WORKING WITH NUMBERS GRADE 7 NAME: CLASS 3 17 2 11 8 22 36 15 3 ( ) 3 2 Left to Right Left to Right + Left to Right
Back 2 Basics Welcome back! Your brain has been on holiday for a whilelet s see if we can wake up a few of those brain cells! Maths calculations can be divided into 1. mental maths 2. pen and paper maths 3. calculator maths. You need to be equally good at all three! Remember, there are always similarities and mental tricks that will make your life easier! mental maths pen and paper maths calculator maths Calculate as quickly as you can: 1. 6 + 5 26 + 5 75 + 6 36 + 5 2. 109 + 1 169 + 1 299 + 1 999 + 1 3. 10 + 20 10 + 70 90 + 10 890 + 10 4. 5 +8 25 + 8 78 + 5 328 + 5 5. 5+9 45 + 9 89 + 5 449 + 5 6. 6 + 7 76 + 7 97 + 6 207 + 6 7. 6 + 8 36 + 8 126 + 8 318 + 6 8. 7 +9 127 + 9 219 + 7 2 129 + 7 9. 8 + 9 218 + 9 568 + 9 2 438 + 9 Calculate as quickly as you can: 1. 58 + s = 64 158 + n = 164 358 + m = 494 2. 74 + j = 80 74 + h = 89 74 + k = 109 3. 81 + a = 90 81 + g = 99 81 + h = 119 4. 97 + s = 100 97 + w = 104 97 + e = 124 5. 63 + d = 80 63 + r = 82 63 + t = 132 6. 72 + f = 87 64 + y = 75 86 + p = 100 1
Calculate as quickly as you can: 1. 14 5 24 5 154 5 104 5 2 13 6 53 6 183 6 203 6 3 15 7 65 7 125 7 305 7 4 13 8 43 8 223 8 403 8 5 17 9 77 9 247 9 507 9 6 16 7 23 5 411 4 202 7 Write down the answers only: 1. 60 + 80 80 + 90 20 + 90 2. 70 + 90 60 + 50 40 + 90 3. 90 + 50 70 + 60 80 + 50 4. 70 + 80 500 + 700 200 + 800 5. 990 + 10 1 190 + 10 2 100 + 900 6. 700 + 700 800 + 700 1 990 + 10 Give the answers only: 1. 99 + 1 599 + 1 999 + 1 2. 98 + 2 898 + 2 998 + 2 3. 97 + 3 697 + 3 1 997 + 3 4. 496 + 7 795 + 8 503 6 5. 511 13 202 8 1 002 7 6. 792 + 9 896 + 7 1 006 9 Give the answers only: 1. 43 5 173 5 150 60 140-90 2. 72 7 262 7 170 80 150-70 3. 93 6 243 6 270 90 310-80 4. 56 7 146 7 340 80 420-90 5. 27 9 377 9 230 50 330 80 2
Do you know your tables? Give the answers only: 1. 7 8 4 9 54 9 45 5 2. 6 5 7 6 56 8 81 9 3. 8 7 9 7 32 4 24 8 4. 9 6 5 7 42 7 36 9 5. 8 4 9 3 72 9 56 8 6. 50 80 70 70 90 70 50 60 7. 630 9 6 300 9 6 300 90 6 300 900 8. 420 6 420 60 4 200 60 4 200 600 9. 350 70 3 500 70 3 500 700 350 7 10. 560 80 7 200 900 540 60 5 400 60 Work Order Look through the 4 sums done by Pumba or Timon carefully and decide who gets the best mark! a) b) c) d) a) b) c) d) In which operations ( ) can the numbers be changed around? Order of Numbers in Addition and Multiplication is the same as is the same as 3
Look at the stories below: Subtraction I go to the shop with R8 and I spend R6. I have R2 left. I go to the shop with R6 and I spend R8. I owe the shop keeper R2. Order of Numbers in subtraction In subtraction we CANNOT change the order of the numbers: is NOT the same as 2 Division 8 worms are to be shared amongst 2 meerkats. 2 worms are to be shared amongst 8 meerkats. Order of Numbers in Division In division we CANNOT change the order of the numbers: is NOT the same as Exercise a. Write in the answer for each sum in Column B. b. Tick the correct way (or ways) of writing each question in Column A in mathematical language from Column B. Column A Column B 1. There were 6 apples in Basket A and 24 apples in Basket B. How many apples were there altogether? 2. I ran 6 laps around the field every day. How many laps did I run in 24 days? 3. Jill ate 6 of the 24 peaches in the box. How many peaches were left? 4. There were 6 cars to carry 24 learners on an outing. How many learners should travel in each car? 4
