Maths Facts Booklet

7/28/2019 Maths Facts Booklet

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MATHS FACTS

Information for parents

Level 3(Grades 3 & 4)

Spensley Street Primary School2009



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MATHS FACTS

VELS Level 3 – Grades 3 and 4

This booklet explains many of the number-based Maths concepts whichare taught to children in Grades 3 and 4. It is intended to help you tounderstand what we teach, so that you can support your child’slearning at home.

It is important for children to realise that Maths is not just about Number and sums. Maths is all around us in shape, counting, measurement, time

and data. Maths arises constantly in our everyday lives, and children will benefit from being involved in purposeful applications of Maths at homewith you, such as when shopping, reading timetables, construction,cooking, looking and maps and doing jigsaws.

In this booklet we have outlined common strategies for solving number problems. Our intention, always, is to teach for understanding, not just tobe able to ‘do’ the process. Bear in mind that children may need to

experience a process many times before they understand it. Every bit of practice they do at home makes a difference.

This booklet focuses only on Number . For your information, we haveincluded the VELS Victorian government expectations for the wholemathematics curriculum at Level 3 (usually students in Grades 3 & 4).

We hope that this booklet will be a useful reference for you and your

child. If you have specific questions or concerns, please don’t hesitate tocontact your child’s teacher.

Spensley Street teaching staff, 2009



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CONTENTS

Useful definitions 3

The four operations – Mental strategies 7

• Addition 7

• Subtraction 8

• Multiplication 9

Times Tables 10

• Division 11

Enjoying Mental Maths together 13

The four operations – Written algorithms 14

• Addition 14

• Subtraction 15

• Multiplication 17

• Division 18

References 19

Note to parents 19

Appendices

1. Victorian Essential Learning Standards (VELS) 20

2. Times Tables Tips 24



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TEACHING FOR UNDERSTANDING: NUMBER

USEFUL DEFINITIONS

What we mean by:

Place Value

A number is made up of several digits or numerals. When we teach placevalue we are careful, in our use of language, to distinguish between thenumber e.g. 576, and the digits or numerals which combine to make thatnumber – 5, 7, and 6.

Our number system is based on multiples of ten. A digit’s value is determinedby where it is placed in a number.

For example, the digit 4 in the number 874 represents 4 ones, the numeral 7 represents 7 tens and the numeral 8 represents 8 hundreds.

M.A.B. – Multibase Arithmetic Blocks

These are also known asBase 10 blocks. Theycan be used to physically

demonstrate place value.



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More Useful definitions:

Odd Numbers are not evenly divisible by 2.

e.g. 1 3 5 7 9 11 13

Even Numbers are evenly divisible by 2.e.g. 2 4 6 8 10 12 14

Rounding (to nearest 10)

Numbers ending in 1, 2, 3, 4, go down to the nearest 10.

Numbers ending in 5, 6, 7, 8, 9 go up to the nearest 10.

e.g. 63→

60 (rounded to the nearest 10)38 → 40

Greater than > Less than <

The ‘greater than’ and ‘less than’ symbols show relationships betweennumbers. They are often used for true and false questions.

e.g. 27 > 15 and 240 < 420

Skip CountingCounting patterns. Learning patterns is very helpful in understanding how our number system works, and helps children to build up their bank of number facts. Skip-count forwards or backwards in multiples of certain numbers.

e.g. 5, 10, 15, 20, 25 or 96, 94, 92, 90, 88

10s, 2s and 5s are the simplest patterns to begin with. It is good to vary thestarting point.

e.g. count in 10s beginning at 8 (8, 18, 28) or 152 (152, 162, 172)

Estimate

A thoughtful guess. We encourage children to estimate an answer beforecalculating in order to check the reasonableness of their answer. This isespecially important when learning about place value. Estimating is a usefulskill which gets better with practice and is important in developing good



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number sense. Learning how to estimate often involves using mental imagesand strategies.

e.g. 307 x 3 can be viewed as 300 x 3 to give an estimate of 900.

Equation

An equation is a statement that two expressions are equal. Both sides of theequation must balance.

e.g. 6 + 3 = 5 + 4

5 x 10 = 93 – 43

125 ÷ 5 = 5 x 5

Basic Facts

We encourage children to build up a bank of basic number facts. These facts

should be learned so that the response is almost automatic. As calculationsbecome more complex, a good number fact bank enables children to focus onthe process without having to work out each little calculation.

