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

Module 1: Numbers

Module 1: Numbers

• After completing this module, you will be able to:– use numbers correctly when working with

problems in a personal and familiar context and in the workplace

– perform calculations accurately / correctly to solve problems in a personal and familiar context and in the workplace

– identify and use appropriate measuring tools and techniques to solve problems in a personal and familiar context and in the workplace

Introduction• The following are two tickets attached to

the outside of cardboard packaging boxes:

Item F1130

GW 4kg

NW 3kg

Meas. 22,5x22,5x32

Batch # C33217-003

Pack 01/19/2004

RBG D70745

Work press 100 bar

Max press 110 bar

Max vol 6l / min

T max 40ºC

2006 No6830001432

230 V 50Hz1.6kW 92dB

Elsewhere on box

360 l/hr

Introduction

• Write down what you consider to represent:– Quantity (cardinal number)

– Order (ordinal number)

– Dimensions

– Code

– Anything else

• What type of product do you think is packed in box b

1. USE NUMBERS CORRECTLY

• At the end of this outcome, you will• be able to count, order and estimate

• know that positive and negative numbers have direction

• work with fractions, decimals and percentages

• understand different time notations

Case study

• Activity regarding: – calculations with numbers – conversion of measurements – ratio of ingredients

• You may not be able to answer these questions before working through the learning activities in the module. Therefore, read the Case Study and questions and come back to it at the end of the module. Do it with a friend.

Case study• Tony and his wife, Emily, have recently started a small catering

firm. They work from home and travel to pre-arranged venues to cater for meetings, weddings, funerals etc.

• They have been asked to provide a light lunch at a small day-long congress. There will be 38 people for lunch. They have experienced energy black-outs and always take their own gas stoves with them. They also try to prepare dishes in their own kitchen ahead of time as far as possible to ensure that the catering process runs smoothly.

• They decide to make Potato Rosti with Salmon. They have a recipe which will serve 4 people (Source: Sunday Times, 16th March 2008). They will supply orange and grape juice as well as water for drinks with the lunch. For dessert they decide to serve Chocolate Cheesecake with the coffee/tea. The recipe serves 12 people.

Case study

1. Make a list of ingredients necessary to cater for 38 people2. Decide how much fruit juice to buy and add this to the list3. In the same list adapt the ingredients for 100 people for

possible future reference4. Find the prices of the ingredients and calculate how much this

lunch will cost Tony and Emily5. A famous British chef also attends the congress and asks Tony

and Emily whether they would mind to give him the recipes if he credits them in his new recipe book. They ask you to quickly convert the measurements to the Imperial system as the British do not work in the metric system

You find out that: – 1 tablespoon = 15ml; 1 teaspoon = 5ml; 1oz = 28,35g; 10 fluid

ounces = 300ml.– List the ingredients with Imperial measurements.

Case study

1. Make a list of ingredients necessary to cater for 38 people

For Potato RostiTo all intents and purposes, we can presume that there are 40 people, otherwise you’ll end up with half ingredients. 30 Potatoes10 Onions10 eggs150ml + extra1kg Salmon10 lemons10 cartons extra cheese

Case study1. Make a list of ingredients necessary to

cater for 38 peopleChocolate cheesecakeHere we have a bit of a dilemma: We could increase the amount to 40 people (and serve everyone slightly more) or decrease the amount to 36 people (and serve everyone slightly less). While its better in this case to round up, it’s easier to round down.750 g toasted cereal167g toasted hazelnuts13 1/3 tbsp butter83 g dark chocolate1167g cream cheese333g castor sugar667 ml castor sugar1 litre double cream3 1/3 sachet gelatin10 tbsp water583.3g dark chocolate583 g white chocolate

Case study

2. Decide how much fruit juice to buy and add this to the list

Assuming that each person would have 2 glasses of fruit juice = 500ml x 40 people=20 litres of fruit juice=10 litres orange and 10 litres grape

Case study

3. In the same list adapt the ingredients for 100 people for possible future reference

For potato rosti•75 potatoes•25 onions•25 onions•375 ml oil•2.5 kg salmon•25 lemons•25 cartons cream cheese

For chocolate cheesecakeHere, we can round up to 108 people and then multiple by 9•2025 toasted cereal•450g toasted hazelnuts•36 tbsp butter•225 dark chocolate•3.15 kg cream cheese•900g castor sugar•1800 ml yoghurt•2700 ml double cream•9 sachets gelatin•27 tbsp water•1,575 kg dark chocolate•1,575 kg white chocolate

Case study5. A famous British chef also attends the congress and asks Tony and Emily

whether they would mind to give him the recipes if he credits them in his new recipe book. They ask you to quickly convert the measurements to the Imperial system as the British do not work in the metric system

You find out that: – 1 tablespoon = 15ml; 1 teaspoon = 5ml; 1oz = 28,35g; 10 fluid ounces = 300ml.– List the ingredients with Imperial measurements.

