150 7 Fractions You already know that you can only subtract fractions when the denominators are the same. If the denominators are different, you must write the fractions as equivalent fractions with a common denominator, then subtract the numerators. Here is a method for subtracting mixed numbers. 1 Change each mixed number into an improper fraction. 2 Subtract the improper fractions and cancel this answer to its simplest form. 3 If the answer is an improper fraction, change it back to a mixed number. 7.3 Subtracting mixed numbers In this section you will … • subtract mixed numbers. Worked example 7.3 Work out a 3 1 1 5 4 5 − b 6 2 1 3 4 9 − Answer a 31 5 16 5 = and 14 5 9 5 = Change both the mixed numbers into improper fractions. 16 5 9 5 7 5 − = Subtract the fractions. They already have a common denominator of 5. 7 5 2 5 = 1 The answer is an improper fraction so change it back to a mixed number. b 6 1 3 19 3 = and 2 4 9 22 9 = Change both the mixed numbers into improper fractions. 19 3 22 9 57 9 22 9 35 9 −=−= Subtract the fractions, using the lowest common denominator of 9. 35 9 8 9 = 3 The answer is an improper fraction so change it back to a mixed number. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
151 7.3 Subtracting mixed numbers Exercise 7.3 1 Copy and complete these subtractions. a 5 2 1 3 2 3 − b 9 3 1 6 5 12 − Step 1: 16 3 8 3 − Step 1: 6 41 12 − Step 2: 16 3 8 3 3 − = Step 2: 6 41 12 12 41 12 12 −= −= Step 3: 3 3 = 2 Step 3: 12 4 4 = = 5 c 5 3 3 4 5 6 − d 4 1 1 4 3 5 − Step 1: 23 4 6 − Step 1: 4 5 − Step 2: 23 4 6 12 12 12 − = − = Step 2: 4 5 20 20 20 − = − = Step 3: 12 12 = 1 Step 3: 20 20 = 2 2 Work out these subtractions. Show all the steps in your working. a 2 1 3 8 5 8 − b 3 1 3 5 7 10 − c 4 1 2 3 11 12 − d 5 3 2 3 1 4 − Think like a mathematician 3 Work with a partner or in a small group to discuss this question. Look at the different methods that Anders and Xavier use to work out 9 3 4 7 6 7 − Anders Change 94 7 into 8+1+ 4 7 =8+ 7 7 +4 7 =8 11 7 So 9 4 7 – 36 7 is the same as 8 11 7 – 36 7 So 8 – 3=5 and 11 7 – 6 7 =5 7 , so answer is 55 7 Xavier Subtract whole numbers: 9 – 3=6 Subtract fractions: 4 7 – 6 7 =–2 7 So 94 7 – 36 7 =6 – 2 7 =55 7 a What are the advantages and disadvantages of: i Ander’s method ii Xavier’s method? b Which method do you prefer: Anders’ method, Xavier’s method or the method in the worked example? Explain why. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
152 7 Fractions 4 Work out these subtractions. Show all the steps in your working. Use your preferred method. a 3 1 3 14 4 7 − b 7 2 1 3 7 12 − c 8 4 2 3 1 4 − d 6 4 7 12 17 18 − 6 Shen has two pieces of fabric. One of the pieces is 13 4 m long. The other is 2 3 8 m long. a estimate, then b calculate, the difference in length between the two pieces of material. 7 Zalika has a length of wood that is 5 1 4 m long. First, Zalika cuts a piece of wood 13 5 m long from the length of wood. Then she cuts a piece of wood 2 9 10 m long from the piece of wood she has left. How long is the piece of wood that Zalika has left over? 8 The diagram shows the lengths of the three sides of a triangle. a estimate, then b calculate, the difference in length between the longest and shortest sides of the triangle. 5 m6 7 7 m3 4 3 m 2 3 Write your answer to part b as a mixed number in its simplest form. 3 4 1 m 3 8 2 m ? m 1 4 5 m 3 5 1 m 9 10 2 m Think like a mathematician 5 Work with a partner or in a small group to answer this question. Marcus is looking at the question 9 3 2 7 8 9 − a Is Marcus correct? Explain your answer. b Choose two mixed numbers of your own but don’t subtract them yet. Write between which two whole numbers your total will be. Check that your answer is correct. c Think of subtracting any two mixed numbers. Write a rule for working out between which two whole numbers the total will be. d How would you change this rule if you were subtracting 3, 4 or 5 mixed numbers? Without subtracting the fractions, I know the answer is going to be between 5 and 7. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
153 7.3 Subtracting mixed numbers 10 Sami drives 16 5 8 km from his home to work. Sami drives 112 5 km from his home to the supermarket. What is the difference between the distance he drives from his home to work and from his home to the supermarket? 11 Fina has two bags of lemons. One bag has a mass of 4 7 10 kg. The other bag has a mass of 2 4 15 kg. What is the difference in mass between the two bags of lemons? 12 This is part of Rio’s homework. He has made a mistake in his solution. Question Work out 43 5 – 9 10 Answer 43 5 =4 6 10 4 6 10 – 9 10 =4 3 10 a Explain the mistake Rio has made. b Work out the correct answer. 13 In this pyramid, you find the mixed number in each block by adding the mixed numbers in the two blocks below it. Complete the pyramid. 3 4 12 2 3 8 4 5 2 5 9 1 3 4 12 2 3 − 8 7 10 4 kg 4 15 2 kg Tip If you cannot see Rio’s mistake, work through the question yourself and then compare your answer with his. Think like a mathematician 9 Work with a partner or in a small group to discuss this question. What is the quickest method to use to work out the answer to 6 3 5 8 1 2 − ? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
154 7 Fractions 14 The perimeter of this quadrilateral is 3513 36 m. 1 9 5 m 5 6 9 m 2 3 8 m Work out the length of the missing side. Tip The perimeter of a shape is the distance around the edge of the shape. Summary checklist I can subtract mixed numbers. You already know how to multiply a fraction by an integer. For example: 2 3 × 12 Solution: 12÷3=4 and 2×4=8 You also know how to multiply two integers together using partitioning. For example: 8×23=8×20+8×3 Solution: 160+24=184 You can now combine these methods to multiply an integer by a mixed number. 7.4 Multiplying an integer by a mixed number In this section you will … • multiply an integer by a mixed number. Key words mean partitioning simplified Worked example 7.4 Work out i an estimate and ii the accurate answer to a 2 16 1 2 × b 4 20 2 3 × We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
155 7.4 Multiplying an integer by a mixed number Exercise 7.4 1 Copy and complete these multiplications. a 3 8 3 8 8 1 2 1 2 ×=×+ × = + = b 2 12 2 12 12 1 4 1 4 × =× + × = + = c 4 9 4 9 9 2 3 2 3 ×=×+ × = + = d 8 10 8 10 10 3 5 3 5 × =× + × = + = 2 This rectangle has length 15m and width 2 1 3 m. Work out a an estimate for the area of the rectangle b the accurate area of the rectangle. 2 m 1 3 15m Continued Answer a i Estimate: 3×16=48 Round the fraction to the nearest whole number. ii 2 16 2 16 16 32 8 40 1 2 1 2 × =× + × = + = Use partitioning to split the multiplication into two parts. 2×16=32 and 1 2 × = 16 8 Add the two numbers together to get the total. b i Estimate: 5×20=100 Round the fraction to the nearest whole number. ii 4 20 4 20 20 2 3 2 3 × =× + × Use partitioning to split the multiplication into two parts. = 80 + 40 3 4×20=80 and 2 3 40 3 × = 20 = 80 13 + 1 3 Change 40 3 into a mixed number. = 931 3 Add the two numbers together to get the total. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
