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Teach Whole numbers & decimals for understanding page 1

Teach whole numbers, integers and decimals for Understanding

Earlier material on Whole number (sections 1-25) may be found on

Teach for Understanding: Number1

Advanced whole numbers, integers & decimals

Whole numbers: advanced operations 26 Order of operations p3 27 Distributive laws (Do it to both) p4 28 Round & estimate

for single-digit multiplication p5 29 Multiply a multiple of 10

by a single digit p6 30 Multiply any number

by a single digit p7 31 Multiply by 10, 20, 30 etc p8 32 Multiply any two 2-digit numbers p9 33 Round & estimate

for two-digit multiplication p10 34 Divide by one digit p11 35 Round & estimate division p13 36 Divide by 2-digit numbers p14

Integers 37 Models of integers and adding p16 38 Subtract integers p17 39 Multiply integers p18

Multiples, factors, primes 40 Multiples p19 41 Common multiples and LCM p20 42 Factors p21 43 Common factors and HCF p22 44 Prime and composite numbers p23 45 Prime factors p24

Decimals linked to fractions and percentages are found in ‘Fractions’.

Decimal operations 46 Add and subtract decimals p25 47 Multiply decimals by single-digit

whole numbers p26 48 Divide decimals by single-digit

whole numbers p27 49 Multiply decimals by ten p28 50 Divide decimals by ten p29 51 Multiplication with powers of ten p30 52 Decimal division with powers of

ten p31 53 Estimate when multiplying

decimals by whole numbers p32 54 Estimate when dividing

decimals by whole numbers p33 55 Multiply tenths by tenths p34 56 Multiply any two decimals p35 57 Divide by tenths p36 58 Divide by any decimals p37

Exponents, base 2 & scientific notation

59 Square and cube numbers p38 60 Exponential notation p39 61 Base 2 numbers, adding

and multiplying p40 62 Scientific notation p42 63 Exponent laws p43 64 Square roots and surds p44


Resources for learning The curriculum described in this section does not use textbooks. Instead it calls on the wealth of high quality learning resources that are available, mainly through MAV.

Lesson plans

Maths300

RIME RIME 5&6

Teaching advice

Continuum Assessment for Common Misunderstandings Scaffolding Numeracy in the Middle Years Guidelines in Number Mental Computation Practical teaching strategies for children with learning difficulties

Problem solving

Working Mathematically Investigations Maths With Attitude Mathematics Task Centre

Worksheets Active Learning (Number & Algebra) Active Learning 2 (Number & Algebra) Maths at Work

Books

Action Numeracy – middle primary Action Numeracy – upper primary

Computers

Interactive Learning Learning Objects from FUSE or Scootle

Resources from MAV may be purchased with a credit card or school order number on-line,

using the MAV’s web site: <www.mav.vic.edu.au/shop>. Click on any of the above to order.

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=maths300+subscription&method=exact

http://www.mav.vic.edu.au/shop/members-price.html

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=RIME%3A+CD+complete&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=RIME+for+Years+5+%26+6&method=exact

http://www.education.vic.gov.au/studentlearning/teachingresources/maths/mathscontinuum/number/default.htm

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=Maths+With+Attitude&method=exact

http://www.blackdouglas.com.au/mathematicscentre/index.htm

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=active+learning&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=active+learning+2&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=maths+at+work&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=inter-active+learning&method=exact

https://fuse.education.vic.gov.au/

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=working+mathematically+-+investigations&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=guidelines+in+number&method=exact

http://www.mav.vic.edu.au/pubs/mental-computation.html

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=practical+teaching&method=exact

http://www.mav.vic.edu.au/cgi-bin/csv/search/new-member-csvsearch.pl?search=action+numeracy&method=exact

http://www.scootle.edu.au/ec/curriculum?learningarea=%22Mathematics%22&menu=3


26 Order of operations

The order of operations is the agreed order in which calculations should be done.

Suggested activities

1. Order of operations

Where a calculation has only adding and subtracting, do them in the order they come, left to right.

Where a calculation has only multiplying and dividing, do them in the order they come, left to right.

Where a calculation has brackets, do what is in the brackets first.

Where a calculation mixes multiplying or dividing with adding or subtracting, do the multiplying or dividing first - in the order they come, left to right..

2. Provide plenty of examples of these rules in action. To work out 5 x (8 + 3) you would do the brackets first, making it 5 x 11, then multiply, to get 55.

To work out 5 x 8 + 5 x 3 you must do the multiplications first, making it 40 + 15 = 55.

3. Use flashcards which combine operations.


27 Distributive laws: ‘Do it to both’

The two Distributive laws say that you can do things in a different order if you are careful. If you want to work out brackets first you can do so, but it is possible to ‘remove’ the brackets if you follow the way numbers work.


1. Examples: Multiplying: To work out 5 x (8 + 3) you can

• work out the brackets first, then multiply. So we get 5 x 11 = 55. • do the multiplication to both 8 and 3, and add the answers. So 5 x 8 + 5 x 3 = 40 + 15 = 55.

The distributive law is the basis of the single digit and two-digit multiplications that follow.

Dividing: To work out (40 – 12) ÷ 4 you can

• work out the brackets first, then divide. So we get 28 ÷ 4 = 7.

• do the division to both 40 and 12 and subtract the answers. So 40 ÷ 4 – 12 ÷ 4 = 10 – 3 = 7.

Subtracting: To work out 20 – (3 + 5) you can

• work out the brackets first, to get 20 – 8 = 12.

• subtract both numbers from 20 to get 20 – 3 – 5 = 12.

2. Play the game ‘Got-it’. Players use a regular pack of cards, either without picture cards, or with them (J = 11, Q = 12, K = 13). They are dealt five each and the next card is turned up.

Each player tries to make an expression (using imagined brackets, +, –, x or ÷ where needed) that has the value of the turned up card. The points scored equal the number of cards used, if the answer is correct. For example, if a 4 is turned up, (3 + 5) ÷ 2 + 7 – 7 = 4 earns five points.

Two or more separate expressions may be used for more points. For example, 7 – 3 = 4 and 7 – 5 + 2 = 4 would also earn five points.

3. Student finds all the ways of making change. For 20 cents, 50 cents and $1, there are patterns to be found.


28 Round and estimate for multiplication Because many people use calculators to get exact answers, we need to concentrate on the skills of estimating and number sense.


1. Rounding The purpose of rounding is to reduce the complexity of the number. It usually means replacing some of the digits by zeros.

We can round two-digit numbers to the nearest multiple of 10.