For potato rosti•3 potatoes•1 Onion•1 egg•0.5 fl oz oil•3.53 oz salmon•1 lemon•1 carton cream cheese

For lemon cheese cake•7.94 oz toasted cereal•1.76 oz hazelnuts•2 fl oz butter•0.88 oz dark chocolate•12.35oz•3.53 oz•6.67 fl oz yoghurt•10 fl oz crean•1 sachet gelatin•1.5 fl oz water•6.17 oz dark chocolate6.17 oz white chocolate

1.1 Count, order and estimate

9 3 8 5 2 7 6 1 4

6 7 5 4 1 8 2 3 9

2 1 4 9 6 3 8 5 7

7 5 9 3 8 4 1 6 2

3 8 2 1 9 6 7 4 5

1 4 6 2 7 5 9 8 3

4 6 7 8 5 9 3 2 1

5 9 1 6 3 2 4 7 8

8 2 3 7 4 1 5 9 6

Activity 2: Count, order and estimate numbers


1. Has the government set aside sufficient funds for the recapitalisation programme?

2. How many taxis will have been scrapped by 2011?3. What percentage of transport trips in SA are not made

by minibus taxi?4. Estimate the daily revenue of the taxi industry in

Johannesburg alone.5. Do you think that it is fair of a taxi operator to demand

R100 000 for replacement of an old taxi?6. Estimate how many taxi’s were sold by Toyota in 20077. Calculate an estimated percentage increase in sales of

Toyota minibus taxis from 2007 to 2008.


1. Has the government set aside sufficient funds for the recapitalisation programme?

R68 million has been set aside, together with another R570 million totals R638 million. This needs to replace 86 600 taxis.This gives R7367 per taxi.If you say that the government has R570 million to replace 25 720 taxis, then each taxi gets R22 161, which still falls short of the R50 000.


2. How many taxis will have been scrapped by 2011?

Total taxis scrapped = 13400 + 25720 = 39 120 taxis projected to be scrapped


3. What percentage of transport trips in SA are not made by minibus taxi?

Percentage of trips not made by taxi = 100% - 35% = 65%


4. Estimate the daily revenue of the taxi industry in Johannesburg alone.

The difficult part of this question is what we do with weekends. One option would be to take only weekdays, which would overestimate the daily value, another option would be to take the whole week, which would underestimate the value earned during a weekday. The proper value would lie somewhere between these two values.For 7 day week: R10.96 million per dayFor 5 day week: R15.33 million per day


5. Do you think that it is fair of a taxi operator to demand R100 000 for replacement of an old taxi?

Considering that these taxis are going to be scrapped, R100 000 is a lot of money, probably far too much.


6. Estimate how many taxi’s were sold by Toyota in 2007

Taxis sold by Toyota: between 900 and 1200. Let’s estimate it to 1000 taxis


7. Calculate an estimated percentage increase in sales of Toyota minibus taxis from 2007 to 2008.

The lower bound increases by 33%, and the upper bound by 20%. Here we can settle on 25%.


2.Order the following fractions from the largest to the smallest i.e. arrange in descending order.

87

54

107

10068

1000538

100007321

20045

4016

100095

30099

65

107

209

87


3. Give a sensible estimate:– Dogs become extremely frightened when crackers are fired.

After the fright they got during a New Year’s celebration in one city, 49 dogs were found straying on the streets. More or less how many dogs were now ownerless?

– An educare student buys a calculator for R36,75; two black ball point pens at R12,99 each; two exam blocks at R5,55 each; and a flip-file for 16,87. First estimate, then calculate how much money she spent.

– If 9 tiles cover 1 square metre of paving, estimate how many tiles will be required for 39 square metres of paving?

– If a caterer knows that most men have three slices of toast with breakfast, an most women have only one slice, estimate how many slices of toast he must prepare if he expects 81 men and 67 women for breakfast in the college hostel.


3. Give a sensible estimate:– Dogs become extremely frightened when crackers are fired.

After the fright they got during a New Year’s celebration in one city, 49 dogs were found straying on the streets. More or less how many dogs were now ownerless?

– An educare student buys a calculator for R36,75; two black ball point pens at R12,99 each; two exam blocks at R5,55 each; and a flip-file for 16,87. First estimate, then calculate how much money she spent.

– If 9 tiles cover 1 square metre of paving, estimate how many tiles will be required for 39 square metres of paving?

– If a caterer knows that most men have three slices of toast with breakfast, an most women have only one slice, estimate how many slices of toast he must prepare if he expects 81 men and 67 women for breakfast in the college hostel.

50 dogs

R90 (actual R90.70)

R351

R310 (actual R310)


4. A 12-chalet guest house in Swellendam had 12 guests per night at two guests per chalet on average during peak season in December of 2008. The manager expects that tourism will pick up by 25% during the next year. – How many chalets were rented out on average

during December 2008?– What chalet occupancy rate per night should he

expect in December 2009?– Name a few factors that could influence this

estimate.

6 chalets

8 chalets

Activity 3• Your friend, Thandi, has to plan the end-of-year party

for the workers in the fruit juice packaging factory where she is a shift boss. The party will be held in a courtyard adjoining the staff kitchen at the factory. However, Thandi has problems at home and on the 1st of December you offer to help. 1. Make a list of questions that you need to ask Thandi2. Make a list of items you want to buy3. Find the prices of these items and set up a budget for

Thandi to approve. Estimate the total price4. You also suggest that each person should receive a small

gift. Make a few suggestions and discuss it with Thandi.5. The party will be held on the Friday closest to the 12th of

December. Organise your preparation activities and meetings with Thandi with dates attached so that you do not run out of time.

