Algebraic Operations

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# Algebraic Operations - PowerPoint PPT Presentation

Algebraic Operations. Int2. Unit 3 Outcome 1. Simplify. Adding and Subtracting fractions. Multiply and Divide fractions. Change the subject of the formula. Harder. Harder. Harder. Relative Frequency &amp; Probability. Starter Questions. Int2. Unit 3 Outcome 1.

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### Algebraic Operations

Int2

Unit 3 Outcome 1

Simplify

Multiply and Divide fractions

Change the subject of the formula

Harder

Harder

Harder

Relative Frequency & Probability

### Starter Questions

Int2

Unit 3 Outcome 1

1. Simplify the following fractions :

3a²b4

9b²

= a² b2

3

4x² + x

4x + 1

= x (4x +1)

(4x +1)

= x

(a)

(b)

2.

Calculate √16

= 4

(4 x 4)

(2 x 2 x 2)

3.

Calculate 3√8

= 2

Calculate 1 +1

3 6

= 1 x2=2 +1 = 3= 1

3 2 6 6 6 2

4.

### Algebraic Operations

Int2

Learning Intention

Success Criteria

• Know the term quartiles.
• To simplify algebraic fractions.
• Calculate quartiles given a frequency table.

### Unit 3 Outcome 1

Int2

Example 1b

Example 1a

4 + 2

ff

1 + 4

7 7

Common denominator

LCM = 7

LCM = f

= 4 + 2

f

= 1 + 4

7

The letter or number is the same on the bottom line

= 6

f

= 5

7

17-Nov-14

### Unit 3 Outcome 1

Int2

Example 2b

Example 2a

8 + 5

ww

4 + 5

11 11

Common denominator

LCM = 11

LCM = w

= 8 + 5

w

= 4 + 5

11

The letter or number is the same on the bottom line

= 13

w

= 9

11

17-Nov-14

### Unit 3 Outcome 1

Int2

When the denominator is different

Example 2b

Example 2a

4 + 5

cw

4 + 5

5 10

Common denominator

Cross multiply

Cross multiply

=4 x w + 5 x c

c x w w x c

=4 x10 + 5 x5

5 x10 10x5

The letter or number is the same on the bottom line

= 4w + 5c

cw wc

= 8+ 5= 13

10 10 10

= 4w + 5c

cw

17-Nov-14

### Unit 3 Outcome 1

Compare with real numbers to help

Int2

Subtracting Algebraic Fractions

Example 3b

Example 3a

7 - 4

dd

6 - 2

9 9

The letter or number is the same on the bottom Line

LCM = 7

LCM = f

= 7 - 4

d

= 6 - 2

9

Common denominator

= 3

d

= 4

9

17-Nov-14

### Unit 3 Outcome 1

Int2

Subtracting Algebraic Fractions

Example 2b

Example 2a

7 - 4

dd

6 - 2

9 9

The letter or number is the same on the bottom Line

LCM = 7

LCM = f

= 7 - 4

d

= 6 - 2

9

Common denominator

= 3

d

= 4

9

17-Nov-14

### Statistics

Int2

Quartiles from Frequency Tables

To find the quartiles of an ordered list you consider its length. You need to find three numbers which break the list into four smaller list of equal length.

Example 1 : For a list of 24 numbers, 24 ÷ 6 = 4 R0

6 number

Q1

6 number

Q2

6 number

Q3

6 number

The quartiles fall in the gaps between

Q1 : the 6th and 7th numbers

Q2 : the 12th and 13th numbers

Q3 : the 18th and 19th numbers.

### Statistics

Int2

Quartiles from Frequency Tables

Example 2 : For a list of 25 numbers, 25 ÷ 4 = 6 R1

1 No.

6 number

Q1

6 number

6 number

Q3

6 number

Q2

The quartiles fall in the gaps between

Q1 : the 6th and 7th

Q2 : the 13th

Q3 : the 19th and 20th numbers.

### Statistics

Int2

Quartiles from Frequency Tables

Example 3 : For a list of 26 numbers, 26 ÷ 4 = 6 R2

6 number

1 No.

6 number

Q2

6 number

1 No.

6 number

Q1

Q3

The quartiles fall in the gaps between

Q1 : the 7th number

Q2 : the 13th and 14th number

Q3 : the 20th number.

