Teacher Professional Enhancement (part 1) PDF

EPS 444:Teacher Professional Enhancement (Numeracy) Lecturer: Japhet K. Osiakwan (Ph.D.) Tel: 0244974711 Unit 6: Numeration System and Algebraic Expressions Real Number System Word problems involving the number system Unit 8: Statistics a...

EPS 444:Teacher Professional Enhancement (Numeracy) Lecturer: Japhet K. Osiakwan (Ph.D.) Tel: 0244974711 Unit 6: Numeration System and Algebraic Expressions Real Number System Word problems involving the number system Unit 8: Statistics and Probability Algebraic expressions Tabular and graphical representation of data Linear equations and inequalities in one variable Measures of central tendencies (mean mode and median) Graphs and tabular representation of linear equations Measures of dispersion (range. Standard deviation and variance) Probability of simple events Unit 7: Measurement principles and geometry concepts Conversion of measurement units Lines, line segments and angles Length, perimeter and area of basic shapes (e.g. square, rectangles, circles, etc) Surface area and volume of rectangular solid 2  Diagnostic Test 1. Zero is an even integer (True/false) 2. Zero is a positive integer (True/false) 3. What is the product of all integers from −𝟏𝟗 to 50 ? 4. How many real numbers are between −𝟖 and 34 ? 5. If 𝑨 = −𝟑 ≤ 𝒙 ≤ 𝟓, 𝒙 ∈ 𝑹 and 𝑩 = 𝟏 ≤ 𝒙 ≤ 𝟔, 𝒙 ∈ 𝑹 , find 𝑨 ∩ 𝑩  Answers: 1. Zero is an even integer (True/false) True 2. Zero is a positive integer (True/false) False 3. What is the product of all integers from −19 to 50 Zero 4. How many real numbers are between −8 and 34. Uncountable 5. If 𝑨 = −𝟑 ≤ 𝒙 ≤ 𝟓, 𝒙 ∈ 𝑹 and 𝑩 = 𝟏 ≤ 𝒙 ≤ 𝟔, 𝒙 ∈ 𝑹 , find 𝑨 ∩ 𝑩 5. If 𝑨 = −𝟑 ≤ 𝒙 ≤ 𝟓, 𝒙 ∈ 𝑹 and 𝑩 = 𝟏 ≤ 𝒙 ≤ 𝟔, 𝒙 ∈ 𝑹 , find 𝑨 ∩ 𝑩 If your answer is 𝟏, 𝟐, 𝟑, 𝟒, 𝟓 , it is NOT correct Correct Answer: 𝑨 ∩ 𝑩 = 𝟏 ≤ 𝒙 ≤ 𝟓, 𝒙 ∈ 𝑹 Note: R is the set of all real numbers A real number” can be an integer (positive and negative whole numbers), a fraction, decimal, or a complex number. As such it is impossible to list all the numbers between −𝟑 and 5 or between 1 and 6 5 Unit 6: Numeration System and Algebraic Expressions Note: The focus of the Numeracy is on Number and Number Sense Let us look at the Real Number System 6  Real Number System 1) Counting/ Natural Numbers (N) 𝑵 = 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, 𝟕, 𝟖, … 2) Whole Numbers (W) 𝑾 = 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, … 7 Example If 𝒏 𝑵 = 𝒙, 𝐭𝐡𝐞𝐧 𝒏 𝑾 𝒊𝒔 ________ A. 0 B. 1 C. 𝒙 + 𝟎 D. 𝒙 + 𝟏 Answer: D 3) Integers (Z) The set of both positive and negative whole numbers is known as integers 𝒁 = … , −𝟐, −𝟏, 𝟎, 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, 𝟔, … 4) Rational Numbers (Q): Any number that can be expressed 𝒂 in the form , where 𝒃 ≠ 𝟎 is called a rational number 𝒃 𝟏 𝟏 𝟏 𝑸= … , −𝟑, −𝟐 , −𝟐, −𝟏 , −𝟏, 𝟎, , 𝟏, 𝟐, 𝟑, 𝟒, 𝟓, … 𝟐 𝟐 𝟐 5) Irrational Number (I): Irrational Number (I): Any number that cannot be expressed 𝑎 in the form is an irrational number. Irrational numbers may 𝑏 also be defined as non-terminating or non-recurring decimals. Examples of irrational numbers 2, 5, etc. The Real Number System (Diagram) R Q Z W N I 11  Other sets of numbers 6) Even Numbers: Natural numbers that are divisible by 2 Even Numbers = 𝟐, 𝟒, 𝟔, 𝟖, 𝟏𝟎, 𝟏𝟐, 𝟏𝟒 … 7) Odd Numbers: Natural numbers that are NOT divisible by 2. Odd numbers: 𝟏, 𝟑, 𝟓, 𝟕, 𝟗, 𝟏𝟏, 𝟏𝟑, 𝟏𝟓, 𝟏𝟕, 𝟏𝟗, 𝟐𝟏, 𝟐𝟑, 𝟐𝟓, 𝟐𝟕, … 8) Prime numbers: Natural Numbers greater than 1 that has only two factors, the number itself and one (1). Note: Such factors should be from the set of natural numbers Prime numbers = 𝟐, 𝟑, 𝟓, 𝟕, 𝟏𝟏, 𝟏𝟑, 𝟏𝟕, 𝟐𝟑, 𝟐𝟗, … 12 9) Composite Numbers Composite numbers are natural numbers that have three or more factors (Numbers with more than two natural factors) Note: Composite simply means “something made by combining things” ✓ Composite numbers are numbers that can be made by multiplying two natural numbers, excluding 1 and the number itself. ✓ A whole number that cannot be made by multiplying other whole numbers is a Prime Number ✓ All whole numbers above 1 are either composite or prime. 