Lessons: Using technology
Introduction
Technology can be very useful as a tool when modelling. For many years now many people have recognised that, particularly when working with real data, calculators are essential. Other technology is equally important when dealing with many problems: in workplaces, for example, workers often use computers with software such as technology. This module will allow participants opportunities to consider how “generic” software such as spreadsheets, graph-plotting and dynamic geometry software can be used in a range of different situations.
It is best if technology is available for participants to work with. A few computers (perhaps one per group) with suitable software loaded and a number of graphic calculators would be useful.
Materials for participantsTeacher diary pages for this sub-module
The module starts by asking participants to audit their current use of technology in mathematics, to think about professional development needs in this regard and to think how technology might be useful to students when modelling.
After having an opportunity to think through some of the issues whilst engaging in some modelling activities where they may use technology themselves they are asked to reconsider this latter point.
Please distribute all of the Teacher Diary pages for this sub-module at the beginning of the session.
Resources L.4.1, L.4.2, L.4.3, L.4.4 (Tasks)
Materials you will need
Power Point file: Lessons_using technology.ppt
Dynamic geometry files: playground race 1, playground race 2, playground race 3, fencing 1, fencing 2, garage door 1, garage door 2
Spreadsheet files: playground race 1.xls, playground race 2.xls, fencing 1.xls; fencing 2.xls, cooling.xls
Introducing the sub-module
Allow about 2 hours
/ This module focuses on how technology (the use of computers and graphic calculators) can enhance mathematical modelling and provide opportunities for students to learn mathematics including modelling competencies and also how to use “mathematical” software/technology.
/ There are many possible ways in which technology can be used in mathematics classrooms. Here we will restrict discussion to how “generic” software and purpose built “mathematical” calculators can be used as tools when modelling.
/ Unusually we start the module by asking teachers to take a little time to consider their current use of technology and to spend a few minutes to think of how they might use technology in, and for, modelling.
You may want to allow some discussion about this in pairs, small groups or the whole group.
/ This slide suggests ways in which technology can be helpful when modelling………
….. perhaps it will be a good idea to return to this at the end of the session as by then participants will have had an opportunity to use technology to do some modelling themselves
Main activities of the sub-module
/ This sub-module can be used in a number of different ways depending on the facilities you have available.
In the ideal situation you will have some technology for participants to use. However, this is not essential. Even without technology participants can work on a problem and try to identify where, and how, technology would have been helpful: perhaps you can then demonstrate this with some of the files provided.
Four problems are suggested: they are available in Resources L.4.1, L.4.2, L.4.3 and L.4.4. Perhaps each group of teachers could work on one of them. Ideally you will have some computers available with spreadsheets, dynamic geometry and possibly graph plotting loaded. A number of graphic calculators could also be made available.
Before setting groups off working on the problem run through the activity details on the next slide.
/ If you wish, you may ask the whole group to work on just one of the problems.
/ Ask each group to produce a poster of their solutions to the problem and to illustrate this with their thoughts about using technology.
These posters can be used to prompt discussion.
You can use the remainder of this powerpoint presentation to illustrate other ideas if you wish.
Each of the problem boxes here hyper-links to the appropriate part of the presentation – at the end of the section you can return to the “problem page” page by clicking on arrows like this:
You will need Resources L.4.1, L.4.2, L.4.3 and L.4.4.
To move to the plenary discussion at the end of the session click on the arrow:
/ This is a modified version of a problem participants may have met in earlier sub-modules. It has been modified here to make it slightly easier to approach using dynamic geometry software.
/ First of all investigate a simple, if not the simplest, possible situation.
/ One way to tackle the problem is to draw the situation a lot of times and measure the distance that pupils in the race will run. The drawings could be to scale.
An alternative is to use technology to investigate the situation lots of times. Dynamic geometry software can be used as shown here (file: playground race1). As the “point” is dragged along the fence the distances are measured and you can investigate to see where the total distance is least.
