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• FAMILY HISTRICK
• Write the number of brothers you have living
• Multiply the brother(s) by 2
• Add 3
• Multiply result by 5
• Add to this number the number of sisters living
• Multiply the total by 10
• Add the number of dead brothers and dead sisters.
Numbers and Numerations systems
• Misconceptions and barriers in teaching and learning number
• Development of real number, up to irrational.
Misconceptions and barriers in teaching and learning number.
• Misconceptions abound in mathematics.
• They are picked throughout the educational life of a child. Some are inherent to the subject, others are the results of teaching techniques that encourage the emergence of such misconceptions.
• Researchers agree that most misconceptions are difficult to overcome.
• Thus it is more important for teachers to make sure that the misconceptions do not arise in the first place.
• First, teachers should be aware of areas that have the potential to generate misconceptions in the minds of the children.
• Then enough work, and examples should be focused on directly contradicting the perceived misconception
• 1. Misconceptions Associated with the Arithmetic Operations
543 subtraction
- 237
=314
An explicit reference to the non-commutativity of subtraction will reduce the occurrence of such errors. Students should be made aware at an early age of the importance of order in subtraction.
• Division: The belief that the divisor should always be smaller than the dividend
• The extent of this problem was investigated by several researchers. Graeber and Baker (1992) put the following question to fifteen nine-year-old and fifteen ten-year-old children:
• QUESTION:
• “Five pounds of trail mix was shared equally by fifteen friends. How many pounds of trail mix did each friend get?”
• Twenty-four out of the group of 30 children responded by performing the operation 15÷5 and giving 3 as the answer. Perhaps more worrying was the fact that 42 percent of a sample of sixty-five trainee elementary school teacher trainees gave the answer 15÷5.
• The source of this misconception lies clearly in the students’ early encounters with division. Such encounters are almost always in situations where a whole number has to be divided by one of its factors.
TRIALS:
• Identify some misconceptions associated with:
• Zero
• Decimals
• Fractions
• Percentages.
• Algebra
Real Numbers
What is the difference between numbers and numerals?
• Number – expresses an idea of quantity. It is abstract. To make a number real we use symbol to represent it.
• The symbols which are used to represent numbers are called
Numerals.
• The numeration system widely used today is the Hindu-Arabic system
• The Hindu-Arabic numerals use only ten symbols called digits to write all the numbers
• 0, 1, 2, 3, 4, 5, 6, 7, 8, 9,
• They are sometimes known as the base ten numerals.
• NB: Number concept is the foundation for the basis of much work in mathematics at the primary years
To begin number work, pupils in the early stages of the school are taken through the following:
• Learning the number names,i.e recital of the number names in the conventional order , one, two, three etc.
• Write the numerals. The child is made to write the symbols of the number names using some activities.
• Tracing the numerals using stencils or cut out.
• Writing in the air
• Writing in the sand (sand tray)
• using my first copybook.
Counting Numbers or Natural Numbers
• What is counting?
• It is assigning number names to a given objects in the right order
• Anytime one is asked to count, there should be object
• Natural Numbers
• Natural numbers are the set of numbers, from 1 to ∞ but it doesn’t include fractional and decimal numbers. They are also known as counting numbers
• The set of all-natural numbers, denoted by N, is the infinite set N= 1,2,3,4 …
• Whole numbers are also a set of numbers starting from 0 to ∞. They also don’t include fractional and decimal parts.
• Whole Numbers: W={0,1,2,3,4,5,6,…}
Natural Numbers and Whole Numbers Properties
• Addition Property
• When two natural numbers are added, it results in natural numbers only.
• Eg: 34+45 = 79
• When two whole numbers are added, it results in the whole number only.
• Eg: 6+0= 6
• Subtraction Property
• When two natural numbers are subtracted it may or may not result in a natural number.
• Eg: 5 – 2 = 3 is natural number
• But 2 – 5 = -3 is not a natural number
• Same is the condition for whole numbers. If two whole numbers are subtracted, then its not necessary it will result in the whole number.
• Multiplication Property
• When two natural numbers or whole numbers are multiplied, then it always results in natural numbers and whole numbers respectively.
• Eg: 2 X 6 = 12 is a natural number
• 4 X 0 = is a whole number, where 4 and 0 are also whole numbers.
• Division Property
• Division of two natural numbers and whole numbers may or may not result in natural and whole numbers respectively. It means if the result is in fraction or decimal, then they are not considered as natural and whole numbers.
• Eg: 4/2 = 2 is natural as well as the whole number
• But 5/2 = 2.5 is neither natural nor whole number
Integers
• The limitation of the set of whole numbers is exposed when a bigger number is subtracted from a smaller one like 3-7. It is, therefore, necessary to expand the set of whole numbers to include negative numbers.
• In Maths, integers are the numbers which can be positive, negative or zero. These numbers are used to perform various arithmetic calculations
• Set of integers are denoted by Z={…,-2,-1,0,1,2,…}
• Types of Integers
• Integers comes in three types:
• Zero
• Postive Numbers(Natural) and
• Negative Numbers(Negatives of Natural Numbers}
• Integers are not just numbers on a paper; they have many real-life applications. The effect of positive and negative numbers in the real world is different. They are mainly used to symbolize two contradicting situations.
