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Past Paper Practice Questions Resource Sheets
Answers
Extra digital questions, in the form of Multiple-choice questions and Flip cards, for all chapters can be found online at Cambridge GO. For more information on how to access and use your digital resource, please see inside the front cover.
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1 Review of number concepts
KNOWLEDGE FOCUS
In this chapter, you will answer questions on:
• identifying and classifying different types of numbers
• finding common factors and common multiples of numbers
• writing numbers as products of their prime factors
• working with integers used in real-life situations
• calculating with powers and roots of numbers
• understanding the meaning of indices
• using the rules of indices
• revising the basic rules for operating with numbers
• performing basic calculations using mental methods and with a calculator
• rounding numbers in different ways to estimate and approximate answers.
EXAM SKILLS FOCUS
In this chapter you will:
• show that you understand the command word ‘give’ and can successfully answer ‘give’ questions
• show that you can use a calculator to work out answers using several operations at once.
In this chapter you will focus on the command word ‘give’. It is important that you recognise how to answer a question that has this command word.
Give produce an answer from a given source or recall/memory.
‘Give’is used when you are asked to produce an answer from a given source or from your own recall or memory. For example, you may be asked to ‘give three different factors of the number 36’, or you may be asked to ‘give a clear reason why Juan is incorrect’.
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You will also focus on how to complete questions which require you to use several calculator operations at once. For example, you may be asked to estimate the value of an expression such as
and then calculate its exact value for comparison with your estimate. The combination of squares, roots and fractions requires care when using your calculator.
1.1 Different types of numbers
1 List the following.
a Four consecutive odd numbers between 10 and 20.
b All even numbers between 121 and 129.
c All prime numbers between 10 and 20.
d Four integers greater than or equal to 10 and less than or equal to 13.
e Three negative integers less than −5.
f Three positive decimals that are less than 0.8.
g Three cube numbers between 5 and 100.
2 State whether the following are odd or even.
a The sum of three odd numbers.
b The product of two even numbers.
c The difference between 18 and 11.
d The difference between two even numbers.
e The product of an even number with an odd number.
f The square of an odd number.
3 Write down three positive decimals that are greater than 100 but less than 100.5.
4 Here is a set of numbers: {−7, −6, −3, −1, 0, 2, 4, 6, 7, 13, 17, 49}.
a Give two square numbers included in the set. [2]
b Give two prime numbers included in the set. [2]
c Give the smallest integer that is larger than the product of the two smallest numbers included in the set. [2]
[Total: 6]
5 Juan says that he is thinking of a number and that the square of his number is 9.
UNDERSTAND THESE TERMS
• Odd number
• Even number
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a Give two possible values of Juan’s number. [2] Juan now says that his number is also less than 2.
b Find Juan’s number and give a clear reason for your answer. [2]
[Total: 4]
• Integer
• Square number
• Cube number
1.2 Multiples and factors
1 List the first five multiples of each of the following.
a 7 b 2 c 11
2 Use your calculator to find and list the first six multiples of each of the following. a 17 b 8 c 12
3 List all multiples of 4 between 61 and 73.
4 Find the lowest common multiple of each of the following.
a 3 and 7 b 5 and 6 c 6 and 8
d 7 and 21 e 24 and 26 f 100 and 125
5 List all the factors of each of the following.
a 3 b 15 c 16
d 25 e 20 f 108
6 Find the highest common factor of each of the following.
a 12 and 15 b 18 and 20 c 20 and 36
d 33 and 66 e 84 and 14 f 36 and 48
7 Estelle and Jotin stand next to each other, facing in the same direction. At the same time, Estelle and Jotin start to spin around on the spot. Estelle takes 5 seconds to complete a single turn and Jotin completes a single turn in 3 seconds.
a Give the first five multiples of 5. [1]
b Give the first five multiples of 3. [1]
Estelle and Jotin stop spinning when they are both facing in the same direction again.
c After how many seconds will Estelle and Jotin stop spinning? [2]
Estelle and Jotin decide to start spinning again, but slower this time. It now takes Estelle 6 seconds to complete a single turn and Jotin completes a single turn in 9 seconds.
d After how many seconds of spinning at new speeds will Estelle and Jotin both be facing in the same direction again? [2] [Total: 6]
RECALL AND CONNECT 1
Explain why both the HCF and LCM of any two even numbers will be even.
