Dividing Fractions - not just KFC!

Is there anything with more potential for pupils to go wrong with in the arena of fractions than division by a fraction? Whether it is turning over the wrong fraction, both fractions, or not even having a clue about it, division by a fraction does seem to be a real stumbling block for a huge number of pupils. So I thought I would share the best 3 approaches I know to dividing by fractions.

1) Multiplying by the reciprocal

This is basically where KFC comes from - although it is really important that pupils do understand the language of reciprocal and can identify reciprocals for areas of maths like functions. I like to build this by looking at unit fractions first, and definitely mixing up dividing both integer and fractional values, i.e.

                                                             6 ÷ ¼

                                                             ½ ÷ ⅓

                                                            ⅚ ÷ ⅛

Showing that these are the same as 6 x 4, ½ x 3 and ⅚ x 8 respectively is an important first step. Once this is secure we would look at dividing by a non-unit fraction as dividing by something x times bigger than the unit fraction, and so needing to divide by the unit fraction and by x i.e.

                                          ⅚ ÷ ⅘ = ⅚ ÷ ⅕ ÷ 4 = ⅚ x 5 x ¼ = ⅚ x 5/4 = 25/24

Highlighting and reinforcing the fact that 4 is the reciprocal of ¼, 3 is the reciprocal of ⅓, etc makes this approach complete.

2) Dividing term by term

Although not an approach used a lot, this can be a really nice link to multiplication provided pupils can work with the fractions within a fraction that result. The idea centres on being able to divide numerators and denominators independently i.e. 

                                                                          ⅚ ÷ ⅘ =

We can then proceed to multiply by 4/4 and by 5/5 (or alternatively simply by 20/20 if pupils will understand the reason for this in one step)

3) Using common denominators

Like addition and subtraction (and particularly if you have already worked out common denominators for addition or subtraction) if fractions are given with a common denominator then dividing them can be quite straightforward.

⅚ ÷ ⅘ = 

The idea here is if you have 25 lots of something and you divide by 24 of the same something then you have 25/24 independently of the something. So

i.e. if we have 25 thirtieths divided by 24 thirtieths you have 25/24 independent of the original thirtieths.

It may be that pupils will take to one method of dividing fractions over others, and that the pupils who grasp the concept quickly can work with all three, showing they are equivalent, choosing the optimum approach for different situations and in general working with all three to achieve true mastery of division by a fraction.

Parallel lines are the same length and other such nonsense!

Recently we have been talking about the messages and misconceptions we convey without meaning to. A colleague of mine (not in my school) put me onto one - when we draw parallel lines we nearly always draw them the same length. A quick google image search suggests that this is not just the maths teachers I know:

We can see that whilst most of the pictures do show horizontal or vertical lines, all of the pictures show parallel lines the same length. Whilst some might say that this isn't really significant, I wonder if it is not something we should be aware of anyway in terms of forcing ourselves to think about the implicit messages that we give to pupils alongside the explicit content or skills we are trying to teach.

Another example that we have come across recently is that the equation 3x = 4 has no solutions because "three doesn't go into four". There was some debate as to whether this shows a general lack of understanding of division, or is a function of the fact that most equations we being to show pupils in there initial introduction to equation solving have whole number solutions. On the subject have you ever noticed that pupils struggle a lot more with equations of the form  compared to 4x - 3 = 5? Could it be that on balance they see many more equations of the second type than the first?

Some other areas of discussion:

• Index laws using a base that is not a single term and powers that are not integers or simple fractions.
• Area of triangles where perpendiculars are horizontal and vertical.
• Fractions - only ever talking about simplification of fractions with a numerator and denominator that are positive integers.

There are lots of other patterns you can find in textbooks and other materials that teachers naturally draw on for their own examples - so my suggestion is to really think about the breadth of examples that are possible with the maths pupils are learning; and not just the typical examples you may have seen before.

Probability without numbers

"There are n sweets in a bag. 6 of the sweets are orange. The rest of the sweets are yellow. Hannah takes at random a sweet from the bag. She eats the sweet. Hannah then takes at random another sweet from the bag. She eats the sweet. The probability that Hannah eats two orange sweets is ⅓.

