Math 4 part 1

9. to the right 10. to the right 11. to the left E. 12. downward 13. upward 14. upward 15. downwardTry this outLesson 1 1. f(x) = (x + 2)2 – 3 Vertex: (-2, -3) Axis of symmetry: x = -2 Direction of opening: upward Table of values x (x + 2) 2 - 3 f(x) 0 (0 + 2) 2 - 3 1 -1 (-1 + 2) 2 - 3 -2 -2 (-2 + 2 ) 2 - 3 -3 -3 (-3 + 2) 2- 3 -2 -4 (-4 + 2) 2- 3 12. f(x) = -(x – 2)2 + 4 vertex: (2, 4) Axis of symmetry: x = 2 Direction of opening: downward Table of valuesx -(x - 2)2 +4 f(x)4 -(4 – 2)2 + 4 03 -(3 – 2)2 + 4 32 -(2 - 2)2+ 4 41 -(1 – 2)2 + 4 30 -(0 – 2)2 + 4 0 14

3. f(x) = -(x + 1)2 + 3 vertex: (-1, 3) Axis of symmetry: x= -1 Direction of the opening: Upward Table of values f(x) x -(x + 1)2 +3 -1 1 -(1 + 1) 2 + 3 2 0 -(0 + 1)2+ 3 3 -1 -(-1 + 1)2+ 3 2 -2 -(-2 + 1)2+ 3 -1 -3 -(-3 + 1)2+ 34. f(x) = (x - 1)2 + 3 vertex: (1, 3) Axis of symmetry: x = 1 Direction of opening: Upward Table of values f(x) x (x - 1)2 + 3 7 3 (3 – 1) 2 + 3 4 2 (2 – 1 ) 2+ 3 3 1 (1 – 1) 2 + 3 4 0 ( 0 – 1) 2 + 3 7 -1 (-1 – 1) 2 + 35. f(x) = 1/3(x – 1)2 + 2 vertex : (1, 2) Axis of symmetry: x = 1 Direction of opening: Upward Table of valuesx 1/3(x–1)2 + 2 f(x)3 1/3(3–1)2 + 2 3.32 1/3(2–1)2+ 2 2.31 1/3(1–1)2 + 2 20 1/3(0 -1)2+ 2 2.3-1 1/3(-1–1)2+ 2 3.3 15

Lesson 2For each set of functions, tell which graph is narrower or wider. 1. f(x) = 2x2 , wider f(x) = 3x2, narrower 2. f(x) = -1 x2, wider 2 f(x) = -2x2, narrower 3. f(x) = 4x2, narrower f(x) = 1 x2, wider 4 4. f(x) = 5x2, narrower f(x) = 4x2, wider 5. f(x) = -3x2, narrower f(x) = -1 x2, wider 3 6. f(x) = -x2, wider f(x) = -3x2, narrower 7. f(x) = 2x2, wider f(x) = 4x2, narrower 8. f(x) = -5x2, narrower f(x = - 2x2, wider 9. f(x) = 2 x2, narrower 3 f(x) = 1 x2, wider 2 10. f(x) = - 1 x2, wider 3 f(x) = - 1 x2, narrower 2 16

Lesson 3:1. moves to the left1. moves to the right2. moves to the right4 moves to the left12. moves to the left13. moves to the right14. moves to the right15. moves to the right16. moves to the left17. moves to the rightLesson 4A. 1. upwards 2. downwards 3. downwards 4. upwards 5. downwards 6. downwards 7. upwards 8. upwards 9. downwards 10. upwardsLesson 5A.1. y = -(x –2) 2 + 1 vertex : (2,1) Axis of symmetry: x + 2 Direction of the graph: Downward Table of valuesx -(x – 2) 2 + 1 f(x)4 -(4 – 2) 2 + 1 -33 -(3 – 2) 2 + 1 02 -(2 – 2) 2 + 1 11 -(1 – 2) 2 + 1 00 -(0 – 2) 2 + 1 -3 17

2. y = 2(x + 2) 2 –3 vertex: (-2 – 3) Axis of symmetry: x = -2 Direction of the graph: Upwards Table of values x 2(x + 2) 2 - 3 f(x) 0 2(0 + 2) 2 - 3 5 -1 2( -1 + 2) 2 -3 -1 -2 2( -2 +2 ) 2 -3 -3 -3 2( -3 + 2) 2 - 3 -1 -4 2( -4 + 2) 2 - 3 53. y = (x – 1) 2 + 2 Vertex: (1, 2) Axis of symmetry: x = 1 Direction of the graph: Upwards Table of valuesx (x – 1)2 + 2 x3 (3 –1)2+ 2 62 (2 – 1)2 + 2 31 (1 – 1)2 + 2 20 (0 – 1)2 + 2 3-1 (-1 – 1)2 + 2 64. y = (x + 1)2 –2 Vertex: (-1, -2) Axis of symmetry: x = -1 Direction of the graph: Upwards Table of valuesx (x + 1)2 –2 f(x)1 (1 + 1)2 - 2 20 (0 + 1)2 - 2 -1-1 (-1 + 1)2 - 2 -2-2 (-2 + 1)2 - 2 -1-3 (-3 + 1)2 - 2 2 18