Order of Operations The value of mathematical expressions is calculated according to a strict order. Certain operations have priority over others. This is called order of operations, more usually called BODMAS. Work Order with Mixed Operations Inside Brackets / Of Division / Multiplication (work from left to right) Addition / Subtraction (work from left to right) ( ) Left to Right + Left to Right EXAMPLE ( ) Simplify inside the bracket Multiply and divide Subtract and add The following exercises 1 6 must be done in your workbook. Write the question, show working and give answers. 1. Use the order of operation rules to calculate the following. Set out your work step by step as shown above in the example. 1.1 (6 + 12) 3 1.8 (11 2) 8 1.2 21 2 8 1.9 (200 120) 4 1.3 6 2 + 8 1.10 20 (5-1) 1.4 20 20 4 1.11 28 (4 +3) 1.5 20 5 1 1.12 (49-21) 4 1.6 48 + 2 3-6 1.13 (12 + 8) 2 1.7 18 3 + 6 1.14 28 4 2 5
2. Rewrite and insert brackets only where they are essential so that each of the following will be a true statement: 2.1 9 2 + 3 = 45 2.4 20 4 + 1 = 6 2.2 16 7 3 = 27 2.5 18 + 9 3 =9 2.3 20 4 + 1 = 4 2.6 64 8 6 = 2 3. Calculate, showing ALL your working: 3.1 ( ) 3.6 3.2 ( ) 3.7 ( ) 3.3 3.8 3.4 3.9 3.5 ( ) ( ) 3.10 4. According to the BODMAS rule the following equations are wrong. Put brackets into each equation to make it correct. 4.1 2 3 + 5 = 16 4.5 12 3 4 + 3 = 4 4.2 20 8 3 = 4 4.6 3 + 6 4 + 5 = 41 4.3 6 + 3 4 = 36 4.7 12 5 5 + 2 = 1 4.4 12 6 + 6 = 0 5. Do these simple fraction calculations MENTALLY, using the word order rules. 5.1 5.3 5.2 5.4 6. See if you can find 10 different ways of using each of these four numbers to make up sums that will give you the answer 12. 3 1 4 2 6
Properties of 1 and 0 The numbers 1 and 0 behave in a special way when we use them in addition, subtraction, multiplication and division calculations. Do the calculations and answer the questions. 1. a) b) c) What happens when we add 0 to a 2. number? a) b) c) d) e) Does the order of numbers in subtraction matter? f) Is the same as? 3. a) b) c) d) e) What happens when we multiply by 0? f) What happens when we multiply by 1? 4. a) b) c) d) What happens when we divide a number by itself? 5. a) b) c) What happens when we divide a 6. number by 1? a) b) c) What happens when we divide 1 by a number? d) Does the order of the numbers in 7. division matter? Use your calculator: a) b) c) Is there a difference in your answer? NOTE: Any number divided by 0 is undefined. 0 divided by any number is 0. 8. a. f. b. g. c. h. d. i. e. j. 7
POWERS AND INDICES 2 2 2 2 can be written as 2 4 index power 2 4 base 1. List the powers of 4 from 4 1 to 4 4 2. List the values of the power of 5 from 5 1 to 5 5 3. Write in index form and then calculate: 2 2 2 2 = = 2 2 2 2 2 2 = = 5 5 5 = = 4 4 4 = = 9 9 = = 3 3 3 3 = = 2 2 2 = = 4 4 4 4 4 4 = = 4 4 = = 8 8 = = 3 3 = = 11 11 11 11 = = 10 10 10 = = 145 145 145 = = 3 3 3 = = = = 4. In your workbook, calculate (without using your calculator) the value of each of the following by writing out the power as a multiplication sum and then giving its value: eg 2 5 = 2 2 2 2 2 = 32 4.1 3 4 4.2 2 7 4.3 100 2 4.4 12 2 4.5 11 2 4.6 1 4 4.7 4 3 4.8 2 3 3 2 4.9 4 3 2 2 4.10 ( ) 5. Write the following sequences in index form: 5.1 3; 9; 27; 81; 243 = 5.2 5; 25; 125; 625; 3 125 = 5.3 2; 4; 8; 16; 32; 64; 128 = 5.4 10; 100; 1 000; 10 000; 100 000 =
