digits value place value

�2 1 Number

54

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 2 1.2 Reading, writing and ordering

whole numbers2 1.3 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.

2

�2 1 Number

56

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….

• Derive and use positive integer complements to 100.

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.

2

�2 1 Number

58

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 or divide any number by powers of 10.

• 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 resourcesAll the rides videoBinding video

Resources

Resource sheet 2 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.

2

rounding

�2 1 Number

60

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 3 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.

2

negative number positive number

�2 1 Number

62

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 interactiveRP KC Number knowledge checkRP PS Number problem solving

Follow upUnit 3 section 3.6 Solving equations with negative coeffi cients

Resources

1.7 Negative numbers 2 1.8 Working with negative numbers2 1.9 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.

2