5. y = (x + 1) – 4 Vertex: (-1, -4) Axis of symmetry: x = -1 Direction of the graph: Upwards Table of values f(x) x (x + 1)2 – 4 0 1 (1 + 1)2 - 4 -3 0 (0 + 1)2 - 4 -4 -1 (-1 + 1)2 - 4 -3 -2 (-2 + 1)2 - 4 0 -3 (-3 + 1)2 - 4B. 1. y = x2 + 5 2. y = x2 -3 3. y = 2x2 + 2 4. y = (x - 4)2 5. y = 3(x + 2)2 6. y = 2(x + 3)2 + 5 7. y = 3(x – 3)2 – 2 8. y = (x + 3)2 – 4 9. y = -2(x – 5)2 + 3 10. y = -3(x + 2)2 + 4What have you learnedA. 1. Upward 2. Upward 3. Downward 19

B. 4. y = 3(x – 1) 2 - 4 5. y = -x2 + 2x – 1C. E. 6. narrower 12. downward wider 13. upward 14. upward 7. narrower 15. downward wider 20D. 8. to the right 9. to the left 10. to the right 11.to the left

Module 2 Statistics What this module is about This module is about finding the measures of central tendency of groupeddata. As you go over this material, you will develop the skills in computing themean, median and mode of grouped data. What you are expected to learn This module is designed for you to find the measures of central tendencyusing grouped data. Specifically, you are to find the mean, median and mode ofgrouped data.How much do you knowUse the frequency distribution table below to answer the questions.Scores of Students in a Mathematics TestClass Frequency46 – 50 141 – 45 236 – 40 231 – 35 326 – 30 721 – 25 1016 – 20 1311 – 15 66 – 10 41–5 21. What is the class size?2. What is the class mark of the class with the highest frequency?3. What is ∑ fX ?4. Find the mean score.5. What is the median class?

6. Determine the cumulative frequency of the median class.7. Solve for the median score.8. What is the modal class?9. Determine the lower boundary of the modal class10. Compute for the modal score. What you will do Lesson 1 The Mean of Grouped Data Using the Class Marks When the number of items in a set of data is too big, items are grouped forconvenience. The manner of computing for the mean of grouped data is given bythe formula: Mean = ∑(fX) ∑f where: f is the frequency of each class X is the class mark of class The Greek symbol ∑ (sigma) is the mathematical symbol for summation. Thismeans that all items having this symbol are to be added. Thus, the symbol ∑fmeans the sum of all frequencies, and ∑fX means the sum of all the products ofthe frequency and the corresponding class mark.Examples:Compute the mean of the scores of the students in a Mathematics IV test. Class Frequency 46 – 50 1 41 – 45 5 36 – 40 11 31 – 35 12 26 – 30 11 21 – 25 5 16 – 20 2 11 – 15 1 2

The frequency distribution for the data is given below. The columns X andfX are added. Class f X fX 46 – 50 1 48 48 41 – 45 5 43 215 36 – 40 11 38 418 31 – 35 12 33 396 26 – 30 11 28 308 21 – 25 5 23 115 16 – 20 2 18 36 11 – 15 1 13 13 ∑f = 48 ∑fX = 1,549 Mean = ∑(fX) ∑f Mean = 1, 549 48 Mean =32.27The mean score is 32.27.Solve for the mean gross sale of Aling Mely’s Sari-sari Store for onemonth. Sales in Pesos Frequency 4,501 – 5,000 3 4,001 – 4,500 4 3,501 – 4,000 6 3,001 – 3,500 5 2,501 – 3,000 7 2,001 – 2,500 3 1,501 – 2,000 1 1,001 – 1,500 1The frequency distribution for the data is given below. The columns X andfX are added. Sales in Pesos f X fX 1,001 – 1,500 1 1,250 1,250 1,501 – 2,000 1 1,750 1,750 2,001 – 2,500 3 2,250 6,750 2,501 – 3,000 7 2,750 19,250 3,001 – 3,500 5 3,250 16,250 3,501 – 4,000 6 3,750 22,500 4,001 – 4,500 4 4,250 17,000 4,501 – 5,000 3 4,750 14,250 3

