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Hamilton Secondary Numeracy Project Numeracy Shining Term 2 www.hsnp.org.uk Published by Hamilton Trust 1A Howard Street, Oxford, OX4 3AY www.hamilton-trust.org.uk [email protected] tel: 01865 253980 Development team Ruth Merttens, Jennie Kerwin Alison Fahey, Deidre Holes, Jeanette Viney, Mike O’Regan Contents Contents i Introduction to HSNP ii Table showing different levels v Content of HSNP vi Weekly activities 1 Homework 43 Homework answers 55 HSNP © Hamilton 2012 Page i Shining Term 2 Introduction to HSNP Structure of HSNP Numeracy - Four levels of proficiency 1. Stepping up – for those who have not achieved the level of numerical fluency expected at the end of Upper Key Stage 2. (Level 3s at entry to Y7) 2. Keeping up – for those who have just about achieved the level expected at the end of Upper Key Stage 2 but who are not secure with this. (Shaky Level 4s at entry to Y7) 3. Simmering – for those who have achieved the standard expected at the end of Upper Key Stage 2 and who are secure with it, but who need to sustain their numerical fluency. (Secure Level 4s) 4. Shining – for those who are good at number work and who need to sustain their proficiency and do a bit of exploration. (Level 5s) Advice as to which pupils do which levels Year 7 Some pupils will need Stepping up – the lowest level of numeracy intervention which provides teaching and practice of basic skills. This will suit pupils who enter Y7 with a level of numeracy no higher than Level 3. Some pupils will require Keeping up – the slightly harder programme for those not far below where we would want them to be in terms of numeracy levels. This provides a little teaching and a great deal of practice of basic numeracy skills. This suits pupils who are operating at high Level 3 or a low Level 4 in relation to numeracy. HSNP © Hamilton 2012 Page ii Shining Term 2 Some pupils will require Simmering – the standard numeracy programme. This recognises that these pupils have achieved a reasonable level of numeracy but that they need to practise it or else they will forget! These pupils are operating at Level 4. Some pupils will need Shining – the advanced programme for pupils who are numerically fluent. This aims to broaden their understanding of number and to encourage deeper exploration of numerical concepts. Year 8 A few pupils may still require the basic level, Stepping up. Many pupils will have moved on to the next level, Keeping up. Most Y8 pupils will hopefully be at the average level, Simmering. A few pupils may be wanting advanced level work, provided in Shining. Year 9 Some Y9 pupils may still not have progressed from the slightly lower than average level of Keeping up. Most will be requiring Simmering or Shining. The following table provides an overview of how the different levels of programme operate. Some sets of pupils will be using the same level of materials for more than one year. This is because their skills, once acquired, are simply being kept ‘on-the-boil’, so to speak. To accommodate this, we shall be providing a second and even third set of these materials so that pupils will not be doing the same activities twice. This table shows which pupils may be using the different levels of programme in successive years. The column headings refer to the different ‘Turns’ pupils may have at the same level of programme. So, for example, some Y7 pupils may have three goes at the Simmering, since doing this programme keeps their numeracy skills honed over the three years of KS3. HSNP © Hamilton 2012 Page iii Shining Term 2 HSNP © Hamilton 2012 Page iv Shining Term 2 Table showing how the different levels operate within HSNP 1st ‘Turn’ 2nd ‘Turn’ 3rd ‘Turn’ Y10/Y11 Stepping up Between 15% and 40% of Y7 Between 5% and 10% of Y8 Keeping up 20% - 30% of Y7 Y8 who were in Steppingup in Y7 Y8 who need another go at this Y9 who cannot move beyond this Simmering Around 50% of Y7 Y8 who were in Keeping-up in Y7 Y9 who were in Steppingup and then in Keeping-up Y8/Y9 who need to keep their skills on the boil Y9 who need to keep their skills simmering Y10 who need to keep their skills simmering Shining 10% - 20% of Y7 Y8/Y9 who have topped out of simmering Y8, Y9 who are very good at numeracy but need something so they don’t forget it. Y9 who are very good at numeracy but need something so they don’t forget it. Between 20% and 40% of cohort HSNP © Hamilton 2012 Page v Shining Term 2 Content of HSNP Wk Term 1 Place Value - Numbers 1 to 10 million, ordering on a line, comparing, even larger numbers – to a googol 2 3 4 5 Addition - Mental addition and written addition; make up numbers to give hard additions to each other. What makes an addition difficult? Subtraction - Mental subtraction & written subtraction. Revise methods Chn explain why a partic strategy is better than others. Make up ‘hard’ subtractions/ why are they hard? Multiplication - Times tables, multiples and factors, mental strategies incl. doubling and halving – divisibility rules Division - Reverse of multiplication = chunking, written division. Explore methods, Create harder divisions – what makes them hard? HSNP © Hamilton 2012 Term 2 Term 3 Fractions - Concept of fraction, find fractions of fractions, divide by a fraction – what happens? Decimal and fraction equivalences incl 1/8 Addition of fractions Explain why we need common denominator Introduce ‘smile then kiss’ – can they explain how it works? Decimals and Fractions - Round whole numbers & decimal numbers, using rounding to estimate answers to complex calculations Addition - Explore consecutive numbers (can each no. be made by adding consec nos?) Diffs between prime nos. Also Dig roots of primes Subtraction – Explore 1089 investigation. Also explore palindromic numbers Subtraction of fractions - Subtract fractions with related denominators (1/2 – ¼, ¼ - 1/8 etc. Or fractions in a sequence, ½ - 1/3, 1/3 – 1.4, etc. Look for patterns in subtraction Multiplication Explore patterns: 7 x 7, 67 x 67, 667 x 667 etc. 99 x 11, 99 x 22, 99 x 33 etc. 999,999 X2, X3, X4 etc. Division - Explore patterns: What 5-digit no ÷ 4 gives an answer which is its reverse? Divide 2521 by 1, 2, …10 write down the remainder in each. Page vi Multiplication - Binary numbers. Explore these – do ‘age trick’. Do Egyptian and Russian multiplication – why do they work. Division - Explore patterns: 22 – 1X3, 32 – 2X4, 42 – 3X5, 42 – 4X6 etc. And (11–2) ÷9, (111–3) ÷9, (1111–4) ÷9 etc. And 56÷11, 78÷11, 34÷11, 122÷11, 89÷11, 37÷11, 49÷11, 60÷11 Shining Term 2 6 Place Value - Decimal numbers to 3-places and beyond, order and compare using a line, PV calculations 7 Addition - Magic squares. Explore patterns include Durer’s and Franklin’s and diabolic squares 8 Subtraction - Subtract decimal numbers with different numbers of digits, include up to 4 places of decimals. Round to estimate first. Subtraction - Find a difference between negative numbers; explore patterns in adding and subtracting negative numbers 9 Multiplication Tables, multiples and factors, written multiplication including of two decimal numbers e.g. 34.57 x 23 10 Division Strategies for written division of decimals. Multiplication Rehearse hard mults written method. Explore factorials. Explore reverse digits 12X42 &21X24 then 12X84, 13X62, 23X96, 24X63. Division - Explore patterns in fractions to decimals: 1/9, 2/9, 3/9 etc. 1/11, 2/11. 3/11 etc. 3/7, 4/7, 5/7 etc. Explore patterns in factors Find prime factors. Perfect numbers and Amicable numbers HSNP © Hamilton 2012 Place value Recurring decimals, non-recurring decimals. Converting fractions to decimals and vice versa Addition - Pascal’s triangle and Fibonacci sequence. Explore patterns Page vii Place value - Natural nos, integers, Rational numbers and irrational numbers – explore these Addition - Test Goldbach’s conjecture Every even no > 4 = sum of 2 odd primes, every odd no = sum of 3 odd primes. Test Chebyshev’s theorem Subtraction - Explore differences: between no & reverse 871 – 178 etc Between square nos / cubic nos Between nos in arithmetic series Multiplication Explore indices. Work out 12 + 22, 12 + 22 + 32, etc. Work out 22, 23, 24, 25 etc. Work out 32, 33, 34, etc. Find ans’s digital roots Division - Explore roots. Find square roots. Explore Chinese method for finding sq roots. Explore triangular nos and square nos. Test Diophantus’ rule: 8T +1 = sq no. Shining Term 2 HSNP © Hamilton 2012 Page viii Shining Term 2 Numeracy Shining Term 2 Weekly Activities HSNP © Hamilton 2012 Page 1 Shining Term 2 Week 1 overview Fractions Objectives Know equivalences between key fractions and decimals Find fractions of fractions Divide by fractions For this week you will need: A3 paper, playing cards, 0-9 dice, Dartboard fractions at http://www.topmarks.co.uk/Flash.aspx?f=DartboardFDP Multiplying Fractions millionaire at http://www.math-play.com/Multiplying-FractionsMillionaire/Multiplying-Fractions-Millionaire.html Link to homework page Watch out for pupils who: do not realise that to find half of a fraction, for example, is the same as multiplying a fraction by one half; do not see the link between fractions and division; when dividing by fractions, divide by the denominator, e.g. if dividing by ¼, just divide by 4. HSNP © Hamilton 2012 Page 2 Shining Term 2 Week 1 Fractions Day 1 Objective: Know equivalences between key fractions and decimals You will need: A3 paper, Dartboard fractions at http://www.topmarks.co.uk/Flash.aspx?f=DartboardFDP Teacher input with whole class Give each pair a sheet of A3 paper and ask them to draw a line as long as they can on