13  Examples: a) Is 5 a prime number or a composite number? Answer 5 is a prime number because we cannot multiply any other whole numbers like 2, 3 or 4 together to make 5 b) Is 6 a prime number or a composite number? Answer 6 is composite because 6 can be made by multiplying two natural numbers (i.e. 2×3) so it is NOT a prime number. 14  Facts ✓1 is neither a Prime number nor a Composite number. ✓ All even numbers except 2 are composite numbers. ✓ 4 is the smallest composite number 15  Sample Questions 1 Which one of the following numbers is prime? A. 18 B. 19 C. 20 D. 21 Answer: 18 is composite because 18 can be written as a product of factors, not including 1 and 18: That is 18 = 2 × 9 or 18 = 3 × 6 19 is prime because 19 doesn't have any factors except 1 and 19 (B is correct) 20 is composite because 20 can be written as a product of factors, not including 1 and 20: That is 20 = 2 × 10 or 20 = 4 × 5 21 is composite because 21 can be written as a product of factors, not including 1 and 21: That is 21 = 3 × 7 16  Sample question 2 Which one of the following numbers is composite? (A composite number is a number greater than 1 that is not prime.) A. 67 B. 69 C. 71 D. 73 Answer: 67, 71 and 73 are all prime numbers, so cannot be composite. 69 = 3 × 23 is composite. Correct Answer is B 17  Sample question 3 What is the smallest 2-digit prime number? Answer Start with the smallest 2-digit number 10. 10 is not a prime number Answer: 11 is a prime number 18  Comparing Integers: When you are given two integers, how will you find out which of them is bigger or smaller? When asked to find which of two numbers is bigger, remember to illustrate or picture the two given numbers on the number line. The number to the right side is always bigger. -15 -10 -5 0 5 10 15 20 19 Now, try these examples: Which of these two numbers is bigger? 1) -100 and -5 2) 0 and -10 3) -30 and 5 20  Sample Question Which two temperatures are closest to each other? A. −𝟐 degrees Celsius and 3 degrees Celsius B. 2 degrees Celsius and 6 degrees Celsius C. −𝟔 degrees Celsius and −𝟑 degrees Celsius D. −𝟑 degrees Celsius and 3 degrees Celsius Answer: C Note: This can easily be done by illustrating the pair of numbers on the number line. 21  Word Problems involving real numbers There are four basic arithmetic operations in mathematics: Addition, Subtraction, Multiplication and Division. Note: i) The result of adding two or more things is the sum ii)The result of subtracting two or more thigs is the difference iii)The result of multiplying two or more things is the product iv)The result of dividing two or more things is the quotient 22  The four basic operations Operation Symbols Result 1.Addition + Sum 2. Subtraction − Difference 3. Multiplication × Product 4. Division ÷ Quotient 23  Examples 1. What is the sum of GH¢250.00 and GH¢150.00 ? The answer is GH¢400.00 (Thus the result of adding 250 to 150 or 250 + 150 = 400) 2. Find the product of 20m and 100m The answer is 2,000m (Thus, the result of multiplying 20 by 100 or 20 × 100 = 200) 3. What is the difference between 50km and 35km? The answer is 15km (Thus, the result of subtracting 35 from 50 or 50 – 35 = 15) 4. What is the quotient of 100 and 25? The answer is 4 (Thus, the result of dividing 100 by 25 or 100 ÷25 = 4) 24  Rules/Order of operations In mathematics, a string of operations must be performed in proper order. One of the orders very common to all of us is the acronym BODMAS, where B – Bracket O – Off D – Division M – Multiplication A – Addition S – Subtraction 25  Other acronym is PEMDAS, where P – Parenthesis (bracket) E – Exponent M – Multiplication D – Division A – Addition S – Subtraction The difference between the two acronyms is the omission of “exponent” in BODMAS 26 Examples: Simplify the following 1) 𝟐 + 𝟑 × 𝟒 𝟐 2) 5 + 𝟓 − 𝟑 𝟑) 𝟔 off 𝟒 + 𝟔 Solution 1) 𝟐 + 𝟑 × 𝟒 = 𝟐 + 𝟏𝟐 = 𝟏𝟒 𝟐 1) 𝟓 + 𝟓 − 𝟑 = 𝟓 + 𝟐𝟐 = 𝟓 + 𝟒 = 𝟗 1) 𝟔 off 𝟒 + 𝟔 = 𝟔 × 𝟏𝟎 = 𝟔𝟎 27  Note: 1) For addition and subtraction, work from left to right 2) For multiplication and division, work from left to write. Example 𝒂) 𝟓 − 𝟑 + 𝟖 Answer = 10 𝒃) 𝟏𝟎 × 𝟒 ÷ 𝟐 Answer = 20 c) 20 ÷ 𝟏𝟎 × 𝟐 Answer = 4  Sample Question 1 What is the sum of the product and quotient of 8 and 8? A. 49 B. 50 Hint: C. 98 (Product) + (quotient) D. 102 (8× 𝟖) + (8÷ 𝟖) = 64 + 1 = 65 29  Sample question 2 If you take the sum of 340 eggs and 520 eggs from 1240 eggs, how many eggs will you have left? Solution Total number of eggs = 1,240 Number of eggs taken out (sum of 340 and 520) = 340 + 520 = 860 Subtract the sum from the total number of eggs. Thus, Number of eggs left = 1,240 – 860 = 380 eggs 30  Sample question 3 A teacher bought three items priced at GH¢38.00, GH¢45.00 and GH¢65.00. He paid with eight GH¢20.00 notes. How much change did he receive? Solution Total cost of the three items = 38 + 45 + 65 = GH¢148 Money paid = 8 × 20 = GH¢𝟏𝟔𝟎 Change received = 160 – 148 = GH¢12.00 31  Sample Questions 4 In a school, a headteacher bought a number of blue pens for his BECE candidates. He gave 135 of the pens to the girls and 90 to the boys and had 275 left. How many blue pens did the headteacher buy? Solution Number of pens given to the girls = 135 Number of pens given to the boys = 90 Number of pens left = 275 Now, you are to find the total number of pens the headteacher bought. Simply add all the pens together 135 + 90 + 275 = 500 pens Hence, the headteacher bought 500 pens 32  Sample question 5 Teachers organised activities for three classes of 25 pupils and five classes of 28 pupils. What was the total number of pupils involved? Solution: Three classes of 25 pupil = 3 × 25 = 75 pupil Five classes of 28 pupil = 5 × 28 = 140 pupil Total Number of pupils involved 𝟕𝟓 + 𝟏𝟒𝟎 = 𝟐𝟏𝟓 pupils 33  Sample question 6 Nine hundred copies of a book have to be packed into boxes. Each box can hold 45 of the books. Find the total number of boxes needed to pack the books. Solution Total number of books = 900 Number of books that can go into one box = 45 How many 45’s can you get in 900? Simply Divide 900 by 45 Thus, 9𝟎𝟎 ÷ 𝟒𝟓 = 𝟐𝟎 boxes Hence, the number of boxes needed is 20 34  Sample question 7 A student bought a story book from a store. He gave the cashier GH¢25.00 and got back GH¢13.00 in change. The student saw that he had gotten too much change and he gave GH¢3.00 back. What was the cost of the story book? Solution The student gave 25 cedis to the cashier and got back 13 cedies. It means that the student paid = 25 – 13 = GH¢ 12.00 to cashier But out of the change of 13.00 cedis, the student gave back 3.00 cedis to the cashier So, the total cost of the book = 12.00 + 3.00 = GH¢15.00 35  Equations and inequalities in one variable Note: a) Any mathematical statement containing “ = ” is called an equation b) Any mathematical statement containing “, ≤ 𝒐𝒓 ≥ is called an inequality. c) Any mathematical statement that does not contain equal to sign and any of the inequality signs is known as an expression 36 Example: Write an algebraic expression for each of the following verbal expressions. a) “7 more than y” Answer: 𝒚 + 𝟕 or 𝟕 + 𝒚 b) “4 less than y” Answer: 𝒚 − 𝟒 NOT 𝟒 − 𝒚 c) “9 less than the sum of x and 2 Answer: 𝒙+𝟐 −𝟗 37  Sample question 1 There are x boys and y girls in a classroom. Each pupil had two pens. Which of these