UNDERSTAND THESE TERMS
• Multiple
• Lowest common multiple (LCM)
• Factor
• Highest common factor (HCF)
1.3 Prime numbers
1 List all the prime numbers that are:
a between 10 and 20 b less than 15
c even d multiples of 7
REFLECTION
How did you know you had written down all the correct answers? What method or checks did you use to feel confident that you hadn’t missed any? What might help you do this more efficiently next time?
2 Express the following numbers as a product of their prime factors.
36
3 Copy and complete the table. For each pair of numbers:
• express each number as the product of prime factors
• find the highest common factor (HCF)
• find the lowest common multiple (LCM).
48 and 3648 = 36 = 120 and 125120 = 125 = 162 and 18162 = 18 =
4 Khaleel completes an assault course every 60 seconds and immediately starts the course again. Shauri does the same but only takes 45 seconds to complete the course.
a Find the lowest common multiple of 45 and 60. [2]
b Find the time it will take for Khaleel and Shauri to reach the finish line at the same time. Give your answer in minutes and seconds. [2]
[Total: 4]
RECALL AND CONNECT 2
Explain why it is not possible to write a prime number as a product of prime factors.
1.4 Working with directed numbers
1 Arrange each of the following lists of numbers in descending order.
a −6; 3; 8; −9; −11; 12; 15
b −8; 8; −9; −7; 6; −10
c (−6 + 6); (−12 + 18); (2 × −6); (18 ÷ −9); (−3 × −3 × −3); (−3 − −4)
2 The water in a lake starts to freeze if the surface temperature falls below freezing point, which is at 0 °C. On Monday, at 6.00 am, the surface temperature is 5 degrees below freezing point.
a Give the temperature on Monday at 6.00 am as a directed number. [1]
During Monday, the surface temperature of the lake increases by 6 °C and then falls by 18 °C by midnight.
b Give the temperature at midnight as a directed number. [2]
[Total: 3]
RECALL AND CONNECT 3
Rewrite each of these statements using mathematical symbols. You may choose from any of =, ≠, ,, <, ., >.
a 7.2 is bigger than 7.1.
b −2 multiplied by −3 is not equal to −6.
c The number of cars, C, in a car park is greater than or equal to 17.
d 10.01 is greater than 10.005.
e The square of 6 is bigger than 27.
f 4 is greater than 20 divided by 6.
REFLECTION
How did you remember which symbol was appropriate for each question? Did the shape or structure of the symbol help you recall its meaning? How might you remember the difference more easily next time?
1.5 Powers, roots and laws of indices
1 Work out the following.
a
e
2 Work out the following, using a calculator if you need to.
a 113 b 133 c 174 d 404
3 Find a value of x to make each of the following statements true. a x4 = 625
x = 13
4 Find the square root of each of the following numbers. a
5 Find the cube root of 3 × 3
6 Calculate each of the following.
7 Write the following numbers in ascending order.
73; 90; 82; 53; 27
8 Evaluate each of the following.
9 Express the following as products of prime factors. Give your answers in index notation.
10 Simplify each of the following, using the laws of indices.
11 Evaluate the following, giving your answers as fractions.
12 Rewrite each of these using the root symbol.
13 Use a calculator to evaluate the following. Give your answers to 3 decimal places.
14 You are given that a = 36 × 58 × 72.
a Give a reason why you know that a is a square number. [1]
b Find the square root of a. Give your answer as the product of powers of prime numbers. [2]
[Total: 3]
RECALL AND CONNECT 4
Find the HCF and LCM for the two numbers shown. Give your answers as products of powers of prime factors.
