chapter 1 number - pearson global...
TRANSCRIPT
digits value place value
Chapter 1 Number
2
GCSE 2010N a Add, subtract, multiply and divide any numberN b Order rational numbers
FS Process skillsSelect the mathematical information to use
FS PerformanceLevel 1 Understand practical problems in familiar and unfamiliar contexts and situations, some of which are non-routine
Specifi cation
ResourcesNumber lines
ActiveTeach resourcesWord numbers quiz Place value animationNumber line addition interactive
Resources
1.1 Understanding digits and place value 1.2 Reading, writing and ordering
whole numbers1.3 The number line
Concepts and skills
• Understand and use positive numbers… both as positions and translations on a number line.
• Writing numbers in words.
• Writing numbers from words.
• Order integers….
Functional skills
• L1 Understand and use whole numbers … in practical contexts.
Prior key knowledge, skills and concepts
• Students should already know addition number bonds to 9 + 9 and times tables to 10 × 10.
Starter
• Would you rather have 5p or £5? Why? (Both have the same face value but the place value is different.)
Main teaching and learning
• Teach students to equate place value with column value.
• Write the column headings shown on p.2 of the Student Book on the board. Include Ten Thousands, Hundred Thousands and Millions if you wish.
• Write 3429 on the board. Point to one of the fi gures in 3429 and ask What is the value of this fi gure?
• Ask students to place each fi gure under the appropriate column heading.
• Write ‘fi ve hundred and seven’ on the board (in words) and ask students to place each fi gure under the appropriate column heading.
• Write the numbers 327, 40, 635, 9003, 500 on the board. Ask students to place the numbers under the column headings and using this, order the numbers from smallest to largest.
• Revise the number line for positive integers
• Ask students for three numbers between 20 and 40. Write the numbers on the board and then, together, construct an appropriate number line (appropriate length and scale divisions) to show these three numbers. Use the numbers to check students can pinpoint their position on the line and ask students to explain how the number line can help them order these three numbers.
• Ask students for 6 pairs of numbers less than 20, write these on the board, numbering the pairs 1 to 6. For even pairs ask students to mentally add the numbers, for odd pairs they should mentally subtract the smaller number from the larger. Ask students to explain their mental strategies for doing this and compare different strategies.
Common misconceptions
• Some students will write 428 for four thousand and twenty-eight. Remind students that if ‘hundreds’ (for example) is not mentioned it still has a place value and needs a place holder of zero.
Enrichment
• Include ten thousands, hundred thousands and millions.
Plenary
• Discuss moves on a number line (see Exercise 1C) in preparation for the next lesson.
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Section 1.1–1.3 Digits, place value and the number line
3
1 Write down the place value of each digit in the number 2376.
.....................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................
2 Write in words, the numbers:
a 326 .......................................................................................................................................................................................................................
b 4152 .....................................................................................................................................................................................................................
c 15 370 ..................................................................................................................................................................................................................
d 2006 .....................................................................................................................................................................................................................
3 Write as numbers:
a eight hundred and thirty seven .........................................
b nine thousand three hundred and twenty-fi ve .........................................
c twenty-two thousand and fi fty-three .........................................
d three thousand six hundred and fi ve. .........................................
4 Put these numbers in order of size. Start with the smallest.
9781 15 361 6542 6452 6524
.....................................................................................................................................................................................................................................
5 Put these numbers in order of size. Start with the biggest.
2371 8711 2731 2317 20 317
.....................................................................................................................................................................................................................................
6 Write down the biggest and the smallest number you can make using each of the digits 8, 1, 9, 3, 4 once only.
.....................................................................................................................................................................................................................................
Guided practice worksheet
Hint Start with the biggest place value number on the left.
Hint There are no hundreds so put 0 in the hundreds column.
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Chapter 1 Number
4
GCSE 2010N a (part) Add, subtract… any number
FS Process skillsRecognise that a situation has aspects that can be represented using mathematicsUse appropriate mathematical procedures
FS PerformanceLevel 1 Understand practical problems in familiar and unfamiliar contexts and situations, some of which are non-routine
Specifi cation
Linkshttp://nlvm.usu.edu/en/nav/frames-asid_188_g_4_t_l/html?open=instructions& from=category_g_4_t_l.html
Resources
1.4 Adding and subtracting
Concepts and skills
• Add, subtract… whole numbers….