• For example, when the temperature is above zero, positive numbers are used to denote temperature whereas negative numbers denote the temperature below zero. They help one to compare and measure two things like how big or small or more or fewer things are and hence can quantify things.
• Some real life situations where integers come into play are player’s scores in golf, football and hockey tournaments, the rating of movies or songs, in banks credits and debits are represented as positive and negative amounts respectively
• Sample Problem:
• Question : Plot the following integers on the number line: -121, -97, -82, -67, -43, -10, 0, 10, 36, 55, 64, 77, 110, 126, 147.
• The rational numbers are numbers which can be expressed as a fraction and also as positive numbers, negative numbers, and zero. It can be written as p/q, where q is not equal to zero. Denoted by Q
• Rational word is derived from the word ‘ratio’, which actually means a comparison of two or more values or integer numbers and is known as a fraction. In simple words, it is the ratio of two integers.
• Example: 3/2 is a rational number. It means integer 3 is divided by another integer 2
• Examples of Rational Numbers
• number 9 can be written as 9/1 where 9 and 1 both are integers.
• 0.5 can be written as ½, 5/10 or 10/20 and in the form of all termination decimals.
• 81−−√ is a rational number, as it can be simplified to 9 and can be expressed as 9/1.
• 0.7777777 is recurring decimals and is a rational number
Irrational Numbers
• The numbers which are not a rational number are called irrational numbers. Now, let us elaborate, irrational numbers could be written in decimals but not in fractions which means it cannot be written as the ratio of two integers.
• Irrational numbers have endless non-repeating digits after the decimal point. Below is an example of the irrational number
• 2 =2.828… The is no special symbol to denote irrational numbers
• We consider the complement of rational numbers i.e
• List of Irrational Numbers
• Similarly, as we have already defined that irrational numbers cannot be expressed in fraction or ratio form, let us understand the concepts with few examples.
• 5/0 is an irrational number, with the denominator as zero.
• π is an irrational number which has value 3.142…and is a never-ending and non-repeating number.
• √2 is an irrational number, as it cannot be simplified.
• 0.212112111…is a rational number as it is non-recurring and non terminating.
• There are a lot more examples apart from above-given examples, which differentiate rational numbers and irrational numbers.
Real Numbers
• Real numbers can be defined as the union of both the rational and irrational numbers. They can be both positive or negative and are denoted by the symbol “R”. All the natural numbers, decimals, and fractions come under this category. See the figure, given below, which shows the classification of real numerals.
UNIT TWO
OPERATIONS AND PROPERTIES ON INTEGERS
Note : Point out to pupils that the meaning of ‘plus’ and ‘minus’ are different from those of ‘positive ‘ and “negative”
“positive” and “negative” tell us whether a number is greater than or less than zero. “plus” and “minus” on the other hand ,tell us to “add” and to “subtract” numbers. Thus ,pupils should be taught -2 as “negative 2” and not “minus 2”.
Pupils should understand that when we write (-6)-(-3) we mean “negative 6 minus negative 3”
TRIALS:
Write down three practical ways you would use to distinguish between 3 and -3.
ANS:
• Three meters high above ground level for 3 and three meters deep (below ground level ) for -3.
• 3 steps forward from a marked destination and three steps backwards from the marked destination (spot).
• Having an amount of GH¢ 3.00 (credit) for 3 and owing GH¢3.00 (debt) for -3
• Temperature 3 degrees above zero and 3 degrees below zero degrees.
Addition of integers
• Using number line:
• We can do addition of integers by associating it with movement on the number line. Pupils are ask to draw number on the floor and move along this model according to the specific direction.
• NOTE:
• In this model the operation of “ addition” is represented by a “forward movement”
• Positive number indicates that you should “face right or positive direction” while negative number means that they should “ face left or the negative direction
Addition of integers
• (-3)+(-4)
• To find this sum, begin at 0, face left or the negative direction, and move to -3. Still facing the negative direction, move forward 4 units, ending at -7
• Hence, pupils discover that (-3)+(-4)=-7
• Consider the sum (-4)+(+3)
• Begin at 0, face negative direction and move to -4. Now, face the positive direction and move forward 3 units, ending at -1.
• Hence pupils discover that (-4)+(+3)=-1
Addition of integers using charged particles
• Using charged particle model.( -2) + 3
• Guide pupils to take 2 negatives charge particles and 3 positives charge particles.
• Guide pupils to pair or combine the negative and positive charge particles to form a circle, these circles represent neutral charges and are considered as zeros
• Guide pupils to count the uncombined / unpaired charge particles
• Pupils realized that we are left with 1 positive charge particle.
• Therefore( -2) + (+ 3) = + 1
• TRY :: – 3 + (-6)
5 +(-4)
-3+6
Subtraction of integers
• Using number line :
• In this case the operation of “ subtraction” is represented by a “backward movement” . A positive number denotes facing the positive direction while a negative number means facing the negative direction on the number line.
