n properties of numbers

Properties of numbers

Previous learningBefore they start, pupils should be able to:

recall addition and subtraction facts for each number to 20

recognise odd and even numbers

recognise multiples of 2, 5 and 10.

Objectives based on NC levels 3 and 4 (mainly level 4)In this unit, pupils learn to:

represent problems using words or diagrams

look for and visualise patterns

manipulate numbers

record, explain and compare methods

engage in mathematical discussion of results

begin to generalise

and to:

identify squares of numbers to 12 3 12

recognise multiples and use simple tests of divisibility

order, add and subtract positive and negative numbers in context.

Lessons 1 Square numbers

2 Multiples and divisibility

3 Positive and negative integers

About this unit Pupils’ confidence in responding to numbers in everyday situations is strengthened by having a good ‘feel for number’. This means being aware of significant relationships between numbers and knowing at a glance which properties they possess and which they do not.

In this unit pupils learn to use patterns to help them to recall number facts and recognise number properties.

Assessment This unit includes: an optional mental test that could replace part of a lesson (p. 00); a self-assessment section (N2.1 How well are you doing? class book p. 00); a set of questions to replace or supplement questions in the exercises

or homework tasks, or to use as an informal test (N2.1 Check up, CD-ROM).

Common errors and misconception

Look out for pupils who: have difficulty in remembering number facts, such as addition and

subtraction facts to 20, or multiplication facts to 10 3 10; confuse squaring and doubling; lack confidence in working in the negative part of the number line, and

who think that 23 1 5 5 8, or that 23 2 5 5 8.

N2.1

� | N4.1 Properties of numbers

Key terms and notation problem, solution, method, pattern, relationship, order, solve, explain, representcalculate, calculation, calculator, add, subtract, multiply, divide, divide exactly, divisible, sum, total, difference, product, greater than (.), less than (,), valuepositive, negative, integer, odd, even, multiple, square, perfect square, square root, digit sumtemperature, degrees Celsius (°C)

Practical resources calculators for pupils individual whiteboards

Exploring maths Tier 2 teacher’s book N2.1 Mental test, p. 00 Answers for N2.1, pp. 00200Tier 2 CD-ROMPowerPoint files N2.1 Slides for lessons 1 to 3 Prepared toolsheets N2.1 Toolsheets 3.1 to 3.3 Tier 2 programs and tools Calculator tool Number grids Number and shape sorter

Tier 2 class book N2.1, pp. 00200 N2.1 How well are you doing?, p. 00Tier 2 home book N2.1, pp. 00200Tier 2 CD-ROM N4.1 Check up, p. 00

Useful websites Multiples an NRICH package of problems and puzzles nrich.maths.org/public/viewer.php?obj_id=5530Grid game www.bbc.co.uk/education/mathsfile/gameswheel.htmlMulti sequencer www.amblesideprimary.com/ambleweb/mentalmaths/supersequencer.html

N4.1 Properties of numbers | 3

Learning points When a number is multiplied by itself the result is a square number.

81 is the square of 9. It can be written as 92.

A square number can be represented by dots arranged in the shape of a square.

1 Square numbers

Starter

4 | N2.1 Properties of numbers

Main activity

Show slide 1.1. to discuss the objectives for this unit. Say that this lesson is about square numbers.

Show slide 1.2. Point to the 3 by 3 pattern of dots.

How many dots are there? How are they arranged?

Write 9 in the box below the dots and 3 3 3 in the box below. Continue up to 6 3 6.

Point to the row of numbers 1, 4, 9, 16, 25, 36. Say that these are called square numbers – each is the result of multiplying a number by itself and can be represented by dots arranged in a square shape.

Say that there is a special way of reading and writing square numbers. Point to 12 and 22 saying ‘one squared, two squared’. Ask pupils to write on their whiteboards 32, 42, 52 and 62. Enter these in the table.

Launch Number grids. Choose a multiplication grid with 10 rows and columns, a start number of 0 and a step of 1. Say that you will highlight the first six numbers in the sequence of square numbers. Click to highlight 1, 4, 9, 16, 25, 36. Point out that 16 or 4 3 4 lies in the fourth column and the fourth row.