156 7 Fractions 3 Lin has 20 containers. The mean amount of water the containers hold is 2 2 5 litres. Lin uses this formula to work out the total amount of water that the containers hold. total amount of water mean amount of water number of co = × ntainers Lin thinks the total amount of water the containers can hold is 46 litres. Is Lin correct? Explain your answer. 4 Copy and complete these multiplications. Use estimation to check your answers. a 4 9 4 9 9 4 1 2 1 2 9 2 1 2 ×=×+ × = + = + = b 3 11 3 11 11 3 4 3 4 33 4 × =× + × = + = + = c 5 7 5 7 7 2 3 2 3 ×=×+ × = + = + = d 2 6 2 6 6 2 5 2 5 ×=×+ × = + = + = 5 The diagram shows a square joined to a rectangle. Work out a an estimate for the area of the shape b the accurate area of the shape. 4 9 12 cm 5cm 5cm Tip For the estimate in part a, round 4 1 2 to 5, then work out 5×9. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
157 7.4 Multiplying an integer by a mixed number Think like a mathematician 7 Work with a partner or in a small group to discuss this question. Look at the different methods that Anders and Xavier use to work out 3 8 2 3 × Anders 3 2 3 × 8=3 × 8+ 2 3 × 8 = 24+ 16 3 = 24+5 1 3 = 29 1 3 Xavier Change 3 2 3 into 11 3 11 3 × 8=88 3 = 29 1 3 a What are the advantages and disadvantages of i Ander’s method ii Xavier’s method? b Use both methods to work out i 2 6 3 7 × ii 6 12 5 9 × c Which method do you prefer, Anders’ or Xavier’s? Explain why. 6 Martha is going to lay paving slabs on part of her garden. This part of her garden is a rectangle with length 33 5 m and width 2m. a Martha estimates that the area of the rectangle is 6m2 . Is Martha correct? Explain your answer. b Work out the area of the rectangle. Martha buys paving slabs that cost $42 per square metre. She can only buy a whole number of square metres. Martha works out that the paving slabs will cost her $294. c Is Martha correct? Explain your answer. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
158 7 Fractions 8 This is how Zara works out 2 15 1 6 × . 2 1 6 × 15=2 × 15+ 1 6 × 15 = 30+ 15 6 = 30+2 3 6 = 30 +2 1 2 = 32 1 2 Sofia says, ‘You changed 15 6 to a mixed number and then simplified 3 6 to 1 2 . I would have simplified 15 6 to 5 2 before changing it to a mixed number.’ a Do you prefer Zara’s method or Sofia’s method? Explain why. b Use your preferred method to work these out. Write your answer in its simplest form. i 3 10 3 8 × ii 4 14 3 4 × iii 2 12 7 10 × 9 Jamal works in a garden centre. It takes him 5 1 4 minutes to plant one tray of seedlings. How long will it take him to plant 50 trays of seedlings? Give your answer in hours and minutes. Tip Seedlings are seeds that are just starting to grow into plants. Think like a mathematician 10 Work with a partner or in a small group to answer this question. a Work out i 3 2 1 5 × ii 3 3 1 5 × b What is the smallest integer that you must multiply by 3 1 5 to get a whole number answer? c What is the smallest integer that you must multiply by 32 5 to get a whole number answer? What about 33 5 and 34 5 ? d What do you notice about your answers to b and c? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
159 7.5 Dividing an integer by a fraction Continued e What is the smallest integer that you must multiply by 3 1 7 to get a whole number answer? What about 32 7 , 33 7 , 34 7 , 35 7 and 36 7 ? What do you notice about your answers? f Try starting with fractions with different denominators such as 6, 8, 9 and 11, for example 2 1 6 or 4 1 8 , etc. Do the patterns you noticed in parts d and e work for these fractions as well? Explain your answers. 11 Work out a an estimate for the area of the blue section of this rectangle b the accurate area of the blue section of this rectangle. 12 m 3 5 4 m 2 3 2m Look back at this section on multiplying an integer by a mixed number. a What did you find easy? b What did you find hard? c Are there any parts that you think you need to practise more? Summary checklist I can multiply an integer by a mixed number. Look at this diagram. It shows three rectangles, each divided in half. When you work out 3 1 2 ÷ , the question is asking you ‘How many halves are in three?’ 7.5 Dividing an integer by a fraction In this section you will … • divide an integer by a proper fraction. Key words reciprocal upside down 1 2 3 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
160 7 Fractions Worked example 7.5 Work out a 4 1 3 ÷ b 10 3 4 ÷ Answer a 4 4 3 12 1 3 ÷ = × = You can use this diagram to work out how many thirds are in four. b 10 10 13 3 4 4 3 40 3 1 3 ÷ = × = = Use the reciprocal method to answer this question. Turn the fraction upside down to write the reciprocal and multiply. The answer is an improper fraction which cannot be cancelled down. Write the answer as a mixed number. You can see that there are six, so 3 6 1 2 ÷ = Another method you can use is to turn the fraction upside down, then multiply by the integer. This is called multiplying by the reciprocal of the fraction. So, 3 3 6 1 2 2 1 6 1 ÷ = × = = Tip The reciprocal of a fraction is the fraction turned upside down. So the reciprocal of 1 2 is 2 1 Exercise 7.5 1 Work out the answers to these calculations. Use the diagrams to help you. a 2 1 3 ÷ b 4 1 2 ÷ c 3 1 4 ÷ d 5 1 5 ÷ 2 Read what Sofia says about dividing an integer by a unit fraction. a Can you explain why Sofia’s method works? b Check your answers to Question 1 using Sofia’s method. c Use Sofia’s method to work out i 12 1 3 ÷ ii 25 1 4 ÷ iii 8 1 9 ÷ The quick way to divide an integer by a unit fraction is to multiply the integer by the denominator of the fraction. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
161 7.5 Dividing an integer by a fraction 3 The area of a rectangle is 16m2 . The width of the rectangle is 1 5 m. What is the length of the rectangle? 4 Kai uses this formula to work out the average speed of a car in kilometres per hour (km/h), when he knows the distance it has travelled and the time it has taken. speed=distance÷time Work out the average speed of these cars. The first one has been done for you. a distance=30 km, time= 1 4 hour So, speed=30 30 4 120 1 4 ÷ = × = km/h b distance=45 km, time=1 2 hour c distance=16 km, time=1 6 hour Tip area of rectangle =length×width, so length=area ÷width. 6 Work out the answers to these calculations. Use the diagrams to help you. a 4 2 3 ÷ b 6 3 4 ÷ c 4 2 5 ÷ d 8 4 7 ÷ Think like a mathematician 5 Work with a partner or in a small group to answer this question. a How can you use this diagram to work out 2 2 3 ÷ ? b How can you use this diagram to work out 3 3 4 ÷ ? c Discuss your methods with other learners in the class. Write the method that you like better. Tip Think of the question as ‘How many 3 4 are in 3?’ Tip Think of the question as ‘How many 2 3 are in 2?’ We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