- If the ones digit is 0, 1, 2, 3 or 4, we can call it 0. So the number goes DOWN to the previous multiple of 10. For example, 23 becomes 20 - If the ones digit is 5, 6, 7, 8 or 9, we can call it 10. So the number goes UP to the next multiple of 10.

For example, 37 becomes 40. Note that numbers that are half-way (such as 75) go up (to 80).

In a similar way we can round three-digit numbers to the nearest multiple of 100. This is useful for estimating single-digit multiplication. Here is an example: 7 x 23

Round 23 DOWN to 20. Then the problem is now simpler: 7 x 20, and the answer is about 140, actually a bit above 140.

Another example: 7 x 78

Round 78 UP to 80. Then the problem is now simpler: 7 x 80, and the answer is about 560, actually a bit less.

2. Use the ‘Estimate and calculate’ game. (Each player needs a calculator.) Two or more students take turns to make up suitable questions. They then have to estimate the correct answer and write down their estimate. Finally they each check the answer on a calculator. The person closest to correct wins the point. (Finding the closest may required some subtraction on the calculator.)


29 Multiply a multiple of 10 by a single digit

This is the essence of place value; multiply by 10 and the digits shift left.


1. Use Base 10 blocks. For example, 6 x 40. Students should work a few of these kinds of problems with Base 10 material until they are convinced that the answer is the same as (6 x 4) x 10. They can count using their number chart three times: two 100s and 40 extra, making an answer of 240.

makes

6 lots of 40

240

t ens oneshundreds


30 Multiply any number by a single digit This uses the idea that you can multiply two parts and add the results. It has been suggested above as providing alternatives for assisting recall, but now it is the basis of the entire computation.


1. Use the 100-chart and number line Here is an example: 3 x 24. It means 3 lots of 24, so we count 3 lots of 20 (60) and 3 lots of 4 (12). So the answer is 72. Students should see it on both a number chart and number line.

2. Use arrays The multiplication above (3 x 24) can be seen on the array below. Students can use base 10 blocks to represent any multiplication this way.


31 Multiply any number by 10, 20, 30 etc.

This idea and skill is fundamental to multiplying by two-digit numbers. It is the reason we use the zero place- holder in long multiplication. It should become automatic, e.g. 40 x 3.


1. Multiplying a single digit by 10 Students should come to realise that, for example, 10 lots of 3 is the same as 3 lots of 10 (30), and that in the process the number has increased from 3 ones to 3 tens. The 3 moves left one place, and 0 is inserted.

This can be clearly shown using Base 10 blocks. For example, 10 lots of 3 can be traded for 30.

2. Multiplying a single digit by a multiple of 10 For example, 20 x 4.

3. Use Base 10 material and number charts (above) Students should work a few of these kinds of problems with Base 10 material until they are convinced that the answer is the same as (2 x 4) x 10. The 8 moves left one place, and 0 is inserted.

20 lots of 4

t ens ones

is the same as

2 lots of 40

t ens ones

4. Multiplying any number by a multiple of 10 By extension, 20 x 23 is (2 x 23) x 10, and so we work out 2 x 23 and move digits left one place, inserting the place-holder 0. This also applies when the second number already has one or more zeros. For example 40 x 300 = 1200 x 10 = 12 000.


32 Multiply any two 2-digit numbers This again uses the idea that you can multiply two parts and add the results. In fact we add four separate parts to get the total answer.


1. Use Base 10 blocks. Here is an example: 53 x 24. It means 53 lots of 24, so we should actually make 53 lots of 24 with the Base 10 material.

To save time so we use the distributive law. 53 lots of 24 is the same as 3 lots of 24 and 50 lots of 24.

a 3 lots of 24: Multiplying by 3 is something we have done already.

We have 3 lots of 20 (60) and 3 lots of 4 (12), so this part of the answer is 72.

b 50 lots of 24: This also needs to be done in two bits and added. 50 lots of 4 (or 4 lots of 50) comes to 200.

50 lots of 20 can be done in stages too.

5 lots of 20 is 100, and 50 is 10 times 5, so it is 10 times 100, or 1000.

Putting these (200 and 1000) together makes 1200. c Finally we can put the lot together: 72 from 3 lots of 24, and 1200 from 50 lots of 24. Answer: 1272.

Here is the whole thing set out as a rectangle: length 53, width 24. So the rectangle area = 53 x 24. However the area can be clearly broken up into four parts, and therefore produces a four-part version of the long multiplication algorithm.

2. Use a grid Here is the same example. The two numbers are broken into tens and ones.

Amount 20 4 Multiplier 50 1000 200 1200 Product

3 60 12 72 1272

The parts of the answer are put into each part of the grid. It shows how the four-part ‘long-multiplication’ can collapse into the more ‘traditional’ two-part method.


33 Round and estimate for two-digit multiplication Because many people use calculators to get exact answers, we need to concentrate on the skills of estimating and number sense.

Rounding is always to a single non-zero digit. For example, 23 becomes 20, and 37 becomes 40. Numbers that are half-way (such as 75), go up (to 80).

This is useful for estimating single-digit multiplication.

Suggested activities Here is an example: 57 x 23

Round 57 to 60 and 23 to 20. Then the problem is 60 x 20, and the answer is about 1200.

Another example: 39 x 78 Round 39 to 40 and 78 to 80. Then the problem is 40 x 80, and the answer is about 3200, or a bit less.


34 Divide by one digit It is helpful to think of divisions as splitting into equal shares. (“How many?” is not very helpful here.)

Division is then seen as the opposite to multiplication.

When dividing into numbers greater than 10 lots of the divisor it is good to learn to pick out ‘10 lots of the divisor’ as many times as possible.


1. Use Base 10 blocks When dividing students will also learn to find 10 lots of the number. For example 30 = 10 lots of 3. For example, 42 ÷ 3.

Students look for 10 lots of 3, and this accounts for sharing 30. They find this using the fact that 10 lots of 3 is the same as 3 lots of 10.

There are then only 12 to be shared. This is done by regrouping the 10 as ten 1s.

t ens ones

3) 4230 1012

14t ens ones t ens ones

2. Use a number grid. Here is an example: 42 ÷ 3 shown as ‘long multiplication’.

This example uses the ‘Distributive law’, which says you can split the number being divided and add the results. Note that single-digit dividing is the reverse of single-digit multiplying. For example, 3 x 12 = 36 is the multiplying that corresponds to 36 ÷ 3 = 12.

