Physical World Sequences TN
When learning something new most techniques/methods/strategies can often be categorised into two broad pedagogical approaches, which can be used in isolation, or in combination (with varying degrees of each - depending on the student/class?):
1. ‘Breaking down and building up’
Break the knowledge down into individual, isolated skills and practise them until each skill can be done confidently and fluently. Integrate different individual skills together to be able to complete sequences of skills with confidence and fluency, and with different variations. Practice using these skills with real world problems until you feel confident and fluent in their use. Use the skills you have acquired to solve problems that you set for yourself!
2. ‘Get a feel for the big picture before zooming in on the individual parts’
Start with a holistic view of what you will be able to do with this knowledge once you are fluent and confident with it – get inspired. See an exploded diagram of the different skills required and how they relate to each other. Zoom in on the different skills you need to acquire and pick them off one by one.
This activity hopes to give students a feel for the big picture, and how each of the component parts of analysis, graphs and sequences fit together. It can be used at the end of these topics, or towards the beginning of sequences and/or equations of lines/functions, depending on the teacher’s preferences.
The powerpoint slides can be printed out for students to use in class, or laminated so as to be re-usable with future classes. The cards should be on the tables (in an easily accessible network area on the computer) ready for students to get started on as soon as they arrive. Alternatively, students can use the resource directly in powerpoint, cutting and pasting cards into groups on new slides (this is why the resource is presented using powerpoint). This can save time if cutting out or laminating pre-session isn't possible.
Give students 10 minutes to match up the cards as they see fit.
I would recommend that the teacher be very active during the first 10 minutes, circulating between all the groups to eavesdrop on conversations as to the different strategies being used by students. It can motivate and help students get started if the teacher writes up a selection of different group’s strategies on the board after about 5 minutes (particularly to act as a stimulant for groups that are having more difficulty).
Once a group feels confident they have found a solution they can use calculators, TiNspire, Geogebra, Autograph or other graphing software to check their solutions - the Matching Sequences activity gives more details on how to do this.
Alternatively, they can try the 'WisWeb' virtual manipulative that enables students to check the nth term, sequence and graph connection (demonstration of how to use the manipulative below):
When I've used this activity I have preferred not to show students how to use calculators, software etc. to check their groupings until they have first reasoned and discussed their way to a reasonable solution for each set of five cards.
If students struggle, their task can be constrained to matching just two or three of the cards each time, rather than all five e.g. the number sequence with the graph, or number sequence, graph and equation etc. If they manage to complete this and produce a display they can then be set the challenge of finding the sequence formula that goes with the ones they have already completed, or the physical world statement etc.
This activity provides a great opportunity for display work. I found particularly effective a display students made showing only one set of five cards, with their reasons written on the arrows from equation to number sequence etc.explaining how they worked out that the cards matched. Another group displayed only the graphs and how they had split these into 3 categories: increasing, decreasing and increasing then decreasing.
The following is a suggesiton for how this activity can be run in class.
One or two people (depending on the size of the group) will stay with their table's cards to explain to visitors from other groups the reasons for matching the cards they have matched. The others will go and visit the work of other groups for about 5mins to see if they agree or disagree, looking carefully at each groups reasons if they don't agree.
After visiting other group's work, students have another 5-10 minutes to change any of their groups of cards. This allows students to refine their ideas and clarify any misconceptions in discussion with their classmates. It also allows the teacher to hear what each group is thinking and better understand which student's need what help.
What to Expect
Students who feel unconfident with graphs, equations and sequences can have a tendency to match up any old way just to “get it done”, and not really have any reasons for the decisions they have made. This is where the teacher’s role is crucial in helping students to take the first few steps in analysing this large quantity of information by:
- Encouraging student’s to have confidence in their thoughts, and that even if their solution isn’t a productive one in this instance, it will help the teacher know where the student is coming from and therefore is more likely to result in a useful explanation that can clarify that groups/students particular misconception.
- Have a range of simple approaches ready to get them started, here are some strategies that I find student’s typically tend to start using e.g. :
- Which number sequences and graphs are increasing and which are decreasing?
- Which set of numbers, graphs, equations etc. do you think first increase and then start to decrease ?
- Read off the coordinates from one of the points on the graph – in which number sequence can you find this coordinate?
- Pick a graph and look at the scale on its X and Y axis, which number sequences do this match? Using your calculator, substitute one of the ‘n’ values from a number sequence into one of the ‘sequence formulae (nth terms)’, does the calculator's answer match the T(n) column?
- Which of those equations represent a straight line (constant gradient/slope)?
I find the use of ‘n’ and T(n) for sequences, which uses set notation, often confuses students. They’ve just got comfortable with y=3x+2 type relationships and now someone’s talking about T(n) = 3n + 2. They get confused as to what this might mean and why the change from ‘x’ to ‘n’ and often won’t make the link, unless helped by the teacher, that these two equations express the same relationship. If they understand one, they can use it to understand the other. One of the aims of this activity is to help students make this link.