row”. Players in turn take two cards from a pack with

1 to 9 in it (A is 1 – K, Q, J and 10 removed) and cover any number they can make

(e.g., 4 and 3 could be 43 or 34) with a counter (can remove opponents counter to

place yours). The first player to get three in a row (row, column or diagonal) wins.

(3) Teaching seriation

Use the board to identify the numbers on left, right, above and below chosen number

– show how left and right is 1 less and 1 more, above and below is 10 less and 10

more.

In the first teaching direction, 3 down and 7

across is given and students find they reach

37. In the second teaching direction, 54 is

given and students fiind the movement (5

down and 4 across) that reach this number.

This is an example of the generic pedagogy

of reversing – teaching in both directions.

These activities practice finding and placing

numbers.

Look at 78, 1 less is 77, 1 more is 79, 10

less in 68 and 10 more is 88.

Start with the number and look at the

left, right, above and below numbers –

use a calculator to determine relation of

these numbers to the original number.

Start with the number and determine

(use a calculator) 1 less, 1 more, 10 less

and 10 more and then find these

numbers on the board and their relation

to the starting number.



Extension ideas:

a. Construct a 99 board window with a hole in middle, place over a number so

can only see that number and write numbers 1 less and 1 more, above and

below.

b. Give 3x3 squares with number in middle and ask for other numbers.

c. Give 3x3 squares with numbers on outside and ask for number in middle

(reversal of (b) above).

d. Give jigsaw pieces from 99 board with only one number written in one square

and ask students to fill in other squares.

2.3 99 BOARD MATERIALS AND ACTIVITIES

The following materials and activities are taken from Numeration materials developed under

the leadership of Dr Annette Baturo for training Indigenous teacher aides. Note the

extension of activities to 3-digit numbers. Copy and laminate this 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 windows

Find the centre worksheet

9_

78

8_ 87

C

82

62

73 71

A

36

16

27 25

B

52

43

D

_5

4_

E

_9

F _7

4_

G



Fill all the spaces worksheet 1

A 15 B 45

26

35 66

C D

79 42

87

63

E

33 F 5

41



Fill the space worksheet 2

A _2 B 3_

5_

72 _6 57

C _8 D 4_

7_

87 63

E

_3 F _3

6_ 5_



Fill the space worksheet 3

363 578

A B 586

385 596

C D

922 235 236

931 932

942 254 255

264

E

839 F 666

847



2.4 NUMBER LINE

The number line represents number as points in order along a line. Each number is in

relation to other numbers before and after it, and in relations to the ten and hundred before

and after it. Activities focus on building understanding of in terms of their position in relation

to other numbers so that can easily determine order and rounding (as well as seriation to a

lesser extent). A sequence of possible activities follows.

(1) Walking number track

Start with a number track made of A4 sheets of paper with consecutive numbers from

0 to 20 on the floor in a line forming a number track. Get the students to walk on the

track stating the numbers as they step on them (start from different numbers) going

forward and backwards. Get watching students to also state the numbers.

(2) Transfer to a picture of a number track

Transfer this activity to a symbolic number track. Give each student a copy of a printed

number track (provided at the end of this section) and a counter. Repeat the activity,

but instead of standing on numbers and stepping, have students place their counter on

a number and move it one square at a time to symbolise “stepping”. Start from

different numbers.

Extension ideas: Play “race-track” games – e.g., snakes and ladders. Focus on students

knowing that they count jumps not numbers.

(3) Transfer to number line

Place a number line under the track. Discuss how numbers are at the end of the

spaces not in the middle of the spaces. Discuss how count along the numbers but that

the count on or the count back is the number of jumps between numbers not the

number of numbers.

Extension ideas: Make sure students can work with number lines with all numbers

marked, 5s and 10s only marked, only some 10s marked, and no numbers marked.

(4) Construction of number lines

Construct a number line from cm graph paper or straws. Cut into 10cm pieces, join

together to make a 100 cm, mark the start, finish and joins with 0, 10, 20, 30, and so

on (see below). Use the number line to: (a) place numbers, and (b) state what number

is in a particular place. Discuss the numbers in terms of ten – e.g., 56 is 5 tens (pieces

10 long) and then a little over half the next piece, while the position which is 7 ten

pieces and a little more along is about 72.