### Statistics

Int2

Quartiles from Frequency Tables

Example 4 : For a list of 27 numbers, 27 ÷ 4 = 6 R3

6 number

1 No.

6 number

6 number

1 No.

6 number

1 No.

Q2

Q1

Q3

The quartiles fall in the gaps between

Q1 : the 7th number

Q2 : the 14th number

Q3 : the 21th number.

### Statistics

Int2

Quartiles from Frequency Tables

Example 4 : For a ordered list of 34.

Describe the quartiles.

34 ÷ 4 = 8 R2

Q2

8 number

1 No.

8 number

8 number

1 No.

8 number

Q1

Q3

The quartiles fall in the gaps between

Q1 : the 9th number

Q2 : the 17th and 18th number

Q3 : the 26th number.

Statistics

Int2

Quartiles from Frequency Tables

Now try Exercise 1

Start at 1b

Ch11 (page 162)

### Starter Questions

Int2

Statistics

Int2

Quartiles from Cumulative Frequency Table

Learning Intention

Success Criteria

• Find the quartile values from Cumulative Frequency Table.
• 1. To explain how to calculate quartiles from Cumulative Frequency Table.

Statistics

Int2

Quartiles from Cumulative Frequency Table

Example 1 :

The frequency table shows the length

of phone calls ( in minutes) made from

an office in one day.

Cum. Freq.

1

2

2

2

3

5

3

5

10

4

8

18

5

4

22

Statistics

Int2

Quartiles from Cumulative Frequency Table

We use a combination of quartiles from a frequency table

and the Cumulative Frequency Column.

For a list of 22 numbers, 22 ÷ 4 = 5 R2

5 number

1 No.

5 number

Q2

5 number

1 No.

5 number

Q1

Q3

The quartiles fall in the gaps between

Q1 : the 6th number Q1 : 3 minutes

Q2 : the 11th and 12th number Q2 : 4 minutes

Q3 : the 17th number. Q3 : 4 minutes

Statistics

Int2

Quartiles from Cumulative Frequency Table

Example 2 :

A selection of schools were asked

how many 5th year sections they have.

Opposite is a table of the results.

Calculate the quartiles for the results.

Cum. Freq.

4

3

3

5

5

8

6

8

16

7

9

25

8

8

33

Statistics

Int2

Quartiles from Cumulative Frequency Table

We use a combination of quartiles from a frequency table

and the Cumulative Frequency Column.

Example 2 : For a list of 33 numbers, 33 ÷ 4 = 8 r1

1 No.

8 number

Q1

8 number

8 number

Q3

8 number

Q2

The quartiles fall in the gaps between

Q1 : the 8th and 9th numbers Q1 : 5.5

Q2 : the 17th number Q2 : 7

Q3 : the 25th ad 26th numbers. Q3 : 7.5

Statistics

Int2

Quartiles from Cumulative Frequency Table

Now try Exercise 2

Ch11 (page 163)

### Starter Questions

Int2

2cm

3cm

29o

4cm

A

C

70o

53o

8cm

B

Quartiles fromCumulative FrequencyGraphs

Int2

Learning Intention

Success Criteria

• Know the terms quartiles.
• 1. To show how to estimate quartiles from cumulative frequency graphs.
• 2. Estimate quartiles from cumulative frequency graphs.

New Term

Interquartile range

Semi-interquartile range

(Q3 – Q1 )÷2 = (36 - 21)÷2

=7.5

Cumulative FrequencyGraphs

Int2

Quartiles

40 ÷ 4 =10

Q3

Q3 =36

Q2

Q2 =27

Q1

Q1 =21

New Term

Interquartile range

Semi-interquartile range

(Q3 – Q1 )÷2 = (37 - 28)÷2

=4.5

Cumulative FrequencyGraphs

Cumulative FrequencyGraphs

Int2

Q3

= 37

Quartiles

80 ÷ 4 =20

Q2

= 32

Q1

=28

Quartiles fromCumulative FrequencyGraphs

Int2

Now try Exercise 3

Ch11 (page 166)

### Starter Questions

Int2

Standard Deviation

Int2

Learning Intention

Success Criteria

• Know the term Standard Deviation.
• 1. To explain the term and calculate the Standard Deviation for a collection of data.
• Calculate the Standard Deviation for a collection of data.

Standard Deviation

For a FULL set of Data

Int2

The range measures spread. Unfortunately any big

change in either the largest value or smallest score

will mean a big change in the range, even though only

one number may have changed.