1.2 Positive and negative numbers

- ... –5 –4 –3 –2 –1 0 1 2 3 4 5 … • The number line is similar to a thermometer which is not vertical but

lies horizontal. The negative (small) numbers always lie to the left of zero and the positive numbers to the right of zero. Note that negative 2 (or –2) is smaller than negative 1 (or –1). Therefore the further to the left on the negative side, the larger the digits but the smaller the value of the digits.

• The symbol at the extreme left and right-hand sides of the line, represent infinity.

• The dots on the sketch indicate that the numbers continue to negative infinity on the left, and to positive infinity on the right.

• To conclude, the number line always runs from the extremely small numbers on the left to the extremely large numbers on the right.

• The numbers printed on this specific number line are all integers or whole numbers.

• Between the whole numbers/integers lie numerous fractions.

Activity 4• Question 1:

a. Calculate the values in the last column

b. Find out what the names in the first column mean

c. What are the given values and should there not be a unit attached?

12 145,42

1315,50

5692,61

12 308,06

6494,61

Activity 4

2. On a winter’s day in Sutherland in the Karoo, the temperature is 15,5 degrees Celsius. The temperature drops 19 degrees during the night. What is the new temperature reading? You have just calculated 15,5 – 19. Do this on your calculator and notice the display method.

-3.5C

Activity 4

• A head-line in the Business Day on 17th March 2008 read: “January retail sales negative.”

• Read these excerpts from the article:– This week is the calm before the storm of local

inflation figures next week, but there maybe gloomy news when retail sales and current account numbers are released on Wednesday.

Activity 4

• ……..retail trade figures for November last year had shown the first contraction in real terms since 2001 (-2% year-on-year) and the December figure reflected a real 0,5% year-on-year decline.

• “We expect retail trade to continue showing a decrease in January figures and to record a 2% year-on-year decline for the first month of this year,” Efficient Research said.

• The rand has been under pressure lately and part of the blame is a swelling current account deficit which reached a level of 8,1% to gross domestic product (on annualised basis) in the third quarter of last year.

Activity 4• We expect the fourth quarter to report a smaller current

account deficit, as the trade deficit narrowed to about R16bn from R23,1bn in the third quarter. Despite this, the annual figure of the current account deficit to gross domestic product, is still expected to be larger than the -6,5% recorded in 2006, possibly nearing -7%,” it said.

• Other important data published in the Quarterly Bulletin will include debt and savings levels. The rate of increase in household debt to disposable income started to rise from 2005 and during the third quarter of last year it reached 77,4%. Efficient Research said they expected further rises in this measure as higher interest rates and general inflationary pressures continued to put households’ budgets under pressure.

Activity 4

• “The flip side of the coin is of course that household saving to disposable income recorded a negative figure of -0,5% (i.e. dis-saving) for 2006. Data for last year will again report a negative figure as all three quarters so far have also recorded negative values,” they said.

• Across the pond the UK is in for a busy week ahead of Easter. Bishop said the UK”s consumer inflation data was likely to lurch higher in February, to 2,6% from 2,2% “but this should reflect the new treatment of the previously announced utility tariff hikes, not an unforeseen worsening in underlying inflation trends”.

Activity 4

• 1. Explain the meaning of:a. “Retail figures…showed contraction”

b. “a 2% year-on-year decline”

c. “a swelling current account deficit”

d. “the trade deficit narrowed”

e. “higher interest rates and general inflationary pressures”

f. “rate of increase of household debt to disposable income”

g. “household saving to disposable income recorded a negative figure”

h. “consumer inflation data likely to lurch higher”

i. “utility tariff hikes”

Activity 4

• Question 2– How can -7% be “larger than” -6,5%?

– Express the rate of increase in household debt to disposable income in the third quarter of the previous year, as a fraction.

– Express the household saving to disposable income in 2006 as a fraction.

1.3 Normal fractions, decimal fractions and percentages

821657.324

800 000 + 20 000 + 1000 + 600 + 50 + 7 + 0.3 + 0.02 + 0.004


Fractions

• Between the whole numbers are fractions of numbers

• A fraction next to a positive number means a bit more than that number

• A fraction next to a negative number mean a bit less than that number


Examples

1 ½ =


• 6 ½ can be written as 13/2 or 6.5

• 6.5 is the same as 6.50 or 6.500


• Converting a common fraction to a decimal fraction– To work out ¾ on your calculator, you can

press 3 ÷ 4. Your answer will be 0.75

• Convert 0.825 to a common fraction– To determine the numerator

– Write the numbers after the decimal fraction

– To determine the denominator

– Write 1 followed by the number of 0s as the number of digits after the comma

0, 825 =


1000

825


• Percentages– A percentage is a fraction of the whole amount,

but always with the denominator equal to 100

– Percent means out of 100

– Percentage is a ratio expressed with respect to 100, and can be written as a fraction with the number under the lines being 100


• How a percentage is calculated– % is given, answer is given in a unit other than

a percentage

– % is asked, answer is given as a percentage


• Converting from a percentage to a number– To work out 25% of 28, type in: 1 ÷ 4 x 28

– The answer you get will be 7

• Converting from a number to a percentage

Activity 5

• Question 1Common fraction Decimal fraction Percentages

3/8

0.625

15/45

16/20

0.75

25/100

1453/2000 0.7265

52.8%

4/5

0.375 37.5%

5/8 62.5

0.333 33.33%

0.8 80%

3/4 75%

0.25 25%

72.65%

0.52866 / 125

0.8 80%

Activity 5

70

3

144

48

1500

25

52

13

350

15

3

1

60

1

4

1

Activity 5

• Question 3

a.You can find 25% of a value by just dividing by 4. Explain this.

b.If you want to find 20% of a value, what must you divide by?