162 7 Fractions 8 Work out the answers to these calculations. Use the reciprocal method. The first two have been started for you. a 11 11 3 4 4 3 3 ÷ = × = = b 9 9 5 6 6 5 ÷ = × = = c 7 4 5 ÷ d 12 7 10 ÷ e 10 4 11 ÷ 9 This is part of Anil’s homework. You can see that he simplified the improper fraction to its lowest terms before he changed it into a mixed number. Question Work out 10÷4 5 Answer 10÷4 5 =10 × 5 4 = 50 4 = 25 2 = 12 1 2 Think like a mathematician 7 Work with a partner or in a small group to answer this question. Read what Zara says. a Use the diagram to show that Zara is correct. b Use the diagram to work out 4 3 4 ÷ c Complete the reciprocal method to check your answer to part c is correct. 4 4 3 4 4 3 16 3 ÷ = × = = If your answers to parts b and c are different, explain the mistake you have made. d Discuss your answers with other learners in the class. e Discuss when you think it is easier to use the diagram method and the reciprocal method. The answer to 3 2 3 ÷ is 4 1 2 We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
163 7.5 Dividing an integer by a fraction 10 Sofia is looking for patterns in the division questions. She has come up with two ideas. Are Sofia’s ideas correct? Explain your answers. Look back at the questions you have done in this exercise to help you explain. 11 a Here is a sequence of calculations. 1 1 6 ÷ , 2 1 6 ÷ , 3 1 6 ÷ , 4 1 6 ÷ , … i Work out the sequence of answers. ii Write the next two terms of the sequence. iii Describe the sequence of answers in words. Read what Marcus says. a Whose method do you prefer, Anil’s or Marcus’s? Explain why. b Work out these calculations. Give each answer as a mixed number in its lowest terms. i 6 4 7 ÷ ii 4 6 11 ÷ iii 12 9 10 ÷ iv 9 12 13 ÷ I use a different method. I change the improper fraction to a mixed number, and then simplify the fraction to its lowest terms like this. 50 4 2 4 1 2 = = 12 12 Activity 7.5 a On a piece of paper, write four division questions: two like those in Question 1 and two like those in Question 8. You must use an integer and a proper fraction. b On a separate piece of paper, work out the answers. c Exchange your questions with a partner and answer their questions. d Exchange back and mark each other’s work. Discuss any mistakes that have been made. 1. When you divide an integer by a proper fraction, the answer is always bigger than the integer you started with. 2. When you divide an integer by two different proper fractions, the larger fraction will give you the larger answer. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
164 7 Fractions b Here is a different sequence of calculations. 1 2 6 ÷ , 2 2 6 ÷ , 3 2 6 ÷ , 4 2 6 ÷ , … i Work out the sequence of answers. ii Write the next two terms of the sequence. iii Describe the sequence of answers in words. c Compare your sequences of answers in parts a and b. What do you notice? Explain why this happens. d Look at your answers to part bi and, without actually completing the calculations, write down the sequence of answers for this sequence of calculations. 1 3 6 ÷ , 2 3 6 ÷ , 3 3 6 ÷ , 4 3 6 ÷ , … Explain how you worked out your answer. e Here is another sequence of answers for a sequence of calculations. Calculations: 1 1 15 ÷ , 2 1 15 ÷ , 3 1 15 ÷ , 4 1 15 ÷ , … Answers: 15, 30, 45, 60, … Use this information to write down the sequence of answers for this sequence of calculations. 1 5 15 ÷ , 2 5 15 ÷ , 3 5 15 ÷ , 4 5 15 ÷ , … Explain how you worked out your answer. Summary checklist I can divide an integer by a proper fraction. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
165 7.6 Making fraction calculations easier In this section you will … • simplify calculations containing fractions. When you are calculating using fractions, you can often make a calculation easier by using different strategies. These strategies will help you to work with fractions mentally. This means you should be able to do simple additions, subtractions, multiplications and divisions ‘in your head’. You should also be able to solve word problems mentally. This section will help you to practise the skills you need. For harder questions, it may help you to write down some of the steps in the working. These workings will help you to remember what you have done so far, and what you still need to do. With all calculations, you must remember the correct order of operations. 7.6 Making fraction calculations easier Key word strategies 3 4 3 8 + = ? Worked example 7.6 Work out mentally a 3 4 3 8 + b 4 5 3 4 − c 6 3 4 ÷ d 2 5 2 3 1 2 × + ( ) Answer a 6 8 3 8 9 8 + = Change 3 4 to 6 8 so you can add it to 3 8 = 11 8 Then change 9 8 to a mixed number. b 4 5 3 4 4 4 3 5 5 4 − = × − × × Numerator: Multiply the diagonal pairs of numbers, shown by the red and blue arrows – so, work out 4×4 and 3×5. Denominator: multiply the denominators, so work out 5×4. = 16 15 − 20 Finally do the subtraction. = 1 20 This gives a numerator of 1, with a denominator of 20. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
166 7 Fractions Exercise 7.6 In this exercise, work out as many of the answers as you can mentally. Write each answer in its simplest form and as a mixed number when appropriate. 1 Work out these additions and subtractions. Some working has been shown to help you. a 1 3 1 6 2 6 1 6 6 +=+= = b 1 8 1 4 1 8 8 + = + = c 4 5 1 10 8 10 1 10 − = − = d 5 6 1 3 5 6 6 − = − = = 2 Work out these additions and subtractions. Use the same method as in part a of the worked example. a 1 2 1 6 + b 3 4 1 8 + c 3 5 1 10 + d 1 2 3 8 + e 3 4 5 12 + f 7 15 4 5 + g 1 3 1 9 − h 1 4 1 8 − i 1 5 1 15 − j 2 3 1 6 − k 4 5 1 10 − l 11 20 2 5 − Continued c 6 3 4 ÷ 6×4=24 Multiply the 6 by the 4. 24÷3=8 Then divide the answer by 3. This is equivalent to turning the fraction upside down and multiplying by 6. d 2 3 1 2 2 2 3 1 3 2 + = × + × × Using the correct order of operations, brackets come first. = 4 3 + 6 Work out the addition. = 7 6 Leave the answer as an improper fraction. 2 5 7 6 14 30 × = Now work out the multiplication. Multiply the numerators and multiply the denominators. = 7 15 Write the answer in its simplest form. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