(5) Cord and pegs

Make pieces of paper with 0 and 100 (or other end point numbers) and numbers in

between. Get two students to hold a cord. Peg 0 and 100 at each end. Give students

10

20 30 40 20



numbers and get them to peg on cord where numbers should be. Discuss each

pegging. Reverse the activity by placing pegs and discussing what numbers should go

there.

(6) Ordering numbers with materials

Give students two numbers, get them to peg on cord, place on straw/cm paper

number lines or mark on picture of number line and work out the larger. Always get

students to predict first and give reasons for predictions.

(7) Ordering numbers in mind

Have students imagine the line in their mind and then use the imagined line to

compare and order numbers.

Extension ideas: Use this imaginary line to play games below

2.5 COMPARING AND ORDERING GAMES AND GAME BOARDS

(1) Chance number – Make a number

Materials: digit cards, large version of boards below, card deck (0-9 only)

Directions:

a. After teacher (or another student) deals 2 to 4 cards (depending on board

being used), use numbers to make smaller/larger number with digit cards on

game board as required.

b. As teacher (or another student) deals 2 to 4 cards one at a time, use first

number to place a digit card on board (have to choose tens or ones), second

number fills the other position. If make higher/lower number, score 1 point, 0

otherwise. Winner is largest score after 5 games.

c. As for (1) or (2) above but win if closest to 50.

d. As for (1) above, but three cards are dealt to choose from.

(2) Chance number – Beat the teacher

Materials: digit cards, large copy of boards in game (1), card deck (0-9 only)

Directions: As for “Make a number” but score/win if beat the teacher (who is also

playing).

(3) Chance number – Risk a card

Materials: digit cards, large copy of boards in game (1), card deck (0-9 only)

Directions:

a. As for “Make a number” (2) but when complete, can give up a number and

take the value of a third dealt card.

b. “Double risk” – can give up two numbers and 4 cards dealt (one at a time) –

can set rule that numbers cannot be risked from the same place value.



(4) Chance order

Materials: digit cards, game boards as below, card deck (0-9 only)

Directions:

a. After teacher (or another student) deals 4 to 6 cards, use numbers to make

the left hand 2-digit or 3-digit number smaller/larger than the right hand

number with digit cards on game board as required. Score 1 point if left hand

2-digit or 3-digit number is correctly larger (smaller) than right hand 2-digit or

3-digit number. Score 2 points if smaller 2-digit or 3-digit number is largest

possible. The winner is who has highest score after 5 games.

b. As teacher (or another student) deals 4 to 6 cards one at a time, use first

number to place a digit card on board (have to choose tens or ones, or

hundreds, tens or ones, in either the left hand or right hand 2-digit number),

continue making choices and placing digits on board before next card called.

Score 1 if correct and 0 if not. The winner is who has highest score after 5

games.

c. As teacher (or another student) deals 4 to 6 cards one at a time, use first

number to place a digit card on board (have to choose tens or ones, or

hundreds, tens or ones, in either the left hand or right hand number),

continue making choices and placing digits on board before next card called.

Score 0 if not correct but score the value in the tens or hundreds place of the

smaller 2-digit or 3-digit number if correct. The winner is who has highest

score after 5 games.

2.6 DIGIT CARDS AND PLACE VALUE CHART

Digit cards are small cards with numbers on them that can fit into the columns on a place

value chart as follows.

They can be used with any size chart to make numbers and to study what happens when

digits change place value position – to make students aware of the multiplicative (x10 or

÷10) relationship between place-value positions.

(1) Using students’ bodies

Make up large copies of digit cards and place value positions, e.g., 3 place values as

below, and give students calculators.

