Section 4. Consolidation lessons

Section 4 Consolidation lessons

Contents Mathematics consolidation lessons 5 Why have the lessons been produced? 5 How should I use the lessons? 7 Previous tests on CD-ROM 7 Preparing pupils for the Year 7 progress tests 7 Consolidation lessons 9 Lesson 1 Place, value, addition and subtraction 9 Lesson 2 Multiplication 15 Lesson 3 Using fractions 27 Lesson 4 Fractions and decimals 33 Lesson 5 Probability 39 Lesson 6 Calculators 49 Lesson 7 Word problems 55 Lesson 8 Interpreting data 63 Lesson 9 Shapes and angles 71 Lesson 10 Coordinates and reflections 81 Lesson 11 Sequences 89 Lesson 12 Perimeter and area 95 3

Targeting level 4 in Year 7 Mathematics consolidation lessons This pack contains: brief guidance; 12 mathematics consolidation lessons targeted at Year 7 pupils working towards level 4. The pack supplements Springboard 7: a mathematics catch-up programme for pupils entering Year 7, issued 06/01, reference DfEE 0049/2001. Springboard 7 provides teaching points and materials to support pupils who entered Year 7 below level 4 in mathematics but who will, with additional support, attain level 4 by the end of the year. Why have the lessons been produced? Following the positive response from schools to Springboard 7, we have produced 12 consolidation lessons that focus on the topics that pupils find difficult and that are key to attaining level 4 in mathematics. These lessons also highlight the applications and understanding of mathematical ideas. All the lessons link with and some make use of Springboard 7 materials. The lessons also refer to the Framework for teaching mathematics: Years 7, 8 and 9. The format and style of lessons is similar to that in the Year 9 booster kit: mathematics, DfES 0015/2002, with which many teachers will be familiar. The lessons are available on the Standards website: www.standards.dfes.gov.uk/keystage3/strands/mathematics/ Copies of the OHTs and resource sheets are supplied on a CD. The objectives for the lessons are drawn from the yearly teaching programmes for mathematics and are listed below. 1 Place value, addition and subtraction Read and write whole numbers in figures and words, and know what each digit represents (Y5) Use known number facts and place value to consolidate mental addition/subtraction (Y6) Use standard column procedures to add and subtract whole numbers (Y7) 2 Multiplication Use informal pencil and paper methods to support, record or explain multiplications. Extend written methods (Y5, Y6) Multiply and divide integers and decimals mentally by 10, 100 and 1000, and explain the effect (Y7) 3 Using fractions Order fractions and position them on a number line (Y6) Calculate simple fractions of quantities and measurements (Y7) Use names and abbreviations of units of measurement (Y7) 5

4 Fractions and decimals Reduce a fraction to its simplest form by cancelling common factors in the numerator and denominator (Y6) Recognise the equivalence between the decimal and fraction forms (Y6) 5 Probability Use vocabulary and ideas of probability, drawing on experience (Y7) 6 Calculators Round positive whole numbers to the nearest 10, 100 or 1000 (Y7) Enter numbers (in a calculator) and interpret the display in different contexts (Y7) Carry out calculations with more than one step (Y7) Check a result by considering whether it is of the right order of magnitude (Y7) 7 Word problems Consolidate the rapid recall of multiplication facts up to 10 10, and quickly derive associated division facts (Y7) Solve word problems (Y7) Solve simple problems about ratio and proportion (Y7) 8 Interpreting data Extract and interpret data in tables, graphs, charts and diagrams (Y6) Interpret diagrams and graphs (including pie charts) and draw simple conclusions (Y7) 9 Shapes and angles Use correctly the vocabulary for lines, angles and shapes (Y7) Know the sum of angles at a point, on a straight line and in a triangle (Y7) 10 Coordinates and reflections Find coordinates of points determined by geometric information (Y7) Understand and use the language and notation associated with reflections (Y7) Recognise transformation and symmetry of a 2-D shape: reflection in given mirror lines and line symmetry (Y7) 11 Sequences Recognise and extend number sequences (Y6) Generate sequences from practical contexts (Y7) Recognise squares of numbers to at least 12 12 (Y7) 12 Perimeter and area Know and use the formula for the area of a rectangle; calculate the perimeter and area of shapes made from rectangles (Y7) Use names and abbreviations of units of measurement (Y7) 6

How should I use the lessons? These are consolidation lessons pupils will have met the topics before. You could use the lessons during the year as a key lesson to finish a topic or you may prefer to use them in the summer term in the run up to the progress test. If you use the lessons as a basis for new teaching and learning, a single lesson, with suitable additional examples, discussion and practice, will need to be spread over two or three sessions. Quick revision is no substitute for sound teaching throughout the year. Remember that none of these lessons can match your pupils needs exactly: they will need at least some modification. It is also important to both maintain and develop pupils mental mathematics skills. These are addressed in the starters in many of the lessons but mental skills should be used in all elements of the lessons. Many of the lessons incorporate Key Stage 3 test questions. It is a good idea to use such questions in your lessons during the year so that pupils become familiar with the style of questions and the standard required to achieve level 4. Previous tests on CD-ROM Previous years end of Key Stage tests are available on the Testbase CD-ROM produced by QCA/Doublestruck. This can be obtained from Testbase, PO Box 208, Newcastle on Tyne, NE3 1FX; tel 0870 9000 402; fax 0870 9000 403; www.testbase.co.uk. The CD-ROM is supplied free of charge. Individual Key Stage subjects are accessed using registration codes, at a cost of 25 per subject. Some LEAs have purchased a licence. Preparing pupils for the Year 7 progress tests The Year 7 progress tests in mathematics are available from the QCA and will consist of two written papers and a mental mathematics paper. Each written paper is allocated 45 minutes and carries 40 marks. Test A is to be done without a calculator; a calculator is allowed for Test B. The mental mathematics test is on audio tape and takes 20 minutes. It carries 20 marks. Pupils are awarded a single level for mathematics. The tests cover all the aspects of the Key Stage 3 programmes of study. They are designed to measure the overall progress that pupils have made in mathematics during Year 7. The new progress tests (from 2003) reflect the changes in end of Key Stage mathematics tests. Using and Applying Mathematics (UAM) is a key aspect. Approximately one eighth of the marks will be for questions that require UAM to get a correct answer or where UAM is assessed directly the most common form being an explain question. As it is impossible to use and apply mathematics without some mathematical content, all UAM marks will contribute to the content balance of each test. Questions testing problem-solving skills will be similar to those already used in Key Stage 2 and Key Stage 3 tests. Some questions will have limited structure, with few intermediate steps indicated, to enable pupils to demonstrate their skills. Topics that are developed in Key Stage 3, for example algebra and probability, will be tested at levels 3 and 4. Pupils should know how long is allowed for each test and should be familiar with the general layout and design, which will be similar to previous tests. 7

