OBJECTIVE

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OBJECTIVE. REVISION MOD 3. G. Check the side of the slide to see what level you are working at!. F. E. D. C. B. A. A*. INTEGERS. INTEGER is a whole number. HCF / LCM simple numbers – C HCF / LCM complex or more than two numbers – B Recognise prime numbers – C

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REVISION MOD 3

G

### Check the side of the slide to see what level you are working at!

F

E

D

C

B

A

A*

INTEGERS
• INTEGER is a whole number.
• HCF / LCM simple numbers – C
• HCF / LCM complex or more than two numbers – B
• Recognise prime numbers – C
• Write a number as product of its prime numbers – C
• Find the reciprocal of a number - C
Multiples

G

• These are all of the integers that appear in your number’s times table!
• 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36.
• 7, 14, 21, 28, 35, 42, 49, 56, 63, 70, 77, 84
Factors

G

• These are all of the integers that will divide into your number and leave no remainder!
• They are usually listed in pairs!

e.g. the factors of 36 are:

1 & 36

2 & 18

3 & 12

4 & 9

6 & 6

Prime Numbers & Prime Factors
• APRIME NUMBERhasTWO DIFFERENT FACTORS

1 & ITSELF.

The prime numbers less than 30 are ….

2, 3, 5, 7, 11, 13, 17, 19, 23, 29

• APRIME FACTOR, is a factor that is also a prime number.

e.g. factors of12are1, 2 ,3 ,4 , 6 & 12of these2 & 3are prime factors.

12can be written as a product of prime factors…

12 = 2 x 2 x 3 in its INDEX FORM = 22 x 3

C

Highest common Factor
• Thehighest common factor (HCF)of two numbers, is the largest factor common to both.

e.g.factors of 18 are 1,2,3,6,9,18

factors of 30 are 1,2,3,5,6,10,15,30

The highest factor common to both numbers is 6.

We use HCF’s when cancelling fractions!!!

C

Lowest Common Multiple
• The Lowest Common Multiple (LCM) of two numbers, is the smallest number that appears in both time tables.
• The example below is for the 9 & 15 times table…..

e.g. the multiples of 9 are 9,18,27,36,45,54,63,….

the multiples of 15 are 15,30,45,60,…

45 is the lowest common multiple of each sequence of numbers

C

Prime factor product trees
• Products of prime numbers can be written as “trees”.

180

2 x 2 x 3 x 3 x 5 = 180

or; in INDEX FORM

22 x 32 x 5 = 180

90

x

2

45

x

C

2

15

3

x

5

x

x

3

x

x

x

HCF and LCM
• We can use prime factors to find the HCF and LCM…

e.g. 504 = 2 x 2 x 2 x 3 x 3 x 7

700 = 2 x 2 x 5 x 5 x 7

HCF is 2 x 2 x 7 = 28

LCM is 2 x 2 x 2 x 3 x 3 x 5 x 5 x 7 = 12600

504 = 23 x 32 x 7

700 = 22 x 52 x 7

HCF is 22 x 7

LCM is 23 x 32 x 52 x 7

B

This is what’s left from BOTH numbers

when you take out the

HCF

Consecutive Numbers
• A set of 5 consecutive numbers will increase by 5 each time, or are divisible by 5.

e.g. 1+2+3+4+5 = 15

2+3+4+5+6 = 20

If n = starting number, then the next is (n+1), etc .

n + (n+1) + (n+2) + (n+3) + (n+4) = 5n +10

= 5(n+2)

C

Thus5 is always factor of a series of five

consecutive numbers

INDICES
• INDEXis another word forPOWER.
• Recall integer squares / square roots to 15 –D
• Recall integer cube / cube roots to 5 –D
• Use index laws for positive powers –C
• Use index laws for negative powers –B
• Use index laws with simple fractional powers –A
• Use index laws with complex fractional powers –A*
Square Numbers & Cube Numbers
• ASQUARE NUMBERis a NUMBER x ITSELF.

1 x 1 = 1, 2 x 2 = 4, 3 x 3 = 9, 4 x 4 = 16 and so on

Remember the first 15 Square Numbers ….

1,4,9,16,25,36,49,64,81,100,121,144,169,196,225.

