Trial, Record and Improve Calculator Activities    blackdouglas.com.au

Trial, Record & Improve with the assistance of computers is the most common way to solve equations in the real world. Most real life situations do not produce the neatly solvable first and second degree equations which we were taught to solve at school. Other numerical methods have to be used which are based on the principle of making an intelligent guess about the solution and then feeding that guess into a set of equations managed by a computer in order to refine the guess. The process continues until the desired accuracy is obtained.

With the assistance of a simple calculator which uses an Algebraic Operating System, this same process can be used by quite young students to investigate and solve and/or create equations. The evidence for this is provided by the examples from classrooms recorded below.

Introduction

To gain a feel for this approach try to solve the following equation yourself without applying the rules you were taught at school. Instead, guess a solution, apply it with the aid of the calculator and then refine your guess based on the information you gain from the calculator answer.

^{x 0.6 + 3.08 = 3.8}

Now make up one of your own. All you have to do is start with a number, say 1.8, operate on it (by hand) and write the equation with its answer, eg:

Now hide one of the numbers on the left in a 'box', eg:

^{1.8 ÷ + 6 = 6.36}

However, be aware that simple four function calculators do not have the order of operations programmed in. The Texas Instruments MathMate does however, so the buttons can be pressed with confidence knowing that the results will obey accepted BOMDAS conventions. To explore this difference consider the following:

We know that this equation:

is correct and is equivalent to the one above because division takes precedence over addition. But simple four function calculators have not been programmed to know this. When you enter:

in that order a simple four function calculator will give the answer 1.56, because it operates in the order the numbers are entered and not according to the agreed hierarchical order of operations. Check what happens on the MathMate.

With this background, a classroom with a calculator culture opens many possibilities for equation work which are not available to a classroom where calculators are not freely available.

Application: Part 1

For example, even very young children create their own equations with the help of concrete materials; to do so is part of every mathematics curriculum. Teachers who take children's created equations and play 'Hide and Seek' by covering a number with a stuck on 'box', eg:

^{7 + + 3 - 5 = 7}

immediately encourage the solution of equations.

Students may solve such an equation in many ways, but encouraging the use of a calculator and:

• Trial (I guess 8 and try it with the calculator)
• Record (Answer is too big)
• Improve (I'll try 6)

allows virtually all students to attempt a solution.

Application: Part 2

Teachers have used the process to encourage children to solve equations with one hidden number (as above) or to solve equations like these:

^{+ = 55}

^{+ = 55}

Examples From Classrooms

REBECCA

Using the calculator to solve:

^{+ = 55}

wanted to start with 38.5 as the first addend.

38.5 + 41.5 = 80 ... That's miles too high!

38.5 + 28.5 = 67 ... That's still too high!

38.5 + 12 = 50.5 ... That's too low - it needs to be a bit higher than 12.

38.5 + 18 = 56.5 ... That's just a bit too high. I can go back by one.

38.5 + 17 = 55.5

She understood 0.5 as the half way point between 2 consecutive whole numbers, eg: 38, 38.5, 39. Went on to solve to 55 by adjusting 38.5 to 38 - needed teacher help.

SCOTT - Grade 2

Using the calculator to solve:

^{+ = 55}

Wrote these on his book.

23.9 + 31.1 = 55

23.7 + 31.3 = 55

48.5 + 7.5 = 56 ... Means make smaller.

48.5 + 6.5 = 55

Teacher:

How did you know that the .9 matched the .1 and the .7 matched the .3 etc.

Scott:

It's easy. It's just like the other columns. They add up to equal 10 and then carry.

ELIZA

Using the calculator to solve:

^{+ = 55}

21 + 25 = 46 ... That's too small.

21 + 35 = 56 ... That's one too many.

21 + 34 = 55 ... I knew it had to be 34.

and

35 + 29 = 64 ... Much too high.

35 + 15 = 50 ... I can work it out now.

35 + 20 = 55 ... You have to add 5 more to the 15 because 55 is 5 more than 50.

SHANNON

Using the calculator to solve:

^{+ = 55}

25 + 26 = 51 ... It has to get bigger.

25 + 29 = 54 ... Now I know it.

25 + 30 = 55 ... I could work it out in my head.

22 + 27 = 49 ... I've got to go bigger.

22 + 34 = 56 ... Oh, just a bit too high.

22 + 33 = 55 ... It was just one too big so I went back by one.

EMMA - Grade 3

This vignette shows Emma readjusting her aims when the problem she begins working on is too difficult.

After seeing others in the room using decimals - mainly 0.5 - to solve the problem:

^{+ = 55}

tried the following:

24.5 + 26 = 50.5

24.5 + 31 = 55.5

24.5 + 30 = 54.5 ... I've gone bigger and smaller. It doesn't make sense.

Emma:

I don't know where to go here. I don't know what the .5 does. I'll do it without it.

24 + 31 = 55

Teacher:

How did you know this so quickly?

Emma:

It's easy - the 4 & 1 make 5 and the 2 & 3 make 50.

In the standard approach children of this age would never see equations like these. Neither would they be able to display the level of number sense displayed in these examples. Have we been letting kids down?

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