1. Year 7 Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk) www.drfrostmaths.com Last modified: 4 th April 2016 Objectives: (a) Solve equations, including with unknowns on both sides and with brackets. (b) Form equations from context (with emphasis on quality of written communication, e.g. "Let x be...").

2. For Teacher Use: Recommended lesson structure: Lesson 1: Solving simple linear equations Lesson 2: When variable appears on both sides/brackets Lesson 3: Forming and solving equations from context. Lesson 4: Introducing variables to solve equations. Lesson 5: Solving Equations Levelled Activity (separate) Lesson 6: Consolidation/Mini-assessment Go >

3. KEY TERMS This is an example of a: Term A term is a product of numbers and variables (no additions/subtractions) Expression An expression is composed of one or more terms, whether added or otherwise. Equation An equation says that the expressions on the left and right hand side of the = have the same value. ? ? ?

4. ? STARTER

5. a 4 2 2 We already know that the ‘=’ symbol means each side of the equation must have the same value. If we added something to one side of the equation, what do we have to do with the other side? +2 Equations must always be ‘balanced’

6. a 4 a 4 a 4 If we tripled the load on one side of the scales, what do we have to do with the other side? ×3×3 ×3 Equations must always be ‘balanced’

7.  To solve an equation means that we find the value of the variable(s). Solving

8. -20 44 44 ? ? ?? ? ? Strategy: Do the opposite operation to ‘get rid of’ items surrounding our variable. -4 33 ×6×6 ? ? ? Solving Bro Tip: Many students find writing these operations between each equation helpful to remind them what they’re doing to each side, but you’ll eventually want to wean yourself off these. Bro Note: You can probably see the answer to this in your head because the equation is relatively simple, but this full method is crucial when things become more complicated

9. +5 33 33 ?? ?? ? ? Test Your Understanding

10. -4 66 66 ?? ?? ? ? Bro Note: In algebra, we tend to give our answers as fractions rather than decimals (unless asked). And NEVER EVER EVER recurring decimals. When the solution is not a whole number Your Go… ?

11. -3 ×5×5×5 ?? ? ? ? ? Dealing with Fractions

12. What step next? Use your planners to vote for the step that would be easiest to do next in solving the equation. ×  +-

13. ×  +- -7

14. ×  +- 33 33

15. ×  +- +1

16. ×  +- ×3×3×3×3 y3y3

17. ×  +-  (-1) Multiplying by -1 or dividing by -1 would have the same effect.

18. Exercise 1 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24  ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

19. What might our strategy be? Collect the variable terms (i.e. The terms involving a) on one side of the equation, and the ‘constants’ (i.e. The individual numbers) on the other side. ? What happens if variable appears on both sides?

20. -3 -2a ?? ? ? 33 33 ? ? ? This is to get rid of the constant term on the left. We could have done these two steps in either order. What happens if variable appears on both sides? Strategy? Collect the variable terms on the side of the equation where there’s more of them (and move constant terms to other side). ?

21. ? More Examples ? Both methods are valid, but I prefer the second – it’s best to avoid dividing by negative numbers, and is less likely to lead to error. Way we’d have previously done it… ? ?

22. Test Your Understanding ? ?

23. Dealing with Brackets If there’s any brackets, simply expand them first! ? ?

24. Test Your Understanding ? ?

25. Exercise 2 Solve the following. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22  ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?

26. Year 7 Forming and Solving Equations Dr J Frost (jfrost@tiffin.kingston.sch.uk)

27. RECAP :: Forming/Solving Process [JMC 2008 Q18] Granny swears that she is getting younger. She has calculated that she is four times as old as I am now, but remember that 5 years ago she was five times as old as I was at that time. What is the sum of our ages now? Worded problem Stage 1: Represent problem algebraically Stage 2: ‘Solve’ equation(s) to find value of variables. We previously learnt how to form expressions given a worded context. We’ll learn how to actually solve these equations formed now! We’ll first focus on problems where the expressions have already been specified.

28. Example Step 1: Think of a sentence which would have the word “is” in it. Write each part of sentence as an algebraic expression, with “is” giving =. Step 2: Solve! Expr 1?Expr 2? Solve! Sentence with “is” in it?

29. Another Example The rectangle and triangle have the same area. Determine the width of the rectangle. Expr 1? Expr 2? Solve! “Area of triangle is area of rect” “is” sentence?

30. Check Your Understanding Expr 1? Expr 2? Solve! Expr 1? Expr 2? Solve!  ? 1 2

31. Exercise 3 1 2 3 4 5 6 7 8 ? ? ? ? ? ? ? ? (See printed sheet)

32. Exercise 3 9 10 11 ? ? ? ? (See printed sheet)

33. Forming the expressions yourself Enoch is 5m shorter than Alex. Hajun is double the height of Alex. Their combined height is 35m. Find Alex’s height. Use the word “Let …” to define your variable(s)! You want a clear narrative while being as concise as possible. ? ? ?

34. [JMC 2013 Q7] After tennis training, Andy collects twice as many balls as Roger and five more than Maria. They collect 35 balls in total. How many balls does Andy collect? More Examples [TMC Regional 2014 Q9] In a list of seven consecutive numbers a quarter of the smallest number is five less than a third of the largest number. What is the value of the smallest number in the list? ? ?

35. Test Your Understanding The length of the rectangle is three times the width. The total perimeter is 56m. Determine its width. Bro Reminder: You should usually start with “Let …” In 4 years time I will be 3 times as old as I was 10 years ago. How old am I? ? ? 1 2

36. Exercise 4 1 2 3 4 5 ? ? ? ? ? 6 ? (See printed sheet)

37. Exercise 4 7 ? (See printed sheet)

38. Exercise 4 (Note: this is not intended to be a full proof!) 8 9 10 11 ? ? ? ? (See printed sheet)

39. Exercise 4 22 33 ? ? (See printed sheet)

40. Exercise 4 44 ? (See printed sheet)