∑f = 30 ∑fX = 99,000 Mean = ∑(fX) ∑f Mean = 99, 000 30 Mean = 3,300The mean gross sale is P3, 300.Try this outSolve for the mean of each grouped data using the class marks.Set A1. Scores of Diagnostic Test of IV-Narra Students Score Frequency 36 – 40 1 31 – 35 10 26 – 20 10 21 – 25 16 16 – 20 9 11 – 15 42. Height of IV-1 and IV-2 Students Height in cm Frequency 175 – 179 2 170 – 174 5 165 – 169 8 160 – 164 11 155 – 159 21 150 – 154 14 145 – 169 17 140 – 144 23. Midyear Test Scores of IV-Newton Score Frequency 41 – 45 1 36 – 40 8 31 – 35 8 26 – 30 14 21 – 25 7 16 – 20 2 4

4. Ages of San Lorenzo High School Teachers Age Frequency 21 – 25 5 26 – 30 8 31 – 35 8 36 – 40 11 41 – 45 15 46 – 50 14 51 – 55 12 56 – 60 5 61 – 65 2 5. Pledges to the Victims of Typhoon Mulawin Pledges in Pesos Frequency 9,000 – 9,999 4 8,000 – 8,999 12 7,000 – 7,999 13 6,000 – 6,999 15 5,000 – 5,999 19 4,000 – 4,999 30 3,000 – 3,999 21 2,000 – 2,999 41 1,000 – 1,999 31 0 – 999 14Set B 1. Scores of Periodic Test of IV-Molave Students Score Frequency 46 – 50 2 41 – 45 9 36 – 40 13 31 – 35 11 26 – 30 10 21 – 25 5 5

2. Height of IV-2 StudentsHeight in cm Frequency 175 – 179 3 170 – 174 4 165 – 169 10 160 – 164 9 155 – 159 24 150 – 154 11 145 – 169 13 140 – 144 63. Midyear Test Scores of Students in EnglishClass Frequency91 – 95 186 – 90 681 – 85 776 – 80 471 – 75 766 – 70 1261 – 65 556 – 80 551 – 55 146 – 50 24. Ages of Sta. Barbara High School TeachersClass Frequency21 – 25 426 – 30 1431 – 35 1536 – 40 1141 – 45 1246 – 50 1051 – 55 956 – 60 361 – 65 3 6

5. Monthly Income of the Families of Fourth Year Students Income in Pesos Frequency 9,000 – 9,999 18 8,000 – 8,999 22 7,000 – 7,999 33 6,000 – 6,999 56 5,000 – 5,999 50 4,000 – 4,999 31Set C 1. Scores of Achievement Test in Filipino of IV-Kamagong Students Score Frequency 86 – 90 2 81 – 85 9 76 – 80 8 71 – 75 13 66 – 60 12 61 – 65 6 2. Weight of First Year Students Weight in kg Frequency 75 – 79 1 70 – 74 4 65 – 69 10 60 – 64 14 55 – 59 21 50 – 54 15 45 – 69 14 40 – 44 1 3. Final Test Scores of IV-Rizal Score Frequency 91 – 95 1 86 – 90 5 81 – 85 9 76 – 80 16 71 – 75 6 66 – 70 3 7

4. Ages of Seniro Factory EmployeesAge Frequency21 – 25 826 – 30 1831 – 35 1136 – 40 1641 – 45 1246 – 50 1051 – 55 256 – 60 261 – 65 15. Average Grades of Students of Engineering Block in the First SemesterAverage Grade Frequency 1.01 – 1.50 4 1.51 – 2.00 10 2.01 – 2.50 18 2.51 – 3.00 26 3.01 – 3.50 24 3.51 – 4.00 16 4.01 – 4.50 7 4.51 – 5.00 5 Lesson 2The Mean of Grouped Data Using the Coded Deviation An alternative formula for computing the mean of grouped data makes useof coded deviation:M=ean A.M. +  ∑(fd)  i  ∑f where: A.M. is the assumed mean f is the frequency of each class d is the coded deviation from A.M. i is the class interval 8

Any class mark can be considered as assumed mean. But it is convenientto choose the class mark with the highest frequency. The class chosen to containA.M. is given a 0 deviation. Subsequently, consecutive positive integers are assigned to the classesupward and negative integers to the classes downward. This is illustrated in the next examples using the same data in lesson 1.Examples: Compute the mean of the scores of the students in a Mathematics IV test.Class Frequency46 – 50 141 – 45 536 – 40 1131 – 35 1226 – 30 1121 – 25 516 – 20 211 – 15 1 The frequency distribution for the data is given below. The columns X, dand fd are added. Class f X d fd46 – 50 1 48 3 341 – 45 5 43 2 1036 – 40 11 38 1 1131 – 35 12 33 0 026 – 30 11 28 -1 -1121 – 25 5 23 -2 -1016 – 20 2 18 -3 -611 – 15 1 13 -4 -4A.M. = 33∑f = 48∑fd = -7 i=5M=ean A.M. +  ∑(fd)  i  ∑f  9