What are the next numbers in the sequence? [49, 64, 81, 100]

How would we write seven squared?

What number do you square to get 81?

What is the square of ten?

What is the next square number after 100? And after that?

Ask the class to chant the sequence of square numbers. Point to the numbers as pupils say them. Click on ‘Hide products’ and chant again.

If you prefer, use slide 1.3 instead of Number grids. You can white out the slide for the last chant by pressing W on the keyboard, or by right-clicking then choosing Screen > White screen. Press W or right-click again to restore the slide.

Use the Calculator tool to show pupils how to use the x2 . key on their calculators. Tell them that they are going to investigate sums of two square numbers.

What is six squared? What is two squared?

What is the sum of six squared and two squared? [40]

Demonstrate the key sequence 6 x2 1 2 x2 5

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N2.1 Properties of numbers | �

Homework

Review

Explain that 40 is an interesting number because it is the sum of two square numbers. Say that 13 is also the sum of two squares. Ask pupils to discuss in pairs what the two squares might be [2 and 3].

Which other numbers up to 30 are the sum of two different squares?

Ask pupils to investigate this in pairs for a few minutes. Discuss how to work systematically, e.g. add 12 to each of 22, 32, 42 and 52. Since 12 added to 62 is too big, now add 22 to each of 32, 42 and 52.

Why don’t we need to try adding two squared to one squared?[same as 12 1 22]

Leave the pairs to work for a few more minutes, then gather the complete set of results: 5, 10, 13, 17, 20, 25, 26, 29.

How do we know that we have found all the possibilities?

Establish that because they have worked systematically through all the possible pairs they must have checked them all.

Say that a mystery number, when squared, has the answer 225.

Is the mystery number less than 10? [No – its square would be less than 100.]

Is the mystery number more than 20?

Confirm with the Calculator tool that 202 5 400, so the number is less than 20.

Could the mystery number be even?

Establish that the product of two even numbers is always even, so the mystery number is odd (and lies between 10 and 20). Write 11, 13, 15, 17 and 19 on the board.

Which of these numbers could it be? Explain why.

Draw out that 15 is the only number which, when multiplied by itself, will result in a number with a units digit of 5. Confirm using the Calculator tool.

Sum up with the points on slide 1.4.

Ask pupils to do N2.1 Task 1 in the home book (p. 00).

Select individual work from N2.1 Exercise 1 in the class book (p. 00).

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� | N2.1 Properties of numbers� | N2.1 Properties of numbers

Learning points A multiple of 5 is a number that divides exactly by 5.

A number is a multiple of 2 if its last digit is even.

A number is a multiple of 3 if its digit sum is a multiple of 3.

A number is a multiple of 4 if half of it is even.

A number is a multiple of 5 if its last digit is 0 or 5.

A number is a multiple of 10 if its last digit is 0.

Starter

Slide 2.1

Main activity

Say that this lesson is about multiples. Remind pupils that a multiple of a number divides exactly by the number.

Draw a large square box on the board. Ask pupils to suggest some numbers below 60. If they are multiples of 5, write them in the box. If not, write them outside.

Once there are at least three numbers in the box, ask:

What is my rule for putting numbers in the box?

Continue asking for numbers until pupils recognise the rule. Repeat with multiples of 7, multiples of 2 and multiples of 11. Invite pupils who think that they know the rule to the board to write another number in the box.

Write 96 on the board.

Is this number odd or even? How do you know?

Point out that the last digit is even, so the whole number is even. An even number is divisible by 2, i.e. divides exactly by 2 with no remainder. It is also a multiple of 2. Say a few numbers and ask pupils to say if the number is divisible by 2.

Return to 96.

Is this number divisible by 3?

Tell the class that there is a quick way to find out by adding up all the digits, i.e. 9 1 6 5 15. Explain that 15 is called the digit sum. Because it is a multiple of 3, the number 96 is a multiple of 3. Test a few more numbers for divisibility by 3.

Return to 96.

Is this number divisible by 4?

Say that there is an easy way to find out. We know that 96 is even so it divides exactly by 2, so find half of 96. Point out that because 48 is even, 96 can be divided exactly by 2 and then exactly by 2 again. So 96 is divisible by 4. Test a few more numbers for divisibility by 4.