/ Dynamic geometry software can be used to quickly graph how one variable varies with another.
Here the distance of the point where the pupils touch the fence from the point on the fence opposite the starting tree has been plotted as x and the total distance of the race as y.
Now perhaps a solution for this simplified situation is available.
(file: playground race 2)
/ Dynamic geometry software also allows tables of data to be collected.
(file: playground race 3)
/ The tabulated data may be transferred into a spreadsheet where it may also be graphed (file: playground race1.xls)
At this point you might like to consider how more advanced pupils might like to tackle this using Pythagoras’ theorem.
They may be able to see if they can find the function that fits the data by considering the spatial situation.
/ Think through how some of the initial simplifying assumptions might now be adjusted.
This is perhaps where dynamic geometry provides a particularly powerful exploratory tool.
You can return to the “problem page” page by clicking on the arrow:
/ First of all investigate the simplest possible situation.
/ One way to tackle the problem is to draw the situation a number of times so that students get a “feel” for what is involved.
Perhaps then the results can be collected in a table and a graph drawn (this can be replicated using a spreadsheet).
An alternative is to use dynamic geometry software to investigate the situation lots of times. A possible construction is shown here (file: fencing 1). As the point P is dragged the lengths of the fencing vary and the area of the run is calculated by the software.
This allows you to see when the area is a maximum.
/ Dynamic geometry software allows you to quickly tabulate how the area varies as the length of one of the sides is varied.
(File: fencing 2)
/ This data can be exported into spreadsheet software and graphed easily allowing a solution to this simplified situation to be reached.
(File: fencing 1.xls)
/ Alternatively this modelling problem can be explored using a spreadsheet from the outset – and this is a useful way to develop algebraic thinking…..
This slide shows how pupils might first of all develop a spreadsheet formula that relates the length of the two sides to the overall length of fencing available.
(File: fencing 2.xls)
/ This slide shows the spreadsheet extended to calculate the area of the run.
/ And finally a graph can be plotted of area against one of the side lengths.
/ Think through how some of the initial simplifying assumptions might now be adjusted.
You can return to the “problem page” page by clicking on the arrow:
/ First of all, the garage door mechanism needs to be understood.
Perhaps students should then try constructing a side - view of the situation.
To do this they will have to make some assumptions about the lengths involved and make their drawing to scale.
Typically a garage door might have a height of 2 metres.
/ Groups of students could work to construct a number of diagrams with the door in different positions.
Dynamic geometry allows you to investigate this with just one construction. (File: garage door:1)
/ The animation facility coupled with tracing the movement of the door allows you to see at a glance the space needed for the door as it is opened and closed.
Of course the solution depends on the height of the car’s bumper.
Depending on the ability / age of pupils this particular problem could be tackled using parametric equations.
(File: garage door:2)
/ Think through how some of the initial simplifying assumptions might now be adjusted.
You can return to the “problem page” page by clicking on the arrow:
/ This slide gives some real data – however, participants may like to collect their own. Graphic calculators can be used to do this if a data logger and temperature probe is available.
/ Here the data is shown entered into a graphic calculator and a graph plotted.
Students might like to consider if they can model the data using a function. Of course, most students will not have knowledge of exponential functions but linear functions may be used to model parts of the data.
Will they be sufficient if you are considering cooling over a relatively short period of time?
/ Another possibility is to explore an iterative process, reducing the temperature by the same proportion over each time interval.
Draw attention to the screenshots here which show how to carry out such an iterative process very quickly.
Plenary discussion
/ Returning to the objectives of the sub-module emphasise that you have only had an opportunity to touch on some of the many ways in which technology can be used as a tool to support mathematical modelling.
/ The next slide returns to suggest some ways in which technology can be useful: this slide suggests three major technology tools that allow such activities to be undertaken.
/ Encourage participants to use their teacher diaries to plan to ensure that their students can use technology to help them with mathematical modelling.
LESSONS ▪USING TECHNOLOGY▪ page 1
© LEMA 2007