6. In your workbook, copy and fill in the missing index: 6.1 64 = 8 6.2 1 000 000 = 10 6.3 16 = 4 6.4 64 = 4 6.5 25 = 5 7. In your workbook, calculate, showing ALL working out: 7.1 1 3 + 4 2 7.11 5 4 9 2 + 3 5 7.2 5 2 + 8 2 7.12 3 4 3 2 1 2 7.3 10 3 + 5 7.13 (5 3 10 2 ) 2 2 7.4 10 2 1 2 7.14 2 7 6 2 + 2 2 7.5 9 2 3 2 7.15 2 3 2 4 7.6 3 2 + 4 2 7.16 8 2 2 1 3 4 7.7 2 5 + 5 2 7.17 4 2 2 3 10 2 7.8 5 3 10 2 7.18 3 2 + 2 2 5 7.9 10 2 2 2 7.19 [2 3 ] 2 7.10 9 2 2 6 + 2 3 8. Which is larger 3 4 or 4 3? Prove your answer in your workbook. 9. Sionné decides to save some money during March. On the first of March, she saves 2c. Each day after that she saves an amount that is double what she saved the previous day. How much money will she save on the 4 th day? How much money will she save on the 10 th day? In your workbook, calculate, showing ALL your working. 10. What if Sebastian starts with 3c and each day after that he saves 3 times the amount he saved the previous day? How much money would he save on the 2 nd day? How much money would he save on the 3 rd day? In your workbook, calculate, showing ALL your working. 9
MULTIPLES and LCM Multiples are the result of multiplying a number by 1, 2, 3 etc. The multiples of 3 are generated as follows: 1 3 = 3 2 3 = 6 3 3 = 9 4 3 = 12 etc We write the multiples of 3 as: M 3 = 3; 6; 9; 12; 15; 18;.. And the multiples of 5 as: M 5 = 5, 10, 15; 20; 25;.. Then the lowest common multiple (LCM) of 3 and 5 is 15. Exercises: 1. List the first six multiples of 9 and 12 M 9 = M 12 = Write down the LCM of 9 and 12 2. List the multiples of 3; 4 and 9 M 3 = M 4 = M 9 = Write down the LCM of 3, 4 and 9 3. Complete: M 4 = M 6 = M 10 =. The LCM of 4, 6 and 10 10
FACTORS and HCF A factor is a number that is multiplied by another number (or numbers) to produce a product. A factor is a number that will divide into another number without leaving a remainder. eg and and therefore 4 and 6 are factors of 24. Factors always come in pairs! The factors of 24 are: 1 24 2 12 3 8 4 6 We write the factors of 24 as F 24 = 1; 2; 3; 4; 6; 8; 12; 24 This plant shows the factor pairs of 20. They are 1 20, 2 10 and 4 5 1. Write in the factor pairs on these plants: 11
PRIME NUMBERS A Prime Number has only two factors, the number itself and the number 1. A Composite Number has more than two factors. The number 1 is neither prime nor composite, as it only has one factor. Search for the Prime Numbers from 1 to 100 Instructions: Cross out the number 1 which is NOT PRIME. Cross out ALL the multiples of 2 EXCEPT 2. Cross out ALL the multiples of 3 EXCEPT 3. Cross out ALL the multiples of 5 EXCEPT 5. Cross out ALL the multiples of 7 EXCEPT 7. Put a ring around all the uncrossed numbers. How many pairs of prime numbers are consecutive (one after the other) natural numbers? Could you get three consecutive numbers which were all prime numbers? Explain. Can the product of 2 prime numbers be a prime number? Write down anything interesting you notice about the pattern of the primes. How would you check to see if the number 10 000 019 is prime or not? Explain. 12
Exercise: 1. Say whether the following statements are true or false. 20 is a multiple of 10 2 is a factor of every even natural number All prime numbers are odd. 12 is a multiple of 1. 22 is a multiple of 10. 6 is a factor of 38. Any even natural number is a multiple of 2. 5 is a multiple of 35. 1 is a multiple of every natural number. 2. Write down the number that is both a factor and a multiple of 6. 13