� Multiples and divisibility

27 55 59 201 35 23


Homework

Review


Return to 96.

Is this number divisible by 5 or by 10?

Confirm that it is not, since it does not end in 5 or 0.

Launch Number and shape sorter. Choose a two-way Carroll diagram and then ‘Is a multiple of 3 / Is a multiple of 4’. Involve pupils in dragging the numbers to the correct part of the diagram. Ask questions such as:

How do you know that 42 is a multiple of 3?

How do you know that 42 is not a multiple of 4?

How do you know that 24 is divisible by 3 and divisible by 4?

Write 96 on the board again.

Is 96 divisible by 6?

Explain that all multiples of 6 divide exactly by 2 and also by 3. We know that 96 divides exactly by 2 because it is even. We also know that it divides exactly by 3 because its digit sum of 15 is a multiple of 3. So 96 is divisible by 6.

Launch Number and shape sorter again. This time choose a two-way Venn diagram, then ‘Is a multiple of 3 / Is a multiple of 6’. Involve pupils in dragging the numbers to the correct region. Ask, for example:

How do you know that 39 is a multiple of 3?

How do you know that 39 is not a multiple of 6?

Sum up with the points on slides 2.1 and 2.2.


SIM


SIM

� | N2.1 Properties of numbers

Starter

Main activity

3 Positive and negative integersLearning points

The negative number 6 is called ‘negative 6’ and written as 26.

Numbers get less as you count back along the number line beyond nought or zero, so 210 is less than 25.

Six degrees below zero is minus six degrees Celsius (26°C).

210°C is a lower temperature than 25°C.

Always include the units when you write a temperature.

Say that this lesson is about positive and negative numbers.

Show the first number line on toolsheet 3.1 Point at random to divisions on the line and ask pupils to say the number. As they do so, write it on the line.

Explain that the line shows the integers from 25 to 5. Integers are positive or negative whole numbers and zero. The negative integers are called ‘negative one’, ‘negative two’, and so on. Say that we usually don’t write 12 for ‘positive two’ but write 2.

Show the number line on toolsheet 3.2. Again, point at random to divisions on the line and ask pupils to say the number. Write each number on the line, then ask questions like:

Tell me a number that is less than 220.

That is more than 230.

That lies between 220 and 10.

Record answers on the board using the < and > signs, e.g.

260 < 220 29 > 230 220 < 0 < 10

Show the number line on toolsheet 3.3. Explain that, just like a number line with only positive numbers, it is possible to add and subtract by counting steps along the line.

What is 5 more than 22? [record as22 1 5 5 3]

What is 6 less than 4? [record as 4 2 6 5 23]

What is the difference between 5 and23?

Stress that a difference is measured by the number of steps or the distance between the numbers, and can be recorded as 5 2 (23) 5 8.

Which pairs of numbers have a difference of 4?

Show slide 3.1. Say that the thermometers show the temperatures in Leeds and Barcelona on the same day in winter. Point out the °C (degrees Celsius) abbreviation. Say that as you move down the scale the temperature is falling.

-3

3 5

0 +5

TO

TO

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Homework

Review

What is the temperature in Leeds? In Barcelona?

How much colder is it in Leeds than Barcelona?

Point out 25°C on each scale. Invite a pupil to show 27°C. Explain that this is read as ‘minus seven degrees Celsius’ not ‘negative seven degrees Celsius’ and that it means that the temperature is seven degrees Celsius below zero.

The temperature falls by 5 degrees in each city. What are the temperatures now?

On another day, Leeds is 3°C. Barcelona is 7 degrees colder. What is the temperature in Barcelona?

Remind pupils that they should include the units when they write a temperature.

Show slide 3.2 and work through the questions.

Show slide 3.3. Ask pupils to write the temperatures in order on their whiteboards from hottest to coldest. Use the temperatures to ask questions such as:

What is the difference between 12°C and 37°C? Between 12°C and –12°C? Between 212°C and –2°C?

The temperature is 22°C. How many degrees must it rise to reach 12°C?

The temperature falls from 37°C to 212°C. How many degrees has it fallen?

After each question, ask pupils how they worked out the answer.