Lesson 1 Place value, addition and subtraction Oral and mental starter 10 minutes Objectives Read and write whole numbers in figures and words, and know what each digit represents (Y5) Vocabulary digit, numeral, figures more than, less than Resources individual whiteboards calculator place-value chart or place-value cards Q Ask pupils to write on whiteboards, in figures, seven thousand and twenty-three. Encourage pairs of pupils to compare their results and to sort out errors themselves. Errors will probably involve the zero digit; many pupils will find 7123 easier than 7023. Demonstrate and explain using a place-value chart or place-value cards. Consolidate using similar examples. In the next questions, say the numbers in words; pupils write their answers in figures. Q What number is 2 more than 199?... 2 more than 999? Q Write in digits the number that is 2 more than 1 999 999. Check understanding of place value through questioning. Extend the activity to include questions such as 5 less than 1003; 7 less than 2 000 004; 0.3 less than 1.2; 0.4 less than 9.1. Focus on bridging across 10, 100 and 1000 as appropriate: 1003 5 = (1003 3) 2 = 1000 2 = 998 Pupils can use a calculator to check results. Main teaching 40 minutes Objectives Use known number facts and place value to consolidate mental addition/subtraction (Y6) Use standard column procedures to add and subtract whole numbers (Y7) Vocabulary sum, difference, complement Resources OHT 1.1 number line or 100 square Springboard 7 Units 1 and 2 (Plenary) Resource sheet 1.2 (Plenary) Pupils should be able to recall addition and subtraction facts within 20 and complements of 100. A common error is 100 32 78. Model 100 32 using a number line or a 100 square: 32 8 40 40 60 100 Extend to complements of 1 with one, then two, decimal places, for example: 1 0.76 0.24 Ask pupils to add and subtract mentally 2 two-digit numbers. If pupils make errors, track back through the progression illustrated by these examples: 46 50 (adding tens) 43 52 (units within 10) 43 58 (units greater than 10) 63 52 (tens greater than 100) 63 58 (units greater than 10 and tens greater than 100). Q How did you work out...? Discuss the methods pupils use: 63 58 (63 50) 8 113 8 63 58 (60 50) (3 8) 110 11 63 58 (63 8) 50 71 50 Similarly, check the progression in subtraction illustrated by these examples: 48 5 46 30 67 42 64 37 Discuss the methods pupils use: 64 37 (64 30) 7 64 37 (64 40) + 3 A number line provides useful support. Springboard 7 Unit 1 pages 53 59 contain further examples if required. Follow up in future lessons as a starter activity. 9

Introduce OHT 1.1. Pupils need to choose an appropriate calculation method. Allow time for pupils to complete the calculations. Focusing on particular examples, discuss their methods of solution. In question 10, for example, changing the order of calculation makes the question easier. Plenary 10 minutes By the end of the lesson pupils should: be able to add and subtract whole numbers. Framework supplement of examples pages 88, 92, 94, 104 (includes decimals) Level 4 Identify and discuss particular errors that pupils have made. The following test questions provide a useful summary: 238 1487 723 154 Check pupils understanding and accuracy. Some pupils will need extra support or time (in lessons or through homework) to become confident with subtraction. Resource sheet 1.2 lists some mental mathematics questions. Springboard 7 Unit 2 pages 79 85 provide consolidation. 10

OHT 1.1 Addition and subtraction Work out these calculations without a calculator. For each question, decide whether you: can do it in your head; need some jottings to help you to get the answer; need to use a written method. 1 523 98 2 436 253 3 345 457 789 4 716 897 5 1076 57 6 674 233 7 547 289 8 1784 98 9 6052 1567 10 7894 8792 2358

Resource sheet 1.2 Mental mathematics test questions 1 Write the number three thousand and six in figures. 2 Write the number four and a half million in figures. 3 Write in figures the number two thousand and two. 4 What is fifty-eight multiplied by ten? 5 Subtract nineteen from sixty-five. 6 What number do I need to add to nine hundred and ninety-four to make one thousand? 7 Subtract eighteen from one hundred. 8 Subtract one hundred from six thousand and three. 9 There are two hundred and fifty people in a cinema. Fifty-five are children. How many are adults? 10 In a group of sixty-three children, twenty-nine are boys. How many are girls?

Lesson 2 Multiplication Oral and mental starter 10 minutes Objectives Multiply and divide integers and decimals mentally by 10, 100, 1000 and explain the effect (Y7) Vocabulary digits, tens, hundreds, thousands Resources Large number cards made from Resource sheets 2.1a h Give the eight large cards made from Resource sheets 2.1a h to individual pupils. Invite them to show the number 423 at the front of the class. Q Multiply 423 by 10. Take pupil suggestions and discuss how each digit is multiplied by 10. Emphasise that the decimal point does not move and that zero acts as a place holder to give 4230. Repeat with 41.2 10. Q Divide 423 by 10. Take pupil suggestions and discuss how each digit is divided by 10. The decimal point does not move. Demonstrate the answer (42.3), with pupils manipulating the cards. Repeat with other two- and three-digit numbers and decimals, and using 100 and 1000 as multipliers and divisors. Discuss results and encourage pupils to explain their reasoning. Ensure pupils are confident with place value when multiplying and dividing by 10, 100 and 1000. Main teaching 40 minutes Objectives Use informal pencil and paper methods to support, record or explain multiplications. Extend written methods (Y5, Y6) Vocabulary partition, product Resources Large place-value cards Springboard 7 Unit 6 OHT 2.2 (Plenary) Remind pupils of doubles and near doubles. Instant recall of multiplication facts up to 10 10 is essential to progress with multiplication. Discuss strategies to help learn multiplication facts, for example: 7 8 (7 7) + 7; 7 8 (5 8) (2 8) 7 8: double 7 (14), double 14 (28) and double 28 56 Doubling the 3 times table gives the 6 times table. Use place-value cards to demonstrate that 28 is made up of a 20 and an 8. Partition other two-digit numbers such as 42 and 79. Q How would you multiply 42 by 7? Draw out the key concept that 42 7 is equivalent to 40 7 plus 2 7. Demonstrate this on a grid: 40 2 7 280 14 Show the sum of 280 14 giving 294; so 42 7 294. Demonstrate other products using this method and then set questions in the form tu u, extending to htu u. Springboard 7 Unit 6 pages 231 and 232 provide examples. Use problems in which some figures are missing from a calculation, for example: 30? or 3 = 228.? 180 48 Framework for teaching mathematics: from Reception to Year 6 section 6 pages 66 and 67 illustrate the progression in multiplication. 15

