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.

Numeracy stops here

History and Maths

If you have ever read the book ' What is History?' by E. H. Carr you may have come across the idea that 'History stops here.' What this mans is that some historians in the past believed that the History that they wrote was the definitive view; there was no other interpretation. Of course History is constantly being rewritten in the light of current ideas, attitudes and data.

The way I was taught is the only way, 'numeracy stops here'. This is the view of many parents and teachers. They believe how they learnt to add, subtract, multiply and divide is the only correct method to use, anything else is ‘strange’, not ‘proper’ or at worst some kind of  trick being played upon their child. None of this is true of course. Each child has his or her own way of learning and remembering things and once confidence is gained using their own method they should continue using it.

 Subtraction

Jim, like many pupils could happily subtract when the sum which looks like this.

However as soon as we introduce something more complicated anxiety sets in, he had convinced himself that he couldn’t and never would be able to remember how to in his own words ‘do it when the number on top was smaller’. Such as in the example below. We have all been there as teachers or parents with our pupils or children wanting to subtract 7 from 3 because they can’t remember or don’t know why they have to do in effect 13 minus 7.

A succession of teachers had tried and failed to teach Jim how to do subtract using the traditional method and failed, his parents despaired also having failed to teach Jim how to subtract. He was now 16. Jim is a bright lad, a visual learner, he wanted to subtract so that he could pass his exams.

 

Open number line

Jim was very comfortable with the open number line, so I decided to show him how to use it to subtract. First I drew the line like the one below and explained that we were at going to go from 17 to 43. I marked on the two numbers.

 

 

 

Now I asked Jim to jump to the nearest 10 from 17, he happily drew a line from 17 to 20.

 

How about jumping to the nearest 10 to 43 I asked. ‘What 40?’ he queried. He then drew the next jump and without prompting he then drew the final leap.

 

So how far have you jumped I asked him to which he replied 26. It was then only a small step to make the connection between this method and the subtraction su  set out in the traditional way. Jim was thrilled at being able to do subtraction with, as he termed it, ‘difficult take aways’. He successfully continued to use this method with ease and success, passed his exams and reduced his fear and anxiety when tackiling Maths.

 

Many of you may dismiss using this method as not being correct but it worked for Jim. It may not work for everyone but he was successful and solved problems so who is to deny him, or anyone that but adherence to other methods just because that was how they were taught.

 

Why not follow me on Twitter at @croftsr1

 

 

Choose three numbers and ...

Ever felt that sinking feeling as you are about to tackle fractions again? You can imagine what the pupils think, well you probably knoe when they say ‘We’ve done this.’ Or ‘Not again’. Here is an idea tht will give a different slant to the fearsome task of teaching factions.

 

Step 1

Ask for 3 numbers between 2 and 9 inclusive. Write them on the board. Now ask the pupils in pairs to make as many different fractions as they can in 2/3 minutes. At the end of the allotted time collect the results on the board, the chances are they’ve forgotten to repeat the numbers with same numerator as denominator. Add any they have missed.

 

Step 2

Again in pairs or groups ask them to put the fractions into groups that appear the same. Collect the results, hopefully you will get top heavy fractions, equivalent fractions and ‘normal’ fractions. Ask them how they could be displayed in a table. Again some paired or group work.

 

Step 3

Now demonstrate that probably the most effective way of showing the results is in a table like that below. So for example if the numbers chosen were 2, 3 and 7 the table would look like this.

 

 

Step 3

You can now start to ask questions such as what is special about the diagonal with the entries 2/2, 3/3, 7/7? What do they equal? What other numbers can equal these?

 

What is different about 3/2, 7/2, 7/3? What are these type of fractions called? Hw can we change them?

 

I am sure you get the idea, the variations are endless and where you go depends on the class, their prior learning and what you want to achieve. This can also be extended to 4 numbers or where ever you want.

 

This idea was taken from the book ‘Starting points’ by Banwell, Saunders and Tahta published in 1972. A vision of how maths education could have gone. It would be interesting to compare it to today’s practice in classrooms throughout the world.

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.

Adding fractions - again and again and again ...

For years I taught adding fractions the traditional way. I taught it to the same pupils aged 11, I then taught it to them again aged 12. Yes you’ve guessed it 13, 14 , 15 and 16. Colleagues I worked with did the same, all with the same degree of success, or should I say failure, what is wrong with these kids we would say they just don’t get it. After a long, long time the ‘penny dropped’, it wasn’t them that didn’t get it, it was us Maths teachers.

 

Last night I taught a 15 year old to add fraction together in 15 minutes. She said ‘Is that it, it’s easy, for four years I was so confused and now … wow.’ I then had to teach her mum to add two fractions together who said why doesn’t everyone do that.

 

Most students can add two fractions together if the denominator is the same, this fact is easy to establish. You then need to tell them this is the only way to add two fractions if the bottom number is the same. Hopefully they agree. You then present them with two fractions like those below.

 

Now highlight one of the denominators and say you will use this to add the fractions. I find it really helpful to use colour, don’t dismiss this simple trick if you want to be successful and more importantly if you want the pupils to be successful.

Now say you are going to multiply the other fraction by the highlighted 4. Like this

 

Now highlight the other fraction.

Follow the same process. Multiply the other fraction by 3.

Finally complete the calculations like this

Point out the denominators are the same and then complete the sum.

 

After a couple of examples the pupils will happily be adding fractions. OK I know it isn’t perfect, such as when you have denominators of 2 and 4 but if you want them to be confident with adding fractions it is a really brilliant method. You have to tech pupils to cancel down so the problem of denominators of 2 and 4 is eventually tackled.

Stand up/Sit down - times tables practice

Times tables - how can they stand it?

Have you ever struggled to find another way to teach multiplication or times tables? I have. It is never easy to find new ways to teach and/or practice multiplication tables. The kids are probably under pressure from home to learn them and in turn it increases the pressure on you, the teacher. Governments also use them as a handy measure of progress that the general public understand.

A fun activity

 This is great fun and can produce lots of laughter in the classroom. It is particularly  useful when the class is a bit sleepy, such as after lunch. (Or even when you are feeling a bit low on energy.) Equipment needed, NONE!

• Tell the class they have to stand. You are going to call out numbers in the three times table (for example). If the number is in the three times table they have to stay standing BUT if its not they have to sit.

• You say the number 9 staring intently at the class. No movement. 3, the same. 10 and the class sits down, or at least a few confident souls do and the rest follow. Once they have the idea you can continue, with the calling out of numbers becoming incresingly rapid. You can of course fool some by a slight bend of your knees as if you are about to sit down, even when you call out  number which is not a multiple of 3. Who follows your lead? Someone will. Continue until you think they, or you, have had enough.

• This is fun and it enables you to see who is secure in their knowledge of the 3 times table, or whatever table you chose; it is an almost instant assessment tool.

Variations

• Of course you can use other multiplication tables, but sticking to our three times table you could start to use 2 X 3, 4 X 3, 6 X 5, 2 X 7, etc. see if they can work out which are in the chosen multiplication table.

Order of teaching

I think there is a distinct order in which times tables should be taught, namely 10, 5, 2, 4, 9, 6, 3, 8 and 7. Experience has shown that these are the most difficult in ascending order.

This excellent book by Steve Chinn, 'What to do if you can't learn your multiplication tables', discusses the problems and strategies for learning the multiplication tables. He is an expert on learing difficulties, especially dyslexia and dyscalculia, who shares his knowledge through a series of books focused on Mathematics.

Why not follow me on Twitter @croftsr1

Please feel free to leave a comment, I would really appreciate it.