Encourage students to use various alternatives to assist in recall.

For example 56 ÷ 8 is (40 ÷ 8) + (16 ÷ 8) = 5 + 2 = 7.

3. Use Base 10 blocks for more complex examples Usually the easier way to do division of hundreds by a single digit number is ‘sharing’. Here is an example.

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4. Play the Remainder game. Two students place their counters on the first number (13) on the grid. They then take turn rolling a die (1 to 6 or 1 to 10). In all cases the player divides the dice number into the number under their counter. Then they move forward by the remainder. (Example 13 ÷ 4 = 3, remainder 1; forward 1.) Note that some numbers will have no remainder, and the player cannot move. First to get to WIN wins.

5. Use division in table format Multiplication in table format is demonstrated above in #1, activity 2. Division is the inverse of multiplication so students are aiming to find the amount such that the result of multiplying the divisor and the amount is the product. This method is used in the spreadsheet with this name.

Here is an example, the same as the one above: 114 ÷ 3. The divisor and product are given.

Amount 30 8 Divisor 3 90 24 114 Product

Students find a multiple of 10 (here 30) that multiples with the divisor to give a number under the product. (In this example, 3 x 30 if 90, and 3 x 40 is too big.) They work out the part of the product still unaccounted for (in this example, 24). Then finally they know that 3 x 8 will solve the problem.


35 Round and estimate division

This time it is more convenient to round the number to the nearest multiple of the divisor. Then the answer is clear.

Examples: 25 ÷ 3 changes to 24 ÷ 3 (8) instead of 30 ÷ 3 (10). 74 ÷ 5 changes to 50 ÷ 5 = 10 74 ÷ 8 becomes 72 ÷ 8 = 9.


36 Divide by two digit numbers

This is the process once known as ‘long division’. The essence of this process is being able to subtract multiples of the divisor, and keep doing this until the process is completed. See the example on the left below. This method retains the meaning of the division, and is recommended for the majority of students.

An efficient user of the ‘long division algorithm’ is able to use estimation (see #4 above) to determine first the best multiple of 10 times the divisor, and then the best one-digit multiple of the divisor. This allows the answer to be placed above the correct digit at the top of the problem. See the example on the right below.

23 )1234 690 30 544 460 20 84 69 3 15 remainder Ans: 53, rem 15

53 23 )1234 115 84 69 15 remainder Ans: 53, rem 1


1. Start with single digit divisors This has been demonstrated in #3 above, activity 5. Base 10 blocks can help students to understand the process.

2. Use division in table format Multiplication in table format is demonstrated above in #1, activity 2. Division in table format is demonstrated above in #3, activity 6. Division is the inverse of multiplication so students are aiming to find the amount such that the result of multiplying the divisor and the amount is the product.

Students could transfer their working to the two formats demonstrated above (in the Teach with understanding section.)

3. Divide by multiples of 10 Introduce dividing by multiples of 10, such as 30. This leads to students searching for the multiple of 30 just under the Here is an example, the same as the one above: 1345 ÷ 30. The divisor and product are given.

Amount 40 4 Divisor 30 1200 120 1320 Product Ans: 44 and 25 remainder 145 left 25 left

Another way to express the answer would be: 1345 = 30 x 44 + 25

Students find a multiple of 10 (here 40) that multiples with the divisor (30) to give a number under the product. (In this example, 30 x 40 is 1200, and 30 x 50 is too big.) They work out the part of the product still unaccounted for (in this example, 145). Then finally they know that 30 x 4 will solve the problem. There will still be 25 remainder.

Students could transfer their working to the two formats demonstrated above (in the Teach with understanding section).

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4. Divide by simple 2-digit numbers These are numbers for which it should be relatively easy to estimate the amounts. Such numbers would be 11, 12, and other teens, 21, 31 and similar numbers.

Here is an example, the same as the one above: 1345 ÷ 30. The divisor and product are given. The first estimnate is 1345 ÷ 30, giving 40. So 31 x 40 = 1240, leaving 105. The second estimate is 105 ÷ 30, giving 3, so 31 x 3 = 93, leaving the final remainder of 12.

Amount 40 3 Divisor 31 1240 93 1333 Product Ans: 43 and 12 remainder 105 left 12 left

Another way to express the answer would be: 1345 = 31 x 43 + 12

Students could transfer their working to the two formats demonstrated above.

5. Divide by any 2-digit numbers This requires students to be able to estimate well. Here is an example: 1345 ÷ 57. The divisor and product are given. The first estimate is 1345 ÷ 60, which rounds to 1200 ÷ 60 = 20. So the first try is 57 x 20 = 1140. This leaves 205.

The next estimate is 205 ÷ 60, which rounds to 180 ÷ 60 = 3. So the next step is 57 x 3 = 171. This leaves a remainder of 34, less than 57, so we are done!

Amount 20 3 Divisor 50 1000 150 1150

7 140 21 161 1140 171 1311 Product

Ans: 43 and 34 remainder Another way to express the answer would be: 1345 = 57 x 23 + 34

Students could transfer their working to the two formats demonstrated above.


37 Models of integers, including ‘opposites add to zero’, and adding The practice with integers can become a rapid degeneration into rules without understanding. Instead, use enjoyable methods to develop understanding that leads to a ‘discovery’ of rules with meaning.


1. Opposites Look at situations with opposites (use raised negative signs to distinguish from subtraction) - some real (e.g. bank balance with bills and cheques, temperatures and temperature changes, levels in a building and movements of a lift, sports scores: ahead/behind)

- some contrived (unit squares with pos and neg sides, opposite walks along number line, walking forwards and backwards)

2. Notation can be helpful (or confusing) Use notation that distinguishes between the position on a number line (e.g. +2, –3) and an operation (adding: +2 + –3, and subtracting: +2 – –3).

3. Adding opposites (the key idea, opposites add to zero)

4. Successive walks along number line, e.g. ‘walk the plank’

5. Derive ‘rules’ and short cuts based on experience


38 Subtract integers

This is not at all obvious, and requires time and care. Be careful not to fall back on simplistic rules.

In particular do NOT use words such as ‘minus times minus is plus’ for the meaning of subtracting a negative.


1. Subtracting positive numbers moves you to the left. Use a number line or thermometer image. Reversing +5 – +2 (= +3) to +2 – +5 (= –3) produces the opposite.

2. Subtracting negative numbers moves you to the right Subtracting +3 moves 3 to the left, but subtracting –3 moves 3 to the right.