it. They mark on 0, ¼, ½ ¾ and 1 above the line. Where does 0.5 belong on this line? What is half of 0.5?, so where will that belong? Agree that 0.25 is half of 0.5 and as ¼ is half of ½, 0.25 belongs in the same place. They write 0.5 and 0.25 in the correct places below the line. Ask pupils to mark the equivalent decimal to ¾ below the line. What is half of a ¼? What is half of 0.25? Paired pupil work Pupils work in pairs to mark eighths on their lines, and work out the equivalent decimals. They then mark tenths on a fresh line, then fifths and any equivalent decimals that they can. Teacher input with whole class Ask pupils to share equivalent pairs of fractions and decimals. Display Dartboard fractions, choosing the ‘equivalent fraction/decimal’ from the left menu. Click on the green segments, pupils write the equivalent fractions. Repeat, this time clicking on the pink sections, pupils write the equivalent decimals. HSNP © Hamilton 2012 Page 3 Shining Term 2 Week 1 Fractions Day 2 Objective: Find fractions of fractions You will need: 0-9 dice, Multiplying Fractions millionaire at http://www.math-play.com/Multiplying-Fractions-Millionaire/MultiplyingFractions-Millionaire.html Teacher input with whole class What is half of a quarter? Record 1/2 × 1/4 = 1/8. What do you notice? The answer is SMALLER than either fraction and the denominator is the product of the two denominators. Ask pupils to find half of the following fractions: 2/5, 2/3, 5/6. Stress that the answer is always smaller than either of the two fractions being multiplied. Pupils discuss in pairs how they might find 1/3 of 3/4 and then 2/3 of 3/4. Draw out dividing each quarter into 3, and so 1/3 of 3/4 is 3/12, i.e. we have multiplied the denominator by 3. So how do we find 2/3? (Multiply the numerator by 2.) Multiplying pairs of fractions is easy as we just multiply the numerators together and the denominators together! Remind pupils how to do this with several pairs of fractions, e.g. 2/3 × 3/4 and 3/10 × 4/5, reducing the fractions of the answer. Paired pupil work Pupils play the following game in pairs. They each roll a dice twice to make a fraction (if they roll 0, roll again). They multiply the fractions, simplifying where they can. How many can they do in three minutes? Teacher input with whole class Write 3/10 × 5/9 on the board and ask pupils to work out the answer We can write this as: 3 × 5 10 × 9 We can multiply the numerators and then the denominators, but to make it easier we can do some cancelling first. Do this to give: 1×1 2×3 Pupils practise this with 3/10 × 5/12 and 4/7 × 14/10. If time, split into two teams and play Multiplying Fractions millionaire. HSNP © Hamilton 2012 Page 4 Shining Term 2 Week 1 Fractions Day 3 Objective: Divide by fractions You will need: playing cards Teacher input with whole class Record 36 ÷ ¼ on the board. How many quarters are in 36? Agree there are four in each whole so 4 × 36 altogether. What do you notice? Pupils divide 36 by 1/9, 1/3 and 1/12. (There are 9 1/9ths in every whole so 36 lots of 9 ...) Repeat with 20 ÷ 2/5. How many fifths in 20? Five in every one whole, so 5 x 20. But if each part is twice as big, i.e. two fifths, how many then? Agree that there will be half as many. What do you notice? Agree the answer is the same as 20 × 5/2. Pupils find 20 ÷ 3/5 and 4/5. Write ½ ÷ ¼ on the board. Say that we can think of this as how many quarters are in ½, two! Repeat with 1/3 ÷ 1/9. What do you notice? This is the same answer as multiplying 1/3 by 9. Repeat with ¾ ÷ 1/8. How many eighths in each quarter? In three quarters? Point out that this gives the same answer as 3/4 × 8/1. So remind pupils that to divide by a fraction, we can invert the second fraction and multiply to give the same answer. And multiplying pairs of fractions is easy! Together work through the stages in working out 4/5 ÷ 2/3, suggesting that they do some cancelling before the multiplication, e.g. 4×3 5×2 =2×3 5×1 = 6/5, 1 1/5. Ask pupils to do the same for 6/7 ÷ 9/14. Paired pupil work Pupils remove Jokers, Jacks, Queens and Kings from a set of playing cards, shuffle the rest and place face down. They each take two cards and use them to make fractions. They record this as a division. They invert the second fraction and multiply the two together, either cancelling first or simplifying afterwards where necessary. Repeat at least nine times. HSNP © Hamilton 2012 Page 5 Shining Term 2 Week 2 overview Addition of fractions Objectives Add pairs of fractions with related denominators Add pairs of fractions with unrelated denominators using ‘smile and kiss’ For this week you will need: 1-10 number cards Link to homework page Watch out for pupils who: add the denominators and numerators to add pairs of fractions (rather like multiplying the numerators and denominators to multiply pairs of fractions); struggle to convert both fractions to ones with the same denominator because their understanding of equivalence is shaky; think that when the numerator and denominator are multiplied by the same number, that the result is a bigger fraction. HSNP © Hamilton 2012 Page 6 Shining Term 2 Week 2 Addition of fractions Day 1 Objective: Add pairs of fractions with related denominators You will need: none required Teacher input with whole class Write the following additions on the board and ask pupils to discuss in pairs how they would solve each: 1 /2 + 1/4, 5/6 + 2/3, 3/5 + 7/10, 5/9 + 1/3 Draw out that we cannot add the pairs of fractions as they have different denominators so first we need to change them so both have the same denominator. So in the first we can change both fractions to quarters, in the second change both to sixths, in the third change both to tenths and the fourth change both to ninths. Ask pupils to work in pairs to do this, and find the totals. Explain that some will give an improper fraction as an answer. So their work is still not done as they should then convert the answer to a mixed number! Take feedback. Paired pupil work Pupils work in pairs to draw a pyramid of boxes, beginning with three on the bottom row. They write 1/2, 5/8 and 3/4 in the bottom row. They add neighbouring fractions and write the answer in the box that overlaps the pair of numbers. They continue up the pyramid until they reach a total at the top, e.g. 20 9 1 /2 /8 Simplify to 5/2, then convert to 2½ 11 /8 /8 5 /8 3 /4 Repeat with fractions 1/2, 5/6 and 1/12 in the bottom row. Take feedback. HSNP © Hamilton 2012 Page 7 Shining Term 2 Week 2 Addition of fractions Day 2 Objective: Add pairs of fractions with unrelated denominators using ‘smile and kiss’ You will need: none required Teacher input with whole class Write 2/3 + 3/4 on the board and ask pupils to discuss in pairs how they would solve it. Take feedback. Agree that we need to convert both fractions to the ones with the same denominator. Discuss what fraction we could convert them to. We are looking for a number that both 3 and 4 will go into. Draw out that we could convert them both to 1 /12s, i.e. find the equivalent numbers of twelfths. Ask pupils to do this, and then convert 3/4 to twelfths. They then find the total. Show (or remind) pupils that there is way to remember how to do this called ‘smile and kiss’. Draw the following on the board: 12 3 5 3 4 15 Talk through the steps: multiply the denominators together (smile) multiply 3 by 4, and 5 by 3 (kiss) then add these together to get 27/20. Ask pupils to discuss in pairs why this works. Take feedback. Agree we are changing each fraction into 20ths by multiplying the denominators. 3 + 3 5 4 = 12 + 15 20 = 27 20 Ask pupils to convert 27/20 to a mixed number. Paired pupil work Pupils use ‘smile and kiss’ to add 2/5 + 2/3, 1/4 + 1/3, 2/7 + 2/3 and three other pairs of fractions of their choice. Remind them to convert to mixed numbers and simplify the answer where possible. HSNP © Hamilton 2012 Page 8 Shining Term 2 Week 2 Addition of fractions Day 3 Objective: Add pairs of fractions with unrelated denominators using ‘smile and kiss’ You will need: 1-10 number cards Teacher input with whole class Write 1/6 + 3/4 on the board. We can work this out using ‘smile and kiss’. But you might also spot that rather than converting both to 1/24s we could convert both to 1/12s. Ask half the class to use 'smile and kiss’ to find the answer, and the other half to convert both fractions to 1/12s, then add them. Point out that both ways give the same answer. The ‘smile and kiss’ group can simplify their answer at the end to give 11/12. Paired pupil work Ask pupils to work in pairs to either spot a common multiple of the denominator and both fractions, or use smile and kiss’. 