Exploring sequences using graphing technology
This year, we’ve taken a detailed look at how graphing technology can support understanding of various mathematical concepts, from probability distributions to statistics. We want to pass on as many tips as possible so teachers – and consequently students – can get the best out of their calculators.
We love talking to experts to get their first-hand insights into how using a graphic calculator can help students get to grips with some of the trickiest topics. In this post, Amarpreet Singh Kular, who recently joined us at Casio after just over 11 years teaching secondary school maths, discusses sequences and the techniques he found useful when exploring this area with his students.
Thoughts on teaching sequences
Ask any teacher and we’re sure they will agree that, on the surface, this doesn’t seem to be the most difficult topic students will encounter in their studies. It often boils down to understanding patterns and notation, and the ability to decipher which of the four main types of sequences they’re dealing with.
However, during his time in the classroom, Amarpreet often found that sequences become more challenging when skills are combined and interleaved with other topics (for example, linear sequences in a fraction), when dealing with modelling problems at A level, and in questions involving two types of sequences.
Your approach to tackling these challenges will depend on your individual teaching style. One method Amarpreet found useful when he was teaching was starting lessons with regular activities to reinforce what defines particular sequence types and how to identify them. These included brainstorming sessions that asked students to list everything they knew about arithmetic and geometric sequences, such as their definitions, notation, differences and the formulae associated with them.
Looking back on his time in the classroom, Amarpreet said if he had the option to do anything differently, he would place a greater emphasis on how calculators can support learning and enable greater efficiency for teachers.
Graphing technology can open up new ways of exploring sequences, broadening the scope for teachers to find ways into the topic that suit their own style and work for their students.
However, he also stressed it was important for students to understand that the calculator is simply a tool for investigation and exploration. To get the best out of it, they need to grasp the underlying concepts first, and also know how to use the calculator in the right way.
Putting the tools to use
“It’s about students knowing what the calculator can and cannot do. It’s important to understand the functionality,” Amarpreet said.
“The calculator can’t distinguish what type of sequence it’s working with, for example, or what the rule of the sequence is. Students have to be able to understand this themselves.”
Where calculators do offer value is in supporting learning, reinforcing understanding and providing visual tools for students to examine problems.
Amarpreet offered some tips on how to make the best use of the fx-CG50 to explore sequences, along with some specific examples showing what the calculator is capable of.
Check your settings
Firstly, it’s important to select the correct setting for the sequence in question. The calculator’s generic setting is to model recursion sequences, so it’s vital to change this when working with normal sequences, which is possible via the F keys.
Another key point for students to remember when using the fx-CG50 graphic calculator is the need to select ‘n’ as the variable in their sequence, rather than the standard variable button they might be used to. They can then model the sequence and find the value of n at any point, which is useful for problem solving and checking answers.
A common challenge students are likely to come up against in exams is finding the sum of a sequence, which they can do in ‘run matrix’ mode on the fx-CG50. Adding a start and end position, followed by the relevant formula, will allow them to calculate the sum of the sequence.
It’s worth noting that this feature is also available on the fx-991CW, but the recursion app is only found on the fx-CG50.
The recursion app has a number of powerful applications when working with sequences. Among them is the ability to find when a sequence exceeds a certain value, by entering the relevant formula, selecting an end point and seeing at what point in the sequence the value is reached.
Alternatively, students can plot a graph of the sequence by entering the formula and choosing the value they want to reach, then look at where the two lines intersect.
This underlines how having access to a calculator can support students’ learning by offering different approaches to problems and providing visual representations of the mathematical outcomes.
We want to help teachers and students alike benefit from graphing technology by passing on as much information as we can about using calculators in the classroom.
In a free Casio training session, you can hear a qualified maths teacher speak about some of the key functions of the fx-CG50 and how they can be applied to an A level syllabus topic.
Sign up here.