Hundreds Tens Ones

0 1 2 3 4 5 6 7 8 9

less

than

less

than

greater

than

greater

than



Give 3 students PV cards and organise them to stand in correct position:

Give another student a digit card, say 6, and get them to stand in front of each

position. Add zero cards to show what each number means. Press buttons to place

numbers on calculator, e.g.: 6 in tens position:

Repeat this for 2 and 3 digit numbers on cards in front of PV cards, e.g. 230, 604, 14,

824, and 615. Move from cards to calculator and calculator to cards (reversing). Say

numbers in terms of 100s, 10s and 1s and properly.

(2) Generalising the movement

Put a digit card in front of PV cards, move card left and right, use calculator x and ÷

buttons to show relationship in moves, e.g. 6 tens going to 6 ones is ÷10 and 6

ones going to 6 hundreds is x100. Put a number in calculator, e.g. 40 and multiply or

divide by 10, move cards to show these multiplications and divisions (note that the

place value cards could be stuck on wall):

e.g. x 10 ÷ 10

Write down patterns in movements and relation to x and ÷ 10 (generalisation). (Note:

the pattern is better seen with more place-value positions. The materials can also be in

virtual form and displayed with a data projector of smart board.)

(3) Translation to pictures

After acting out the above in front of class, all students could be given their own small

digit cards and place value chart materials, or given this in a form of a slide rule where

the digits are pulled across under the place-vale positions (note that this material can

Hundreds Tens Ones

Hundreds Tens Ones

6

0

60

Hundreds Tens Ones

4

Hundreds Tens Ones

4



also be in virtual form). See the example of a slide rule, for one to million place-value

positions, on the following page.

(4) Ones to millions slide rule

Cut out slide rules and slides. Cut along dotted lines. Insert slides. Use as with digit

cards with a calculator to look at relationships as digits move left and right.

MIL

LIO

N

HU

ND

RE

D

TH

OU

SA

ND

TE

N

TH

OU

SA

ND

TH

OU

SA

ND

HU

ND

RE

D

TE

N

ON

E

1 7

0 7

4

8 2

5

3 8

6



(5) Pattern of threes

The digit cards can build the important pattern of threes (e.g., for 356 872 913, break

digits into threes, e.g., 356/872/913, recognise that right hand 3 digit are ones, middle

3 are thousands and left hand 3 digits are millions. Say as a series of 3 digit numbers

(covering other 3 digit groupings as say the 3 digits in focus) - e.g., 356 millions, 872

thousands and 913 ones.

Construct following cards and organise students to stand in order with PV cards. Get

another student (or students) to move in front of PV cards with digit cards. Using zero

cards where needed, get students to state numbers shown. Also reverse - start with

numbers and ask students to show these with PV and digit cards

Place cards on wall, get students to place digits in front of PV and then move left (L) or

right (R) one or more spaces. Use calculators to follow these movements by x, ÷ by

10/100/etc. as appropriate. Ensure work is both ways: show movement L/R and then

find x/÷ by 10/100/etc., and show x/÷ by 10/etc., and find movement L/R (reversing).

Propose general rule for relating PV positions in terms of x, ÷ (generalising).

2.7 PLACE VALUE AND COMPARING GAMES

(1) Wipe-out (place value)

Materials: Calculator, worksheet (if wanted).

Number of players: 2

Directions:

a. One student calls out a number, e.g. 673, 56 782, 24.875. Other students put

in calculator then 1st student calls out a digit. Other students have to change

number on calculator (wipe the 7) with a single subtraction, e.g. 603. 56 802,

24.805.

b. Can be used as a worksheet as below:

Number Digit Subtraction Result

284 8 −80 204

745 892 5 −1 000 740 892

c. Do examples with 2 digit positions to wipe (e.g., 347.642 – wipe out both 4s).

Millions

H

Millions

T

Thousands

H

Ones

O

Millions

O

Ones

T

Thousands

T

Ones

H

Thousands

O

4

Millions

H

Millions

T

Thousands

H

Ones

O

Millions

O

Ones

T

Thousands

T

Ones

H

Thousands

O



(2) The Big One (place value)

Materials: Calculator


Directions:

a. 1st player chooses a number between 9 and 100 (does not reveal this to

opponent) and enters on his/her calculator choice ÷ choice = (this give 1 on

the calculator).

b. The calculator is then given to the 2nd player. The 2nd player puts guess =,

guess = until 1 appears (has guessed the number).

c. Players take turns being the 1st and 2nd player. The winner is the player with

the lowest number of guesses after 5 games.

d. Players can set numbers between 9-100, with up to 2 decimal places (e.g.,

can choose a decimal number).