7/10 + 3/4; 3/8 + 5 /6; 5/6 + 4/9. Teacher input with whole class Take feedback on which way pupils found easier. Paired pupil work Pupils shuffle a set of 1 to 10 number cards, and place upside down. They take the top four and use in any order they like to make two fractions both less than 1. They add the two fractions together. If the answer is less than 1, they score 1 point. If the answer is greater than 1 they score 2 points. They carry on adding pairs of fractions until you ask them to stop. Did any pair score more than 20 points? HSNP © Hamilton 2012 Page 9 Shining Term 2 Week 3 overview Subtraction of fractions Objectives Subtract pairs of fractions with related denominators Subtract pairs of fractions with unrelated denominators using ‘smile and kiss’ Identify patterns, and make predictions For this week you will need: 1-9 digit cards Link to homework page Watch out for pupils who: subtract the denominators and numerators to subtract pairs of fractions; struggle to convert both fractions to ones with the same denominator because their understanding of equivalence is shaky; think that when the numerator and denominator are multiplied by the same number, the result is a bigger fraction. HSNP © Hamilton 2012 Page 10 Shining Term 2 Week 3 Subtraction of fractions Day 1 Objective: Subtract pairs of fractions with related denominators You will need: none needed Teacher input with whole class Write 1/3 – 1/6 on the board. Say that it is difficult to subtract a third from a sixth until we use the fact that one third is the same as two sixths. Repeat with 3/4 - 5/8. If we convert both fractions to eighths, then this subtraction is easy. Ask pupils to find the answer Write 7/10 – 1/5 on the board. Ask pupils to convert the fractions to fractions with the same denominator and then to work out the subtraction. Take feedback. Paired pupil work Write the following fractions on the board: 1 /2, 1/4, 5/6, 2/3, 7/12, 3/8, 5/9 Pupils work in pairs to choose two fractions where they can see they can convert one fraction to have the same denominator as the other fraction. They subtract the smaller from the larger. They carry on, trying to find as many different subtractions as they can. How many different subtractions are possible where they can convert one fraction to the same denominator as the other? (10) HSNP © Hamilton 2012 Page 11 Shining Term 2 Week 3 Subtraction of fractions Day 2 Objective: Subtract pairs of fractions with unrelated denominators using ‘smile and kiss’ You will need: none required Teacher input with whole class Write 2/3 – 1/4 on the board and ask pupils to discuss in pairs how they would solve it. Take feedback. Agree that we need to convert both fractions to ones with the same denominator. Discuss what fraction we could convert them to. We are looking for a number that both 3 and 4 will go into. Draw out that we could convert them both to 1/12s. Ask pupils to so this and then to work out the subtraction. Remind pupils how they used ‘smile and kiss’ to add pairs of fractions with different denominators. Draw the following on the board: 8 2 3 1 4 3 Use ‘smile’ to convert both fractions to 1/12s. What do we do next? (‘Kiss’, multiply each numerator by the opposite denominator to give the equivalent number of 1/12s). We then subtract the numerators. 2 – 1 3 4 =8-3 12 =5 12 Write 3/4 – 3/10 on the board. We can work this out using ‘smile and kiss’. But you might spot that rather than converting both to 1/40s we could convert both to 1/20s. Half the class use 'smile and kiss’ to find the answer and the other half convert both fractions to 1/20s. The ‘smile and kiss’ group simplify their answer to give 9/20. Paired pupil work Pupils work in pairs to spot a common multiple of the denominators, or use smile and kiss’ to solve: 5/6 – 3/4; 7/8 – 5/6; 5/6 – 2/9, then four of their own pairs of fractions. HSNP © Hamilton 2012 Page 12 Shining Term 2 Week 3 Subtraction of fractions Day 3 Objectives: Identify patterns, and make predictions You will need: none required Teacher input with whole class Write the following subtractions on the board and ask pupils to work them out: 1 – 1/2, 1/2 – 1/3, 1/3 – 1/4, 1/4 – 1/5, 1/5 – 1/6 What do you notice about these pairs of fractions? Can you spot a pattern in the answers? What will the next subtraction be? And the answer? Take some predictions for the answers. Pupils then find the answer to 1/6 – 1/7. Repeat for 1/7 – 1/8. (Pupils should spot that the denominator is the product of the two denominators, and the numerator is always one. The answers are 1/2, 1 /6, 1/12, 1/20, 1/30, 1/42, i.e. denominators increase by 4, 6, 8, 10, 12… each time. The denominator could be generalised to n(n+1) where n is the number of the term in the sequence.) Paired pupil work Ask pairs to work out subtractions in the following sequences, to make and test their predictions: 1 – 1/3, 1/2 – 1/4, 1/3 – 1/5, 1/4 – 1/6… and 2 /3 – 1/2, 3/4 – 2/3, 4/5 – 3/4, 5/6 – 3/4, 7/8 – 5/6… Teacher input with whole class Take feedback. HSNP © Hamilton 2012 Page 13 Shining Term 2 Week 4 overview Multiplication Objectives Use mental, written and calculator methods to multiply numbers together, including finding squares Look for patterns and rules, make and test predictions Solve puzzles using factors For this week you will need: Calculators, Sudoku grids printed out from Product doubles Sudoku at http://nrich.maths.org/6434/index Link to homework page Watch out for pupils who: do not know their times tables and so do not spot multiples of numbers; when they spot a pattern, do not test out their theory with further examples; are not estimating so don’t spot obvious arithmetical mistakes. HSNP © Hamilton 2012 Page 14 Shining Term 2 Week 4 Multiplication Day 1 Objectives: Use mental, written and calculator methods to multiply numbers together, including finding squares; Look for patterns and rules, make and test predictions You will need: Calculators Teacher input with whole class Ask pupils to work out 99 × 11, 99 × 22 and 99 × 33. they discuss in pairs what they notice and try and predict the next answer 99 × 11 = 1089 99 × 22 = 2178 99 × 33 = 3267 Take feedback. What patterns have they spotted? Can they explain this pattern? Discuss how the answers differ from 100 × 11, 100 × 22 and 100 × 33. What will the answer be to 99 × 77? How could you work this out? Pupils discuss in pairs, take feedback and discuss continuing the pattern on from 99 × 33 to 99 × 77, and also working out 100 × 77 and subtracting 77. Paired pupil work Pupils work in pairs to work out 999 × 2, 999 × 3, 999 × 4… until they spot a pattern and can predict the next product. 999 × 2 = 1998 999 × 3 = 2997 999 × 4 = 3996… They investigate the patterns in sequences of similar calculations such as 9999 × 2, 9999 × 3, 9999 × 4 and/or 999,999 × 2, 999,999 × 3, 999,999 × 4… Teacher input with whole class Ask pairs of pupils to report back what they found. Can they predict what the answers to 9999 × 2, 9999 × 3 and 9999 × 4 might be? HSNP © Hamilton 2012 Page 15 Shining Term 2 Week 4 Multiplication Day 2 Objectives: Use mental, written and calculator methods to multiply numbers together, including finding squares; Look for patterns and rules, make and test predictions You will need: Calculators Paired pupil work Pupils work out 112, 1112, 11112… until they spot a pattern and can predict the next answer in the sequences of squares. 112 = 121 1112 = 12321 11112 = 1234321 Teacher input with whole class Take feedback. What would 111,1112 be? How do you know? What digit will be in the middle? What about 111,111,1112? The answer is too big to fit on our calculators! But if we are sure of the pattern, we can predict the answer. Paired pupil work Ask pupils to find 152, 252 and 352. 152 = 225 252 = 625 352 = 1225 They discuss in pairs what they notice about the first two digits and last two digits. Teacher input with whole class Can they predict what 452 would be? How could they work out 652? Ask them to come up with a trick for squaring two-digit multiples of 5, e.g. multiply the first digit by the next number, e.g. if 35 multiply 3 by 4, then square 5 and put this on the end, e.g. to give 1225. Ask pupils to find out if their rule works for three-digit multiples of 5, e.g. 1252. (Yes!) HSNP © Hamilton 2012 Page 16 Shining Term 2 Week 4 Multiplication Day 3 Objective: Solve puzzles using factors You will need: Product doubles Sudoku at http://nrich.maths.org/6434/index click on ‘printable page’ and print out a page for each pair Teacher input with whole class Display Product doubles Sudoku. Explain that the numbers on the lines are products of numbers on either side. Explain how Sudoku grids work, namely that numbers 1 to 9 are used once in each square, and in each row and column. Help pupils to begin solving the puzzle, e.g. as below. Point to the right hand square on the middle row. What is the only number which could go between 8 and 18? Agree that 1 and 2 are common factors, but only 2 is possible. Write 4, 2 and 9 in this square. Point to 20 in the square below. What numbers from 1 to 9 have a product of 20? Agree that 5 must go on the left and 4 on the right as there is already a 4 in the column to the left. Discuss what numbers must lie on either side of 8 and 1 in the row below (1 and 8, not 4 and 2 as 4 is already in this square, and 1 will go with 4 on the left). Fill in 4 to the left of 4 on the line then 1 and 8 on either side of 8. Paired pupil work Ask pupils to continue working on the Sudoku in pairs. Teacher input with whole class Pause pupils in their work to take feedback. E.g. discuss the first square, how 5 and 6 need to go either side of 30, but as one must also be a factor of 18, 6 must go in the middle, so 5, 6 and 3 belong in the bottom row of that square in that order. Discuss how 1 and 9 belong in the first row of the bottom square on the left (we can’t have 3 × 3 as 3 would be repeated) and 9 must be in the middle as we need a factor of 18. Discuss the knock on effect of where 2 and 1 must lie on either side of 2 in the top left square. Paired pupil work Pupils continue to work in pairs. What numbers are left to go in the left three columns? Where can these go, given the numbers already in each of the three squares? HSNP © Hamilton 2012 Page 17 Shining Term 2 Week 5 overview Division Objectives Solve number puzzles Use mental, written and calculator methods to divide numbers Find prime numbers For this week you will need: Calculators, 100 squares (see resources), Remainder game at http://nrich.maths.org/6402 Link to homework