3. Work with opposite pairs, leading to ‘subtracting negative is the same as adding the positive’

4. Walks on number line: if doing a walk to the left is adding a negative, then undoing a walk to the left is subtracting a negative...,

5. Images involving time Think of helicopter taking off, (+) then run the video backwards (–). Now think of making a video of it landing (–) and run that video backwards (+)

6. Postman stories: You are wrongly delivered a cheque; taking away the positive makes you feel worse off.

You are wrongly delivered a bill; taking away the negative makes you feel better off!


39 Multiply integers

This idea should come as late as possible. (There are some who maintain it should be early because ‘it is the easiest rule to remember’, but it still makes no sense.) It is no explanation that ‘the calculator does it so it must be right’.


1. Think about possible meanings ‘Start at 0 and add 3 lots of +2’ is the meaning of +3 x +2.

Think of what it could mean (e.g. you find 3 lots of $2 = $6).

‘Start at 0 and add 3 lots of –2’ is the meaning of +3 x –2.

(e.g. you receive 3 IOUs of $2 and owe $6).

‘Start at 0 and subtract 3 lots of +2’ is the meaning of –3 x +2.

(e.g. you lose 3 lots of $2 and are down $6).

‘Start at 0 and subtract 3 lots of –2’ is the meaning of –3 x –2.

(e.g. you are forgiven 3 IOUs for $2 and are up $6)


40 Multiples


1. Multiples are the ‘answers’ to the multiplication tables. The multiples of 7 are 7, 14, 21, etc.

2. Start with a 100 square. Ask students to pace a transparent counter on every third number (the multiples of 3); they will quickly see the pattern, and should be able to explain it. Repeat for many other sets of multiples.

3. Ones-digit patterns Set the digits 0 to 9 in a ring. Then find the pattern of ones-digits as the multiples unfold.

For example for multiples of 3 the pattern is:

3, 6, 9, 2 (i.e. 12), 5, 8, 1, 4, 7, 0 then repeat.

Explore this for patterns with other multiples.

This is related to the pattern on the 100-chart. Some students might be able to explain this.


41 Common multiples and lowest common multiple

Common multiples are numbers that appear in the lists of the multiples of at least two numbers; they are multiples that are common to both. For example 12 is a common multiple for 2, 3, 4 6 and 12.

Of the many common multiples of any two numbers, one is the lowest: the lowest common multiple of LCM. It is useful in many calculations, including addition of fractions where it forms the common denominator for the addition.


1. Use the 100 square again Shade in two or more colours. Common multiples will quickly appear.

2. Left-overs A certain number of things can be grouped into 3s and regrouped into 4s. What numbers of things could there be? (Encourage use of counters.) The answers are common multiples of 3 and 4, i.e. multiples of 12.

3. Common multiples Students should become familiar with common multiples of single digit numbers; this requires nothing more than familiarity with their ‘table facts’.

- Repeating cycles

- Events that occur regularly will repeat after each period of their LCM. This is the basis of the two spreadsheets.


42 Factors

Many students confuse the words ‘multiples’ and ‘factors’. Factors are multiplied to give multiples. For example, there are many sets of factors that give 12: e.g. 1 x 12, 2 x 6, 3 x 4, 2 x 2 x 3. The factors of a number will always include 1 and the number itself. A list of the factors of 12 is 1, 2, 3, 4, 6, 12.

Another word for factors is divisors; divisors are the numbers that divide exactly into the number. So all divisors must be factors!


1. Factors are divisors The only real way to check whether a number is a factor is to divide by it. Sorting out a list of all the factors of a number is good practice with tables.

2. Divisibility tests However divisibility tests are helpful to save time.

Divisibility tests: even numbers divide by 2; numbers ending in 0 or 5 divide by 5; numbers whose digits add to a multiple of 3 (or 9) divide by 3 (or 9); if the right two (three) digits are divisible by 4 (8) the number is divisible by 4 (8). There are others, often found in textbooks.


43 Common factors and highest common factor

Common factors are just that, factors that are common to two or more numbers. Clearly 1 is a common factor of all numbers, and is therefore not listed.


1. Common factors The simplest way to find common factors is to list the factors of each number and search for numbers in both lists.

Examples: 12 (factors 1, 2, 3, 4, 6, 12) and 15 (1, 3, 5, 15) have only one common factor: 3; 36 (factors 1, 2, 3, 4, 6, 9, 12, 18, 36) and 48 (1, 2, 3, 4, 6, 8, 12, 16, 24, 48) have common factors of 2, 3, 6, 12.

2. Finding larger factors Numbers that divide by 2 and 3 must also divide by 6. This is because 6 = 2 x 3. The number does not necessarily divide by 12 (for which 3 and 4 are needed). Use of this idea adds greatly to the divisibility tests.

3. Dividing by factors The idea above also means that division by large numbers with simple factors can be done in stages. For example, you can divide by 24 by first dividing (by 3 and then by 8), or (by 4 and then by 6), or (by 2, by 2, by 2 and then by 3).

4. ‘Mutually prime’ numbers Some pairs of numbers have no common factor except 1, e.g. 4and 5. They are called ‘mutually prime’. Neither of them need be a prime number, so that 14 and 15 are mutually prime.

5. The highest common factor (HCF)

It is often useful to be able to identify the highest common factor, HCF. The most common context is for ‘reducing fractions to lowest terms’, often called ‘cancelling’.

The HCF is easy to find from a list of common factors. It is a good exercise for students to try to name HCFs for pairs of numbers as it gets them used to the factors of numbers that will be often used in later work. For example, the HCF of 36 and 48 is 12.

6. For any two numbers, HCF x LCM = product. Example: for 8 and 12, HCF = 4, LCM = 24, and product = 96. 4 x 24 = 96.


44 Prime and composite numbers

Whole numbers (other than 1) are of two kinds, those that are prime numbers and those that are products of prime numbers. The second kind are called ‘composite numbers’. Because prime numbers are the building blocks of numbers, they are important and useful.

Prime numbers cannot be replaced by a product of smaller numbers. They have only two factors: themselves and 1. For these reasons 1 is not defined as a prime, but it is also not a composite.


1. A dice game A simple dice and counter game reinforces the basic idea. A number is a prime if that number of counters cannot be arranged into a filled-in rectangle at least two rows wide. If it can, then you have found two factors of the number (length x width), which proves it is a composite number. Take turns rolling a die and adding that number of counters to the pile. You win a point if you can make a composite number; the more rectangles you make the more points you win.

