1. Trial and Improvement - Foundation Tier 1 Trial and Improvement

2. Trial and Improvement - Foundation Tier 2 In today’s lesson you will: Understand that some equations are difficult (or even impossible!) to solve using the techniques you already know [ALL of you]; Learn how to get a rough (approximate) answer to an equation using the method known as trial and improvement [ALL of you]; Use this method on more difficult problems [MOST of you].

3. Trial and Improvement - Foundation Tier 3 The basic idea of T & I …… We will start by using Trial and Improvement on a problem that we can actually solve using algebra: Solve the equation: 20x + 16 = 60

4. Trial and Improvement - Foundation Tier 4 The basic idea of T & I ……2 We begin by guessing what we think the answer to the equation 20x + 16 = 60 might be. So choose a reasonable guess (or trial ) for x. Let’s try x = 4 So work out the part of the equation with x in it (the left hand side), changing x to equal 4: We get: 20 x 4 + 16 Now this equals 80 + 16 = 96. In the equation we need it to equal 60. This means our original guess for x was too big……. So try a smaller guess…….maybe x = 3 ?

5. Trial and Improvement - Foundation Tier 5 The basic idea of T & I ……3 Now it is a lot easier to put all our working into a table like this:20x+16=60 Trial x20x + 16Too high/too low 480+16 = 96Too High 3

6. Trial and Improvement - Foundation Tier 6 The basic idea of T & I ……5 Now we will use the method on a problem that we would find much more difficult to solve using algebra: Solve the equation: x 2 + x = 26 Correct to 1 d.p.

7. Trial and Improvement - Foundation Tier 7 The basic idea of T & I ……6 It’s much easier to put all our working into this table:x 2 + x = 26 Trial xx 2 + xToo high/too low 4

8. Trial and Improvement - Foundation Tier 8 The basic idea of T & I ……7 x = 4 is a good guess to begin with: x 2 + x = 26 Trial xx 2 + xToo high / too low 4 16 + 4 = 20Too low 5 25 + 5 = 30Too high 4.5 20.25 + 4.5 = 24.75Too low 4.6 21.16 + 4.6 = 25.76Too low 4.7 22.09 + 4.7 = 26.79Too high 4.65 21.6225 + 4.65 = 26.2725Too high

9. Trial and Improvement - Foundation Tier 9 The basic idea of T & I ……7 x 2 + x = 26 (to 1 d.p.) So our guess for x is now between 4.6 and 4.65 – this means it could be: 4.61 or 4.62 or 4.63 or 4.64 or anything in between….. But all of these will be 4.6 if rounded to 1 d.p. So x = 4.6 is our answer. Trial xx 2 + xToo high / too low 4.6 21.16 + 4.6 = 25.76Too low 4.7 22.09 + 4.7 = 26.79Too high 4.65 21.6225 + 4.65 = 26.2725Too high

10. Trial and Improvement - Foundation Tier 10 The basic idea of T & I ……8 Now you will use the method on a problem similar to the previous one: Solve the equation: x 2 – x = 66 Correct to 1 d.p.

11. Trial and Improvement - Foundation Tier 11 The basic idea of T & I ……9 It will be much easier to put all your working into this table:x 2 – x = 66 Trial xx 2 – xToo high/too low 10

12. Trial and Improvement - Foundation Tier 12 The basic idea of T & I ……10 The first three lines of your working should be: x 2 – x = 66 Trial xx 2 – xToo high/too low 10100-10 = 90Too high 981 – 9 = 72Too high 8 64 – 8 = 56 Too low 8.5

13. Trial and Improvement - Foundation Tier 13 The basic idea of T & I ……11 x 2 - x = 66 (to 1 d.p.) Your last guess for x should be between 8.6 and 8.65 – this means it could be: 8.61 or 8.62 or 8.63 or 8.64 or anything in between….. But all of these will be 8.6 if rounded to 1 d.p. So x = 8.6 is our answer.

14. Trial and Improvement - Foundation Tier 14 The basic idea of T & I ……12 Now you will use the method on another problem: Solve the equation: x 2 + 2x = 58 Correct to 1 d.p.

15. Trial and Improvement - Foundation Tier 15 In today’s lesson you should have: Understood that some equations are difficult (or even impossible!) to solve using the techniques you already know [ALL of you]; Learnt how to get a rough (approximate) answer to an equation using the method known as trial and improvement [ALL of you]; Used this method on more difficult problems [MOST of you].

16. Trial and Improvement - Foundation Tier 16 Typical GCSE Exam question: Use the method of trial and improvement to find a solution, to 1 decimal place, of the equation x³ = 100.(4 marks) Trial, xx3x3 Too high/too low 55x5x5 = 125Too high 44x4x4 = 64Too low