At level 3, these should be very familiar:

• basic addition facts to 20,

• basic subtraction facts from 20,

• multiplication facts to 10 x 10 (times tables) and related division facts

Algorithm

The written procedure for setting out and calculating a mathematical problemin a certain way.

e.g. 324 874 364+207 - 23 x 30

Fractions

Fractions show parts of a whole:

The top number is the numerator , and tells us howmany parts of the whole.The bottom number is the denominator , and tells us

how many parts make up the whole.

3

4



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Equivalent fractions

Fractions of equal value are equivalentfractions.

e.g. 1 = 22 4

Adding or subtracting like fractions

When the denominators are the same, simply add or subtract the numeratorstogether and keep the denominator the same.

eg 1 + 2 = 34 4 4

Decimal fractions

Fractions of a whole number, based on tens. Commonly used decimalfractions are tenths and hundredths

e.g. 1.2 is one whole and 2 tenths

Factors

Any whole number that can be multiplied with another to make a givennumber.

e.g. the factors of 12 are 1, 2, 3, 4, 6 and 12.

Product

The result of numbers multiplied together is called the product.

Multiple

A multiple is the product of two or more factors.

e.g. 2 x 8 = 16, so 16 is a multiple of 2 and 8.



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THE FOUR OPERATIONS - MENTAL STRATEGIES

Most of the mathematics that we will do in our lives is mental maths. Theseare some strategies which we teach to help children to be more efficient intheir calculations.

ADDITION

Counting on

Count on from the larger number

e.g. 7 + 4 = (count up from 7…8, 9, 10, 11) =11

Making 10

Ten is an easy number to calculate with, so…make tens first.

e.g. 9 + 5 = 10 + 4 = 14

Adjusting numbers

This is a development of the ‘making ten’ strategy. Adjust the numbers tomake them easier to add.

e.g. 38 + 29 = 40 + 27 (add 2 to 38 to make 40, an easier ten, then take the

added 2 away from 29 = 27) = 67

Jump strategy

This strategy is best demonstrated using a number line. Jumps of tens andones are added along the number line. This gives a helpful visual model.Once they are sure of the process, children can then visualize the processmentally, without the number line in front of them.



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This strategy is best demonstrated using a number line. Jumps of tens andones are subtracted along the number line.

Making 10

Break up the second number and subtract to make a (multiple of) ten first,

then subtract the leftover amount.

e.g. 35 - 7 = 35 - 5 - 2 = 30 - 2 = 28

MULTIPLICATION

Multiplying numbers together is the same as repeated addition.

3 x 6 = 6 + 6 + 6 = 18

Representing multiplication visually helps children to develop multiplicativethinking, clearly seeing the difference between:

the array formed by 3 x 4 = 12

and the linear 3 + 4 = 7

Groups or ‘lots’ of

Make groups of objects

2 lots of 3 = 6

Skip Counting

Counting forwards or backwards in multiples of a given number, beginningwith the number pattern being repeated (the multiplier). This helps children torecognise that particular number’s pattern.

The array alsoshows that 3 x 4 isthe same as4 x 3.



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e.g. skip counting by 5s

5 10 15 20

Doubling

This strategy is used when multiplying numbers by 2, 4 or 8.

x 2 double the number 2 x 5,….10

x 4 double and double again, 4 x 5,….10…..20

x 8 double, double, double, 8 x 5,….10…..20…..40

Multiplication Facts – ‘Times Tables’

VELS expectations

End of Grade 3 – 2s, 5s, and 10s

End of Grade 4 – all multiplication facts to 10x10

We teach times-tables and expect children to learn them “off by heart”. They

form an important bank of number facts for all future maths work. Automaticrecall of multiplication facts enables students to be more accurate in their calculations and allows them to focus on the process and problem-solving of the maths problem, rather than all the little calculations on the way. It isquicker to recall them than to use a calculator every time.

It is important for children to understand what times-tables are so they canuse known facts to work out unknown facts, e.g. if they know that 5 x 5 = 25,they can work out that 6 x 5 = 30.

Children should know their tables as well as being able to skip-count. Skip-counting shows them the number’s pattern, but times tables are needed toknow how many of that number make the answer, and are essential whenworking with division.



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We know that it is ambitious to expect students to know all of their tables bythe end of Grade 4, and that’s why it is essential that they practice regularly athome (see Appendix 2 for ‘Times Tables Tips’).