The semi-interquartile range is less sensitive to a single number changing but again it is only really based on two of the score.

Standard Deviation

For a FULL set of Data

Int2

A measure of spread which uses all the data is the

Standard Deviation

The deviation of a score is how much the score differs from the mean.

Step 5 :

Take the square root of step 4

√13.6 = 3.7

Standard Deviation is 3.7 (to 1d.p.)

Step 1 : Find the mean

375 ÷ 5 = 75

Step 2 : Score - Mean

Step 4 : Mean square deviation

68 ÷ 5 = 13.6

Step 3 : (Deviation)2

Standard Deviation

For a FULL set of Data

Int2

Example 1 : Find the standard deviation of these five

scores 70, 72, 75, 78, 80.

-5

25

-3

9

0

0

3

9

5

25

0

68

Step 1 : Find the mean

180 ÷ 6 = 30

Step 5 :

Take the square root of step 4

√160.33 = 12.7 (to 1d.p.)

Standard Deviation is £12.70

Step 2 : Score - Mean

Step 4 : Mean square deviation

962 ÷ 6 = 160.33

Step 3 : (Deviation)2

Standard Deviation

For a FULL set of Data

Int2

Example 2 : Find the standard deviation of these six

amounts of money £12, £18, £27, £36, £37, £50.

-18

324

-12

144

-3

9

6

36

7

49

20

400

962

0

Standard Deviation

For a FULL set of Data

Int2

When Standard Deviation

is HIGH it means the data

the MEAN.

When Standard Deviation

is LOW it means the data

values are close to the

MEAN.

Mean

Mean

Standard Deviation

Int2

Now try Exercise 4

Ch11 (page 169)

### Starter Questions

Int2

Standard Deviation

For a Sample of Data

Int2

Learning Intention

Success Criteria

• Construct a table to calculate the Standard Deviation for a sample of data.
• 1. To show how to calculate the Standard deviation for a sample of data.
• 2. Use the table of values to calculate Standard Deviation of a sample of data.

Standard Deviation

For a Sample of Data

We will use this version because it is easier to use in practice !

Int2

In real life situations it is normal to work with a sample

of data ( survey / questionnaire ).

We can use two formulae to calculate the sample deviation.

s = standard deviation

∑ = The sum of

x = sample mean

n = number in sample

Q1a. Calculate the mean :

592 ÷ 8 = 74

Step 2 :

Square all the values and find the total

Step 3 :

Use formula to calculate sample deviation

Q1a. Calculate the sample deviation

Step 1 :

Sum all the values

Standard Deviation

For a Sample of Data

Int2

Example 1a : Eight athletes have heart rates

70, 72, 73, 74, 75, 76, 76 and 76.

4900

5184

5329

5476

5625

5776

5776

5776

∑x = 592

∑x2 = 43842

Q1b(ii) Calculate the sample deviation

Q1b(i) Calculate the mean :

720 ÷ 8 = 90

Int2

Standard Deviation

For a Sample of Data

Example 1b : Eight office staff train as athletes.

Their Pulse rates are 80, 81, 83, 90, 94, 96, 96 and 100 BPM

6400

6561

6889

8100

8836

9216

9216

10000

∑x = 720

∑x2 = 65218

Q1b(iii) Who are fitter the athletes or staff.

Compare means

Athletes are fitter

Q1b(iv) What does the deviation tell us.

Staff data is more spread out.

Standard Deviation

For a Sample of Data

Int2

Athletes

Staff

Standard Deviation

For a Sample of Data

Int2

Now try Ex 5 & 6

### Starter Questions

Int2

33o

Scatter Graphs

Int2

Construction of Scatter Graphs

Learning Intention

Success Criteria

• Construct and understand the Key-Points of a scattergraph.
• To construct and interpret Scattergraphs.

2. Know the term positive and negative correlation.

Scatter Graphs

Write down height and weight of each player.

Int2

Construction of Scatter Graph

Bob

Tim

Joe

Sam

Gary

Dave

Jim

x

x

x

x

x

x

x

x

x

x

x

x

Scatter Graphs

Construction of Scatter Graph

Int2

When two quantities are strongly connected we say there is a

strong correlation between them.

Best fit line

Best fit line

Strong positive

correlation

Strong negative

correlation

Scatter Graphs

Int2

Construction of Scatter Graph

Key steps to:

Drawing the best fitting straight line to a scatter graph

• Plot scatter graph.
• Calculate mean for each variable and plot the
• coordinates on the scatter graph.
• 3. Draw best fitting line, making sure it goes through
• mean values.

