Tricks and Tips 2: Sequences, Linear Graphs and Surds

I recently presented a workshop at the National Mathematics Teacher Conference (#mathsconf2015) entitled 'Tricks and Tips: Clever Methods for Explaining Mathematical Concepts'. This is the second in a series of posts summarising the content of that workshop for those who were unable to attend. The aim of my workshop was to encourage people to reflect on their subject knowledge and the effectiveness of their explanations. I also hoped that delegates would learn new methods that they might consider using at school. In today's post I'm covering sequences, linear graphs and surds. My previous post was on methods for finding a Highest Common Factor.

Linear Sequences

What's the n^th term of the sequence below?

5, 8, 11, 14, 17, ...

It might only take you a second to work it out - how did you do it? This is another one of those topics where there's lots of different approaches. Ask the same question to a random sample of teenagers from across the UK and you'll see a wide range of methods used.

Here I'm going to describe four different methods (there may be more!). As you read them, consider whether you're going to stick with the method you currently use, or try something new.

1. Zeroth term
Inspired by this post on Don Steward's blog, I taught the 0^th term method for the first time this year. To work out the value of q in the n^th term U_n = pn + q, we simply step back from the first term in the sequence (ie q = U₀).

In the example 5, 8, 11, 14, 17..., subtract 3 from the first term to get U₀ = 2. So the n^th term is 3n + 2. This method is so quick that it's now my preferred method for working out n^th terms.

When I taught this method it went well, although a misconception did surface later on. When students were asked to 'write down the first 3 terms of the sequence 4n + 3', some of them gave the answer 3, 7, 11 (ie they started their sequences from n = 0 instead of n = 1). This is something to be aware of next time I teach this method.

Jumping along a line - Median Don Steward

2. Shifting times tables
'Shifting times tables' is a popular method. The idea is that we compare our sequence to a times table.  In the example below, compare the sequence to the 3 times table. Ask your students how to shift the three times table to get the sequence and they'll spot that they need to add two, so the  n^th term is 3n + 2.

If you plan to use this method for the first time then I recommend this NRICH article Shifting Times Tables, which comes with an interactive tool. If you subscribe to MathsPad, they also have an interactive tool for identifying shifts.

This method works well for quadratic sequences too (a topic on the new GCSE syllabus). For example to find the n^th term of the sequence 4, 7, 12, 19, 28 we can compare it to n² and notice that it is shifted up by 3 (ie the n^th term is n² + 3).

3. Formula
It's fairly straightforward to derive the following formula for the n^th term:

Why do we restrict this formula to A level? I once took on a GCSE class in Year 11 and asked them to find some n^th terms in a revision lesson. I was surprised to see them all using 'the A level formula' - their previous teacher had taught them it, and why not? They seemed quite happy with it. I wouldn't use it with my Year 7s though - their algebra skills are very basic when they first meet sequences. In our example, this is what they'd have to do to find the n^th term:

Perhaps this is more effort than necessary. The 0^th term method is considerably quicker.

4. Substitution
This is the way I taught sequences for years. To work out the value of q in the n^th term U_n = pn + q, we substitute a value for U_n, p and n, then solve for q. For example in the sequence below, we know that U_n = 3n + q. We substitute the first term to get 5 = 3 + q. Therefore q = 2.

Thinking about it, this is the same method I'd use to find the value of c in y = mx + c if I know the gradient and a point on the line. In fact given that linear graphs are simply graphical representations of linear sequences, any methods for finding the equation of a straight line work for finding the n^th term of a sequence. I don't do enough to make this connection with my students.

Linear Graphs

A straight line through the point (5, 7) has gradient 4. How would you find the equation of the line?

As I said in my post about linear graphs, I teach this differently at GCSE and A level. Wouldn't it be better to pick my preferred approach and stick with it? 

At GCSE my students write down y = mx + c and substitute values for y, m and x, then solve for c (let's call this Method 1). At A level my pupils use the formula y - y₁ = m(x - x₁) (let's call this Method 2). Look at the steps involved - for this question, the methods are equally efficient.

My current Year 10s had an American teacher last term - she covered linear graphs with them. When we came to revise this topic, they took a while to figure out where to start. Evetually I heard one say, "is this the point-slope thing? What was that formula again?". 

OK, so they'd been taught to call gradient 'slope', that's easily fixed. But they'd also been taught Method 2 and they couldn't remember the formula. This is a problem. Given that there's less to memorise, I think they'd be better off with Method 1. 

It's worth reading @srcav's post The Straight Lines Debate for more analysis and opinion on the two methods.

I find that all my students really struggle with linear graphs. It's one of those topics that frustrates maths teachers because it's hard to see exactly what it is that students find so difficult. I teach it over and over again from Year 8 to Year 12 and my students never seem to remember it from the previous year - I'm clearly doing something wrong!

One suggestion (thanks to @letsgetmathing for this) is to approach this topic from a less algebraic perspective - instead, focus on a table of coordinates. This is demonstrated in the example below. Students have to recognise that the y-intercept is at x = 0 and the gradient is the 'step' in the y values.

If your students struggle with the algebra-heavy Method 1 and Method 2, perhaps this table method might work.

Finally, I recommend that all teachers watch this video from James Tanton. He makes the concept of the equation of a line really clear.

3. Surds
Let's finish with three methods for simplifying surds.

In the book Nix the Tricks (essential reading for all maths teachers), we're shown a teaching trick called 'Jailbreak Radicals'. Thankfully I've never seen this 'zero conceptual understanding method' used in the UK.

Instead, most teachers tell students to identify a square factor and then split the surd accordingly eg √45 = √9√5 = 3√5.

Sometimes students struggle to spot square factors. In this case they might prefer to simplify using prime factors.

A geometric method is explained in the extract from Nix the Tricks below. Try to simplify a few surds like this yourself and see what you think. Notice that you still need to identify a square factor.

Source: Nix the Tricks

The feedback from my conference workshop suggests that some people find this unnecessarily complicated. But if you're going to try it, I recommend this post by @ChrisHunter36 and this resource (pages 5 - 6) by @314Piman.

That's it for today's post. I hope you've found it helpful. Please comment below or tweet me if you know of any alternative methods for teaching these topics. My next post will be all about quadratics - sketching, expanding and factorising.