Teaching about functions with real world problems and a graphic approach and with optional CAS supplements
Recommended resource
Asp,G., Dowsey, J., Stacey, K. & Tynan, D. (1995) Graphic Algebra: Explorations with a function grapher. Carlton,VIC: Curriculum Corporation.
About GRAPHIC ALGEBRA
Linear Functions
Quadratic Functions
Exponential Functions
Recommendations for teaching with Graphic Algebra and with technology
About Graphic Algebra
This book provides guidance and suitable problems for approaching the teaching and learning of a range of algebra topics through explorations with a function grapher. There are chapters on linear functions, quadratic functions, exponential functions and reciprocal functions, as well as chapters treating issues of scale.
Key Curriculum Press publishes Graphic Algebra: Explorations with a graphics calculator as a series of blackline masters providing complete curriculum units, or activities to supplement other programs. Graphic Algebra is available in Australia from Objective Learning Materials.
Using Graphic Algebra for Linear Functions
Graphic Algebra is written for students with access to computer/calculator graphing, but a supplement which provides introductory CAS activities has been prepared. The first introduces some simple CAS use into the Graphic Algebra text, and the second is a standalone activity focussing on solving equations by "do the same to both sides (zipped rtf) " with the aid of CAS.
We have used this approach to linear functions from Chapter 1 successfully at the year 8/9 level, with students who had class access to TI83+ calculators.
We evaluated the success of the program using a pretest and posttest and an article reporting on the students' learning has been prepared by Caroline Bardini, Robyn Pierce and Kaye Stacey.
Using Graphic Algebra for Quadratic Functions
Chapter 2 provides a series of problems to explore quadratic functions. The fencing problem can be supplemented by this dynamic geometry file.
Using Graphic Algebra for Exponential Functions
The exponential chapter takes students through a variety of tasks begining with a simple double function in a paper tearing context and moving through to looking at population doubling times around the world.
A supplement to introduce the use of the symbolic features of the TI89 is provided here for Paper Tearing and Ozone Layer problems.
A pretest and posttest are also available.
Recommendations for teaching with Graphic Algebra and with technology
Advice on using all chapters
Graphic Algebra can be used independently by students, to work at their own pace. However, to keep momentum of learning in this sequence of work, we suggest ensuring regular whole class review sessions, including using view screen to model appropriate graphic calculator use.
If students mainly work at their own pace, it is a good idea to do some problems ‘together' as a class, with input from students. Students may, in turn, be responsible for using the graphic calculator attached to the view screen and so demonstrate both their maths and their use of the technology.
Some students may feel that they should not use worksheets when parents have paid for text books. However, there is a need for extra ‘word problems and also for some attention to abstract problems, not set in a context to practice traditional algebraic methods.
Advice on Teaching with Technology
Students were able to learn about using the new technology (in this case TI_83+) at the same time as learning new mathematics, although the progress on the new mathematics was somewhat slower than otherwise. Both boys and girls were initially excited about using calculators, but the novelty does wear off!
It is best if students can have access to calculators outside class, as well as during all lessons. Using the calculators only in class is less than ideal but may be a necessity for many classes. In this case, we recommend clear expectations are set for what work students have to achieve during class time, since setting homework to catch up will not be possible without the graphics calculators.
If students must use the same calculator each class, this requires organization and extends time for distribution.
Learning to use graphics calculators is not trivial. It requires targeted teaching because having students learn only by trial and error is inefficient
Teacher needs poster or overhead to indicate appropriate buttons to press in common sequences.
Teacher needs view screen available for demonstration at all stages of the program.
Students had some difficulties – their explorations and ‘play' led to changed calculator settings that caused errors for graphing. Students need to be sure they know how to cleanup, clear memory and reset defaults.
Students had difficulty setting appropriate graph windows. This is a major theme of the Graphic Algebra program. Students many need help with strategies for doing this.
Students need to be made aware of the range of menu options – i.e. scrolling beyond the immediately visible list.
Students need simple ‘how to use calculator' notes for reference. The appendices of Graphic Algebra are designed to be handed out for this purpose.
Students needed at least one guided stepbystep example for each major activity.
Advice on using problems in context
Students were generally initially interested in each example but needed to be kept moving. Students seemed to ‘slow down' as the interest waned.
By learning in context, students seemed to be better than normal at solving other problems in context, including the standard word problems in the textbook. In addition, they would have seen situations where linear functions are useful and likely to arise in everyday life.
These context problems provide little ‘abstract' practice – we recommend that this be added. A suitable selection of standard text book problems could complement these context problems.
Most students enjoyed the change of style and the challenge of the ‘transformation creations'. The lack of visible scale on graphics calculator graphics means that unless students are guided to ‘trace' or find ‘intercepts', then the details of transformations and position of graph in relation to values of coefficients is not transparent. However students should focus on general patterns, not necessarily making exact reproductions of those images in book.
There is a need to take care with the armspan versus height problems. These may not adequately demonstrate the role of coefficients in producing a graph. The purpose of this exercise is demonstrate that a better fit may be obtained by altering either m or c . However, if the domain is small, several different visual effects may prevent students seeing the generalisation. Students saw a value 0< m <1 as ‘lowering' the graph not as changing the gradient. They did recognize that c<0 lowered the graph.
