MATHEMATICAL LITERACY

MATHEMATICAL LITERACY
JIT

FINANCE (INTEREST) INTEREST RATE AND INTEREST VALUES KEY WORDS/PHRASES
BORROWER - a person or organization that takes out a loan from a bank or financial institution under an agreement to pay it back later, typically with interest. A 'LENDER' - A lender is an individual, a public group, a private group or a financial institution that makes funds available to another with the expectation that the funds will be repaid, in addition to any interest and/or fees, either in increments (as in a monthly bond payment) or as a lump sum.

FINANCE (INTEREST) 'PRINCIPAL' - Principal is most commonly used to refer to the amount borrowed or the amount still owed on a loan, separate from interest. Principal is also used to refer to the original amount of investment, separate from any earnings accrued.

FINANCE (INTEREST) INTEREST – Interest is the cost of borrowing the principal. The little extra money paid for borrowing the principal is what is called “interest” or interest amount. 'INTEREST RATE' is the amount charged, expressed as a percentage of principal, by a lender to a borrower for the use of the money.

FINANCE (INTEREST) Further illustration of key phrases/words
Today Next Year R1 000 Mr Ibrahim Bank R1 000 Mr Ibrahim Bank R100 Principal Borrower Repayment Interest Loan Lender

FINANCE (INTEREST) TYPES OF INTEREST
There are two kinds of interest, namely SIMPLE INTEREST: This is the amount of interest earned per annum (per year) based on the original amount. The same amount is added on the original amount every year. It is an easy and quick method of calculating an interest. Simple interest is money that you pay on a loan, or that you earn on deposits. Simple interest (S.I.) is determined by multiplying the principal (P) with rate of interest (R) and time Interval (T).

FINANCE (INTEREST) COMPOUND INTEREST Compound interest is the interest you earn each time interval, but added on to your initial investment. The interest on the investment for each successive time interval is calculated on the balance from the previous time interval. When working with simple interest calculations, always bear in mind what the CAPS document specifies.

CAPS DOCUMENT SPECIFICATION
Section Level 1: Knowing Level 2: applying routine procedures in familiar contexts Level 3: Applying multi-step procedures in a variety of contexts Level 4: Reasoning and reflecting Interest Explain the meaning and difference between “interest” and the “interest rate”. Identify interest rate values quoted on bank statements. Perform simple interest calculations manually (that is, without the use of a calculator) over multiple time periods. Read values off graphs showing simple and compound investment scenarios Perform compound interest calculations manually (that is, without the use of a formula) over multiple time periods. Complete a table that models a loan scenario and include consideration of a monthly interest calculation, monthly repayment, and monthly amount outstanding on the loan. Draw graphs from given tables of values to represent loan scenarios. Construct a model of a loan or investment scenario without scaffolded or guiding questions. Investigate and describe the impact of increasing the monthly repayments on the total cost of the loan/investment. Investigate and describe the impact of making a lump sum payment into a loan/investment during the first half of the loan/investment period on the total cost of the loan/investment.

TYPES OF INTEREST HINTS/TIPS/BASIC SKILLS:
For easier calculations, convert the percentage to decimals. An easy to remember how is to think of the word "percent" as "per 100." Then, you can convert a percentage into its decimal form by dividing by 100. 5 divided by 100 = 0,05 Note that it is not 5% divided by 100, it's simply 5, so: 5 / 100 = 0,05

COMPOUND INTEREST (C.I)
The concept of compound interest is that interest is added back to the principal sum so that interest is earned on that added interest during the next compounding period. Thus the amount (Principal + Interest) at the end of the interval becomes the principal of the next interval. The total interest over all the interests, calculated in this way is called the Compound Interest or C.I. C.I. at the end of a certain specified period is equal to the difference between the amount at the end of the period and the original principal. C.I. = Final Amount – Principal

TIPS/HINTS FOR WORKING WITH VARYING TIME INTERVALS
Remember that CAPS requires calculations to be done manually, so keep that in mind. When working with compound interest, remember to convert the interest rate to decimal and also convert the time period to the appropriate compound interval.

