A4 fraction addition
Try adding fractions using A4 paper.
Problem
Find a partner and start with a sheet of A4 paper and a pair of scissors each. Have a go at the activity below and compare what you've made!
The whole sheet of A4 paper represents 1 whole.
Take your sheet of paper and fold it in half into two rectangles. Unfold it and cut along the fold line. You might end up with two pieces which look like the two red rectangles below. Each of these rectangles represents $\frac{1}{2}$.
Take one of your rectangles and cut it in half into two smaller rectangles in the same way. By halving this you now have two rectangles which each represent $\frac{1}{4}$, like the blue rectangles below:
Depending on how you chose to cut the rectangle in half, your $\frac{1}{4}$ rectangles might look different to this - they might be long and thin, like the rectangles below:
Now, take one of your $\frac{1}{4}$ rectangles and cut it in half. You will then have two new rectangles which both represent $\frac{1}{8}$ like the green rectangles below.
Halve one of your green rectangles to get two $\frac{1}{16}$ rectangles like the orange one below. Then throw away one of these $\frac{1}{16}$ rectangles so that you have one rectangle of each size.
At this point, stop and compare your rectangles with your partner's rectangles. Do they look the same? Are they different to the ones in the picture above?
Imagine that you made your $\frac{1}{2}$ rectangle the same as the red rectangle above, but then you cut all your other rectangles to be long and thin. Your rectangles might now look like this:
Are there any different rectangles that we could have chosen?
Now, get four pieces of A4 paper and attach one to each of your fraction pieces. Remember that the full piece of A4 paper is counted as "one whole".
You should end up with combinations that look a bit like the ones below. These are mixed numbers of pieces of A4 paper now, because they have a whole number part and a fraction part.
Can you work out what mixed number is represented by each?
Take one of your mixed number shapes and put it with one that your partner has made. Write down the addition calculation and the answer.
For example, the addition below would give you 1$\frac{1}{4}$ + 1$\frac{1}{2}$ = 2$\frac{3}{4}$.
How many different addition number sentences can you make?
You might also like to try a subtraction version of this task.
Student Solutions
Well done to everybody who had a go at this problem.
The children from Ridgeway Primary School sent in an image of their solution. You can click on the image to make it bigger:
Olive from Cranham Primary School had a systematic approach to find all of the possible solutions:
I started working out this problem by adding all the fractions I had created to 1$\frac{1}{2}$. I started off adding 1$\frac{1}{2}$ as the largest fraction to 1$\frac{1}{2}$. I carried on with the fractions getting smaller, so 1$\frac{1}{2}$ + 1$\frac{1}{4}$, 1$\frac{1}{2}$ + 1$\frac{1}{8}$ and 1$\frac{1}{2}$ + 1$\frac{1}{16}$.
Then, I did the same with 1$\frac{1}{4}$, taking care not to use 1$\frac{1}{2}$ again, otherwise I would have the same sum twice. I repeated that with 1$\frac{1}{8}$, avoiding using 1$\frac{1}{4}$ or 1$\frac{1}{2}$, and I repeated it again with 1$\frac{1}{16}$ without using 1$\frac{1}{8}$, 1$\frac{1}{4}$ or 1$\frac{1}{2}$.
With just a few simple steps, I worked out this problem carefully and methodically. I knew that the answer I had found was correct by going over all the possible solutions.
Good ideas!
We received a few solutions from the children at Aston Rowant C of E Primary School in England. George explained a similar method to Olive's:
First I did the half and there was 4 different sums; then the quarters and there were 3 sums; then the eighths, there were 2 sums; and the sixteenths, there was 1 sum left.
Harvey wrote out all of the possible solutions:
1. First I added the wholes which in every case the answer was 2
2. Then I added the fractions together to get my answers and I used all the options that I found
3. When I had finished I found 10 possible sums. They were:
1$\frac{1}{4}$ + 1$\frac{1}{2}$ = 2$\frac{3}{4}$
1$\frac{1}{2}$ + 1$\frac{1}{16}$ = 2$\frac{9}{16}$
1$\frac{1}{2}$ + 1$\frac{1}{2}$ = 3
1$\frac{1}{2}$ + 1$\frac{1}{8}$ = 2$\frac{5}{8}$
1$\frac{1}{8}$ + 1$\frac{1}{4}$ = 2$\frac{3}{8}$
1$\frac{1}{8}$ + 1$\frac{1}{16}$ = 2$\frac{3}{16}$
1$\frac{1}{8}$ + 1$\frac{1}{8}$ = 2$\frac{1}{4}$
1$\frac{1}{16}$ + 1$\frac{1}{4}$ = 2$\frac{5}{16}$
1$\frac{1}{16}$ + 1$\frac{1}{16}$ = 2$\frac{1}{8}$
1$\frac{1}{4}$ + 1$\frac{1}{4}$ = 2$\frac{1}{2}$
James from Hamstel Junior School in the UK sent in these images, which you can click on to enlarge:
The first part of this challenge involved thinking about the different ways to fold and cut a piece of paper in half to show a half, a quarter, an eighth and two sixteenths.