• ACUBE NUMBERis a NUMBER x ITSELF x ITSELF.

1 x 1 x 1 = 1, 2 x 2 x 2 = 8, 3 x 3 x 3 = 27, and so on

Remember the first 5 Cube Numbers ….

1, 8, 27,64,125.

D

Square Root

TheNUMBERthat isSQUAREDto make9is3.

3is called theSQUARE ROOTof9and is written√9.

Remember the square roots as the reverse of the

square numbers.

SO √1,√4,√9,√16,√25,√36,√49,√64,√81,√100,√121,√144,√169,√196,√225

are the numbers from 1 to 15.

D

What are Indices?
• An Index is often referred to as a power

For example:

= 53

5 x 5 x 5

= 24

2 x 2 x 2 x 2

= 75

7 x 7 x 7x 7 x 7

5 is the INDEX

7 is the BASE NUMBER

75 & 24 are numbers in INDEX FORM

Rule 1 : Multiplication

26 x 24

= 210

x

2x2x2x2x2x2

2x2x2x2

35 x 37

= 312

x

3x3x3x3x3

C

3x3x3x3x3x3x3

General Rule

am x an = am+n

Rule 2 : Division

44÷42

= 42

÷

4x4x4x4

4x4

26÷23

= 23

÷

2x2x2x2x2x2

C

2x2x2

General Rule

am÷ an = am-n

Rule 3 : Brackets

(26)2

= 26 x 26

= 212

(35)3

= 35 x 35 x 35

= 315

C

General Rule

(am)n = am x n

Rule 4 : Index of 0

How could you get an answer of 30?

35÷ 35

= 35-5

= 30

30 =

1

C

General Rule

a0 = 1

Combining numbers

x 2 x 2 x 2 x 2

5 x 5 x 5

= 53

x 24

We can not write this any more simply

Can ONLY do that if BASE NUMBERS are the same

Putting them together?

26 x 24

23

= 210

23

= 27

35 x 37

34

= 312

34

= 38

C

25 x 23

24 x 22

= 28

26

= 22

..and a mixture…

2a3 x 3a4

= 2 x 3 x a3 x a4

= 6a7

8a6÷ 4a4

= (8 ÷ 4) x (a6 ÷ a4)

= 2a2

2

2

8a6

4a4

C

Works with algebra too!

a6 x a4

= a10

b5 x b7

= b12

c5 x c3

c4

= c8

c4

= c4

C

a5 x a3

a4 x a6

= a8

a10

= a-2

Summary of rules.

1. am x an = am+n

2. am÷ an = am-n

3. (am)n = am x n

4. a1 = a

5. a0 = 1

1

2

General Rule

a-n =

1

an

More rules….. Rule 6 negative indices

25

32

24

16

23

8

22

4

21

2

20

1

2-1

2-2

B

General Rule

a = √a

n

Rule 7 – Fractional Indices

9 x 9 = 91 =9

From Rule 1 & 4

So 9 = √9

A

Rule 8 – Complex Fractional Indices

81= (4√81)³ = (3)³ = 27

General Rule

Treat the bottom as a fractional index so find

root, then use top part as a normal index.

A*

Standard Index Form
• SIF is a way of writing big or small numbers using indices of 10.
• Convert numbers to and from SIF – C
• Use SIF in simple number problems – B
• Use SIF in complex word problems – A

Why is this number very difficult to use?

999,999,999,999,999,999,999,999,999,999

Too large to comprehend

Too large for calculator

To get around using numbers this large, we use standard index form.

But it still not any easier to handle!?!

Look at this

100,000,000,000,000,000,000,000,000,000

At the very least we can describe it as 1 with 29 noughts.

So, 800,000,000,000 = 8 x 1011 in standard index form

Let’s investigate!

Converting large numbers

We can break numbers into parts to make it easier,

e.g. 80 = 8 x 10 and 800 = 8 x 100

C

800,000,000,000 = 8 x 100,000,000,000

And 100, 000,000,000 = 1011

Standard Form (Standard Index Form)

5.3 x 10n

There will also be a power of 10

C

The first part of the number is between

1 and 10

But NOT 10 itself!!

The first number must be a value between

1 and 10

One of the most important rules for writing numbers in standard index form is:

But NOT 10 itself!!