• Example : (+4) - (+6)
• Ask pupils to begin at 0 on the number line, face the positive direction and move to +4 , while still facing the positive direction, the pupils move backward 6 units, ending at -2.
• Try
• Example : 2
• (+6)–(-3)
• We begin at 0, and since 6 is positive we face the positive direction on the number line and move to +6. Now, -3 is negative, so face negative direction and since we are expected to subtract this number we move backward 3 units, ending at +9.
• Group Assignment
• Try : 3-5 using charge particles
• -2+(-6) using charge particles
-3 – (-5) using charge particles
Multiplication of an integer by a positive integer
• We can use the number line model to help explain integer multiplication to pupils. (4) × +2 we can think of this as “(+2) four times” or (+2)+(+2)+(+2)+(+2).
• Using number line
• Start at 0, face the positive direction and move two units to +2; still facing the positive direction, move forward again two units to 4, then two more units and another two units, ending at +8. Hence 4× 2=8
Using charge particles or positive and negative counters
• 4×2
• Guide pupils to take 4 groups of two charged positive particles.
• Guide pupils to realize that all together, there are 8 positive charge particles.
• Multiplying a negative and a positive number ,
• -3 × 4 :: by the commutative property of multiplication
• This can be interpreted as four(4) groups of three (3) negative charge particles
• Pupils put together the particles and get 12 negative charges.
Therefore -3 × 4 =-12
• use number line ..(Assignment)
Multiplying negative integer by a negative integer
• Using an intuitive method or pattern to solve -3 × -4
• Children already have a fair idea of multiplication of a positive and a negative integer.
• Lead pupils through this activity :
• -3 × 4=-12
• -3 ×3=-9
• -3 × 2=-6
• -3 × 1=-3
• -3 × 0=0
• -3 × -1=3
• -3×-2=6
• -3 × -3=9
• -3 ×-4=12
• Let pupils study the pattern and they can realize a pattern that emerges in which each product is three more than its predecessor.
• This pattern yields the conclusion that the product of two negative integers is positive.
• Group assignment:
• How will you convince a basic child that any number multiply by one gives the same example (hint: use pattern making) and
• Multiply any number by zero, you will get zero.
Division of integers
• To teach division of integers, revise with the children, grouping (repeated subtraction ) and partition (sharing) approaches to whole numbers with pupils.
• Example 8÷2
• Ask pupils to form a collection of eight positive charged particles and use the sharing approach to determine the quotient.
• Sharing 8 among 2 will produce 4 in each group.
• This means 8÷2=4
• NB: use diagram to show the charge particles
• You can also use the grouping approach to get the same result.
• -12÷3
• Ask pupils to collect twelve negative charged particles and use sharing approach o determine.
• Guide pupils to interpret the problem as three pupils sharing 12 negative charged particles. How many will each get?
• Pupils will realize that each will get four negative charge particles (-4)
• Therefore -12÷3=-4
• Group assignment:
• using charge particles describe the steps you will use to guide a pupil work
8÷(-2)
-6÷-2
• Commutative Property
• If we m and n are the numbers, then the general form will be m + n = n + m for addition and m.n = n.m for multiplication.
• Addition: m + n = n + m. For example, 5 + 3 = 3 + 5, 2 + 4 = 4 + 2
• Multiplication: m × n = n × m. For example, 5 × 3 = 3 × 5, 2 × 4 = 4 × 2
• Test subtraction and division
• Associative Property
• If we m, n and r are the numbers. The general form will be m + (n + r) = (m + n) + r for addition(mn) r = m (nr) for multiplication.
• Addition: The general form will be m + (n + r) = (m + n) + r. An example of additive associative property is 10 + (3 + 2) = (10 + 3) + 2.
• Multiplication: (mn) r = m (nr). An example of a multiplicative associative property is (2 × 3) 4 = 2 (3 × 4).
• Distributive Property
• For three numbers m, n, and r, which are real in nature, the distributive property is in the form of :
• m (n + r) = mn + mr and (m + n) r = mr + nr.
• Example of distributive property is: 5(2 + 3) = 5 × 2 + 5 × 3. Here, both sides will yield 25.
• Identity Property
• There are additive and multiplicative identities.
• For addition: m + 0 = m. (zero is additive identity)
• For multiplication: a × 1 = 1 × a = a. (1 is multiplicative identity)
• Discuss: inverse property
• closure property
UNIT THREE
SET OF NUMBERS
The concept of set :
• We often use the word ‘set’ to describe a collection of objects, quantities or numbers
• A set is a well-defined collection of objects, quantities or numbers that have a given property or properties in common
• For example ‘ set of beautiful girls in a class ‘ is not well defined, since beauty they say lies in the eyes of the beholder.
• In other words, the word beautiful is relative.
• Hence, set must always be well defined. For example
• ‘set of even numbers less than 12’ is well defined
• The objects that make up a set are called members or elements of the set. The elements of a set may be named in a list or may be given by a description enclosed in braces { }.