TIPS/HINTS FOR WORKING WITH VARYING TIME INTERVALS

GRAPHS OF SIMPLE INTEREST AND COMPOUND INTEREST
The graph below shows the result of R1 000 invested over 20 years at an interest rate of 10%. The principal figure is in green. The blue part of the graph shows the result of 10% interest without compounding. Finally, the purple part demonstrates the benefit of compound interest over those 20 years.

GRAPHS OF SIMPLE INTEREST AND COMPOUND INTEREST

The rule of 72 Divide the 72 by the interest your savings are earning. For example, let's say it's 3%. Divide 72 by 3 which will give you 24. So, in 24 years your initial investment will have doubled. If you're receiving 6% then your money will have doubled in 12 years.

C. HIRE PURCHASE Many ‘car finance loans’ offered by garages or car dealerships and some lenders are actually hire purchase agreements. Hire purchase is different from a personal loan because you don't own the car until you have made the last repayment. The main reason you might choose a hire purchase agreement is convenience, as the garage selling you a new car may also arrange your finance. So, it saves you having to visit your bank, building society or credit union to arrange a personal loan.

HOW HIRE PURCHASE WORKS
With hire purchase, the garage acts as an agent for a finance company and earns commission to arrange the finance for you. In this case, the garage is acting as a credit intermediary and must be authorised to act on behalf of the finance company.

COMPARING A HIRE PURCHASE AGREEMENT WITH A PERSONAL LOAN
The main difference between using a personal loan and a hire purchase agreement to buy a car is that with a personal loan you borrow money, pay for your car, and own it immediately. With a hire purchase agreement, you don’t own the car until you make the last repayment. This means you cannot sell the car if you run into problems making your repayments.

Fees and charges You are entitled to a list of all additional charges and fees, so ask the garage for this before you sign any agreement.

RESIDUAL/BALLOON PAYMENT
A residual or balloon payment is a lump sum payment that is attached to a loan. The payment, which has a higher value than your regular repayment charges, can be applied at regular intervals or, as is more usual, at the end of a loan period. Typically, any loan agreement you have that comes with a balloon payment is known as a ‘balloon loan’, which runs over longer terms (although this isn’t always the case, just think, ‘big loan - big final payment’).

RESIDUAL/BALLOON PAYMENT
The amount of your final repayment is very important because: a large final payment could be more than the value of the car at the end of the agreement you will have to have that money available to pay at the end of the agreement if you want to own the car you could end up in a new hire-purchase agreement if you cannot afford to pay it off

MONTHLY REPAYMENTS The monthly repayments on a loan are dependent on three things: 1) Size of the loan; 2) Interest Rate; 3) Length of the loan. NOTE: To calculate the monthly repayments on a loan agreement, the following method can be used: Monthly repayments = (Loan amount ÷ 1 000) × factor

MONTHLY REPAYMENTS The “factor” is a value that is dependent on the number of years that the bond is spread over, as well as the interest rate. The higher the interest rate the higher the factor and, hence, the higher the monthly repayments. Similarly, the shorter the number of years over which the bond is to be repaid, the higher the factor will be and, hence, the higher the monthly repayments. The table below gives the various factors as dependent of the length of the bond and the interest rate:

MONTHLY REPAYMENTS

Activity 1 In groups of 4-6 members
Develop the marking guideline for the questions below; Indicate the level of each item Draw the grid to verify compliance against the CAPS

Activity

ACTIVITY Cont.

ACTIVITY 2 60 MINUTES Form groups as instructed by the presenter
Use the given resources below to develop a task All taxonomy levels must be catered for as well as the mark allocation indicated appropriately. Prepare a suggested marking memorandum for the task At the end of the task, each group will present their solutions

Resource A

RESOURCE B

INFLATION Inflation is the process whereby the general price of goods and services increases over a period of time. The rate of inflation is given as a percentage per annum and represents the average increase in the cost of goods and services from one year to the next. The rate of inflation represents the average increase in the prices of goods and services from one year to the next. This means that the prices of individual items will not necessarily increase at the rate of inflation and will often increase at a different (higher or lower) rate to the inflation rate.