Activity 5

• Question 4

• In a department store, you want to buy a jacket which is priced at R589.99 but the computer tag has been crossed through will red ball-point ink and reads R294,99. What is the % discount? When reaching the till to pay, you are told that another 25% discount is valid for the day. What will you be charged?

50%; R221,24

Activity 5

• Question 5: Increasing density of the system:

a.Draw the number line between 8,1 and 8,2 and place hundredths in between.

Activity 5

• Question 5: Increasing density of the system:

b.Draw the number line between 8,11 and 8,12 and place thousandths in between.

Activity 5

• Question 6:– Write the following whole numbers/integers in

fraction form: 129; 45; 467; 1528

129/1; 45/1; 467/1; 1 528/1.

95324/716243

30440/716243

42730/716243

12050/716243

10105/716243

121087/716243

121087/716243

75492/716243

105309/716243

14784/716243

37772/716243

13,30%

4,25%

5,97%

1,68%

1,41%

49,49%

16,91%

10,54%

14,70%

2,06%

5,27%

165213/716243

16775/716243

5651/716243

14624/716243

2775/716243

71281/716243

54107/716243

40302/716243

655283/716243

54960/716243

6000/716243

716243/716243

23,07%

2,34%

0,79%

2,04%

0,39%

9,95%

7,55%

5,63%

91,49%

7,67%

0,83%

100%

1.4 Time notation

Time zones

Mercator projection

World maps and international time zones

• The world map has a grid system. This grid system consists of lines of longitude and lines of latitude

• A full circle rotation equals 360 degrees• Lines of longitude also equal 360 degrees• From zero longitude, lines are measured at regular

intervals of 60 nautical miles (at the equator)• Each degree consists of 60 minutes and each minute

consists of 60 seconds• Lines of longitude converge at the poles• As the earth revolves towards the East, it means that

Australia gets sun before South Africa

World maps and international time zones

• The world map with time zones displays vertical bars which represent each time zone

• From the time zone map, you can see that Dunedin (New Zealand) is 10 hours ahead of South Africa.

• The time zone map has funny shapes in places. This is to keep a whole country with one time zone.

• Each time zone is 15 degrees longitude wide

Activity 5

• Question 1. From the time zone map, if it is 12:00 in South Africa, what time is it in:– Chicago

– Tokyo

– Moscow

– Alaska

– Nairobi

– Beijing

04:00

19:00

13:00

0:00

13:00

18:00

Activity 5

• Question 2. How many time zones are there for the people of the USA?

8 time zones

Activity 5

• Question 3: From the Mercator’s world map, give the estimated grid co-ordinates of:– England

– Tasmania

– The Southern tip of Madagascar

– The Northern tip of Antarctica

– The tip of the horn of Africa

N45 E0º

S35; E150

S30; E50

S65; E60

N10; E50

Activity 5

• Question 4:– A cake has to bake for 2,75 hours. It has to be taken out

of the oven at 15:00. At what time was it put in the oven?

– To make your own chicken stock for soup, the chicken has to be boiled for 1 hour and 20 minutes. If you start at 10:00, at what time can you switch the stove off?

– A Cape Town passenger en route to London has to take a connecting flight at Johannesburg International Airport. The Cape Town flight lands at 9:46 and the flight to London departs Johannesburg at 10:33. How much time does the passenger have to reach the departure gate for the London flight?

12:15

49 minutes

11:20

Activity 6• You work at a travel agent and have made a return booking to

New Zealand for a customer on Singapore Air. The itinerary is as follows:

03 Mar 08 Depart CT Int 120004 Mar 08 Arrive Singapore 052504 Mar 08 Depart Singapore 210505 Mar 08 Arrive Christchurch 115529 May 08Depart Christchurch 105029 May 08Arrive Singapore 174530 May 08Depart Singapore 013030 May 08Arrive Cape Town 07:35

Activity 6• Questions:

1. Calculate the length of the four flight stages

2. Calculate the length of time spent at Singapore airport both

ways

3. Calculate the total time spent to reach the destination – both

ways

Stage 1: 5hr 25; Stage 2: 14hr50; Stage 3: 6hr55; 6hr05

15h40 + 7hr45 = 23hr25

Cape Town – Christchurch = 47hr55min

2. ACCURATE ANSWERS TO PROBLEMS / SOLVE PROBLEMS ACCURATELY

• At the end of this outcome, you will be able to:• Calculate using pen and paper or in your

head

• Estimate and round off numbers

• Add and multiply to simplify calculations where possible

• Use ratio and proportion in problems

2.1 Estimation and approximation / rounding

• An estimate is an informed guess. It is an approximation of a quantity which has been decided by judgement rather than by counting, measuring or doing accurate calculation


• Examples:– The number of TB patients seen at a clinic per year might be 597.

If you want to calculate the average number seen per month, the calculation will be 597 divided by 12, which is 49,75. But this is not an appropriate answer, as you cannot deal with 0,75 of a person. Therefore, the answer should be approximated to the nearest whole number i.e. 50 people.

– An estimate is therefore a rough judgment, and can apply to any of the measurements mentioned earlier, such as temperature, the grain harvest or number of bottles of wine produced per day in a bottling plant.