Trial:
• Explain to Primary 6 pupils why ‘set of numbers is not well defined
• What will be your response if a primary 6 pupil asks you ‘sir/madam is
{a,a,a,a,b,b} a set’
Types of sets
• Finite and infinite set
• a set is finite if the first and the last members can be found.
• A = {2, 4, 6, 8}, (ii) B = {a, b, c, d, …, x, y, z},
• (iii) C = {red, yellow, green, blue, black}.
• Infinite set : A set is infinite if the first or last members or both cannot be found.
• For example, the following sets are infinite:
• (i) N = {1, 2, 3, 4,…}, (ii) Z = {…,–3, –2, –1, 0, 1, 2, 3,…},
• (iii) P = {…,7, 9, 13, 15}
Relations on set
• Equal set and equivalent sets
• The sets A = {6, 2, 1, 5, 3} and B = {3, 1, 2, 5, 6} contain the same elements.
• Two sets are equal if they contain identical elements
Equivalent sets
• The two sets C = {a, b, c, d, e} and D = {1, 2, 3, 4, 5} are not equal. It can be seen, however, that C and D has the same number of elements, that is n(C) = n(D) = 5.
• Two sets are equivalent if they have the same number of elements
• Subsets
A set P is said to be the subset of the set Q if every elements of P belong to the set Q
listing subset of a given set
• QUESTION :
• Show and explain clearly how you will guide JHS pupils to determine the number of all the subset which could be formed from a set containing n members
• Suggested solution:
•: I will guide the child to write down all the subjects of the set containing 1, 2, 3, 4, etc., members as shown below:
• I will guide the pupils to write the number of members in each set and number of members in the subset
No_ of Element. | No_ of subsets | Relations | |
• 0 | 1 | = | 20 |
• 1 | 2 | = | 21 |
• 2 | 4 | = | 22 |
• 3 | 8 | = | 23 |
• 4 | 16 | = | 24 |
. .
. .
. .
• n | 2n |
• I will guide the pupils to write the number of subsets in the index form
• I will guide the pupils to observe the exponents; pupils will now realize the exponents are the same as the number of the elements.
• I will lead the pupil to state in general, if a set has n elements, then its number of subsets is given by 2n
• Group discussion: ‘Universal set’
• Properties of A and ′
• ∪ ′=∪
• ∩ ′=∅ {}
• ′ ′=A
• ∪ ′’=∅
• ∅′ =∪
Operations on set
• Explain intersection of sets with examples to JHS pupil.
• Considering P = {2, 3, 4, 5}
• I will guide the pupil to form a set of members that can be found in P and Q that is = {3, 4}
• I will explain to the pupils, that, the set {3 , 4} is the intersection of set
P and Q written as P∩Q = { 3, 4}
Assignment
• Properties of operations on sets
Venn diagram
Group assignment
Word problem
•
Using set to solve problems
UNIT FOUR
EVERYDAY AND COMMERCIAL ARITHMETIC
RATIO
▪ Ratio is a pair of positive numbers that are used to compare two quantities or two sets of the same unit or dimension
▪ Example :
▪ Ama is 9months and Mary is 3 years .The comparison could be 9months : 36 months
Assignment
Proportion
• An equality of ratios is called Proportion which is useful in problem solving
• Note: We have two types of proportion:
• Direct proportion
• For example, if 1 pencil costs GH₵ 2.0
•2 pencils cost GH₵ 4.00
•3 pencils cost GH₵ 6.00
• This proportion shows that, if there is increase in the number of pencils, we will have increase in the total amount of money.
• Everyday examples:
• The more hours a car spend on the road, more distance it cover
• The more we write with pen, the more ink we use.
• Inverse proportion (indirect)
For example, if we have a piece of work and every worker puts in the same amount of energy, more people will take less time to do the work. A small number of people will take more time to do the same work.
• Everyday examples:
• The faster a lorry runs, the less time it will take to cover the given distance.
• The slower the vehicle, the more the time it will take to cover the given distance
Solving problems on direct proportion
• We can employ two methods
• Unitary methods to solve the problem
• Use ratio method to solve the problem
• Example:
• Q. 8 pencils cost GH₵ 400.00. Find the cost of 5 pencils if the cost of a pencil is the same for any number of pencils
• Unitary method 1
• Explain to pupils that for this method, we always find the value of a unit or just one item after which we then multiply the result by the number of the items that need to be worked out.
• 8 pencils cost GH₵ 400.00, therefore 1 pencil cost = 400Τ8 = GH₵ 50
• Hence, the cost of 5 pencils = 50 × 5 = GH₵ 250.00
• Method 2 (Ratio method)
• Use ratio to find the unknown
• Set the problem as two equal ratios with the unknown which we later find
• So here, 8 pencils cost GH₵ 400.0
• Let 5 pencils cost =
• In the ratio form, we have 8:5 = 400:
• | 8 | = |
| 400 |
|
| ||
5 |
|
| ||||||
• |
|
|
| |||||
8( ) = 5 (400) | ||||||||
• | 8 | = | 5 (400) |
| ||||
8 |
|
| ||||||
• |
| 8 |
| |||||
= 5 (50) | ||||||||
• | = GH₵ 250.00 | |||||||
• Therefore the cost of 5 pencils = GH₵ 250.00
Assignment
RATES
• Rates is the comparison of one measure with another of a different dimension. For example , the rate of speed of a car is the distance traveled compared to the time used to cover the distance, and it is usually measured in kilometres per hour.