Start with the easiest tables 2×, 5×, 10×, which have a regular and predictablepattern, then move onto 3×, 4×, 9× and then 6×, 8×, 7×. Don't forget 1× and

0×

. Make links where possible: e.g. relate 5×

to reading the minute hand on aclock, 7× to days in weeks, 6x to footy scores, etc.

Build Up and Build Down

Build upon known multiplication facts to solve problems.

e.g. Build Up:

To solve 5 x 6 = , you might know that 5 x 5 = 25 plus another 5 = 30.

Extend Basic Facts

e.g. 8 x 5 = 40

8 x 50 = 400

8 x 500 = 4000

8 x 5000 = 40 000

Multiplying by 10 and 100

To multiply a number by 10, move every digit up one place value position andput a zero in the ones column.

e.g. 45 x 10 = 450

To multiply a number by 100, move every digit up two place value positions

and put zeros in the ones and tens columns.

e.g. 54 x 100 = 5400

Double one number, halve the other

e.g. 3 x 16 becomes 6 x 8 = 48



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DIVISION

Division is sharing into equal parts or groups.There are two forms of division, partition (sharing) and quotition (grouping).

Partition (sharing)e.g. If nine toys are shared among three students, how many does eachperson get?

Quotition (grouping)

e.g. How many groups of 3 can be made from a bag of nine lollies?

It is important for children to experience division using real objects andsituations so that they really understand the concept before moving on to

abstract algorithms. The inverse of multiplication, division is based onrepeated subtraction of the same divisor. Hence:

12 ÷ 4 12 – 4 – 4 – 4 = 0

4 was subtracted 3 times, so, 12 ÷ 4 = 3

Number lines can be used to demonstrate repeated subtraction.



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Car Number Plates

See how many different answers you can make using the numbers on a car number plate. For example, if the number plate is 152, the child might say 1 x 5 + 2 is 7, or 1 + 5 + 2 is 8 and so on. Or you could add the digits like this: 157becomes 1 + 5 + 7 which is 13, then 1 + 3 = 4. See which number plate will give a special target number.

THE FOUR OPERATIONS – WRITTEN ALGORITHMS

There are different methods that can be used to solve written algorithms.Children may be exposed to these alongside the standard processeswhich are outlined in this booklet. Some children may prefer to use thesealternative methods to solve algorithms and, provided they are efficient,

they are equally acceptable.

ADDITION

Renaming’ or ‘trading up’

Example 268 + 49

STEP 1

• 8 and 9 is 17

• 17 consists of 1 ten and 7 ones

• 7 is placed on the answer line in the ones column

• 1 ten is traded up to the tens place

HTO

1268

+ 497

STEP 2

• 1 ten is added to the 6 tens

• That is 7 tens so far

• 7 tens are added to the 4 tens•

There are 11 tens• 11 tens consist of 1 hundred and 1 ten

• 1 is placed on the answer line in the tens place

• 1 hundred is traded up to the hundreds place

HTO

1 1

268

+ 4917

STEP 3

• 1 hundred is added to the 2 hundreds

• That is 3 hundreds

• 3 is placed on the answer line in the hundreds place

HTO

1 1

268+ 49

317

(H = hundreds, T = tens, O = ones)



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SUBTRACTION

In addition algorithms, the numbers can be set out in any order; 256 + 132 isthe same as 132 +256. In subtraction this is not so, and the setting out of thealgorithm requires particular care. The largest number must be on the top.

‘Regrouping’, ‘renaming’ or ‘trading down’

Trading is based on the fact that one ten consists of ten ones.



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DecompositionWe no longer teach ‘borrowing’ and paying back, we use decomposition.Decomposition is the name given to a method of subtraction that uses re-grouping or trading.



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MULTIPLICATION

Multiplication by 1 digit – extended form

e.g. HTO

43 STEPSx 5

15 3 ones x 5 = 15

200 4 tens x 5 = 200

215 Add 15 + 200

215

Multiplication by 1 digit – contracted form

e.g. HTO STEPS

53x 1 4212

3 ones x 4 = 12. Write the 2 in the ones column and carry the1 to the tens column, where it is placed below the 5.

5 tens x 4 = 20 tens, plus the 1 ten that was traded = 21 tens.Write 1 in the tens column and place the 2 in the hundredscolumn.