Mea=n 33 +  −7  (5)  48  Mean = 33 + (-0.73) Mean = 32.27The mean score is 32.27.Solve for the mean gross sale of Aling Mely’s Sari-sari Store for onemonth. Sales in Pesos Frequency 1,001 – 1,500 1 1,501 – 2,000 1 2,001 – 2,500 3 2,501 – 3,000 7 3,001 – 3,500 5 3,501 – 4,000 6 4,001 – 4,500 4 4,501 – 5,000 3 The frequency distribution for the data is given below. The columns X, dand fd are added. Sales in Pesos f X d fd 1,001 – 1,500 1 1,250 -3 -3 1,501 – 2,000 1 1,750 -2 -2 2,001 – 2,500 3 2,250 -1 -3 2,501 – 3,000 7 2,750 0 0 3,001 – 3,500 5 3,250 1 5 3,501 – 4,000 6 3,750 2 12 4,001 – 4,500 4 4,250 3 12 4,501 – 5,000 3 4,750 4 12 A.M. = 2,750 ∑f = 30 ∑fd = 33 i = 500 M=ean A.M. +  ∑(fd)  i  ∑f  M=ean 2, 750 +  33  (500)  30  Mean = 2,750 + 550 Mean = 3,300The mean gross sale is P3,300. 10

Try this outSolve for the mean of each grouped data using coded deviation.Set A 1. Scores of Diagnostic Test of IV-Narra StudentsScore Frequency36 – 40 131 – 35 1026 – 20 1021 – 25 1616 – 20 911 – 15 42. Height of IV-1 and IV-2 StudentsHeight in cm Frequency 175 – 179 2 170 – 174 5 165 – 169 8 160 – 164 11 155 – 159 21 150 – 154 14 145 – 169 17 140 – 144 23. Midyear Test Scores of IV-NewtonScore Frequency41 – 45 136 – 40 831 – 35 826 – 30 1421 – 25 716 – 20 2 11

4. Ages of San Lorenzo High School Teachers Age Frequency 21 – 25 5 26 – 30 8 31 – 35 8 36 – 40 11 41 – 45 15 46 – 50 14 51 – 55 12 56 – 60 5 61 – 65 2 5. Pledges to the Victims of Typhoon Mulawin Pledges in Pesos Frequency 9,000 – 9,999 4 8,000 – 8,999 12 7,000 – 7,999 13 6,000 – 6,999 15 5,000 – 5,999 19 4,000 – 4,999 30 3,000 – 3,999 21 2,000 – 2,999 41 1,000 – 1,999 31 0 – 999 14Set B 1. Scores of Periodic Test of IV-Molave Students Score Frequency 46 – 50 2 41 – 45 9 36 – 40 13 31 – 35 11 26 – 30 10 21 – 25 5 12

2. Height of IV-2 StudentsHeight in cm Frequency 175 – 179 3 170 – 174 4 165 – 169 10 160 – 164 9 155 – 159 24 150 – 154 11 145 – 169 13 140 – 144 63. Midyear Test Scores of Students in EnglishClass Frequency91 – 95 186 – 90 681 – 85 776 – 80 471 – 75 766 – 70 1261 – 65 556 – 80 551 – 55 146 – 50 24. Ages of Sta. Barbara High School TeachersClass Frequency21 – 25 426 – 30 1431 – 35 1536 – 40 1141 – 45 1246 – 50 1051 – 55 956 – 60 361 – 65 2 13

5. Monthly Income of the Families of Fourth Year Students Income in Pesos Frequency 9,000 – 9,999 18 8,000 – 8,999 22 7,000 – 7,999 33 6,000 – 6,999 56 5,000 – 5,999 50 4,000 – 4,999 31Set C 1. Scores of Achievement Test in Filipino of IV-Kamagong Students Score Frequency 86 – 90 2 81 – 85 9 76 – 80 8 71 – 75 13 66 – 60 12 61 – 65 6 2. Weight of First Year Students Weight in kg Frequency 75 – 79 1 70 – 74 4 65 – 69 10 60 – 64 14 55 – 59 21 50 – 54 15 45 – 69 14 40 – 44 1 3. Final Test Scores of IV-Rizal Score Frequency 91 – 95 1 86 – 90 5 81 – 85 9 76 – 80 16 71 – 75 6 66 – 70 3 14

