 Trial, Record & Improve Years 2 - 6

with a photo from your classroom.
doug@blackdouglas.com.au

### Summary

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 below. Suitable for threading.

### Materials

• One calculator for each person

### Procedure

A classroom with a calculator culture opens many possibilities for equation work which are not available in a classroom where calculator use is restricted by teacher control.

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 & Seek' by covering a number with a pasted on 'box', eg:

7 + 2 + 3 - 5 = 7

becomes

7 + + 3 - 5 = 7

immediately encourage the solution of equations. Collect children's written work one day and give it back the next day with the 'box' over one number. Can they work out the missing number? Can they work out the missing number in someone else's work?

### Content

• decimal calculations
• decimal interpretation
• division
• equations: creating/solving
• multiplication
• operations - whole number
• order of operations
• place value
• problem solving
• recording - calculator
• recording - written
• subtraction
• times tables

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.

Teachers have used the process to encourage children to solve equations with one hidden number (as above) or to solve equations like these: + = 55     (two unknown numbers both the same) + = 55     (two different unknown numbers)

### Examples From Classrooms

Children recording and teachers questioning are important elements of these examples.

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.

 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.

 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 to mathematics teaching 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?

### Extensions

• Calculating Changes Members can extend this trial, record and improve strategy using the investigation of square roots from first principles contained in the Squares & Square Roots activity. After all, what does anyone actually learn by pressing the square root button on a calculator?
• Maths300 Members can extend Trial, Record & Improve by exploring Lesson 94, which has the same title and acknowledges that it is sourced from Calculating Changes. The lesson plan includes several other games which assist with understanding the creation and solution of equations. The Classroom Contributions section includes the work above.
Lesson 19, Backtracking, is another excellent support for understanding - and becoming quite competent at - creating and solving equations. ### Mathematical Note

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 and write the equation with its answer, eg:

1·8 ÷ 5 + 6 = 6·36

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

1·8 ÷ + 6 = 6·36

NOTE: Be aware that most simple four function calculators do not have the order of operations programmed in, so you must not mix + and - with x and ÷ ... otherwise you will get wrong answers. Any calculator will evaluate 1·8 ÷ 5 + 6 as 6·36. However the equivalent calculation 6 + 1·8 ÷ 5 will not give the answer 6·36 on all calculators.

A calculator such as those preferred for Calculating Changes will give the correct answer but a calculator without the order of operations built in will not, because it evaluates 6 + 1·8 first and then divides. Calculating Changes ... is a division of ... Mathematics Centre