# An illustrative way of adding fractions

I have recently taken on a middle year 11 set who have not been able to get to grips with how to add and subtract fractions with different denominators so I decided to try an illustrative way of adding fractions. They’re mostly aiming for grade C’s so I am sure you can imagine good arithmetic skills with fractions and mixed number is essential.

I decided to try an approach different to the traditional written method of finding a common denominator and adding or subtracting the numerators.

### Using Diagrams

To give an example. We were trying to find the sum of 1/3 and 1/5.

Start by drawing a simple rectangle and split the length up into fifths and the width up into thirds. This creates 15 individual rectangles as shown.

It is clear to see that 1/3 of the rectangle represents 5 individual rectangles. So shade them in. 1/5 is three, shade them in too.

It is clear to see that 1/3 + 3/5 = 8/15.

### Subracting Fractions

For subtractions we tried 3/4 – 2/3. Again, start with a simple rectangle. Split the length in to quarters and the width into thirds. Shade in 3/4 of the rectangle which equates to three of the four columns.

We can see from the rectangle 1/3 is 4 cells so 2/3 equals 8 cells. Simply take away 8 cells from the 8 already shaded.

From the 12 green cells we have taken away 8 with red ones. Therefore we have one 1/12 left so ¾ – 2/3 = 1/12.

### Mixed Numbers

For mixed numbers we used this example. 2 2/5 – 1 5/6.

Convert both mixed numbers to top heavy fractions so we have 12/5 – 11/6. Shade in 2 whole rectangles (or 10 fifths) plus 2 more columns for the 2/5 and subtract from that 1 whole and 5 sixths (or 11 rows of 6).  The result of 2 2/5 – 1 5/6 = 17/30.

Afterwards, we worked through a few more examples on mini-whiteboards where I would pose a problem similar to the above and ask the class to work in pairs to present their workings and solutions to me so I could feedback. After about 6 problems the vast majority were able to begin working independently on a series of past exam questions. Homework was to complete more exam type questions on adding and subtractig fractions.

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