Plenary 10 minutes By the end of the lesson pupils should: understand how to multiply and divide by 10, 100 and 1000; be able to multiply a two-digit number by a single-digit number. Framework supplement of examples page 88 Level 4 Show OHT 2.2. Allow a few minutes for pupils to work out the answers in pairs, then discuss their explanations. Summarise the explanations of the effect of multiplying and dividing by 10 and 100. Invite a pupil to demonstrate how they would multiply 743 by 6. Discuss the partitioning used to break down the question. 16

Resource sheet 2.1a 1

Resource sheet 2.1b 2

Resource sheet 2.1c 3

Resource sheet 2.1d 4

Resource sheet 2.1e 0

Resource sheet 2.1f 0

Resource sheet 2.1g 0

Resource sheet 2.1h

OHT 2.2 Missing numbers Complete each equation to make it correct. Example: 300 10 30 100 1 40 150 400 2 140 6 14 3 37 9 (30 9) (7 ) 4 67 7 ( ) (... ) 5 160 10 16 6 7000 100 700

Lesson 3 Objectives Order fractions and position them on a number line (Y6) Vocabulary gauge, litre numerator, denominator order Resources OHT 3.1 individual whiteboards (optional) Using fractions Oral and mental starter 15 minutes Show OHT 3.1. Ask pupils to read the question. Wait a few minutes then take responses. Q How did you work out the answer? Some pupils may consider the gauge to be a 0 to 60 number line and work out 3 45. Others may say that the tank is three-quarters full, and then calculate 4 of 60. Repeat for tanks of different sizes. Clarify the meaning of a fraction as a number, with a position on a number line, and a fraction as an operator a fraction of a quantity. 1 On the board draw a 0 to 1 number line and ask a pupil to estimate where 8 lies on it. Repeat for other unitary fractions (numerator = 1) and then fractions such as 5 7 8, 10. You may prefer pupils to do this individually, displaying their answers on whiteboards. Using a 0 to 10 number line, repeat the process with examples of fractions greater 1 2 than 1 (e.g. 2, 3 ). 4 5 Main teaching 35 minutes Objectives Calculate simple fractions of quantities and measurements (Y7) Order fractions (Y6) Use names and abbreviations of units of measurement (Y7) Vocabulary kilogram kg, gram g kilometre km hour h, minute min order Resources OHT 3.2 Springboard 7 Unit 13 OHT 3.3 (Plenary) Begin by chanting fractions: 1 1 1 2 1 1 1 3 of 1 = 3 3 of 2 = 3 3 of 3 = 1 3 of 4 = 1 3 1 2 1 1 1 1 2 3 of 5 = 1 3 3 of 6 = 2 3 of 7 = 2 3 3 of 8 = 2 3 1 1 1 1 2 1 3 of 9 = 3 3 of 10 = 3 3 3 of 11 = 3 3 3 of 12 = 4 Explain how to find a fraction of a quantity. 4 Q Find 5 of 20 litres. 7 Q Find 10 of 200 metres. Take pupils suggestions. Check that they have a reliable method for calculation, 1 7 1 for example find 5 of 20, then multiply by 4; to find 10 of 200, find 10 of 200, then 7 multiply by 7, or find 10 of 100, then multiply by 2. Emphasise that the answer should be expressed in the correct units. OHT 3.2 contains a set of similar questions. Emphasise the need to calculate the numerical answer and state the appropriate units. Discuss which are mental calculations and which require some written working. Explain the difference between the question types: find [fraction] of and what fraction of... is...? Springboard 7 Unit 13 pages 425 426 provide further examples involving money, time and measures. Plenary 10 minutes By the end of the lesson pupils should be able to: order simple fractions; work out fractions of quantities and measurements. Framework supplement of examples pages 66 69 Level 4 Use the test question on OHT 3.3. Ask pupils to order the fractions, starting with the smallest. Rectify any errors, distinguishing between careless errors and misconceptions. A common misconception is for pupils to order the denominators ignoring the numerators. Use a list of fractions with the same denominator but different numerators to check pupils understanding. 1 1 1 Ask pupils to draw a picture to explain that of is. Share their explanations. 2 4 8 27

OHT 3.1 Fractions A car s petrol tank holds 60 litres when it is full. How much petrol is in the tank now?

OHT 3.2 More fractions 1 What is of 600 m? 2 What is 10 of 5 m? Give your answer in centimetres. 3 What fraction of a leap year is 1 week? 4 What is of 6 litres? 5 What is of 12 m? 6 What fraction of 1 kilogram is 400 grams? 7 What is of 8.80? 8 Find of 90. 2 3 7 9 9 24 3 10 3 4 5 3 4 3 8 10 What fractions of these shapes are shaded? a b c

OHT 3.3 Ordering fractions Put these fractions in order. Explain your reasoning. 5 1 3 5 1 8 2 8 16 4