4. Ages of Seniro Factory EmployeesAge Frequency21 – 25 826 – 30 1831 – 35 1136 – 40 1641 – 45 1246 – 50 1051 – 55 256 – 60 261 – 65 15. Average Grades of Students of Engineering Block in the First SemesterAverage Grade Frequency 1.01 – 1.50 4 1.51 – 2.00 10 2.01 – 2.50 18 2.51 – 3.00 26 3.01 – 3.50 24 3.51 – 4.00 16 4.01 – 4.50 7 4.51 – 5.00 5 Lesson 3The Median of Grouped Data The median is the middle value in a set of quantities. It separates an orderedset of data into two equal parts. Half of the quantities found above the medianand the other half is found below it.In computing for the median of grouped data, the following formula is used:  ∑f − cf   2 Med=ian lbmc +   i  fmc where: lbmc is the lower boundary of the median classf is the frequency of each classcf is the cumulative frequency of the lower class next tothe median class 15

fmc is the frequency of the median class i is the class interval The median class is the class that contains the ∑ f th quantity. The 2computed median must be within the median class.Examples:1. Compute the median of the scores of the students in a Mathematics IV test. Class Frequency 46 – 50 1 41 – 45 5 36 – 40 11 31 – 35 12 26 – 30 11 21 – 25 5 16 – 20 2 11 – 15 1 The frequency distribution for the data is given below. The columns for lband “less than” cumulative frequency are added. Class f lb “<” cf 46 – 50 1 45.5 48 41 – 45 5 40.5 47 36 – 40 11 35.5 42 31 – 35 12 30.5 31 26 – 30 11 25.5 19 21 – 25 5 20.5 8 16 – 20 2 15.5 3 11 – 15 1 10.5 1 Since ∑ f = 48 = 24, the 24th quantity is in the class 31 – 35. Hence, the 22median class is 31 – 35. lbmc = 30.5 ∑f = 48 cf = 19 fmc = 12 i=5 16

 ∑f − cf   2  Med=ian lbmc +   i  fmc  Med=ian 30.5 +  48 −19  (5)  2     12  Med=ian 30.5 + 2.08 Median = 32.58The median score is 32.58.2. Solve for the median gross sale of Aling Mely’s Sari-sari Store for one month.Sales in Pesos Frequency1,001 – 1,500 11,501 – 2,000 12,001 – 2,500 32,501 – 3,000 73,001 – 3,500 53,501 – 4,000 64,001 – 4,500 44,501 – 5,000 3 The frequency distribution for the data is given below. The columns for lband “less than” cumulative frequency are added.Sales in Pesos f lb “<” cf1,001 – 1,500 1 1,000.5 11,501 – 2,000 1 1,500.5 22,001 – 2,500 3 2,000.5 52,501 – 3,000 7 2,500.5 123,001 – 3,500 5 3,000.5 173,501 – 4,000 6 3,500.5 234,001 – 4,500 4 4,000.5 274,501 – 5,000 3 4,500.5 30Since ∑ f = 30 = 15, the 15th quantity is in the class 3,001 – 3,500. 22Hence, the median class is 3,001 – 3,500. 17

lbmc = 3,000.5 ∑f = 30 cf = 12 fmc = 5 i = 500  ∑f − cf   2 Med=ian lbmc +   i  fmc   30 − 12   2 5 M=edian 3, 000.5 +   (500)   Median = 3,000.5 + 300 Median = 3,300.5The median score is 3,300.5.Try this outSolve for the median of each grouped data using coded deviation.Set A 1. Scores of Diagnostic Test of IV-Narra StudentsScore Frequency36 – 40 131 – 35 1026 – 20 1021 – 25 1616 – 20 911 – 15 4 18

2. Height of IV-1 and IV-2 StudentsHeight in cm Frequency 175 – 179 2 170 – 174 5 165 – 169 8 160 – 164 11 155 – 159 21 150 – 154 14 145 – 169 17 140 – 144 23. Midyear Test Scores of IV-NewtonScore Frequency41 – 45 136 – 40 831 – 35 826 – 30 1421 – 25 716 – 20 24. Ages of San Lorenzo High School TeachersAge Frequency21 – 25 526 – 30 831 – 35 836 – 40 1141 – 45 1546 – 50 1451 – 55 1256 – 60 561 – 65 2 19

5. Pledges to the Victims of Typhoon MulawinPledges in Pesos Frequency9,000 – 9,999 48,000 – 8,999 127,000 – 7,999 136,000 – 6,999 155,000 – 5,999 194,000 – 4,999 303,000 – 3,999 212,000 – 2,999 411,000 – 1,999 310 – 999 14Set B 5. Scores of Periodic Test of IV-Molave StudentsScore Frequency46 – 50 241 – 45 936 – 40 1331 – 35 1126 – 30 1021 – 25 52. Height of IV-2 StudentsHeight in cm Frequency 175 – 179 3 170 – 174 4 165 – 169 10 160 – 164 9 155 – 159 24 150 – 154 11 145 – 169 13 140 – 144 6 20

