YUMI DEADLY CENTRE
School of Curriculum
Enquiries: +61 7 3138 0035
Email: [email protected]
http://ydc.qut.edu.au
Basic Mathematics
Mathematics behind Whole-Number
Numeration and Operations
Booklet VB2: Using 99 Boards, Number Lines, Arrays,
and Multiplicative Structure
YuMi Deadly Maths
Past Project Resource
Acknowledgement
We acknowledge the traditional owners and custodians of the lands in which the mathematics ideas for this resource
were developed, refined and presented in professional development sessions.
YuMi Deadly Centre
The YuMi Deadly Centre is a Research Centre within the Faculty of Education at Queensland University of Technology
which aims to improve the mathematics learning, employment and life chances of Aboriginal and Torres Strait Islander
and low socio-economic status students at early childhood, primary and secondary levels, in vocational education and
training courses, and through a focus on community within schools and neighbourhoods. It grew out of a group that,
at the time of this booklet, was called “Deadly Maths”.
“YuMi” is a Torres Strait Islander word meaning “you and me” but is used here with permission from the Torres Strait
Islanders’ Regional Education Council to mean working together as a community for the betterment of education for
all. “Deadly” is an Aboriginal word used widely across Australia to mean smart in terms of being the best one can be in
learning and life.
YuMi Deadly Centre’s motif was developed by Blacklines to depict learning, empowerment, and growth within
country/community. The three key elements are the individual (represented by the inner seed), the community
(represented by the leaf), and the journey/pathway of learning (represented by the curved line which winds around
and up through the leaf). As such, the motif illustrates the YuMi Deadly Centre’s vision: Growing community through
education.
More information about the YuMi Deadly Centre can be found at http://ydc.qut.edu.au and staff can be contacted at
[email protected].
Restricted waiver of copyright
This work is subject to a restricted waiver of copyright to allow copies to be made for educational purposes only,
subject to the following conditions:
1.
All copies shall be made without alteration or abridgement and must retain acknowledgement of the copyright.
2.
The work must not be copied for the purposes of sale or hire or otherwise be used to derive revenue.
3.
The restricted waiver of copyright is not transferable and may be withdrawn if any of these conditions are
breached.
© QUT YuMi Deadly Centre 2008
Electronic edition 2011
School of Curriculum
QUT Faculty of Education
S Block, Room S404, Victoria Park Road
Kelvin Grove Qld 4059
Phone: +61 7 3138 0035
Fax: + 61 7 3138 3985
Email: [email protected]
Website: http://ydc.qut.edu.au
CRICOS No. 00213J
This material has been developed as a part of the Australian School Innovation in Science, Technology and
Mathematics Project entitled Enhancing Mathematics for Indigenous Vocational Education-Training Students, funded
by the Australian Government Department of Education, Employment and Workplace Training as a part of the
Boosting Innovation in Science, Technology and Mathematics Teaching (BISTMT) Programme.
Queensland University of Technology
DEADLY MATHS VET
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
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 EducationTraining 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 ii
ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
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
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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., 6528 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).
Page 1 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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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.
Page 2 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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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.
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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
Real world situations
Models
Sequence
Physical
Virtual
Pictorial
Language
Models
Patterns
Symbols
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.
Child constructs or sees
Tens
Child says
Child writes
Ones
T Ones
forty-three
43
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
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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.
Concrete materials to represent place value and size relationships between places
Pictorial representation
Tens
Unifix cubes
Bundling sticks
Thousands Hundreds
Sliding the shape from one
position to another
(4)
Ones
Base-10 blocks
Tens
Ones
1
3
Place value chart used in conjunction with
concrete materials or digit cards to represent
position and order of the places
Rotating the shape from one
position to another
Flipping the shape from one
position to another
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).
Start with ...
… copy and tessellate.
Start with ...
… copy and tessellate.
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© 2008, 2011 QUT YuMi Deadly Centre
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.
Use the blocks to show these numbers: 23; 35; 41
Tens
(2)
Ones
Polygons
Students “click and drag” shapes to assess the extent to which they understand the
polygon concept.
Drag the shapes to the correct box.
Polygons
Not polygons
Polygons are
(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.
Make the clock show
4 o’clock
Page 6 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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(4)
© 2008, 2011 QUT YuMi Deadly Centre
Flips, slides, and turns
Flips, slides and turns can be introduced through tangrams (as follows).
USE THE TWO SMALL TRIANGLES ...
USE ALL THE TANGRAM PIECES
TO MAKE THE DOG!
…TO MAKE THE SQUARE AND THE RHOMBUS.
(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.
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© 2008, 2011 QUT YuMi Deadly Centre
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 placevalue 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.
Page 8 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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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.
Page 9 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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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).
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.
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.
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.