2. Make a list Students should make a list of the prime numbers at least as far as 100. Divisibility tests save time.


45 Prime factors

Composite numbers not only have at least two factors, but can be written as the product of ‘prime factors’ – factors that are only primes. So 12 not only is 2 x 6, and 3 x 4, but also 2 x 2 x 3.


1. Factor trees A ‘factor tree’ will always produce the prime factors as at each step you replace any composite by two factors. Since the order of factors makes no difference, it doesn’t matter how you do it.

This uses exponents: 22 x 3.

2. Factorgrams The factorgram is a method for showing all the factors of a number in the same diagram.

The horizontal arrow shows ‘divide by 2’ (one of the two prime factors), and the vertical arrow shows ‘divide by 3’ (the other prime factor).

Decimal operations, particularly adding and subtracting, use the properties of place value – extended to tenths, hundredths etc. – and involve similar ideas to those operations with whole numbers. For that reason they appear in this document. The relationship between decimals, percentages and fractions – conversions, place value ideas etc. – are found in Teach for Understanding: Fractions.


46 Add and subtract decimals

This will be very straightforward once a child has really grasped the place value ideas. If not, persist with material until it ‘clicks’.


1. Demonstrate For example: 4.5 + 2.7. (Each row of the number chart shows 1.) Combine the 4 and 2, and also the 0.5 and the 0.7. The 0.5 and 0.7 make 1.2, so we have 7.2.

For example: 4.5 – 2.7 Each row of the number chart shows 1.

Break the 4.5 into 3 and 1.5. Subtract 2 from the 3 (1), and 0.7 from the 1.5 (0.8). Put these together: 1.8.

Here is an example using hundredths: 0.5 – 0.15

We now call the number chart 1. (It’s like $1.00.) This means the ‘ten-strips’ are each one tenth ($0.10), and each single is one hundredth ($0.01).

This example also shows two numbers with unequal numbers of decimal places (digits after the point).

2. Use flashcards for adding and subtracting.


47 Multiply decimals by single digit whole numbers


1. Stress understanding The only successful way to handle this is to help students to realise that it works just like multiplying whole numbers; the decimal point makes no difference!

Thus 4 x 0.3 = 1.2, and 4 x 0.03 = 0.12 and so on.

2. Show how it works with money: 4 x $0.30 is $1.20, and 4 x $0.05 is $0.20.

3. Show how it works with length: 4 x 0.3 metres (30 cm) is 1.2 metres, and 4 x 0.03 metres (3 cm) is 0.12 m (12 cm). Show how it works with the base ten material.

Call the number chart (100 spots) the 1. Then each strip is 0.1 and each individual spot is 0.01. Then four lots of three strips represent 4 x 0.3, and has the value of 1.2. Also ten lots of two spots represent 4 x 0.03, and has the value of 0.12.

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4. Use flashcards for speed practice.


48 Divide decimals by single digit whole numbers

These ideas are necessary to be able to use estimation successfully. It is helpful to think of divisions as splitting into equal shares. (“How many?” is not very helpful here.) Division is then seen as the opposite to multiplication.


1. Show that this works just like dividing whole numbers Share the amount equally a number of ways. Thus 3 ÷ 2 = 1.5, and 0.3 ÷ 2 = 0.15 and so on.

2. Show how it works with money: $3 ÷ 2 is $1.50, and $0.30 ÷ 2 is $0.15.

3. Show how it works with length: 3 m ÷ 2 = 1.5 m, and 0.3 m (30 cm) ÷ 2 is 0.15 m (15 cm).

4. Show how it works with the base ten material. Call the number chart (100 spots) the 1. Then each strip is 0.1 and each individual spot is 0.01. Then 3 ones shared two ways represents 3 ÷ 2, and has the value of 1.5. Also 3 strips shared 2 ways represents 0.30 ÷ 2, and has the value of 0.15.

5. It works when dividing by a larger number – only the answer is a decimal. For example 3 ÷ 6 = 0.5. (Money: split $3 into 6 parts is $0.50; Length: 3 metres split into 6 parts is 0.5 m).



49 Multiply a decimal number by ten To multiply by the base number of the system (ten) always has the effect of moving all digits to the left one place. It is far better to speak about the changing place value of the digits than ‘moving the decimal point to the right’. They might be equivalent in their effect, but one refers to the meaning and the other to a memorised ‘trick’.


1. Demonstrate The only successful way to handle this is to help students to realise that it works just like multiplying whole numbers by 10: the digits move one place to the left, because their value increases to the next ‘level’ up.

Thus 10 x 0.3 = 3, and 10 x 0.03 = 0.3 and so on.

2. Show how it works with money: 10 x $0.50 is $5, and 10 x $0.05 is $0.50.

3. Show how it works with the base ten material.

4. Number chart Call the number chart (100 spots) the 1. Then each strip is 0.1 and each individual spot is 0.01. Then ten lots of two strips represents 10 x 0.2, and has the value of 2.

Also ten lots of two spots represents 10 x 0.02, and has the value of 0.2.



50 Divide by ten

To divide by the base number of the system (ten) always has the effect of moving all digits to the right one place. It is far better to speak about the changing place value of the digits than ‘moving the decimal point to the left. They might be equivalent in their effect, but one refers to the meaning and the other to a memorised ‘trick’.


1. Stress understanding Show that this works just like dividing whole numbers by 10: the digits move one place to the right, because their value decreases to the next ‘level’ down. Thus 0.3 ÷ 10 = 0.03, and 3 ÷ 10 = 0.3 and so on.

2. Show how it works with money: $0.50 ÷ 10 is $0.05, and $5 ÷ 10 is $0.50.

3. Show how it works with the base ten material. Call the number chart (100 spots) the 1. Then each strip is 0.1 and each individual spot is 0.01. Then two ones shared ten ways represents 2 ÷ 10, and has the value of 0.2.

Also two strips shared ten ways represents 0.2 ÷ 10, and has the value of 0.02.


51 Decimal multiplication with powers of ten

Multiplying by a number of tens, or hundreds, means being aware of the meaning. You will probably find that money examples are the most useful.

Some examples of this type of problem are:

20 x 30 20 x 0.3 20 x 0.03


1. Think of these with the help of money. 20 lots of $30 20 lots of $0.30 (30 cents) 20 lots of $0.03 (3 cents) is $600 is $6.00 (600 cents) is $0.60 (60 cents)

2. It is useful to think of these as the basic multiplication (i.e. 6) and the ways the tens are involved. 20 x 30 20 x 0.3 20 x 0.03 20 is 2 x 10 and 30 is 3 x 10 20 is 2 x 10 and 0.3 is 3 ÷ 10 20 is 2 x 10 and 0.03 is 3 ÷ 100

So 20 x 30 20 x 0.3 20 x 0.03

= 6 x 10 x 10 = 6 x 10 ÷ 10 = 6 x 10 ÷ 100 = 600 = 6 = 6 ÷ 10 = 0.6

Here are the last two of these illustrated with base 10 strips.

