Less Than Fractions
Years 4 - 10

 

Preparation

  • Print this two part investigation board.
  • You need nine tiles numbered 1 to 9 that fit in the squares on this board.
    You can fold and tear scrap paper to fit.
    Or you can print this grid and cut what you need.
  • Write the title of this challenge and today's date on a fresh page in your maths journal.

Getting Started

  • Choose any two tiles, for example, 9 and 7.
  • Place them on the table one above the other to look like a fraction.
    There are always two ways to do this.
    For 9 and 7 the ways are 7/9 and 9/7.
  • Write both fractions in your journal in figures and words. For the example you would write:
    7/9 = 7 ninths
    9/7 = 9 sevenths
  • Sometimes a fraction can be worth more than one.
    If your fractions are more than one write them that way too. For the example you would write:
    9/7 = 9 sevenths = 12/7 = 1 whole and 2 sevenths
  • Repeat for 4 more pairs of tiles.

Optional:
You have made ten (10) fractions so far. If you made all the possible fractions like this from the digits 1 to 9, how many would there be?

  • How do you know?
  • Can you check it another way?

Part A: Less Than Fractions

In the Getting Started, you could make any fraction you wanted. Now you can only make fractions worth less than 1. The more you find, the more successful you are.
Have fun exploring Less Than Fractions
The Starter tells you that there are 36 ways to make fractions < 1 using the tiles.
How does a mathematician know that?

  • If you want to try to find all of them, open your Learning to Work Like a Mathematician page and ask What might a mathematician do?.
  • Are there any questions or strategies listed that might help?

Help from the Ancient Egyptians
When the Ancient Egyptians were learning to work with fractions, they could only do them if the numerators (top line) were 1.

  • What happens if you put 1 in the top box? Using the other tiles can you make any fractions that are less than one.

You are not an Ancient Egyptian. So you can try other numerators.
When you find all the < 1 fractions that have 1 as numerator, what will you use as the next numerator ...and the next ...and the next ...

Part B: Adding to make Less Than Fractions

The more you find, the more successful you are ... and you can stop whenever you have found enough...

...but of course a mathematician would want to know:

  • How many solutions are there?
  • How do I know when I have found them all?
You could begin this problem by breaking it into smaller parts.
For example:
Make the first fraction 1/2.
Now only the tiles 3 to 9 are left for the second fraction.
AND the second fraction has to be less than 1/2.
Explain why in your journal.
Let's try the possibilities for 3 as the second numerator.
  • If the denominator is 4 ... 3/4 is bigger than 1/2. Doesn't work.
  • If the denominator is 5 ... 3/5 is bigger than 1/2. Doesn't work.
  • If the denominator is 6 ... 3/6 is the same as 1/2. Doesn't work.
  • If the denominator is 7 ... 3/7 is less than 1/2. Does work. Yay!
  • If the denominator is 8 ... 3/8 is less than 1/2. Does work. Yay!
  • If the denominator is 9 ... 3/9 is less than 1/2. Does work. Yay!
3 solutions so far. Write them in your journal.

 

Let's try the possibilities for 4 as the second numerator.
  • Now it's your turn. Write out the possibilities for this one to find the solutions
? solutions so far. Write them in your journal.

Then try the possibilities for 5, 6, 7, 8, 9 as the second numerator?
(Hint: It gets easier as you move along.)

  • When you finish you will know how many solutions there are IF 1/2 is the first fraction.
  • But there are still 35 other < 1 fractions from Part A that could be the first fraction!
That's the way it goes when you are working like a mathematician.
When you are using the strategy 'Try every possible case' you really do have to try every case.
At least there aren't an infinite number of cases in this problem!
You don't have to do all the remaining 35 - unless you really want to practise being a mathematician and feel the buzz when you find all the solutions.
  • If you do them all and send your work to us we will show it in the Gallery below.
However, please do try repeating the reasoning above for just one other of the < 1 fractions.

Comparing Fractions

In Parts A and B you needed to compare fractions to find out which is bigger and which is smaller.
For example, you might have to find out which is smaller 4/7 or 3/5.
This is actually a dumb question.
For example, in one of these diagrams 4/7 is greater than 3/5 and in the other diagram 4/7 is less than 3/5!
  • What is the different between the two diagrams?

Comparing fractions only makes sense if the wholes they come from are the same.

You can compare fractions when you carefully rule a diagram with the wholes the same. Now you can easily slide a piece of paper across the screen and see which one is bigger and which one is smaller.

But you won't know how much bigger or smaller.

If you want to know how much bigger or smaller you have to find a way to rule up the whole so its parts match both the sevenths and the fifths.

The whole has been divided into 35 parts to make a whole with thirty-fifths.
Now all the marks on the sevenths line and the fifths line match marks on the whole. Slide the piece of paper across the screen again to see that this is true.
  • How did the mathematician know to divide the whole into 35 parts?
If the two fractions are compared using new parts of the same whole you know which one is bigger and which one is smaller.
  • 4 sevenths = ?? thirty-fifths
  • 3 fifths = ?? thirty-fifths
And you can count how much bigger or smaller.

In your journal explain which one is less and copy and complete this sentence:
If the wholes are the same, the difference between 4/7 and 3/5 = ...

It would be a LOT of work to do drawings for all the less than fractions in Part B.
But you really should try to do it just once.

Do a drawing in your journal to find out which one is less: 3/4 or 2/3

  • Draw a whole divided into fourths (quarters).
  • Exactly underneath it, draw the same whole divide into thirds.
Now you can find out which one is less.
  • Draw the same whole again (lined up with the first two) and divide it into new parts.
Now you can count how much less.

Mathematicians don't want to do a lot of work if they can find a better way by using something else they know.

Mathematicians know that...
  • ...if you multiply any number by 1, the answer is still the same number. Even if it looks different.
Mathematicians also know that...
  • ...every whole is worth 1, no matter how many equal parts it is divided into.
So, instead of taking all that time drawing to compare fractions, mathematicians can calculate like this:
4/7 = 4/7 x 5/5 = 20/35 ... (which is still 4/7 just like in the drawing above)

3/5 = 3/5 x 7/7 = 21/35 ... (which is still 3/5 just like in the drawing above)

Therefore 4/7 is less than 3/5 and the difference between them is 1/35.

Mathematicians thought they had this figured.
But they didn't know about Darren who worked out his own method in Year 5.

Darren's Method for Comparing Fractions

Step 1


Write the fractions beside each other.

Step 2


Multiply along the cross.

Step 3


Write the answers above.

Now you know which one is less and by how much.
Multiply the two denominators to know the denominator of the difference.

Therefore 4/7 is less than 3/5 and the difference between them is 1/35.

  • Choose five pairs of fractions from the 36 in Part A.
  • Write them in your journal one pair at a time.
  • Use any of the methods to work out which is less and
    work out the difference between the two.
  • Check your work using one of the other methods.

Just Before You Finish

For this part you need your maths journal and your Working Like A Mathematician page.
  • Explain all the ways you have worked like a mathematician through this activity.
  • The main skill in this activity is learning to compare fractions to see which is smaller or bigger.
    Give yourself a score out of 10 to show how good you think you are at this.
  • Is there anything else you know now that you didn't know when you started Less Than Fractions?

 

Answers & Discussion

These notes were originally written for teachers. We have included them to support parents to help their child learn from Less Than Fractions.

 

Send any comments or photos about this activity and we can start a gallery here.

 

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