For example, 39 x 106 does have a value but it’s not written in standard index form.

The first number, 39, is greater than 10.

3.9 x 107 is standard index form.

C

Indices of Ten

Notice that the number of zeros

matches the index number

2

100

10

10

10

10

3

1,000

4

10,000

5

100,000

10

So, 45,000,000,000 = 4.5 x 1010

Quick method of converting numbers to standard form

For example,

Converting 45,000,000,000 to standard form

Place a decimal point after the first digit

4.5000000000

Count the number of digits after the decimal point.

C

This is our index number (our power of 10)

And numbers less than 1?

How can we convert 0.067 into standard index form?

0.067 = 6.7 x 0.01

0.01 = 10-2

C

0.067 = 6.7 x 10-2

And numbers less than 1?

How can we convert 0.000213 into standard index form?

0.000213 = 2.13 x 0.0001

0.0001 = 10-4

C

0.000213 = 2.13 x 10-4

56

567

5678

56789

0.56

0.056

0.0056

Write the following in standard form.

0.00056

23

234

4585

4.6

0.78

0.053

0.00123

How to write a number in standard form.

Place the decimal point after the first non-zero digit then multiply or divide it by a power of 10 to give the same value.

= 5.6 x 10 = 5.6 x 101

= 5.67 x 100 = 5.67 x 102

= 5.678 x 1000 = 5.678 x 103

= 5.6789 x 10 000 = 5.6789 x 104

= 5.6  10 = 5.6 x 10-1

C

= 5.6  100 = 5.6 x 10-2

= 5.6  1000 = 5.6 x 10-3

= 5.6  10 000 = 5.6 x 10-4

2.3x 101

2.34x 102

4.585x 103

4.6x 100

7.8x 10-1

5.3x 10-2

1.23x 10-3

Examples:

Exp/EE?

Calculate: 4.56 x 108x 3.7 x 105

+/-

Sharp

Standard Form on a Calculator

You need to use the exponential key (EXP or EE) on a calculator when doing calculations in standard form.

4.56

Exp

8

x

3.7

Exp

5

=

1.6872 x 1014

1.7 x 1014(2sig fig)

Calculate: 5.3 x 10-4 x 2.7 x 10-13

C

5.3

Exp

- 4

x

2.7

Exp

- 13

=

1.431 x 10-16

1.4 x 10-16 (2 sig fig)

Calculate: 3.79 x 1018 9.1 x 10-5

3.79

Exp

18

9.1

Exp

- 5

=

4.2 x 1022(2 sig fig)

### Calculations Using SIF

B

Multiply two numbers

4 x 1018 x 3 x 104

Numbers

Powers of 10

4 x 3

x 1018 x 104

= 12

x 1022

NOT Std Form!

B

= 1.2 x 101

x 1022

= 1.2 x 1023

Complex word problems involving SIF

The mass of the Earth is approximately

6 000 000 000 000 000 000 000 000 kg. Write this number in standard form.

6.0 x 1024

The mass of Jupiter is approximately

2 390 000 000 000 000 000 000 000 000 kg. Write this number in standard form.

2.39 x 1027

A

How many times more massive is Jupiter than Earth?

398

2.39 x 1027 / 6.0 x 1024 =

Complex word problems involving SIF

The mass of a uranium atom is approximately

0. 000 000 000 000 000 000 000 395 g.

Write this number in standard form.

3.95 x 10-22

The mass of a hydrogen atom is approximately

0. 000 000 000 000 000 000 000 001 67 g.

Write this number in standard form.

1.67 x 10-24

How many times heavier is uranium than hydrogen?

A

237

3.95 x 10-22/ 1.67 x 10-24 =

Complex word problems involving SIF

Writing Answers in Decimal Form (Non-calculator)

Taking the distance to the moon is 2.45 x 105 miles and the average speed of a space ship as 5.0 x 103 mph, find the time taken for it to travel to the moon. Write your answer in decimal form.

D

245 000

S

49

hours

S =

so T =

=

=

T

5 000

D

A

Rounding to nearest integer (whole number). G.Rounding to nearest 10 or 100. G.Rounding to given number of decimal places. F.Rounding to given number of significant figures. E.

Rounding.