Writing large numbers as products of their prime factors. Example: Write 504 as a product of prime factors Always start with the lowest prime number that can divide into the number. 2 504 2 252 2 126 3 63 3 21 7 7 1 Exercises In your work books, using the above method, write the following numbers as products of their prime factors. 1. 336 2. 255 3. 210 4. 1225 5. 209 6. 322 14
Finding the HCF of 3 digit numbers Examples: Find the HCF of 120, 300 and 900 2 120 2 300 2 900 2 60 2 150 2 450 2 30 3 75 3 225 3 15 5 25 3 75 5 5 5 5 5 25 1 1 5 5 1 To find the HCF we take each factor with the lowest exponent and multiply them together. Exercises In your work books, using the above method, find the HCF of the following sets of three numbers. 1. 160; 200; 240 2. 180; 300; 540 3. 325; 520; 975 4. 255; 270; 360 5. 136; 187; 425 15
Squares and Square Roots 4 2 is read as four squared and 4 4 = 16 1 1 = 1 2 2 = 4 3 3 = 9 4 4 = 16 Complete the two unfinished tables. Table of Squares Table of Square Roots 1 2 = 1 1 = 1 = 1 2 2 = 2 2 = 4 = 2 3 2 = 3 3 = 4 2 = 5 2 = 5 6 2 = 7 2 = 7 8 2 = 8 9 2 = 10 2 = 10 11 2 = 11 12 2 = 12 20 2 = 20 50 2 = 50 16
1. Calculate the following in your work book, show all steps of working: Example 1 Example 2 = = = = 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 1.10 [ ] 17
Cubes and Cube Roots 2 3 is read as two cubed and 2 2 2 = 8 1 1 1 = 1 2 2 2 = 8 1. Complete the two unfinished tables. 3 3 3 = 27 4 4 4 = 64 Table of Cubes Table of Cube Roots 1 3 = 1 1 1 = 1 = 1 2 3 = 2 2 2 = 8 = 2 3 3 = 3 3 3 = 4 3 = 4 5 3 = 5 10 3 = 10 11 3 = 11 2. Calculate the following in your work book, show all steps of working: Example 1 Example 2 = = = = 2.1 2.2 2.3 2.4 2.5 2.6 18
Order of Operations 1 Follow the rules for the order of operations to calculate the following in your work book, show all steps of working. 1. (2 + 6) 2 + (3 1) 2 2. 2 + 6 2 + 3 1 2 3. (5 + 3) 4 3 4. 5 + 8 4 3 5. (2 1) + 2 2 3 2 ( ) Left to Right Left to Right + Left to Right 6. 2 (1 + 2) 3 + 6 2 7. 2 + 1 6 2 + 1 8. 2 + 8 4 3 + 6 3 9. (2 + 8) 5 2 + (6 2) 2 10 2 + 6 2 4 11. ( ) 12. 13. 6 + 8 2 10 5 14. ( ) 15. 3 16. 1 7 2 2 17. 43 2 9 18. 36 (8 5) 7 19
Order of Operations 2 Follow the rules for the order of operations to calculate the following in your work book, show all steps of working. 1. 8 4 2 2. 48 4 2 3 3. 3 + 4 5 + 7 1 4. 3 4 2 5 5. 144 4 + 16 2 3 2 ( ) Left to Right Left to Right + Left to Right 6. ( ) 7. ( ) 8. 24 4 + 4 2 9. 21 x 3 (10 3) 16 10 ( ) 11. 81 4 15 2 3 12. ( ) 13. ( ) 14. ( ) 2 3 15. 5 27 2 16. 17. 1+ 2 7 2 32 14 15 12 18. 43 2 3 64 20
*WHODUNNIT? Solve the clues to the GUILTY NUMBERS and the write the answers in the magnifying glasses. In one case there are 2 guilty numbers. More than 8 Less than 15 Odd Two digits Not lucky Even Multiple of 4 Factor of 24 More than 10 Multiple of 3 Multiple of 7 More than 20 Less than 40 Multiple of 5 Digit sum of 9 Less than 80 Multiple of 5 Three digits Less than 130 Digit sum of 7 Odd Prime Two consecutive digits Less than 50 Even Square Two digit number 9 factors Digit sum of 9 Factor of 24 Factor of 40 Factor of 52 Not 2 Prime Less than 50 2 digit number Both digits the same 21