Functional skills
• L1 Add, subtract… whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
• Number bonds to 9 + 9
• Place value
• Number differences to 19 – 1
Starter
• Practise instant recall of number bonds.
• Extend to 70 + 50 etc.
Main teaching and learning
• Teach formal column addition that has no carrying.
• Move on to addition that has carrying.
• Teach subtraction with decomposition from tens column.
• Teach fully general column subtraction.
Common misconceptions
• Giving the answer 254 when asked to subtract 89 from 235.
• Reinforce decomposition of 1 from previous column into 10 for the column being considered.
Enrichment
• Write four consecutive numbers into the boxes at the bottom.
• Get the number for the box above by adding the two numbers in the boxes below.
• Continue doing this until you get the top number.
• Can you see any link between the top number and the numbers at the bottom? (The top number is twice the sum of the bottom numbers.)
• You might need to do several before you see the pattern.
• Does your rule work if the bottom numbers are all the same? (Yes.)
• Does it work for consecutive odd numbers? (Yes.)
• Does it work for consecutive even numbers? (Yes.)
Plenary
• Use the darts match context from Exercise 1E question 6. Change the scores to James 423, Sunita 402 and Nadine 364. Repeat the questions using these scores. Discuss methods.
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Section 1.4 Adding and subtracting
5
1 Work out:
a 23 b 274 c 97 + 41 + 142 + 24
........ ........... ........
d 107 e 37 + 15 + 48 f 129 + 207 + 96 89 + 22
........ ............. .............
2 Work out:
a 57 b 307 c 121 – 32 – 95 – 89
........ ........ ........
d 403 e 5000 – 76 – 187
........ ..........
3 Sharon drives a van.
On Monday she drove from York to Leeds, then from Leeds to Doncaster and fi nally back to York.
How far did she drive?
....................................................................................................................................
4 When Peter goes from A to B he either goes through P or Q.
Which is the shorter route and by how much?
....................................................................................................................................
5 There were 23 people on a bus.
At the fi rst stop, 19 people got on and 6 people got off. At the second stop 12 people got off and just 6 people got on.
How many people are now on the bus?
.....................................................................................................................................................................................................................................
Hint Keep digits in correct column.
Hint Check distances are in the same units.
Doncaster
Leeds
York
57 km
44 km
41 km
29 km
55 km
37 kmP B
Q
A
54 km
Guided practice worksheet
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Chapter 1 Number
6
GCSE 2010N a (part) … multiply and divide any numberN q Understand and use number operations and the relationships between them, including inverse operations and hierarchy of operations
FS Process skillsRecognise that a situation has aspects that can be represented using mathematicsUse appropriate mathematical procedures
FS PerformanceLevel 1 Understand practical problems in familiar and unfamiliar contexts and stituations, some of which are non-routine
Specifi cation 1.5 Multiplying and dividing
Concepts and skills
• … multiply and divide whole numbers….
• Multiply and divide numbers using the commutative, associative, and distributive laws and factorisation where possible, or place value adjustments.
Functional skills
• L1 … multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
• Tables to 10 × 10
• Number bonds to 9 + 9
Starter
• Practise instant recall of products.
Main teaching and learning
• Teach students to multiply a 3-digit number by a 1-digit number.
• Ask students to explain two or more different methods for multiplication, for example 157 × 3.
• Extend to 3-digit by 2-digit.
• Teach general multiplication using formal column procedures.
• Teach students to divide a 3-digit number by a 1-digit number.
• Ask students to explain two or more different methods for division, for example 352 ÷ 8.
• Teach long division by a 2-digit number.
Common misconceptions
• Not appreciating column importance, which is seen as ×20 becoming ×2, etc. Stress that although you are only multiplying by a single digit, that digit has place value.
• Mistakes are commonly made with: 7 × 9, 7 × 8, 8 × 9, 6 × 9. Write 7 × 9 as 7 × 10. As this is 10 sevens which is 7 too many, 7 × 9 = 70 – 7 = 63. This is a useful check if you are not sure and works for any product with 9. 7 × 8 = 7 × 2 × 2 × 2 =14 × 2 × 2 = 28 × 2 = 56.
