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
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
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.
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
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:
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 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
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
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.