• Rates can also be expressed as a percentage as found in simple interest, commission, and discount.
• Example , if kofi deposited $200 in her saving account at a simple interest rate of 10% , it means that the bank will pay her 10% of $200 each year in addition to her principal amount.
• During reduction sales and promotion sales, prices of goods are normally reduced by a given percentage of the original price.
• Which means sales price is 90% of the original price if rate of reduction is 10%
Work examples:
• It takes a water pump 18 hours to fill a 2,025-litre capacity polytank. How many litres of water were pumped into the polytank in each hour?
• Suggested solution: number of litre in 1 hour = |
|
| |||
ℎ | |||||
• | = | 2,025 |
|
| |
|
|
| |||
|
| 18 |
|
| |
• |
| = 112.5 litres per hour | |||
SCALES
• Drawing an accurate but different-sized image of an object is called scale drawing.
• In scale drawing, the ratio of the length on the drawing to the corresponding length of the actual object is called the scale.
• Thus, if the scale of a map is 1:100; it means that 1 unit on the map is the equivalent of 100 units on the actual site.
Group Assignment
Percentages
• Reduction in the price of goods and services. ( DISCOUNT)
Trials
A shop owner gives 20% discount on the marked price of all woolen carpets in his shop. Mr. Brew paid GH₵ 250 for one such carpet . How much was the marked price of the carpet?
• A wholesaler allows a total trade discount of 13% on goods purchased up to GH₵ 6900.00 by a retailer. How much did a retailer pay for
goods worth: a). GH₵ 3400.00 b). GH₵ 12000.00 c). GH₵ 9550.00
• a.) if the marked price of an article is GH₵ 25,000 ; find the total amount saved by a distributor for paying promptly
• b.) if a distributor pays GH₵ 11,700 promptly for an article , find the marked price
Commission
• Commission is an amount paid to shop assistant, salesmen or representatives for his/her services. The commission paid to the agent is usually calculated in percentages based on total sales or quantity of goods sold/purchased by the agent.
• Example:
• A salesgirl has a commission of 5% on every GH₵ 10.00 worth of items sold. Calculate her commission on a particular day, when she made sales of GH₵ 570.50
• Commission= 1005 × 570.50= 28.525
• ℎ GH₵ 28.53
• An airline paid its travel agents GH₵ 72 for each plane ticket sold for GH₵ 600. What was their rate of commission?
• Suggested solution:
• Let x% be the rate of commission paid by the airline.
• 100 × GH₵600= GH₵ 72
• X= 72600×100 = 12%
• The airline’s rate of commission is 12%
• Evelyn is a fee-paying university student and wants to raise GH₵ 72 to pay for her second semester fees. She decides to work for Aunt Julie , who pays 7.5% commission to her sales staff. How much money must Evelyn make on sales in order to earn her fees?
• Suggested solution:
• Let GH₵ y be the total amount of money that Evelyn must make .
• Then the GH₵ 72 commission she wants will be earned by:
• GH₵72=1007.5 ×
• = 72 ×100= GH₵ 960
7.5
• Evelyn must make sales worth GH₵ 960 to earn her fees.
Assignment:
• Salesman of JOKA supermarket receives a basic salary of GH₵
1,100,000.00 . In addition , he receives 5% commission on sales up to
GH₵ 10,000,000.00 . For sales over and above GH₵ 10,000,000.00 he receives 8% commission on the excess sales.
• Calculate the total income of a salesman who makes sales of GH₵16,800,000.00
Simple Interest
• The money paid by banks on amounts saved is called interest
• Interest is also paid on money borrowed by account holders from banks or credit unions or from moneylenders.
• The amount saved or borrowed by a person is called the principal.
• The interest calculated on an initial principal at the end of a given period of time is called simple interest .
•SI=PXR%XT
• SI = × ×
• Amount = Principal + Interest
Example:
• Mr. Mensah invested GH₵ 2,500 at 21% per annum simple interest
• a.) Find the interest earned at the end of 4 years
• b) The total amount accrued at the end of four years.
Example 2
• A man borrowed a sum of money from a bank at an interest rate of 19% . After 1 year , he paid GH₵ 89.25 to settle the loan and the interest. How much did he borrow from the bank?
Hire Purchase
• This is a system under which someone may buy an item and pay by instalments.
• Usually, the one buying the item is made to pay some deposit and pay the rest in regular installments until the item is fully paid.
• Example:
• A man sold his car at GH₵ 8,800 to a friend under the following hire purchase terms.
• An initial payment of 20% of the price and the balance paid at 15% simple interest per annum in twelve monthly installments.