167 7.6 Making fraction calculations easier 3 Work out these additions and subtractions. Use the same method as in part b of the worked example. a 1 3 1 5 + b 1 4 1 7 + c 2 9 1 5 + d 3 4 2 3 + e 5 8 1 5 + f 1 4 5 6 + g 1 2 1 3 − h 4 5 1 4 − i 5 7 1 2 − j 3 4 2 7 − k 7 12 3 8 − l 8 9 3 4 − Think like a mathematician 4 a Work with a partner. Discuss the best method to use to work out the answer to this question. In a box of chocolates, 1 5 of the chocolates are white chocolate, 1 2 are milk chocolate, and the rest are dark chocolate. What fraction of the chocolates are dark chocolate? b Compare your methods with those of other learners in the class. Do you think your method was the best method? 5 In a hockey squad, 1 3 of the players are short, 1 4 of the players are medium height and the rest are tall. What fraction of the squad are tall? 6 In a box of fruit, 2 5 are apples, 1 6 are guavas and the rest are coconuts. What fraction of the fruit in the box are coconuts? 7 Work out these calculations. Use the same method as in part c of the worked example. Some working has been shown to help you with the first two. a 4 4 3 2 2 3 ÷ = × ÷ = b 8 8 5 4 4 5 ÷ = × ÷ = c 9 1 2 ÷ d 6 2 5 ÷ e 9 3 4 ÷ f 10 5 6 ÷ Think like a mathematician 8 With a partner, work out how to use the fractions button on a calculator. The fractions button looks like this. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
168 7 Fractions Continued Work out the answer to 18 5 7 ÷ . Write your answer as an improper fraction. Use the calculator to turn the improper fraction into a mixed number. You will need to use this button. S⇔D b c ⇔ d c a 9 a Work out mentally i ii 7 3 5 ÷ iii 11 2 3 ÷ iv 8 5 7 ÷ b Use a calculator to check your answers to part a. c Did you get your answers to part a correct? If not, what mistakes did you make? 10 The diagram shows a path. The area of the path is 10m2 . The width of the path is 3 4 m. What is the length of the path? 11 This is how Marcus mentally works out 1 3 5 6 1 2 × − ( ) First, I work out 5 6 1 2 − which equals 4 4 which cancels down to 1. Then I work out 1 3 ×1, which equals 1 3 a Explain the mistake Marcus has made. b Work out the correct answer. 12 Work out these calculations. If you cannot do them mentally, write down some workings to help you. a 6 5 6 1 6 × − ( ) b 4 1 3 1 3 ÷ + ( ) c 11 12 3 4 1 2 − − ( ) d 11 12 1 2 1 3 − + ( ) e 1 4 1 2 5 9 2 9 + ( ) ( ) × − f 4 1 3 9 2 3 2 5 3 10 + ÷ + ( ) ( ) 9 4 5 ÷ length m 3 4 Tip Remember brackets come first, then indices, then division and multiplication, then addition and subtraction. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
169 7.6 Making fraction calculations easier The answers to these cards can be rearranged to form two different sequences of numbers. 13 Zara works out the answers to these calculation cards. A 3 3 4 3 4 × + ( ) B 2 7 10 1 5 − − ( ) C 4 2 1 2 3 1 6 × − ( ) D 6 1 34 9 7 9 ÷ − ( ) Read what Zara says. Is Zara correct? Write the first term and the term-to-term rule of the sequences you can find. In this exercise, you have used mental methods to work out fraction calculations. Look back at the worked example and the types of question shown in parts a, b, c and d. a Which type of questions have you found i the easiest ii the hardest to work out mentally? b Which type of questions are you confident working out mentally? c Which type of questions do you need more practice with working out mentally? Summary checklist I can use different methods to make fraction calculations easier. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
Check your Progress Nimrah thinks of a number, n . Write an expression for the number Nimrah gets each time. a She multiplies the number by 4. b She subtracts 6 from the number. c She multiplies the number by 3 d She divides the number by 6 then adds 5. then subtracts 1. Copy the number line and show the inequality on the number line. Write down the inequality that this number line shows. Work out the value of each expression. Loli lives with 3 friends. They share the electricity bill equally between the four of them. Write a formula to work out the amount they each pay, in: i words ii letters Use your formula in part a ii to work out the amount they each pay when the electricity bill is $96. 6 Simplify these expressions. a n + n + n b 3c + 5c c 9x − x 7 Simplify these expressions by collecting like terms. a 5c + 6c + 2d b 6c + 5k + 5c + k c 3xy + 5yz − 2xy + 3yz 8 Work these out. a 3 + (x × 2) b 6(3 − w) c 4(3x + 2) d 3(7 − 4v) 9 Solve each of these equations and check your answers. a n + 3 = 8 b m − 4 = 12 c 3p = 24 d x = 3 5 10 Shen has set a puzzle. Write an equation for the puzzle. Solve the equation to find the value of the unknown number. 170 7 Fractions Check your progress 1 Use a written method to convert these fractions into decimals. Write if the fraction is a terminating or recurring decimal. a 3 8 b 4 9 2 Write the fractions −12 5 , − 27 10 , −17 6 and − 38 15 in order of size, starting with the smallest. 3 Work out a 6 2 3 4 5 6 − b 4 9 5 6 × 4 Work out a 9 3 8 ÷ b 8 3 5 ÷ 5 Work out mentally a 2 5 3 10 + b 6 7 2 3 − c 4 2 7 ÷ d 9 3 4 1 3 × − ( ) We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
171 Getting started 1 a Write the number of lines of symmetry for each of these shapes. b Write the order of rotational symmetry for each of these shapes. i ii iii iv 2 a The diagram shows a cuboid. Write the number of i faces ii edges iii vertices of the cuboid. b Draw the top view, front view and side view of this cuboid. 3 Make h the subject of each formula. a x=t+h−p b x h = 4 c y=3xh 4 Label the parts of the circle shown. All the words you need are in the cloud. centre chord diameter tangent radius circumference 8 Shapes and symmetry We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
172 8 Shapes and symmetry Continued 5 Match each 3D shape with its name. a b c d e f g h cylinder tetrahedron sphere equilateral triangular prism cone cube square-based pyramid cuboid 6 A scale drawing of a building uses a scale of 1 :20 a The height of the building on the drawing is 25 cm. What is the height, in metres, of the building in real life? b The length of the building in real life is 12m. What is the length, in centimetres, of the building on the drawing? Wherever you look, you will see objects of different shapes and sizes. Many are natural, but many have been designed by someone. An architect is a person who plans and designs buildings. Architects make scale drawings, and often scale models too, of the buildings they plan. They make sure their designs follow local rules and regulations, and they make sure the people who build their buildings follow the plans correctly. Towns and cities all over the world have buildings designed to meet the needs of the people who live and work there. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
8 Shapes and symmetry 173 The Burj Khalifa in Dubai is the tallest building in the world (as of 2018). It is over 828 metres tall and contains 163 floors. It holds the record for having an elevator with the longest travel distance in the world. Construction began in September 2004 and the building was officially opened in January 2010. The Burj Khalifa cost $1.5 billion to construct. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
