RISPS were something that had been mentioned to me as a good resource for problem solving at A-Level. It can be a daunting experience when A-Level is not something you feel you have a lot of experience in, but I have reached that point were I want to implement more problem solving resources into lessons.
I have started teaching sequences and series to my year 12 group and I thought this would be a great time to implement problem solving beyond that of the textbook. I had done a few lessons on arithmetic sequences and had panned for the next lesson to be a lesson of practise, I decided this would make a great opportunity for problem solving. I looked for resources and remembered RISPS and found a question I thought would be suitable.
Risp 20: When does Sn = un?
Given the sequence u1, u2, u3…, let’s say the sum of the first n terms is Sn.
Consider the sequence 11, 9, 7, 5, … When does Sn = un for this sequence?
Find all values of n so that this is true.
Still with arithmetic sequences:
If you pick any natural number n, and any first term a, can you always find a common difference d so that Sn = un?
If you pick any natural number n, and any common difference d, can you always find a first term a so that Sn = un?
What if we look at the geometric sequence defined by its first term a and its common ratio r?
When does Sn = un here?
Experiment with other sequences/series:
when can Sn = un?
What if un is periodic?
Try the Fibonacci.
What happens if you run this sequence backwards?
What if un is a polynomial function of n?
To see the teacher notes on this RISP go to: http://www.s253053503.websitehome.co.uk/risps/risp20.html
When my students came into the room I set them to work on the appropriate chapter in the textbook and allowed for time for these to be completed. Half way through the lesson I placed the RISP on the board and told the students that if they wanted to attempt something different then they may want to attempt the question on the board. Some of my students decided to attempt the question.
It was brilliant to see students working together and working their way through the problem. One of my students has joined us from the further Maths group (they have decided to go down to Maths) and they were getting stuck into the problem. Two of my students decided to go back to the questions in the textbook and informed me that they wanted time to do some research and investigating into the RISP so they wanted to do it during independent study. Others took the question away with them to complete in independent study. As the RISP mentions geometric sequences and I hadn’t taught that yet, I saw it as an opportunity to encourage them to do some prior learning and look into geometric sequences.
The RISP teacher notes mentions ‘A HA’ moments which I must admit I did see. One of my students did use algebra to answer the question and through this ‘A-Ha’ moments did indeed appear. It was a great opportunity to ask questions to the students to get them to think about what they had found. One of my students made it to r^(x -1) = 1 and decided to take logs of both sides to try and solve this problem. From this, they encountered other problems and decided they needed to go back to r^(x-1) = 1 and remembered that anything to the power of 0 is 1. There was a great lesson in there to be learnt; that basic knowledge needs to be remembered and utilised in A-Level maths.
I am looking forward to seeing how many of my students have attempted this question and seeing what methods they have used. One of the things that I liked about this task was that there were many different ways that this could be attempted. Students can jump straight in with algebra or they can write number sequences and investigate.
I would recommend using this RISP with your students.