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DIVISION

Division is often the most difficult operation for children to understandbecause the symbolic form does not fit with their understandings of placevalue. In order to help children make sense of division, we introduce children

to the estimation method. It is more accommodating of errors than longdivision, and children can complete it in few or more steps depending on their number sense.

Estimation method ___ 6)756 estimate, say 100 100 x 6 = 600

600 -_______________________________________________ 156 estimate, say 10 10 x 6 = 60

__ 60 -_______________________________________________

96 estimate, say 10 10 x 6 = 60 _____ 60 -_______________________________________________

36 I know 6 x 6 = 36 6 _____ 36 -_______________________________________________

00 add up these estimations: 100 + 10 + 10 + 6 = 126

Later, they are introduced to the symbolic form.

Symbolic Form



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References

Atkinson, Sue (1994), Help Your Child With Maths. Hodder & Stoughton, London.

Clarke, D. Helping Your Children With Mathematics. Australian Catholic University.

O’Brien, H & Purcell, G (2005) The New Primary Mathematics Handbook . OxfordUni Press, Aust.

VCAA Victorian Essential Learning Standards http://vels.vcaa.vic.edu.au.

S.S.P.S. Maths Program (2004) Number .

Ymer, M (2008) Notes from teacher professional development session

Note to parents

Many of the diagrams in this booklet have been taken from The New Primary Mathematics Handbook. You may find this a useful reference if you want moreinformation.

This is the first edition of our Maths Facts booklet. We would love to hear your suggestions for improvements.

We hope you find this a helpful resource.

Sue Werner, Annie Neville, Karen South, Jan Whitham, Kate Fay, Mary PortesiS.S.P.S. Maths Committee, 2009



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Appendix 1

VICTORIAN ESSENTIAL LEARNING STANDARDS:MATHEMATICS

This material is an extract from the Victorian Essential Learning Standards by theVictorian Curriculum and Assessment Authority (VCAA), Australia. For more

information visit http://vels.vcaa.vic.edu.au.

Students are considered to be ‘established’ at Level 3 when they candemonstrate most of the following mathematical understandings.

NUMBER

At Level 3, students:• use place value (as the idea that ‘ten of these is one of those’) to

determine the size and order of whole numbers to tens of thousands,and decimals to hundredths.

• round numbers up and down to the nearest unit, ten, hundred, or thousand.

• develop fraction notation and compare simple common fractions suchas 3 /4 > 2 /3 using physical models.

• skip count forwards and backwards, from various starting points usingmultiples of 2, 3, 4, 5, 10 and 100.

• estimate the results of computations and recognise whether these are

likely to be over-estimates or under-estimates.• compute with numbers up to 30 using all four operations.• provide automatic recall of multiplication facts up to 10 × 10.• devise and use written methods for:

• whole number problems of addition and subtraction involvingnumbers up to 999

• multiplication by single digits (using recall of multiplication tables)and multiples and powers of ten (for example,5 × 100, 5 × 70)

• division by a single-digit divisor (based on inverse relations inmultiplication tables).

• devise and use algorithms for the addition and subtraction of numbersto two decimal places, including situations involving money.

• add and subtract simple common fractions with the assistance of physical models.



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SPACE

At Level 3, students:• recognise and describe the directions of lines as vertical, horizontal or

diagonal.• recognise angles are the result of rotation of lines with a common end-point.• recognise and describe polygons.• recognise and name common three- dimensional shapes such as spheres,

prisms and pyramids.• identify edges, vertices and faces.• use two-dimensional nets, cross-sections and simple projections to represent

simple three-dimensional shapes.• follow instructions to produce simple tessellations (for example, with

triangles, rectangles, hexagons) and puzzles such as tangrams. • locate and identify places on maps and diagrams.• give travel directions and describe positions using simple compass directions

(for example, N for North) and grid references on a street directory.

MEASUREMENT, CHANCE & DATA

At Level 3, students:• estimate and measure length, area, volume, capacity, mass and time using

appropriate instruments.• recognise and use different units of measurement including informal (for

example, paces), formal (for example, centimetres) and standard metric

measures (for example, metre) in appropriate contexts.• read linear scales (for example, tape measures) and circular scales (for

example, bathroom scales) in measurement contexts.• read digital time displays and analogue clock times at five-minute intervals.• interpret timetables and calendars in relation to familiar events.• compare the likelihood of everyday events (for example, the chances of rain

and snow).• describe the fairness of events in qualitative terms.• plan and conduct chance experiments (for example, using colours on a

spinner) and display the results of these experiments.• recognise different types of data: non-numerical (categories), separate

numbers (discrete), or points on an unbroken number line (continuous).• use a column or bar graph to display the results of an experiment (for

example, the frequencies of possible categories).

