Mathematical literacy 4

Module 2: Patterns and relationships

Module 2: Patterns and relationships At the end of this module, you will be able to: identify and extend patterns for different relationships in your daily life. identify and use information responsibly from different representations of relationship patterns to solve problems in your daily life. translate between different representations of relationships found in your daily life.

At the end of this module, you will be able to:

identify and extend patterns for different relationships in your daily life.

identify and use information responsibly from different representations of relationship patterns to solve problems in your daily life.

translate between different representations of relationships found in your daily life.

1. IDENTIFY AND EXTEND PATTERNS FOR DIFFERENT RELATIONSHIPS IN THE WORKPLACE At the end of this outcome, you will be able to: investigate and extend numerical and geometric patterns and identify trends in data describe patterns and trends in words and/or through formulae generate patterns from descriptions of them

At the end of this outcome, you will be able to:

investigate and extend numerical and geometric patterns and identify trends in data

describe patterns and trends in words and/or through formulae

generate patterns from descriptions of them

1.1 Numerical and geometric patterns investigated and extended y = 2x Input value (x) 1 2 3 4 Output value (y) 2 4 6 8

Example If 6 people can sit at one table, then 12 people can sit at 2 tables and 18 people at three tables. Let’s tabulate the information, and then you write the formula! Input value 1 2 3 4 Output value 6 12 18 Formula y =

If 6 people can sit at one table, then 12 people can sit at 2 tables and 18 people at

three tables.

Let’s tabulate the information, and then you write the formula!

Activity 1: Arithmetic patterns A ticket on Busliner, a long distance inter-city bus service is worked out by asking R25 per ticket as well as R0,75 per kilometre. Complete the following table and write a formula for the calculation of the output value, which is the cost of the ticket. Formula: How do you think this formula was determined? In your answer mention factors that would influence the Busliner’s cost determination; and experiment with different combinations of basic cost and cost per kilometre. Complete the table: Cost of ticket = R25 + 0,75 x (no of km) Km travelled 100 200 300 400 500 600 1000 Cost (R) R100 R175 R250 R325 R400 R475 R550

A ticket on Busliner, a long distance inter-city bus service is worked out by asking R25 per ticket as well as R0,75 per kilometre. Complete the following table and write a formula for the calculation of the output value, which is the cost of the ticket.

Formula:

How do you think this formula was determined? In your answer mention factors that would influence the Busliner’s cost determination; and experiment with different combinations of basic cost and cost per kilometre.

Complete the table:

Activity 1: Arithmetic patterns A paver needs 12 face bricks per square metre to build a pavement pattern that he has designed. Face bricks cost R2,95 each Formula for number of face bricks per m2: Number of bricks = Formula for cost per square meter of paving: Cost = Complete the table The paver can complete five square metres of paving per day. He has to complete 125 square metres of paving. How long will it take him to complete the job? He is given ten days to complete the job. Draw a table which shows how many workers he will need to help him complete the 125 m 2 within the given time. 12 x m 2 R35,4 x m 2 It will take him 25 days. He will need 2 additional helpers, which will allow them to complete it in 8 1/3 days Number of sq metres of pavements 1 2 3 4 5 10 50 Number of face bricks Cost per square metre 12 24 36 48 60 120 600 R35,40 R70,80 R106,20 R141,60 R177 R354 R1770

A paver needs 12 face bricks per square metre to build a pavement pattern that he has designed. Face bricks cost R2,95 each

Formula for number of face bricks per m2: Number of bricks =

Formula for cost per square meter of paving: Cost =

Complete the table

The paver can complete five square metres of paving per day. He has to complete 125 square metres of paving. How long will it take him to complete the job? He is given ten days to complete the job. Draw a table which shows how many workers he will need to help him complete the 125 m 2 within the given time.

Activity 1: Arithmetic patterns Currency conversion: $1 = R7,66 Formula: Value in Rand: __________ Complete the table: $0.1305 76,60 766,00 7 660 76 600 766 000 7 600 000 Value in US$ 1 10 100 1000 10 000 100 000 1000 000 Amount in ZAR 7.66

Currency conversion: $1 = R7,66

Formula: Value in Rand: __________

Complete the table:

Geometric patterns What is a geometric pattern? A geometric pattern is where there is a constant ratio involved between two consecutive numbers of the pattern. This means that every next number in the sequence divided by the previous number will give the same constant value. Example: 2; 4; 8; 16; 32; 64; 128 Formula y = 2 x

What is a geometric pattern?