Roundingto the nearest whole number

G

• Is the arrow nearer to 6, 7 or 8?
• If it is halfway between, then round UP

6 7 8

Roundingto the nearest 10

G

• Is the arrow nearer to 20, 30 or 40?
• If it is halfway between, then round UP

20 30 40

Roundingto the nearest 100

G

• Is the arrow nearer to 400 or 500?
• If it is halfway between, then round UP

400 500

### Decimal Places

F

Round the following number to 1dp

6.348

F

If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything else and we have our answer.

Now look at the number immediately after where we stopped highlighting

Firstly, highlight the number to the first number after the decimal point

So we have 6.3

Round the following number to 1dp

9.2721

F

If this number is a 5, 6, 7, 8 or 9 we increase the last digit by one.

Now look at the number immediately after where we stopped highlighting

Firstly, highlight the number to the first number after the decimal point

So we have 9.2

So 9.2 becomes 9.3

Round the following number to 2dp

If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything else and we have our answer, but it is not, so we round up the number in the second decimal place to give us our answer.

7.456

F

Firstly, highlight the number to the second number after the decimal point

Now look at the number immediately after where we stopped highlighting

7.46

Round the following number to 2dp

If this number is a 0, 1, 2, 3 or 4 we don’t have to do anything else. In this case it is so we have our answer highlighted.

3.992

F

Firstly, highlight the number to the second number after the decimal point

Now look at the number immediately after where we stopped highlighting

3.99

Round the following number to 1dp

9.2721

F

If this number is a 5, 6, 7, 8 or 9 we increase the last digit by one.

Now look at the number immediately after where we stopped highlighting

Firstly, highlight the number to the first number after the decimal point

So we have 9.2

So 9.2 becomes 9.3

Decimal Places (Rounding)

Numbers can be rounded to 1,2, 3 or more decimal places.

Rounding to 1 d.p

4 . 8 3 2 5

5 or bigger ?

5 or bigger ?

5 or bigger ?

4.8

4.8

4.9

F

4. 8 5 2 5

4. 8 4 2 5

No

No

Yes

For example if a case of wine containing 6 bottles costs £25 then you could price a single bottle by calculating £25  6 = £4.166666667. It would be pointless to write out all the numbers on your calculator display. Since we are dealing with money (pounds and pence) we only need 2 decimal places (2 d.p.) So it would be much better to write down £4.17.

Decimal Places

It is often necessary/convenient/sensible to give approximations to real life situations or as answers to certain calculations.

F

Rounding to 1 d.p

4 . 8 3 2 5

4. 8 5 2 5

4. 8 4 2 5

No

No

Yes

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

4.9

4.9

4.9

4.8

4.8

4.9

F

4. 8 6 2 5

4. 8 7 2 5

4. 8 9 2 5

Yes

Yes

Yes

0.29

0.40

1.43

0.56

0.61

5.84

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

Rounding to 2 d.p

1. 4 2 6 1

5. 8 4 2 5

0. 6 0 8 3

F

No

Yes

Yes

0. 2 9 4 3

0. 5 5 5 0

0. 3 9 7 0

Yes

No

Yes

6.295

0.400

1.426

5.401

0.608

5.843

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

5 or bigger ?

Rounding to 3 d.p

1. 4 2 6 1 8

5. 8 4 2 5 4

0. 6 0 8 3 4

F

Yes

No

No

6. 2 9 4 7 1

5. 4 0 0 9 7

0. 3 9 9 7 7

Yes

Yes

Yes

Take Care!
• Round 3.48 to 1 d.p

F

3.5

• Round 3.48 to the nearest whole number

3 (not 4)

E

### Example

E

Round

235440

To 2 significant figures

235440

Underline the 1st 2 digits

Now look at the next digit

E

If this is 5 or more then you must round up

The 3 is changed to a 4

240000

All other digits are changed to zero

2350

2400

437900

440000

69723

70000

43490

43000

• What are these numbers to 2 significant figures?