Page 10 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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© 2008, 2011 QUT YuMi Deadly Centre
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
Page 11 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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99-board windows
Find the centre worksheet
A
62
71
B
73
82
C
16
25
27
78
87
36
8_
9_
E
D
43
52
4_
_5
F
G
_7
_9
Page 12 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
4_
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
Fill all the spaces worksheet 1
A
15
B
45
26
35
66
C
D
79
42
87
63
E
33
F
5
41
Page 13 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
Fill the space worksheet 2
A
_2
B
3_
5_
72
C
_6
_8
D
57
4_
7_
87
63
E
_3
6_
F
_3
5_
Page 14 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
Fill the space worksheet 3
363
578
A
B
385
586
596
C
D
922
931
235
236
932
942
254
255
264
E
839
F
666
847
Page 15 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
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.
10
20
(5)
20
30
40
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
Page 16 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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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.
Page 17 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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(4)
© 2008, 2011 QUT YuMi Deadly Centre
Chance order
Materials: digit cards, game boards as below, card deck (0-9 only)
less
than
greater
than
less
than
greater
than
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.
0
1
2
3
4
5
6
7
8
9
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
Page 18 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
Give 3 students PV cards and organise them to stand in correct position:
Hundreds
Tens
Ones
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:
60
Hundreds
Tens
Ones
6
0
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):
Hundreds
Tens
Ones
Hundreds
Ones
4
4
e.g. x 10
Tens
÷ 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
Page 19 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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© 2008, 2011 QUT YuMi Deadly Centre
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.
MILLION
HUNDRED
TEN
3
THOUSAND
8
7
THOUSAND
1
8
THOUSAND
6
2
0
5
HUNDRED
7
TEN
4
ONE
Page 20 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
(5)
© 2008, 2011 QUT YuMi Deadly Centre
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
Millions
Millions
Millions
Thousands
Thousands
Thousands
Ones
Ones
Ones
H
T
O
H
T
O
H
T
O
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).
Millions
Millions
Millions
Thousands
Thousands
Thousands
Ones
Ones
Ones
H
T
O
H
T
O
H
T
O
4
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).
Page 21 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
(2)
© 2008, 2011 QUT YuMi Deadly Centre
The Big One (place value)
Materials: Calculator
Number of players: 2
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.
Number of players: 2
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
Number of
guess
guesses
(a)
(b)
Page 22 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
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© 2008, 2011 QUT YuMi Deadly Centre
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
H T O
Step 1
0
+ 3 8 6
1
7 1 5
1 3 (7+6)
(7 hundreds and 15 ones)
- 2 6 8
1 2 0 (40+80)
7
5 0 0 (200+300)
(15 ones – 8 ones = 7 ones)
6 3 3
Steps 2 & 3
6
10 1
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
547
627
633
+ 300
+ 80
+ 6
715
515
455
447
- 200
- 60
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
+ 386
247
+ 400
647
- 14
633
715
- 268
415
+ 32
447
715
- 300
Page 23 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
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
34 is 3 tens and 4 ones
3 down
4 right
23 is 2 tens and 3 ones
2 down
3 right
(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
78 is 7 tens and 8 ones
7 down
8 right
42 is 4 tens and 2 ones
4 up
2 left
Page 24 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
(3)
© 2008, 2011 QUT YuMi Deadly Centre
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
17 is 1 ten and 7 ones
1 down
7 right
26 is 2 tens and 6 ones
2 down
6 right (3+3)
b. 83-25=58
83 is 8 tens and 3 ones
8 down
3 right
25 is 2 tens and 5 ones
2 up
5 left (3+2)
(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).
26 is 2 tens and 6 ones
2 down
6 right
39 is 4 tens less 1 one
4 down
1 left
Page 25 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
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).
85 is 8 tens and 5 ones
8 down
5 right
28 is 3 tens less 2 ones
3 up
2 right
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 678, because adding 2 refers to counting on 2
jumps from the starting number: 67 and 78.
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.
Page 26 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
a. 23+45=68
Start on 23 and move 4 tens forward, then 5 ones forward.
10
0
10
20
10
30
10
40
10
50
5
60
23
70
80
90
100
90
100
63 68
b. 87-34=53
Start on 87 and move 3 tens backwards then 4 ones backwards.
4
0
10
20
30
40
10
50
10
60
70
10
80
53 57
87
Addition and subtraction with carrying
(3)
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.
10
0
10
20
30
10
40
10
50
28
2
5
60
58
70
80
90
100
65
60
b. 73-46=27
Start on 73 and move 4 tens backwards then 3 ones backwards and another 3
ones backwards.
3
0
10
20
3
30
27 33
30
10
40
10
50
10
60
10
70
80
90
73
Page 27 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
100
YuMi Deadly Maths Past Project Resource
(4)
© 2008, 2011 QUT YuMi Deadly Centre
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
200
80
356
8
556
636
644
b. 734-268=466
8
60
474
466
(5)
200
534
734
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
42
40
40
3
82
3
85
42
43
82 85
b. 82-45=37
Make both numbers then build from smallest to largest in the simplest way
45
5
5
30
2
50
30
80
45
50
80 82
2
82
37
c. 717-368=349
368
2
2
30
300
17
370
30
400
368 400
370
700
717
300
700
17
717
349
Page 28 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
(6)
© 2008, 2011 QUT YuMi Deadly Centre
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
40
50
3
Regular strategy, see (4)
368
408
458
461
40
50
3
Compensation strategy
368
b. 428+386=814
408
461
468
300
80
6
Regular strategy, see (4)
428
728
400
808
814
14
Compensation strategy
428
828
814
Page 29 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
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
Array
3 rows of 4
10
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.