A geometric pattern is where there is a constant ratio involved between two consecutive numbers of the pattern. This means that every next number in the sequence divided by the previous number will give the same constant value.

Example: 2; 4; 8; 16; 32; 64; 128

Formula y = 2 x

Example If you put R50 000 in a fixed deposit account in the bank, you are actually lending your money to the bank and it is in your interest to be paid for this action. The bank pays you interest on your money. One South African bank pays 10% per year compound interest. All banks have slightly different interest rates. The calculation of this compound interest forms a geometric pattern. The full starting amount is R50 000; this is called 100%. Add the interest of the year to this full starting amount: 100% + 10% = 110% = which is the same as 1,1. Then, each year the amount in the bank must be multiplied by 1,1 The 1,1 is the constant ratio Year 1: R5 000 x 1,1 = R5 500 Year 2 : R5 500 x 1,1 = R6 050 Year 3: R6 050 x 1,1 = R6 655

If you put R50 000 in a fixed deposit account in the bank, you are actually lending your money to the bank and it is in your interest to be paid for this action. The bank pays you interest on your money.

One South African bank pays 10% per year compound interest. All banks have slightly different interest rates.

The calculation of this compound interest forms a geometric pattern.

The full starting amount is R50 000; this is called 100%.

Add the interest of the year to this full starting amount: 100% + 10% = 110% = which is the same as 1,1.

Then, each year the amount in the bank must be multiplied by 1,1

The 1,1 is the constant ratio

Year 1: R5 000 x 1,1 = R5 500

Year 2 : R5 500 x 1,1 = R6 050

Year 3: R6 050 x 1,1 = R6 655

Activity 2 – constant difference patterns and constant ratio patterns Write down five numbers that you think should logically follow on the numbers in the following sets. 250; 500; 750; 1000; ______; _____; _____; _____ 250; 500; 1000; ____; ____; ____; _____; _____ 250; 275; 302.50; 332.75; _____; _____; ______; _____ 10 000; 11 000; 12 100; _____; ____; ______; _______; ______ _______; _______; 161 051; 16 641; 1 331; _____; ____; _____ Write down what you see as the constant difference or the constant ratio in each of the above sets of numbers 1250 1500 1750 2000 2000 4000 8000 16 000 32 000 336.025 402,628 442,890 487,179 13 310 14 641 16 105,1 17 715,61 19 487,17 19 487 171 1 715 561 121 11 1

Write down five numbers that you think should logically follow on the numbers in the following sets.

250; 500; 750; 1000; ______; _____; _____; _____

250; 500; 1000; ____; ____; ____; _____; _____

250; 275; 302.50; 332.75; _____; _____; ______; _____

10 000; 11 000; 12 100; _____; ____; ______; _______; ______

_______; _______; 161 051; 16 641; 1 331; _____; ____; _____

Write down what you see as the constant difference or the constant ratio in each of the above sets of numbers

Activity 2: Constant difference patterns and constant ratio patterns Order the following numbers to form a patterned sequence of numbers from the smallest number first to the largest number last 0,5358; 0,5433; 0,5458; 0,5383; 0,5483; 0,5508; 0,5408 6375 thousand; 51 million; 0,102 billion; 12 750 000; 2,55 x 10 7 c. 11H30; 10H10; 12H50; 12H10; 10H50; 13H30 0,5358; 0,5383; 0,5408; 0,5433; 0,5458; 0,5483; 0,5508 6,375 million; 12,750 million; 12,500 million; 25,500 million; 51million; 102 million 10:10; 10:50; 11:30; 12:10; 12:50

Order the following numbers to form a patterned sequence of numbers from the smallest number first to the largest number last

