Long division uses the same processes as short division, but the steps are written down. Use long division when the divisor is greater than 12.

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1 Place value Numbers are organised by place value. The first place value is the units. Each place value to the left of this column has a place value 10 times as great as the value in the previous column. Numbers can be placed in order by comparing their place values. Adding and subtracting positive integers Addition and subtraction are the inverse of each other. When we are adding and subtracting, it is important to line up the numbers vertically. When we are adding numbers, the order in which they are added is not important. Large numbers are subtracted by decomposing the larger number into parts. Multiplying and dividing positive integers The basis for all multiplication work is to know your multiplication tables up to 12. Some multiplications can be done mentally using various strategies. Division is the inverse of multiplication. Use short division when dividing by numbers up to 12. When dividing numbers that are multiples of 10, write the question as a fraction, cancel as many zeroes as possible and then divide. Addition and multiplication are commutative and associative. Multiplication is commutative, associative and distributive. Subtraction and division are not commutative, associative or distributive. Long division Long division uses the same processes as short division, but the steps are written down. Use long division when the divisor is greater than 12. Order of operations The operations inside brackets are always calculated first. If there is more than one set of brackets, calculate operations inside the innermost brackets first. Multiplication and division operations are calculated in the order that they appear. Addition and subtraction operations are calculated in the order that they appear. Factors and multipliers A multiple of a number is the answer obtained when that number is multiplied by another whole number. A factor is a whole number that divides exactly into another whole number with no remainder. The factors of a number can be found using factor pairs. Lowest common multiple and highest common factor If two or more numbers have the same multiple, it is called a common multiple. The smallest number that 2 or more numbers can divide into is called the lowest common multiple of those numbers. If two or more numbers have the same factor, it is called a common factor. The highest common factor or HCF of 2 or more numbers is the largest factor that divides into all of the given numbers, with no remainder. 60

2 Estimation When rounding, if the second digit is 0, 1, 2, 3 or 4, the first digit stays the same. If the second digit is 5, 6, 7, 8 or 9, the first digit is rounded up. Homework Book MAPPING YOUR UNDERSTANDING Using terms from the summary, and other terms if you wish, construct a concept map that illustrates your understanding of the key concepts covered in this chapter. Compare your concept map with the one that you created in What do you know? on this chapter s opening page. Have you completed the two Homework sheets, the Rich task and two Code puzzles in your Maths Quest 7 Homework book? Positive integers 61

3 1 State the place value of the digit shown in red in each of the following. a b c d Write each of the following numbers using expanded notation. a 392 b 4109 c d List the numbers 394, 349, 943, 934, 3994, 3499 in ascending order. 4 List the numbers 1011, 101, 110, 1100, 1101 in descending order. 5 Add these numbers. a b c d Calculate each of the following. a b c d Complete these subtractions. a b c d e f Use mental strategies to multiply each of the following. a 2 ì 15 ì 5 b 4 ì 84 ì 25 c 62 ì 20 d 56 ì 300 e 67 ì 9 f 31 ì 19 9 Calculate each of these using short division. a 4172 ó 7 b ó 12 c ó 3 10 Calculate each of these. a 6 ì 4 ó 3 b 4 ì 9 ó 12 c 49 ó 7 ì 12 d 81 ó 9 ì 5 e 6 ì 3 ó 9 ó 2 f 12 ó 2 ì 11 ó 3 11 Divide these multiples of 10. a ó 120 b 4900 ó 700 c ó Evaluate the following using long division: a 4706 ó 13 b 6867 ó 21 c ó 17 d 3762 ó Evaluate the following using long division: a 5702 ó 17 b 6932 ó 23 c ó 13 d ó Write the rules for the order of operations. 15 Follow the rules for the order of operations to calculate each of the following. a 35 ó (12-5) b 11 ì c 8 ì 3 ó 4 d 5 ì ì 5 e (6 + 4) ì 7 f ì 7 g 3 ì (4 + 5) ì 2 h 5 + [21 - (5 ì 3)] ì 4 16 By rounding each number to its first digit, estimate the answer to each of the calculations. a b c 5304 ó 143 d 5706 ì 68 e f g 9480 ó 2559 h 289 ì List the first 5 multiples of each number. a 11 b 100 c 5 d 20 e 13 f 35 62