a = 34 × 57 × 113
b = 25 × 36 × 54 × 11 × 13
1.6 Order of operations
1 Work out each of the following, showing all steps in your calculations.
a (4 + 3) × 6
(16 − 12) × 3
c 25 ÷ (24 − 19) d 18 − (14 − 3)
e (48 − 16)(48 − 16) ÷ (16 − 12) f (13 − 8) × (12 − 7)
2 Work out each of the following, showing all steps in your calculations. a 13 − 4 × 6 + 12
3 Use your calculator to find each of the following. a 12 − 4
REFLECTION
When you used your calculator, how did you make sure the operations were done in the right order? Did you break the question into steps or check your working to avoid mistakes? What part of the process could you improve?
4 a = 3; b = 2.5; c = −4.23
Use your calculator to find
Give the full answer as shown on your calculator. [2] [Total: 2]
1.7 Rounding and estimating
1 Round each number to the nearest 10.
44
2 Round each number to the nearest 1000.
3 Round the number 4.567 75.
a to 1 significant figure
b to 2 significant figures
c to 5 significant figures
4 Estimate the value of each of the following. Show the rounded values you use in your working.
UNDERSTAND THIS TERM
• Estimate
REFLECTION
How did you use estimation to check that the answer showing on your calculator made sense? How does this help you feel more confident that your answer is close to being correct?
5 Janice is trying to find an approximation to 5. 5 2
a Calculate the value of 5.
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. Give your answer to 5 decimal places. [2]
b Janice finds her approximation by rounding all of the numbers in the question to 1 significant figure and leaves her answer as a fraction in its lowest term. Find the fraction Janice calculated. [2]
c Use your calculator to find the value of Janice’s approximation. Give your answer to 5 decimal places. [1]
d State whether Janice’s approximation is an underestimate or an overestimate. [1]
[Total: 6]
RECALL AND CONNECT 5
Use your calculator to evaluate
a to the nearest whole number
b to the nearest 10
c to 1 decimal place
d to 3 significant figures.
REFLECTION
and round your answer:
How can you use estimation to check the answer shown on your calculator is sensible? How can this make you feel confident that you are at least close to the correct answer?
SELF-ASSESSMENT CHECKLIST
Let's revisit the Knowledge and Exam skills focus for this chapter. Decide how confident you are with each statement.
1identify and classify different types of numbers
2find common factors and common multiples of numbers
3write numbers as products of their prime factors
4work with integers used in real-life situations
List all the even, odd, square, cube and prime numbers from 1 to 30.
Complete Question 6 in section 1.2. Then write your own HCF/LCM question and swap with a partner to solve and mark.
Create sets of flashcards with a number, its prime factorisation and with a factor tree.
A room starts at 10 °C. Find the temperature if the room cools by 19 degrees. Now write your own real-life example involving negative numbers.
5calculate with powers and roots of numbers
Complete Q1 and Q4 in section 1.5. Then explain the difference between a square root and a cube root to someone else.
CONTINUED
Now I can Show it
6understand the meaning of indices Work out 45. Write a sentence to explain what the index means.
7use the rules of indices
8revise the basic rules for operating with numbers
9perform basic calculations using mental methods and with a calculator
10round numbers in different ways to estimate and approximate answers
Simplify 38 × 39, writing your answer as a power of 3.
Evaluate 3 × (4 + 7) + 52. Now write down three number sentences and identify which operation comes first in each.
Mentally work out 25 × 4 and 150 ÷ 6. Then use a calculator to work out (6.5 × 4.3) ÷ 2 and explain each step to a friend.
Round 4.2567 to 2 significant figures. Then estimate the value of √ 51 × 9 by rounding and comparing it with your calculator result.
11answer questions that include the command work ‘give’
12use a calculator to work out answers using several operations at once.
Explain to a partner how to answer a ‘give’ question clearly and completely.
Calculate 3. 2 7 − 2. 3 8 3.6 × 7.1 . Give your answer to 3 significant figures. Then reflect on how you checked the steps were in the right order.
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