0,5358; 0,5433; 0,5458; 0,5383; 0,5483; 0,5508; 0,5408

6375 thousand; 51 million; 0,102 billion; 12 750 000; 2,55 x 10 7

c. 11H30; 10H10; 12H50; 12H10; 10H50; 13H30

Activity 2: Constant difference patterns and constant ratio patterns 4. If a certain bank decides that annual interest will be paid at a compound rate of 10% of the amount deposited, calculate the amounts in the bank after one, two, three and four years. 11 000 12 100 13 310 14 641 16500 18150 19965 R41772,48 R21961,5 R62658,72 22000 24200 26620 29282 83554,96 Amount of deposit Amount after one year Amount after two years Amount after three years Amount after four years 15 th term 10 000 1,1 x 10 000 1.1 2 x 10 000= 1.1 3 x 10 000= 1.1 4 X10 000 15 000 1,1 x 15 000 1.1 2 x 15 000= 1.1 3 x 15 000= 1.1 4 X 15 000 20 000 1,1 x 20 000 1.1 2 x 20 000= 1.1 3 x 20 000= 1.1 4 X 20 000

4. If a certain bank decides that annual interest will be paid at a compound rate of 10% of the amount deposited, calculate the amounts in the bank after one, two, three and four years.

Activity 2: Constant difference patterns and constant ratio patterns A friend tells you that he will loan you money at a simple interest rate of 10% p.a. Fill in the table: R1000 R2000 R3000 R15000 R1500 R3000 R4500 R22500 R2000 R4 000 R6 000 R 30 000 Loan amount Cost of loan for 1 year Cost of loan for 2 years Cost of loan for three years Cost of loan for 15 years 10 000 0,1 X 10 000 x 1 0,1 X 10 000 x 2 0,1 x 10 000 x 3 15 000 20 000

A friend tells you that he will loan you money at a simple interest rate of 10% p.a. Fill in the table:

Activity 3 – Patterns in Asanda’s working day Remember Asanda has a Shiny Car valet service? Well, Asanda is severely stressed by his not-too-strong finances. He starts smoking. He finds that he needs a cigarette every two hours. He knows that cigarette smoking is bad for his health and he thinks that smoking “lights” is perhaps better for him than smoking the stronger cigarettes. How many cigarettes does he smoke per week? How many packets of cigarettes does he smoke per year? Find out what a packet of “lights” costs and then calculate how much Asanda spends per year on cigarettes? Draw up a table of the weekly number of packets against the total cost of the packets. 56 per week 2920 per year R3066 per year Week number 1 5 10 15 20 25 50 Total

Remember Asanda has a Shiny Car valet service? Well, Asanda is severely stressed by his not-too-strong finances. He starts smoking. He finds that he needs a cigarette every two hours. He knows that cigarette smoking is bad for his health and he thinks that smoking “lights” is perhaps better for him than smoking the stronger cigarettes.

How many cigarettes does he smoke per week?

How many packets of cigarettes does he smoke per year?

Find out what a packet of “lights” costs and then calculate how much Asanda spends per year on cigarettes?

Draw up a table of the weekly number of packets against the total cost of the packets.

Activity 3 – Patterns in Asanda’s working day If Asanda rather takes this yearly cigarette money and month by month places it in an investment account at 10% interest, calculate how much he will have at the end of one year. Draw up a table of this savings exercise. After one year of not smoking, Asanda decides to start smoking again, but he leaves the saved money in the investment account. He forgets about it for 15 years. Draw up a table to show how his money grows during the 5 years if the interest is compounded on a monthly basis. Look at module on finance for information on compound interest.

If Asanda rather takes this yearly cigarette money and month by month places it in an investment account at 10% interest, calculate how much he will have at the end of one year.

Draw up a table of this savings exercise.

After one year of not smoking, Asanda decides to start smoking again, but he leaves the saved money in the investment account. He forgets about it for 15 years. Draw up a table to show how his money grows during the 5 years if the interest is compounded on a monthly basis. Look at module on finance for information on compound interest.

1.1.2 Direct and inverse relationships and trends Examples of direct relationships As the volume of petrol that you put into the car’s tank increases, so does the amount of money that you have to pay. We say that the cost is directly related or dependent on the litres of petrol put into the tank. As the distance that you travel increases, so both the time, as well as the cost of the journey increase. Examples of an inverse relationship As the petrol price shows a steady increase over the years, the volume of petrol that you can purchase for the same amount, shows a steady decrease. The pattern is: The input value times the output value gives a constant number.

Examples of direct relationships

As the volume of petrol that you put into the car’s tank increases, so does the amount of money that you have to pay. We say that the cost is directly related or dependent on the litres of petrol put into the tank.