– Rounding up: If 52 people need to be transported, how many 10-seater buses are needed? (52 10 = 5,2 ). Round up to 6 buses otherwise some people will not have a seat.

– 16,4 bottles of 750ml capacity are needed to bottle 12 300ml of wine. Round down to 16 bottles. You cannot sell 0,4 of a bottle of wine.


• Approximation in mathematics has a different meaning to estimation. An approximation is also a value that is close to the true value of that number. It is also known as rounding off. A number can be rounded off to the nearest hundred, for example the number of people attending a rugby match might be counted as 34 768 but if rounded off to the nearest hundred the number would be 34 800; rounded off to the nearest thousand, the number would be 35 000; and rounded off to the nearest ten thousand, the number would be

How to round off:

• If you have to round off/approximate 35 768 to the nearest hundred, put your pencil on the hundreds digit, and establish whether the digit immediately to its right is a five, a six, a seven, an eight or a nine. If so, then this five or larger than five number, forces the hundreds number, seven in this example, to become one larger, namely eight. Therefore 35 768 becomes 35 800. If the number to the right is smaller than five, then there will be no change to the digit you are interested in, but zero’s will be placed to the right of this number.

Activity 7• A municipality sent the following estimated information

with regard to water usage to its rate payers:• In the garden: Hose pipe = 30L per minute• Car wash = 1 000L per wash.• Bathroom: 1 toilet flush = 10L

1 bath = 100 – 200L• Bathroom sink: 1 sink = 30L• Investigate whether these four estimates are close. If

not, calculate the percentage differences.

Activity 81. Estimate the following:

a. How long will it take to travel 1 500 km by car if your average speed is 100km/h?

b. If you walk one kilometer in 10 minutes, how long will it take to complete a 20-km walk?

c. If a typist types at a speed of 60 words per minute, how long will she take to type a document of 6 000 words?

d. If a tourist has R600 to spend per day, how many days can he travel on R10 000.

e. Estimate the monthly income of a business which operates five days per week, at an estimated daily income of R900.

f. If four sheets of corrugated iron cover roughly 8 square metres of roof, estimate the number of sheets necessary to cover a roof of 96 square metres

15 hours

3hr20min

1hr40min

16 days

day x R900 = R19 800

48 sheets

Activity 82. Approximate / round off the following figures

65 000 000

126 000 000

100 000

400 000

70 000

670 000

498 625

32 453

567,896

64,490

Activity 8

3. One way to calculate your safe exercise heart rate is to subtract your age from 220 and take of the difference, rounded to the nearest whole number.a. Calculate your safe exercise heart rate.b. Calculate the safe exercise heart rate of a 40-year-

old person.c. After running you find that your heart rate is 160.

You are 20 years old. Is that a safe heart rate?

135bpm

No

Activity 8

4. You have R220 which you can use to buy clothes. You buy three items of clothing at a sale at a discount of 25%

a. Estimate whether you have enough money to buy the items:

– a T-shirt priced at R64,99– a pair of shorts priced at R84,95– a cap priced at R58,55

b. Calculate your actual cost and change if you hand the teller R220.

Yes

R63.63

Activity 8

5. Decide whether you should round up or down and then give the rounded figure:

a. If 79 people have to be transported, how many 9-seater buses are needed?

b. 20 cookies fill a packet. How many packets can be filled with 98 cookies?

Up; 9 busses

Up; 4 packets

Activity 8

6. In the tourism industry, there were 20 000 visitors to Cape Town in the month of July. The airport company estimated that the visitors for September will be 25% more. Give this estimate.

25 000 visitors

1.2.2 Calculations

1.2.2 Calculations

• If you press the wrong number, use your calculator’s “clear entry” key.

• If you press the incorrect operation key, simply press the correct key and the calculator will only obey the latter instruction.

• If you press the wrong operation followed by a number before you realise your mistake, you will have to correct the mistake with the opposite calculation procedure.

Example

• If you want to calculate 6 × 8, but you mistakenly press 6 + 8, then all you have to do to cancel the + 8 is to subtract 8, which is called the opposite or reverse calculation procedure. Then carry on with the sum:

• 6 + 8 – 8 × 8 = 48

Memory keys

• If you want to calculate:

• 10 x 20 – 5 x 10, do the following:

• 10 x 20 → M+ (the calculator adds 200 to the memory).

• 5 x 10 → M– (the calculator subtracts 50 from the memory).

• Then press MRC to obtain the answer. You should see 150 on your screen.

Memory keys• The memory of the calculator can also be used as follows:• Do the following calculations on your calculator:• 52 860 – 5 238 =• 52 860 + 3 325 =• 52 860 x 13 =• 52 860 ÷ 12 =• This is the required keystroke sequence for a short-cut method:• 52 860 → M+ →• MRC → – → 5 238 → = (you will see the answer to the first

calculation).• MRC → + → 3 325 → = (you will see the answer to the second

calculation).• MRC → x → 13 → = (you will see the answer to the third calculation).• MRC → ÷ → 12 → = (you will see the answer to the fourth

calculation).

Constant functions

• Your calculator can add, subtract, multiply or divide by a constant number.

• Press: 2 + = = = = = = =• Press: 2 x = = = = = = =• You have programmed your calculator to be a

2 + machine or a 2x machine.• Press: 2 x = 16 384 = = = =• Press: 2 – = 48 = = = = =• You have programmed your calculator to be a

2 ÷ machine or a 2 – machine.