expressions represents the total number of pens carried to the classroom? A. 2(x + y) B. 2 + (x + y) C. 2x + y D. 2xy Answer = A 38  Sample question 2 Twenty gallons of petrol were poured into two containers of different sizes. If the amount of petrol poured into the larger container is y, which of the following represents the amount poured into the smaller container? A. 20 ÷ y B. y + 20 C. y – 20 D. 20 – y Answer = D 39  Take note of the following terms as they appear in word problems English Words Mathematical meaning 1. Is equal to, is the same, is, was, will be, Equals (=) 2. More than, older than, increased by, exceeds, farther Addition (+) than 3. Fewer than, less than, difference, minus, Subtraction ( - ) decreased by, younger than, lost, 4. Times, of, product, multiply by Multiplication (X) 5. Divide by, quotient, per, for, Division (÷ ) Inequalities i) More than, greater than > ii) Fewer than, less than < iii) At least, ≥ ≤ iv) At most, 40  Examples involving inequalities 1. Kofi has fewer than GH¢150.00. Write down an inequality for the statement Answer: Let the amount Kofi has be = y, then Kofi has 𝒚 < 𝟏𝟓𝟎 2. The sum of y and 35 is at least 200. Write an inequality for the statement. Answer 𝒚 + 𝟑𝟓 ≥ 𝟐𝟎𝟎 41  Examples involving equations 3) The product of 2 and 7 more than a certain number is 16 times that number. What is the number? Solution Let the number be y Product of 2 and 7 more than y is 𝟐 × 𝒚 + 𝟕 Also, 16 times the number y, is 𝟏𝟔𝒚 𝟏𝟒 = 𝟏𝟔𝒚 − 𝟐𝒚 Note that the word “is” means equal to. 𝟏𝟒 = 𝟏𝟒𝒚 The equation is, 𝟐 × 𝒚 + 𝟕 = 𝟏𝟔𝒚 𝒚=𝟏. 𝟐𝒚 + 𝟏𝟒 = 𝟏𝟔𝒚 Hence, the number is 1 42  More examples on equations Kofi has 3 times as much money as Ama. If Kofi gives Ama GH¢50.00, Ama will then have 3 times as much money as Kofi. How much money do the two of them have altogether? Solution Draw a simple table Present amount Action Ama 𝒙 𝒙 + 𝟓𝟎 Kofi 𝟑𝒙 𝟑𝒙 − 𝟓𝟎  Solution Present amount Action Ama 𝒙 𝒙 + 𝟓𝟎 Kofi 𝟑𝒙 𝟑𝒙 − 𝟓𝟎 After the action, Ama will then have 3 times as much money as Kofi The equation is 𝒙 + 𝟓𝟎 = 𝟑 𝟑𝒙 − 𝟓𝟎 𝒙 + 𝟓𝟎 = 𝟗𝒙 − 𝟏𝟓𝟎 𝟐𝟎𝟎 = 𝟖𝒙 𝒙 = 𝟐𝟓 𝑨𝒎𝒂, 𝒙 = 𝟐𝟓, 𝑲𝒐𝒇𝒊, 𝟑𝒙 = 𝟕𝟓 ∴ 𝑻𝒐𝒕𝒂𝒍 = 𝟐𝟓 + 𝟕𝟓 = 𝑮𝑯¢𝟏𝟎𝟎  Question 2 In 5 years, Jojo will be twice as old as he was 9 years ago. How old is Jojo now? Solution Present age Age in 5 years Age 9 years ago Jojo 𝒙 𝒙+𝟓 𝒙−𝟗  Solution Present age Age in 5 years Age 9 years ago Jojo 𝒙 𝒙+𝟓 𝒙−𝟗 In 5 years, Jojo will be twice as old as he was 9 years ago The equation is 𝒙+𝟓=𝟐 𝒙−𝟗 Next solve for x 𝒙 + 𝟓 = 𝟐𝒙 − 𝟏𝟖 𝟐𝟑 = 𝒙 𝒙 = 𝟐𝟑 𝒚𝒆𝒂𝒓𝒔  Question 3 John had 36 mangoes and Alex had 48. How many mangoes should John give to Alex so that Alex would have three times as many mangoes as John? Solution  Solution After the action, Alex would have three times as many mangoes as John 𝟒𝟖 + 𝒙 = 𝟑 𝟑𝟔 − 𝒙 Next, solve for x 𝟒𝟖 + 𝒙 = 𝟏𝟎𝟖 − 𝟑𝒙 𝒙 + 𝟑𝒙 = 𝟏𝟎𝟖 − 𝟒𝟖 𝟒𝒙 = 𝟔𝟎 𝒙=𝟓  Percentages Percent means out of 100 or per 100 20 For example 20% is the same as 100 40 Also, 40% is the same as 100 1) What is 20% of 200 students? Solution: 𝟐𝟎 20% × 200 = × 𝟐𝟎𝟎 𝟏𝟎𝟎 2) What is 60% of GH₵300 ? = 𝟒𝟎 𝒔𝒕𝒖𝒅𝒆𝒏𝒕𝒔 Solution: 𝟔𝟎 60% × 300 = × 𝟑𝟎𝟎 𝟏𝟎𝟎 = 𝑮𝑯₵𝟏𝟖𝟎. 𝟎𝟎  Sample question 3) What is 20% of 60% A. 10% B. 12% C. 15% D. 25% Solution: 20 60 20% × 60% = × 100 100 12 = = 𝟏𝟐% 100  Sample Question Twenty-five percent of the length of a rope is 55cm. What is the full length of the rope? A.110 cm B. 220 cm C. 250 cm D. 500 cm Solution Let the length of the rope = 𝑥 25𝑥 = 55 × 100 25 It means that × 𝑥 = 55 100 100×55 𝑥= = 220 25𝑥 25 = 55 100 Sample Question What percentage of the letters of the word “MATHEMATICS” are M? A.0.09 % B. 0.18 % C. 9.09 % D. 18.18 % Solution Number of M’s in the word MATHEMATICS = 2 Total number of letters in the word = 11 𝟐 Percentage of M = × 𝟏𝟎𝟎 𝟏𝟏 = 18.18%  Quantity left or fraction left ✓If you are working