1 is the identity number for multiplication. This means that 1 multiplied by any number will equal the number. e.g. 1 7 = 7 Natural numbers begin at 1 and continue for infinity. N = 1; 2; 3;4 ; If we multiply any number by 0 the answer is 0. 7 0 = 0 0 is the identity number for addition. This means that 0 + any number will equal the number. e.g. 0 + 7 = 7 Whole numbers begin at 0 and continue to infinity. N 0 = 0; 1; 2; 3; O divided by any number is 0. e.g. 0 134= 0 Factors are the numbers by which a number is divisible without leaving a remainder. e.g. The factors of 6 are 1; 2; 3; 6. i.e. F 6 = 1; 2; 3; 6 We cannot divide by 0 as the answer is undefined Multiples are answers to the multiplication tables. e.g. The multiples of 4 are 4; 8; 12; 16; 20; 24; or M 4 = 4; 8; 12; 16; 20; Prime numbers are numbers with only two factors: 1 and the number itself. The number 1 can therefore not be a prime number because it only has one factor! Composite numbers are numbers that have more than two factors Order of operations: Brackets, Exponents, Of Division and Multiplication Addition and Subtraction 22
Estimation and Rounding Off Estimation is an important skill, as it is a check of accuracy. It is easy to press the wrong key on a calculator or make a silly mistake and end up with the wrong answer. Rounding off is one type of estimation. Exercise 1 1.1 Round off the following numbers to the nearest ten and to the nearest hundred. Nearest 10 Nearest 100 411 93 17 55 157 589 385 472 1.2 Round off the following numbers to the nearest the nearest hundred and to the nearest thousand. Nearest 100 Nearest 1 000 1 818 605 895 999 9 357 49 993 995 8 395 23
Estimating to check calculations Example Will R100,00 be enough to buy the following grocery items? Marked cost Estimate Cheese R24,67 Milk R9,69 Bread R8,45 Newspaper R6,90 Coffee R45,80 Total Exercise 2 In your work book, write a number sentence for each of the following. Estimate each answer and finally accurately calculate. 2.1 Find the sum of 7 597 and 986. 2.2 What is the total of 4 975 and 3 097 2.3 Calculate the difference between 4 096 and 5 802. 2.4 By how much is 417 less than 1 000? 2.5 Increase 407 by 589. 2.6 What must be added to 429 to get 1 010? 2.7 The difference between 537 and another number is 208. Find this number. 2.8 By how much does 923 exceed the number 777? 24
Story sums Example A carpenter must saw a plank of wood, that is 6 800 mm long, into 8 equal sections. He wastes 40 mm. How long will each section be? Estimate = 800 mm Length per section = (6 800 40) 8 = 845 mm Exercise 3 3.1 A machine cuts and labels 624 cartons a week. How many will it make in a year if it stopped for a week a year? 3.2 58 similar trucks had to carry a total of 1 276 crates of vegetables. How many crates should be loaded on to each truck? 3.3 In a city of 115 634 people, 50 236 were women, 32 709 were men and the rest were children. How many children were there? 3.4 Peter bought 8 bottles of orange juice at R6,92 per bottle and 8 bottles of lemonade of R7,75 each. 3.4.1 Find the total cost. 3.4.2 Now set out your sum in another way and calculate the total cost. 25