• Calculate
• a. the amount paid, every month by the friend
• b. the total amount the friend paid for the car.
Assignment:
• The cash price of a freezer was GH₵ 600.00 . A man paid 30% of the cash price as deposit. He then paid GH₵ 65.00 a month , for eight months.
• a. How much did he pay altogether for the freezer?
• b. find the interest charged.
• c. find , also the approximate rate of interest
• A female teacher bought an electric sewing machine GH₵ 350.00 from a company on hire purchase. She made an initial payment of 35% of the price and the balance paid at 18% simple interest per annum in twelve monthly instalments. Calculate :
• a. the amount she paid every month
• b. the total amount she paid for the machine.
Compound interest
• Practical Examples:
• Calculate the compound interest on GH₵2,000, 000.00 for 2years at 3 12% per using .
Profit and loss
• Basic concepts:
• Example :
• A man makes a profit of 15% by disposing of an article at $1725. What is his actual profit ?
Assignment:
• Bala sold an article for $6,900.00 and made a profit of 15%. If he sold it for $6,600 how much will be the profit percent ?
• A man invested a certain amount of money in two separate projects in the ratio 3:2 . His profit was calculated on interest rates of 5% and 4% simple interest, respectively . If after two years , he received a sum of $9,200.00 as his profit for the two projects. Calculate the total amount he invested in the two projects.
Taxation( income Tax)
• In dealing with income tax , some basic definition must be well understood .
• Example :
• Suppose this year :2008, income tax will be calculated according to the following schedule:
Taxable (chargeable) income | Rate of tax |
|
|
First GH₵ 50.00 | Free |
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Next GH₵ 30.00 | 5% |
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Next GH₵ 30.00 | 10% |
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Next GH₵ 60.00 | 15% |
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Next GH₵ 60.00 | 20% |
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Next GH₵ 60.00 | 30% |
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Next GH₵ 60.00 | 40% |
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Next GH₵ 350.00 | 50% |
• Mr. Odum’s annual salary is GH₵ 520.00 . Calculate
• a. his chargeable income b. the annual tax he will pay
• c. the percentage tax he pays
Suggested solution :
Amount GH₵ | Rate of tax | Tax | Remaining taxable |
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| income GH₵ |
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First GH₵ 50.00 | Free | 0 | 470 (i.e 520-50) |
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Next GH₵ 30.00 | 0.05 × 30 | GH₵ 1.50 | 440.00 (i.e 470-30) |
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Next GH₵ 30.00 | 0.10 × 30 | GH₵ 3.00 | 410.00 |
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Next GH₵ 60.00 | 0.15 × 60 | GH₵ 9.00 | 350.00 |
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Next GH₵ 60.00 | 0.20 × 60 | GH₵ 12.00 | 290.00 |
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Next GH₵ 60.00 | 0.30 × 60 | GH₵ 18.00 | 230.00 |
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Next GH₵ 60.00 | 0.40 × 60 | GH₵ 24.00 | 170.00 |
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Rest GH₵ 170.00 | 0.50 × 170 | GH₵ 85.00 | - |
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Assignment:
UNIT FIVE
NUMBER BASES AND MODULAR ARITHMETIC
INTRODUCTION
• Number bases is a system of counting, where one could only count up to a certain number known as the base.
• The system of numbers which are used today is based on ten symbols namely zero, one, two,…nine. This system is called the ‘decimal numeral’ because it is based on ten.
• In this system the values of a symbol depends on the position or place in the numeral. This is known as ‘place value’
• Counting with only one hand, one might count in base five instead of ten.
• Most electronic circuit operate with two state ‘on and of’ , the base two or binary system
• Basically, if we want to work in base three, we use three symbols 0,1,2. Base four : four symbols: 0,1,2,3 etc.
• It can therefore seen that in any base the highest digit is one less than the base. In base eleven and twelve , we use ‘T’ and ‘E’ to represent ten and eleven respectively.
• OPERATIONS ON NUMBERS
• USING BUNDLES OF TEN AND SINGLE STICKS (BASE TEN)
• We describe below how we can use this to find the sum of 37 and 25
• Children are guided to represent 37 by taking three bundles of ten sticks and seven single (loose) sticks.
- guide the pupil to represent 25 by also taking two bundles of ten sticks and 5 single sticks
• Guide pupils to put them together to obtain five bundles of sticks and twelve single stick
• Guide pupils to regroup the twelve single sticks to obtain one bundle of ten sticks and two single sticks
• We now have altogether six bundles of ten sticks and two single sticks
Explain to the pupil that the six bundles of ten sticks represent
60 and the two single sticks represent 2 ones altogether giving
us 62 the result
Therefore 37 + 25 = 62
• Block = 1000
• Note: 10 cubes = 1 long
• 10 longs = 1 flat
• Guide pupils represent 39 by three longs and nine cubes
- Guide pupils to put them together to obtain seven longs and eleven units
- Guide pupils to regroup the eleven units to obtain one long and one unit’
- We now have altogether eight longs and one unit
Together
- Explain to pupils that the eight long represent 8tens and one unit represents one which gives us 81
- Therefore 42 + 39 = 81
Group Assignment:
• Using base five multi-base material
• Find the sum of the following in base five
• 12 +24
• 13+33
• 14+11
SUBTRACTION OF TWO OR MORE DIGIT WHOLE NUMBERS
• Problem: 34 – 18
using concrete material :: Dienes’ base ten blocks (materials).