page Watch out for pupils who: do not know their multiplication facts and so can’t make use of them to list or spot multiples; don’t realise that when they divide one number by another on a calculator and get a whole number, then the first number is divisible by the second; when they spot a pattern, do not test out their theory with further examples; are not estimating so don’t spot obvious arithmetical mistakes. HSNP © Hamilton 2012 Page 18 Shining Term 2 Week 5 Division Day 1 Objective: Solve number puzzles You will need: Remainders game at http://nrich.maths.org/6402 Teacher input with whole class Play the Remainders game. The computer will think of a number less than 100. Ask pupils to suggest what you might divide the number by to find out more about it. Enter the divisor and click ‘divide’. You will be given the remainder. Discuss what this tells you about the mystery number (e.g. odd or even if you divided by 2, what the units digit is if you divided by 10). Ask pupils to suggest another number to divide by. Be careful not to waste questions! For example if you divide by 6 and were told the remainder was zero, there is no point dividing by 3! Carry on until you can form a manageable list of possible numbers and then discuss what to divide by to eliminate some more. Carry on until pupils are sure what the mystery number is. Enter it and click ‘submit. Repeat. What numbers were good to divide by last time? (e.g. 10, as this cuts down the possibilities to just 10). Paired pupil work Pupils play the same game in pairs, taking it in turns to think of a number less than 100. Their partner asks them to divide it by suggested numbers, and they give the remainder. HSNP © Hamilton 2012 Page 19 Shining Term 2 Week 5 Division Day 2 Objectives: Use mental, written and calculator methods to divide numbers; Find prime numbers You will need: Calculators, 100 squares (see resources) Teacher input with whole class Explain that if pupils divide 6,469,693,230 by the first 10 prime number in turn, they will obtain a very special number! What’s the first prime number? And the next? How can we find the first 10 prime numbers? Paired pupil work Give each pair a 100 square and ask them to find the first 10 prime numbers, ringing them. Teacher input with whole class Display a 100 square and ask pupils up to the board to ring the first 10 prime numbers. Ask them to share how they came up with them. Paired pupil work Ask pupils to divide 6,469,693,230 by each prime number in turn. What answer do they get? (1!) 6469693230 ÷ 2 = 3234846615 3234846615 ÷ 3 = 1078282205 1078282205 ÷ 5 = 215656441 215656441 ÷ 7 = 30808063 30808063 ÷ 11 =2800733 2800733 ÷ 13 = 215441 215441 ÷ 17 = 12673 12673 ÷ 19 = 667 667 ÷ 23 = 29 29 ÷ 29 = 1 Teacher input with whole class How do you think someone came up with this amazing large number which when divided by the first 10 prime numbers gives one? Does it matter if we divide by the smallest prime number first or the largest? Work with a partner to come up with a number that when divided by the first three prime numbers gives an answer of 1. Divide to check. HSNP © Hamilton 2012 Page 20 Shining Term 2 Week 5 Division Day 3 Objective: Use mental, written and calculator methods to divide numbers You will need: Calculators Paired pupil work Ask pupils to divide 2521 by numbers 2 to 10 and find the remainder each time. What do they notice? Pupils discuss in pairs how they think someone came up with this number (e.g. finding 2520 as a common multiple of 2 to 10, then adding 1, the lowest common multiple in fact). Ask pupils to use this to come up with other numbers which always give a remainder of 1 (e.g. double 2520, then add 1 to see if that works.) They divide them to test them out. Take feedback. Teacher input with whole class Choose any 3 digits and make a 6 digit number by repeating the 3 digits in the same order (e.g. 265,265). Whichever digits you choose, the number will always be divisible by three prime numbers less than 20. Paired pupil work Pupils work in pairs to find out what the three numbers are! (7, 11 and 13). They use a calculator to help. Ensure that they realise that if a number is divisible by another, then the answer will be a whole number. Of course they will make numbers which are divisible by other numbers but must find the three that are in common. HSNP © Hamilton 2012 Page 21 Shining Term 2 Week 6 overview Decimals Objectives Know common fractions and decimals equivalents Use division on a calculator to convert fractions to decimals Understand terminating, recurring and non-recurring decimals Investigate patterns For this week you will need: Calculators, Decimals four in a line grid (see resources), an IWB calculator, e.g. the one at http://www.crickweb.co.uk/ks2numeracytools.html#Toolkit%20index2a, Link to homework page Watch out for pupils who: think that 0.2 is equivalent to one half, 0.3 to one third, 0.4 to one quarter, 0.5 to one fifth etc; write 3.450 when multiplying 3.45 by 10 saying that we add a zero when multiplying by 10; encourage them to see how the value of each digit changes according to its place in a number and that whilst adding a zero is a useful shortcut, it only applies to whole numbers. HSNP © Hamilton 2012 Page 22 Shining Term 2 Week 6 Decimals Day 1 Objectives: Know common fractions and decimals equivalents; Use division on a calculator to convert fractions to decimals; Understand terminating, recurring and non-recurring decimals You will need: Calculators, Decimals four in a line grid (see resources) Teacher input with whole class Ask pupils to use their calculators to divide 1 by 2. Before they do, what do they think will be answer and why? Agree that 0.5 is the decimal equivalent to 1/2, and the answer to 1 divided by 2. Repeat for 1/3. Discuss the recurring nature of this decimal and remind pupils that recurring decimals repeat forever. Paired pupil task Pupils divide 1 by 4, 5…10 predicting the answers where they can. They record the equivalent fractions and decimals in a list. Teacher input with whole class Take feedback and discuss which are terminating decimals and which are recurring decimals. Remind pupils how to record dots or lines above the recurring digits. Say how one third and two thirds can be rounded to 0.33 and 0.67 as approximate equivalents. Explain that decimal equivalents to fractions are always either terminating decimals or recurring decimals. But some numbers (irrational numbers which can’t be expressed as fractions) such as √2 and Pi are non-recurring decimals, i.e. they go on forever, but there is no repeating pattern of digits. Display the Decimals grid and play Four in a line. Divide the class into 4 teams, assign a colour to each. Each team take turns to choose a decimal from the grid, and say its fraction equivalent, using their list of unit fractions and decimal equivalents to help. If necessary, use a calculator to check. If correct, ring the chosen decimal in their colour. Carry on playing until one team has four ringed numbers in a line. 0.125 1.25 0.375 1.8 0.6 1.5 0.5 0.25 0.67 0.1 0.75 0.875 0.9 1.75 0.8 0.7 0.4 0.33 0.2 0.625 HSNP © Hamilton 2012 Page 23 Shining Term 2 Week 6 Decimals Day 2 Objectives: Use division on a calculator to convert fractions to decimals; Understand terminating, recurring and non-recurring decimals You will need: Calculators Teacher input with whole class Divide 13 by 11 and write the answer on the board, without telling pupils what division you did to give this decimal. 1.1818181818…. Discuss what sort of decimal this is (recurring). Say that this decimal is the equivalent decimal to a fraction, and to give this decimal, you divided the numerator by the denominator, both being whole numbers less than 20. Ask pupils to discuss in pairs what numbers they might be and what they couldn't be. Take feedback. Agree that as it is a recurring decimal, you did not divide by 2, 4, 5, 8 or 10 as these denominators give terminating decimals. Also it wasn’t a single digit denominator as we don’t recognise the decimal part of the number as one of those we looked at on Day 1. Paired pupil work Ask pupils to try out some numbers on their calculators. Teacher input with whole class Take feedback about what they have found out so far. What can you say about the numerator compared to the denominator? E.g. the numerator is more than denominator, but not as much as twice the denominator as the number begins with 1. It looks as if the numerator is only a little more than the denominator. Paired pupil work Pupils carry on and work out the equivalent fraction. If they find the answer, they choose another pair of numbers and ask another pair to work out what numbers they divided. HSNP © Hamilton 2012 Page 24 Shining Term 2 Week 6 Decimals Day 3 Objectives: Use division on a calculator to convert fractions to decimals; Investigate patterns You will need: Calculators Paired pupil work Ask pupils to divide 1, 2, 3, 4, 5 and 6 by 7, and to write the equivalent fractions and decimals. They work in pairs to investigate what they notice about the patterns of recurring digits in each. For example, they may see that each has the same repeating pattern of digits, but starting in a different place. These could be re-ordered as follows to make the pattern more obvious: 1/7 = 0.142857… 3/7 = 0.428571… 2/7 = 0.285714… 6/7 = 0.857142… 4/7 = 0.571428… 5/7 = 0.714285… Is there anything interesting about the sum of the recurring digits? They have a digit sum of 9. Also the 1st and 4th digits have a digit sum of 9, the 2nd and 5th, 3rd and 6th. Or the difference between consecutive digits? (3, 