(3) Target (order, estimation)

Materials: Calculator, worksheet if necessary.


Directions:

a. Give students a starting number and a target number, e.g. 37 and 9176.

Enter 37 x in calculator. Then press guess =, guess =, until get the target.

(No pressing of “clear all”).

b. Students take turns being the starting number provider. After 5 goes each,

the winner is the student with the lowest number of guesses.

c. Can be done with worksheet, e.g.

Number Target Too high Too low Current

guess

Number of

guesses

(a)

(b)



3. WHOLE NUMBER ADDITION AND SUBTRACTION

3.1 METHODS FOR ADDING AND SUBTRACTING

There are three strategies for adding and subtracting:

(1) Separation (Place Value)

This method is based on considering number as separate place-value positions. The

computations are undertaken by separating the number into place-value positions,

operating on each position and then combining. For example, 247+386 and 715-268:

2 4 7

+ 3 8 6

1 3 (7+6)

1 2 0 (40+80)

5 0 0 (200+300)

6 3 3

Step 1 H T O

7 1 5 (7 hundreds and 15 ones)

- 2 6 8

7

(15 ones – 8 ones = 7 ones)

Steps 2 & 3

7 1 5 (6 hundreds 10 tens and 15 ones)

- 2 6 8

4 4 7

(15 ones – 8 ones = 7 ones)

(10 tens – 6 tens = 4 tens)

(6 hundreds – 2 hundreds = 4 hundreds)

The method acts on the hundreds, tens and ones separately – it uses place-value

oriented materials.

(2) Sequencing

This method is based on a rank or number line orientation to number. The

computations keep one number as is and add/subtract bits of the other number in

sequence. For example, 247+386 and 715-268

247 715

547 + 300 515 - 200

627 + 80 455 - 60

633 + 6 447 - 8

The 300, the 60 and the 8 are added separately and the 200, 60 and 8 are subtracted

separately. It is not necessary to have any order – the numbers could be added or

subtracted in reverse order, or in a mixed up order.

(3) Compensation

This method is again based on rank. Numbers near the ones in the computation are

chosen to make the operation easy and then the changes compensated for. For

example:

247 247 715 715 + 386 + 400 - 268 - 300 647 415 - 14 + 32 633 447

6 10 1

0 1



For 247+386, we added 14 too much and so we had to remove it. For 715-268, we

took off 32 too much, so we had to add it.

These three methods are all available for students to use. However, for this booklet and the

virtual materials, we will be focusing mainly on the sequencing and compensation methods.

3.2 99 BOARD

Adding and subtracting on the 99 board is based on up being subtracting 10 and down being

adding 10. Left or back is subtracting1, and right or forward is adding 1. This allows the

sequencing and compensation strategy to be used.

(1) Adding with no carrying

To add 34+23, start at 0, move 3 down and 4 right to give 34, and then 2 down and 3

right to give the total.

34+23=57

(2) Subtracting with no carrying

To subtract 78-42, start at 0, 7 down and 8 right to give 78, then move 4 up and 3 left

to give the total remaining.

78-42=36

34 is 3 tens and 4 ones

3 down

4 right


2 down

3 right


7 down

8 right


4 up

2 left




2 down

6 right

39 is 4 tens less 1 one

4 down

1 left

(3) Adding and subtracting with carrying

To do this, repeat steps in (1) and (2) but with the ones, have to move down to the

next line for addition and up to the above line for subtracting. See examples (a)

17+26 and (b) 83-25.

a. 17+26=42

b. 83-25=58

(4) Building compensation strategy

Adding 9 becomes adding 10 and subtracting 1, and subtracting 8 becomes subtracting

10 and adding 2.

a. 26+39=65

Start at 0, move 2 down and 6 right, and 4 down and 1 left (instead of 3 down

and 9 right).