As the distance that you travel increases, so both the time, as well as the cost of the journey increase.

Examples of an inverse relationship

As the petrol price shows a steady increase over the years, the volume of petrol that you can purchase for the same amount, shows a steady decrease.

The pattern is: The input value times the output value gives a constant number.

Inverse relationship

Activity 4 State whether the following tables show a direct or an inverse relationship. Complete the tables. 10 20 Price of house by estate agent in one year (R) R8 500 000 R4 250 000 R1 700 000 R850 000 R425 000 Number of the specific houses sold per year 1 2 5

State whether the following tables show a direct or an inverse relationship. Complete the tables.

Activity 4 Distance travelled = 1200km Inverse 60 30 20 15 10 Speed (km/h) 20 40 60 80 100 120 Time of journey 12

Distance travelled = 1200km

Activity 4 R148,50 R445,50 R594,00 Direct litres of petrol 1 15 45 60 Total cost 9,90

Activity 4 Direct R79,20 R99,00 R118,80 R138,60 Petrol consumption 8 10 12 14 Cost per 100km

Activity 4 Look at the table on cash loans and answer the questions: Cash loan amount R4 000 R6 000 R8 000 Monthly loan repayment options Within 12 months R466 R679 R892 Within 24 months R290 R421 R551 Within 36 months R234 R339 R443

Look at the table on cash loans and answer the questions:

Activity 4 Look at the table on cash loans and answer the questions: Is there a constant difference or a constant ratio between the loan repayment amounts for the three different payment options? Is there a direct or inverse relationship between the loan repayment amounts with regard to the repayment periods? Calculate how much a loan of R8 000 will cost if repaid over 3 years. Calculate the % increase of the cost of the loan with regard to the loan amount. Sketch three graphs Constant difference Inverse R443 x 36 months = R15948 R15948 ÷ 8000 x 100 = 199.35

Look at the table on cash loans and answer the questions:

Is there a constant difference or a constant ratio between the loan repayment amounts for the three different payment options?

Is there a direct or inverse relationship between the loan repayment amounts with regard to the repayment periods?

Calculate how much a loan of R8 000 will cost if repaid over 3 years.

Calculate the % increase of the cost of the loan with regard to the loan amount.

Sketch three graphs

Activity 4 Look at the following table on a cash loan of R10 000. Monthly payment options dependent on repayment period. R13 260 R16 334 R19 692 R3 260 R6 334 R9 692 Repayment period 12 months 24 months 36 months R10 000 loan R1 105 R681 R547 Actual repayment amount = months x monthly repayment amount Cost of the loan

Look at the following table on a cash loan of R10 000. Monthly payment options dependent on repayment period.

Activity 4 Is there a positive or negative relationship between the repayment period and the monthly repayment amounts? Complete the table with the actual repayment amount as well as the cost of the loan for different time periods. Is it better to repay within 12 months or 36 months? Give a reason for your answer. d. Get a recent micro loan repayment table from a bank and from cash loan business. Compare the two options on a graph. Negative 12 months, the loan is cheaper since you pay less interest

Is there a positive or negative relationship between the repayment period and the monthly repayment amounts?

Complete the table with the actual repayment amount as well as the cost of the loan for different time periods.

Is it better to repay within 12 months or 36 months? Give a reason for your answer.

d. Get a recent micro loan repayment table from a bank and from cash loan business. Compare the two options on a graph.

Activity 5 In the following sketch, there are four graphs. Describe the trends of each graph.

In the following sketch, there are four graphs. Describe the trends of each graph.

1.1.3 Putting patterns into words an the other way around A simple flow diagram: Input value -> × 0,2 -> + 50 = Output value This can also be written as an algebraic formula: y = 0,2 x + 50 The value that you put into the formula is the independent variable (usually called the x-value) and the value that is produced by the formula, is the dependent variable (usually called the y-value). Its value depends on the value that is fed into the formula

A simple flow diagram:

Input value -> × 0,2 -> + 50 = Output value

This can also be written as an algebraic formula:

y = 0,2 x + 50

The value that you put into the formula is the independent variable (usually called the x-value) and the value that is produced by the formula, is the dependent variable (usually called the y-value). Its value depends on the value that is fed into the formula

1.1.3 Putting patterns into words an the other way around Put into words y = 0,2x + 50 would read: A certain number must be multiplied by 0,2 and 50 must be added to the answer to obtain the output value or answer. Put in other words it could mean: A waitress at a 5-star restaurant earns a basic amount per day of R50 plus 20% commission on the total income that she generated.

