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Introducing Algebra 14th November 2006 Algebra involves … … developing pupils’ understanding of “generalised arithmetic” … constructing and transforming expressions and formulae … representing problems and their solutions in tabular, graphical or symbolic form. … developing reasoning skills Algebra is … … a puzzle Engaging pupils Engaging pupils Engaging pupils Engaging pupils Two Principles Use ‘concrete’ examples in contexts familiar to pupils 24 chocolates altogether Base new work on prior mathematical understanding Spilt Ink I’ve knocked a paint pot over my homework and I can’t remember the number underneath the stain. What number was it? 6 + 3 = 14 Hidden number What number am I hiding beneath my finger? 2 x + 1 = 13 Find the number What number does 3 x stand for? - 2 = 19 Number sentences with holes in What numbers could + and stand for? 6 + = 14 Puzzle? Maths? Nrich example On a farm there were some hens and sheep. Altogether there were 8 heads and 22 feet. How many hens were there? A = 1 B = 2 C = 3 D = 4 E = 5 etc… Crack the Code H 8 Code Key 13 M V 22 1 A O 15 +5 20 T C 3 8 H N 14 19 S Cryptarithmetic O N E + T W O S I X Cryptarithmetic S E N D + M O R E M O N E Y Year 6 lesson Number Sequences with holes in Maths Pack 1 Number Sequences 4 7 10 13 16 1st 2nd 3rd 4th 5th 20th ? GCSE Write down the next term in the following sequence. 5 8 12 17 ……… Maths Foundation Tier June 2004 Paper 1 Q. 4(a) Next term? 1 2 … 1 2 3 … 1 3 6 … 0 1 1 2 O E T 2 T G F 6 E … F 8 S G S 5 E E Nine … 7 … V Odd One Out Function jigsaw Number machines Number machines Maths Pack 2 Number machines Number machines Maths Pack 2 Simple formulae expressed in words I think of a number, then multiply it by 2. I get the answer 14. What number did I think of? Number Magic Think of a number between 1 and 10. Add 7 to it. Multiply the answer by 2. Subtract 4 from the result. Divide the answer by 2. Take away the number you first thought of. If A =1, B = 2 etc. Convert your answer into a letter. Think of an animal whose name starts with this letter. Number Magic : How does it work? Think of a number between 1 and 10. n Add 7 to it. n+7 Multiply the answer by 2. 2n + 14 Subtract 4 from the result. 2n + 10 Divide the answer by 2. Take away the number you first thought of. n+5 5 If A =1, B = 2 etc. Convert your answer into a letter. E Think of an animal whose name starts with this letter. . . Guess the Rule My rule multiplies by something then takes something away What answer does 3 give with your rule? 11 Try another number or guess my rule Guess my Number My rule multiplies by 3 then takes 2 away What is your answer? 13 Guess the number I started with Finding the rule Input X3 Output + 2 1 3 5 2 6 8 3 9 11 4 12 14 5 15 17 3 3 3 3 Finding the rule Input X3 Output + 2 1 3 5 2 6 8 3 9 11 4 12 14 5 15 17 3 3 3 3 An activity This is a 3-snake. It takes seven straws to make this snake. How many straws would it take to make a 100-snake? Working through the activity 1-Snake 3 Straws 4-Snake 9 Straws 2-Snake 5 Straws 3-Snake 7 Straws 5-Snake 11 Straws Snake X2 Straws + 1 1 2 2 4 5 3 6 7 4 8 9 5 10 11 100 X2 200 + 1 3 201 2 2 2 2 Another Investigation Solving equations I think of a number, then add 3 to it. I get the answer 10. What number did I think of? + 3 = 10 Solving equations I think of a number, then add 3 to it. I get the answer 10. What number did I think of? + 3 = 10 Solving equations I think of a number, then add 3 to it. I get the answer 10. What number did I think of? ? + 3 = 10 ? Solving equations I have a large stone to weigh on a balance. When I add 5kg to the stone, they balance with 12kg. What does the stone weigh? S + 5= 12 Solving equations I think of two numbers. When I add 5 to the first number I get the answer 