## Teaching for understanding – adding fractions

This year I have the privilege of teaching top set year 11 – a group of very bright mathematicians. Occasionally though, they’ll come out with something that makes me cringe. This week it was their method for adding fractions. At the start of the lesson I put this on the board:

and asked the students to add it together. Very quickly they held up their whiteboards with the correct answer:

‘Great’ I thought – no problems with that, and just to check I asked a student how they’d done it.

“Easy,” he replied, “just use the upside-down picnic-table”.

For those who are unfamiliar with this idea, the upside-down picnic-table is the algorithmic method of adding two fractions together by simply multiplying top-left by bottom-right added to bottom-left multiplied by top-right, all over the product of the denominators – so-called because when you draw what you’re multiplying, you end up with what resembles an upside-down picnic-table; we’d probably refer to it as cross-multiplying!

Now, I despise this method, and told the students as much. They asked why (fantastic – I like students to question what I’m telling them!).

I explained to them that I disliked it for a couple of reasons – one, that students often don’t understand why it works, and two, that it often results in students doing more work than necessary. They weren’t convinced by the first reason – they seemed happy that they knew how to add or subtract fractions, so I gave them an example and asked them to add:

They soon showed me whiteboards with:

I then asked them to make sure they simplified their answer. A couple of moments later 29 students held up whiteboards with this answer:

When I asked them if they could simplify it, they said “no, because we can’t factorise it”.

I then suggested that they looked at the original question and thought about what they needed to do to the two denominators to make them equal. One student very quickly pointed out that they right-hand denominator was the difference of two squares, and so if he multiplied (x+1) by (x-1) the denominators would be equal.

I then asked them to do that with the left-hand term:

They then all simplified it to:

We then did several questions similar to this, and within ten or fifteen minutes the students were convinced that this method was the best method.

Cross-multiplication, the upside-down picnic-table, whatever you call that method, is an easy method to teach, but does it promote understanding? When taught that way, students can very quickly gain procedural understanding of how to add fractions, but I don’t think they truly understand what they are doing – instead they are simply applying a method that they have learnt parrot fashion. This method will very quickly fade with many students – and with those where it doesn’t, they’ll find themselves getting stuck when faced with harder questions, as illustrated above.

The same is true for other “quick” methods, for example:

Methods like this do have a use – for last minute revision when all other methods have failed. However, when teaching fractions lower down the school – in year 7 and 8 for example, I think it is really important to teach methods that develop the understanding.

I would start off using diagrams and a fraction wall and making sure that students were equally happy seeing fractions such as one half being equivalent to two quarters, or three sixths etc, before showing them an addition method that relies on them finding a common denominator first.

Whichever methods you do use – ask yourself the question: “Am I teaching for understanding, or am I teaching a quick fix?”

## Piet Mondrian Inspired Mathematics

Piet Mondrian (1872-1944) was a Dutch artist who is famous for creating very bold, grid style paintings that only use blue, red, yellow and black on a white background.

I’ve used his paintings to inspire a couple of different lessons – graphs of vertical and horizontal lines and inequalities & regions on a graph.

## Graphs of horizontal and vertical lines

In the past, when I’ve taught this lesson I’ve done plenty of work on mini-whiteboards first, asking pupils to name the equation of various different horizontal and vertical lines. Once they are all confident I’ve shown them a selection of Piet Mondrian’s work and told them that they are going to recreate some of this. I’ve then given them a piece of 1cm squared A4 paper and asked them to draw a set of axis. Once they’ve done that (don’t underestimate the difficulty some lower ability pupils will find doing this!), they are instructed to draw several horizontal and vertical lines on their axes before labelling them. After checking with me that they are correctly labelled, they have then used colour to turn their plain looking graph into a piece of Mondrian inspired artwork.

## Regions and Inequalities

When I’ve done this with higher ability pupils in the context of regions and inequalities, I’ve again shown them pieces of Piet Mondrian’s work, and then given them a set of inequalities that they have to plot and colour on a graph.

The instructions below work best when they are plotted on a set of axes drawn on 1cm squared A4 paper in a landscape orientation. The axes need to be drawn on axes with a scale going  from -7 ≤ x ≤ 7 and -5 ≤ y ≤ 5. If you’re wanting to create display work with it, get the pupils to leave a 2cm gap between each number on their axes.

## Area, Perimeter and Algebra

Updated – thanks to @JulieToddMath for spotting the mistake!

On Sunday evening, whilst following one link to another, I came across the Parent Guide PDF’s published on the College Preparatory Mathematics website. One of the things that caught my eye when I was looking through them was this problem:

I really liked the idea of combining algebra work with area and perimeter, so I decided to make a couple of my own. This is something that works at so many different ability levels.

The first one I made for my year 11 class so they could practise their factorisation and expanding skills.

Solution.

The second would be more appropriate for a year 12 class, as well as practising factorisation and expansion they also need polynomial division.

Solution.

If you like them, try making some yourself. How else could this idea be used?

Update:

Mr Calculator on Twitter suggested an alternative question to ask for the first problem: “If the area is 82, find x.”

With year 11 today I gave them them the top diagram, told them the area was 82, and asked them to find the perimeter.