UGRC 120 Numeracy Skills

Transcription

1 UGRC 120 Numeracy Skills Session 1 REVIEWS OF BASIC ALGEBRAIC MATHEMATICS II Lecturer: Dr. Ezekiel N. N. Nortey/Mr. Enoch Nii Boi Quaye, Statistics Contact Information: ennortey@ug.edu.gh/enbquaye@ug.edu.gh College of Education School of Continuing and Distance Education 2014/ /2017

2 Session Overview OVERVIEW Session 1 builds on Session 0. Here, we seek to introduce students to the concept of approximations and standard forms, solving problems involving rates and linear equations involving one or two unknown variables. Dr. E N N Nortey, Statistics Slide 2

3 Session Outline The key topics to be covered in the session are as follows: Approximations, Ratios, Proportions, Percentages and Rates Solving Linear Equations (One or Two Unknown Variables) Formulating and Solving Linear Equations Dr. Ezekiel N N Nortey, Statistics Slide 3

4 Goals and Objectives At the end of the session, the student will be able to: Understand and compute basic measures of rates such as ratios, fractions, proportions and percentages. Perform basic approximations of numbers e.g. decimals and significant figures. Solve linear equations in one or two unknown variables. Formulate and solve linear equations from real world problems. Dr. Ezekiel N N Nortey Slide 4

5 Topic One APPROXIMATIONS, RATIOS, PROPORTIONS, PERCENTAGES & RATES Dr. E N N Nortey, Statistics Slide 5

6 Section 1 In this section we shall discuss the following: Rates and Approximations Solving Linear Equations (One or Two unknown variables) Formulating and Solving Linear Equations (Real world problems) Dr. Ezekiel N N Nortey Slide 6

7 Rates and Approximations Dr. Ezekiel N N Nortey Slide 7

8 Rates and Approximations Dr. Ezekiel N N Nortey Slide 8

9 RATES AND APPROXIMATIONS Dr. Ezekiel N N Nortey Slide 9

10 RATES AND APPROXIMATIONS Dr. Ezekiel N N Nortey Slide 10

11 RATES AND APPROXIMATIONS Dr. Ezekiel N N Nortey Slide 11

12 Rates and Approximations Dr. Ezekiel N N Nortey Slide 12

13 RATES AND APPROXIMATIONS Dr. Ezekiel N N Nortey Slide 13

14 RATES AND APPROXIMATIONS Dr. Ezekiel N N Nortey Slide 14

15 RATES AND APPROXIMATIONS Direct Proportion and Inverse Proportion A direct proportion indicates a relationship between two quantities such that an increase in one leads to a corresponding increase in the other and a decrease in one leads to a corresponding decrease in the other. e.g. consider the relationship between the age of a child and the corresponding age of the father. As age of son increases so does age of father, therefore there is a direct relationship between the age of a son and the corresponding age of a father. Dr. Ezekiel N N Nortey Slide 15

16 Rates and Approximations An inverse proportion indicates a relationship between two quantities such that an increase in one leads to a corresponding decrease in the other and vice versa. E.g. the larger the number of competitors, the lower the price of the commodity. This means that there is an inverse relationship between number of competitors and the price of a commodity. This is true of course in general, but up to a certain limit. Dr. Ezekiel N N Nortey Slide 16

17 Rates and Approximations Refer to session 1 examples and activities NOW and take a look at Example 1.2 Refer to session 1 examples and activities NOW and complete Activity 1.2 Dr. Ezekiel N N Nortey Slide 17

18 Rates and Approximations Dr. Ezekiel N N Nortey Slide 18

19 RATES AND APPROXIMATIONS Dr. Ezekiel N N Nortey Slide 19

20 RATES AND APPROXIMATIONS Refer to session 1 examples and activities NOW and complete Activity 1.3 Dr. Ezekiel N N Nortey Slide 20

21 RATES AND APPROXIMATIONS Dr. Ezekiel N N Nortey Slide 21

22 RATES AND APPROXIMATIONS Refer to session 1 examples and activities NOW and complete Activity 1.4 Dr. Ezekiel N N Nortey Slide 22

23 RATES AND APPROXIMATIONS Rounding of Numbers and Significant Figures It is frequently desirable to abbreviate, or round numbers to make them easier to understand and use. When doing this, however, we must be cautious about the results and their implications. For example, if we are out shopping and we see some item on sale for GHC 9.99 most of us would view this as GHC 10, mentally, we round the original number to the nearest whole number. Dr. Ezekiel N N Nortey Slide 23

24 Rates and Approximations The principle of rounding numbers is based largely on how many significant figures we require. The term significant figures relates to the number of digits in the number that are precise and accurate. Dr. Ezekiel N N Nortey Slide 24