Lesson 4 Fractions and decimals Oral and mental starter 15 minutes Objectives Reduce a fraction to its simplest form by cancelling common factors in the numerator and denominator (Y6) Recognise the equivalence between decimal and fraction forms (Y6) Vocabulary fraction, equivalent, numerator, denominator Resources OHT 4.1 3 4 Show OHT 4.1 with written in the centre. Ask pupils to suggest equivalent 3 fractions to 4. Through a series of questions lead pupils to the idea of a multiplier for both numerator and denominator to obtain equivalent fractions. 3 9 Q How do you get from 4 to 12? Q If the numerator is 15, what will the denominator be? How do you know? Extend so that pupils see the equivalence between fraction and decimal forms. 3 Q Can you write 4 in another way? as a decimal? (0.75) Repeat with 0.1 written in the centre of OHT 4.1, looking for fraction equivalents. 1 10 Ensure pupils see that 0.1 is equivalent to 10 and 100. 24 Ask pupils, working in pairs, to do the same with 60. 2 Draw this together and show that the key equivalent fractions are 5 and 0.4. Discuss how pupils arrived at these. Check that pupils can obtain equivalent fractions by dividing numerator and denominator by the same divisor. Main teaching 35 minutes Objectives Recognise the equivalence between decimal and fraction forms (Y6) Vocabulary fraction, equivalent, numerator, denominator Resources 4 counting sticks Springboard 7 Unit 5 OHTs 4.2 and 4.3 (Plenary) Use three counting sticks divided into ten sections. Ask pupils to arrange the following sets of fractions on separate sticks. 1 2 3 4 5 6 7 8 9 0, 10, 10, 10, 10, 10, 10, 10, 10, 10, 1 1 2 3 4 0, 5, 5, 5, 5, 1 0, 0.2, 0.4, 0.6, 0.8, 1 Compare the counting sticks by aligning them beneath each other. 4 Q What do you notice about 10 and 0.4? Q Is any other fraction the same as 0.4? 3 Q What do you notice about 10? Between which two values does it lie? 3 Q What is the decimal equivalent of 10? Repeat for other values. 1 1 3 Ask a pupil to place the numbers 0, 4, 2, 4, 1 on a fourth counting stick. Line up this stick with the others. 1 Q Which values are the same as 2? Q Between which two decimal values does a quarter lie? 1 3 Q What is the decimal equivalent of 4?... of 4? Ask pupils to imagine a 0 to 1 number line with 100 divisions. Count from 0 to 0.12 together (0, 0.01, 0.02... 0.12). Q What is the first number after zero on the line? (0.01) Q What is the middle number? (0.5) Q What is the number on the line immediately before the middle? (0.49) 19 Q Where would 100 be? 7 Q Where would 10 be? Repeat for other fractions. Springboard 7 Unit 5 pages 189 and 190 provide further practice examples. 33

By the end of the lesson pupils should be able to: find equivalent fractions; convert between decimals and fractions. Framework supplement of examples pages 60 65 Level 4 Plenary 10 minutes Use OHT 4.2 to make connections between equivalent fractions. Pupils should learn key conversions and order simple fractions. Use OHT 4.3, which is based on a test question on equivalent fractions. Ask pupils to answer the questions, then discuss their responses. What errors did pupils make? Correct any misconceptions. 34

Equivalent fractions OHT 4.1

OHT 4.2 More equivalent fractions 1 2 1 2 1 3 1 3 1 3 1 5 1 6 1 4 1 8 1 10 1 12 1 24

OHT 4.3 Fraction problems 1 Look at these fractions. 1 1 5 2 3 6 Mark each fraction on the number line. The first one is done for you. The first one is done for you. 0 1 1 2 1 mark 2 Fill in the missing numbers in the boxes. 2 12 6 1 2 12 1 6 24 2 marks

Lesson 5 Probability Oral and mental starter 15 minutes Objectives Use vocabulary and ideas of probability drawing on experience (Y7) Vocabulary certain, uncertain, good chance, no chance impossible, even chance equally likely, possible likely, unlikely, not likely, fair unfair, random, risk Resources Resource sheet 5.1a d, possibly made into word cards Objectives Use vocabulary and ideas of probability drawing on experience (Y7) Vocabulary as above Resources OHTs 5.2a and b and 5.3 Springboard 7 Unit 7 By the end of the lesson pupils should: understand equally likely events; be able to use correctly the vocabulary of probability. Framework supplement of examples pages 276 277 Level 4 You may want to cut out the words on Resource sheet 5.1a d or enlarge and mount them on card. Show pupils one of the words. In pairs, invite them to prepare an explanation or example to illustrate the meaning of the word. Discuss pupils suggestions. Explore, clarify and consolidate the meaning of each word. Ideally you should build up this vocabulary through a series of small inputs before the lesson. You can then use this starter to revise the vocabulary and meanings. Main teaching 40 minutes Introduce OHT 5.2a Cards, which has been adapted from a test question and builds on the vocabulary and explanations in the starter. Pupils might attempt the question individually. Encourage pupils to read each part and produce clear written explanations. Discuss the written explanations in pairs, and consider how to improve them; emphasise that marks are awarded for giving clear written explanations. Explain and clarify any issues that arise. Explore the common misconception of equally likely events through discussion of this example: John says: To throw a six or not to throw a six, each has a probability 1 of as either you do it or you don t. 2 Q Is he correct? Ask pupils to explain their answer. OHT 5.3 Tokens develops this understanding. Encourage pupils to produce clear written explanations. Springboard 7 Unit 7 pages 253 254 provide further examples. Plenary 5 minutes Ask pupils to give examples of: events that are equally likely; events that are not equally likely. Q Is it equally likely that the next baby born in the local hospital is a boy or girl? Discuss the question and answer. You might go on to use these ideas in simple probability calculations. Springboard 7 Unit 7 page 255 introduces the probability scale. 39

Resource sheet 5.1a Probability words certain uncertain good chance no chance

Resource sheet 5.1b Probability words impossible even chance equally likely possible

Resource sheet 5.1c Probability words likely unlikely not likely fair

Resource sheet 5.1d Probability words unfair random risk

OHT 5.2a Cards (a) Joe has these cards: 8 3 9 4 5 2 7 9 Sara takes a card without looking. Joe says: On Sara s card, is more likely than. Explain why Joe is wrong. 1 mark (b) Here are some words and phrases: impossible not likely certain likely Choose a word or a phrase to fill in the gaps below. It is... that the number on Sara s card will be smaller than 10. 1 mark It is... that the number on Sara s card will be an odd number. 1 mark

OHT 5.2b Cards (c) Joe still has these cards: 8 3 9 4 5 2 7 9 5 Joe mixes them up and puts them face down on the table. He turns over the first card, like this: Joe is going to turn the next card over. Complete this sentence: On the next card,... is less likely than... 1 mark The number on the next card could be higher than 5 or lower than 5. Which is more likely? Tick the correct box. higher than 5 lower than 5 cannot tell Explain your answer. 1 mark