174 8 Shapes and symmetry In this section you will … • identify the symmetry of regular polygons • identify and describe the hierarchy of quadrilaterals. 8.1 Quadrilaterals and polygons You already know how to describe the side length and symmetry properties of a regular polygon. For example, a regular pentagon has: • 5 sides the same length • 5 lines of symmetry • rotational symmetry of order 5. A quadrilateral is a 2D shape with four straight sides. These are the seven quadrilaterals you need to know. square rectangle parallelogram rhombus kite trapezium isosceles trapezium You can describe quadrilaterals using the properties of their sides and angles. For example, a square has: • all sides the same length • two pairs of parallel sides • all angles 90°. Key words hierarchy lines of symmetry quadrilateral regular polygon rotational symmetry We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
175 8.1 Quadrilaterals and polygons Worked example 8.1 a Sketch a regular octagon. Describe the side length and symmetry properties of the octagon. b Sketch a parallelogram. Describe the side and angle properties of a parallelogram. Answer a A regular octagon has: • 8 sides the same length • 8 lines of symmetry • rotational symmetry of order 8. The octagon is regular, so all sides are the same length. The diagram shows the lines of symmetry. In one complete turn, the octagon will fit onto itself exactly 8 times. b A parallelogram has: • two pairs of sides the same length • two pairs of parallel sides • opposite angles that are equal. Exercise 8.1 1 This diagram shows some regular polygons. a b c d e f We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
176 8 Shapes and symmetry a Copy and complete this table. Name of regular polygon Number of sides Number of lines of symmetry Order of rotational symmetry pentagon 5 hexagon heptagon 7 octagon nonagon 9 decagon 10 b What do you notice about the number of sides, the number of lines of symmetry, and the order of rotational symmetry for each of the polygons? c Copy and complete these sentences. The number of sides of a regular polygon is ........................... the number of lines of symmetry. The number of sides of a regular polygon is ........................... the order of rotational symmetry. d Use your answers to part c to answer these questions. i A hendecagon is a regular polygon with 11 lines of symmetry. How many sides does it have? ii A dodecagon is a regular polygon with order of rotational symmetry 12. How many sides does it have? 2 Look at rectangle ABCD. Write true or false for each statement. If the statement is false, write the correct statement. a AC is the same length as BD. b AB is parallel to AC. c BD is parallel to AB. d All the angles are 90°. A B C D We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
177 8.1 Quadrilaterals and polygons 3 Copy and complete the side and angle properties of these four quadrilaterals. Choose from the words in the box. a A square has: sides the same length pairs of parallel sides angles are 90°. b A rectangle has: pairs of sides the same length pairs of parallel sides angles are 90°. c A rhombus has: sides the same length pairs of parallel sides angles are equal. d A parallelogram has: pairs of sides the same length pairs of parallel sides angles are equal. opposite two all Think like a mathematician 4 Work with a partner or in a small group to discuss these questions. a Is a square a rectangle? Is a rectangle a square? b Is a square a rhombus? Is a rhombus a square? c Is a parallelogram a rectangle? Is a rectangle a parallelogram? d Is a parallelogram a rhombus? Is a rhombus a parallelogram? Discuss your answers with other groups in the class. 5 Zara is describing a square to Marcus. Has Zara given Marcus enough information for him to work out that the quadrilateral is a square? Explain your answer. My quadrilateral has two pairs of parallel sides and all the angles are 90º. What is the name of my quadrilateral? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
178 8 Shapes and symmetry 6 Look at isosceles trapezium ABCD. Write true or false for each statement. If the statement is false, write the correct statement. a AC is the same length as CD. b AB is parallel to CD. c Angle CAB is the same size as angle ACD. d Angle BDC is the same size as angle ACD. 7 Copy and complete the side and angle properties of these three quadrilaterals. The missing words are all numbers. a A trapezium has: pair of parallel sides. b An isosceles trapezium has: pair of sides the same length pair of parallel sides pairs of equal angles. c A kite has: pairs of sides the same length pair of equal angles. A This is angle CAB C D B This is angle BDC Think like a mathematician 8 Work with a partner or in a small group to discuss these questions. a Is a trapezium always an isosceles trapezium? Is an isosceles trapezium always a trapezium? b Is a kite a rhombus? Is a rhombus a kite? c Is a parallelogram a trapezium? Is a trapezium a parallelogram? Discuss your answers with other groups in the class. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
179 8.1 Quadrilaterals and polygons 9 Marcus is describing a kite to Zara. My quadrilateral has two pairs of sides the same length and two pairs of equal angles. What is the name of my quadrilateral? Has Marcus given Zara the correct information for her to work out that the quadrilateral is a kite? Explain your answer. 10 Follow this classification flow chart for each quadrilateral. Write the letter where each shape comes out. a square b rectangle c parallelogram d kite e trapezium f rhombus g isosceles trapezium Only one pair of parallel sides? START Two sides the same length? All sides the same length? All sides the same length? Two pairs of equal angles? K no no yes yes yes All angles are 90º? no no no no yes yes yes M N H J L I We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
180 8 Shapes and symmetry 11 This diagram shows the hierarchy of quadrilaterals. In the diagram, a quadrilateral below another is a special case of the one above it. For example, a square is a special rectangle but a rectangle is not a square. Use the diagram to decide if these statements are true or false. a A parallelogram is a special trapezium. b A kite is a special rhombus. c A trapezium is a special quadrilateral. Look back at this exercise. a How confident do you feel in your understanding of this section? b What can you do to increase your confidence? Summary checklist I can identify the symmetry of regular polygons. I can identify and describe the hierarchy of quadrilaterals. Activity 8.1 On a piece of paper, write four statements like the ones in Question 11. Two of them must be true and two of them must be false. Exchange statements with a partner. Write if your partner’s statements are true or false. Exchange back and mark each other’s work. Discuss any mistakes. Quadrilateral Trapezium Isosceles Trapezium Rectangle Parallelogram Square Rhombus Kite We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