Plenary
• Write questions based on Exercise 1F questions 7 and 8, using your students’ names and the local football club.
CD ResourcesResource sheet 1.5
Linkshttp://www.bbc.co.uk/schools/gcsebitesize/maths/number/multiplicationdivisionact.shtml
ActiveTeach resourcesThe audience videoQualifying heats 1 videoAll rides videoBinding video
Resources
Resource sheet 1.5Opposite cornersThe diagram shows a 100 square.A rectangle has been shaded on the 100 square.
1 2 3 4 5 6 7 8 9 10
11 12 13 14 15 16 17 18 19 20
21 22 23 24 25 26 27 28 29 30
31 32 33 34 35 36 37 38 39 40
41 42 43 44 45 45 47 48 49 50
51 52 53 54 55 56 57 58 59 60
61 62 63 64 65 66 67 68 69 70
71 72 73 74 75 76 77 78 79 80
81 82 83 84 85 86 87 88 89 90
91 92 93 94 95 96 97 98 99 100
The numbers in the opposite corners are 24 and 36 and 26 and 34.
Rules. 1. Multiply the numbers in the opposite corners
24 × 36 = 864 26 × 34 = 884
2. Find the difference between these products 884 – 864 = 20
Explore the result of finding the difference between the products of the opposite corners for 2 × 3 rectangles and rectangles (or squares) of different sizes.
You can also explore what happens if the number grid is an 81 square, 64 square etc.
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Section 1.5 Multiplying and dividing
7
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rounding
Chapter 1 Number
8
GCSE 2010N u (part) Approximate to specifi ed or appropriate degrees of accuracy including a given power of ten….
FS Process skillsSelect the mathematical information to use
FS PerformanceLevel 1 Identify and obtain necessary information to tackle the problem
Specifi cation
ResourcesJar of beads
Linkshttp://www.bbc.co.uk/schools/gcsebitesize/maths/number/roundestimateact.shtml
ActiveTeach resourcesReading scales quizRounding interactive
Follow upChapter 5 Decimals and Rounding
Resources
1.6 Rounding
Concepts and skills
• Round numbers to a given power of 10.
Functional skills
• L1 Understand and use whole numbers … in practical contexts.
Prior key knowledge, skills and concepts
• Place value
Starter
• Ask for estimates of the number of beads in a jar (or similar). The important thing is to get an estimate.
• Discuss whether or not exact suggestions make sense.
• Discuss whether nearest 10 or nearest 100 is most sensible.
Main teaching and learning
• Give students an exact value, and ask them to tell you the nearest 10. (Students’ own answers.)
• Extend to nearest 100, 1000, million.
• Use some or all of the following questions and ask students to say whether numbers should be left as they are, or rounded up, or down, to the nearest 10, 100, 1000 or other degree of accuracy:(a) 43 people lift a train!(b) 6845 people attended a football match.(c) 24 572 people live in Hightown.(d) 285 people turned up to a public meeting.(e) 2348 people attended a pop concert.(f) 405 people were watching the game.(g) The distance is 2014 km.(h) He walked for 10 206 m.(i) The weight of the van was 3803 kg.
Common misconceptions
• Students often truncate numbers instead of rounding.
Enrichment
• Ask what is the smallest number which, when rounded to the nearest 100, gives 700? (650).
• Repeat using other numbers if necessary.
Plenary
• Use Exercise 1B question 5 and ask students to round the fi gures to the nearest 1000, 10 000, 100 000.
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Section 1.6 Rounding
9
When you give your age, you give it to the nearest year because this is sensible.
The editor of a magazine might ask for articles of 300 words.
Anything more accurate would not be sensible.
1 Round these numbers to the nearest 10.
a 58 ...................... b 22 ...................... c 79 ...................... d 35 ......................
e 7 ...................... f 234 ...................... g 359 ...................... h 762 ......................
i 293 ...................... j 307 ...................... k 1003 ...................... l 3204 ......................
m 2995 ...................... n 555 ......................
2 Round these numbers to the nearest 100.
a 204 ...................... b 390 ...................... c 83 ...................... d 5430 ......................
e 445 ...................... f 649 ...................... g 12 381 ...................... h 53 807 ......................
i 100 093 ...................... j 230 988 ...................... k 45 ......................
