Topic: Algebra

LO: Being able to do Trial and Improvement.

AGREE LEARNING OBJECTIVES

PREPARE FOR LEARNING

STARTER CAN YOU SOLVE THIS EQUATION…?

x3 + 2x = 110

TRIAL ANDIMPROVEMENT

PRESENT NEW INFO

When x=4,

x3 + 2x = 110

TOO HIGH!

x3 + 2x = 43 + 8 = 64 + 8 = 72TOO LOW!

x3 + 2x = 53 + 10 = 125 + 10 = 135

When x=5,

So x must be between 4 and 5!

Solve x2 + x = 53 correct to 1 decimal place

This means we need to find a number with

one decimal place which gives the

closest answer to 53

To get all the marks in the exam we need to:

• First find the two whole number values of x that will give the closest answers to 53: one will be a bit too small, the other will be a bit too big

• Then ZOOM in between these two numbers to find the two numbers with one decimal place that give the closest answers to 53: again one will be a bit too small, the other will be a bit too big

• To work out which one of those to numbers is the BEST answer you need to ZOOM in again to HALFWAY between them, then you will be able to see which one was closest to 53 – and sometimes you may be surprised

• This last step is the most commonly missed out part because sometimes it seems unnecessary, but missing it out will lose you marks in the exam

Let’s do it!

CONSTRUCT MEANING

x x2+x

6 42 too small

7 56 too big

Solve x2 + x = 53 correct to 1 decimal place

First find the two whole number values of x that will give the closest answers to 53

6 7

5642

100

x must be between 6 and 7

Solve x2 + x = 53 correct to 1 decimal place

x x2+x

6 42 too small

7 56 too big

ZOOM in between these two numbers to find the two numbers with one decimal place that give the closest answers to 53

Let’s zoom in!

6 7

5642

Solve x2 + x = 53 correct to 1 decimal place

x x2+x

6 42 too small

7 56 too big

6.5 48.75 too small

6.8 53.04 too big

6.7 51.59 too small

Now we try sensible values to find the two best values of x with one decimal place.

Let’s try halfway for

starters

6 7

5642

6.5

48.75

Let’s try a bit higher but still

just one decimal place

6.8

53.04

6.7

51.59

The best answer to 1dp will be

either 6.7 or 6.8, but which one?

Solve x2 + x = 53 correct to 1 decimal place

x x2+x

6 42 too small

7 56 too big

6.5 48.75 too small

6.8 53.04 too big

6.7 51.59 too small

6.75 52.3125

ZOOM in between these two numbers to HALFWAY – then we’ll be able to see which side the true answer lies

Let’s zoom in!

53.04

6.7

51.59

6.75

52.3125

53 is this side of half way

So 6.8 is the best value of

x to 1dp6.8

We write:

x = 6.8 to 1dp

It may seem like 6.8 was obviously the closest answer, but without this final checking stage you will lose marks!

So In Summary…

• Find the best two whole number solutions

• Zoom in to find the best two solutions to 1dp

• Zoom in halfway to find which one of them was the closest overall

x x2+x

6 42 too small

7 56 too big

6.5 48.75 too small

6.8 53.04 too big

6.7 51.59 too small

6.75 52.3125

Answer: x = 6.8 to 1dp

You write this:

DEMONSTRATE UNDERSTANDING

Some questions for you to do…

WORK IN YOUR BOOKS – COPY DOWN THE QUESTION

MORE TRIAL & IMPROVEMENTQUESTIONS:

The equation

x3 – 6x = 72also has a solution between 4 and 5Use a trial and improvement method to find this solution.Give your answer correct to one decimal place.You must show all your working.

x3 − 3x = 15 (has a solution between 2 and 3)

x3 + 2x = 99 (has a solution between 4 and 5)

CONSTRUCT MEANING

Reading x and y values from a graph – An example

Extension: What line is this?

DEMONSTRATE UNDERSTANDING

A questions for you to do…

DRAW THE GRAPH IN YOUR BOOKS – COPY DOWN THE QUESTIONS

DEMONSTRATE UNDERSTANDING – EXTENSION WORK

Level ladders… algebra

So what have we learned?REVIEW

LO: Being able to do Trial and Improvement.

So what are the steps to follow to solve anEquation using Trial and Improvement?