Put into words y = 0,2x + 50 would read:

A certain number must be multiplied by 0,2 and 50 must be added to the answer to obtain the output value or answer.

Put in other words it could mean: A waitress at a 5-star restaurant earns a basic amount per day of R50 plus 20% commission on the total income that she generated.

1.1.3 Putting patterns into words an the other way around Put into words y = 0,2x + 50 would read: A certain number must be multiplied by 0,2 and 50 must be added to the answer to obtain the output value or answer. Put in other words it could mean: A waitress at a 5-star restaurant earns a basic amount per day of R50 plus 20% commission on the total income that she generated.

Put into words y = 0,2x + 50 would read:

A certain number must be multiplied by 0,2 and 50 must be added to the answer to obtain the output value or answer.

Put in other words it could mean: A waitress at a 5-star restaurant earns a basic amount per day of R50 plus 20% commission on the total income that she generated.

1.1.3 Putting patterns into words and the other way around Input values 0 300 600 900 1200 1500 1800 2100 2400 Output values 50 110

Activity 6 For each table in this activity identify the independent and dependent variables. Complete the flow diagrams and write in words what the flow diagrams mean: R310 R370 R430 R490 R730 Input value (R) = customers amount Output value (r) = earnings of waitress 300 x0.2 +250 = 600 900 1200 2400

For each table in this activity identify the independent and dependent variables.

Complete the flow diagrams and write in words what the flow diagrams mean:

Activity 6 Metro bus ticket: 1,75 - 4,25 4,25 – 5,75 5,75 – 7,25 7,25 – 8,75 8,75 – 10,25 0-5 km = zone 1 x0.5 +1.75 = 5-8 km = zone 2 8-11km = zone 3 11-14km = zone 4 14-17km = zone 5

Activity 6 Complete the following tables according to the given formulae or written instructions y=25x + 5 Also write this instruction in words (think of an example where this formula could have been applied) 30 55 80 105 130 155 180 205 230 input 1 2 3 4 5 6 7 8 9 output

Complete the following tables according to the given formulae or written instructions

y=25x + 5

Also write this instruction in words (think of an example where this formula could have been applied)

Activity 6 Complete the operating instructions, give the correct formula and then complete the tables: Divide the x-value by _______ and add ___ to the answer: Formula: _______________ 1/3 5 5x + 3 110 140 170 200 230 260 x-value 5 15 25 35 45 55 65 75 85 y-value 20 50 80

Complete the operating instructions, give the correct formula and then complete the tables:

Divide the x-value by _______ and add ___ to the answer:

Formula: _______________

Activity 6 Multiply the x-value by ________ and subtract _________ from the answer. Formula: _______________ If you cannot get the formula, think of a way to extrapolate, i.e. predict what the other y-values must be. 5 3 5x + 3 73 88 103 118 133 x-value 3 6 9 12 15 18 21 24 27 y-value 13 28 43 58

Multiply the x-value by ________ and subtract _________ from the answer.

Formula: _______________

If you cannot get the formula, think of a way to extrapolate, i.e. predict what the other y-values must be.

Activity 7 Sketch the five graphs of the information in activity 6. If you have forgotten how to sketch graphs, look at section 3.2.

Sketch the five graphs of the information in activity 6. If you have forgotten how to sketch graphs, look at section 3.2.

1.1.4 Make representations of the relationships Continuous relationships Discrete relationships Step-relationship Piece-wise linear relationship

Continuous relationships

Discrete relationships

Step-relationship

Piece-wise linear relationship

Activity 8: Sketch the relationship Here is a typical tax table. It explains how individuals must calculate their tax liability. Taxable income Tax rate <0 – 112 500 18% of the amount 112 501 – 180 000 20 250 + 25% of the amount above 112 500 180 001 – 250 000 37 125 + 30% of the amount above 180 000 250 001 – 350 000 58 125 + 35% of the amount above 250 000 350 001 – 450 000 93 125 +38% of the amount above 350 000 450 001 and above 131 125 + 40% of the amount above 450 000

Here is a typical tax table. It explains how individuals must calculate their tax liability.