3 Round these numbers to the nearest 1000.
a 5500 ...................... b 842 ...................... c 3200 ......................
d 6455 ...................... e 9786 ...................... f 24 488 ......................
4 Write the number 4117 to the nearest hundred.
......................
Note If the number ends in 5 it is usual to round up.
Hint Make sure number keeps same order of magnitude.
G
Guided practice worksheet
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negative number positive number
Chapter 1 Number
10
GCSE 2010N a Add, subtract, multiply and divide any numberN b Order rational numbers
FS Process skillsRecognise that a situation has aspects that can be represented using mathematicsFind results and solutions
FS PerformanceLevel 2 Apply a range of mathematics to fi nd solutions
Specifi cation
ResourcesThermometers with negative values
LinksSubstitution in expressions and formulaehttp://www.bbc.co.uk/schools/gcsebitesize/maths/number/negativenumbersact.shtml
ActiveTeach resourcesScales 2 interactiveDirected numbers (addition and subtraction) quizNumber lines interactive
Follow up21.6 Solving equations with negative coeffi cients
Resources
1.7 Negative numbers 1.8 Working with negative numbers1.9 Calculating with negative numbers
Concepts and skills
• Understand and use positive and negative integers both as positions and as translations on a number line.
• Add, subtract, multiply and divide … negative numbers….
Functional skills
• L2 Understand and use positive and negative numbers … in practical contexts.
Prior key knowledge, skills and concepts
• The number line.
• All number bonds.
Starter
• Establish the need for negative numbers through temperature, height above sea level, etc.
Main teaching and learning
• Extend the number line to include negative integers.
• Teach students how to add with negative and positive numbers.
• It may help some students to work out the difference between two numbers starting with positive numbers. This naturally leads on to the difference between a positive and a negative number as adding the sizes of the two numbers together. Another way is to consider subtraction as the inverse operation to addition; otherwise the answer always seems to be positive.
• Teach students how to fi nd the difference when negative numbers are involved (this is best done within the context of temperature using a thermometer as the number line).
• Teach students the rules for multiplying and dividing negative numbers.
Common misconceptions
• The difference between a negative number and subtracting a positive number is sometimes misunderstood.
• Emphasise the distinction between a minus sign that belongs to a number and the minus sign that signals ‘do a subtraction’.
Plenary
• Ask students to explain the rules for multiplying and dividing negative numbers.
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Section 1.7–1.9 Negative numbers
11
1 Write these numbers in order of size. Start with the smallest.
3 −4 −2 5 −7 −12
.....................................................................................................................................................................................................................................
2 The table gives information about the temperature at midnight in fi ve cities.
a Write down the lowest temperature. ...........................................................
b Work out the difference in temperature between Cardiff and Plymouth.
..........................................................................................................................................
c Work out the temperature which is half way between −1°C and 5°C.
..........................................................................................................................................
3 At midnight, the temperature was −5°C.
By 9 am the next morning, the temperature had increased by 3°C.
a Work out the temperature at 9 am the next morning.
.................................................................................................................................................................................................................................
At midday, the temperature was 7°C.
b Work out the difference between the temperature at midday and the temperature at midnight.
.................................................................................................................................................................................................................................
4 Work out:
a −3 + 2 ...................... b 3 + −1 ...................... c −7 + −3 ......................
d −11 + (+4) ...................... e (+3) + (−5) ......................
5 Work out:
a (−4) − (+2) ...................... b (+5) − (−2) ...................... c (−3) − (−5) ......................
d 8 − −3 ...................... e −6 − 2 ......................
6 Work out:
a (−2) × (6) ...................... b (+3) × (+5) ...................... c (−4) × (−3) ......................
d (+3) × (−7) ......................
7 Work out:
a −15 divided by 3 b 16 ÷ (−4) c −80 ÷ −5 d 40 divided by −8
...................... ...................... ...................... ......................
8 Simplify:
a –123
...................... b 147–
...................... c ––248
......................
d ++243
...................... e –486
...................... f 357–
......................
