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?