L3 Numeracy Week 4 : Equations and Algebra Expressions and (mainly linear) equations 12

Task 1 In pairs report to your partner a) One word that describes how you feel about the last session. b) All the things you have done since last week’s session to do with this course 12-1210

Line Up Take a card and arrange yourselves in numerical order to the value of your card. 1210-1225

Outcomes Represent algebraic expressions in multiple ways (words, symbols, diagrams, tables) Compare a range of methods of solving an algebraic problem and identify mistakes and misconceptions Form and solve simultaneous equations Relate the approaches to your own teaching and context

Write down an algebraic expression that means. Multiply n by 3, then add 4 Add 4 to n and then multiply the answer by 3 Add 2 to n and then divide the answer by 4 Subtract 4 from n Square n and multiply the answer by 4 Add 4 to n and then square Multiply n by 6 and then square Add 2 to n; add three to n; and then multiply these expressions together. 1225-1235 – miniwhiteboards?

Words – Symbols Card set A – Algebraic expressions Card set B – Explanations in words In small groups Take turns in matching cards Always explain your thinking Challenge when you don’t understand or need clarification 1235-50

Tables of Numbers Card set A –Algebraic expressions Card set B – Explanations in words Card set C Tables of numbers Now match card set C to the others 1250-1255

Areas Card set D – Areas of shapes Card set A – Algebraic expressions In your small group, Take turns in matching card set D to card set A Always explain your thinking Challenge when you don’t understand or need clarification 1255-110

Extension 4x + 12 x2 - 5x – 6 4(x + 3) (x + 1)(x – 6) 2x( x – 1) Can you draw a diagram to represent any of these expressions? 4x + 12 4(x + 3) 2x( x – 1) 2x2 - x x2 + 5x + 6 (x + 3)(x +2) x2 - 5x – 6 (x + 1)(x – 6) x2 – x – 6 (x – 3)( x + 2) x2 - 36 (x + 6)(x – 6) Extension 110-120 Break 20mins

Plenary Malcolm Swan (2005) Improving Learning in Mathematics. Interpreting multiple representations Learners match cards showing different representations of the same mathematical idea. They draw links between different representations and develop new mental images for concepts. Which maths topics could be taught using multiple representations?

Solving linear Equations There are two common ‘methods’ for solving linear equations…. ‘change the side, change the sign’ or ‘you always do the same to both sides’. When used without understanding, such rules result in many errors. 140

Common Errors For example: What is wrong with these examples

Understanding the method ‘Doing the same to both sides’ is the more meaningful method, but there are two difficulties: Knowing how to change both sides of an equation so that equality is preserved Knowing which operations lead towards the desired goal.

Solving Linear Equations Video explanations Khan Academy https://www.khanacademy.org/math/algebra-basics/alg-basics-linear-equations-and-inequalities/alg-basics-variables-on-both-sides/v/why-we-do-the-same-thing-to-both-sides-multi-step-equations You tube https://youtu.be/vkhYFml0w6c Up to 150

Simultaneous Equations Find the numbers…. 1. Two numbers which when added give the value 5, and when subtracted give the value 1. 2. Two numbers multiply together to give 10 and add together to give -7 Can you write down the related equations? 150-155

Application Can you think of some everyday examples where simultaneous equations may be useful? Try these http://www.ehow.com/info_8710568_10-can-used-everyday-life.html 155-200

A problem A man buys 3 fish and 2 chips for £2.80 A woman buys 1 fish and 4 chips for £2.60 How much are the fish and how much are the chips? Can you form two equations to represent this problem? 200-205

Simultaneous Equations When you have two unknowns you need two equations 3f + 2c = 280    (1) f + 4c = 260      (2) f = price of the fish in pence c = price of the chips in pence There are a few ways to solve these equations here is one

Elimation: Step 1 Make one of the unknowns equal by multiplying. We know that: 3f + 2c = 280    (1) f + 4c = 260      (2) Multiply (1) by 2 gives: 6f + 4c = 560 (3) Then (3)-(2) gives 5f = 300 Then solve this: f = 300/5 = 60 Therefore the price of fish is 60p

Substitution: Step 2 Substitute this value into either (1) or (2): 3f + 2c = 280    (1) 3(60) + 2c = 280 180 + 2c = 280 2c = 280 – 180 = 100 c = 100/2 = 50 Therefore the price of chips is 50p

Other Methods Graphical: Where two Straight lines cross http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/simultaneoushirev2.shtml Substitution: Rearrange one of the original equations to isolate a variable, then substitute into other equation (useful for quadratics) http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/simultaneoushirev1.shtml

Exam Papers It is sometimes useful to work with past papers to explore a topic. If there are different possible methods, one approach is to compare them…….. Consider…… 5x + 2y = 11 4x – 3y = 18 Ali, Xo, Edward and Kristina all attempted to solve the equations in different ways 205-225

Marking Students Work In your small group, discuss their answers: What did you like about the answer? What method was used? Was the method clear, accurate, efficient? What errors were made? How might the work be improved? Are there any specific teaching strategies that might be useful in addressing any problems highlighted by the answer?

Plenary – Swan Again Analysing reasoning and solutions Learners compare different methods for doing a problem, organise solutions and/or diagnose the causes of errors in solutions. They begin to recognise that there are alternative pathways through a problem, and develop their own chains of reasoning. When do you show and compare different methods for a problem? Examples……

L3 – A0N (past paper) A charity collection bottle contained 1 494 two-pence and five-pence coins with a total value of £47.46 Use this information to form two equations about the number of two pence coins and the number of five-pence coins in the bottle. 1 mark Calculate the number of two-pence coins and the number of five-pence coins in the bottle. 2 marks Show how you can check your answers to part b 1 mark 230-240

Practise Simultaneous Equations: for each question Use the information to form two equations Solve the equations to find two unknowns Show how you can check your answers For help and examples see: http://www.bbc.co.uk/schools/gcsebitesize/maths/algebra/simultaneoushirev1.shtml More Practice: CIMT online exercises

More online help Video with elimination method https://corbettmaths.com/2013/03/05/simultaneous-equations-elimination-method/ Practise questions with answers http://www.mathsgenie.co.uk/resources/87_simultaneous-equationsans.pdf

Check Outcomes How well can you…. Represent algebraic expressions in multiple ways (words, symbols, diagrams, tables) Compare a range of methods of solving an algebraic problem and identify mistakes and misconceptions Form and solve simultaneous equations Relate the approaches to your own teaching and context