City Temperature
Cardiff −2°C
Edinburgh −4°C
Leeds 2°C
London −1°C
Plymouth 5°C
Guided practice worksheet
F
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factor multiple prime common factor common multiple prime factor prime number
Chapter 1 Number
12
GCSE 2010N c (part) Use the concepts and vocabulary of factor (divisor), multiple, common factor,… prime number and prime factor decomposition
FS Process skillsSelect the mathematical informationto useExamine patterns and relationships
FS PerformanceLevel 1 Select mathematics in an organised way to fi nd solutions
Specifi cation
CD ResourcesResource sheet 1.10
Linkshttp://nrich.maths.org/public/search.php?search=factorshttp://www.bbc.co.uk/schools/gcsebitesize/maths/number/primefactorsact.shtml
ActiveTeach resourcesWord problems quiz 2HCF and LCM interactiveLadder method interactive
Follow up8.2 Equivalent fractions
Resources
1.10 Factors, multiples and prime numbers
Concepts and skills
• Recognise even and odd numbers.
• Identify factors, multiples and prime numbers.
• Find the prime factor decomposition of positive integers.
• Find common factors and common multiples of two numbers.
Functional skills
• L1 … multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
• Tables up to 10 × 10
Starter
• Counting aloud in 2s starting from, say, 30. Then from, say, 250. Would it be any more diffi cult from, say, 2310? What is the pattern?
• Continue similarly from an odd number start. What is the pattern?
• Although this is the introduction to odd and even numbers, it can also be used for multiples.
Main teaching and learning
• Even numbers always end in 2, 4, 6, 8 or 0.
• Odd numbers always end in 1, 3, 5, 7 or 9.
• Multiples are obtained by multiplying the number by a positive whole number.
• The factors of a number are whole numbers that divide exactly into the number, including 1 and the number itself.
• Prime numbers only have 1 and the number itself as factors.
• Numbers can be written as products of prime numbers.
• Explain the tree method of decomposition.
Common misconceptions
• Students sometimes confuse factors and multiples. (Tell them that multiples come from multiplying.)
• Thinking that 1 is a prime number.
Enrichment
• Most numbers have an even number of factors. There are some numbers that have an odd number of factors. Find some of these and say what is special about them.
• Factors – Packing and tiling problems.
• Multiples – How many biros costing 19p can I buy with £5? (Multiples of 19 are not easy, but using multiples of 20 you can buy 25 biros and get 25p change. Enough to buy another biro. Answer: 21 biros.)
Plenary
• Ask students to give examples of: prime numbers, factors of a given number (say 72 (1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36), multiples of, say, 4 (e.g. 8, 12, 16, 20), and to write a number such as 144 as a product of its prime factors. (144 = 2 × 2 × 2 × 2 × 3 × 3 or 144 = 24 × 32)
Resource sheet 1.10The factor gameThere are various ways of playing this game but the class elimination version illustrates the principles. As there is an element of luck with when the question reaches you there is no real shame in dropping out.
1. Choose a number. Although any number will do, one with several factors is best.
2. The first student names a factor of that number.
3. The next student names a different factor of the number.
4. Once all the factors have been exhausted the only acceptable answer from the next student is ‘no further factors’ or words to that effect.
5. Any student unable to offer a correct answer is eliminated.
6. Continue until a winner is found. As you get towards the end students may be difficult to eliminate, so have some numbers with tricky factors like 13, 17 up your sleeve.
An alternative version is to play as a team game, scoring points for correct answers. You may prefer to use nominated team members to answer rather than throwing it open. This avoids the game being dominated by able individuals.
Another possibility if you wish to emphasise that factors occur in pairs is to require answers as factor pairs. In this instance the class will need to be advised about square numbers as having a repeated factor.
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Section 1.10 Factors, multiples and prime numbers
13
1 Using only the numbers 1, 4, 7, 2 write down as many even numbers as possible.
.................................................................................................................................................................................
.................................................................................................................................................................................
2 A postwoman is going to walk up the side of a street with the even numbered houses and come back down the side with odd numbers. She has letters for houses 34, 17, 3, 20, 14, 9, 31, 22, 20, 3, 28, 6.
Put the house numbers in the order in which she will deliver the letters.
.....................................................................................................................................................................................................................................