• Guide the pupil to represent 34 by taking three longs and four cubes/units
• Explain to the pupil that we cannot take away 8 cubes from the four cubes, so there is the need to decompose (breakdown) one of the longs in order to make it possible to take the 8 cubes
• Guide the pupil to note that, after the decomposition we now obtain two longs and fourteen cubes
• Now guide the pupil to subtract 18 from 34 by taking 8 cubes from the 14 cubes and one long from the two longs to obtain one long and six cubes at the end
- Guide the pupil to note that, the one long represent 10 and 6 cubes represent 6 ones
- Therefore 34-18=16
• After developing the concept of number bases using the manipulatives, we now guide them to come out with the algorithm.
Converting any numerals to base ten numerals
• Numbers written in any base use the place value system.
• This idea helps us to convert any number in any base to a number in base 10, by expressing the number in expanded form.
Converting a base 10 numeral to other bases
• Any base 10 number can be changed into a number in another base by applying repeated division by the number base required.
• CONVERT 128 as a number in each of the following bases.
• 1. base 2 2. base 5 3. base 8
• 4. base 12
Conversion between non-decimal bases
• To convert between non-decimal bases, first convert to base ten and then convert your results to the required base
• EQUATIONS INVOLVING NUMBER BASES
• This approach is the same as converting from other base to base ten, except here, the base itself may not be a numeral.
• After expansion, we may get linear equation or a quadratic equation and then solve for the unknown
MODULAR ARITHMETIC
• Some events in real life are repetitive. For example , each week has seven days , one day has 24 hours and every year has 12 months.
• We can think of this events as a cycle
• Using the example of the days of a week, the number of days in the cycle is called its modulus and each day is called a cyclic variable.
• This type of circular arithmetic is called modular arithmetic
• Application :
• If we take Sunday as the starting point (0),
• Starting at Sunday, what day will it be in 15 days?
• We can think of modular arithmetic as a special base arithmetic in which only the remainder is reported.
• The modular arithmetic base 7 is written as mod 7.
• In general, in any mod n arithmetic , the remainder will form a set with elements {0, 1 , 2 , 3, … n-1}
• Trials :
• What day of the week is the 39th day after Friday.
• List all the elements of mod 5
• Simplify : 29 mod 5
Operations in modular arithmetic
• In modular arithmetic , you can also add, subtract and multiply
• Example :
• Simplify 57 + 28 mod 6 ≡ 85 mod mod 6
• ≡ 1 mod 6
• Simplify 18 – 7 mod 4 ≡ 11 mod 4
• ≡ 3 mod 4
• Simplify -27mod 8
• NB: If a number is negative, add the simplest multiple of the modulus that will make the result positive when added and simplify to arrive at the answer
• Simplify -27 mod 8 ≡ -27 +(8× 4)mod 8 (the simplest multiple is 32)
• ≡ -27 + 32 mod 8
• ≡ 5 mod 8
Assignment
UNIT SIX
RELATIONS AND FUNCTIONS
In Chapter 1, you learnt that a set is any well defined collection of objects or numbers. In this Chapter, we shall examine the connection or association between the elements of one set with the elements of another set.
Consider the sets X = (a, b, c, d, e) and associated to the element of Y using arrow as shown in Fig. 7.1. The arrow diagram suggests the relation ‘is double’.
Suppose the headmaster of Method High School, asked 10 teachers to stay after school for a special meeting. Fig. 9.2 shows the names of the teachers and six soft drinks.
The arrows associate teachers with the soft drinks they prefer. For example, Mr. Kingsley Osei and Mr. James Tetteh prefer Coca Cola Mr. Carl Kotei, does not like any of the six
soft drinks. It can | be seen that there is a relation between the sets | X of teachers and Y of | |
soft drinks. In Fig. | 9.1 and | Fig. 9.2, a relation is shown by a “one way” arrow elements of | |
another set Y | |||
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Example 7.10
The sets A and B are defined by A = {cat, dog, lion, bird, cock, cow} and B ={purrs, whistles, barks, crows, lows, roars, talks}
(a) State the relation from the first set A to the second set B.
(b) Use an arrow diagram to march the members of set A to that of set B.
(c) Write down the domain, the co-domain, and the range of each relation
Example 7.12
Determine the domains and ranges of the following relations.