E

E

For example:

Round

0.004367

to 2 significant figures

0.004367

Underline the 1st 2 digits which are not zero

Look at the next digit along

E

You change the 3 to a 4

If it is 5 or more you add 1 to the previous digit

0.0044

You can ignore any number after the 1st 2 digits which are not zeros

Round the following to 2 significant figures

0.05475

0.055

0.00475

0.0048

0. 45475

0.45

E

Significant figures
• When first identifying significant numbers, zeros at the beginning or end don’t usually count, but zeros ‘inside’ the number do.
• Digits of a number kept in place by zeros where necessary.
• The rounded answer should be a suitable reflection of the original number e.g.

24,579 to 1 s.f could not possibly be 2

24,579 to 1 s.f is 20,000

E

49382.95 to 2 s.f. and1dp =

0.05961 to 1 s.f. and2dp =

374.582 to 3 s.f. and 1dp =

0.0009317 to 2 s.f. and 3dp =

E

49000 49383.0

0.06 0.06

375 374.6

0.00093 0.001

Objective:
• Share a quantity into a given ratio.C.
• Find an unknown number that fits a given ratio.C.

Sharing a quantity into a given ratio

For example, share 36 into the ratio 2 : 7

First ADD the ratio 2 + 7 = 9

Second DIVIDE this answer into the quantity to be shared 36 ÷ 9 = 4

Third MULTIPLY the ratio by this answer 2 X 4 : 7 X 4

This is the answer 8 : 28

C

A

D

M

and

+

÷

X

Share 32 into the the ratio 3 : 5

3 + 5 = 832÷ 8 = 4

3 X 4 = 12 : 5 X 4 = 20

C

Finding an unknown number that fits a given ratio

Example – If the ratio of red beads black beads is 3 : 5, how many black beads will I need for 21 red beads?

Red : Black

3 : 5

21 : ?

C

Red : Black

3 : 5

21 : 35

C

X 7

X 7

Another Example

Red : Black

2 : 7

12 : ?

2 X 6 is 12 so you

multiply 7 by 6 to get the ?

C

X 6

Red : Black

2: 7

12 : 42

X 6

C

X 6

FRACTIONS

Top number is the NUMERATOR, bottom number is the

DENOMINATOR

Find equivalent fractions. F

Simplify a fraction to its lowest form.E

Add and subtract fractions with common denominator.D

Multiply and divide fractions.D

Add and subtract fractions with different denominator.C

Convert to and from fractions, decimals and percentages.D

Be able to convert a recurring decimal to a fraction.C

Part

NUmerator

Denominator

o

Whole

n

What makes a fraction?

1

1

+

3

4

What Happens if
• The two bottom numbers are different

C

Find LCM (Lowest Common Multiple)

Find multiples of 3 and 4

X table shows the multiples of 3

3,6,9,12,15,18,21,……..

X table shows the multiples of 4

4,8,12,16,20,24,…………

C

7

12

Change the denominators into 12

• 3

3

4

C

1/3

1/4

+

+

= ?

C

=

+

3/12 + 4/12 = 7/12

Another example but TAKING AWAY
• The two bottom numbers are different

C

2

1

-

3

6

Find LCM (Lowest Common Multiple)

Find multiples of 3 and 6

X table shows the multiples of 3

3,6,……..

X table shows the multiples of 6

6,…………

C

2

1

-

3

6

2

2

C

1/3

1/3

1/6

1/6

4/6 - 1/6 = 3/6 = 1/2

2/3 - 1/6 = ?

C

2 X 2

1

1

3

-

=

=

4

+

1

=

2

2 X 3

6

6

6

6

Dividing Fractions - Method
• Invert the second fraction and then multiply

D

If it’sfractionsyou’ve got tosum,

the first thing to do ischeckitsbum.

if they’re not, then it’s a pain.

Equal bumsis what youneed,

usetimes tables, yourbumstofeed.

times and divide are not so bad.

Fortimesdo thebottomandthenthetop,

dividedo thesamewiththe2nd bottom up.

.

How can we write 0.3 as a fraction.

.

Let n = 0.3

.

So 10n = 3.3

.

.

So 10n - n = 3.3 – 0.3

B

So 9n = 3

So n = 3 = 1

9 = 3

..

How can we write 0.3451 as a fraction.

..

Let n = 0.3451

..

So 10000n = 3451.451

..

..

So 10000n - 10n = 3451.451 – 3.451

B

So 9990n = 3448

So n = 3448 = 1724

9990 4995