Activity 9

1. Do the following calculations using the M+ and MRC keys on the calculator. Write down the complete keystroke sequence.– 164 578 + 8 429 =

– 164 578 – 3 562 =

– 164 578 x 18 =

– 164 578 ÷ 8 =

173 007

161 016

2 962 404

20572.25

Activity 9

2. A second-hand car salesman adds R1 478 to the price of each vehicle sold, to compensate for damages suffered during a burglary. Calculate the prices of the following cars by programming your calculator to do a constant addition

Activity 9

3. A bill for 22 of each of four items has to be calculated. Work out the bill, item by item. Then write the total for the bill. Work out the bill again, but this time use the short-cut methods of your calculator. Then write down the keystroke sequence that you could use with your short-cut method

Price of one item Price of 22 of these items

R1 235,68

R252,75

R398,52

R549,87

Total amount owed

Keystroke sequence:

R27 184,96

R5560,50

R8635,44

R12 097.14

R53 478.04

Examples of using BODMAS calculations

1. 7 x (3 + 2) = 35 Any calculations within brackets must be done first.

2. 6 + ¼ of 100 = 31 You have to do the “of” which implies multiplication, before the addition.

3. 2 x 7 + 9 x 4 = 14 + 36 = 50 You have to do the two multiplication calculations before the addition. Also: 26 + 4 x 8 = 58

4. 14 ÷ 2 – 6 ÷ 2 = 4 You have to do the two division calculations before the subtraction.Also: 48 – 6 + 99 ÷ 3 = 75

5. 6 x 2 ÷ 3 = 4 If only multiplication and division calculations need to be done, they

are done from the left to the right.6. 24 – 8 + 12 – 4 = 24 If only addition and subtraction

calculations need to be done, they are done from left to right

Activity 10

1. Are these equivalent calculations – give answers in each case:

Answer Answer

148 – 12 x 7 + 28 (148 -12) x (7 + 28)

6222 +148 ÷ 2 -25 6222 + (148 ÷ 2) - 25

1600 ÷ (135 -25 -23) 1600 ÷ 135 – (25-23)

400 – (60 – 18) 400 – 60 - 18

1000 x (30 + 65) 100 x 30 + 65

5 x (8 + 5) 5 x 8 + 5 x 5

(7 + 3) x 13 7 x 13 + 3 x 13

92

6271

21.33

358

95 000

65

130

4760

6271

9,85

322

30 065

65

130

Activity 10

2. Two adjacent rooms (A & B) are five metres long and six metres long respectively. They are both 4 metres wide. Sketch the floor space of these two rooms from above and then calculate the area of wall to wall carpet that needs to be ordered to cover the floors of both rooms. Do the calculation by two methods

Activity 10

3. A builder has to budget for 7 of the same bathrooms. Each bathroom will have:– one basin + pedestal @ R139,00,– one geyser @ R 1 359,00,– one bath @ R199,00,– one shower @ R149,00.

• First estimate, then calculate the total cost of the 7 bathrooms.

• Once again, do the calculation by two methods.

Activity 10

4. A baker has to bake 25 cakes. For each cake he will need:– 2,5 cups of flour,

– 1,5 cups of sugar,

– half a cup of butter.

• Calculate by two methods how much of each ingredient he will need.

Activity 10

• Four friends decide to spend eight days in Zimbabwe. They want to visit a National Park called AZambezi in Zimbabwe (cost = R2 390 per person for 4 days) and then go to the Victoria Falls Safari Lodge (cost = R2 790 per person for 4 days). First estimate and then calculate by two methods the total cost.

Activity 10

• Interpret the news: Mathematical indices– A valuable mathematical calculation method is

that of using an index.

– An index is just a single number derived from a bunch of other numbers.

– Look at the Breakfast Index as explained by Seiter in his book: Everyday Math for Dummies. The elements of the breakfast index are:

Breakfast items Price in 1995 Price in 2000

One dozen eggs $1,29 $1,36

One pound of bacon

$2,49 $2,59

Half a gallon of milk $0,89 $0,91

A stack of pancakes $3,19 $4,59

Can of frozen orange juice

$1,17 $1,37

Total $9,03 $10,82

5,43%

4,01%

2.25%

43,89%

17.90%

19,82%

Activity 10• In an indexing scheme you set the original total equal to the number one by

dividing the total by itself. Then for other years, you get an indexed number by dividing the new total by the original total.

• In other words:– Index for 1995 = 1– Index for 2000 = 1,22

• You can make a graph of these numbers over the years and use the graph to track an approximate price of breakfast. One of the conveniences of such an index is that just by looking at the figures you can see the % change, 22% in this case.

• The inflation index also known as the CPIX is made up of a longer list of goods and services, but basically it is the same idea.

– Convert the dollar prices to rand and the Imperial measurements to metric. (Conversion table in Activity 9).

– In the last column add the present prices and calculate the present breakfast index.• However, the word index derives from indicator, and some indices are calculated

on a different basis.• Look at the index for obesity in the following Case study.

Activity 11

• The table of BMI (Body Mass Index) gives an indication of obesity. Obesity is associated with hypertension, heart disease and adult-onset diabetes, as well as back and joint problems.