181 8.2 The circumference of a circle You already know the names of the parts of a circle. Did you know there is a link between the circumference of a circle and the diameter of a circle? This table shows the circumference and diameter measurements of four circles. Circle Circumference (cm) Diameter (cm) Circumference ÷diameter A 9.1 2.9 B 19.8 6.3 C 25.1 8 D 37.1 11.8 Copy the table and fill in the final column. Give your answers correct to two decimal places. What do you notice? You should notice that all the answers are 3.14 correct to 2 decimal places. This means that the ratio of the diameter to the circumference of a circle is approximately 1 :3.14 The number 3.14... has a special name, pi. It is written using the symbol π. π is the number 3.141 592 653 589…, but you will often use 3.14 or 3.142 as an approximate value for π. You now know that circumference diameter = π, so you can rearrange the formula to get: C=πd where: C is the circumference of the circle d is the diameter of the circle 8.2 The circumference of a circle In this section you will … • know and use the formula for the circumference of a circle. Key words accurate approximate value circumference diameter pi (π) radius semicircle Tip C=πd means C=π×d We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
182 8 Shapes and symmetry Exercise 8.2 1 Copy and complete the workings to find the circumference of each circle. Use π=3.14. Round your answers correct to 1 decimal place (1 d.p.). a diameter=6cm C=πd =3.14×6 = = cm (1 d.p.) b diameter=25cm C=πd =3.14× = = cm (1 d.p.) c diameter=4.25m C=πd = × = = m (1 d.p.) 2 Copy and complete the workings to find the circumference of each circle. Use π=3.142. Round your answers correct to 2 decimal places (2 d.p.). a radius=7cm d=2×r =2×7 =14 cm C=πd =3.142×14 = = cm (2 d.p.) b radius=2.6cm d=2×r =2× = cm C=πd =3.142× = = cm (2 d.p.) c radius=0.9m d=2×r =2× = m C=πd = × = = m (2 d.p.) Worked example 8.2 Work out the circumference of a circle with a diameter 3 cm b radius 4m. Use π=3.14. Round your answers correct to 1 decimal place (1 d.p.). Answer a C =πd =3.14×3 =9.42 =9.4 cm Write the formula you are going to use. Substitute π=3.14 and d=3 into the formula. Work out the answer. Round your answer to 1 d.p. and remember to write the units, cm. b d=2×r=2×4=8m C=πd =3.14×8 =25.12 =25.1m You are given the radius, so work out the diameter first. Write the formula you are going to use. Substitute π=3.14 and d=8 into the formula. Work out the answer. Round your answer to 1 d.p. and remember to write the units, m. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
183 8.2 The circumference of a circle Think like a mathematician 3 So far in this unit you have used approximate values for π. You have used π=3.14 and π=3.142. There is another approximate value you can use: π=22 7 A more accurate value for π is stored on your calculator. Can you find the button with the π symbol on it? a Use the π button on your calculator to work out the accurate circumference of a circle with diameter 12cm. Write all the numbers on your calculator screen. b Now work out the circumference of the same circle using approximate values for π of: i 3.14 ii 3.142 iii 22 7 c Compare your answers to parts a and b. Which approximate value for π gives the closest answer to the accurate answer? d When you answer questions and you need to use π, which value of π do you think it is best to use? Explain why. 4 Work out the circumference of each circle. Use the π button on your calculator. Round your answers correct to 2 decimal places (2 d.p.). a diameter=9cm b diameter=7.25m c radius=11cm d radius=3.2m Think like a mathematician 5 Work with a partner or in a group to answer this question. So far in this unit you have used the formula, C=πd In questions 2 and 4, you found the circumference when you were given the radius. Can you write a formula to find the circumference which uses r (radius) instead of d (diameter)? Test your formula on Question 4, parts c and d. Does it work? Compare your formula with other groups in the class. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
8 Shapes and symmetry 184 For questions 6 to 11, use the π button on your calculator. 6 Fu and Fern use different methods to work out the answer to this question. Work out the diameter of a circle with circumference 16.28 cm. Give your answer correct to 3 significant figures. This is what they write. Fu Fern Step 1: Make d the subject of the formula. C=π× d C π =d d= C π Step 2: Substitute in the numbers. d= 16.28 π =5.18208... =5.18 cm (3 s.f.) Step 1: Substitute in the numbers. C=π× d 16.28=π× d Step 2: Solve the equation. 16.28=π× d 16.28 π =d 5.18208...=d d=5.18 cm (3 s.f.) a Look at Fu and Fern’s methods. Do you understand both methods? Do you think you would be able to use both methods? b Which method do you prefer and why? c Use your preferred method to work out the diameter of a circle with: i circumference=28cm ii circumference=4.58m d Make r the subject of the formula C r = 2π e Use your formula from part d to work out the radius of a circle with: i circumference=15cm ii circumference=9.25m 7 The circumference of a circular disc is 39 cm. Work out the diameter of the disc. Give your answer correct to the nearest millimetre. Tip Remember, C r =2π means C r = ×2 π × We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
185 8.2 The circumference of a circle 8 A circular ring has a circumference of 5.65 cm. Show that the radius of the ring is 9mm, correct to the nearest millimetre. 9 This is part of Ahmad’s homework. Question Work out the perimeter of this semicircle. Answer: perimeter=half of circumference+diameter P= πd 2 +d = π × 16 2 +16 =25.13+16 =41.13 cm 16cm a Use Ahmad’s method to work out the perimeter of a semicircle with: i diameter=20cm ii diameter=15m iii radius=8cm iv radius=6.5m Round your answers correct to 2 d.p. b Imagine you have a friend who does not know how to work out the perimeter of a semicircle. Are you confident you could explain to them how to work it out? Can you use your knowledge to explain how to work out the perimeter of a quarter-circle? Make a sketch of a quarter-circle to help you. 10 The diagram shows a semicircle and a quarter-circle. Read what Zara says. 15m 10m Is Zara correct? Show working to support your answer. I think the perimeter of the semicircle is greater than the perimeter of the quarter-circle. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
186 8 Shapes and symmetry 11 Work out the perimeter of each compound shape. Give your answers correct to two decimal places. a 12 cm 8cm 14.4cm b 4.5m 4.5m c 28mm 28mm d 3.6cm 3.6cm 4.5 cm 4.5 cm Summary checklist I know that π is the ratio between the circumference and the diameter of a circle. I can use the formula for the circumference of a circle. Tip Remember, the perimeter is the total distance around the outside of the whole shape. Make sure you include all the different parts of the perimeter. In this section you will … • find the connection between the number of vertices, faces, and edges of 3D shapes • draw front, side, and top views of 3D shapes to scale. You already know how to describe a 3D shape using the number of faces, vertices and edges. You also know how to draw the top view, front view and side view of a 3D shape. The top view is the view from above the shape. It is sometimes called the plan view. The front view is the view from the front of the shape. It is sometimes called the front elevation. The side view is the view from the side of the shape. It is sometimes called the side elevation. You also need to be able to draw the top view, front view, and side view of a 3D shape to scale. 8.3 3D shapes Front Side Top Key words front view, front elevation side view, side elevation top view, plan view We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