Find the mean for theAge and Prices values.

Draw in the best fit line

Price

(£1000)

Age

1

9

1

8

2

8

3

7

3

6

3

5

4

5

4

4

5

2

Mean Age = 2.9

Mean Price = £6000

Scatter Graphs

Int2

Construction of Scatter Graph

Is there

a correlation?

If yes, what kind?

Strong negative correlation

Scatter Graphs

Construction of Scatter Graph

Int2

Key steps to:

Finding the equation of the straight line.

• Pick any 2 points of graph ( pick easy ones to work with).
• Calculate the gradient using :
• Find were the line crosses y–axis this is b.
• Write down equation in the form : y = ax + b

Crosses y-axis at 10

Scatter Graphs

Int2

Pick points

(0,10) and (3,6)

y = 1.38x + 10

Scatter Graphs

Construction of Scatter Graph

Int2

Now try

Int2

### Probability

Int2

Learning Intention

Success Criteria

• Understand the probability line.
• To understand probability in terms of the number line and calculate simple probabilities.
• Calculate simply probabilities.

0

0.5

1

Int2

Impossible

Evens

Certain

Not very

likely

Very

likely

Seeing

a butterfly

In July

School

Holidays

Winning the

Lottery

Baby Born

A Boy

Go back

in time

0

0.5

1

Int2

Impossible

Evens

Certain

Not very

likely

Very

likely

It will

Snow in winter

Homework

Every week

Everyone getting

100 % in test

Toss a coin

That land

Going without

Food

for a year.

### Probability

Int2

We can normally attach a value

to the probability of an event happening.

To work out a probability

P(A) =

Probability is ALWAYS in the range 0 to 1

### ProbabilityNumber Likelihood Line

1

2

3

4

5

6

8

7

0

0.5

1

Int2

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

Impossible

Evens

Certain

8

P =

= 1

Q. What is the chance of picking a number between 1 – 8 ?

8

4

Q. What is the chance of picking a number that is even ?

= 0.5

P(E) =

8

Q. What is the chance of picking the number 1 ?

1

= 0.125

P(1) =

8

### ProbabilityLikelihood Line

0

0.5

1

Int2

52 cards in a pack of cards

0.1

0.2

0.3

0.4

0.6

0.7

0.8

0.9

Impossible

Evens

Certain

Not very

likely

Very

likely

26

= 0.5

P (Red) =

Q. What is the chance of picking a red card ?

52

13

Q. What is the chance of picking a diamond ?

= 0.25

P (D) =

52

4

Q. What is the chance of picking ace ?

P (Ace) =

= 0.08

52

Probability

Int2

Now try Ex 8

Ch11 (page 177)

Int2

### Relative Frequencies

Int2

Learning Intention

Success Criteria

• Know the term relative frequency.
• To understand the term relative frequency.
• Calculate relative frequency from data given.

Relative Frequencies

Relative Frequency always added up to 1

Int2

Relative Frequency

How often an event happens compared

to the total number of events.

Example : Wine sold in a shop over one week

0.5

180 ÷ 360 =

0.25

90 ÷ 360 =

0.25

90 ÷ 360 =

1

360

### Relative Frequencies

Int2

Example

Calculate the relative frequency for boys and girls

born in the Royal Infirmary hospital in December 2007.

Relative Frequency adds up to 1

500

1

0.4

0.6

Relative Frequencies

Int2

Now try Ex 9

Ch11 (page 179)

Int2

### Probability from Relative Frequency

Int2

Learning Intention

Success Criteria

• Know the term relative frequency.
• To understand the connection of probability and relative frequency.
• Estimate probability from the relative frequency.

### Probability from Relative Frequency

When the sum of the frequencies is LARGE the relative frequency is a good estimate of the probability of an outcome

Int2

Example 1

Three students carry out a survey to study left

handedness in a school. Results are given below

### Probability from Relative Frequency

Who’s results would you use as a estimate of the probability of a house being alarmed ?

Megan’s

Int2

Example 2

What is the probability that a house is alarmed ?

Three students carry out a survey to study peoples

favourite colours. Results are given below

0.4

Probability from Relative Frequency

Int2

Now try Ex 10

Ch11 Start at Q2 (page 181)