Activity 11• The BMI in the top row is calculated with the formula BMI = mass divided by

length squared (calculations of the index are in kg/m²). However, in this table the heights as well as body weights, are given in Imperial units. a. Convert the “inches” column into metre readings. Approximate each calculation to a

logical answer (Use one inch = 2,54 cm)b. Convert the selected row of body weight readings into kilograms. Approximate each

calculation to a logical answer. (Use one pound = 0,4536 kg).c. Select your own height and weight and read your own BMI from table 1. Use Table 2 to

decide in which category of disease risk you fall. d. Abdominal fat impacts on glucose metabolism and on cholesterol function. Therefore,

waist circumference is sometimes an even better measure of obesity.e. abdominal obesity is defined as follows:– waist circumference of an adult female > 80 cm.– waist circumference of an adult male > 94 cm.

• Using the answers obtained in c. and d. describe your own level of obesity. (You have just interpreted numerical/mathematical information, approximated answers logically, drawn your own conclusions and verbally conveyed your conclusion.)

2.3 Ratio and proportion

• Ratio– Comparison between two or more similar

quantities can be given as a ratio.

– The ratio of x to y can be expressed as or x : y.

– Dividing x and y by a common factor will simplify a ratio.

– E.g. 3 : 2 is the simplest form of the ratio 6 000: 2 000.

– A ratio does not have units.


• Example of ratio between more than two items:– Rosie, Lily and Jane earn R2 400 and share it in the

ratio 1 : 5 : 4.

– The total number of parts = 10.

– Rosie will get one part of this or: 1/ 10 of R2 400 = R240

– Lily will get five parts or: 5/10 of R2 400 = R1 200

– And Jane will get 4 parts or: 4/10 of R2 400 = R960

Activity 121. Express the following ratios in the simplest form:

a. 64 : 800b. 39 : 930c. 60/144 d. 250mm/75cme. 20c : R1,20f. 40 min : 1 H 40 ming. 1 H 20 min : 10 minh. 40c : R3,60i. 1 km : 250 mj. R1,25 : R6,00k. 72 kg : 144 kg

2:2513:310

125

mmmm310

1c : 6c2 min : 5 min8 min : 1 min

1c : 9c4m : 1m1c : 4c1kg: 2kg

Activity 12

2. R48 550 has to be shared between Warren and Patricia. Calculate Warren’s share if the money is divided in the ratio:

a. 3 : 7

b. 5 : 8

c. 4 : 5

R14 565R18 783.08

R12 127.50

Activity 12

3. The ratio of different kinds of flour in a gluten-free bread is: corn flour : pea flour : oats to the ratio of 1:2:3– If you use 500 g of corn flour for two loaves,

which amount of pea flour and oats will you have to add to the dough?

1kg pea flour; 1,5 kg oats

Activity 12

4. On a map of a hiking trail in the Limietberg region in the Cape, a hiker measures that he still has to walk 8 cm of trail.

• The scale of the map is 1 : 50 000

• How many kilometres does he still have to walk?

4km

Activity 12

5. Divide R11 780 between three labourers. The one labourer worked for two days, the second one for three days and the third one for six days

Worker 1: R2 141,81; Worker 2: R3212,72; Worker 3: R6425.45

Activity 12

6. A silk screen artist is creating a design for material to be made up for a dress designer. She decides to print rectangles and uses the golden ratio between the breadth and the length of the rectangles - two parts divided into the ratio 1 : 1,618 because this ratio presents a pleasing image to the human eye. If the rectangles that she sketches have a side length of 35mm, what must the breadth of the rectangle be?

21.63cm

Activity 12

7. Water leaks from a tap at a rate of 17 drops every 20 seconds. Calculate how much water has gone to waste in:

a. 24 hoursb. one weekc. one monthd. one year

• Discuss the possible cost of the lost water to the home owner.

73440 drops

514 080 drops

15 422 400 drops

187 639 200 drops


• Proportion– A proportion is a statement where two ratios

are equal. Four quantities are in proportion if the ratio of the first to the second equals the ratio of the third to the fourth.

– Example• 4; 8; 6; and 12 are in proportion because 4 : 8 = 1 : 2

and 6 : 12 = 1 : 2 therefore: 4 : 8 = 6 : 12


• Direct proportion– Two quantities are said to be directly

proportional if, as one quantity decreases (or increases), the other quantity also decreases (or increases).

• Example 1: If one loaf of bread costs R6,50 then two loaves of bread will cost R13, three loaves of bread will cost R19,50 etc.

• Example 2: The set (12; 20; 32 ) is in direct proportion to the set (3 ; 5 ; 8 ) and the constant quotient is 4, i.e. 12 ÷ 3 = 4; 20 ÷ 5 = 4 and 32 ÷ 8 = 4.


• Indirect proportion– Two quantities are said to be indirectly

proportional if, as one quantity decreases (or increases), the other quantity increases (or decreases).

• Example 1: As the price of petrol increases, the number of litres that you purchase, becomes less.

• Example 2: The set (40; 24; 15 ) and the set (3; 5; 8) are in indirect or inverse proportion to each other since 40 x 3 = 120; 24 x 5 = 120 and 15 x 8 = 120.


• Rate– Unlike ratio, rate is used to compare different kinds

of quantities, e.g. speed is the rate at which you travel in kilometers per hour or metres per second.

– Example 1: A typist can type 650 words in 10 minutes. Her rate of typing is 65 words per minute.

– Example 2: A sprinter can run a distance of 100 metres in 10 seconds. This means he has an average speed of 100m / 10s = 10 m / s.