STRUCTURE



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At Level 3, students:• recognise that the sharing of a collection into equal-sized parts (division)

frequently leaves a remainder.• investigate sequences of decimal numbers generated using multiplication or

division by 10.• understand the meaning of the ‘=’ in mathematical statements and

technology displays (for example, to indicate either the result of acomputation or equivalence).

• use number properties in combination to facilitate computations (for example,7 + 10 + 13 = 10 + 7 + 13 = 10 + 20).

• multiply using the distributive property of multiplication over addition (for example, 13 × 5 = (10 + 3) × 5 = 10 × 5 + 3 × 5).

• list all possible outcomes of a simple chance event.• use lists, Venn diagrams and grids to show the possible combinations of two

attributes.•

recognise samples as subsets of the population under consideration (for example, pets owned by class members as a subset of pets owned by allchildren).

• construct number sentences with missing numbers and solve them.

WORKING MATHEMATICALLY

At Level 3, students:• apply number skills to everyday contexts such as shopping, with appropriate

rounding to the nearest five cents.• recognise the mathematical structure of problems and use appropriate

strategies (for example, recognition of sameness, difference and repetition) tofind solutions.

• test the truth of mathematical statements and generalisations. For example,in:• number (which shapes can be easily used to show fractions)• computations (whether products will be odd or even, the patterns of

remainders from division)•

number patterns (the patterns of ones digits of multiples, terminating or repeating decimals resulting from division)

• shape properties (which shapes have symmetry, which solids can bestacked)

• transformations (the effects of slides, reflections and turns on a shape)• measurement (the relationship between size and capacity of a container).



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• use calculators to explore number patterns and check the accuracy of estimations.

• use a variety of computer software to create diagrams, shapes, tessellationsand to organise and present data.



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Appendix 2

TIMES TABLES TIPSSome ideas for learning the ‘times tables’ at home

Remember that:

• Children learn in many different ways, and it is important to try a range of practice strategies until you find the ones which suit your child.

• Rote learning can be boring, so try using games to help relieve the monotony.

• It takes lots of repetition for the facts to be learned reliably.

• It is a good idea to have short, frequent bursts of practice rather than anoccasional long frenzy on a Sunday evening!

• A times-tables poster is a useful reference to have around the house.

Some useful resources:

CHANTS AND SONGS (on CD – several available from the ABC shop)

BOOKS from the newsagent, e.g. Times Tables Practice

FLASHCARDS – download or make your own. These can be used for revision of known facts, or to play:

SnapUse the question and answer cards. Matches can include a question and ananswer (eg. 15 and 3x5) or a question and a question (3x4 and 2x6). The player with the most cards wins.

Memory This is a quieter game and can be played independently or with a partner. When

thinking takes over 15 seconds it becomes associated with long term memory. Setup the question and answer cards in two separate grids, face down. Turn over oneof each to find pairs.N.B. In both games it may be appropriate to include only some sets of tables.

PLAYING CARDSTables War



Use a deck of playing cards. The picture cards all equal ten. Place all cards in apile face down. Players take turns to flip over two cards at a time. The first player to call out the multiple wins that pair. When all the cards have been played, theplayer with the most cards wins. Initially, you may prefer to use a tables grid as anaid.

COMPUTER GAMES e.g. Mathletics (all children have a code and can access this

at home)

Other ideas:

LOOK FOR PATTERNSWrite out the times tables you are learning, remembering to keep the ones and tensin vertical columns.Look for patterns in the answers, especially in the ones column.

Eg. 2x double the factor 4x double and double5x always has 5 or 0 in the ones place6x answers to even factors end in that digit (i.e. 6x4=24, 6x8=48)9x the tens column counts forwards and the ones column counts

backwards10x just add zero!

DRILL PRACTICE. You can use flash-cards or the computer (e.g.

http://www.teachingtables.co.uk/)

There are also some fun games and drill-practise activities in the “Maths – TablesPractice” section of the websitehttp://www.commercialps.vic.edu.au/links/Maths%20links.htm

Maths Facts Booklet

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