with fractions and you want to find the fraction left, simply subtract the fraction from 1. ✓It means that in terms of fractions, the whole or total is 1 ✓If you are working with percentages and you want to find the percentage left, simply subtract the percentage from 100. ✓In the case of percentage, the total is always 100. ✓Next, apply simple proportion to solve 53  Examples 1) A student spends two-thirds of her money on books. What fraction of the money is left? Solution: Fraction spent = 𝟐 𝟑 2 𝟏 Fraction left = 1 − = 3 𝟑 2) A man gave 20% of his salary to the mother and 35% to the father. What percentage of the money is left? Solution: Percentage given out = 20% + 35% = 55% Percentage left = 100 – 55 = 45%  Sample question 1) A lady teacher gave 20% of her salary to the mother and 15% to the father. If she still had GHȻ520 left, how much was her salary? Solution Percentage given out = 20% +15% = 35% Percentage left =100 – 35 = 65% If 65% = GHȻ520 Now, apply simple proportion Then, 100% = ?? 100 Thus, × 520 = GHȻ800 65 55  Sample question 1) Joan used two-thirds of her weekly savings to buy a textbook. If she realized that she has GHȻ10 left, how much was the weekly savings? Solution If 1Τ3 = GHȻ10 𝟐 Fraction spent = 𝟑 Then, 1 = ? 𝟐 𝟏 Fraction left = 𝟏 − = Thus, 1 × 10 = 30 𝟑 𝟑 1Τ 3 Money left = GHȻ10 Hence, Joan’s weekly savings is GHȻ30.00 Now apply simple proportion 56  Measurement Unit 7: Measurement principles and geometry concepts Conversion of measurement units Lines, line segments and angles Length, perimeter and area of basic shapes (e.g. square, rectangles, circles, etc) Surface area and volume of rectangular solids 57 Let us begin with a Context: A boy has two mobile phones. One of them beeps every 3 seconds; the other beeps every 5 seconds. If the two phones are switch on at exactly the same time, how many times during the next hour will both phones beep at the same time?  What did you get? First question to ask yourself is when will both phones beep together for the first time? Did you get 15 seconds? Next, how many “15 seconds” are in 1 hour? Remember, 1 hour = 60 × 60 = 3600 𝑠𝑒𝑐𝑜𝑛𝑑𝑠 That is 3600 ÷ 15 = 240 Hence, in 1 hour, the phones will beep together 240 times.59 How old are you? How much do you weigh? How tall are you? How much water can be filled in your water bottle? How hot is it today? To answer the above questions; what we need is to measure. Thus, To find how old you are, you need to measure time To know how much you weigh, you must weigh yourself. To know how tall you are, you need to measure your height (length) To know how much water you can fill in your water bottle, you need to measure the capacity of your bottle. To find out how hot it is today, you need to measure the temperature. So, what exactly is “measurement”? 60  Measurement Measurement is a process of finding a number that shows the amount of something. The number that shows the size or amount of something is called “unit of measurement”. Usually the number is in reference to some standard measurement, such as a meter or kilogram. There are two measurement systems: 1. The Metric System: Metric system is based on the SI. This system is based on the meter, liter, and gram as units of length (distance), capacity (volume), and mass (weight) respectively. 