Solution |
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a) | The domain = | {1,2,3,4} |
| and | the range = {2,4,6,8} |
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b) | The domain = | {-3,-2,2,3} | and | the range={4,9} |
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c) | The domain = | {1,6} |
| and | the range = {-2, -1, 1, 2} |
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Exercise 7.1
For each of the following pairs of sets, the relation from the first set to the second set is stated. Match the numbers and identify the type of relation.
a) A = {father, man, pig, dog, lion, hen}, B = {pup, chicken, piglet, child, cub}
Creature → The young.
b) M = {convict, king, priest, soldier, monk}, N = {barracks, palace, temple, monastery, prison}
Person → Traditional home.
c) K = {army, band, bench, board, choir}, L = {musicians, ants, singers, soldiers, bishops, magistrates, directors}
Term → ‘Group of’
d) P = {-2, -1, 0, 1, 2, 3}, Q = {1, 1, 4, 9}
Number → Square of number
e) S = {36, 48, 52}, T = {3, 4, 6, 9, 12}
Number → Factor of number.
f) U = {Kojo, Kwabena, Kwaku, Yaw, Kofi, Kwame, Kwasi},
V = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
7.21 is {-2, -1, 0, 1, 2, 3} and the range is {-4, -2, 0, 2, 4, 6}. Write the domain and range of the mapping in Fig. 7.22.
Example 7.20
Make a table of the mapping defined by the rule x → 3x – 2 on the domain {-3, -2, -1, 0, 1, 2, 3}
Solution
The image of x = -3, for instance, is given by y = 3(-3) -2 = -11. The following is the table of the mapping relating the values in the domain to that of the co-domain.
x | -3 -2 | -1 | 0 | 1 | 2 | 3 |
↓ | ↓ ↓ | ↓ | ↓ | ↓ | ↓ | ↓ |
3x – 2 -11 -8 | -5 | -2 1 4 7 | ||||
Example 7.21
Make a table of the mapping defined by each of the following rules on the domain {-2, -1, 0, 1, 2, 3}
Find the rule of a mapping
Determining the rule of a mapping depends on whether the mapping is linear or exponential.
1) Linear mapping
A mapping is said to be linear if the difference between consecutive images is constant. Consider the two sets X = {0, 1, 2, 3, 4} and Y = {2, 5, 8, 11, 14}. The first term of the set X is denoted by a whilst that of the set Y is denoted by b. Each element in X (denoted by x) can be associated to the elements of Y (denoted by y) using arrows as follows:
4 | 5 | 6 | 7 | d) | 3 | 4 | 5 | 6 | |
| ↓ | ↓ | ↓ | ↓ |
| ↓ | ↓ | ↓ | ↓ |
| 7 | 5 | 3 | 1 |
| 1 | 7 | 13 | 19 |
e) | 0 | 1 | 2 | 3 | f) | 1 | 2 | 3 | 4 |
| ↓ | ↓ | ↓ | ↓ |
| ↓ | ↓ | ↓ | ↓ |
| 1 | 2 | 4 | 8 |
| 3 | 6 | 12 | 24 |
g) | 3 | 4 | 5 | 6 | h) | 4 | 5 | 6 | 7 |
| ↓ | ↓ | ↓ | ↓ |
| ↓ | ↓ | ↓ | ↓ |
| 5 | 10 | 20 | 40 |
| 12 | 6 | 3 | 1 ½ |
i) | 5 | 6 | 7 | 8 | j) | 0 | 1 | 2 | 3 |
| ↓ | ↓ | ↓ | ↓ |
| ↓ | ↓ | ↓ | ↓ |
| -4 | -1 | 2 | 5 |
| 7 | 14 | 28 | 56 |
k) | 0 | 2 | 4 | 6 | l) | 1 | 3 | 5 | 7 |
| ↓ | ↓ | ↓ | ↓ |
| ↓ | ↓ | ↓ | ↓ |
| 1 | 3 | 5 | 7 |
| 2 | 5 | 8 | 11 |
Definition of a Function
Given a relation in x and y, we say “y is a function of x” if for every element x in the domain, there corresponds exactly one element y in the range.
Relation Vs. Function
Recall: A relation is a set of ordered pairs. A relation produces one or more output numbers for every valid input number. eg. {(–2, 3), (3, 4), (4, 7), (–2, 6)}
A function is also a set of ordered pairs, however, for every valid input number, there is only ONE output number. eg. {(0, 2), (2, 6), (3, 8), (4, 10)
Solution:
a. This relation is defined by the set of ordered pairs {(1, 4), (2, -1), (3, 2)}.
Notice that for each x in the domain there is only one corresponding y in the range. Therefore, this relation is a function.
When x = 1, there is only one possibility for y: y = 4
When x = 2, there is only one possibility for y: y = -1
When x = 3, there is only one possibility for y: y = 2
b. This relation is defined by the set of ordered pairs
When x = 1, there are two possible range elements: y = 3, and y = 4. Therefore, this relation is not a function.
c. This relation is defined by the set of ordered pairs {(1, 4), (2, 4), (3, 4)} When x = 1, there is only one possibility for y: y = 4
When x = 2, there is only one possibility for y: y = 4
When x = 3, there is only one possibility for y: y = 4
Because each value of x in the domain has only one corresponding y value, this relation is a function.
Inverse of a given functoin
GOOD LUCK FOR YOUR EXAMS
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