I liked working in different colours and laying them out to clearly see the different combinations that could be made. It is interesting to see the different shapes that still represent the same fraction amount. I found 8 combinations.
The next part involved using just two of the combinations and making mixed fractions using a whole piece of paper and a fraction. I had to identify the value of each combination.
Next, I made a list of all the different addition number sentences that are possible from adding one from each combination.
Thank you all for sharing your ideas about this task with us.
Teachers' Resources
A4 Fraction Addition
Find a partner and start with a sheet of A4 paper and a pair of scissors each. Have a go at the activity below and compare what you've made!
The whole sheet of A4 paper represents 1 whole.
Take your sheet of paper and fold it in half into two rectangles. Unfold it and cut along the fold line. You might end up with two pieces which look like the two red rectangles below. Each of these rectangles represents $\frac{1}{2}$.
Take one of your rectangles and cut it in half into two smaller rectangles in the same way. By halving this you now have two rectangles which each represent $\frac{1}{4}$, like the blue rectangles below:
Depending on how you chose to cut the rectangle in half, your $\frac{1}{4}$ rectangles might look different to this - they might be long and thin, like the rectangles below:
Now, take one of your $\frac{1}{4}$ rectangles and cut it in half. You will then have two new rectangles which both represent $\frac{1}{8}$ like the green rectangles below.
Halve one of your green rectangles to get two $\frac{1}{16}$ rectangles like the orange one below. Then throw away one of these $\frac{1}{16}$ rectangles so that you have one rectangle of each size.
At this point, stop and compare your rectangles with your partner's rectangles. Do they look the same? Are they different to the ones in the picture above?
Imagine that you made your $\frac{1}{2}$ rectangle the same as the red rectangle above, but then you cut all your other rectangles to be long and thin. Your rectangles might now look like this:
Are there any different rectangles that we could have chosen?
Now, get four pieces of A4 paper and attach one to each of your fraction pieces. Remember that the full piece of A4 paper is counted as "one whole".
You should end up with combinations that look a bit like the ones below. These are mixed numbers of pieces of A4 paper now, because they have a whole number part and a fraction part.
Can you work out what mixed number is represented by each?
Take one of your mixed number shapes and put it with one that your partner has made. Write down the addition calculation and the answer.
For example, the addition below would give you 1$\frac{1}{4}$ + 1$\frac{1}{2}$ = 2$\frac{3}{4}$.
How many different addition number sentences can you make?
You might also like to try a subtraction version of this task.
Why do this activity
This activity enables pupils to help to develop their concept of fractions and begin to add fractions with the same denominator and denominators that are multiples of the same number. It provides a context within which pupils can explore and reason about properties of fractions.
Possible approach
Begin with a large piece of paper and split this into fractions by repeated halving (as in the first part of the activity). Discuss with pupils the size of the pieces that you're making. Ask children to suggest how you could split the paper in half into two rectangles - there are two ways of doing this at each stage!
Stick some of the parts to a fresh piece of paper (as in the second part of the activity) and ask them to describe what each is (1$\frac{1}{2}$, 1$\frac{1}{4}$, 1$\frac{1}{8}$ and 1$\frac{1}{16}$).
Working in pairs, pupils can then repeat the activity with their own pieces of paper. Then set them the task of doing the additions using their fractions, and recording as many number sentences as they can.
Key questions
Tell me about the size of this rectangle.
How are you adding these two?
How do you know that your solutions are correct?
Could you have chosen a different rectangle to represent that fraction?
Are there other ways of adding them?
Possible extension
When pupils have made the 1$\frac{1}{2}$, 1$\frac{1}{4}$, 1$\frac{1}{8}$ and 1$\frac{1}{16}$, they can then make 1$\frac{3}{4}$, 1$\frac{3}{8}$ and 1$\frac{3}{16}$. Then children can see how many addition number sentences they can make including those. They could also create different representations for the same addition, choosing differently shaped rectangles that represent the same fraction.
Possible support
Pupils could cut the fractions out of a sheet with the lines already marked.