3 List all the factors of:
a 32 .................................................................................................... b 54 ....................................................................................................
c 27 .................................................................................................... d 36 ....................................................................................................
4 Write down three multiples of:
a 12 ........................................................... b 6 ...........................................................
c 20 ........................................................... d 29 ...........................................................
5 From the numbers in the cloud write down:
a two numbers that are factors of 72 ................................................................................................
b a number that is a multiple of 15 ......................................................................................................
c all the prime numbers. ..........................................................................................................................
6 Why is it not possible to make an odd number from the digits 4, 2, 0, 8?
.....................................................................................................................................................................................................................................
7 Write down a prime number that is greater than 20.
.....................................................................................................................................................................................................................................
Hint There are 12 possible numbers and they must end in 2 or 4.
Hint Try 1×, then 2×, then 3×, then 5× and 7×.
105316 8
11144
13
6559
F
Guided practice worksheet
M01_MSAF_TG_GCSE_0877_C01.indd 13 17/05/2010 13:30
lowest common multiple (LCM) highest common factor (HCF)
Chapter 1 Number
14
GCSE 2010N c (part) Use the concepts and vocabulary of … Highest Common Factor (HCF), Least Common Multiple (LCM)….
FS Process skillsSelect the mathematical information to useExamine patterns and relationships
FS PerformanceLevel 1 Select mathematics in an organised way to fi nd solutions
Specifi cation
Linkshttp://www.bbc.co.uk/education/mathsfi le/shockwave/games/gridgame.html
ActiveTeach resourcesMultiples and factors quiz
Follow up8.7 Adding and subtracting fractions
Resources
1.11 Finding lowest common multiple (LCM) and highest common factor (HCF)
Concepts and skills
• Find common factors and common multiples of two numbers.
• Find the LCM and HCF of two numbers.
Functional skills
• L1 … multiply and divide whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
• Mental division by 2, 3, 5 and 7.
Starter
• A stamp measures 6 cm by 3 cm. There are to be 60 stamps to a sheet, which has to be rectangular. Work out possible sizes for the sheet. (3 cm by 360 cm, 6 cm by 180 cm, 12 cm by 90 cm, 9 cm by 120 cm, 18 cm by 60 cm, 24 cm by 45 cm, 15 cm by 72 cm, 30 cm by 36 cm.)
Main teaching and learning
• Explain how to fi nd the LCM by listing families of multiples. The product of the numbers is always a common multiple but not necessarily the least. (List the fi rst few multiples of the numbers. Pick out the smallest that is in all lists. This is the LCM.)
• Ask students to write out lists of up to twenty multiples of each of the following numbers: (a) 2 and 3 (b) 5 and 7 (c) 6 and 4 (d) 12 and 8 (e) 9 and 6.
• For each pair of numbers, circle the smallest number that appears in both lists of multiples. Is the circled number always the product of the two original numbers? For (d) and (e) ask What are the factors? What are the common factors?
• Explain how to fi nd the HCF. (List all the factors of the numbers. Pick out the smallest that is in all lists. This is the HCF.)
• Some students might benefi t from knowing that the HCF of any two numbers must be a factor of the difference between the two numbers.
Common misconceptions
• Confusing HCF with LCM.
Enrichment
• Question 2 of the guided practice worksheet may be used here.
Plenary
• Discuss question 5 on the guided practice worksheet.
• Discuss question 32 of the Review exercise. Explore the possibilities if the cartons contain 144 bottles.
M01_MSAF_TG_GCSE_0877_C01.indd 14 17/05/2010 13:30
Section 1.11 Finding lowest common multiple (LCM) and highest common factor (HCF)
15
1 a Workout all the factors common to 18 and 20.
.................................................................................................................................................................................................................................
b Which is the highest of these common factors?
.................................................................................................................................................................................................................................
2 a Write down all the common factors of:
i 18 and 30 .......................................................................................................................................................................................................
ii 42 and 60 .......................................................................................................................................................................................................
iii 84 and 108 .....................................................................................................................................................................................................
b Examine the link between these factors and the number you get by subtracting the two numbers.
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
3 Alan goes up some stairs two at a time. Peter goes up the same stairs three at a time.
They start together with their right foot and go up at the same speed.
a How often do they tread on the same stair?