3. Use flashcards, for example 300 x 20, 300 x 0.2, 0.3 x 0.2.


52 Decimal division with powers of ten

Again, dividing by tens or hundreds will produce a decimal number, and the meaning is important.

Some examples of this type of problem are:

6 ÷ 200 0.6 ÷ 2 0.6 ÷ 20


1. Think of these with the help of money. It is helpful to think of divisions as splitting into equal shares. (The division idea of “How many?” is not helpful here.)

$6 shared by 200 $0.6 shared by 2 $0.6 shared by 20

is 600 cents shared by 200 is 60 cents shared by 2 is 60 cents shared by 20

= 3 cents ($0.03) each = 30 cents ($0.30) = 3 cents ($0.03) So $6 ÷ 200 = $0.03 So $0.60 ÷ 2 = $0.30 So $0.6 ÷ 20 = $0.03

It is useful to think of these in terms of the basic division (6 ÷ 2 = 3) and the ways the tens are involved.

6 ÷ 200 0.6 ÷ 2 0.6 ÷ 20

20 is 2 x 10 x 10 0.6 is 6 ÷ 10 0.6 is 6 ÷ 10 and 20 is 2 x 10 So 6 ÷ 200 So 0.6 ÷ 2 So 0.6 ÷ 20

= 6 ÷ (2 x 10 x 10) = (6 ÷ 10) ÷ 2 = (6 ÷ 10) ÷ (2 x 10)

= 3 ÷ 100 = 3 ÷ 10 = 3 ÷ 100

= 0.03 = 0.3 = 0.03

Here are these problems illustrated with base 10 strips. (The ‘flat’ shows 1.)

2. Use flashcards, such as 60 ÷ 200, 6 ÷ 200, 6 ÷ 20, 0.6 ÷ 2, 0.6 ÷ 20


53 Estimate when multiplying decimals by whole numbers

The essence of this is to use single digit rounding, and a knowledge of how to handle zeros after the non-zero digit and zeros after the decimal point.


1. Discuss these examples: 3456 x 890 276 rounds to 3000 x 900 000. Now 3 x 9 = 27, so the answer is about 27 with eight additional zeros. So 3000 x 900 00 ~ 2 700 000 000.

3456 x 0.065 rounds to 3000 x 0.07. Now 3 x 7 = 21. Now move the digits in the 21 left three places (for the 1000) to get 21 000.

Finally move the digits of 21 000 right by two places (because of the 0.07).

So 3000 x 0.07 = 210. 30 x 0.0005 will be 30 x 5 = 150 but with the digits moved right four places.

This gives 0.0150, or 0.015.

At this stage, answers should be checked with a calculator.

At a later stage, the estimating forms a check on the correct key pressing with the calculator.

2. Use flashcards.


54 Estimate when dividing decimals by whole numbers

There are three steps: rounding, changing the divisor to a single digit whole number, and finally dividing.


1. Rounding The essence of this is to use single digit rounding for the divisor, and to round the number being divided to the nearest multiple of the single digit divisor. Example: 35.87 ÷ 63. The 63 rounds to 60. We need to round the 35 in the first number to a multiple of 6. The nearest one is 36, so the question becomes 36.00 ÷ 60 or 36 ÷ 60.

2. Changing the divisor to a single digit whole number Since we have a single non-zero digit in the divisor, we can divide both numbers by ten the same number of times to make the divisor into a single digit.

Example: 36 ÷ 60 changes into 3.6 ÷ 6.

3. Dividing now is simple. 3.6 ÷ 6 is just 36 tenths ÷ 6, so the answer is 6 tenths or 3.6 ÷ 6 = 0.6. At this stage, answers should be checked with a calculator.

At a later stage, the estimating forms a check on the correct key pressing with the calculator.

4. Use flashcards.


55 Multiply tenths by tenths It is tempting to just tell students the rule (“add the number of decimal places”) and forget to teach for understanding. To do this will lead to disaster. Instead we focus on the meaning of problems such as 0.1 x 0.1, and lead students to work out the rule for themselves based on their own understanding. The secret is to give them at least one way of thinking about the problem that makes sense. The more models they have the better.

So what does 0.1 x 0.1 mean?

(a) It is one tenth of one tenth. Start with one tenth of a dollar ($0.10), and get one tenth of that ($0.01). So 0.1 x 0.1 = 0.01.

(b) Start with one tenth of a metre (0.1 m or ten cms), and get one tenth of that (0.01 m or 1 cm).

So 0.1 x 0.1 = 0.01.

(c) It is the area of a square that is one tenth of a metre on each side. The diagram shows a square with its sides divided into ten parts.

The little square has an area of 0.1 x 0.1. Its area is one hundredth of the total square.

So 0.1 x 0.1 = 0.01 again. Note that multiplying by 0.1 has the same effect as dividing by 10.


1. Special cases There are some special cases that can look confusing; these are the cases where the answer has a final 0. Examples: 0.2 x 0.5 = 0.1 (actually 0.10), 0.4 x 0.5 = 0.2, and so on.

2. Multiplying can make a number smaller Some students will be confused because they have a clear rule in their heads that tells them that multiplying will always make a number larger. This is clearly not true when you multiply by a number (fraction, decimal or percentage) less than 1. They need to realise that the simple rule that has served them up to now needs to be modified.

3. Estimating A quick mental estimate can be very helpful. Also remember that you can do a multiplication in any order; choose the easier one.

4. Flashcards Use flashcards that recall table facts and include use of the rule for tenths x tenths. For example 0.1 x 0.3, 0.5 x 0.6, 1.1 x 0.7 etc.


56 Multiply any two decimals

This section aims to lead students towards a general rule that they will understand.


1. Invite students to compete this table and explore the patterns it shows. x 300 30 3 0.3 0.03 0.003 0.2 60 6 0.6 0.06 0.006 0.000 6 0.02 6 0.6 0.06 0.006 0.000 6 0.000 06 0.02 0.6 0.06 0.006 0.000 6 0.000 06 0.000 006 0.002 0.06 0.006 0.000 6 0.000 06 0.000 006 0.000 000 6

The rule is generally expressed this way:

Count the number of decimal places (digits after the point) in the numbers being multiplied, and add them. If there are digits left of the point, call these negative. For example, 0.2 x 300 has 1 right and 3 left, adding to 2 left, so the answer is 60.