3. Midyear Test Scores of Students in Filipino Class Frequency 73 – 75 1 70 – 72 6 67 – 69 7 64 – 66 4 61 – 63 7 58 – 60 12 55 – 57 5 52 – 54 5 49 – 51 1 46 – 48 2 4. Ages of Tagkawayan High School Teachers Class Frequency 25 – 28 4 29 – 33 14 33 – 36 15 37 – 40 11 41 – 44 12 45 – 48 10 49 – 52 9 53 – 56 3 57 – 60 2 5. Monthly Income of the Families of Fourth Year Students Income in Pesos Frequency 9,000 – 9,999 18 8,000 – 8,999 22 7,000 – 7,999 33 6,000 – 6,999 56 5,000 – 5,999 50 4,000 – 4,999 31Set C 1. Final Grades in Filipino of IV-Kamagong Students Score Frequency 89 – 91 2 86 – 88 9 83 – 85 8 80 – 82 13 77 – 79 12 74 – 76 6 21

2. Weight of First Year StudentsWeight in kg Frequency 93 – 99 1 86 – 92 4 79 – 85 10 72 – 78 14 65 – 71 21 58 – 64 15 51 – 57 14 44 – 50 13. Final Grades of IV-Rizal Students in MathematicsScore Frequency93 – 95 190 – 92 587 – 89 984 – 86 1681 – 83 678 – 80 34. Ages of IRSO Foods Company WorkersAge Frequency27 – 22 833 – 28 1839 – 34 1145 – 40 1651 – 46 1257 – 52 1063 – 58 25. Average Grades of Students of Engineering Block in the First SemesterAverage Grade Frequency 1.01 – 1.50 4 1.51 – 2.00 10 2.01 – 2.50 18 2.51 – 3.00 26 3.01 – 3.50 24 3.51 – 4.00 16 4.01 – 4.50 7 4.51 – 5.00 5 22

Lesson 4 The Mode of Grouped Data The mode of grouped data can be approximated using the followingformula: Mo=de lbmo +  D1  i  D1 + D2 where: lbmo is the lower boundary of the modal class.D1 is the difference between the frequencies of the modal class andthe next lower class.D2 is the difference between the frequencies of the modalclass andthe next upper class.i is the class interval. The modal class is the class with the highest frequency. If binomialclasses exist, any of these classes may be considered as modal class.Examples:1. Compute the mode of the scores of the students in a Mathematics IV test. Class Frequency 46 – 50 1 41 – 45 5 36 – 40 11 31 – 35 12 26 – 30 11 21 – 25 5 16 – 20 2 11 – 15 1 23

he frequency distribution for the data is given below. The column for lb isadded. Class f lb 46 – 50 1 45.5 41 – 45 5 40.5 36 – 40 11 35.5 31 – 35 12 30.5 26 – 30 11 25.5 21 – 25 5 20.5 16 – 20 2 15.5 11 – 15 1 10.5Since class 31 – 35 has the highest frequency, the modal class is 31 – 35.lbmo = 30.5 D1 = 12 – 11 = 1 D2 = 12 – 11 = 1 i=5Mo=de lbmo +  D1  i  D1 + D2 Mo=de 30.5 +  1  (5) 1 + 1 Mode = 30.5 + 2.5 Mode = 33The mode score is 33.2. Solve for the median gross sale of Aling Mely’s Sari-sari Store for one month.Sales in Pesos Frequency1,001 – 1,500 11,501 – 2,000 12,001 – 2,500 32,501 – 3,000 73,001 – 3,500 53,501 – 4,000 64,001 – 4,500 44,501 – 5,000 324

The frequency distribution for the data is given below. The column for lb isadded. Sales in Pesos f lb 1,001 – 1,500 1 1,000.5 1,501 – 2,000 1 1,500.5 2,001 – 2,500 3 2,000.5 2,501 – 3,000 7 2,500.5 3,001 – 3,500 5 3,000.5 3,501 – 4,000 6 3,500.5 4,001 – 4,500 4 4,000.5 4,501 – 5,000 3 4,500.5 Since the class 2,501 – 3,000 has the highest frequency, the modal classis 2,501 – 3,000. lbmo = 2,500.5 D1 = 7 – 3 = 4 D2 = 7 – 5 = 2 i = 500 Mo=de lbmo +  D1  i  D1 + D2  =Mode 2, 500.5 +  4 4 2  (500)  +  Mode = 2,500.5 + 333.33 Mode = 2,833.83The mode score is 2,833.83.Try this outSolve for the mode of each grouped data.Set A 1. Scores of Diagnostic Test of IV-Narra Students Score Frequency 36 – 40 1 31 – 35 10 26 – 20 10 21 – 25 16 16 – 10 9 1–5 4 25