17 is 1 ten and 7 ones

1 down

7 right


2 down

6 right (3+3)


8 down

3 right


2 up

5 left (3+2)



b. 85-28=57

Start at 0, move 8 down and 5 right, and 3 up and 2 right (instead of 2 up and 8

left).

3.3 NUMBER LINE

The number line is simpler for addition and subtraction than the 99 board – movements are

right for addition and left for subtraction.

(1) Adding and subtracting on a number track

a. Begin with a large number track on the floor.

Show 6+3 as walking forward 6 and walking forward another 3.

Show 12-5 as walking forward 6 and walking back 5.

Adding and subtracting involves jumps forward and backward, not numbers

forward and back, e.g. 6+2 is 678, because adding 2 refers to counting on 2

jumps from the starting number: 67 and 78.

b. Next, move onto the drawn number track, moving to calculate addition and

subtraction exampled by jumping forward and back.

Extension Ideas: Play games where digits are obtained by chance (cards, die), which

have to be added along a number track or line.

(2) Adding and subtracting larger numbers on a number line

Relate the number track to the number line as below:

Initially, use low numbers such as 7+8 and 14-9 by jumping forward and backward.

Then move onto larger numbers by moving forward and backward in 10s and 1s as the

examples below show.


8 down

5 right

28 is 3 tens less 2 ones

3 up

2 right



a. 23+45=68

Start on 23 and move 4 tens forward, then 5 ones forward.

b. 87-34=53

Start on 87 and move 3 tens backwards then 4 ones backwards.

(3) Addition and subtraction with carrying

This is the same as (2) but goes past the 10 as the examples below show.

a. 28+37=65

Start on 28 and move 3 tens forward then 2 ones + 5 ones (7) forward.

b. 73-46=27

Start on 73 and move 4 tens backwards then 3 ones backwards and another 3

ones backwards.

10 10 10 10 5

0 10 20 30 40 50 60 70 80 90 100

23 63 68

0 10 20 30 40 50 60 70 80 90 100

53 57 87

4 10 10 10

0 10 20 30 40 50 60 70 80 90 100

30

33 73 27

3 3 10 10 10 10

60

28 58

0 10 20 30 40 50 60 70 80 90 100

65

10 10 10 2 5



(4) Number line with no markings

This is the same as (2) again but with a lot more flexibility in the diagram, as the

examples below show.

a. 356+288=644

b. 734-268=466

(5) Additive subtraction

The method whereby numbers are subtracted by looking at the difference between the

two numbers is a very useful strategy. See the examples below.

a. 85-42=43

b. 82-45=37

Make both numbers then build from smallest to largest in the simplest way

c. 717-368=349

474 534 734 466

8 60 200

356 556 636 644

200 80 8

40 3

42 40 82 3 85

43

42 82 85

5 30 2

45 50 80 82

45 5 50 30 80 2 82

37

368 2 370 30 400 300 700 17 717

349

2 30 300 17

368 400 700

370 717



(6) Compensation strategy

If adding something near 100, like 95, it is easier to add 100 and subtract 5. So, the

number line method can be used to represent the compensation strategy as the

following examples show.

a. 368+93=461

Regular strategy, see (4)

Compensation strategy

b. 428+386=814

Regular strategy, see (4)

Compensation strategy

40 50 3

368 408 458 461

40 50 3

368 408 461 468

400 14

428 828 814

300 80 6

428 728 808 814



4. WHOLE NUMBER MULTIPLICATION AND DIVISION

4.1 METHODS FOR MULTIPLYING AND DIVIDING

Multiplication is primarily combining equal groups. There are three models for this as

follows for the example 3x4.

Set 3 groups of 4

Number line 3 hops of 4

0 5 10

Array 3 rows of 4

The array model can become 3 rows of 4 squares, e.g.

4

3

So, the array model can be viewed in terms of area.

Division is both sharing and grouping.

(1) Sharing

Sharing is when you take a number (the product) and partition it equally amongst the

groups. It is when the number of groups is known not the number in the group. For

example, in the problem 12÷3:

I have a jar of 12 lollies and need to put equal number of lollies into each of 3 bags.

How many in each bag?

