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# Algebraic Operations

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1. 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 & Probability

2. 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.

3. Algebraic Operations Int2 Learning Intention Success Criteria • Know the term quartiles. • To simplify algebraic fractions. • Calculate quartiles given a frequency table.

4. Unit 3 Outcome 1 Int2 Adding Algebraic Fractions 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 27-Oct-14

5. Unit 3 Outcome 1 Int2 Adding Algebraic Fractions 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 27-Oct-14

6. Unit 3 Outcome 1 Int2 Adding Algebraic Fractions 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 27-Oct-14

7. 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 27-Oct-14

8. 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 27-Oct-14

9. 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.

10. 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.

11. 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.

12. 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.

13. 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.

14. Statistics Int2 Quartiles from Frequency Tables Now try Exercise 1 Start at 1b Ch11 (page 162)

15. 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.

16. 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

17. 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

18. 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

19. 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

20. Statistics Int2 Quartiles from Cumulative Frequency Table Now try Exercise 2 Ch11 (page 163)

21. Starter Questions Int2 2cm 3cm 29o 4cm A C 70o 53o 8cm B

22. 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.

23. 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

24. 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

25. Quartiles fromCumulative FrequencyGraphs Int2 Now try Exercise 3 Ch11 (page 166)

26. 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.

27. 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.

28. 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.

29. 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

30. 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

31. Standard Deviation For a FULL set of Data Int2 When Standard Deviation is HIGH it means the data values are spread out from the MEAN. When Standard Deviation is LOW it means the data values are close to the MEAN. Mean Mean

32. Standard Deviation Int2 Now try Exercise 4 Ch11 (page 169)

33. 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.

34. 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

35. 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

36. 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

37. 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

38. Standard Deviation For a Sample of Data Int2 Now try Ex 5 & 6

39. Starter Questions Int2 33o

40. 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.

41. This scattergraph shows the heights and weights of a sevens football team Scatter Graphs Write down height and weight of each player. Int2 Construction of Scatter Graph Bob Tim Joe Sam Gary Dave Jim

42. 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

43. 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.

44. 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

45. 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