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
b How often do they tread on the same stair each with the right foot?
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
4 On a long main road lamp posts are placed at 56-metre intervals. On the same road the drains are spaced at 84-metre intervals. At a point on the road there is a lamp post and a drain together.
How far down the road is it to the next point where drain and lamp post are together?
.....................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................
.....................................................................................................................................................................................................................................
5 Three lemons are showing on a fruit machine.
The lemon symbol in the fi rst position appears every 4th turn. The lemon symbol in the second position appears every 5th turn. The lemon symbol in the third position appears every 6th turn.
a Explain why all the lemon symbols will show on the 120th turn.
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
b Do all three lemons appear together before this? If so, when?
.................................................................................................................................................................................................................................
.................................................................................................................................................................................................................................
Guided practice worksheet
C
F
M01_MSAF_TG_GCSE_0877_C01.indd 15 17/05/2010 13:30
square number cube number
Chapter 1 Number
16
GCSE 2010N d Use the terms square, positive and negative square root, cube and cube root
FS Process skillsUse appropriate mathematical procedures
FS PerformanceLevel 1 Use appropriate checking procedures at each stage
Specifi cation
CD ResourcesResource sheet 1.12
ActiveTeach resourcesRP KC Number knowledge checkRP PS Number Travel problem solving
Follow up10.4 Using a calculator – working out powers and roots
Resources
1.12 Finding square numbers and cube numbers
Concepts and skills
• Recall integer squares up to 15 × 15 and the corresponding square roots.
• Recall the cubes of 2, 3, 4, 5 and 10.
• Find squares and cubes.
Functional skills
• L1 … multiply … whole numbers using a range of strategies.
Prior key knowledge, skills and concepts
• Long multiplication
Starter
• Use the Alice in Mathsland resource sheet as a class activity.
Main teaching
• Teach students about square numbers.
• Show the number 9 represented by nine dots in a square. Ask students to say what other numbers, represented by dots, would form a square. Without drawing squares, how can we work out the fi rst 10 square numbers? (2 × 2; 3 × 3 etc).
• Teach students about squaring.
• Teach students about cube numbers.
• If we calculate square numbers by multiplying a number by itself, how would we calculate a cube number? (Write the number down three times then multiply.) Ask students to calculate the fi rst 5 cube numbers.
• Teach students about cubing.
Common misconceptions
• Students sometimes double numbers instead of squaring them.
• Students sometimes multiply by 3 instead of cubing.
Enrichment
• 1 000 000 = 10002 = 1003 1 million is a square number and a cube number. Can you fi nd another number which is both a cube number and a square number? There are nine of them less than 1 million. (1, 64, 729, 4096, 15 625, 46 656, 117 649, 262 144, 531 441)
Plenary
• Devise a quiz based on the chapter review. For example, ask students to defi ne a factor.
Resource sheet 1.12Alice in MathslandThis is an investigative activity that looks at the link between square numbers and geometry. The way you use it could be as an investigation or as a class teaching exercise.
Alice was making patterns with tiles.The patterns she was making were stair patterns.
4 tiles 9 tiles 16 tiles
She knew that 4, 9 and 16 were all square numbers and wondered if all the patterns would use a number of tiles that was a square number.
Launch investigationPlenaryShe made some more patterns and then had a brainwave.
If she cut off the right-hand side and turned it upside down it would exactly fit the left-hand side of her pattern and make a square.This would always be true. Her result had been proved.
She then experimented some more and added tiles in a different way to make squares.
1 + 3 (1 + 3) + 5 (1 + 3 + 5) + 7
She realised that she had found another interesting result.The sum of consecutive numbers starting with 1 always made a square number.
Z02_MSAF_TG_GCSE_0877_RES.indd 4 27/04/2010 14:38
M01_MSAF_TG_GCSE_0877_C01.indd 16 17/05/2010 13:30
Section 1.12 Finding square numbers and cube numbers
17
Sec
tio
n 1
.12
Fin
din
g s
qu
are
nu
mb
ers
and
cu
be
nu
mb
ers
17A
1
Find
the
fi rst
15
squa
re n
umbe
rs. T
he fi
rst t
wo
have
bee
n do
ne fo
r you
.