The unknown is 3x?=12.

(2) Grouping

For grouping, the product is partitioned into a number of equal groups. It is when the

number in the group is known. For example, in the problem 12÷3:

I have a jar of 12 lollies and I need to put 3 lollies into each bag. How many bags?

The unknown is ?x3=12.

The traditional set model algorithm of booklet VB1 is sharing. The array method of division

(see Section 4.3) will be grouping which will give rise to a new grouping algorithm.



Similar to addition and subtraction, there are three methods for multiplying and dividing 2

and 3 digit numbers:

(1) Separation

This is the traditional method where things are done by separating into place values (sharing), for example:

(2) Sequencing

Here one number (the larger for 2 x 1 digit) is held unseparated and parts of the other number are done in sequence, for example:

Multiplying Dividing

Separating 8 into 2x2x2: Building up to the product (grouping)

(3) Compensation

A simple multiplication or division is found and then compensated for, for example:

For this booklet, we focus on methods (1) and (2) (separation and compensation). This

work is based on the array/area model of multiplication. For the set model (lots of, groups

of) and bundling sticks, MAB and money, see booklet VB1. The number line is not used.

4.2 ARRAYS AND MULTIPLICATION

The area/array model is a powerful tool for multiplication, particularly because it shows the

distributive law. That is that 34x7 is 30x7 plus 4x7.

(1) Arrays lead to area

Have students to think of 3x5 as 3 rows of 5 and construct arrays using counters for various multiplications. Then move onto making the arrays with squared and finally to shading squares

37

X 8

74

148

296

double (37x2)

double (74x2)

double (148x2)

37

x 8

37

x 10

370

- 74

296

37

x 10

370

- 37

333

- 37

296

(2 groups of 37 too many)

(Less one group of 37)

(Less another group of 37)

(10x37)

- ( 2x37)

8x37

9 747

450

297

180

117

90

27

27

0

(50 groups of 9)

(20 groups of 9)

(10 groups of 9)

( 3 groups of 9)

83 groups of 9 make 747

83 9 747

72 27 27 0

37

x 8

56

240

296

sharing tens sharing ones

(7 x 8)

(30 x 8)



Counters Squares

Shading

Relate this to area as length x width, so construct area for various multiplications as follows.

Extension ideas: Make a drawing of 6x4. Rotate the diagram 90° to show that 6x4 =

4x6 (the commutative law).

(2) Distributive law and place value

Construct 2 digit by 1 digit multiplication, for example:

4 x 23 7 x 49

Make these with 2mm graph paper. See that they can be divided as follows.

4 x 23 7 x 49

Show that 4x20 + 4x3 is the same as 4x23 and 7x40 + 7x9 is the same as 7x49 (count

squares, use a calculator).

8 5

19 12

6 4

6 4

23

4

49

7

20 3

4

40 9

7



(3) Multiple of 10 multiplication

Show that if 3x4=12, then 3x40=120. Use graph paper or calculators to see this. Keep

going till students see the pattern, e.g., 5x60 = 5x6 tens = 30 tens = 300.

Extension ideas: You can move on to show that 30x40 = 1200 and widen the pattern

in multiplying by multiples of 20.

(4) Using Arrays to Multiply 2d x 1d

Construct areas, for example, break examples into parts based on place value,

calculate the parts and add to get the answer.

a. 4x23

92

b. 7x86

602

Extension ideas: Can change the diagram in other ways, for example:

a. 4x23

92

23

4

80 12

23 x 4 12 (4x3) 80 (4x20)

92

20 3

4

86 x 7 42 (7x6) 560 (7x80) 602

86

7

80 6

7

560 42

23

4

23 x 4 46 (2x23) 46 (2x23)

92

23 2 2

46 46



b. 7x86

602

(5) Using Arrays in more Complicated Multiplications

Consider 2d x 2d and 3d x 2d examples.

a. 23x47

1081

b. 68x382

4.3 ARRAYS AND DIVISION

The idea with arrays and division is to reverse the direction of arrays and multiplication. It

uses a grouping idea of division not a sharing idea.