2. The US Standard Units: This system uses inches, feet, yards, and miles to measure length or distance. Capacity or volume is measured in fluid ounces, cups, pints, quarts or gallons. Weight or mass is measured in ounces, pounds and tons. We shall focus on the metric system of measurement 61  The metric system The metric system is a system of measuring in mathematics. It has three main units: Meters (m) for Length Kilogram (kg) for Mass Seconds (s) for Time. In fact, with these three simple measurements, we can measure nearly everything in the world. We can also combine the meter, kilogram and second to make new units. We shall look at a few common units that are based on the meter, kilogram and second: 62  Measurement of Length The most common metric units of length are Kilometres (km), Metres (m), Centimetres (cm) and Millimetres (mm) There are: 10 mm in 1 cm 100 cm in 1 m 1000 m in 1 km For instance, When you have something that is 10 millimeters, it can be said that it is 1 centimeter long.  Measurement of Length For example: ✓ Your fingernail is about 1 cm wide. ✓ ATM card is about 0.76 mm tick ✓ From one goal post of a football field to the other post is about 100 meters ✓ A4 paper ranges from 0.05 mm to 0.10 mm in thickness 64  A meter is equal to 100 centimeters. 100 centimeters = 1 meter Meters might be used to measure the length of a house, or the size of a playground or football field etc. We can use millimeters or centimeters to measure how tall we are, or how wide a table is, but to measure the length of football field it is better to use meters. What do you think is the distance from Cape Coast to Accra in meters?  The distance from one city to another or how far a plane travels are measured using kilometers. When you need to get from one place to another, you will need to measure that distance using kilometers. A kilometer is equal to 1,000 meters. 1,000 meters = 1kilometer E.g. What is 500 meters in kilometers? Answer is half kilometers E.g. What is 2 kilometers in meters? Answer is 2,000 meters 66 Summary 10 millimeters = 1 centimeter 100 centimeters = 1 meter 1,000 meters = 1 kilometer 67  Sample question Which one of the following do you think measures about 0.3 km? A.The perimeter of a football field B.The distance run in a marathon race C.The height of the highest mountain D.The distance from the Earth to the Moon The correct answer is A Reasons: The distance run in a marathon race is about 42 km. The height of the highest mountain (Mount Everest) is 8.8km, and the distance to the Moon is much more. The perimeter of a football field is about 300m or 0.3km 68  Sample question How many meters are there in 50¼ km ? A.5,025 m B.5,250 m Solution C.50,025 m 1 km = 1,000 m D.50,250 m So, 50¼ km = 50.25 × 1,000 m = 50,250 m The correct answer is D 69  Weight/Mass ✓The amount of matter a thing consists of is called its mass. ✓Mass is generally measured by how much something weighs. Measuring mass means to measure the heaviness of a thing ✓The more matter something has, the more it will weigh. ✓The standard unit of measure for mass comes from the metric system and is either grams or kilograms There are: 1,000 grams in 1 kilogram 70 That is 1000 grams = 1 kilogram and 1,000 kilograms = 1 tonne For example, A bag of cement is = 50kg This is the same as = 𝟓𝟎𝒌𝒈 × 𝟏, 𝟎𝟎𝟎 = 𝟓𝟎, 𝟎𝟎𝟎𝒈 A small bag of rice is = 5kg This is the same as = 𝟓 × 𝟏, 𝟎𝟎𝟎 = 𝟓, 𝟎𝟎𝟎𝒈  Note: (i)To convert from kilograms to grams, multiply by 1,000. (ii)To convert from grams to kilograms, divide by 1,000 Examples: Convert the following 1) 500g 1) Half of a kg to grams 2) 5,000g 2) 5kg to grams 3) 500g to kg 3) 0.5 kg 4) 4000g to kg 4) 4 kg  Sample question Which one of the following do you think weighs about 500g? A. An egg B. A cell phone C. A box of sugar D. A sofa The correct answer is C Reason The egg and the cell phone weigh much less than 500g A sofa weighs much more than 500g. The box of sugar is about 500g 73  Sample question How many grams are there in ½ tonne? A.500 g Solution B.5,000 g 1 tonne = 1,000 kg C.500,000 g So ½ tonne = ½ X 1,000 = 500 kg D.5,000,000 g Also, 1 kg = 1,000 g The correct answer is C So 500kg = 500 X 1,000 g = 500,000 g 74  Volume: Volume is length by length by length. So, the Unit of volume is meters × meters × meters, which is written m3 (Cubic meter) A cube that is 1 meter on each side is also equal to 1,000 liters. 