2. Height of IV-2 StudentsHeight in cm Frequency 175 – 179 2 170 – 174 5 165 – 169 8 160 – 164 11 155 – 159 21 150 – 154 14 145 – 169 17 140 – 144 23. Midyear Test Scores of IV-NewtonScore Frequency41 – 45 136 – 40 831 – 35 826 – 30 1421 – 25 716 – 20 24. Ages of San Lorenzo High School TeachersAge Frequency21 – 25 526 – 30 831 – 35 836 – 40 1141 – 45 1546 – 50 1451 – 55 1256 – 60 561 – 65 2 26

5. Pledges to the Victims of Typhoon MulawinPledges in Pesos Frequency9,000 – 9,999 48,000 – 8,999 127,000 – 7,999 136,000 – 6,999 155,000 – 5,999 194,000 – 4,999 303,000 – 3,999 212,000 – 2,999 411,000 – 1,999 310 – 999 14Set B 1. Scores of Periodic Test of IV-Molave StudentsScore Frequency46 – 50 241 – 45 936 – 40 1331 – 35 1126 – 30 1021 – 25 52. Height of IV-2 StudentsHeight in cm Frequency 175 – 179 3 170 – 174 4 165 – 169 10 160 – 164 9 155 – 159 24 150 – 154 11 145 – 169 13 140 – 144 6 27

3. Midyear Test Scores of Students in EnglishClass Frequency91 – 95 186 – 90 681 – 85 776 – 80 471 – 75 766 – 70 1261 – 65 556 – 80 551 – 55 146 – 50 24. Ages of Sta. Barbara High School TeachersClass Frequency21 – 25 426 – 30 1431 – 35 1536 – 40 1141 – 45 1246 – 50 1051 – 55 956 – 60 361 – 65 15. Monthly Income of the Families of Fourth Year StudentsIncome in Pesos Frequency 9,000 – 9,999 18 8,000 – 8,999 22 7,000 – 7,999 33 6,000 – 6,999 56 5,000 – 5,999 50 4,000 – 4,999 31 28

Set C 1. Scores of Achievement Test in Filipino of IV-Kamagong StudentsScore Frequency86 – 90 281 – 85 976 – 80 871 – 75 1366 – 60 1261 – 65 62. Weight of First Year StudentsWeight in kg Frequency 75 – 79 1 70 – 74 4 65 – 69 10 60 – 64 14 55 – 59 21 50 – 54 15 45 – 69 14 40 – 44 13. Final Test Scores of IV-RizalScore Frequency91 – 95 186 – 90 581 – 85 976 – 80 1671 – 75 656 – 70 34. Ages of Seniro Factory EmployeesAge Frequency21 – 25 826 – 30 1831 – 35 1136 – 40 1641 – 45 1246 – 50 1051 – 55 256 – 60 261 – 65 1 29

5. Average Grades of Students of Engineering Block in the First Semester Average Grade Frequency 1.01 – 1.50 4 1.51 – 2.00 10 2.01 – 2.50 18 2.51 – 3.00 26 3.01 – 3.50 24 3.51 – 4.00 16 4.01 – 4.50 7 4.51 – 5.00 5Let’s summarize1. When the number of items in a set of data is too big, items are grouped forconvenience. The manner of computing for the mean of grouped data is given bythe formula: Mean = ∑(fX) ∑f where: f is the frequency of each class X is the class mark of class 2. An alternative formula for computing the mean of grouped data makesuse of coded deviation: M=ean A.M. +  ∑(fd)  i  ∑f  where: A.M. is the assumed mean f is the frequency of each class d is the coded deviation from A.M. i is the class interval Any class mark can be considered as assumed mean. But it is convenientto choose the class mark with the highest frequency. The class chosen to containA.M. is given a 0 deviation. Subsequently, consecutive positive integers areassigned to the classes upward and negative integers to the classes downward. . 30

3. In computing for the median of grouped data, the following formula isused:  ∑f − cf   2  Med=ian lbmc +   i  fmc  where: lbmc is the lower boundary of the median class f is the frequency of each class cf is the cumulative frequency of the lower class next to the median class fmc is the frequency of the median class i is the class interval The median class is the class that contains the ∑ f th quantity. The 2computed median must be within the median class 4. The mode of grouped data can be approximated using the followingformula: Mo=de lbmo +  D1  i  D1 + D2  where: lbmo is the lower boundary of the modal class D1 is the difference between the frequencies of the modal class and the next upper class D2 is the difference between the frequencies of the modal class and the next lower class i is the class interval The modal class is the class with the highest frequency. If binomialclasses exist, any of these classes may be considered as modal class 31