2, 6, 3, 2, 6…) Treat the first three digits of 1/7 as a number and add to the second three, i.e. 142 + 857 = 999. Take each pair and add, 14 + 28 + 57 = 99. Pupils may want to carry on and divide 8, 9, 10, 11, 12 and 13 by 7 and see if the patterns continue. Suggest they add the decimal equivalents of fractions with a total of 1, e.g. 0.142857… + 0.857142. What do they notice? Why is this the case? Teacher input with whole class Ask pairs to feedback and share their findings. HSNP © Hamilton 2012 Page 25 Shining Term 2 Week 7 overview Addition patterns Objectives Add several numbers mentally Identity patterns For this week you will need: A3 paper, calculators Link to homework page Watch out for pupils who: are not fluent in adding two-digit numbers; when they spot a pattern, do not test out their theory with further examples; are not concerned with accuracy as one mistake will mean the rest of the sequence is out! HSNP © Hamilton 2012 Page 26 Shining Term 2 Week 7 Addition patterns Day 1 Objective: Add several numbers mentally; Identify patterns You will need: Calculators Teacher input with whole class Tell pupils a little about Pascal’s triangle, how it was named after Pascal, although discovered much earlier in various places around the works including India and China. In the western world it is usually named after Pascal who wrote much about it. Draw the first few lines of the triangle on the board and explain how the two numbers above give the numbers between below (think of 0 off the triangles to give 1 on the outer edges): 1 1 1 1 2 1 1 3 3 1 Paired pupil work Ask pupils to continue the triangle, recording at least 10 rows. They discuss in pairs what patterns they notice. Teacher input with whole class Discuss what patterns pupils can see in the triangle, e.g. symmetry, diagonal line of numbers 1, 2, 3… diagonal lines of triangular numbers. Paired pupil work Ask pupils to look at the total of each row and discuss what they notice. Teacher input with whole class Take feedback and discuss the results, number totals 1, 2, 4, 8, 16… and point out if necessary that not only do the totals double each time but there are all powers of 2 (20, 21, 22, 23…). You could also ask pupils to work out powers of 11 and see what they notice. HSNP © Hamilton 2012 Page 27 Shining Term 2 Week 7 Addition patterns Day 2 Objectives: Add several numbers mentally; Identify patterns You will need: A3 paper, calculators Paired pupil work Ask pupils to work in pairs to draw Pascal’s triangle on a sheet of A3 paper. Emphasise the importance of accuracy both in terms of arithmetic, but also in trying to keep the shape for the triangle. Neatness will be important! They should complete as many rows as will fit on the paper. Ask pupils to add 1 and 3 from the diagonal sequence of triangular numbers, jot down the totals, then add 3 and 6, 6 and 10, etc. What do they notice? Ask pupils to draw a diagonal line from the outside of the triangle, find the totals of the numbers and look to see if they can see this total nearby in the triangles. These lines are sometimes called hockey sticks or Christmas stockings! Pupils could also ring all the even numbers in the triangle. What pattern is formed? Teacher input with whole class Take feedback. Pupils share what they discovered. HSNP © Hamilton 2012 Page 28 Shining Term 2 Week 7 Addition patterns Day 3 Objectives: Add several numbers mentally; Identify patterns You will need: Calculators Teacher input with whole class Tell pupils a little about Fibonacci (nickname for Leonardo de Pisa) and the history of the Fibonacci sequence as a result of solving a classic maths problem about rabbit breeding, an idealised problem in which a pair of rabbits give birth to a pair each month. The question was how many rabbits there would be after one year. The problem assumes no rabbits die; all females have two rabbits, one of each sex each month. You could show them diagrams for this, e.g. at http://www.maths.surrey.ac.uk/hostedsites/R.Knott/Fibonacci/fibnat.html. Write the first few terms of the sequence and ask pupils to discuss how each term is generated: 0, 1, 1, 2, 3, 5, 8, 13, 21… Paired pupil work Ask pupils to continue the sequence to at least the 20th term, using mental arithmetic to begin with, and then written methods/calculators when the numbers become substantially larger. They discuss any patterns that they notice. Suggest that they look at the pattern of odd and even numbers. What if they start the sequence and two other consecutive numbers, e.g. 3, 4 giving 3, 4, 7, 11, 18…? Do they get the same pattern of odd and even numbers? Why? Pupils could also look at where the multiples for 3, 5 and 8 lie in the sequence (i.e. multiples of the numbers in the sequences.) Teacher input with whole class Take feedback. Pupils share what they discovered. HSNP © Hamilton 2012 Page 29 Shining Term 2 Week 8 overview Positive and negative integers Objectives Add positive and negative integers Find the difference between negative integers Explore patterns in adding and subtracting negative numbers For this week you will need: -10 to 10 number line (see resources), playing cards, Connect 3 at http://nrich.maths.org/5911 Link to homework page Watch out for pupils who: are confused by adding and subtracting negative numbers. Use images such as the number line to help them find difference and use contexts such as temperature or debt; subtract the absolute value when subtracting pairs of negative numbers, rather than thinking of finding the difference between the two numbers. HSNP © Hamilton 2012 Page 30 Shining Term 2 Week 8 Positive and negative integers Day 1 Objective: Add positive and negative integers You will need: -10 to 10 number line (see resources), playing cards Teacher input with whole class Display the -10 to 10 number line. Write -5 + 8 underneath, remind pupils how they can start at -5 and add on 8 to reach 3. Write 8 + (-5) and discuss how this has the same answer as addition is commutative and also has the same answer as 8 – 5 as adding a negative number is the same as subtracting its (absolute) value. Repeat with 3 + (-5) and agree that this time we end up with a negative number, the same answer as that to -5 + 3. Record -3 + -2 and agree if we add these two negative numbers we end up with another negative number, -5. If you owe someone £3 and someone else £2, you owe £5, so if you don’t have any pocket money left you have minus or negative £5! Paired pupil work Pupils shuffle a set of playing cards having removed the Jokers, Jacks, Queens and Kings. They turn over the top two cards. Black cards are positive numbers and red are negative. The first to say the total wins the pair of cards. Continue until all cards are gone. Who won most cards? Teacher input with whole class Remind pupils of the Fibonacci sequence: 0, 1, 1, 2, 3, 5, 8… begin to extend this in the other direction, i.e.: 3, -2, 1, -1, 0, 1, 1, 2, 3, 5, 8, 11 Discuss how you are thinking what must be added to the new first term to the next, e.g. what must be added to 3 to make -2, answer -5., So what must be added to -5 to make 3? Paired pupil work Challenge pupils to work in pairs to extend the Fibonacci sequence through zero. HSNP © Hamilton 2012 Page 31 Shining Term 2 Week 8 Positive and negative integers Day 2 Objective: Find the difference between negative numbers You will need: -10 to 10 number line (see resources), playing cards, Connect 3 at http://nrich.maths.org/5911 Teacher input with whole class Display the -10 to 10 number line. Ring -8 and -5. What is the difference between these two numbers? LOOK carefully at the line. Record -8 – (-5) = 3. If the overnight temperature in Glasgow was -10°C and in London was -2°, what was the difference in temperature? Record -10 – (-2) = 8. Repeat with other pairs of negative numbers and ask pupils to record the subtraction each time. Ring one positive number and one negative number, e.g. 3 and -6. What is the difference between these two numbers? Look at the line, What subtraction can we write? 3 –(-6) Repeat with other pairs of numbers. Paired pupil work Pupils shuffle a set of playing cards having removed Jokers, Jacks, Queens, Kings. They turn over the top two cards. Black cards are positive numbers and red are negative. The first to say the difference wins the pair of cards. Continue until all cards are gone. Who won most cards? Teacher input with whole class Play Connect 3, choosing to play against the computer or one team against the other. Click on roll the dice, then drag the numbers and subtraction sign into the equation. When you are happy, click on ‘place counter’. The aim is to connect three counters. HSNP © Hamilton 2012 Page 32 Shining Term 2 Week 8 Positive and negative integers Day 3 Objective: Explore patterns in adding and subtracting negative integers You will need: none required Teacher input with whole class As a class, list all the combinations of adding and subtracting positive and negative integers: Positive add positive Negative add negative Positive add negative Negative add positive Positive subtract positive Negative subtract negative Positive subtract negative Negative subtract positive Paired pupil work Ask pupils to work in pairs to investigate the possible answers to each, assigning one possibility to each group, except the first. We don’t need a group to work on positive add positive! They should try out different numbers to see if it is possible to get negative answers, positive answers or either, and if either what affects the answer. Teacher input with whole class Take feedback from each group and record their