1,000 milliliters = 1 Liter 1,000 Liters = 1 cubic meter ( m3) So a liter is actually one-thousandth of a cubic meter. 𝟏 Liter = m3 𝟏,𝟎𝟎𝟎 75  Sample questions How many milliliters are there in 4½ liters? A.450 ml Solution B.4,050 ml 1 liter = 1,000 ml C.4,500 ml So 4½ liters = 4½ × 1,000 ml D.45,000 ml = 4.5 × 1,000 ml The correct answer is C = 4,500 ml 77  Reading of Dates: How to Count days, weeks, months and Years In counting of days, weeks, months or years, we make use of the concept of “remainder”. In mathematics, remainder is simply “left over” in a division problem. Note: If the remainder is 0, it means the same time, day, week, month or year. If the remainder is 1, it means the next hour, day, week, month or year If the remainder is 2, it means the next two hours, days, week, month or year) If the remainder is 3, it means the next three hours, days, week, month or year, etc 78 Recall the following: 7 days make a week 4 weeks make a month 12 months make a Year.  Example 4: We are in July, 2021, what month will it be after 14 months? Solution: Simply divide 14 by 12 months (since we have 12 months in a year). Thus, 12 will go into 14 once with a remainder 2 Thus 14/12 = 1 Remainder 2, Since the remainder is 2, the answer is the next two months after July. Thus September. 80  Example 5 Given that we are in July, 2023, what month will it be after 31 months? What about the year? Solution 31/12 = 2 remainder 7 Since the remainder is 7 it means count 7 months after September. That will be February The year will be 2026  Sample examination question The fence of a drinking spot is made up of bamboo slats. The slats of the fence are painted in four colours, which appear in a fixed order, such as red, yellow, blue, green, red, yellow, blue, green, …. If the first slats is red, what colour is the 120th slat? Solution The planks are: red, yellow, blue, green, red, yellow, blue, green etc. Since only 4 colours are involved, we divide 120 by 4 and note the remainder. Thus, 120/4 = 30 remainder 0 Since the remainder is 0, the correct answer is the starting plank. Hence, the 120th plank is painted red 82  Time Reading Note: For any time reading: 60 second = 1 minutes 60 minutes = 1 Hour 24 hours = 1 day  Time Reading What is the time on each of the clocks below? 3:38 10:12 Which of them is morning time?  Note: There are two common time readings: These are the 12-hour Time and 24-hour Time (known as military time in USA) a) For the 24-hour time, there is no need to attach AM or PM to the time. It is not necessary at all, because its on 1 to 24 scale. b) For the 12-hour time, it is compulsory to attach AM or PM to the time. This is because, for example, 2 o’clock can be in the afternoon or morning 85  3 o’clock on the 24-hour 11 12 time is “morning” NOT Morning afternoon. Evening 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Morning Afternoon Evening  Examples i) 3 o’clock on the 24-hour time is “morning” NOT afternoon. ii) 2 o’clock on the 24-hour time is “morning” NOT afternoon iii) 14 o’clock is “afternoon (thus, 2 pm on the 12-hour clock) iv) 19 o’clock is “evening” (thus, 7 pm on the 12-hour clock) v) 24 o’clock (or 00 o’clock) is “mid-night” 87  Example 2 Write these times in 24-hour clock time: (a) 3:06 a.m. (b) 8:14 p.m. Solution (a)Since it is AM, it means it is 3 hours from the beginning of the day, so the 24-hour time is 03:06. (also written as 0306) 88  (b) 8:14 p.m. Since it is PM, it means it is 8 hours from 12 (noon). So simply add 12 to it. Thus 12 + 8:14 = 20:14. So the 24-hour time is 20:14. (also written as 2014) NOTE: The 24-hour clock time must always contain exactly 4 digits.  