Comparing fractions (part 2)



Comparing fractions - how many segments?

How do you compare fractions with different denominators? Comparing fractions is a nightmare for pupils and teachers. This is a very difficult subject to teach and probably an even worse topic to learn. Put this on an examination paper and you will separate the wheat from the chaff, and mostly it will be chaff. How can we be more successful and not have to teach it year after year to the same pupils.

Fractions – is that pizza for me (slice 2)?

What fraction of the pizza am I getting?

 Who has not had problems teaching or learning fractions?  It is a painful experience for educator and learner. Despite our best intentions we do not always succeed. How can we make it easier? (if you have not done so have a look at my previous post Fractions - is that pizza for me (slice 1)?

  

Comparing fractions (part 1)

Comparing fractions (part 1)

Comparing fractions

How do you compare fractions? Who has not struggled with learning or teaching comparing fractions? Ask a child (or even an adult) which is bigger 3/4 or 5/7 and you’ll be met with a blank stare and a shrug of the shoulders, lets be honest it is difficult.

Why do people have such difficulty with fractions and even more so comparing them?  Perhaps they do not comprehend that a fraction is just part of a whole and have not had enough practical experience beyond 1/2, 1/4, etc. They need to involve themselves in dealing uncommon fractions such as 5/7, 4/9 and so on.

Below is how I tackle this problem, there is no rushing this activity and it could take several lessons to achieve good results but it is worth it. Once established it will provide a firm foundation for further work.

Fractions – is that pizza for me (slice 1)?



Another fraction of a pizza

Have you ever wanted to improve the pupils learning and your teaching of fractions? I have and I have probably taught fractions the same way as everyone else, but I still had children who did not fully understand what a fraction is, just like everyone else. How many metaphorical pizzas have been used each day during Maths lessons up and down the land? Yet our pupils have the same misconceptions year after year, despite using their favourite food. What is the problem? What was I doing wrong? What is to be done?

 

The problem

If you give a diagram like the one below and ask the question what fraction is shaded? What fraction is unshaded? How many children would give the answers 2/8 and 6/8, probably most, but can they make the leap to ¼ and ¾? They find this really difficult and need lot of questioning and prompting to see the equivalence.

 

 

Another issue is when you ask a student to share 4 chocolate bars between say 5 friends and ask how much do they each get. It takes some time before a 12 year old for example realises it is 4/5. (Perhaps I am being a bit optimistic there.)

 

As the pupil get older they are introduced to ratio. Do they ever see the link between a ratio of 2:3 and the fractions 2/5 and 3/5?

 

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.