Area, equations and that frog again

Those of you who read my post about how to use a frog to solve equations might want to use this before that lesson. It can also be used independently before starting to find the areas of complex shapes. It is deceptively simple, but powerful.

 

All you need to do I draw two lines and on one put its length, say 10cm, and on the other x cm and 8cm. Explain that the two lines are the same length. Two frogs have a jumping competition, they agree to jump over the same course of 10 cm. The first jumps the full 10 cm. Fred the frog jumps but only a distance of  x cm but then covers the rest of the course, which is 8 cm. How far is x cm?

As you know this is really x + 8 = 10. Depending on the level of pupils you can continue with further examples, make the link to algebra or move on.

 

The next step (or jump if you are a frog) is to draw similar lines  such as one which is 10 cm and the other which is x cm, x cm and 3 cm or whatever appeals to you. Again you can use the story of the two frogs one jumping the full 10 cm the other Fred covering x, cm then x cm then 3 cm. Emphasise that both x cm are the exactly  the same distance. Then ask how far is x cm?

 

Of course the pupils are solving 2x + 3 = 10. You can judge how far to take this idea it really does depend on the class or pupil. Use in its simplest form before doing complex shapes as pupils often fail to grasp how to find a missing dimension.

 

If you are wondering about previous mentions of Fred the frog see my previous post ‘solving equations with a frog’.

 

 

If anyone has used the ideas that I have given you over these last few blogs how did it go? Were they successful? How did you improve them? (I'm sure you can). Please leave a comment.

Solving equations

When planning to teach a class how to solve equations I would normally follow the steps outlined in my previous blogs which are

1. Solving equations, a staring activity

2. Solving equations with a frog.

The class should now have all the skills and knowledge to be able to solve equations. I reintroduce the frog story but this time I show them we cn write it as 3x +1 = 16. It is important to now make it clear what your objective is, to find x. Link it back to your diagram by asking 'what did you do first when trying to find how far Fred jumps?' Hopefully they will repond subtract 1, which you now show them.



Once again link back to the Fred the frog story, reminding them about the three jumps covering 15 metres and ask 'so how long was one jump?' To which they should answer 5. 'How do you know?' This leads on to division by 3. You then show them usual notation used with the arrows by the side.

I am confident that hould you follow the three stages I have outlined in thee blogs your success rate in teaching solving equtions will soar. Don't forget to use a different colour for the arrows and operations as this help pupils to distinguish what is happening to the equations form the equation itself.



Solving equations - life from both sides

Equation problem

Solving equations can be taught time and time again, but what percentage of a class can perform the complex operations involved. What percentage could repeat what they have done a day later? To us it seems obvious that 'what we do to one side we to the other', how frustrating then that this simple procedure seems to be so difficult for many students to grasp.

I have often been bewildered by students who could solve equations one day but not have a clue the next. I would ask myself what was wrong with them? Followed by what was wrong with my teaching? What else could I do? As a young teacher I asked my colleagues what they did they reassured me the same as me and not to worry most kids don't understand equations.

The root of the problem

It is not so much being able to reach an answer that causes the problem more it is the lack of appreciation that the equation has two distinct sides. They are also equal. Once this fact is conquered the next issue is the. Realisation that what you do to one side you do to the other.

The solution

This article is about tackling the both sides problem. When this was first suggested to me I was very cynical. Why should I try this I've always taught solving equations the same way, with the same amount of success as everyone else. But I decided to give it a try. Many pupils have a problem balancing an equation. Many Maths books have  pictures of weighing scales, some letters and numbers are placed in either tray and successive numbers and letters are removed until a satisfactory result is arrived at. But it doesn't mean much to pupils, how any of them have seen scales in this digital age?. This analogy doesn't seem to help much. This is what I tried and it was really successful.


First I wrote a simple equation on the board like this.

Most pupils will be able to tell you that x is 2. But that is not the object of the exercise, you want to get them to realise that what ever you do to one side you do to the other. I know you can tell them that but it doesn't not always register. I now ask them for a number any number which I then add to both sides. For xample lets say they chose 5 I would write

Putting the arrows on and in different colours has proved to me to be very important. It places the emphasis on what we are doing to both sides not on why we are doing it, that will come later. I then ask for another number. Lets ssume someone says 100.

I quite happily write +100 on both sides and continue. I might then suggest or it could come from the students subtracting. I then give an example of subtraction.

It depends upon the class how long you continue this. You could demonstrate a few more examples on the board, or put a starting point and get pupils to come up to the board and take suggestions from the class and develop the equations that way. It is not important to find x your objective is to get them to understand how to balance equations. I would follow this up with individual work, giving them starting points and letting their creative juices flow. 

 

Why not read the accompanying post

How to solve an equations with a frog

 

 

 

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Solving equations with a frog




Fred, ready to solve an equation

 

 Solving equations with a frog

Solving equation can be a slow and painful experience for many pupils and teachers. It just seems to be a complete mystery for far too many kids despite our best efforts that is why I have found using Fred the Frog such a powerful tool, you don’t even have to teach them about equations.

I start by telling them they are going to solve an equation butt I’m not going to tell them how to do it because they already know. This causes quite a few puzzled looks. Here is the story.

‘Fred the frog is resting on his Lilly pad not having had breakfast and feeling quite hungry when he spotted a bug in the distance his favourite food.’

 I then draw a picture of Fred and the bug. As you can see no special artistic skills are needed for this.



‘Fred jumps a distance we don’t know how far. Let’s give it a letter, what letter would you like?’

‘d’ says our enthusiastic pupil. I then draw d on the diagram like this.



I then say, ‘Fred jumps again and being an exceptional frog he jumps exactly the same distance again’. Further drawing now takes place.

 ‘Surprise, surprise he jumps the same distance again.’ More drawing takes place.

Finally he does one more jump but this time we know it is exactly one metre and Fred has breakfast  he eats the bug. Yet more drawing.





I then explain that Fred is truly exceptional, not only can he jump the same distance he can also read. ‘He looks up from his breakfast and sees a sign which says he is 16 metres from where he started.

Now ask ‘How far is one jump?’ Very quickly you will get the answer 5 metres. Now the killer question to ask ‘how do you know?’ Pupils will tell you to take 1 off the 16 so you are the last d that make 15 . Because there are 3 ds them you divide 15 by 3 to get 5. Its now up to you how you develop this. You can tell more Fed the frog stories with different numbers or you can go straight into ‘you’ve just solved 3d + 1 = 16’ and show them what they have done using traditional algebra.  I taught one girl who loved Fred so much she solved all her equations using this technique, never progressed to conventional notation and passed all her maths exams. You can also extend it to solving simultaneous equations, but that’s another story.