• Scale on sketches and maps:– When a draughtsman sketches a house, he

cannot make the sketch of the same size as the house. The sketch has to be much smaller.

– A scale or multiplication factor is included on sketches and maps.

– The scale states that a certain ratio exists between the lengths on the sketch and the lengths of the actual building.

Activity 13

1. Measure the sketch of the floor plan of the house:

a. The width of the windows.

b. The width of the doors onto the stoep.

c. The floor area of the stoep.

Activity 13

• The scale for the plan sketch is 1 : 100, which implies that for each one millimetre measured on the sketch, you will measure 100 mm on the actual building. And for 1 cm measured on the sketch, you will measure 100 cm on the actual building. For 2 cm on the sketch you will measure 2 x 100 cm on the actual building etc… Lengths on the building are 100 times that on the sketch. The scale is a conversion factor – you are converting the measurements on the sketch to the measurements on the actual building.

Activity 13

• Now calculate the actual lengths of the following:a. The width of the windows.b. The width of the doors onto the stoep.c. The floor area of the stoepd. If the windows are 1,5 m high and the doors are 2

m high, calculate the interior wall space that has to be painted.

e. Organise the information in a table i.e. tabulate the data.

1200mm = 1,2 metres

1000mm = 1,0 metres17,38m2

Activity 13

2. Convert the following imperial measurements to metric measurements

Imperial measurement Metric measurement

56 inches

1 900 yards

650 miles

32 pints

74 gallons

85 ounces

25 tons

142.24

1737.36

1045.85

18,1856

280.12 litres

2,409 kg

25 tons

Currency rates

• Currency rate is the exchange ratio between different monetary units– Examples:– A South African fruit farmer who exports

apples, pears, grapes or oranges to the UK, knows the exchange rate

– A South African traveller wants British Pounds before boarding the flight to London. He exchanges rands for pounds at the foreign exchange section of a bank

Monetary unit R1 equals One foreign unit equals R

14/10/05 20/03/08 14/10/05 20/03/08

US $ 0.1515 0.1227 6.6019 8.1520

British £ 0.0863 0.0619 11.5889 16.1549

Euro € 0.1262 0.0794 7.9223 12.5957

Activity 14

• With the above exchange rate table, complete the table:

Price per kg Quantity Price in ZAR Price in GBP Price in EUR Price in US$

Apples @ R2.50

2t

Pears @ R3 1,5t

Grapes @ R4 750 kg

Oranges @ R1.50

3t

R5 000

R4 500

R3 000

R4 500

£309.50

£278.55

£185,70

£278,55

€397.00

€357.30

€238.20

€357.30

$613.50

$522.15

$368.10

$522.15

Activity 14

• South African travellers want foreign currency before boarding their flights. They can exchange Rands for Dollars at the foreign exchange section of a bank. Calculate the Dollar value that the following travellers will get at the exchange rate as in the table given above.

Rands USD GBP EUR

R10 000

R4 509

R 6 558

$1 515

$683,11

$993,54

£619

£279,10

£405,94

€794

€358.01

€520.71

Activity 15

• On a road map of Cape Town city: The 38 mm side of one map grid box represents more or less 850 m on the ground.a. Work out the scale of this map. Approximately how

long would it take you to walk this distance?b. It is 48 mm – the shortest distance along the roads –

from the SA Museum in the Gardens to the historical Koopman’s De Wet house. Calculate the actual distance.

c. It is 42 mm directly, as the crow flies, between the above-mentioned two points. Calculate the actual distance

1:22 368; 10 min

1,07 km

939m

Activity 16 – Case study• A certain cell phone contract consists of a connection fee of R99,00, and

a monthly subscription fee of R135,00.• For this they will supply per month:

– 85 off-peak minutes– 15 peak minutes– 15 local SMS’s.

• Questionsa. If you buy this cell phone on the 1st February, how much will you have

spent by the end of October?b. What is the simplified ratio between off-peak minutes: peak minutes: local

SMS’s?c. Estimate whether the time that you spend on your cell phone is in direct

proportion to the time that you are allowed on your contract. Explain your estimate

(R99 + R135 x 9) = R1 314

5,66: 1:1

Activity 16 – Case study

d. The contract further states that the charges are for the first minute, thereafter in units of 30 seconds. Look at the following table and then explain what is meant by the previous statement.


e. Calculate what the 85 off-peak minutes + 15 peak minutes + 15 local SMSs would cost you if you phone from MTN to other networks. How much would you save by using the contract?Cost of calls = 85 calls x R1,15 x 1 minute = R97.75Cost of peak calls = 15 calls x 2,75 x 1 minute = R41.25Cost of SMS = 0.75 x 15 = R11,25Total cost = R150,25 Saving = 150,25 – 135,00 = R15,25


f. The following information is supplied for “Pay as you Go” clients:

Calculate what the same calls as in (e) would cost.Cost of calls = 85 calls x R1,40 x 1 minute = R119Cost of peak calls = 15 calls x 2,85 x 1 minute = R42,75Cost of SMS = 0.75 x 15 = R11,25Total cost = R173


g. *ITRR stands for International Telkom Retail Rates. To make a call to Cuba the ITRR is R20.00 per minute. How much would it cost to make a call lasting 6.5 minutes.

Cost per minute = R20 + R1,80 = R21,80Cost per call = R21,80 x 6,5minutes = R141,70

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

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