OHT 5.3 Tokens A class has some gold tokens and some silver tokens. The tokens are all the same size. (a) The teacher puts four gold tokens and one silver token in a bag. Leah is going to take one token out of the bag without looking. She says: There are two colours, so it is just as likely that I will get a gold token as a silver token. Explain why Leah is wrong. 1 mark (b) (c) How many more silver tokens should the teacher put in the bag to make it just as likely that Leah will get a gold token as a silver token? 1 mark Jack has a different bag with 8 tokens in it. It is more likely that Jack will take a gold token than a silver token from his bag. How many gold tokens might there be in Jack s bag? 1 mark

Lesson 6 Calculators Oral and mental starter 10 minutes Objectives Round positive whole numbers to the nearest 10, 100 or 1000 (Y7) Check a result by considering whether it is of the right order of magnitude (Y7) Vocabulary approximate, rounding, estimate Resources OHT 6.1 Show the grid on OHT 6.1. Ask pupils to round the numbers on the first row to the nearest 10. Q How did you arrive at your answer? Repeat by asking pupils to round the numbers on the second row to the nearest 100 and those on the third row to the nearest 1000. Each time ask pupils to explain their answers. Extend the activity by rounding any of the numbers to the nearest 10. Ensure pupils understand which are the critical values when rounding to the nearest multiple of 10, and that they do not get confused when rounding 149 to the nearest 100, or 25 to the nearest 10. (Pupils may round 149 to 150 and then 200.) Main teaching 40 minutes Objectives Enter numbers in a calculator, and interpret the display in different contexts (Y7) Carry out calculations with more than one step (Y7) Vocabulary display, recurring decimal Resources OHTs 6.2 and 6.3 OHP calculator class set of calculators Springboard 7 Units 2, 10 and 15 Introduce the grid on OHT 6.2. Ask pupils to approximate the numbers to get an estimate for the calculation. Use this to decide which of the questions have the correct answer and which are wrong. Ask pupils to explain their reasons. Use the OHP calculator to check the answers. For each of the problems on OHT 6.3 ask pupils to: read the question; write down the calculation; decide if it is appropriate to use a calculator; use the correct key sequence; write down the answer to the question. Ask pupils to give you their answers. Do each calculation on an OHP calculator. Make sure pupils can interpret the result. Discuss any points needed to support pupils methods and use of the calculator. Further consolidation can be found in Springboard 7 Unit 2 Star challenge 11 page 93, Unit 10 page 351 and Unit 15 pages 485 and 486. Plenary 10 minutes By the end of the lesson pupils should be able to: use a calculator efficiently; interpret the display in a variety of contexts and match the answer to the original problem. Framework supplement of examples pages 108 109 Level 4 Select examples from Springboard Unit 15 page 491. For each question ask pupils to: write down their calculation; explain their method of calculation; give their answer; explain how they would check to make sure the answer is correct. Use pupils responses to rectify misconceptions and to re-emphasise the key points. Remind pupils always to ask: Can I do the calculation in my head? 49

Rounding numbers OHT 6.1 78 184 996 224 1176 3436 7149 9056 3456 75 621 809 023 142 499

Estimating OHT 6.2 32 49 8990 31 98 123 1568 29 1205.4 3950 79 65 312 3600 500 2028 600

OHT 6.3 Problems to solve 1 Janine buys 18 books at 4.95 each. Work out the total cost. 2 Gareth has 456.25 in his bank. He withdraws 654.32. What is his bank balance now? 3 Start with 41. Divide by 7 and then multiply your answer by 35. What is the final answer? 4 I have 60 bags each containing 24 sweets. The sweets are shared equally among 36 people. How many sweets would each person get? 5 Indra buys five bulbs at 85p each, three plugs at 1.05 each and an adaptor at 5.20. How much change does she get from a 20 note?

Lesson 7 Word problems Oral and mental starter 10 minutes Objectives Consolidate the rapid recall of multiplication facts up to 10 10, and quickly derive associated division facts (Y7) Vocabulary product Practise the recall of multiplication facts up to 10 10. Build on the main teaching of lesson 2. Discuss with pupils strategies to help them learn multiplication facts, for example: 7 8 (7 7) 7; 7 8 (5 8) (2 8) 7 8: double 7 (14), double 14 (28) and double 28 56 Doubling the 3 times table gives the 6 times table. Demonstrate patterns, links between related facts and squares. Extend questioning to include division, drawing out links: 8 7 56 7 8 56 56 8 7 56 7 8 Q How many sixes are there in 54? Ask further similar questions. Apply this knowledge to simple mental problems. Q It costs 15p to park a car for 8 minutes. How much will it cost to park for 16 minutes?... 24 minutes?... 40 minutes? Q Six eggs cost 70p. How much will 30 eggs cost? Main teaching 35 minutes Objectives Solve word problems (Y7) Solve simple problems about ratio and proportion (Y7) Vocabulary prime, consecutive litre, metre euro Resources OHTs 7.1a and b and 7.2 (Plenary) class set of calculators Springboard 7 Unit 15 Resource sheet 7.3 (Plenary) OHT 7.1a and b list a set of word problems. Questions 1 4 can be solved in one step. The rest are multi-step problems. Select a problem. Ask pupils to read the question. Clarify any vocabulary. Through questioning, help pupils to build up a strategy to solve the problem. Q What am I being asked to do/calculate? Q What information am I given? Encourage pupils to summarise this by writing down or marking key words or numbers. Q What calculation do I need to do? Insist that pupils write down the calculation, for example 56.7 9. Q How will I do that calculation: in my head, by writing or using a calculator? Q What is the answer? Does it make sense? How can I check it? To give pupils confidence, ask them to work in pairs and to pick two problems to solve. Clarify any vocabulary. Invite a pair to explain their solution. Sort out any errors or misconceptions using other pupils responses. Move on to further questions. Problems involving money and other real-life situations are included in Springboard 7 Unit 15 pages 491 and 492. The Framework supplement of examples pages 2 20 list a range of suitable problems. Plenary 15 minutes By the end of the lesson pupils should be able to: solve a range of word problems. Framework supplement of examples pages 2 20 Level 4 Pick one of the multi-step questions on OHT 7.1a and b. Invite pupils to describe the process of solving the problem. Emphasise the need to: write down the calculation before completing it, especially when using a calculator; interpret the answer in the context of the question; check the answer; include the correct units in the answer. You may want to use the adapted test question on OHT 7.2 to consolidate the work. Resource sheet 7.3 lists some questions taken from mental tests. 55

OHT 7.1a Problems 1 In a school hall there are 38 chairs in a row. How many chairs are there in 23 rows? 2 A group of 534 people is going on a coach trip. Each coach can carry 52 people. How many coaches are needed? 3 Find the cost of 208 bottles of cola at 35p per bottle. 4 I have cut 65 cm from a 3.5 m length of rope. How much rope is left? 5 How many 28p stamps can I buy for 5? How much change will I get? 6 Six friends went to a restaurant. The total cost of the set menu for the group was 75. How much would the set menu cost for eight people? 7 For every eight biscuits in a box, five of them are chocolate. There are 40 biscuits in the box. How many of them are chocolate biscuits?