ROUNDING OFF Examples: ROUND OFF 1 625,3995 to the nearest thousand 2 000 to the nearest hundred 1 600 to the nearest ten 1 630 to the nearest unit / to the nearest whole number 1 625 to the nearest tenth / to 1 decimal place 1 625,4 to the nearest hundredth / to 2 decimal places 1 625,40 to the nearest thousandth / to 3 decimal places 1 625,400 Exercise 1 1. Round off to the nearest thousand: 2. Round off to the nearest hundred 2 648,356 2 648,356 5 499,499 5 499,499 9 872,198 2 9 872,198 2 1 369,983 1 369,983 12 958,63 12 958,63 789,764 789,764 3. Round off to the nearest ten: 4. Round off to the nearest unit: 2 648,356 2 648,356 5 499,499 5 499,499 9 872,198 2 9 872,198 2 1 369,983 1 369,983 12 958,63 12 958,63 789,764 789,764 5. Round off to 1 decimal place: 6. Round off to 2 decimal places: 2 648,356 2648,356 5 499,499 5 499,499 9 872,198 2 9 872,198 2 1 369,983 1 369,983 12 958,63 12 958,63 789,764 789,764 26
Exercise 2 1. Round off 547,896 2. Round off 1 208,906 to the nearest hundred to the nearest ten to the nearest whole number to the nearest tenth to the nearest hundredth to the nearest hundred to the nearest ten to the nearest whole number to 1 decimal place to 2 decimal place 3. Round off to the nearest hundred: 4. Round off to the nearest ten: 2 568 2 568 394,6 394,6 12 659,24 12 659,24 130 678 130 678 89,25 89,25 999 999 5. Round off to the nearest whole number: 6. Round off to the nearest unit: 2 568 46,382 394,6 127,457 12 659,24 3,249 130 678 58,605 2 89,25 0,078 999 3,996 7. Round off to 1 decimal place: 8. Round off to 2 decimal places: 46,382 46,382 127,457 127,457 3,249 3,249 58,605 2 58,605 2 0,078 0,078 3,996 3,996 27
INTEGERS Where have you seen numbers less than 0? We call numbers less than 0 negative numbers. They are not minus numbers; minus refers to the subtraction of two numbers. Numbers greater than 0 are therefore called positive numbers. EXERCISE 1 Write the following as positive or negative numbers: 1. 1000m below sea level 2. A temperature 8 below zero 3. A loss of R100 4. A surplus of R50 5. 32 C on Tuesday 6. The height 8863,2m of Mount Everest 7. A profit of R150 8. A debt of R4 9. The year you were born 10. 1000 BCE The numbers... 3 ; 2 ; 1 ; 0 ; +1 ; +2 ; + 3... are called integers. They can be shown on a number line: 5 4 3 2 1 0 +1 +2 +3 +4 +5 x If there is no sign in front of a number it is positive. 28
EXERCISE 2 1. Write > or < between the following pairs of numbers: 1.1 +2 +6 1.6 1 +1 1.2 0 3 1.7 3 1 1.3 2 7 1.8 5 11 1.4 3 +3 1.9 99 30 1.5 +6 0 1.10 54 134 2. Write down the next three numbers in each sequence: 2.1 +4 ; +3 ; +2 ; +1 ; 2.2 1 ; 2 ; 3 ; 4 ; 2.3 9 ; 7 ; 5 ; 3 ; 2.4 +15 ; +11 ; +7 ; +3 ; 2.5 2 ; 5 ; 8 ; 11 ; 2.6 8 ; 6 ; 4 ; 2 ; 2.7 200 ; 150 ; 100 ; 50 ; 2.8 90 ; 70 ; 50 ; 30 ; 2.9 3 ; 6 ; 9 ; 12 ; 2.10 5 ; 4 ; 2 ; 1 ; 3. Arrange in ascending order (smallest to largest) 3.1 7 ; 9 ; 1 ; +4 ; 0 ; 8 ; +3 3.2 10 ; +15 ; 20 ; +12 ; 2 ; 0 : 13 3.3 5 ; 12 ; 3 ; 4 ; 0 ; 2 ; 7 3.4 45 ; 36 ; 42 ; 31 ; 12 ; 16 ; 8 3.5 3 ; 7 ; 11 ; 2 ; 1 ; 5 ; 18 29
4. 4.1 Which would you prefer: R12 or R8 20 or 12 above 100 marks 4.2 Which is warmer: 20 C or 20 C 38 C or 26 C 4.3 Which is deeper: 20m or 40m 5. Write down an integer which is: 5.1 seven greater than 12 5.2 four less than 12 5.3 six less than +1 5.4 eight more than 2 5.5 seven less than 3 6. Write down the integers which are 6.1 greater than 3 6.2 less than 2 6.3 greater than 4 but smaller than +3 6.4 less than 0 but greater than 4 6.5 greater than 6 but less than 4 7. Write down the values of x if x is an integer: 7.1 x > 2 7.2 x < +2 7.3 x < 4 7.4 x > 3 7.5 6 < x < 2 30