(a) Show that n² – n – 90 = 0

(b) Solve n² – n – 90 = 0 to find the value of n."

Look familiar? This question caused massive controversy when it was released in summer 2015 as it was seen as too much like things to come - many felt that it was more like the sort of question we might expect in 2017 when the new '1-9' GCSE is first examined and had no place in the current GCSE. Whether you believe this or not, the point is clear that pupils need to understand the ideas of probability and apply them outside the realms of numerical chance. With that in mind I thought I would share some ideas about developing probability without giving (too many) values.

Probability and Proportion

I am surprised we do not see more links between probability and proportion as ultimately probability is a proportional idea, the chance of something happening is measured as a proportion of the things that are possible or as a proportion of a number of trials in an experiment. In the past proportionality has generally be pretty limited to calculating an expected number of trials that would satisfy the given condition. I think it is clear though that with anything up to 25% of the new GCSE paper content being linked to ratio and proportion I think that we will see a lot more questions linking these two topics in the future. Questions like the ones below could become much more common:

1) A packet of sweets has orange, blackcurrant, strawberry and lemon sweets in the ratio 4:3:2:1. James and Sarah both buy packets of the same number of sweets. James doesn't like strawberry and so gives all of his strawberry sweets to Sarah. Sarah gives James all of her lemon sweets in return. If James takes a sweet at random from his bag, work out the probability that James take a lemon sweet.

2) A childs' shape sorter has red, green, blue and yellow shapes. The number of red shapes is twice the number of green shapes. The number of blue shapes is twice the number of yellow shapes. In total the number of red and green shapes is twice the number total number of blue and yellow shapes. Work out the probability of a child selecting a red shape if the shape is taken at random.

To be fair it strikes me that a lot of ratio and proportion question can be adapted to give a probability question - question (1) above could just as easily be "write down the ratio" rather than "work out the probability" and there are lots of ratio and proportion questions out there that could be adapted to this vein.

Probability and Algebra

Hannah and her sweets have given us a pretty clear indication that this will be a rich source of links for examiners to mine and again it makes perfect sense: if you understand the ideas of probability and algebraic expressions/equations then there should be no reason why you cant apply the two ideas together. We have also seen in the SAMs at least one question that has purely algebraic expressions inside a Venn diagram linked to probability for pupils to work with and I am sure we will see more examples in the coming years.
1)

2) A bag of counters contains red, blue and green counters. There is one more red counter than green, and one more green counter than blue. Stefan takes a counter out of the bag and puts it on the table, followed by a second. The probability that Stefan takes a blue followed by a red is 1/9. Calculate the probability that Stefan takes two greens.

It strikes me that replacing lots of the numbers in current probability questions with letters will generate questions of this type, and so would be well worth some time in faculty meetings designing.

No doubt at this point people will be thinking "yes but probability and statistics will only be 15% or so of the content..." and of course they are right, but don't forget that 15% of 240 marks is a good 36 marks, so there is plenty of space for one of two questions of this type to creep in, particularly as they can also count towards the 20%  to 30% Algebra content or 25% to 20% Ratio content so I would suggest it is well worth building questions like this into your GCSE schemes.

Dimension and Pythagoras

My Year 10 have recently been working with Pythagoras in 3-D objects, and quite typically in my experience they were having difficulty identifying suitable triangles to calculate some of the lengths; particularly those lengths that go through a shape requiring multiple applications of Pythagoras. Being ready for this I decided to try an approach that I had been considering that links the number of applications of Pythagoras' Theorem to the number of dimensions that the line moves through. The approach met with some success and I can see how it might have real potential in linking to dimensional analysis so I thought I would outline it here.

The first and one of the key points was to ensure that pupils understood that Pythagoras' Theorem is a relationship concerning area. Although we often use Pythagoras' Theorem to solve for missing lengths, the actual essence of the relationship is between the areas of three squares where two of them meet to form a right angle. The image below is one that is typically used to illustrate this (and one I have used lots in the past).

Once my pupils understand that this is a relationship area, the discussion is then turned to dimensions. What I am hoping to show pupils is that the two shorter sides of a right triangle are lines that only move in one dimension, whereas the diagonal moves in two (as the two sides are at right angles they can be considered to be two independent dimensions). So Pythagoras' Theorem can be thought of as a relationship that starts with lines moving in a single dimension, and relates them to a line moving in two dimensions (linking to area being a two dimensional concept).