OHT 7.1b Problems 8 When a travel agent changes money he charges 2 and then gives 1.5 euros for each 1. I have 50 to change. How many euros will I receive? 9 A soup recipe uses four large carrots in each litre of soup. How many large carrots do I need to make 3 litres of soup? 1 2 10 The sum of two prime numbers is 45. What are the numbers? 11 Find two consecutive numbers with a product of 1406. 12 In a school, three classes each have 28 pupils, one class has 29 pupils and four classes each have 30 pupils. How many pupils are there altogether? What is the mean class size?

OHT 7.2 Coaches (a) A club wants to take 3000 people on a journey to London. The club secretary says: We can go in coaches. Each coach can carry 52 people. How many coaches do they need for the journey? Show your working. 2 marks (b) Each coach costs 420. How much is each person s share of the cost? 2 marks

Resource sheet 7.3 Mental mathematics questions 1 Multiply twelve by thirty. 2 What is forty-two divided by six? 3 What is twenty-one divided by three? 4 What is one quarter of thirty-two? 5 How many five-pence coins make forty-five pence? 6 A pen costs three pounds forty-nine. I buy two pens. How much change do I get from ten pounds? 7 What is the cost of four birthday cards at one pound and five pence each? 8 What is the cost of five cassettes at one pound ninety-nine pence each? 9 A tape costs three pounds ninety-nine. How much would five of these tapes cost? 10 Two tickets cost eight pounds. How much do five tickets cost? 11 A bag of oranges costs one pound forty-nine pence. How many bags could you buy with ten pounds? 12 Gary collects ten-pence coins. Altogether he has twelve pounds. How many ten-pence coins is that?

Lesson 8 Interpreting data Oral and mental starter 15 minutes Objectives Interpret diagrams and graphs, and draw simple conclusions (Y7) Vocabulary bar chart mode, maximum Resources OHT 8.1 Objectives Extract and interpret data in tables, graphs, charts and diagrams (Y6) Interpret diagrams and graphs (including pie charts), and draw simple conclusions (Y7) Vocabulary pie chart Resources OHTs 8.2 8.4 Springboard 7 Unit 12 OHT 8.5 (Plenary) Display OHT 8.1. Ask pupils to discuss the questions in pairs. Each pupil has to convince their partner that their answer is correct. Allow pupils time and encourage them to check and reconsider their answers. Distinguish between questions that require: reading information from the graph; deductions from the information; interpretation and explanation. Main teaching 35 minutes Display OHT 8.2. Demonstrate how to find the distance from Hull to Exeter. Ask pupils to answer the questions on the sheet and to compare their answers with their partner s. Take feedback to ensure pupils can find the correct distances. Model a written answer to the last part of the question. Show OHT 8.3. Explain that it shows the number of different types of ticket sold. Q How many different types of ticket are sold? Ensure pupils understand that there are nine different types. Q How many First-class saver tickets are sold? Q How many return tickets are sold? Q How many more Standard tickets than First-class tickets are sold? Q Why do you think a lot of student tickets are sold? Extend with further questions. Show OHT 8.4. Q Are there more men passengers than women passengers? Q How many child passengers are there? What other information would we need to know to find the answer? Q When is it better to use a pie chart than one of the other types of graph? Explain your answer. Consolidate using Springboard 7 Unit 12 pages 398 400. Plenary 10 minutes By the end of the lesson pupils should be able to: extract data from a variety of sources; interpret data and give a reason for any conclusion. Framework supplement of examples pages 268 271 Level 4 Display OHT 8.5. Ask pupils to discuss the six responses and decide whether they agree with them. Take feedback, ensuring pupils give correct reasons for their comments. 63

OHT 8.1 Goals Goals scored by Harriers This bar chart shows the number of goals scored by Harriers in last season's matches. frequency 1 2 3 4 5 6 7 0 0 1 2 3 4 5 number of goals scored 1 What was the highest number of goals Harriers scored in a match? 2 How many matches in total did Harriers play? 3 In how many matches did Harriers score more than three goals? 4 What was the most common number of goals scored (mode)? 5 How likely are Harriers to score seven goals in a match when they play in the same league this season?

OHT 8.2 Distances This table shows the distances between towns. Distances in miles Hull Exeter 305 Bangor 199 289 Dover 261 248 331 Hull Exeter Bangor Dover Which two towns are the shortest distance from each other? Mrs Davis drove from Bangor to Exeter. What is the distance between Bangor and Exeter? Mrs Davis then drove from Exeter to Dover. What is the distance between Exeter and Dover? How far did Mrs Davis drive altogether? Why is this answer different from the distance from Bangor to Dover in the table?

OHT 8.3 Train tickets The ticket office at Exeter station keeps a record of the type of ticket it sells. The tickets sold on a Friday in December are shown in the table below. First class Standard Student Single 7 19 1 Return 12 24 53 Saver 11 37 36

OHT 8.4 Travellers The ticket office also records whether each traveller is a man, a woman or a child. The results are shown in the pie chart. Travellers children men women

OHT 8.5 Favourite colour John and Sandip collected some data on favourite colour from a group of children. 30 John s bar chart 25 frequency 20 15 10 5 0 red green blue yellow colour Sandip s pie chart yellow red blue green Children s comments I think the pie chart is best as you can easily see which is the most popular. Yellow and green are about the same in popularity. I think the bar chart is better as you can compare easily. I could work out how many children were asked from the pie chart. About half of the children chose blue. Red is the least popular colour and it is easy to see from both diagrams.

