Visual Sequences TN
Arithmetic sequences can be wonderfully intuitive and lend themselves nicely to wanting to generalise. Attaching a physical representation to them gives a helpful meaning. The practical task and the blank canvas allows everyone in at the beginning. Students tend to bounce of each other and get more and more creative as they build an understanding of what they are learning. Adding visual aid to this concept helps no end! Photographs help to make this a memorable experience that students can always draw upon to help remember how they first understood the concept.
The following is some practical advice about how the activity might be run.
The whole activity can is written on the associated worksheet.
Multilink cubes and a little bit of space. A camera is also handy for recording and sharing the work. If computers are available then students can add photographs to their records which is nice.
As usual much depends on time available. The activity can be done in hour if there is not too much dwelling! The start is where time can go quickly.
Starting and finishing
- The activity can be directed from the front without the worksheet in the beginning introducing the worksheet later or it can be simply presented at the beginning. This depends on teacher preference. My preference is to present the simple question and at some point during the subsequent activity hand out the worksheet and ask students to record some of their thoughts.
- The first key turning point is classifying the models into those that use a number of cubes corresponding to the term of the sequence and those that don't, and then into those that grow systematical and those that don't. The trick is to turn the emphasis onto the systematic ones without crushing any creativity!
- The significance of the system is that it reinforces the idea or origin of 'the common difference'
- The worksheet suggest that students should repeat with a different sequence. The teacher should decide if there is time or if they want to do this. It may be preferable to move on to generating different terms of the sequence.
- The focus should shift on to generalising for the nth term of a sequence and defining the associated variables.
This activity can be a good one to keep records of and the worksheet offers a perfect opportunity to do so. Students should try to complete all of the questions. If the facility exists to take and share photographs then this powerful visual aid can be added to great advantage.
This paragraph talks about the the sorts of things to expect and watch out for during this activity and the possibilities that exist within in it to change or extend the task.
- As suggested above, if you offer an open task like this you should expect students to be creative and not necessarily to do what you might choose. Hopefully some students will offer the sorts of models that help them to make progress and you can steer the rest of the class in this direction.
- Not surprisingly, there is a temptation for some students to be less constructively creative with the cubes and start building guns! Its worth stopping to talk about why that is such an instinctive thing to do! Teachers need to find their own way to manage this, but the task usually sucks everybody in.
- There is usually a nice moment when students stop wanting to build and start wanting to calculate and it is great to stop and point this out.
- Its also nice to stop and carefully define variables as students start to use them. Introduce the IBs preferred variables as they come up.
- There is an interesting confusion between the different ways of expressing the nth term. Students are often exposed to the nth term concept through a different method where it is expressed in the form y = mx + c where m is the common difference and c is the associated translation. Working out the nth term as the first term plus a multiple of the common difference is generally more intuitive. Its good to take the opportunity to marry these two together at this point to reinforce that they are essentially the same thing. A section of the worksheet covers this.
This activity can be followed by 'Arithmetic Sum' which also makes use of multilink cubes and aims at deriving general formulae. There is also a worksheet on arithmetic sequences.