Activity 8: Sketch the relationship The tax threshold is R43 000 for people under 65 years of age in the year 2008. What does this mean? Use the tax table to calculate the amount of tax paid by the different salary amounts. Represent this piece wise linear relationship in your table on graph paper. Note that each tax bracket will have its own unique line graph. People who earn less than R43 000 do not pay tax

The tax threshold is R43 000 for people under 65 years of age in the year 2008. What does this mean?

Use the tax table to calculate the amount of tax paid by the different salary amounts.

Represent this piece wise linear relationship in your table on graph paper. Note that each tax bracket will have its own unique line graph.

0 R12 600 R18 000 R24 625 R32 125 R40 125 R49 125 R58 125 R68 625 R79 125 R89 625 R100 725 R112 125 R123 525 R135 125 R147 125 R271 125 R551 125 R751 125 Annual earnings of individual Income tax amount (R) 40 000 70 000 100 000 130 000 160 000 190 000 220 000 250 000 280 000 310 000 340 000 370 000 400 000 430 000 460 000 490 000 800 000 1 500 000 2 000 000

Activity 9 Complete the following in a group, but each learner should include all three questions in his / her portfolio of evidence. Decide within the group how to delegate work.

Complete the following in a group, but each learner should include all three questions in his / her portfolio of evidence.

Decide within the group how to delegate work.

Activity 9 Investigate the cost of touring by train long distance in South Africa . a. Shosholoza Meyl Fares One way fares Sleeper-6 Sleeper 4 Cape Town to Johannesburg R320 R495 Cape Town to Kimberly R210 R325 Cape Town to Durban R210 R325 Johannesburg to Durban R155 R250 Johannesburg to Kimberly R110 R170

Investigate the cost of touring by train long distance in South Africa .

a. Shosholoza Meyl Fares

Activity 9 b. Find the distance between the cities and represent the information on graph paper for both the sleepr-6 and sleeper-4 options. KMB CPT DBN EL JHB PE KMB 968 811 780 476 743 CPT 968 1753 1079 1402 769 DBN 811 1753 674 557 984 EL 780 1079 674 982 310 JHB 476 1402 557 982 1075 PE 743 769 984 310 1075

b. Find the distance between the cities and represent the information on graph paper for both the sleepr-6 and sleeper-4 options.

Activity 9 Describe any noticeable trend How do you think the fares were calculated?

Describe any noticeable trend

How do you think the fares were calculated?

Activity 9 Ask the following people how they calculate the cost of their services or products: Sale of glass cut per area and designated thickness at a local glass cutter A clothing shop with cheaper clothes and large turnover Withdrawing cash at an ATM of your own bank Telkom landline costs.

Ask the following people how they calculate the cost of their services or products:

Sale of glass cut per area and designated thickness at a local glass cutter

A clothing shop with cheaper clothes and large turnover

Withdrawing cash at an ATM of your own bank

Telkom landline costs.

Activity 9 A certain cell phone operator, called TKS, has a R75 per month contract called Select75 as well as a R150 per month contract called Select150. a. Asanda decides to take the Select75 contract. Draw a table and write a formula for using this contract for one year. 324 399 474 549 624 699 849 999 Months 0 1 2 3 4 5 6 7 8 10 12 Cost 99 174 249

A certain cell phone operator, called TKS, has a R75 per month contract called Select75 as well as a R150 per month contract called Select150.

a. Asanda decides to take the Select75 contract. Draw a table and write a formula for using this contract for one year.

Activity 9 b. Inspect the table, represent the information on a graph and make a few predictions about cell phone usage of the group. Voice Calls Select75 (Peak) Select75 (Off peak) Select150 (Peak) Select150 (Off peak) TKS to TKS 2.10 0.95 1.85 0.90 TKS to landline 2.10 0.95 1.85 0.95 TKS to other cell phone 2.75 1.15 2.70 1.15

b. Inspect the table, represent the information on a graph and make a few predictions about cell phone usage of the group.

Activity 9 c. Represent this information on a graph and make a few predictions about the cell phone usage of the group.

c. Represent this information on a graph and make a few predictions about the cell phone usage of the group.