findings by the side of each group. For example, negative add positive depends on the absolute value of each number, so -3 + 8 gives a positive answer but -8 + 3 gives a negative answer. HSNP © Hamilton 2012 Page 33 Shining Term 2 Week 9 overview Multiplication Objectives Multiply two-digit numbers by two-digit numbers using a written method Multiply three and four-digit numbers by two-digit numbers using a written method Multiply numbers with one or two decimal places using a written method Explore factorials Explore patterns and find rules For this week you will need: Calculators, an IWB calculator, e.g. the one at http://www.crickweb.co.uk/ks2numeracy-tools.html#Toolkit%20index2a Link to homework page Watch out for pupils who: do not know their times tables. This lack of knowledge will really slow down their work in multiplication and division so use Simmering Term 1 or 2, week 4, day 1’s activities with tables as necessary; encourage them to turn the multiplication round, e.g. if they don’t know nine 6s, to use six 9s, or to use doubling, e.g. double four 6s to find eight 6s; make place value errors when multiplying decimals, e.g. 6 × 0.7 = 0.42, or 0.3 × 0.2 = 0.6; encourage them to estimate the answer first; they might find it helpful to multiply 0.3 by 2 for example, and then divide by 10 to find 0.3 × 0.2. HSNP © Hamilton 2012 Page 34 Shining Term 2 Week 9 Multiplication Day 1 Objectives: Multiply two-digit numbers by two-digit numbers using a written method; Explore patterns and find rules You will need: none required Teacher input with whole class Ask pupils to work in pairs to use a written method, e.g. the grid method, one to work out 12 × 42 and the other to reverse the digits in each number and work out 21 × 24. What do they notice? Repeat with 12 × 84 and 21 × 48. Ask them to compare their grid layouts with each other. What do they notice? Now ask them to try 25 × 63 and 52 x 36. Oh, it doesn’t work this time. I wonder why…. Paired pupil work Pupils work in pairs to try other examples and try and work out why some pairs of numbers have the same product when their digits are reversed and some do not. Encourage them to compare the numbers in their grid to give them a clue. Teacher input with whole class Take feedback about what pupils have found. Draw out that if the product of the tens digits and the product of the ones digits are the same then the product of the numbers will be the same too. E.g. 12 × 42, 1 × 4 = 2 × 2. show 23 × 96 = 32 × 69: × 20 3 90 1800 270 6 120 18 1920 + 288 2208 × 30 2 60 1800 120 9 270 18 2070 + 138 2208 Paired pupil work Ask pupils to work in pairs to find other pairs of numbers which have the same product when reversed. They should use their written method to check. Teacher input with whole class Together as a class make a list of examples, and use it to generate others. HSNP © Hamilton 2012 Page 35 Shining Term 2 Week 9 Multiplication Day 2 Objective: Multiply three- and four-digit numbers by two-digit numbers using a written method; Multiply numbers with one or two decimal places using a written method You will need: an IWB calculator, e.g. the one at http://www.crickweb.co.uk/ks2numeracy-tools.html#Toolkit%20index2a Teacher input with whole class Write the following multiplications on the board and ask pupils to agree a ranking from easy to hard, writing them in order on their whiteboards: 68 × 37, 4200 × 30, 4.6 × 23, 7868 × 57, 2.6 × 4.2, 5,000,000 × 3, 65.34 × 37, 347 × 6, 2.111 × 4, 6.7 × 2 Paired pupil work Each pair compares their list with another pair and justifies their order. The four pupils divide the calculations between them and work out the multiplications using the grid method. Teacher input with whole class Take feedback and discuss what makes multiplication easier or harder, which could pupils work out in their heads, and for which did they need to use jottings or decomposition. Draw out that it is not necessarily just the number of digits (as 5,000,000 × 3 is easy!) or those with decimals which are more difficult (6.7 × 2 is easy too!) although they may find those with decimal places in both numbers more difficult, and also those with more digits in both numbers, particularly with higher digits making the multiplication and addition a bit more tricky. Ask each group of four to come up with a really easy multiplication with lots of digits that they think the rest of the class can work out before you can work out the answer on a calculator. They show the multiplication on a whiteboard to you and the class. If you can use a calculator to work out the answer first, you win a point. If the class work out the answer first, they win a point. Repeat for each four. Who won, teacher or class?! Challenge each four to come up with a multiplication where they would really prefer to use a calculator! They must explain why. HSNP © Hamilton 2012 Page 36 Shining Term 2 Week 9 Multiplication Day 3 Objective: Explore factorials You will need: calculators Teacher input with whole class Remind pupils that 5! five, factorial, is shorthand for 5 × 4 × 3 × 2 × 1. Work out 1!, 2!, 3! and, so on. How quickly do you think the multiplication will get to the stage where you need to use a calculator? Do you think the answers will get too big to fit all the digits in the display? If so, when do you think that will happen? Take suggestions. Paired pupil work Pupils work out factorials 1! 2! 3!... until all the digits no longer fit in the display. Teacher input with whole class Take feedback. Were pupils surprised at how quickly the answers became very large? Ask three pupils to stand at the front. Let’s find out how many ways three people can stand in a row. We have a choice of three people to stand on the left. Ask one person to stand on the left. Now we have a choice of two people to stand next in line, and then we will on have one person left, so there is no choice who goes third. So with that person first, we have two ways of ordering people. We will have the same number of ways with each of the other two people first, so we have 3 × 1 ways of ordering 3 people. Record their initials to show the six ways, e.g. ABC ACB BAC BCA CAB CBA Discuss how the problem changes with four pupils. How many choices for the first spot? And then the possibilities for the remaining spots are as above. So there are four lots of the above, i.e. 4 × 3 × 2 × 1, 4! How many ways do you think there are of ordering five pupils? HSNP © Hamilton 2012 Page 37 Shining Term 2 Week 10 overview Division Objectives Find factors of numbers Find prime factors Find patterns and rules Explore perfect and amicable numbers For this week you will need: Calculators, Factor tree at http://www.softschools.com/math/factors/factor_tree/ Link to homework page Watch out for pupils who: do not know their multiplication facts and so don’t recognise multiples; when they spot a pattern, do not test out their theory with further examples; do not recognise small prime numbers. HSNP © Hamilton 2012 Page 38 Shining Term 2 Week 10 Division Day 1 Objectives: Find factors of numbers; Find patterns and rules You will need: none required Teacher input with whole class Remind pupils how to find factors of 16, in an ordered way, e.g. 1, 16; 2, 8; 4, 4. Remind them that they can stop trying to find factors when they get to the square root of a number. Individual practice Ask pupils to work out the factors of numbers 1 to 20, recording their results in a table. They then ring numbers whose factors are half even and half odd, e.g. 14 with factors 1, 2, 7 and 14. Teacher input with whole class What do you notice about the numbers whose factors are half odd and half even? Which number do you think will be the next to have factors which are half odd and half even? Paired pupils work Pupils predict and test out other numbers which they think might have factors which are half odd and half even. When they are able to predict which numbers have factors like this, challenge them to come up with a general rule so that given any number, they could say whether or not it will have factors which are half even and half odd. Teacher input with whole class Take feedback, asking pairs to explain their rules to the rest of the class. For example they may say that the numbers are all multiples of 4 plus 2, multiples of 4 – 2, (2, 6, 10, 14, 18), some may describe this using an algebraic expression, 4n + 2 or 4n - 2. So do you think 100 will have half odd and even factors? 101? 102? 