INFLATION Cont. Since inflation refers to the increase in prices, increases in the inflation rate means that people are able to buy fewer goods with their money. In other words, inflation results in reduced buying power.

INFLATION There are many ways to measure inflation, the best know of which is the Consumer Price Index (CPI). The CPI is a value that represents the change in the price of a “basket of goods” in a month in one year to the price of the same basket of goods in the same month of the previous year. This CPI value is then used to determine the inflation rate as a percentage. The “basket of goods” is not a real basket but rather the “idea” of a basket of groceries that we might buy at a supermarket or general dealer. The basket of goods for the CPI consists of about items that are either goods or services. The goods are items such as groceries, clothing, alcohol, tobacco and vehicles. The services include housing, transport, water and electricity rates.

INFLATION

THE IMPACT OF INFLATION:
The main impact of inflation is that it reduces buying power. STOKVEL This is a savings or investment society to which members regularly contribute an agreed amount and from which they receive a lump sum payment. NB: Stokvels usually operate on monthly basis. So remember to convert the interest rate to monthly rate.

FUNERAL PLANS Planning a funeral is always a difficult time for any family. It should be a dignified and memorable final farewell for a loved one, but it can become very expensive. Because of the high costs involved, many financial institutions offer Funeral Cover, where the costs are taken care of. You choose the amount of cover you want, for a predetermined monthly payment. The payment attracts compound interest.

FUNERAL PLANS Cont. The calculations are done in the same manner stokvel calculation is done. At the time of death of a covered member, the amount is paid out after all necessary documentation is received by the financial services provider.

LOANS Sometimes in life, the things we want are so expensive that we cannot afford to pay them cash. In such cases, a loan becomes the alternate option to afford what we want. A LOAN may be explained as a kind of arrangement in which a lender agrees to give money or property to a borrower and the borrower agrees to repay the money or return the property, with interest at some point in time.

LOANS Cont. For instance, when you buy a house, you (borrower), borrow money from a financial institution such as a bank (lender). The loan is expected to be repaid in equal monthly instalments over a given period of time. Usually, there is a predetermined time frame for repaying the loan. As a guideline, your monthly payments should not exceed 30% of your monthly income. We use the formula

LOANS M = P i(1 + i)n (1 + i)n – 1

LOANS Where M is the monthly repayment, P is the amount borrowed, i is the interest rate n is the number of months the loan has to be paid back to calculate the monthly payments on a house. When the money is borrowed, the money is available immediately and the interest is calculated monthly on the outstanding balance of the loan. This is called paying instalment on a reduced balance.

BASIC CALCULATOR SKILLS

EXCHANGE RATES KEY PHRASES
An importer is a person or organization who buys goods in another country to sell in their own country An exporter is a person or organization that sells goods produced in one country to another country. With regards to exchange rates, If more rands are needed for one American dollar, it means the rand is Weak or has depreciated; if less rand is needed for one American dollar, it means the rand is Strong or has appreciated. Using the exchange rate between the British Pound and the South African Rand; EXCHANGE RATES

With regards to exchange rates,
If the exchange rate in Rand per Pound increases over time, then: One needs to pay more rands for one pound The rand becomes weaker against the pound Exporting goods to Britain becomes more favourable, because you will get more rands. Importing goods from Britain becomes more expensive, because you will need more rands to import the goods.

However, if the exchange rate in Rand per Pound decreases over time, then:
Fewer rands must be paid for one British Pound The rand becomes stronger against the Pound Importing goods from Britain becomes less expensive Exporting goods to Britain becomes less favourable.

EXCHANGE RATES Cont. Exchange rates change every day and throughout the day, so it is very important to check the most current rates during that day if you want to determine the latest rates. Below is the exchange rate of the Rand to other major currencies as at 10/04/2017 taken from

HISTORICAL EXCHANGE RATES CHART

INTERPRETING AND ANALYSING GRAPHS
BASIC SKILLS TO FOLLOW IN ANALYZING AND INTERPRETING GRAPHS STEP 1: DESCRIPTION What kind of graph (line graph, bar chart, pie chart) is it? What do the title, key, axes, labels, sectors tell you? What are major changes/differences you can see?