(1) Using Arrays for Division

Look at 24 ÷ 6. Construct an area as below. 6 on the left hand side and 24 for the

area. The question is what is the top length? So, 6 x ? = 24

That is, how many of the divisor is in the quotient?

This guides the answer: 24÷6=4

86

7

86 1 2 4

86 172

344

86 x 7 86 (1x86) 172 (2x86)

344 (4x86)

602

382

68

300 80 2

60

8

1800 4800 120

2400 640 16

382 x 68 16 (8x2) 640 (8x80)

2400 (8x300) 120 (60x2)

4800 (60x80) 18000 (60x300) 25976

?

6 24

47

23

40 7

20 800 40

3 120 21

47 x 23 21 (3x7) 120 (3x40)

140 (20x7) 800 (20x40)

1081



Are there 10 6’s? 20 6’s? And so on....

Underestimate

Then look again at 10 6’s and so on...

Finally, look at 6 x ? = 42

Then add the parts

20 + 20 + 7 = 47

So 282 ÷ 6 = 47

(2) Using Area for Division 2d/3d ÷1d

We now look at how many of the divisor is in the quotient but start to estimate using

place value. That is, is there 10 of the divisor? Is there 20 of the divisor? Examples will

explain it better.

a. 72 ÷ 3

Step 1: Construct the area

Step2: Ask: Are there 10 threes in 72?

Are there 20 threes in 72? And so on...

This can be done step by step or all

together.

Step 3: Continue on looking at

remaining unknowns.

So, the answer is 24.

Step 4: Check by multiplying 3 x 24.

b. 282 ÷ 6

(3) Introduce a Recording Procedure

The array way of doing division involves underestimating the divisor – so it leads to a

new grouping algorithm (grouping because you are always thinking, how many groups

of the divisor?). Let us look at some examples.

a. 136 ÷ 4

?

3 72

?

6 282

20 ?

6 120 162

20 20 ?

6 120 120 42

20 20 7

6 120 120 42

10 10 ?

3 30 30 12

or 20 ?

3 60 12

20 4

3 60 12

? 4 136

20 ?

4 80 56

20 10 ?

4 80 40 16

20 10 4

4 80 40 16

4 136 - 80 (20 lots of 4) 56 - 40 (10 lots of 4)

16 - 16 ( 4 lots of 4) 0 34 So, 136÷4=34



b. 783 ÷ 9

c. 1081 ÷ 23

d. 13824 ÷ 54

Special Note: This method is excellent for estimation. In example d., you can see that

200 54’s will be a little over 10,000 with about another 3000, which is about 100 54’s,

i.e., estimate 250.

? 9 783

50 ?

9 450 323

50 20 ?

9 450 180 153

50 20 10 ? 9 450 180 90 63

50 20 10 7

9 450 180 90 63

9 783 - 450 (50 lots of 9) 333 - 180 (20 lots of 9)

153 - 90 (10 lots of 9) 63 - 63 ( 7 lots of 9) 0 97

So, 739÷9=97 So, 136÷4=34

?

54 13824

100 100 ?

54 5400 5400 3024

100 100 50 ? 54 5400 5400 2700 324

100 100 50 6

54 5400 5400 2700 324

?

23 1081

20 ?

23 460 621

20 20 ? 23 460 460 161

20 20 5 ? 23 460 460 115 46

20 20 5 2

23 460 460 115 46

23 1081 - 460 (20 lots of 23) 621 - 460 (20 lots of 23)

161 - 115 ( 5 lots of 23) 46 - 46 ( 2 lots of 23) 0 47

So, 1081÷23=47 So, 136÷4=34

54 13824 - 5400 (100 lots of 54) 8424 - 5400 (100 lots of 54)

3024 - 2700 ( 50 lots of 54)

324 - 324 ( 6 lots of 54) 0 256

So, 13824÷54=256 So, 136÷4=34

Basic Maths Booklet VB2: Mathematics behind Whole-Number ... · DEADLY MATHS VET Basic Mathematics...

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Transcript of Basic Maths Booklet VB2: Mathematics behind Whole-Number ... · DEADLY MATHS VET Basic Mathematics...