Mathematical Literacy Level 3 Module 14

Mathematical Literacy NQF 3 Downloaded from hlayiso.com Maths Lit NQF Level 3

Identify relationships and complete patterns to solve problems in a workplace context Module 14 Downloaded from hlayiso.com Maths Lit NQF Level 3

Overview 14.1 Describing features of patterns and relationships in words 14.2 Use given information to establish a specific pattern 14.3 Finding missing terms in a pattern 14.4 Finding the formula for a linear pattern 14.5 Constructing and extending patterns from given formulae Downloaded from hlayiso.com Maths Lit NQF Level 3

Describing features of patterns and Unit 14.1 relationships in words Independent Variables Dependent Variables An independent variable stands A dependent variable depends on alone and is not changed by other the independent variable. A change variables. This is the input value. in the independent variable affects the dependent variable. This is the output value. Downloaded from hlayiso.com Maths Lit NQF Level 3

Describing features of patterns and Unit 14.1 relationships in words Independent variable causes a change in dependent variable, and the dependent variable does not cause a change in independent variable. • For example, the number of items sold causes a change in income, and the income does not cause a change in the number of items sold. Look at what happens to the dependent variable as the independent variable increases. • The dependent variable may increase. • Example: The total cost of items increases as you buy more items. • The dependent variable may decrease. • Example: The number of accidents in the workplace decrease as the number of safety training sessions increases. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.1 page 266 Unit 14.1 1. Philisile is analysing the cost of buying stationery for the company. She looks at the cost of paper and puts the values into a table. She finds a clear pattern. Number of reams of 1 2 3 4 5 paper Cost of paper R150 R300 R450 R600 R750 a) Which variable is dependent and which is independent? Explain. b) Describe whether the pattern is decreasing or increasing. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.1 page 266 Unit 14.1 1. a) The number of reams changes the cost of the paper, and the cost of the paper does not cause a change in the number of reams. Number of reams of paper is independent. The cost of the paper is dependent. It changes as more paper is bought. b) The cost of the paper increases as the number of reams increases. It increases by R150 for each ream of paper added. Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 14.1 Unit 14.1 Complete Exercise 14.1 on page 267 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Direct and inverse proportion Unit 14.1 Direct proportion Indirect proportion Cost Cost Number Number of cans of cans As one value increases, the other As one value increases, the other also increases in the same decreases. The two values proportion. If one value decreases, multiplied together give the same so does the other value, in the same constant value. proportion. • Not all number patterns show direct or inverse proportion. • Increasing or decreasing patterns are not always in proportion. Downloaded from hlayiso.com Maths Lit NQF Level 3

How to test whether relationships Unit 14.1 are proportional Direct proportion: Number of reams of paper 1 2 3 4 5 Cost of paper R150 R300 R450 R600 R750 • The number of reams of paper is doubled from 2 to 4, and the cost of paper is also doubled, from R300 to R600. • So these two variables change in the same proportion. • So the relationship is directly proportional. Downloaded from hlayiso.com Maths Lit NQF Level 3

How to test whether relationships Unit 14.1 are proportional Inverse proportion: Length (m) 1 2 3 4 5 6 Breadth (m) 6 3 2 1,5 1,2 1 • The table shows the possible length and breadth of a room: • Length × breadth for each value is 6. • So the relationship is inversely proportional. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.2 page 269 Unit 14.1 For each of the following examples: a) Complete the sentence. ‘(Independent variable) causes a change in (dependent variable).’ b) Are the following relationships increasing or decreasing? c) Does the relationship show a direct or an inversely proportional relationship between the variables? Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.2 page 269 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Unit 14.1 The image cannot be displayed. Your computer may not have enough memory to open the image, or the image may have been corrupted. Restart your computer, and then open the file again. If the red x still appears, you may have to delete the image and then insert it again. Call time 10 15 20 25 30 35 40 (minutes) Cost (R)) 18 27 36 45 54 63 72 Number of painters 1 2 3 4 Number of days 9 27 3 45 Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.2 page 269 Unit 14.1 Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.2 page 269 Unit 14.1 Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 14.2 Unit 14.2 Complete Exercise 14.2 on page 267 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Use given information to establish Unit 14.2 a specific pattern Constant difference patterns • They are linear patterns. • The difference between terms is a constant number. Example 1: +3 +3 +3 ” 2; 5; 8; 11 An increasing pattern -3 -3 -3 Example 2: 12; 9; 6; 3 A decreasing pattern Downloaded from hlayiso.com Maths Lit NQF Level 3