12. When I add 3 to the second number I get the answer 12. What numbers did I think of? ? + 5 = 12 ? + 3 = 12 Solving equations I think of two numbers. When I add 5 to the first number I get the answer 12. When I add 3 to the second number I get the answer 12. What numbers did I think of? a + 5 = 12 b + 3 = 12 Solving equations I think of a number, then add 3 to it. I get the answer 10. What number did I think of? a + 3 = 10 a Loop Cards Solving equations S + 3 = 10 Solving equations S + 3 = 10 7 Take 3 from each side Solving equations 2S + 3 = 15 Solving equations 2S + 3 = 15 12 Take 3 from each side Solving equations 2S = 16 Take 3 from each side Halve each side http://nlvm.usu.edu/en/nav/vm1_asid_324.html Solving equations 2a + 3 = 15 a x by 2 2a +3 2a + 3 6 by 2 12 -3 15 Solving equations : Trial & Improvement 2a + 3 = 26 Try a = 10 x by 2 20 +3 23 Too small Try a = 11 x by 2 22 +3 25 Too small Try a = 12 x by 2 24 +3 27 Too big Try a = 11.5 x by 2 23 +3 26 Number Relationships I think of two numbers. When I add them together, I get the answer 10. What numbers did I think of? x + y = 10 “Y to the sky” Connect 4 Coordinates 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Connect 4 Coordinates 10 9 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 My table is 4s100 90 Tables Grid My table is 3s 80 70 60 50 40 7 x 3 is 21 6 x 4 is 24 30 20 10 0 0 1 2 3 4 5 6 7 8 9 10 Odd One Out (3,6) (2,5) (5,10) (4,8) (1,2) Odd One Out … Coordinates 10 (3,6) (2,5) 9 8 7 6 (5,10) 5 4 (4,8) (1,2) 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Odd One Out … Coordinates 10 (3,6) (2,5) 9 8 7 6 (5,10) 5 4 (4,8) (1,2) 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Partner numbers 6 4 13 3 11 19 5 16 Partner numbers : Rule ? x by 3 then + 1 n 3n + 1 x + y = 10 10 9 8 Could be … 7 2 and 8 6 2nd Number y 5 4 3 2 1 0 0 1 2 3 4 5 1st Number 6 x 7 8 9 10 x + y = 10 10 9 8 Could be … 7 6 and 4 6 2nd Number y 5 4 3 2 1 0 0 1 2 3 4 5 1st Number 6 x 7 8 9 10 x + y = 10 10 9 8 Could be … 7 9 and 1 6 2nd Number y 5 4 3 2 1 0 0 1 2 3 4 5 1st Number 6 x 7 8 9 10 x + y = 10 10 9 8 Could be … 7 … etc 6 2nd Number y 5 4 3 ½ and 6 ½ 3 7.4 and 2.6 1 2 0 0 1 2 3 4 5 1st Number 6 x 7 8 9 10 Patterns in Coordinates 10 9 (3,8) (5,8) (3,4) (5,4) 8 7 6 5 4 3 2 1 0 0 1 2 3 4 5 6 7 8 9 10 Coordinates & Translations 10 9 (4,8) 8 7 6 5 4 (6,5) (1,4) (4,5) 3 2 (3,1) 1 (1,1) 0 0 1 2 3 4 5 6 7 8 9 10 Graphs and Equations 3 x x– 2= y 3 x x– 2 3 4 2= y 10 9 x= 1 y= 1 8 7 6 5 4 3 2 1 0 0 1 5 6 7 8 9 10 3 x x– 2 3 4 2= y 10 9 x= 2 y= 4 8 7 6 5 4 3 2 1 0 0 1 5 6 7 8 9 10 3 x x– 2 3 4 2= y 10 9 x= 3 y= 7 8 7 6 5 4 3 2 1 0 0 1 5 6 7 8 9 10 a = 4 b = 3 “What else do I know?” 3a + 3b 2a + 2b 3a + b = 21 = 14 = 15 a + b = 7 a + 2b = 10 a - b = 1 2a = 8 3a - b = 9 Year 6 Plenary Sum and Product Substitution Dice White Dice Operation 2x 7x 3x + 1 4x -1 5x + 2 8x - 6 On your whiteboard … I think of a number. When I multiply it by itself I get the answer 42. Estimate the number I thought of. Developing Thinking Find me a formula that has the value 7 when a = 2 and b = 3 a b =7 24 Game Four 4s + - x Estimation Golf 1st Hole : Par 4 273 < 19 x b < 275 2nd Hole : Par 5 160 < x2 < 162 What’s It Worth? Maths Pack 3 Simplifying Which expressions are the same as 7b? Algebra Loop Card Game Algebra Addition Squares Algebra Addition Squares Algebra Addition Squares Algebra Addition Squares Follow the rule Follow the rule Find the rule nrich http://www.nrich.maths.org.uk/public/index.php “Real Life Graphs” Like terms I have 3 apples and 2 bananas in one bag and 4 apples and 3 bananas in another bag. How many apples and bananas do I have altogether? Like terms I have 3 apples and 2 bananas in one bag and 4 apples and 3 bananas in another bag. How many apples and bananas do I have altogether? 