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

2. 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

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

4. 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

5. 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.
– miniwhiteboards?

6. 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

7. 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

8. 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

9. 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 Break 20mins

10. 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?

11. 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

12. Common Errors
For example:
What is wrong with these examples

13. 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.

15. Solving Linear Equations
Video explanations
Khan Academy
You tube
Up to 150

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

17. Application
Can you think of some everyday examples where simultaneous equations may be useful? Try these

18. 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?

19. 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

20. 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

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

22. Other Methods Graphical: Where two Straight lines cross
Substitution: Rearrange one of the original equations to isolate a variable, then substitute into other equation (useful for quadratics)

23. 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

24. 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?

25. 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……

26. L3 – A0N (past paper)
A charity collection bottle contained 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

27. 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:
More Practice: CIMT online exercises

28. More online help
Video with elimination method Practise questions with answers

29. 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