Page 30 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
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: 37
83
9
x 8
56
(7 x 8)
240 (30 x 8)
296
(2)
747
72
27
27
0
sharing tens
sharing ones
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)
9
37
747
X8
450
74 double (37x2)
148 double (74x2)
297
296 double (148x2)
117
180
90
(50 groups of 9)
(20 groups of 9)
(10 groups of 9)
27
27
0
(3)
( 3 groups of 9)
83 groups of 9 make 747
Compensation
A simple multiplication or division is found and then compensated for, for example:
37
x 8
37
x 10
37
(2 groups of 37 too many)
370
-
37
(Less one group of 37)
333
-
37
x 10
370
(10x37)
- 74
- ( 2x37)
296
8x37
(Less another group of 37)
296
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
Page 31 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
Counters
Squares
Shading
Relate this to area as length x width, so construct area for various multiplications as
follows.
8
19
5
12
Extension ideas: Make a drawing of 6x4. Rotate the diagram 90° to show that 6x4 =
4x6 (the commutative law).
6
6
4
(2)
4
Distributive law and place value
Construct 2 digit by 1 digit multiplication, for example:
4 x 23
7 x 49
49
23
7
4
Make these with 2mm graph paper. See that they can be divided as follows.
4 x 23
7 x 49
20
4
40
3
9
7
Show that 4x20 + 4x3 is the same as 4x23 and 7x40 + 7x9 is the same as 7x49 (count
squares, use a calculator).
Page 32 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
(3)
© 2008, 2011 QUT YuMi Deadly Centre
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
23
4
20
23
x 4
12
80
92
3
4
80
(4x3)
(4x20)
12
92
b. 7x86
86
7
80
86
x 7
42
560
602
6
7
560
(7x6)
(7x80)
42
602
Extension ideas: Can change the diagram in other ways, for example:
a.
4x23
23
4
2
2
23
23
x 4
46
46
92
(2x23)
(2x23)
46
46
92
Page 33 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
b.
© 2008, 2011 QUT YuMi Deadly Centre
7x86
86
7
86
x 7
86
172
344
602
86
1
2
4
86
172
(1x86)
(2x86)
(4x86)
344
602
(5)
Using Arrays in more Complicated Multiplications
Consider 2d x 2d and 3d x 2d examples.
a. 23x47
47
47
x 23
21
120
140
800
1081
23
40
7
20
800
40
3
120
21
(3x7)
(3x40)
(20x7)
(20x40)
1081
b. 68x382
382
68
300
60
80
2
1800
4800
120
2400
640
16
8
382
x 68
16
640
2400
120
4800
18000
25976
(8x2)
(8x80)
(8x300)
(60x2)
(60x80)
(60x300)
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?
6
24
This guides the answer: 24÷6=4
Page 34 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
(2)
© 2008, 2011 QUT YuMi Deadly Centre
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
3
Step 1: Construct the area
72
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.
3
10
10
?
30
30
12
or
20
?
3
60
12
20
4
60
12
3
So, the answer is 24.
Step 4: Check by multiplying 3 x 24.
b. 282 ÷ 6
6
6
6
(3)
Are there 10 6’s? 20 6’s? And so on....
?
6
282
Underestimate
20
?
120
162
Then look again at 10 6’s and so on...
Finally, look at 6 x ? = 42
20
20
?
120
120
42
Then add the parts
20
20
7
20 + 20 + 7 = 47
120
120
42
So 282 ÷ 6 = 47
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
4
4
4
4
4
?
136
20
?
80
56
20
10
?
80
40
16
20
10
4
80
40
16
136
- 80
56
- 40
16
- 16
0
(20 lots of 4)
(10 lots of 4)
( 4 lots of 4)
34
So, 136÷4=34
Page 35 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009
YuMi Deadly Maths Past Project Resource
© 2008, 2011 QUT YuMi Deadly Centre
b. 783 ÷ 9
9
?
9
-
783
9
9
9
9
50
?
450
323
-
50
20
?
450
180
153
-
50
20
10 ?
450
180
90 63
50
20
10 7
450
180
90 63
-
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
c. 1081 ÷ 23
?
23
23
23 1081
- 460
621
- 460
161
- 115
46
- 46
0
1081
20
?
460
621
23
20
20
?
460
460
161
23
20
20
460
460
20
20
23
460
460
d. 13824 ÷ 54
54
5
?
115 46
5
2
?
13824
100
100
?
5400
5400
3024
54
100
100
5400
5400
100
100
54
5400
5400
50
?
2700 324
50
(20 lots of 23)
( 5 lots of 23)
( 2 lots of 23)
47
So, 1081÷23=47
So, 136÷4=34
115 46
54
(20 lots of 23)
6
2700 324
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
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.
Page 36 ASISTEMVET08 Booklet VB2: Using 99 Boards, Number Lines, Arrays, and Multiplicative Structure, 08/05/2009