Sum up the lesson with the points on slide 3.4.

Round off the unit by referring again to the objectives. Suggest that pupils find time to try the self-assessment problems in N2.1 How well are you doing? in the class book (p. 00).



Read each question aloud twice. Allow a suitable pause for pupils to write answers.

1 Write the next odd number after twenty-nine. 1998 KS2

2 Which is the lowest of these temperatures? [Write on board: 2°C 25°C 5°C 0°C 21°C] 2005 KS2

3 What is three times three added to four times four? 2003 KS2

4 Write three even numbers that add to twenty. 2004 KS3

5 What temperature is ten degrees lower than seven degrees Celsius? 2006 KS2

6 What number is nine squared? 1997 KS3

7 Write down an even number that is a multiple of seven. 2005 KS3

8 What is the smallest whole number that is divisible by five and by three? 2004 KS3

9 The temperature on Monday was minus eight degrees Celsius. [Write on the board 28°C.] 2005 PT On Tuesday, it was ten degrees higher. What was the temperature on Tuesday?

10 What is the next square number after thirty-six? 2005 PT

11 Write a number that is a multiple of ten and also a multiple of twelve. 2006 PT

12 What number multiplied by eight equals forty-eight? 2005 KS3

Key:KS3 Key Stage 3 Mental test PT Progress test KS2 Key Stage 2 Mental testQuestions 1 to 3 are at level 3. Questions 4 to 12 are at level 4.

Answers 1 31 2 25°C

3 25 4 One of these sets of three numbers: 2, 2, 16; 2, 4, 14; 2, 6, 12; 2, 8, 10; 4, 4, 12; 4, 6, 10; 4, 8, 8; 6, 6, 8

5 23°C 6 81

7 e.g. 14, 28 8 15

9 2°C 10 49

11 e.g. 60, 120 12 6

N�.1 Mental test


N�.1 Check up and resource sheets

Answer these questions by writing in your book.

Properties of numbers (no calculator)

1 1995 level 3

Ali drew a picture to show what there is above and below the sea at Aber.

The anchor is at about �40 m.

a What is at about �10 m?

b What is at about �10 m?

c What is about 30 m higher than the chest?

2 2005 KS2 level 3

Which three of these numbers add to make a multiple of 10?

11 12 13 14 15 16 17 18 19

3 1997 KS2 level 3

One of these numbers when multiplied by itself gives the answer 49.Which number is it?

2 3 4 5 6 7 8 9

4 2002 KS2 level 4

Write all the multiples of 8 in this list of numbers.

18 32 56 68 72

Check up

Pearson Education 2008

+20 m

0 m

20 m

40 m

diver

anchor

boat

bird

�sh

chest

hotel

N2.1

Tier 2 resource sheets | N2.1 Properties of numbers | N2.1 Pearson Education 2008 Tier 2 resource sheets | N2.1 Properties of numbers | N2.1

Check up [continued]

Properties of numbers (calculator allowed)

5 2003 Progress Test level 4

The 4th square number is 16.

What is the 5th square number?


Here is a grid with some numbers shaded.

The grid continues.Will the number 35 be shaded?Write Yes or No.Explain your answer.


The diagram shows what is above and below sea level.

a What is about 50 m lower than the bird?

b An octopus is at about �40 m. About how many metres higher is the diver than the octopus?

30

20

10

10

20

30

0 boat

diver

�sh

eel

butter�y

kite

bird

metres

1 2 3 4

5 6 7 8

9 10 11 12


N�.1 Answers

1� | N2.1 Properties of numbers

Class bookEXERCISE 11 a 100 b 225 c 400 d 1225

2 a 8 3 8 5 64 b 9 3 9 5 81 c 13 313 5 169 d 12 3 12 5 144 e 14 3 14 5 196 f 22 3 22 5 484

3 a 72 b 50 c 98 d 45 e 34 f 21 g 4 h 100

4 Other solutions may be possible. a 25 5 16 1 9 b 50 5 25 1 25 c 17 5 16 1 1 d 29 5 25 1 4 e 85 5 49 1 36 f 52 5 36 1 16 g 61 5 25 1 36 h 125 5 100 1 25 i 16 5 25 2 9 j 20 5 36 2 16 k 15 5 16 2 1 l 40 5 49 2 9 m 77 5 81 2 4 n 21 5 25 2 4 o 64 5 100 2 36 p 35 5 36 2 1