What have you learnedUse the frequency distribution table below to answer the questions. Scores of Students in a Mathematics IV TestClass Frequency46 – 50 141 – 45 236 – 40 331 – 35 1026 – 30 621 – 25 916 – 20 511 – 15 66 – 10 41–5 21. What is the class size?2. What is the class mark of the class with the highest frequency?3. What is ∑ fX ?4. Find the mean score.5. What is the median class?6. Determine the cumulative frequency of the median class.7. Solve for the median score.8. What is the modal class?9. Determine the lower boundary of the modal class10. Compute for the modal score. 32

Answer KeyHow much do you know1. 502. 183. 1,0854. 21.75. 16 – 206. 257. 20.58. 16 – 259. 15.510. 19Try this outLesson 1Set A Set B Set C 1. 24.6 1. 34.7 1. 73.8 2. 156.75 cm 2. 156.88 cm 2. 57.5 kg 3. 30 3. 71.9 3. 79.25 4. 42.38 years 4. 39.63 years 4. 36.75 years 5. P4,019.50 5. P6,589.98 5. 2.97Lesson 2 Set C 1. 73.8Set A Set B 2. 57.5 kg 1. 24.6 1. 34.7 3. 79.25 2. 156.75 cm 2. 156.88 cm 4. 36.75 years 3. 30 3. 71.9 5. 2.97 4. 42.38 years 4. 39.63 years 5. P4,019.50 5. P6,589.98 Set C 1. 81.12Lesson 3 2. 67.83 kg 3. 85.56Set A Set B 4. 34.31 years 1. 24.25 1. 35.05 5. 3.39 2. 156.17 cm 2. 156.58 cm 3. 29.43 3. 60.5 4. 43.17 years 4. 39.05 years 5. P3,666.17 5. P6,428.07 33

Lesson 4Set A Set B Set C 1. 23.19 1. 37.17 1. 71.33 2. 156.56 cm 2. 156.82 cm 2. 56.81 kg 3. 28.19 3. 68.42 3. 78.44 4. 44.5 years 4. 30.5 years 4. 28.44 years 5. P2,332.83 5. P6,206.40 5. 2.905What have you learned1. 472. 333. 1,1594. 24.155. 21 – 256. 267. 24.398. 31 – 359. 30.510. 32.32 34

Module 2 Triangle Trigonometry What this module is about This module is about law of sines. As you go over this material, you willdevelop the skills in deriving the law of sines. Moreover, you are also expected tolearn how to solve different problems involving the law of sines. What you are expected to learn This module is designed for you to demonstrate ability to apply the law ofsines to solve problems involving triangles. How much do you know1. Find the measure of ∠ B in ABC if m ∠ A = 8o and m ∠ C = 97o.2. Find the measure of ∠ A in ABC if m ∠ B = 18o14’ and m ∠ C = 81o41’.3. Two of the measures of the angles of ABC are A = 8o3’ and B = 59o6’. Which is the longest side?4. What equation involving sin 32o will be used to find side b? B 93o50’ c4 A 32o bC5. Given a = 62.5 cm, m ∠ A = 62o20’ and m ∠ C = 42o10’. Solve for side b.6. Solve for the perimeter of ABC if c = 25 cm, m ∠ A = 35o14’ and m ∠ B = 68o.7. Solve ABC if a = 38.12 cm, m ∠ A = 46o32’ and m ∠ C = 79o17’.

8. Solve ABC if b = 67.25 mm, c = 56.92 mm and m ∠ B = 65o16’.9. From the top of a building 300 m high, the angles of depression of two street signs are 17.5o and 33.2o. If the street signs are due south of the observation point, find the distance between them.10. The angles of elevation of the top of a tower are 35o from point A and 51o from another point B which is 35 m from the base of the tower. If the base of the tower and the points of observation are on the same level, how far are they from each other? What you will do Lesson 1 Oblique Triangles An oblique triangle is a triangle which does not contain a right angle. Itcontains either three acute angles (acute triangle) or two acute angles and oneobtuse angle (obtuse triangle). Acute triangle Obtuse triangle If two of the angles of an oblique triangle are known, the third angle canbe computed. Recall that the sum of the interior angles in a triangle is 180o.Examples:Given two of the angles of ABC, solve for the measure of the third angle.1. A = 30o, B = 45o.A + B + C = 180o.30o + 45o + C = 180o. C = 180o – (30o + 45o) C = 180o – 75o C = 105o 2