Use given information to establish Unit 14.2 a specific pattern Constant ratio patterns • You can multiply by the same constant number between terms. • So the ratio between terms is constant. Example 1: 3; 9; 27; 81; … ×3 ×3 ×3 ” An increasing pattern Example 2: 1000; 800; 640; 512; … ×0,8 ×0,8 ×0,8 A decreasing pattern The constant ratio is a decimal or a fraction. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.4 page 272 Unit 14.2 1. Dikeledi is learning to process and input invoices in a new role in her company. During the learning process she notices that she is able to process the following number of invoices per day. Day 1 2 3 4 5 Number of invoices 1 3 9 27 81 a) How do we get each number from the one before it? b) What kind of pattern is this? Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.4 page 272 Unit 14.2 Day 1 2 3 4 5 Number of invoices 1 3 9 27 81 1. a) The difference between terms is not constant, so this sequence does not have a constant difference. We can get each term in the sequence by multiplying the previous term by 3. b) This pattern has a constant ratio between the terms of 3. We can write the ratios of each term to the previous term: 3 = 9 = 27 = 81 = 3 1 3 9 27 Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.4 page 272 Unit 14.2 2. In 1990, when cellphones were first introduced into South Africa, the average monthly bill for cellphone service was R500. Since then, the average monthly bill has decreased every five years: a) How do we get each number from the one before it? b) What kind of pattern is this? Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.4 page 272 Unit 14.2 2. a) This is not a constant difference pattern. If we take the ratio of pairs of consecutive terms we see that: 270 = 243 = 0,9 300 270 You can check this for all the pairs of consecutive terms. There is a constant ratio of 0,9 between the terms. b) This is a constant ratio pattern. (The pattern is obtained by subtracting 10% from the previous term in each case, which is the same as multiplying by 0,9.) Downloaded from hlayiso.com Maths Lit NQF Level 3

Patterns which show trends but no mathematical Unit 14.2 relationship Many situations show repeating patterns and growth trends. For example: • Businesses may base their sales forecasts on trends observed. • Electricity providers know which times of day have the highest demand on electricity supply. • Textbook sales are highest in the first term of the year, and lowest in the middle of the year. • Most shoes sold in the ladies’ footwear department are size 5. • The number of customers at a restaurant per day changes over the week according to a predictable pattern. Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 14.3 Unit 14.2 Complete Exercise 14.3 on page 274 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Finding missing terms in a pattern Unit 14.3 Methods for finding missing terms Work from one term of the pattern to the next, by adding a constant 1 difference or multiplying by a common ratio. If you are given a formula, find the value of the dependent variable 2 from the independent variable. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.6 page 275 Unit 14.3 1. Dimakatso deposits money in his savings account each month according to the following pattern. How much will he deposit in the 6th and 7th months, if he continues the pattern? Month 1 2 3 4 5 6 7 Amount 200 220 242 266,20 292,82 ? ? deposited (R) Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.6 page 275 Unit 14.3 1. Common difference or common ratio? • This is a constant ratio pattern. • 220 ÷ 200 = 1,1 242 ÷ 220 = 1,1 • 266,20 ÷ 242 = 1,1 200 × 1,1 = 220 • 220 × 1,1 = 242 242 × 1,1 = 266,20 • 6th month will be 292,82 × 1,1 = R322,10 • 7th month will be R322,10 × 1,1 = R354,31 Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 14.4 Unit 14.3 Complete Exercise 14.4 on page 276 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Finding the formula for a linear pattern Unit 14.4 Examples of formulae for simple linear patterns: 1. A taxi driver charges R10 for every km travelled. Formula: Total cost of a trip = 10 × number of km C = 10n 2. Chicken pieces cost R90 per kg. Cost = 90 × number of kg C = 90n 3. Palesa makes 15 picture frames per day. Total number of picture frames = 15 × number of days Downloaded from hlayiso.com Maths Lit NQF Level 3

Finding the formula for a linear pattern Unit 14.4 Note (refer to example C = 10n) • The dependent variable is on the LHS on its own. We calculate the dependent variable by putting the value of the independent variable into the formula. • The formulae are all direct proportions. There is no fixed value added to the independent variable. • The constant of proportionality is the number used to multiply the independent variable. • The formulae can be written in words with letters to represent the variables. Remember 90n means the same as 90 × n Downloaded from hlayiso.com Maths Lit NQF Level 3

Formulae with a constant added Unit 14.4 Examples: 1. A taxi driver charges a fixed cost of R16 plus R10 for every km travelled. Total cost of a trip = 16 + (10 × number of km) C = 16 + 10n 2. You run a business making dog kennels to sell. You spend R1 600 to start your business, and it costs you R45 to make each kennel. Total cost for a given number of kennels. Cost = R1 600 + (45 × number of kennels) C = 1 600 + 45n Downloaded from hlayiso.com Maths Lit NQF Level 3