4 18 Find the lowest common multiple (LCM) of the following pairs of numbers. a 3 and 12 b 6 and 15 c 4 and 7 d 5 and 8 19 Find all the factors of each of the following numbers. a 16 b 27 c 50 d 42 e 36 f List the factor pairs of the following numbers. a 24 b 40 c 48 d 21 e 99 f Uluru is a sacred Aboriginal site. The map below shows some roads between Uluru and Alice Springs. The distances (in kilometres) along particular sections of road are indicated. To Darwin Finke River Simpsons Gap Hermannsberg Stanley Chasm 127 Alice Springs 195 Wallace Rockhole 132 Palmer River Kings Canyon resort Henbury Meteorite Craters Ayers Rock resort 83 Curtin Springs Mt Ebenezer 56 Uluru Map not to scale sealed road unsealed road To Adelaide Kulgera a How far is Kings Canyon resort from Ayers Rock resort near Uluru? b What is the shortest distance by road if you are travelling from Kings Canyon resort to Alice Springs? c If you are in a hire car, you must travel only on sealed roads. Calculate the distance you need to travel if driving from Kings Canyon resort to Alice Springs. Positive integers 63

5 2 In summer, an ice-cream factory operates 16 hours a day and makes 28 ice-creams each hour. a How many ice-creams are produced each day? b If the factory operates 7 days a week, how many ice-creams are produced in one week? c If there are 32 staff who run the machines over a week, how many ice-creams would each person produce? 3 Hung and Frank are cyclists who are riding around a track. They ride past the finish line together, and from then Hung takes 25 seconds to complete a lap and Frank takes 40 seconds. a How long will it be until they next pass the finish line together? b How many laps will each have ridden when this occurs? 4 When you add two even positive integers, the answer is even; the sum of an even and an odd positive integer is an odd integer; two odd positive integers add to give an even integer. What happens if you perform multiplication on these types of integers? Give examples to support your statements. Write a general statement to summarise what would result when you multiplied more than two positive integers. 5 Joe, Claire and Daniela were having races up and down a flight of stairs. Joe took the stairs three at a time, Daniela took the stairs two at a time while Claire took the stairs four at a time. In each case, they reached the bottom with no steps left over. a How many steps are there in the flight of stairs? List three possible answers. b What is the smallest number of steps there could be? c If Joe can also take the stairs five at a time with no steps left over, what is the smallest number of steps in the flight of stairs? 6 Place the numbers 1 to 6 in these circles so that the number in every circle below two higher circles shows the difference between the two numbers in the higher circles 7 A perfect number is one whose factors (all except the number itself) add to the number. For example, 6 is a perfect number because (the sum of its factors, excluding 6) is equal to 6. Show why 496 is a perfect number. 64

6 8 Complete this subtraction by filling in the missing numbers in the boxes At a confectionary factory, a machine dispenses 760 lollies into 8 jars. Assuming that an equal number of lollies is dispensed into each jar, find the number in each jar. 10 Arrange the digits 0, 2, 4, 5, and 8 to form the smallest possible 5-digit number. You must use each digit once and only once. 11 In a race, one dirt bike rider completes each lap in 40 seconds while another completes it in 60 seconds. How long after the start of the race will the two bikes pass the starting point together? 12 There are sets of numbers known as Hailstone numbers. Here s an example. Pick any integer and enter it into your calculator. If it is an even number, divide by 2. If it is an odd number, multiply by 3, then add 1. Repeat the process with the new number over and over. Explain what happens in the long run, and why these numbers would be called Hailstone numbers. 13 Julie sells 8 bottles of soft drink for $3 each and 12 bottles of water for $2 each. a Write a calculation that will find the total value of Julie s sales. b Find the total value of Julie s sales. 14 At a football match, Richard estimates the crowd to be people. If Richard s estimate is correct to the nearest 1000 people, what is the greatest number of people that could possibly be at the football match. ebookplus Interactivities Test yourself Chapter 2 int-1814 Word search Chapter 2 int-2762 Crossword Chapter 2 int-2763 Positive integers 65