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.

Efficient Calculator Methods for Percentages Changes

Using efficient calculator methods for percentages changes can be a tricky experience for students. Sometimes because the numerical methods involved can overwhelm the visual concept but also because students often see two distinct methods.