Lesson 9 Shapes and angles Oral and mental starter 15 minutes Objectives Use correctly the vocabulary for lines, angles and shapes (Y7) Vocabulary scalene triangle, isosceles triangle, parallelogram, pentagon, square, rhombus parallel, equal prism, sphere, cube Resources OHTs 9.1 9.3 (you may want to make card cut-outs from OHT 9.3) 3-D shapes cones, spheres and cubes Use OHT 9.1 to show shapes in less familiar orientations. Ask pupils to name the shapes: rectangle, isosceles triangle, square. Pupils can check their answers as you rotate the OHT to show the shape in a more familiar orientation. Discuss and establish the properties of each of the shapes. Note that a square satisfies the criteria for both a rectangle and a rhombus. OHT 9.2 shows 2-D representations of some 3-D shapes: cone, sphere and cube. Check that pupils recognise them. Have solid shapes available to emphasise the link. Ask pupils to sketch some other 3-D shapes, for example a square-based pyramid. Use OHTs 9.3a d for a hide and reveal activity. Cover one of the shapes on the OHP then gradually reveal the shape. Pupils attempt to recognise and name the shape as it is revealed. Q Why do you think it is a? Q Why can it not be a? Main teaching 35 minutes Objectives Use correctly the vocabulary for lines, angles and shapes (Y7) Know the sum of angles at a point, on a straight line and in a triangle (Y7) Vocabulary as above Resources OHTs 9.4a and b Select one of the 2-D shapes from the lesson starter. Ask pupils to write down the name of the shape and describe its properties using the key mathematical vocabulary angles, lengths and diagonals. Discuss pupils responses. Clarify any misunderstandings. Move on to talk about the sizes of the angles. Q Is this angle greater or less than a right angle? Spend a short time honing pupils mental calculation skills using: pairs of angles that make 90 (right angle); pairs of angles on a straight line that add up to 180. Extend the work using OHTs 9.4a and b, which includes multi-step calculations involving angles at a point and angles in a triangle. Model how to set out a calculation and state the reasons for an answer. Plenary 10 minutes By the end of the lesson pupils should: know how many degrees there are at a point, in a quarter turn, and on a straight line; be able to solve simple problems involving the calculation of angles. Framework supplement of examples page 182 Level 4 Clarify any misunderstandings or errors from the previous exercise. Ask pupils to visualise and name a shape that you describe. Take suggestions after each piece of information. The shape I can see has five faces. Three of the faces are rectangles. Two of the faces are parallel. The two parallel faces are triangles. A particular kind of chocolate comes in a box of this shape. What is it? (a triangular prism) 71

OHT 9.1 Shapes

OHT 9.2 Solid shapes

OHT 9.3a Hide and reveal

OHT 9.3b Hide and reveal

OHT 9.3c Hide and reveal

OHT 9.3d Hide and reveal

OHT 9.4a Angles Work out each of the angles. 1 AB is a straight line. 2 AB is a straight line. A A a 85 B 40 a =... b =... c =... b 126 c B 3 4 AB is a straight line. A 82 e f 38 B d 70 d =... e =... f =... 5 AB is a straight line. h B 26 A g 120 i g =... h =... i =...

OHT 9.4b Angles Work out each of the angles. AB is a straight line. 1 2 A A a B b c 58 a =... A B b =... c =... 3 4 d 36 e A 36 126 B B d =... e =... 5 6 B 18 A g h 70 B f A f =... g =... h =...

Lesson 10 Coordinates and reflections Oral and mental starter 15 minutes Objectives Find coordinates of points determined by geometric information (Y7) Understand and use the language and notation associated with reflections (Y7) Vocabulary point, coordinate, reflection Resources OHT 10.1 Objectives Recognise transformation and symmetry of a 2-D shape: reflection in given mirror lines and line symmetry (Y7) Vocabulary lines of symmetry reflection, mirror lines Resources OHTs 10.2 and 10.4 (Plenary) Handouts 10.3a and b tracing paper, mirrors square tiles if available Springboard 7 Unit 14 By the end of the lesson pupils should be able to: plot points and read coordinates in the first quadrant; reflect 2-D objects in mirror lines; complete the reflection of a shape in a given mirror line. Framework supplement of examples pages 202 206 Level 4 Using the grid on OHT 10.1, invite pupils to plot the points (1,2) (6,2) and (6,7). Indicate the points in turn and check the coordinates. Q What are the coordinates of the fourth corner if the shape is a square? Invite a pupil to plot the points (1,5), (3,1), (3,9). Q What are the coordinates of the fourth point that makes this a square? Pupils should discuss and then compare their answers. Reflect the squares in the vertical and horizontal lines on the diagram. Discuss where the new squares will be and the coordinates of their vertices. Main teaching 35 minutes Using OHT 10.2, demonstrate how to use tracing paper to find the image of trapezium A after reflection in the line OT. Ask pupils to trace shapes C and D and to find the mirror line so that shape C is a reflection of shape D. Q Can you always find the reflection of a shape in a given mirror line? Q If you have two identical shapes, can you always find a mirror line that will reflect one onto the other? Ask pupils to draw examples to demonstrate their responses. Using tracing paper and mirrors (and square tiles if available), allow pupils to investigate the problems on Handouts 10.3a and b. Consolidate with Springboard 7 Unit 14 pages 452 and 453. Plenary 10 minutes Show OHT 10.4. Q Which triangles are reflections of triangle A? Q Find pairs of triangles that will reflect onto each other. Q Are there any triangles that cannot be reflected onto any of the others? Q How can you tell if you cannot reflect a given triangle onto another one? 81

OHT 10.1 Coordinates y 20 18 16 14 12 10 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 x

OHT 10.2 Reflections T A C D O

Handout 10.3a Symmetry 1 For each question start with the shape above. Move one square only to make each of the shapes described. Make a shape with: a b c d e a horizontal line of symmetry only a vertical line of symmetry only a diagonal line of symmetry only diagonal, horizontal and vertical lines of symmetry no lines of symmetry (different from the original shape).