BASIC SKILLS TO FOLLOW IN ANALYZING AND INTERPRETING GRAPHS
STEP 2: INTERPRETATION What are the reasons for changes/differences you described? What are main points/aspects you get from the chart? STEP 3: CONCLUSION What do the results tell you about the topic? Are there any missing information?

Activity 3 - In groups of 4-6 members - Develop the marking guideline for the questions below; - Indicate the level of each item Draw the grid to verify compliance against the CAPS

Study the graph and answer the questions that follow.

60 MINUTES Form groups as instructed by the presenter
Use the given resources below to develop a task All taxonomy levels must be catered for as well as the mark allocation indicated appropriately. Prepare a suggested marking memorandum for the task At the end of the task, each group will present their solutions

RESOURCE A

RESOURCE B

RESOURCE C

RESOURCE D

RESOURCE E

RESOURCE F

RESOURCE G

RESOURCE H

MEASUREMENT PRESENTER : GLADYS MOGOROGA
mmmeam

GROWTH CHARTS Percentiles are measures of spread which divide the data into 100 equal portions. Percentiles are used to analyse the spread of large sets of data which are then recorded as percentages. E.g. In census, the data collected, analysed and is then divided into 100 portions and the report is given as a percentage

GROWTH CHARTS (Cont…) The value at the 5th percentile implies that 5% of values lie below 5th percentile and 95% of the values lie above the 5th percentile. The value at quartile 1 implies that 25% of the values lie below 25th percentile and 75% of the values lie above the 25th percentile. The concept of the percentiles is used when the data is large. This concept will be used in growth charts.

GROWTH CHARTS (Cont…) The curve on the growth chart represents the percentile values of the collected data from different age groups .E.g. height, length, weight, circumference of the head.

GROWTH CHARTS (Cont…) The growth chart is used to compare the BMI of an individual versus the one of their age group. This is also used to determine the health status of individuals.

GROWTH CHARTS (Cont…) Mass and height are used to determine the BMI. The formula for BMI is : BMI = 𝑤𝑒𝑖𝑔ℎ𝑡 ℎ𝑒𝑖𝑔ℎ𝑡²

GROWTH CHARTS (Cont…)

Types of questions should be covered

Reading information form the chart
From the weight vs age growth chart, 2 values will be given and the question will be based on the reading of the third value. If age and percentile curve are given: Draw the vertical line from the age until it touches the percentile curve. Then draw the horizontal line until it touches the required weight. If weight and percentile curve are given: Draw the horizontal line until it touches the percentile curve, then draw the vertical line until it touches the required age

GROWTH CHARTS (Cont…) 2. Understanding significance of the curve
Example: what does it mean if a girl has a weight that places her at the 75th percentile? The concept of quartiles may be used to address this question. The inverted box and whisker plot may be used to explain the significance of the curve as indicated below. At 85th percentile, 15% of the data lies above this curve and 85% of data lies below this curve.

It is important to emphasise the fact that the curve represents the common characteristics of a specific age and gender; as boys and girls don’t grow at the same rate. Using the graph below, we will then say 15% of girls her age have the weight greater than hers and 85% of girls her age have weight less than hers.

3.Understanding significance of positioning on the chart
The health status is used to interprete the significance of positioning on the chart. The BMI vs age growth chart is used to determine the status of an individual. If the age and BMI are given, use the skills used in type 1 to determine the percentile curve and then the percentile curve may be used to determine the status according to the status chart.

EXAMPLE 1 Mr Mkhari the director at Tshwane South district went to OR Tambo international airport to welcome 5 exchange students from Australia. The 3 girls and 2 boys were accommodated at Manhattan hotel in Pretoria for the first 3 days. On the second day, they all had fever .Mr Mkhari took the students to Medforum hospital for medical check-up. Before any medical examination, the nurse took the readings of their mass ,height ,blood pressure and temperature 1.1 Give the possible reason why they all had fever on the second day? (2) 1.2 Ian has the BMI-for-age value that positioned him on the 60th percentile, what does this mean? 1.3 How old is Bruce, if his BMI of 25 kg/m2 positions him on the 85th percentile? 1.4 Joy is a 16 year old girl with a BMI-for-age value that positioned her on the 50th percentile, determine her BMI. 1.5 The nurse said Celine one of the girls has the average BMI. What does this mean? [10]