103? Pupils try out the numbers which they think fit the pattern. HSNP © Hamilton 2012 Page 39 Shining Term 2 Week 10 Division Day 2 Objective: Find prime factors You will need: Factor tree at http://www.softschools.com/math/factors/factor_tree/ Teacher input with whole class Click on ‘user number’ and enter 36 into the Factor tree. Ask a pupil to enter a factor, e.g. 9. Press enter and the corresponding factor will be given and boxes to enter factors of 9 and 4 for example. Ask different pupils up to the board to enter factors until there are only prime factors. Ask pupils to check that 3 × 3 × 2 × 2 does equal 36. Repeat for 36, but asking a pupil to enter a different first factor, e.g. 6. Continue the process to show that the factor tree will give the same prime factors, 2, 3, 3, 2, just in a different order. Paired pupil work Ask pupils to work in pairs to choose other numbers less than 100 and draw their own factor trees. Challenge them to find numbers with other prime factors than 2, 3 and 5 which are very common! Teacher input with whole class Take feedback. HSNP © Hamilton 2012 Page 40 Shining Term 2 Week 10 Division Day 3 Objective: Explore perfect and amicable numbers You will need: Calculators Teacher input with whole class Ask pupils to list the factors of 6 less than itself and to add them. What do they notice?! Explain that because the sum of the factors less than itself is 6, 6 is called a perfect number. Paired pupil practice Challenge pupils to work in pairs to find the next perfect number. Say that it is less than 40 and recommend that they divide the work up between them. Teacher input with whole class Did they find it? 28= 1 + 2 + 4 + 7 +14. Paired pupil practice Challenge pupils to find all the factors of 220 and 284 less than themselves, and then to add them up. (They may find it helpful to use calculators to check divisibility.) Explain that these numbers are called amicable numbers. Can they find out why? Teacher input with whole class Take feedback. Factors of 220 less than 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, and the sum is 284. Factors of 284 less than 284 are 1, 2, 4, 71 and 142, and the sum is 220. So they are the total of each other’s factors! Say that the next pair of numbers like this are 1184 and 1210! Amicable numbers were known to the Pythagoreans, who credited them with many mystical properties! HSNP © Hamilton 2012 Page 41 Shining Term 2 HSNP © Hamilton 2012 Page 42 Shining Term 2 Hamilton Secondary Numeracy Project Shining Term 2 Homework Name ___________________________ HSNP © Hamilton 2012 Page 43 Shining Term 2 Week 1 Fractions Play Math matching at http://www.harcourtschool.com/activity/con_math/con_math.html. Click on matching fractions and decimals to reveal the pictures in the least number of moves that you can. Use a calculator to divide the numerator by the denominator to find the decimal equivalents of 1/9, 1/99, 1/999… What do you notice? Can you predict the next in the pattern? Solve the following by thinking of how many halves are in 20 for example: 20 ÷ 1/2 6 ÷ 1/4 3 ÷ 1/8 7 ÷ 1/6 30 ÷ 1/5 Roll a 0-9 dice twice to create a fraction (if you roll 0, roll again). Multiply the fraction by 2/3. □ × 2 □ 3 How many can you do in two minutes? Repeat, this time dividing each fraction by 1/4. HSNP © Hamilton 2012 Page 44 Shining Term 2 Week 2 Addition of fractions Play Fractone at http://www.coolmath-games.com/0- fractone/index.html. Choose ‘pretty good’ or ‘I’m going for it’! Click on pairs of fractions with a total of 1 as quickly as you can. Sometimes you will need to click on pairs with different denominators, e.g. 4/8 and 1/2. What was your time? Take the Jacks, Queens, Kings and Jokers out of a deck of playing cards. Shuffle what’s left. Take two and make a fraction less than one. Repeat to make another fraction. Find the total using ‘smile and kiss’. Repeat until you have eight additions with a total of between 1 and 2. Play Fruit shoot fractions additions at http://www.sheppardsoftware.com/mathgames/fractions/FruitShootF ractionsAddition.htm. Choose level 3 and relaxed mode. Add the pair of given fractions and click on the fruit with the answer. Record the addition and answer in your homework book. What was your score? HSNP © Hamilton 2012 Page 45 Shining Term 2 Week 3 Subtraction of fractions Play Fruit shoot fractions subtractions at http://www.sheppardsoftware.com/mathgames/fractions/FruitShootF ractionsSubtraction.htm. Choose level 3 and relaxed mode. Subtract the pair of given fractions and click on the fruit with the answer. Record the subtraction and answer in your homework book. What was your score? Roll two dice (or roll one twice) to generate a fraction less than one. Use ‘smile and kiss’ to subtract this fraction from 8/9. How many can you do in five minutes? 8 − 4 9 6 Play Adding and subtracting fractions challenge at http://www.math- play.com/adding-and-subtracting-fractions-game.html. Click to roll the dice, then draw a card. Add or subtract the fractions. You win or lose points for each right or wrong answer. Carry on until you reach the finish. What was your score? HSNP © Hamilton 2012 Page 46 Shining Term 2 Week 4 Multiplication Play the video about Alex’s number plumber at http://nrich.maths.org/8387. Click on the picture below and enter the same number as on the video. Keep pressing ‘drop’ so that the last output becomes the next input. Click on ‘results table’ on the far right. What do you notice about the final digits of each number? Choose your own number to enter and see what happens. Keep reentering the output as the next input and look at the results table. Can you predict the pattern for a new number? Record what you find. Find the squares of the following numbers: 12, 23, 34, 45 and 56. Find the digital root of each answer, e.g. for 1156 add 1, 1, 5 and 6 to give 13, then add 1 and 3 to give the single-digit number 4. What do you notice about the digital roots of these numbers? Try other two-digit numbers with consecutive digits. Does it matter whether the larger digit is first or second? Try three-digit numbers with consecutive digits, e.g. 234. HSNP © Hamilton 2012 Page 47 Shining Term 2 Week 5 Division Work to find the biggest four-digit number that you can that is divisible by each of its digits. Each digit must be different. E.g. 1236 is divisible by 1, 2, 3 and 6, but I’m sure you can do better than that! Two people are thinking of the same number less than 100. One divides it by 3 and gets a remainder of 1; the other divides it by 20 and gets a remainder of 3. What is the number? Make up a similar puzzle to try on someone else, try and make sure that there is only one answer. HSNP © Hamilton 2012 Page 48 Shining Term 2 Week 6 Decimals Play Fruit shoot at http://www.sheppardsoftware.com/mathgames/fractions/FractionsTo Decimals.htm. Start with level 4. Click on fruits with decimal equivalents to the given fractions. Record your score. Now have a go at level 5! Use a calculator to find the decimal equivalents for 1/13, 2/13, 3/13… 11/13. Is there a pattern of recurring digits? Which fractions have the same pattern? What do you notice about the sum of the digits? Can you find any other interesting digit sums? Explore any other patterns you find! Write up what you discovered, your teacher will be interested to read what you found! HSNP © Hamilton 2012 Page 49 Shining Term 2 Week 7 Addition patterns Carry on Pascal's triangle so that you have at least 12 rows. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 Look at the second number in each row. Does this go into each number in the row apart from 1? Look to see if there is pattern for this rule. Use the internet to research Fibonacci sequences and spirals in nature. Write about one fact which you found interesting. This sequence is a twist on Fibonacci sequence. The fourth number is the sum of the first three numbers, the fifth number is the sum of the previous three numbers and so on. Continue the sequence so that you write at least 15 numbers. 0, 1, 1, 2, 4, 7, 13, 24… What patterns can you find? Is there a pattern of odd and even numbers? HSNP © Hamilton 2012 Page 50 Shining Term 2 Week 8 Positive and negative integers Josh says if you subtract a positive number from a positive number, you will always get a positive answer. What do you think? Explain your thinking with some examples. Roll a 0 to 9 dice and flip a coin to determine whether the number is positive (heads) or negative (tails). Repeat, then find the difference between the two numbers. Draw a number line jotting if it helps. Record five subtractions. 7 -7 5 0 5 5 – (-7) = 12 Play Walk the plank at http://www.math-play.com/integersgame.html. Choose attributes for the person you want to walk the plank. You will be asked a question, click on each pirate to view their answers and choose the one you think is right. If correct, you’ll be asked to click on the dice to move the person forward on the plank. Click for the next question. Carry on until the game is complete. Were you successful in getting the person to walk the plank?! HSNP © Hamilton 2012 Page 51 Shining Term 2 Week 9 Multiplication Work out 1! 2! 3!...10! and record the answers. 1! 1 2! 2 × 1 = 3! 3 × 2 × 1 = 4! 4 × 3 × 2 × 1 = … Remember to use your previous answer to help work out the next one. Are the answers odd or even? Why? Are you convinced that all further factorials will be even? Look at the digital roots, e.g. for 9! 3 + 6 + 2 + 8 + 8 + 0 = 27, 2 + 7 = 9, so the digital root is 9. What do you notice about the digital roots? What do you think will happen for further factorials? Why? What do you notice about the digital roots of multiples of 9? Does this help? Use a written method, e.g. the grid method, to work out the following multiplications: 12 × 21 23 × 32 34 × 43 45 × 54 56 × 65 67 × 76 78 × 87 89 × 98 Work out the digital root for each. Can you spot any patterns? HSNP © Hamilton 2012 Page 52 Shining Term 2 Week 10 Division The factors of 48 are: 1 and 48 2 and 24 3 and 16 4 and 12 6 and 8 If we add all the factors less than 48, we get 76: 1+2+3+4+6+8+12+16+24=76 48 is called an abundant number because it is less than the sum of its factors (without itself). 