Example 2 Write these times using digits and a.m. or p.m. (a) 1428 (b) 0742 (c) 1731 91  Solution: (a)Write the 24-hour clock time 1428 using a.m. or p.m. This is 14 hours from the beginning of the day, which means it must be in the afternoon. Subtracting 12 hours from 14 gives 2 hours. So the answer is 2:28 p.m. (b) Write the 24-hour clock time 0742 using a.m. or p.m. This is 7 hours from the beginning of the day, which means it must be in the morning, so the correct time is 7:42 a.m. (c) Write the 24-hour clock time 1731 using a.m. or p.m. Answer is 5:31pm  Sample examination question Abigail leaves home at 09:00 and returns 7 hours later. What time does she get home? (a)Write your answer in 24-hour clock time. (b) Write your answer in 12-hour clock time using a.m. or p.m. 93  Solution: Abigail leaves home at 09:00 and returns 7 hours later. What time does she get home? (a) 09:00 is 9 o’clock in the morning So 7 hours later is (9+7) = 16. Hence, the correct time is 16:00 (b) For the 12-hour clock, it is (16 – 12) = 4. Hence, the correct time is 4:00pm  Sample examination question 2 A car left Accra at 9:45am. It arrived in Kumasi in 3 hours 17 minutes later. What time did the car get to Kumasi? Give your answer using the 24-hour clock. 95 A car left Accra at 9:45am. It arrived in Kumasi in 3 hours 17 minutes later. What time did the car get to Kumasi? Give your answer using the 24-hour clock. Solution Start time is 9:45 Add 3 hours to 9 hours, this gives us 9 +3 = 12 Also Add the 45 minutes to the 17 minutes, that gives us 45 + 17 = 62 So, when we combine the two together, we have 12: 62 But from the 62 minutes we can get 1 hour leaving 2 minutes Add the 1 hour to 12 to get 13 hours Hence, the right time is 13:02 Sample question Start of meeting End of meeting What is the duration for the meeting? The duration is 6 hours 30 minutes Sample question 3 The morning session in a school began at 09:25. There were three lessons of 50 minutes each and one break of 20 minutes. At what time did the morning session end? Give your answer using the 24- hour clock. Solution 3 lessons of 50 minutes = 3×50 = 150 minutes But 150 minutes = 60 +60 +30 = 2 hours 30 minutes So when we add 2 hours 30 minutes to 09:25, we shall get 11:55 But the session includes 20 minutes break So, Add the 20 minutes break to the 11:55 The final time is 11:55 + 20 = 11:75 = 11: 60+15 = 12:15 98  Reading of temperature Liquid in glass Thermometer Clinical Thermometer 30 60 20 50 10 40 0 30 -10 20 -20 10 -30 0 For the liquid in glass thermometer, all numbers below zero are negative values. To determine an increase in temperature for any thermometer, always subtract the smaller reading from the larger reading (thus final reading minus the initial reading).  Examples 1 If the temperature of a body in the morning was 30 degrees Celsius, and increases to 50 degrees Celsius in the afternoon, what was the rise in temperature? Answer: ✓ Larger reding = 50 ✓ Smaller reading = 30 ✓ Rise in temperature = 50 – 30 = 20 degrees Celsius  Example 2 In the morning, the temperature of a room was –20 degrees Celsius, but in the afternoon, it was 15 degrees Celsius. What was the temperature rise? Answer: ✓ Larger value = 15 ✓ Smaller value = -20 ✓Rise in temperature = 15 − −20 = 35 degrees Celsius  Sample examination question At 5am, the temperature was 11 degrees below zero. By noon it had risen to 24 degrees. What was the average (per hour) hourly increase in temperature? Solution: Larger reading = 24 Smaller reading = -11 (negative since its below zero) Temperature increase = 24 – (-11) = 35 degrees Celsius Next, from 5am to noon is 7 hours So the temperature per hour = 35/7 = 5 degrees. Hence, the average hourly increase is 5 degrees It means that the temperature increases by 5 degrees every hour