187 8.3 3D shapes Worked example 8.3 The diagram shows a cuboid. 35cm 15 cm 10cm a Write the number of faces, vertices and edges of the cuboid. b Draw accurately the top view, front view and side view of the cuboid. Use a scale of 1 :5 Answer a 6 faces, 8 vertices, 12 edges The faces are the flat surfaces, the vertices are the corners, and the edges are where two faces meet. b Top view Front view Side view Use the scale to work out the dimensions of the cuboid for the drawing. The scale is 1:5, so 1 cm on the drawing represents 5cm in real life. Length: 35÷5=7cm Height: 10÷5=2cm Width: 15÷5=3cm So, the top view is a rectangle 7cm by 3cm. The front view is a rectangle 7 cm by 2 cm. The side view is a rectangle 3 cm by 2 cm. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
188 8 Shapes and symmetry Exercise 8.3 Think like a mathematician 1 Work with a partner or in a small group to answer these questions. a Copy and complete this table showing the number of faces, edges, and vertices of these 3D shapes. 3D shape Number of faces Number of vertices Number of edges cube cuboid 6 8 12 tetrahedron square-based pyramid triangular prism trapezoidal prism b What is the connection between the number of faces, vertices and edges for all of the 3D shapes? c Write a formula that connects the number of faces (F), vertices (V) and edges (E). d Compare your formula with other groups in your class. Do you have the same formula or a different formula? Is your formula the same, just written in a different way? e Does your formula work for shapes with curved surfaces, or does it only work for shapes with flat faces? Explain your answer. Tip You could start your formula E=… Tip Make a sketch of each shape to help you. Tip Remember, a tetrahedron is a triangular-based pyramid. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
189 8.3 3D shapes 2 Copy and complete the workings and scale drawings for this question. Draw the top view, front view, and side view of these shapes. Use a scale of 1 :2 a cube 6 cm b cuboid 8 cm 5 cm 3cm c cylinder 10 cm 14cm 7 cm Dimensions for drawing: 6÷2= cm Dimensions for drawing: 8÷2= cm 3÷2= cm 5÷2= cm Dimensions for drawing: 7÷2= cm 10÷2= cm 14÷2= cm Top view: Tip Draw a square of side length 3cm. Top view: Tip Draw a rectangle 4 cm by 2.5 cm. Top view: Tip Draw a circle of radius 3.5cm. Front view: Tip Draw a square of side length 3cm. Front view: Tip Draw a rectangle cm by cm. Front view: Tip Draw a rectangle cm by cm. Side view: Tip Draw a square of side length 3cm. Side view: Tip Draw a rectangle cm by cm. Side view: Tip Draw a rectangle cm by cm. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
190 8 Shapes and symmetry Think like a mathematician 3 Work with a partner or in a small group to answer these questions. Li and Seb are drawing the plan view of this shape. 4 cm 2 cm 1 cm 5 cm 3 cm This is what they draw. Li Seb 2 cm 3 cm 3cm 4 cm 4cm 1cm a Have either of them, or both of them, drawn the correct plan view? b Have they drawn their plan views to scale? c Discuss your answers to parts a and b with other groups in the class. d Draw the front elevation of the shape. e Draw the side elevation of the shape from i the left ii the right. Are your drawings for parts i and ii the same? f Discuss and compare your drawings in parts d and e with other groups in the class. Tip Remember, the plan view is the same as the top view. Tip Remember, the side elevation is the same as the side view. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
191 8.3 3D shapes 4 Draw the plan view, the front elevation and the side elevation of this 3D shape. Use a scale of 1 :10 30 cm 15 cm 20 cm 25 cm 60 cm 35cm 5 The diagram shows the dimensions of a shipping container. 6m 2.4m 2.6m Ajani makes a house from three shipping containers. The containers are arranged as shown in the diagram. Draw the plan view, the front elevation, and the side elevation of his house. Use a scale of 1 :100 3m 3m Tip If the views from the left side and from the right side are the same, you only need to draw one side elevation. Tip A shipping container is a very large metal box used to move goods by lorry, train or ship. Think like a mathematician 6 Work with a partner or in a small group to answer these questions. The diagram shows two triangular prisms, A and B. A 3cm 4 cm 5cm 6cm B 24cm 16 cm 18cm 20cm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
192 8 Shapes and symmetry 7 The diagram shows the dimensions of a village hall. The roof is an isosceles triangular prism. Draw the plan view, the front elevation, and the side elevation of the village hall. Use a scale of 1 :200 12m 20m 4m 3m Continued A is a right-angled triangular prism. B is an isosceles triangular prism. a Will the side elevation of prism A be the same from the left side and the right side? Explain your answer. b Draw the plan view, the front elevation, and the side elevation of prism A. Use the actual dimensions shown. Discuss the methods you could use to accurately draw the triangle. Which is the best method? c Will the side elevation of prism B be the same from the left side and the right side? Explain your answer. d Draw the plan view, the front elevation, and the side elevation of prism B. Use a scale of 1:4 Discuss the methods you could use to accurately draw the triangle. Which is the best method? Tip Convert all the dimensions from metres to centimetres before using the scale to work out the dimensions of the scale drawings. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
193 8.3 3D shapes 8 The diagram shows a shape drawn on dotty paper. The shape is made from 1 cm cubes. This diagram shows the plan view, front elevation and side elevation for the shape. A C B The diagrams have been drawn accurately on 1 cm squared paper. a Which diagram, A, B or C shows the i plan view ii front elevation iii side elevation? b Is it possible to have a shape made from a different number of 1cm cubes which has the same plan view as the shape above? Explain your answer. c Is it possible to have a shape made from a different number of 1 cm cubes which has the same plan view, front elevation, and side elevation as the shape above? Explain your answer. 9 The diagram shows four shapes drawn on dotty paper. The shapes are made from 1 cm cubes. Draw accurately the plan view, front elevation and side elevation for each of the shapes. Use 1 cm squared paper. The arrows in part a show the directions from which you should look at the shapes for the plan view (P), front elevation (F) and side elevation (S). Plan Side Front P S F a b c d We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