Steps to find the formula for a situation Unit 14.4 Calculate the constant difference. 1 Part of your formula will be the dependent variable multiplied by this difference. Draw up a table of values for at least four values. 2 Try to use consecutive values 1, 2, 3, 4, for the independent variable. Is the relationship directly proportional? 3 If it is, then you can write down the formula. If it is not, carry on with the next step. Calculate the constant difference. 4 Part of your formula will be the dependent variable multiplied by this difference. Downloaded from hlayiso.com Maths Lit NQF Level 3

Steps to find the formula for a situation Unit 14.4 5 The rest of the formula will be the number that is fixed in the situation. 6 Write down the whole formula. Downloaded from hlayiso.com Maths Lit NQF Level 3

Finding a formula from the number sequence Unit 14.4 • Sometimes you only have a number sequence, and the context does not give you the value of the constant. • Use x for the independent variable and y for the dependent variable. • Make a table of values, with x values in the first row y values in the second row. 1 • The input numbers (x) are the positions of the terms in the sequence. • The output numbers (y) are the numbers in the pattern. Downloaded from hlayiso.com Maths Lit NQF Level 3

Finding a formula from the number Unit 14.4 sequence • Calculate the difference between the output terms. 2 • The difference times the input number is part of the formula. • For example, if the difference is 2, then 2x is part of the formula. • Use this as a starting point. Write a formula to link the input number to the output number. 3 • Try it out on a few terms and see if you need to add or subtract a constant number. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.7 page 279 Unit 14.4 Find the formula for each of the relationships described below. 1. Saul borrows R5 000 from his father, and he plans to pay back the debt at R500 per month. Write a formula for the amount still owing after a certain number of months. 2. Thabang starts with 3 push-ups on day 0 and adds 2 push-ups every day. Write a formula for the number of push-ups in relation to the number of days. Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.7 page 279 Unit 14.4 1. Number of 0 1 2 3 4 5 months Amount still 5 000 4 500 4 000 3 500 3 000 2 500 owing(R) Downloaded from hlayiso.com Maths Lit NQF Level 3

Example 14.7 page 279 Unit 14.4 2. Number of days 0 1 2 3 4 5 Number of push-ups 3 5 7 9 11 13 Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 14.5 Unit 14.4 Complete Exercise 14.5 on page 282 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Constructing and extending patterns Unit 14.5 from given formulae • We can use a formula to generate a pattern of numbers. • With a given formula, you can find the value of any term in the pattern without writing down the whole pattern. > Example 14.9 page 283 question 2: The cost of an internet connection is 450 + 350n where n is the number of months. Set up a table showing the cost from 0 to 5 months. Number of people 10 20 30 40 50 Number of cups of coffee 22 42 62 82 102 Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 14.6 Unit 14.5 Complete Exercise 14.6 on page 283 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

Finding unknown values and solving Unit 14.6 problems > Example 14.10 page 284 1. A cellular phone provider calculates cell phone bills on a contract according to the following formula: • C = 148 + 1,5n. • C is the cost in rand while n is the number of minutes used. a) What kind of pattern is this? b) Set up a table of values to answer the question: What will the bill be if the call time for a month is 130 minutes? Downloaded from hlayiso.com Maths Lit NQF Level 3

Finding unknown values and solving Unit 14.6 problems 1. C = 148 + 1,5n. a) This is a linear pattern because each term will increase by 1,5. b) Table of values: Number of 0 10 20 30 130 minutes Cost 148 163 178 193 343 Downloaded from hlayiso.com Maths Lit NQF Level 3

Exercise 14.7 Unit 14.6 Complete Exercise 14.7 on page 285 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

VIDEO: Applications in the workplace Downloaded from hlayiso.com Maths Lit NQF Level 3

Module assessment Module 14 Complete Module assessment on page 286 of your Student’s Book Downloaded from hlayiso.com Maths Lit NQF Level 3

TERMS AND CONDITIONS This PowerPoint Presentation has been developed by Macmillan Education South Africa (Pty) Ltd. All texts, images, videos, animations, audio and vector simulations contained in the slides are property of Macmillan South Africa. Reproducing, reselling and redistributing this material without the written permission of Macmillan South Africa is prohibited. Lecturers are granted permission to: (i) modify the slides by adding and removing content; (ii) print copies of the presentation; and (iii) download and save the slides to a computer or local server. Nothing in this copyright notice constitutes permission to assert or imply that your use of the materials is sponsored or endorsed by Macmillan South Africa. Downloaded from hlayiso.com Maths Lit NQF Level 3