1
squa
red
= 12
= 1
× 1
= 1
2
squa
red
= 22
= 2
× 2
= 4
3
squa
red
= 32
= ...
......
......
......
......
......
......
. =
......
......
......
....
4
squa
red
= 42
= ...
......
......
......
......
......
......
. =
......
......
......
....
5
squa
red
= 52
= ...
......
......
......
......
......
......
. =
......
......
......
....
6
squa
red
= 62
= ...
......
......
......
......
......
......
. =
......
......
......
....
7
squa
red
= 72
= ...
......
......
......
......
......
......
. =
......
......
......
....
8
squa
red
= 82
= ...
......
......
......
......
......
......
. =
......
......
......
....
9
squa
red
= 92
= ...
......
......
......
......
......
......
. =
......
......
......
....
10
squ
ared
=
102
= ...
......
......
......
......
......
......
. =
......
......
......
....
11
squ
ared
= .
......
......
......
... =
...
......
......
......
......
......
......
. =
......
......
......
....
12
squ
ared
= .
......
......
......
... =
...
......
......
......
......
......
......
. =
......
......
......
....
13
squ
ared
= .
......
......
......
... =
...
......
......
......
......
......
......
. =
......
......
......
....
14
squ
ared
= .
......
......
......
... =
...
......
......
......
......
......
......
. =
......
......
......
....
15
squ
ared
= .
......
......
......
... =
...
......
......
......
......
......
......
. =
......
......
......
....
2
Writ
e ea
ch o
f the
num
bers
bel
ow a
s th
e su
m o
f tw
o sq
uare
num
bers
ta
ken
from
que
stio
n 1.
a 5
......
......
......
......
......
......
......
......
.....
b 8
......
......
......
......
......
......
......
......
.....
c
13 ..
......
......
......
......
......
......
......
......
...
d 58
.....
......
......
......
......
......
......
......
......
e
20 ..
......
......
......
......
......
......
......
......
...
f 29
.....
......
......
......
......
......
......
......
......
g 61
.....
......
......
......
......
......
......
......
......
h
80 ..
......
......
......
......
......
......
......
......
...
i 90
.....
......
......
......
......
......
......
......
......
3
Find
the
fi rst
6 c
ube
num
bers
. Tw
o ha
ve b
een
done
for y
ou.
1
cube
d =
13
= 1
× 1
× 1
= 1
2
cube
d =
23
= 2
× 2
× 2
= 8
3
cube
d =
33
= ...
......
......
......
......
......
......
. =
......
......
......
......
......
......
....
4
cube
d =
......
......
......
......
......
......
....
= ...
......
......
......
......
......
......
. =
......
......
......
......
......
......
....
5
cube
d =
......
......
......
......
......
......
....
= ...
......
......
......
......
......
......
. =
......
......
......
......
......
......
....
6
cube
d =
......
......
......
......
......
......
....
= ...
......
......
......
......
......
......
. =
......
......
......
......
......
......
....
Gu
ided
pra
ctic
e w
ork
shee
t
Hin
t T
he s
um o
f two
num
bers
can
be
the
two
num
bers
add
ed t
oget
her,
or
one
subt
ract
ed fr
om t
he o
ther
Hin
t W
ork
out
each
mul
tipl
icat
ion
sepa
rate
ly. 2
× 2
× 2
= 4
× 2
= 8
G F
M01
A_M
SAF_
TG_G
CSE
_087
7_C
DC
01.in
dd
417
/05/
2010
13
:29
Sec
tio
n 1
.12
Fin
din
g s
qu
are
nu
mb
ers
and
cu
be
nu
mb
ers
17B
4
Wha
t is
the
bigg
est s
quar
e nu
mbe
r sm
alle
r tha
n 25
0?
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.
5
Giv
en th
at 9
3 = 7
29, h
ow m
any
cube
num
bers
are
ther
e le
ss th
an 7
20?
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
......
.
Gu
ided
pra
ctic
e w
ork
shee
t
F
M01
A_M
SAF_
TG_G
CSE
_087
7_C
DC
01.in
dd
517
/05/
2010
13
:29
M01_MSAF_TG_GCSE_0877_C01.indd 17 17/05/2010 13:30