# Getting across what an nth term rule is

On Friday I had one of those ‘this is why I went into teaching lessons’ where everything seemed to come together. What made the experience even more satisfying was the fact that if you’d have been there for the first 10 minutes you’d probably have thought that the lesson was well down the path towards crashing and burning.

The learning objective was to understand what an nth term rule is, what we can use it for and, if possible to get a glimpse into how you find it. I started off on the board with an arithmetic sequence and the term numbers above each term. We developed the discussion around noticing that the sequence ‘went up in threes’, so wrote down the three times table which also goes up in threes. We then discussed how the actual sequence was always one less than the same term in the three times table so the rule for getting the number in the sequence from the term number must be times by three and subtract one.

Blank faces. Silence. Kids deliberately trying to not make eye-contact.

After two more minutes of struggling to get the message across using this explanation I threw the lesson plan in the bin just as a bit of inspiration hit me.

From that moment on we went very kinaesthetic. I got 12 of them lined up at the front of the class holding mini-whiteboards up. The first person in the line had ‘n = 1’ written on their board, the next ‘n = 2’ and so on up to ‘n = 12’ on the board of the last person in the line.

Then I stood at the side of the room so everyone could see me and I held up a white board on which I wrote nth term rules. I started with ‘2n’, then ‘4n’ etc to get things off gently. The pupils had to substitute their number into the nth term rule in their heads and then we went down the line with each person shouting out their answer to generate the sequence. Progress was slow for the first few but within ten minutes they were doing ‘5n -3’ and the like absolutely perfectly.

Once they got to this stage I thought we could really nail the concept of the nth term rule by me jumping on the end of the line and saying ‘I want you to imagine that there are more than twelve people in the line, I want you to imagine there are one hundred. Imagine I’m at the end of the line holding a board that says ‘n = 100′. What number would I be shouting out?’ Instantly the kids all had that lightbulb moment. They all did the substitution using the nth term rule in their heads and a unanimous correct answer rang out! ‘What if I was the fiftieth, two hundredth, ten thousandth person in the line?’ followed as the idea of the nth term rule being able to tell you the answer for any term in the sequence really sank in. It was a lovely moment!

To add cream on top, one bright young lad then said ‘I’ve noticed that when there is a four infront of the n the sequence goes up in fours, and when there is a two infront of the n it goes up in twos’. We tested out his idea with a few more sequences before we discussed how we could mathematise the point i.e. the coefficient of n being the common difference.

During my teacher training, because I was being observed so often, there is no way I’d have deviated from the lesson plan so much and taken such a risk. Now I’m teaching day-to-day with my own classes I feel able to go for it in these situations which, on this particular occasion worked out great. It’s got me thinking about how I can go more kinaesthetic in other topics…

Excellent! Getting the students used to the phrase “nth term” is what I always find difficult.

I tried a different approach the other day as a reminder with my year 8 class, after using sequences to solve a number puzzle.

I asked them to name some numbers that, when divided by 4, gave a remainder of 3. So the students came up with 19, 27, 35, etc etc. When I rearranged the numbers into order, the students noticed that …wow! the sequence goes up in 4’s, but is one less or three more than multiples of four…..

Agreed about the lesson plan! As a student some of my best lessons were the less rigidly planned lessons. I recently taught average from a frequency table using shoe sizes and the students could see how I was finding the median. I then stood them up at the front in a human bar chart by shoe size and then all became clear!

I am a year 7 and I have got abit stuck on my homework can any of you help me plz

Nice story!