Handout 10.3b Symmetry 2 On each of the diagrams below add two more squares to the shape so that the dashed line becomes a line of symmetry. a b

OHT 10.4 Reflecting triangles A B C D F E

Lesson 11 Sequences Oral and mental starter 15 minutes Objectives Recognise and extend number sequences (Y6) Vocabulary odd, even, triangular, square Resources individual whiteboards Springboard 7 Unit 1 Give pupils practice counting on and back in equal steps from and to zero this is one way of building up multiplication facts. Move on to counting on, for example, from 8 in steps of 7: 8 15 22 29...; point out the link with the 7 times table. Pupils will need more practice at counting back. Q Count back from 41 in steps of 5. Give similar examples. Springboard 7 Unit 1 pages 50 51 provides further examples that can be done orally. Revise even numbers. Q Think of the even numbers 2, 4, 6, 8 What picture do you see? Draw it. Make sure that pupils have a mental image of even numbers. Pupils need a picture similar to: Ask pupils to explain the link between the numbers and the pictures. Do the same for odd numbers, square numbers and triangular numbers (see Framework supplement of examples page 146). Emphasise the link between the picture and the numbers.. Main teaching 35 minutes Objectives Generate sequences from practical contexts (Y7) Recognise squares of numbers to at least 12 12 (Y7) Vocabulary pattern, position, term Resources OHTs 11.1 and 11.2 OHT 11.3 (Plenary) Introduce the task on OHT 11.1, which is based on a test question. Discuss and establish ways of recording the information, for example: pattern 1 2 3 number of tiles 4 For each question, establish how pupils worked out their answers. Encourage pupils to move on from describing the pattern as +4 to seeing the link between the pattern number and the number of tiles ( 4). Note, however, that this could lead pupils to a false conclusion: a 4 link between terms does not always lead to position-to-term relation of 4. Use the example on OHT 11.2 to clarify this. Plenary 10 minutes By the end of the lesson pupils should be able: to generate sequences from spatial situations; understand links between a numerical sequence and a spatial pattern. Framework supplement of examples page 154 Level 4 Investigate Growing steps on OHT 11.3. Q How many tiles will there be in the 5th, 6th and 10th patterns? How did you calculate the values? Draw out the link with square numbers. Q How many squares will there be in the 20th, 60th and 76th patterns? Q How many squares will there be on the 3rd, 5th and 10th rows? These two questions provide opportunities to revisit mental and/or written methods of multiplication. 89

OHT 11.1 Patterns Owen has some tiles like these: He uses the tiles to make a series of patterns. pattern number 1 pattern number 2 pattern number 3 pattern number 4 1 Each new pattern has more tiles than the one before. The number of tiles goes up by the same amount each time. How many more tiles does Owen add each time he makes a new pattern? 2 How many tiles will Owen need altogether to make pattern number 6? 3 How many tiles will Owen need altogether to make pattern number 9? 4 Owen uses 40 tiles to make a pattern. What is the number of the pattern he makes?

OHT 11.2 Growing patterns Investigate this growing pattern. pattern 1 pattern 2 pattern 3 How many tiles will be in pattern 6? If my pattern uses 29 tiles, which pattern number is it?

OHT 11.3 Growing steps pattern 1 pattern 2 pattern 3

Lesson 12 Perimeter and area Oral and mental starter 15 minutes Objectives Calculate the perimeter and area of shapes made from rectangles (Y7) Vocabulary area, perimeter, cm, cm 2 Resources centimetre-squared paper or tiles OHT 12.1 Show OHT 12.1, pointing out that the sides of each square tile are 1 cm in length. Q What is the area of one square? Make sure pupils give the correct units. Q What is the perimeter of one square? Make sure pupils give the correct units. Make sure that pupils understand perimeter and area. Q What are the area and perimeter of Figure 1? Q What are the area and perimeter of Figure 2? Using tiles or centimetre-squared paper, ask pupils to draw other shapes with twelve 1 cm 2 tiles and to calculate the perimeter and area of each shape. Explain that all sides must fully touch another side. Q What can you say about the areas of the shapes? Q What is the largest/smallest perimeter you found? Emphasise that for a fixed area the perimeter may vary. Main teaching 35 minutes Objectives Know and use the formula for the area of a rectangle; calculate the perimeter and area of shapes made from rectangles (Y7) Use names and abbreviations of units of measurement (Y7) Vocabulary area, perimeter, minimum, maximum, cm, cm 2 Resources centimetre-squared paper Springboard 7 Unit 3 OHT 12.2 OHT 12.3 (Plenary) Ask pupils to visualise a rectangle with an area of 36 cm 2. Q What could the length and width of the rectangle be? Make sure that pupils identify all the integer pairs. Q Is a 3 12 rectangle different from a 12 3 rectangle? Q What is the perimeter of a 3 12 rectangle? Q What is special about a 6 6 rectangle? Q Which one has the smallest perimeter? Which one has the largest perimeter? Using centimetre-squared paper, ask pupils to draw different shapes with a perimeter of 24 cm. Make sure they do not use diagonal lines. Q What shape with a perimeter of 24 cm has the maximum area? Q What shape with a perimeter of 24 cm has the minimum area? Pupils work in pairs and have to convince their partner that they have found the correct shapes. (You will need to check final results.) Consolidate with questions from Springboard 7 Unit 3 pages 108 112. Extension: Show OHT 12.2. Ask pupils to work out the perimeters of the shapes, giving their answers in the simplest form. Take feedback and rectify any misconceptions. Plenary 10 minutes By the end of the lesson pupils should be able to: work out the area and perimeter of a rectangle work out the area and perimeter of compound shapes based upon rectangles. Framework supplement of examples 234 and 236 Level 4 Show OHT 12.3. Ask pupils to work out the area of the octagon, making sure they give the correct units. Ask pupils to explain how they worked out the area, in particular how they worked out the area for the halved squares. Ask them to consider the second problem. Take feedback and discuss why the diagonal sides must be longer than 1 cm. Ask pupils to write an explanation. 95

OHT 12.1 Area and perimeter Figure 1 Figure 2

OHT 12.2 Perimeters s s s and t are lengths of the sides in centimetres t t The perimeter p of this shape is 3t 2s. t p 3t 2s Work out the perimeters of these shapes. 1 2 b a a c c a b b p = p = e 3 d d 4 5 3 7 p = e e f f f f p = e

OHT 12.3 Shapes 1 Look at the octagon. It is drawn on a 1 cm square grid. What is the area of the octagon? 2 Explain how you know that the perimeter of the octagon is more than 8 cm.