194 8 Shapes and symmetry 10 This is part of Marcus’s homework. Question Accurately draw the outline of the plan view, front elevation and side elevation of this shape. Do not include any internal lines. P S F 20cm 30cm Use a scale of 1: 5 and use 1cm squared paper. Answer The length of the shape is 20 cm, so it is made from 10 cm cubes. The scale is 1: 5, so the length of 20 cm needs to be 20÷5=4 cm The height of 30 cm needs to be 30÷5=6 cm The width of 10 cm needs to be 10÷5=2 cm Plan view Front elevation Side elevation Which of Marcus’s drawings is incorrect: the plan view, front elevation or side elevation? Explain the mistake he has made. We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
8.3 3D shapes 195 11 The diagram shows a shape drawn on dotty paper. The shape is made from cubes. The measurements of the shape are shown in the diagram. Accurately draw the plan view, front elevation, and side elevation for this shape. Use a scale of 1 :2 and use 1 cm squared paper. 12 The diagram shows a shape drawn on dotty paper. The shape is made from cubes. The width of the shape is shown in the diagram. Accurately draw the plan view, front elevation and side elevation for this shape. Use a scale of 1 :3 and use 1 cm squared paper. P S F 16 cm 8 cm 4cm Summary checklist I can draw plan, front and side views of 3D shapes to scale. Top Side 18 cm Front We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
Check your Progress Nimrah thinks of a number, n . Write an expression for the number Nimrah gets each time. a She multiplies the number by 4. b She subtracts 6 from the number. c She multiplies the number by 3 d She divides the number by 6 then adds 5. then subtracts 1. Copy the number line and show the inequality on the number line. Write down the inequality that this number line shows. Work out the value of each expression. Loli lives with 3 friends. They share the electricity bill equally between the four of them. Write a formula to work out the amount they each pay, in: i words ii letters Use your formula in part a ii to work out the amount they each pay when the electricity bill is $96. 6 Simplify these expressions. a n + n + n b 3c + 5c c 9x − x 7 Simplify these expressions by collecting like terms. a 5c + 6c + 2d b 6c + 5k + 5c + k c 3xy + 5yz − 2xy + 3yz 8 Work these out. a 3 + (x × 2) b 6(3 − w) c 4(3x + 2) d 3(7 − 4v) 9 Solve each of these equations and check your answers. a n + 3 = 8 b m − 4 = 12 c 3p = 24 d x = 3 5 10 Shen has set a puzzle. Write an equation for the puzzle. Solve the equation to find the value of the unknown number. 196 8 Shapes and symmetry Check your progress 1 Copy and complete this sentence. A regular pentagon has sides of equal length. It has lines of symmetry and rotational symmetry of order . 2 Write True or False for each statement. a A square is a special rectangle. b A trapezium is a special parallelogram. c A rhombus is a special parallelogram. d A rhombus is a special kite. 3 Work out the circumference of these circles. Use the π button on your calculator. Round your answers correct to 2 decimal places (2 d.p.). a diameter=13cm b radius=2.7m 4 The circumference of a circle is 27 cm. Work out the diameter of the circle. Give your answer correct to the nearest millimetre. 5 Draw the plan view, front elevation and side elevation of each shape. Use a scale of 1 :4 and use 1cm squared paper. a 16cm 12 cm 6 cm b P S F 8cm 24 cm We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
197 Project 3 Quadrilateral tiling This picture shows how you can tile an area with rectangles. They fit together with no gaps, because at each point, there are four 90° angles, which add up to 360°. A tiling pattern like this is called a tessellation. Here is a kite, with a rectangle drawn around it. Draw a kite of your own, and a rectangle to surround it in the same way. Then tessellate your rectangle, keeping it in the same orientation. What do you notice? Can you use this to prove that all kites tessellate? Next, let’s investigate parallelograms. Draw a parallelogram, cut it out, then draw around it to make a tessellation pattern. Can you use what you know about the angles in a parallelogram to prove that they fit together without leaving any gaps? Here is a trapezium. Can you find a way to put two identical trapezia together to make a parallelogram? Can you use this to prove that trapezia will tessellate? Can you find some irregular quadrilaterals that tessellate? You might like to use a dotty grid to explore different options. Look for ways to arrange your quadrilaterals so the four angles that meet at each point add up to 360°. Are there any quadrilaterals that do not tessellate? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
198 Getting started 1 For each of these sequences, work out i the term-to-term rule ii the next two terms. a 4, 7, 10, 13, ..., ... b 28, 26, 24, 22, ..., ... 2 Write the first four terms of the sequence that has a first term of 3 and a term-to-term rule of ‘Multiply by 2’. 3 This pattern is made from squares. Pattern 1 Pattern 2 Pattern 3 a Draw the next pattern in the sequence. b Copy and complete the table to show the number of squares in each pattern. Pattern number 1 2 3 4 5 Number of squares 3 5 c Write the term-to-term rule. d How many squares will there be in Pattern 10? 4 Work out the first four terms in each of these sequences. a nth term=6n b nth term=n−1 Tip Substitute n=1, 2, 3 and 4 into the nth term formulae. 9 Sequences and functions We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE
199 9 Sequences and functions Throughout history, mathematicians have been interested in number patterns and sequences. Look at this pattern of dots. 1 dot 3 dots 6 dots 10 dots 15 dots Continued 5 Copy these function machines and work out the missing inputs and outputs. 2 input output __ __ 16 + 45__ a 7 input output __ 5 __ __ – 7 15 b 4 input output __ __ 30 5 × 10 __ c 8 input output __ 6 9 __ ÷ 4 __ d The number of dots in each pattern forms the sequence 1, 3, 6, 10, 15, … The numbers 1, 3, 6, 10, 15, … are called the triangular numbers, because the dots can be arranged in the shape of a triangle as shown in the pattern above. You can see how the sequence is formed: + 2 + 3 + 4 + 5 1 3 6 10 15 The next two terms in the sequence will be: 15+6=21 and 21+7=28 Can you work out the next three triangular numbers? We are working with Cambridge Assessment International Education towards endorsement of this title. Original material © Cambridge University Press 2021. This material is not final and is subject to further changes prior to publication. SAMPLE