2. Estimating is still based on rounding to one non-zero digit. This is the most important skill.

For example 0.00682 x 0.07391 (estimate 0.007 x 0.07 = 0.000 49) 0.83 x 53.6 (estimate 0.8 x 50 = 40)

3. Equivalent forms of expressions With problems such as the second one above, it is useful to be aware of equivalent forms of the problem, such as transferring a factor of 10 from one number to the other.

For example, 0.8 x 50 = 8 ÷ 10 x 5 x 10 = 8 x 5 = 40

4. Use flashcards that recall table facts and include use of the general rule. For example 0.01 x 0.003, 500 x 0.6, 1.1 x 0.07 etc.


57 Divide by tenths Again the only successful way to teach this is to provide examples and use good questioning techniques to help students to understand what is going on.


1. Money How many 50 cent coins can you get for $3? We can show this problem as 3 ÷ 0.5. In this case division means ‘how many?’ The answer is clearly 6.

Similarly how many 10 cent coins could you get for $3? This is 3 ÷ 0.1, and the answer is 30. So dividing by 0.1 has the same effect as multiplying by 10!

Just as with multiplying by tenths makes numbers smaller, so also dividing by tenths produces a surprise for beginners. It makes the answer bigger! The first two examples above show that the smaller the number you are dividing by, the larger is the answer!

2. Equal multiplication by ten There is a useful idea that can help, particularly for estimation. It is equal multiplication by 10. We know that 3 ÷ 0.1 is 30. We can also multiply both numbers in the division by 10 to get 30 ÷ 1, and the answer is still the same!

3. Flashcards Use flashcards that recall table facts and include use of the rule for dividing by tenths. For example 12 ÷ 0.3, 300 ÷ 0.6, 3.5 ÷ 0.7 etc.


58 Divide by any decimals

There are some general rules that students should be able to discover for themselves and then reinforce.


1. When you divide by a smaller number, the answer gets bigger. The smaller the divisor, the larger the answer. If the first number is bigger than the divisor, the answer will be bigger than 1.

For example 6 ÷ 0.3 = 20, 6 ÷ 0.03 = 200, 6 ÷ 0.003 = 2000

• If the first number is smaller than the divisor, the answer will be smaller than 1. Again, the smaller the divisor, the larger the answer.

For example 0.0006 ÷ 0.3 = 0.002, 0.0006 ÷ 0.03 = 0.02, 0.0006 ÷ 0.003 = 0.2

2. Here are some more examples of equal multiplication by 10: Work out 0.4 ÷ 0.05 using money (‘how many 5 cent coins can you get for 40 cents?’). You will get 8. However multiply both by 10 (4 ÷ 0.5) and by 10 again (40 ÷ 5) and you will also get 8! It is more difficult to understand the meaning of a problem like 0.3 ÷ 0.6.

However, equal multiplication by 10 will turn it into a problem of dividing by a whole number: 3 ÷ 6. The answer is 0.5.

3. Estimating Estimation requires division fact rounding, where the divisor (second number) is rounded to a single

non-zero digit and the first number is rounded be a nearby multiply of that non-zero digit. Here are some examples:

0.38 ÷ 0.075 rounds to 0.4 ÷ 0.08.

Then equal multiplication by 10 (twice) leads to 40 ÷ 8 = 5.

0.069 ÷ 0.0043 rounds to 0.08 ÷ 0.004.

Then equal multiplication by 10 (three times) gives 80 ÷ 4 = 20.

4. Use flashcards that recall table facts and include use of the general rule. For example 0.012 ÷ 0.003, 180 ÷ 0.06, 2.1 ÷ 0.07 etc.


59 Square and cube numbers

Square numbers are called ‘square’ because that number of small cubes may be arranged into a ‘filled-in’ square. Cubic numbers are similarly called because that number of small cubes (e.g. minis) may be arranged into a ‘solid’ cube.


1. Squares Make all the square numbers up to 100 (10 x 10) with minis. Explore the square numbers and patterns in them. Find them in the 100 chart.

Explore unit digits (1, 4, 9, 5, 6, 9, 4, 1, 0, 1,...) Why?

Explore step sizes between squares (1, 3, 5...). They are odd numbers. Why? Explore step sizes between every second square (8, 16, 24,...). They are multiples of 8. Why?

Look for any two square numbers that have a difference (or a sum) that is a square number.

2. Cubes Make all the cubic numbers up to 125 (5 x 5 x 5) with minis. Use a calculator to explore unit digits of cubes. Explore the differences between cubes, and the differences between the differences.


60 Exponential notation

The major task here is to find simple models that will enable students to have a mental picture of the meaning of exponential notation. The obvious one is doubling, and ‘cutting and stacking’ is the way we will do it.


1. Doubling Imagine a very large sheet of paper. It is 1 thickness only. Now cut it in half and stack the two pieces on top of each other (i.e. ‘cut and stack’).

After one ‘cut and stack’ there are 2 thicknesses.

Continue and note the pattern in the numbers of thicknesses: 1, 2, 4, 8, 16 etc. The number of thicknesses is 1 x 2 x 2 .. for the number of times you have ‘cut and stacked’.

This is written as 1 x 2c.

Note that the meaning of 20 is contained in this model. When c = 0 (no ‘cut and stacks’), 2c must be 1.

Tower of Hanoi and Paper folding The well-known tower puzzle has numbers of moves (m) that are one less than powers of 2 (see the Maths a Work reference): m = 2c– 1.

This is also the formula for the number of creases (c) you get when you fold a strip of paper a number of times (f); see the spreadsheet. c = 2f– 1.

2. Powers of 10 greater than 1 Students are familiar with the idea that big powers of 10 have large numbers of 0s. All they have to do at this stage is to recognise the number of 0s as the number of times the 1 has been multiplied by 10. So 100 means 1 x 10 x 10 = 1 x 102, and 1000 000 means 1 x 106.

3. Exponential notation Introduce the idea that the base number (b) can be any positive number at all; in general p is the power, b the base and e the exponent: p = be. - Use repeated multiplication on a calculator to see powers of 3, 4, etc.

- Use numbers such as 1.1 as the base; this will later relate to exponential growth and compound interest.

