NCV 4 Mathematical Literacy Hands-On Support Slide Show - Module 2 Part 1

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.

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

1.1 Numerical and geometric patterns investigated and extended

y = 2xInput 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 =

Activity 1: Arithmetic patterns

1. 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.a. Formula: b. 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.

c. Complete the table:

Km travelled

100 200 300 400 500 600 1000

Cost (R)

Cost of ticket = R25 + 0,75 x (no of km)

R100 R175 R250 R325 R400 R475 R550

Activity 1: Arithmetic patterns2. A paver needs 12 face bricks per square metre to build a pavement

pattern that he has designed. Face bricks cost R2,95 eacha. Formula for number of face bricks per m2: Number of bricks = b. Formula for cost per square meter of paving: Cost = c. Complete the tabled. 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 m2 within the given time.

Number of sq metres of pavements

1 2 3 4 5 10 50

Number of face bricks

Cost per square metre

12 x m2

R35,4 x m2

12 24 36 48 60 120 600

R35,40 R70,80 R106,20 R141,60 R177 R354 R1770

It will take him 25 days. He will need 2 additional helpers, which will allow them to complete it in 8 1/3 days

Activity 1: Arithmetic patterns

• Currency conversion: $1 = R7,66– Formula: Value in Rand: __________

– Complete the table:

Value in US$ 1 10 100 1000 10 000

100 000

1000 000

Amount in ZAR 7.66

$0.1305

76,60 766,00 7 660 76 600 766 000

7 600 000

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 = 2x

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

Activity 2 – constant difference patterns and constant ratio patterns

1. Write down five numbers that you think should logically follow on the numbers in the following sets.a. 250; 500; 750; 1000; ______; _____; _____; _____b. 250; 500; 1000; ____; ____; ____; _____; _____c. 250; 275; 302.50; 332.75; _____; _____; ______; _____d. 10 000; 11 000; 12 100; _____; ____; ______; _______;

______e. _______; _______; 161 051; 16 641; 1 331; _____;

____; _____2. Write down what you see as the constant difference

or the constant ratio in each of the above sets of numbers

1250 1500 1750 20002000 4000 8000 16 000 32 000

336.025 402,628 442,890 487,17913 310 14 641 16 105,1 17 715,61

19 487,17

19 487 171 1 715 561 12111 1

Activity 2: Constant difference patterns and constant ratio patterns

3. Order the following numbers to form a patterned sequence of numbers from the smallest number first to the largest number lasta. 0,5358; 0,5433; 0,5458; 0,5383; 0,5483; 0,5508;

0,5408b. 6375 thousand; 51 million; 0,102 billion; 12

750 000; 2,55 x 107

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


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.

Amount of deposit

Amount after one year

Amount after two years

Amount after three years

Amount after four years

15th term

10 000 1,1 x 10 000 1.12 x 10 000= 1.13 x 10 000= 1.14 X10 000

15 000 1,1 x 15 000 1.12 x 15 000= 1.13 x 15 000= 1.14 X 15 000

20 000 1,1 x 20 000 1.12 x 20 000= 1.13 x 20 000= 1.14 X 20 000

11 000 12 100 13 310 14 641

16500 18150 19965

R41772,48

R21961,5R62658,72

22000 24200 26620 2928283554,96


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

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

R1000 R2000 R3000R15000

R1500 R3000 R4500 R22500

R2000 R4 000 R6 000 R 30 000

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.a. How many cigarettes does he smoke per week?b. How many packets of cigarettes does he smoke per year?c. Find out what a packet of “lights” costs and then calculate how

much Asanda spends per year on cigarettes?d. Draw up a table of the weekly number of packets against the

total cost of the packets.

Week number 1 5 10 15 20 25 50

Total

56 per week2920 per year

R3066 per year

Activity 3 – Patterns in Asanda’s working day

e. 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.

f. Draw up a table of this savings exercise. g. 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 relationships1. 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.

2. 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.

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 10 20

Activity 4

• Distance travelled = 1200km

Speed (km/h) 20 40 60 80 100 120

Time of journey 12

Inverse

60 30 20 15 10

Activity 4

litres of petrol 1 15 45 60

Total cost 9,90 R148,50 R445,50 R594,00

Direct

Activity 4

Petrol consumption 8 10 12 14

Cost per 100km

Direct

R79,20 R99,00 R118,80 R138,60

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

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

Activity 4

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

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

R13 260 R16 334 R19 692

R3 260 R6 334 R9 692

Activity 4

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

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

c. 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

Activity 5

• 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 = 0,2x + 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.

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:

Input value (R) = customers amount

Output value (r) = earnings of waitress

300

x0.2 +250 =

600

900

1200

2400

R310

R370

R430

R490

R730

Activity 6

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

Metro bus ticket:

1,75 - 4,25

4,25 – 5,75

5,75 – 7,25

7,25 – 8,75

8,75 – 10,25

Activity 6

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

a. y=25x + 5

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

input 1 2 3 4 5 6 7 8 9

output 30 55 80 105 130 155 180 205 230

Activity 6

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

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

Formula: _______________

x-value 5 15 25 35 45 55 65 75 85

y-value 20 50 80

1/3 5

5x + 3

110 140 170 200 230 260

Activity 6

d. 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.

x-value 3 6 9 12 15 18 21 24 27

y-value 13 28 43 58

53

5x + 3

73 88 103 118 133

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.

1.1.4 Make representations of the relationships

• 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



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

Activity 8: Sketch the relationship

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

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

c. 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

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

0R12 600R18 000R24 625R32 125R40 125R49 125R58 125R68 625R79 125R89 625R100 725R112 125R123 525R135 125R147 125R271 125R551 125R751 125

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.

Activity 9

1. 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

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

Activity 9

c. Describe any noticeable trend

d. How do you think the fares were calculated?

Activity 9

2. Ask the following people how they calculate the cost of their services or products:a. Sale of glass cut per area and designated

thickness at a local glass cutterb. A clothing shop with cheaper clothes and large

turnoverc. Withdrawing cash at an ATM of your own

bankd. Telkom landline costs.

Activity 9

3. 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.

Months 0 1 2 3 4 5 6 7 8 10 12

Cost 99 174 249 324 399 474 549 624 699 849 999

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

Activity 9

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

NCV 4 Mathematical Literacy Hands-On Support Slide Show - Module 2 Part 1

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Transcript of NCV 4 Mathematical Literacy Hands-On Support Slide Show - Module 2 Part 1