Now let us consider a cuboid like the one below:

An early job of work to do with pupils here is to make sure they can identify lines that move in one, two or three dimensions. Generally for me this leads to lots of gesturing around the room and drawing imaginary lines along walls and floors, as well as between corners across the room. Once pupils understand how these lines are moving we can start looking at which distance can be solved with only one application of Pythagoras' Theorem and which cannot. For example in the cuboid above the distances AF, BD, FC etc can all be solved directly using a single application of Pythagoras' Theorem as they are all lines that move in two dimension. 

Now let us consider a line that moves in three dimensions, for example EC. A tip from me on this, before drawing the triangle in the shape, try drawing the rectangle first (as below). For some reason pupils see this more easily that just the triangle.

The discussion we had here is that we cannot expect to solve for lengths like this with one application of Pythagoras, because this length moves in all three dimensions and Pythagoras' Theorem is a two dimensional relationship. In order to solve this problem we are going to need two applications of Pythagoras, one to relate two of the one-dimensional lines with a line that moves in two dimensions (in the above case either EG or AC will do the job) and then to use this line moving in two dimensions with the line moving in the third dimension to relate to the line moving in all three dimensions.

To some this might seem like overkill, but what it does is give pupils an objective test as to whether a line can be solved using given information - if it moves in more than two dimensions it cannot be solved by a single application of Pythagoras' Theorem using lines that only move in one dimension. Of course from here you can complicate things and look at other three dimensional shapes, begin to make judgements about whether line can be considered to move in only one dimension (right-angled to each other), two dimensions or three dimensions. I also think it reinforces area as a key concept and will provide a nice link to dimensional analysis of different formulae when it comes to looking at that concept in more depth.

So in the future I think I will definitely be talking about Pythagoras' Theorem as an area relationship, and definitely be talking about dimensions that lines move in more formally with pupils; if your pupils are having trouble applying Pythagoras' Theorem to three dimensions why not try it as well?

Proportion and straight line graphs

By now pretty much everyone involved in delivering the new Maths GCSE course in England are aware of the increased emphasis on pupils having proportional reasoning skills. Ratio, Proportion and rates of change as a separate strand of the GCSE is worth up to 25% of the assessed content of the qualification, and will appear linked with lots of other areas of maths. Already we are used to the idea of ratio and proportion appearing in trigonometry, scale diagrams, recipes, value for money, many different contexts; I want to look specifically at a couple of ways proportion appears in straight line graphs.

Most teachers of the new GCSE (and quite possibly old) will be familiar with the obvious relationship that proportion has with straight line graphs; namely the graph of two variables that vary directly with each other. Graphs of the form y = kx are a fairly straight-forward link between proportionality and straight line graphs.

In this graph, the y values and x values are proportional, with the ratio x:y being 3:5. This kind of proportion should be relatively straight forward for any pupil that really understands proportion as an idea, and for those more graphically minded may even help with being able to visualise proportion. So what about this graph:

Clearly this is not a 'proportion' graph in the sense that y and x are not in proportion to each other. However if we take a closer look...

Clearly there is a proportion going on here, but what is it? Of course in this case it is not the variables that are in proportion; rather it is the rates of changes in the variables that are in proportion. Specifically in this case that the change in y is half of the change in x (leading of course to the gradient of ½). 

This proportionality is often overlooked, or at least not made explicit, but given that rates of change is now part of the new GCSE I think it will be worth highlighting the idea of a straight line as a line where the rate of change of y is proportional to the rate of change of x and that this proportionality is where we get the concept of gradient. This may well help pupils when it comes to rates of changes of curves by applying tangents; if pupils are already familiar with the idea of gradient at rate of change because it has been made explicit when working with straight lines the the concept should come more readily when moving on to rates of change of curves.

So in order to ensure your pupils are ready for rates of change at GCSE, consider introducing them not just to graphs where the variables are in proportion, but also where the rate of change of one variable with respect to another is proportion: for if pupils can gain a deep understanding of how gradient links to proportionality then the beginnings of calculus are well within their grasp.