EXAMPLE 2 Mr Sepeng and Ms Mogoroga, the soccer and netball coaches at Reitumetse Secondary school have taken the 2 girls and 2 boys who were selected to take part in the provincial competition to Dr Bodiba the local doctor, to check their health status. Medical examinations were done and results were given .Use the table below to answer the questions that follows BMI FOR AGE PERCENTILE RANGE WEIGHT STATUS < 5th percentile Underweight 5th to < 85th percentile Healthy 85th to < 95th Risk of overweight ≥95th percentile Overweight 2.1 Determine Lethabo’s health status who is 14 years old, and her BMI is 21kg/m2 2.2 What advise do you think Dr Bodiba will give the educators about Thato, the 16 years old boy who has the BMI of 16kg/m2.Give 2 advises? 2.3.1 Mahlogonolo the 15 years old grade 8 girl, is 150cm tall and weighs 60kg.Determine her health status. 2.3.2 How much weight must she lose for her to be classified under the ‘healthy’ category?

Activity 1: Duration 1 hour 30 minutes
Formulate groups of 8 members each. Formulate the context suitable for the resource. Each group will be allocated the resources to use. Develop a 20 marks task All taxonomy levels should be covered, balance the taxonomy levels. Compile the memorandum and grid analysis for the task. Each group should present.

RESOURCE A

RESOURCE B

MEASUREMENT PART 2 THEMBI MODUBU

Face: Flat surface that forms part of solid 3D object
Surface area: Sum of all surfaces Diameter: A line that divides the circle into 2 equal parts Radius: A line drawn from the centre of the circle to the circumference of the circle. Half of a diameter.

Circumference of a circle= 𝟐𝝅𝒓 where r = 𝑑𝑖𝑎𝑚𝑒𝑡𝑒𝑟 2

Circumference of a circle= 𝟐𝝅𝒓 where r

Area of a rectangle = ℓ x b

Area of a triangle= ½ b x h

Rectangular box Surface area of a rectangular box =2(ℓ x w) + 2(ℓ x h) + 2(w x h)

Surface Area To calculate the surface area ,we may calculate the area of each flat surface separately: Surface area= Area of A + Area of B + Area of C + Area of D + Area of E + Area of F =(3cm x6cm) + (3cm x 8cm) + (6cm x8cm) +(3cm x 8cm) +(3cm x6cm) +6cm x8cm) = 180cm2 Which can also be written as : Surface area= 2 (3cm x6cm) + 2(3cm x 8cm) + 2(6cm x8cm) = 36cm2 + 48cm2 + 96cm2

Volume of a rectangular box
= area of base x h = ℓ x w x h

Volume of a rectangular box = area of base x h= ℓ x w x h

Cylinder A cylinder is a geometric solid that is very common in everyday life, such as a can. If you take it apart you find it has two ends, called bases, that are usually circular. The bases are always congruent and parallel to each other. If you were to 'unroll' the cylinder you would find the side is actually a rectangle when flattened out

Surface area of a cylinder = 2𝛑r2 +2𝛑rh

The surface area of a cylinder can be found by breaking it down into three parts:

The two circles that make up the ends of the cylinder.
The side of the cylinder, which when "unrolled" is a rectangle The cylinder is made up of two circular disks and a rectangle that is like the label unrolled off a can.

Surface Area

The area of each end disk can be found from the radius r of the circle.
The area of a circle is πr2, so the combined area of the two disks is twice that, or 2πr2. The area of the rectangle is the width times height. The width is the height h of the cylinder, and the length is the distance around the end circles. This is the circumference of the circle and is 2πr. Thus the rectangle's area is 2πr × h.