3 a and 2 b and 4 a and 3 b 7 a and 5 b Like terms I have 3 apples and 2 bananas in one bag and 4 apples and 3 bananas in another bag. How many apples and bananas do I have altogether? 3 a + 2 b + 4 a + 3 b 7 a + 5 b Algebra Magic Square Formulae in symbolic form I bought 2 boxes of sweets from one shop and 3 boxes of the same sweets from another shop. How many boxes of sweets did I buy? 2 boxes + 3 boxes = 5 boxes If I then had 45 sweets altogether, how many sweets were in each box? 5 boxes = 45 1 box = 45 5 = 9 Formulae in symbolic form I bought 2 boxes of sweets from one shop and 3 boxes of the same sweets from another shop. How many boxes of sweets did I buy? 2 boxes + 3 boxes = 5 boxes If I then had 45 sweets altogether, how many sweets were in each box? 5 boxes = 45 1 box = 45 5 = 9 Formulae in symbolic form The cost of hiring a limousine is £30 plus £2 for every mile travelled. What would be the total cost of hiring the limousine if you travelled 6 miles in it? Total cost = £30 plus £2 for each of the 6 miles = £30 plus £12 = £42 Formulae in symbolic form The cost of hiring a limousine is £30 plus £2 for every mile travelled. What would be the total cost of hiring the limousine if you travelled 6 miles in it? C = 30 + 2 x 6 = 30 + 12 = £42 Formulae in symbolic form The cost of hiring a limousine is £30 plus £2 for every mile travelled. What would be the total cost of hiring the limousine if you travelled 6 miles in it? C = 30 + 2 x 6 = 30 + 12 Cost = £42 Formulae in symbolic form The cost of hiring a limousine is £30 plus £2 for every mile travelled. What would be the total cost of hiring the limousine if you travelled 8 miles in it? C = 30 + 2 x 8 = 30 + 16 Cost = £46 Formulae in symbolic form The cost of hiring a limousine is £30 plus £2 for every mile travelled. What would be the total cost of hiring the limousine if you travelled m miles in it? C = 30 + 2 x m Formulae in symbolic form The area of a rectangle is calculated by multiplying the length times the width. What would be the area of a rectangle of length 6 cm and width 4 cm? Area = = = Length x Width 6 cm x 4 cm 24 cm2 Formulae in symbolic form The area of a rectangle is calculated by multiplying the length times the width. What would be the area of a rectangle of length 6 cm and width 4 cm? Area = = = Length x Width 6 cm x 4 cm 24 cm2 Formulae in symbolic form The area of a rectangle is calculated by multiplying the length times the width. What would be the area of a rectangle of length 6 cm and width 4 cm? Area = Length x Width Formulae in symbolic form The cooking time for a chicken is ½ hour per kg plus an extra ½ hour. It is also suggested to allow a further ¼ hour before serving. At what time should I put the chicken into the oven if I want to eat at 1 p.m. and the chicken weighs 4 kg? Cooking Time = ½x4+½+¼ = 2 +½+¼ = 2¾ hours hours hours etc… Formulae in symbolic form The cooking time for a chicken is ½ hour per kg plus an extra ½ hour. It is also suggested to allow a further ¼ hour before serving. At what time should I put the chicken into the oven if I want to eat at 1 p.m. and the chicken weighs w kg? Cooking Time = ½xw+½+¼ hours etc… Brackets I’ll have 3 fish and 2 lots of chips, please. 3 fish + 2 chips 3 fish + 2 chips I’ll have fish and chips twice, please. 2(fish + chips) 2(fish + chips) Grid method for 34 x 27 30 4 20 600 80 680 7 210 28 238 + = 918 Grid method for (x + 2)(x + 3) x +2 x 2 x +2x +3 +3x +6 = x2 +5x +6 MicroSMILE : Algebra Equation Buster Flag Diagrams Another sequence What is the next number in this list : 1 Another sequence What is the next number in this list : 1 2 Another sequence What is the next number in this list : 1 2 4 Another sequence What is the next number in this list : 1 2 4 7 Another sequence What is the next number in this list : 1 2 4 7 23 …