trial and improvement - foundation tier 1 trial and improvement

Trial and Improvement - Foundation Tier1

Trial and ImprovementTrial and Improvement


In today’s lesson you will: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].


The basic idea of T & I ……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


The basic idea of T & I ……2The basic idea of T & I ……2We 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 + 16Now 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 ?


The basic idea of T & I ……3The basic idea of T & I ……3Now it is a lot easier to put all our working into a

table like this: 20x+16=60

Trial x 20x + 16 Too high/too low

4 80+16 = 96 Too High

3


The basic idea of T & I ……5The basic idea of T & I ……5Now we will use the method on a problem

that we would find much more difficult to solve using algebra:

Solve the equation:

x2 + x = 26

Correct to 1 d.p.


The basic idea of T & I ……6The basic idea of T & I ……6It’s much easier to put all our working into this

table: x2 + x = 26

Trial x x2 + x Too high/too low

4


The basic idea of T & I ……7The basic idea of T & I ……7x = 4 is a good guess to begin with:

x2 + x = 26

Trial x x2 + x Too high / too low

4 16 + 4 = 20 Too low

5 25 + 5 = 30 Too high

4.5 20.25 + 4.5 = 24.75 Too low

4.6 21.16 + 4.6 = 25.76 Too low

4.7 22.09 + 4.7 = 26.79 Too high

4.65 21.6225 + 4.65 = 26.2725 Too high


The basic idea of T & I ……7The basic idea of T & I ……7 x2 + 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 x x2 + x Too high / too low

4.6 21.16 + 4.6 = 25.76 Too low

4.7 22.09 + 4.7 = 26.79 Too high

4.65 21.6225 + 4.65 = 26.2725 Too high


The basic idea of T & I ……8The basic idea of T & I ……8Now you will use the method on a problem

similar to the previous one:Solve the equation:

x2 – x = 66

Correct to 1 d.p.


The basic idea of T & I ……9The basic idea of T & I ……9It will be much easier to put all your working into

this table: x2 – x = 66

Trial x x2 – x Too high/too low

10


The basic idea of T & I ……10The basic idea of T & I ……10The first three lines of your working should be:

x2 – x = 66

Trial x x2 – x Too high/too low

10 100-10 = 90 Too high

9 81 – 9 = 72 Too high

8 64 – 8 = 56 Too low

8.5


The basic idea of T & I ……11The basic idea of T & I ……11 x2 - 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.


The basic idea of T & I ……12The basic idea of T & I ……12Now you will use the method on another

problem:Solve the equation:

x2 + 2x = 58

Correct to 1 d.p.


In today’s lesson you should have: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].


Typical GCSE Exam question: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, x x3 Too high/too low

5 5x5x5 = 125 Too high

4 4x4x4 = 64 Too low

trial and improvement - foundation tier 1 trial and improvement

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