25 RATES AND APPROXIMATIONS The term significant figures relates to the number of digits in the number that are precise and accurate. A budget figure of GH 126, 784 contains six significant digits (all six numbers are accurate). GH 127, 000 contains only three significant figures (the last three zeros are not). Having decided we want the number to be rounded to three significant figures, we then take the last four digits of the original number and round to the nearest whole number: thus becomes Dr. Ezekiel N N Nortey Slide 25

26 Rates and Approximations Dr. Ezekiel N N Nortey Slide 26

27 Rates and Approximations Always round or approximate values after completing the arithmetic and not before and to ensure that the results which have been rounded are acknowledged as such. Dr. Ezekiel N N Nortey Slide 27

28 Rates and Approximations Refer to session 1 examples and activities NOW and complete Activity 1.5 Dr. Ezekiel N N Nortey Slide 28

29 Refer to session 1 examples and activities NOW and take a look at Example 1.3 Refer to session 1 examples and activities NOW and complete Activity 1.6 Dr. Ezekiel N N Nortey Slide 29

30 Topic Three SOLVING LINEAR EQUATIONS IN ONE AND TWO UNKNOWN VARIABLES Dr. E N N Nortey, Statistics Slide 30

31 SOLVING LINEAR EQUATIONS IN ONE & TWO UNKNOWN VARIABLES Introduction Relations between social and economic quantities can be expressed in the form of equations. A better understanding of social systems can be obtained by a study of these relations usually expressed as systems of equations. A study of one or two simultaneous equations will be the subject of interest. Dr. Ezekiel N N Nortey Slide 31

32 SOLVING LINEAR EQUATIONS IN ONE AND TWO UNKNOWN VARIABLES Dr. Ezekiel N N Nortey Slide 32

33 SOLVING LINEAR EQUATIONS IN ONE AND TWO UNKNOWN VARIABLES If an equation is true for all values of a variable, then it is called an identity. We say the identity is true for all values of x. If an equality is not true for any value whatsoever of the variable, it is a null equation. If two equations have the same solution set, then they are said to equivalent equations. Dr. Ezekiel N N Nortey Slide 33

34 SOLVING LINEAR EQUATIONS IN ONE AND TWO UNKNOWN VARIABLES ax b Dr. Ezekiel N N Nortey Slide 34

35 Solving Linear Equations in One and Two Unknown Variables Dr. Ezekiel N N Nortey Slide 35

36 Solving Linear Equations in One and Two Unknown Variables Refer to session 1 examples and activities NOW and take a look at Example 1.4 Refer to session 1 examples and activities NOW and complete Activity 1.7 Dr. Ezekiel N N Nortey Slide 36

37 SOLVING LINEAR EQUATIONS IN ONE AND TWO UNKNOWN VARIABLES Dr. Ezekiel N N Nortey Slide 37

38 Solving Linear Equations in One and Two Unknown Variables An equation with two unknown variables may have no solution or any number of solutions. If there are two unknown variables, two equations are necessary to arrive at the solution. Dr. Ezekiel N N Nortey Slide 38

39 Solving Linear Equations in One and Two Unknown Variables Dr. Ezekiel N N Nortey Slide 39

40 Solving Linear Equations in One and Two Unknown Variables Methods of Solving Simultaneous Equations Elimination by Addition and Subtraction Refer to session 1 examples and activities NOW and take a look at Example 1.5 Elimination by Substitution Refer to session 1 examples and activities NOW and take a look at Example 1.5 Dr. Ezekiel N N Nortey Slide 40

41 Solving Linear Equations in One and Graphical Method Two Unknown Variables If there are two simultaneous equations in two unknown variables; we can draw their graphs on the same rectangular coordinate system. The point of intersection of the two graphs gives the solution to the simultaneous equations. Dr. Ezekiel N N Nortey Slide 41

42 Solving Linear Equations in One and Two Unknown Variables Refer to session 1 examples and activities NOW and complete Activity 1.7 Dr. Ezekiel N N Nortey Slide 42

43 Topic Four FORMULATING AND SOLVING LINEAR EQUATIONS Dr. E N N Nortey, Statistics Slide 43

44 FORMULATING AND SOLVING LINEAR EQUATIONS Formulating Linear Equations The first stage is to translate the problem into a mathematical formulation. Refer to session 1 examples and activities NOW and take a look at Example 1.6, 1.7 Dr. Ezekiel N N Nortey Slide 44

45 Reading List Refer to students to relevant text/chapter or reading materials you will make available on Sakai Dr. Ezekiel N N Nortey Slide 45