- Use numbers less than 1 such as 0.9 as the base and let students see that the powers get smaller. (They should be able to explain this.) This relates to exponential decay and depreciation.


61 Base 2 numbers, adding and multiplying

The purpose of this topic in the course is three-fold:

Hopefully it can develop an understanding of the use of base-2 to represent and manipulate numbers, and thus demonstrate how computers store and calculate with numbers – without going too deep.

It can reinforce the importance of place value in the base-10 number system, by providing an example in a different base.

By developing and extending the use of exponential ideas and notation it can prepare students for the exponent laws – see #6 below.

Base 2 notation uses the powers of 2 as the basis for grouping the numbers. This example shows how it is done.

Firstly the powers of 2 are 1, 2, 4, 8, 16 etc.

• • • • • • • • • • • • •

In this set we cannot find 16, but there is a group of 8.

(• • • • • • • •) • • • • •

Within the rest there is a 4.

(• • • • • • • •) (• • • •) •

There is no 2, and only a 1 is left. So, reading from the left, the groups are 8, 4, no 2 and a 1.

8 4 2 1

1 1 0 1

If this were a number in base-ten notation the groups would be 1000, 100, no 10 and a 1. But instead it uses base-2, and the only possible digits are 0 and 1. This can correspond to ON and OFF, and enables base-2 to be used by computers. The other name for numbers using two digits only is ‘binary numbers’.


1. Work with the powers of 2 Students enjoy doubling. The famous story about the inventor of chess (in India) gives plenty of practice. The Maths300/RIME lesson “Odds & evens” shows that all chains end in powers of 2.

How many doublings of a piece of paper (say 1 mm) do you need to get to the moon?

2. Work with powers of 2 as collections An old post office used only four separate masses to balance all the number of pounds from 1 to 15. How could they do this? Using 8, 4, 2 and 1. For example, 13 is 8 + 4 + 1, see above.

3. Formalise a process for converting from base 10 to base 2 List the powers of 2 that are likely to be present. Take away these in order. See the example for 13 above. Here is another example. 315 could include 256, 128, 64, 32, 16, 8, 4, 2, or 1

315 – 256 = 59. Because there is a 256, the number starts with 1.

Neither 128 nor 64 are present, so the number has two zeros for those places: 100

59 – 32 = 27. Because there is a 32, the number has a 1 at this place: 1001 27 – 16 = 11, so the number is 10011

11 is made up of 8, no 4, a 2 and a 1. So the number is 100111011.

MORE >


4. Count in base 2, forwards and backwards The process becomes quite simple and mechanical once it is understood. You need to keep adding one object to a pile, and showing that you can form a new larger group at increasingly widespread intervals. The spreadsheet (Base 2) shows this in a diagram where the area of the rectangles keeps growing. 1, 10, 11, 100, 101, 110, 111, 1000, 1001, 1010, 1011, 1100, 1101, 1110, 1111, 10000 and so on.

5. Simple additions using base 2 only These are best set out vertically, so we are emphasising the place-value/regrouping ideas, and showing the similarity to base-ten addition. It is not required to convert these back to base 10, but some students will want to check that the additions are correct when expressed in base-ten.

1 11 101 1101 1 10 111 1110 10 101 1100 11011

One point is to show the basis of how computers work. An electronic circuit operates to perform these additions. The other is to remind them of the fundamentals of how addition works

6. Multiplication in base 2, by single digits This is the simplest multiplication table of all! 0 x 0 = 0, 0 x 1 = 0, 1 x 0 = 0 and 1 x 1 = 1.

7. Multiplication in base 2, by several digits The process is just like base-ten multiplication: multiply by the separate digits and add the partial products. But for the multiplying there is no carrying! Here is an example:

1101 In base 10 13 x101 x5 1101 0000 This line can be omitted 110100 1000001 (64 + 1) 65

Note that the base-ten conversion and checking is NOT required.


62 Scientific notation

A major way of writing large (and small) numbers uses ‘scientific notation’, with a number between 1 and 10, multiplied by a power of 10. Students must first learn to understand this notation and then learn to calculate with it. We start here with numbers greater than 1. For example 123 = 1.23 x 102, 1230 = 1.23 x 103.


1. Notation Demonstrate the notation and ask students to explain to you how it works. It is important that they generate the explanations, so you are sure they really understand.

2. Conversion between notations Make sure students can readily convert both ways, and also learn to read and enter numbers in their calculator in this form.


63 Exponent laws (for multiplying and dividing powers)

The exponent laws (often wrongly called ‘index laws’) simply summarise what happens naturally. The models we have created should now assist in understanding this. Spend plenty of time working with numbers (not algebraic symbols) so students can be sure of what is happening.


1. Work with the doubling model For all of these, provide many examples using numbers. Then get students to generalise in words. The mental image of ‘cut&stack’ leads naturally to the ‘first law’: 2m x 2n = 2m+n, by performing m cut&stacks and then a further n of them.

If we say that m + n = p, then we can ‘undo’ n cut&stacks to leave m of them: 2p ÷ 2n = 2m = 2p – n

Cut&Stacks 0 1 2 3 4 5 6 7 8 9 10 Thicknesses 1 2 4 8 16 32 64 128 256 512 1024

2. Work with powers of 10 The same rules will always apply when multiplying or dividing powers of 10. Students may be familiar with short cuts like ‘add the numbers of zeros’ or ‘subtract the numbers of zeros’.

Examples: 103 x 105 = 1000 x 100 000 = 100 000 000 = 108

108 ÷ 105 = 100 000 000 ÷ 100 000 = 1000 = 103


64 Square roots & surds

We are a long way from whole numbers in general, but we should start there to get the meaning clear. The square root of a number is the number that multiplies by itself to form the number.

The square root of 9 is 3. So this idea is just the inverse of square numbers. Similarly the cube root multiplies by itself three times to form the number. The cube root of 8 is 2.


1. Calculator trial and error Students should use a calculator to try to find the square roots of numbers that do not have exact roots. The best method at this stage is trial and error.

2. Square root as an exponent Because bm x bm = b2m we can write b0.5 x b0.5 = b1, showing that the meaning for b0.5 is the number that multiplies twice to make b, the square root. Check this on a calculator using exact squares and then any positive number.

3. The square root as a function Explore the concept of the ‘square root function’ giving the square root of any positive number. This is the basis of the spreadsheets below.

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Advanced whole numbers, integers & decimalsregistration.mav.vic.edu.au/tm4u/images/files/TM4U/... · Advanced whole numbers, integers & decimals Whole numbers: ... • do the division