Combining these parts we get the final formula:
∴Surface Area of a cylinder =2πr2+2πrh

Where: π is Pi, approximately 3
Where: π is Pi, approximately r is the radius of the cylinder h height of the cylinder

Volume of a rectangular box
= area of base x h = ℓ x w x h

Volume of a cylinder = area of base x h = 𝜋r2xh

TIPS Always make sure that the units of measurement are the same before doing any calculations. Convert units of measurement between different systems (imperial – metric and vice versa). Convert units of measurement between different scales. Change the diameter to radius before doing any calculations by dividing the diameter by 2.

TIPS (Cont…) Choose the suitable formula
Substitute values on the formula and simplify Diameter should always be divided by 2 to be converted to a radius Use the value of 𝜋 given: 3.142 When dealing with semi-circle, remember to divide the formula of circle by 2

When dealing with complex figures, always divide it into smaller figures, calculate the segments and add the answers together To determine the perimeter you add all sides and the units thereof are similar to that of the length Area determined by multiplying 2 sides, the units are always squared The volume is determined by multiplying 3 sides, therefore the units are always cubed

TIPS (Cont…)

Converting from the unit on the left (big) to the one on the right (small), we multiply by the conversion factor. Converting from the unit on the right (small) to the one on the left(big), we divide by the conversion factor.

ACTIVITY 2: Duration : 1 hour
Answer the following questions • Allocate marks and show explanation of marks. • Compile the grid analysis

ACTIVITY 2(Cont..) Mapula is planning a birthday party for her 22 year old daughter, Gayle. She bought 2 cakes, the bigger one with the radius of 20 cm and a height of 5cm, the smaller one with a diameter of 18 cm and a height of 45mm. She then bought the ready-made icing paste she will use to cover the cakes, she used food colourant and icing gun to decorate the cake as indicated below. She did not cover the top part of the big cake; as she used flowers to decorate it. Use 𝝅= 3.142

1.1. Determine the area that will be covered with icing .
ACTIVITY 2(Cont..) 1.1. Determine the area that will be covered with icing . 1.2. The icing paste is sold at R12.45 per 100 cm² in a container. Mapula claimed that the cost of the icing paste was R Verify through calculation if her claim is correct. 1.3. Determine the total volume of the cakes. Use the formula :Volume of a cylinder = 𝝅r2xh

ACTIVITY 2(Cont..)

You may use the following information:
1 gallon = 3,78541 litres 1 inch = 2,54 cm 1 ml = 1 cm3 Volume = π × r2 × h, let π = 3,142 Surface area of cylinder with a closed lid and base = (2 × π × r2) + (2 × π × r × h)

ACTIVITY 2(Cont..) 2.3.1 Determine the radius of a barrel (drum) in centimetres. 2.3.2 Show, by calculations, that the height of the barrel of oil is 96,82 cm 2.3.3 Calculate the surface area of this barrel in m2

3.2.1 A dairy truck has a cylindrical tank that is used to transport juice. The tank has a radius of 1.5m and a length of 9m

ACTIVITY 2(Cont..) a) Calculate the volume of the tanker using the formula: 𝑽=𝝅 𝒓 𝟐 𝒉, where r is the radius and h is the height (in this case, length). Use the formula to calculate the capacity of the tank in litres. Note 1m3= 1kℓ = 1000ℓ(5) b) Hence determine how many juice cartons can be filled from one tanker truck? (2)

ACTIVITY 3 : Duration : 1 hour 30 minutes
Formulate 4 groups. Each group will be given 2 resources. Group 1 will use resources F and G; group 2 will use resources H and I; group 3 will use resources J and K and group 4 will use resources L and M. Develop a task based on the two resources allocated. Your task should cover all taxonomy levels Jig-saw method will then be applied to formulate new groups The new group should compile a memorandum and show mark allocation for each question. Taxonomy levels allocated should be verified and taxonomy level grid should be completed. Groups should present

RESOURCE F

RESOURCE G

RESOURCE H

RESOURCE I

RESOURCE J

RESOURCE K

RESOURCE L

RESOURCE M

CONCLUSION REMEMBER: WE ARE MAKING MARKS IN THE MINDS OF THE LEARNERS THAT WILL NEVER BE ERASED BY THE DUST OF AGES TO COME.

THANK YOU!

MATHEMATICAL LITERACY

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