5 1 5 12

1 1 3 5 22

1 1 3 1 5 5 32

1 1 3 1 5 1 7 5 42

1 1 3 1 5 1 7 1 9 5 52

1 1 3 1 5 1 … 1 19 5 102 5 100

Extension problem6 There are 8 ways to write 150 as the sum of four squares: 144 1 4 1 1 1 1 121 1 16 1 9 1 4 100 1 25 1 16 1 9 81 1 64 1 4 1 1 81 1 49 1 16 1 4 64 1 49 1 36 1 1 64 1 36 1 25 1 25 49 1 49 1 36 1 16

EXERCISE 21 a 10, 20, 30 b 3, 6, 9 c 6, 12, 18 d 9, 18, 27 e 21, 42, 63

2 a True b True c False d True e False f True g False h True

3 18, 56, 72

4 30, 45, 60

5 a No b Yes c No d Yes e Yes f No

6 54 1 36 or 34 1 56

7 a 15 b 36 c 63 d 25 e 22 f 36 or 72

Extension problem8

EXERCISE 31 a 22°C, 27°C, 21°C, 25°C b 21°C c 27°C 25°C 22°C 21°C 3°C

2 The temperature rose 10 degrees

3 5 degrees colder

Numbers from 30 to 60

Multiples of 5

41 49 47 43 31 37

53 59

35 55 33 51 57

52 46

45

30 6036

5442 48

39

32 34 44 38 58 56

Multiples of 3

Multiples of 2


4 City Temperature

difference (degrees)

Belfast 9

Liverpool 10

Cardiff 10

Newcastle 9

London 10

Plymouth 9

York 9

5 11 cm

6 a 6 degrees b 6 degrees c 12 degrees

7 a 23 b 1 c 0 d 21 e 4 f 25 g 24 h 25

8 a 24 1 2 5 22 b 23 1 4 5 1 c 5 2 7 5 22 d 22 2 4 5 26 e 2 2 4 5 22 f 21 1 3 5 2

9

23 4 21

2 0 –2

1 24 3

How well are you doing?1 28 and 35

2 A 219°C B 16 degrees colder C 222°C

3 25

4 A 4, 16, 36 or 64 B 1, 9, 25, 49 or 81 C Any even number that is not a square number D Any odd number that is not a square number

5 a 11 b 36

6 a 5°C b 29°C, 23°C, 0°C, 6°C

7 a Any multiple of 10 that does not divide exactly by 20, e.g. 10, 30, 50, 70, 90, 110, … b Any multiple of 20 must also be a multiple of 10, so it is not possible to put a number in section B

Home bookTASK 11

Numbers 3 6 7 5 4 9 11 15 12

Squares 9 36 49 25 16 81 121 225 144

2 a 32 1 42 5 52 b 62 1 82 5 102

c 92 1 122 5 152 d 52 1 122 5 132

3 150 5 1 1 49 1 100 150 5 4 1 25 1 121 150 5 25 1 25 1 100

TASK 21 a 5, 10, 25 b 7, 14, 21 c 8, 16, 24 d 31, 62, 93

2 a 40, 50, 60 b 40, 45, 50, 60, 75 c 45, 63, 72, 81 d 16, 24, 32, 40, 72 e 24, 60, 72

3Numbers from 30 to 50

Multiples of 3 Multiples of 4

41 43 46 47 49 50

31 34 35 37 38

3033

394542

443240

36

48


TASK 31 a 22°C,

2 A fall of 9 degrees

3 a 26°C, 24°C, 2°C, 4°C b i 10 degrees ii 2 degrees iii 8 degrees iv 8 degrees

4 50 degrees

Tier 2 CD-ROMCHECK UP1 a Bird b Diver c Fish

2 Any three numbers from 11 to 19 inclusive that sum to a multiple of 10

3 7

4 32, 56, 72

5 25

6 No. 35 will not be shaded because all the shaded numbers are even

7 a Fish b About 30 metres

n properties of numbers

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