32 has factors 1, 2, 4, 8 and 16 (apart from 32) and the sum of these factors is 31, so 32 is not an abundant number See if you can find some more abundant numbers! Use a factor tree to find the prime factors of five two-digit numbers. Try and make the biggest tree that you can! 24 6 3 HSNP © Hamilton 2012 4 2 2 Page 53 2 Shining Term 2 HSNP © Hamilton 2012 Page 54 Shining Term 2 Hamilton Secondary Numeracy Project Numeracy Shining Term 2 Homework answers HSNP © Hamilton 2012 Page 55 Shining Term 2 Week 1 Fractions Play Math matching at http://www.harcourtschool.com/activity/con_math/con_math.html. Click on matching fractions and decimals to reveal the pictures in the least number of moves that you can. This is a computer game. Use a calculator to divide the numerator by the denominator to find the decimal equivalents of 1/9 0.111111, 1/99 0.01010101, 1/999 0.001001001001… What do you notice? Can you predict the next in the pattern? Solve the following by thinking of how many halves are in 20 for example: 20 ÷ ½ = 40 6 ÷ ¼ = 24 3 ÷ 1/8 = 24 7 ÷ 1/6 = 42 30 ÷ 1/5 = 150 Roll a 0-9 dice twice to create a fraction (if you roll 0, roll again). Multiply the fraction by 2/3. □ × 2 □ 3 How many can you do in two minutes? Repeat, this time dividing each fraction by 1/4. Examples:- 2/5 x 2/3 = 4/15; 3/6 x 2/3 = ½ x 2/3 = 2/6 = 1/3 2/5 ÷ ¼ = 8/5; 3/6 ÷ ¼ = 12/6 = 2 HSNP © Hamilton 2012 Page 56 Shining Term 2 Week 2 Addition of fractions Play Fractone at http://www.coolmath-games.com/0- fractone/index.html. Choose ‘pretty good’ or ‘I’m going for it’! Click on pairs of fractions with a total of 1 as quickly as you can. Sometimes you will need to click on pairs with different denominators, e.g. 4/8 and 1/2. What was your time? This is a computer game. Take the Jacks, Queens, Kings and Jokers out of a deck of playing cards. Shuffle what’s left. Take two and make a fraction less than one. Repeat to make another fraction. Find the total using ‘smile and kiss’. Repeat until you have eight additions with a total of between 1 and 2. Examples:- 4/5 + 2/6 = 24/30 + 10/30 = 34/30 = 1 2/15 2/9 + 4/5 = 10/45 + 36/45 = 46/45 = 1 1/45 3/6 + 7/8 = 24/48 + 42/48 = 66/48 = 11/8 = 1 3/8 Play Fruit shoot fractions additions at http://www.sheppardsoftware.com/mathgames/fractions/FruitShootF ractionsAddition.htm. Choose level 3 and relaxed mode. Add the pair of given fractions and click on the fruit with the answer. Record the addition and answer in your homework book. What was your score? This is a computer game. HSNP © Hamilton 2012 Page 57 Shining Term 2 Week 3 Subtraction of fractions Play Fruit shoot fractions subtractions at http://www.sheppardsoftware.com/mathgames/fractions/FruitShootF ractionsSubtraction.htm. Choose level 3 and relaxed mode. Subtract the pair of given fractions and click on the fruit with the answer. Record the subtraction and answer in your homework book. What was your score? This is a computer game. Roll two dice (or roll one twice) to generate a fraction less than one. Use ‘smile and kiss’ to subtract this fraction from 8/9. How many can you do in five minutes? Examples:- 8/9 – 4/6 = 48/54 – 36/54 = 12/54 = 2/9 8/9 – 3/5 = 40/45- 27/45 = 13/45 8/9 – 1/6 = 48/54 – 9/54 = 39/54 = 13/18 Play Adding and subtracting fractions challenge at http://www.math- play.com/adding-and-subtracting-fractions-game.html. Click to roll the dice, then draw a card. Add or subtract the fractions. You win or lose points for each right or wrong answer. Carry on until you reach the finish. What was your score? This is a computer game. HSNP © Hamilton 2012 Page 58 Shining Term 2 Week 4 Multiplication Play the video about Alex’s number plumber at http://nrich.maths.org/8387. Click on the picture below and enter the same number as on the video. Keep pressing ‘drop’ so that the last output becomes the next input. Click on ‘results table’ on the far right. What do you notice about the final digits of each number? This is a computer game. Find the squares of the following numbers: 12, 23, 34, 45 and 56. Find the digital root of each answer, e.g. for 1156 add 1, 1, 5 and 6 to give 13, then add 1 and 3 to give the single-digit number 4. What do you notice about the digital roots of these numbers? Try other two-digit numbers with consecutive digits. Does it matter whether the larger digit is first or second? Try three-digit numbers with consecutive digits, e.g. 234. 12² = 144, digital root = 9 21² = 441, digital root = 9 23² = 529, digital root = 7 32² = 1024, digital root = 7 34² = 1156, digital root = 4 43² = 1849, digital root = 4 45² = 2025, digital root = 9 54² = 2916, digital root = 9 56² = 3136, digital root = 4 65² = 4225, digital root = 4 67² = 4489, digital root = 7 76² = 5776, digital root = 7 HSNP © Hamilton 2012 Page 59 Shining Term 2 Week 5 Division Work to find the biggest four-digit number that you can that is divisible by each of its digits. Each digit must be different. E.g. 1236 is divisible by 1, 2, 3 and 6, but I’m sure you can do better than that! 9864 is divisible by 9, 8, 6 and 4, this is the largest possible. Two people are thinking of the same number less than 100. One divides it by 3 and gets a remainder of 1; the other divides it by 20 and gets a remainder of 3. What is the number? The answer is 43. HSNP © Hamilton 2012 Page 60 Shining Term 2 Week 6 Decimals Play Fruit shoot at http://www.sheppardsoftware.com/mathgames/fractions/FractionsTo Decimals.htm. Start with level 4. Click on fruits with decimal equivalents to the given fractions. Record your score. Now have a go at level 5! This is a computer game. Use a calculator to find the decimal equivalents for 1/13, 2/13, 3/13… 11/13. 0.0769230769 0.1538461538 0.2307692307 0.3076923076 0.3846153846 0.4615384615 0.5384615384 0.6153846153 0.6923076923 0.7692307692 0.8461538461 Is there a pattern of recurring digits? Which fractions have the same pattern? What do you notice about the sum of the digits? Can you find any other interesting digit sums? Explore any other patterns you find! Write up what you discovered, your teacher will be interested to read what you found! HSNP © Hamilton 2012 Page 61 Shining Term 2 Week 7 Addition patterns Carry on Pascal's triangle so that you have at least 12 rows. 1 1 1 1 2 1 1 3 3 1 1 4 6 4 1 1 5 10 10 5 1 1 6 15 20 15 6 1 1 7 21 35 35 21 7 1 1 8 28 56 70 56 28 8 1 1 9 36 84 126 126 84 36 9 1 Look at the second number in each row. Does this go into each number in the row apart from 1? Look to see if there is a pattern for this rule. Yes, Yes, No, Yes, No, Yes, No, No etc Use the internet to research Fibonacci sequences and spirals in nature. Write about one fact which you found interesting. This will depend on what the pupils discovered. This sequence is a twist on Fibonacci sequence. The fourth number is the sum of the first three numbers, the fifth number is the sum of the previous three numbers and so on. Continue the sequence so that you write at least 15 numbers. 0, 1, 1, 2, 4, 7, 13, 24, 44, 81, 149, 274, 504, 927, 1705 HSNP © Hamilton 2012 Page 62 Shining Term 2 Week 8 Positive and negative integers Josh says if you subtract a positive number from a positive number, you will always get a positive answer. What do you think? Explain your thinking with some examples. This will be true if the second number is smaller than the first number, if the second number is larger, a negative answer will be produced. Roll a 0 to 9 dice and flip a coin to determine whether the number is positive (heads) or negative (tails). Repeat, then find the difference between the two numbers. Draw a number line jotting if it helps. Record five subtractions. Examples:- 8 – (-5) = 13, 7 – 2 = 5 (-6)- 4 = -10….. Play Walk the plank at http://www.math-play.com/integersgame.html. Choose attributes for the person you want to walk the plank. You will be asked a question, click on each pirate to view their answers and choose the one you think is right. If correct, you’ll be asked to click on the dice to move the person forward on the plank. Click for the next question. Carry on until the game is complete. Were you successful in getting the person to walk the plank?! This is a computer game. HSNP © Hamilton 2012 Page 63 Shining Term 2 Week 9 Multiplication Work out 1! 2! 3!... 10! and record the answers. 1! 1 2! 2 × 1 = 2 3! 3 × 2 × 1 = 6 4! 4 × 3 × 2 × 1 = 24 5! 5 x 4 x 3 x 2 x 1 = 120 6! 6 x 5 x 4 x 3 x 2 x 1 = 720 7! 7 x 6 x 5 x 4 x 3 x 2 x 1 = 5040 8! 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 40 320 9! 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 362 880 10! 10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x 1 = 3 628 800 Remember to use your previous answer to help work out the next one. Are the answers odd or even? Why? Are you convinced that all further factorials will be even? They will all be even as they include x2 Look at the digital roots, e.g. for 9! 3 + 6 + 2 + 8 + 8 + 0 = 27, 2 + 7 = 9, so the digital root is 9. What do you notice about the digital roots? What do you think will happen for further factorials? Why? What do you notice about the digital roots of multiples of 9? Does this help? Use a written method, e.g. the grid method, to work out the following multiplications: 12 × 21 = 252 9 23 × 32 = 736 7 34 × 43 = 1462 4 45 × 54 = 2430 9 56 × 65 = 3640 4 67 × 76 = 5092 7 78 × 87 = 6786 9 89 × 98 = 8722 1 Work out the digital root for each. Can you spot any patterns? HSNP © Hamilton 2012 Page 64 Shining Term 2 Week 10 Division The factors of 48 are: 1 and 48 2 and 24 3 and 16 4 and 12 6 and 8 If we add all the factors less than 48, we get 76: 1+2+3+4+6+8+12+16+24=76 48 is called an abundant number because it is less than the sum of its factors (without itself). 32 has factors 1, 2, 4, 8 and 16 (apart from 32) and the sum of these factors is 31, so 32 is not an abundant number See if you can find some more abundant numbers! 12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100 are the abundant numbers up to 100 Use a factor tree to find the prime factors of five two-digit numbers. Try and make the biggest tree that you can! 24 6 3 4 2 2 2 Answers will depend on the two digit numbers chosen. HSNP © Hamilton 2012 Page 65 Shining Term 1
