Graphing and Models
One of the current trends in mathematics is ‘models’, often connected as ‘functions and models’. What do students bring from their work on graphing in beginning algebra (often linear graphs) to this broader work? Is this an easy transition? Do we face challenges or hidden dangers in this work?
One thing I have noticed is that we often assume facility with basic graphing based on the linear function graphing included in a beginning algebra course. A student can generate a table of values and use those to graph; a student can graph the y-intercept and use slope to find more points on graph to create the line. I suggest that we face a significant gap in knowledge when we present a model to graph on their own.
This is the type of thing I am talking about:
A company finds that it costs $2.50 per glass, in addition to a basic set up cost of $80. Write the linear function for the total cost based on the number of items (glasses). Graph this function for a domain 0 to 100.
The typical beginning algebra class does not prepare students for this work. Here are some of the gaps:
| Typical Beginning Algebra has … | Modeling has … |
| Slope in beginning algebra given just as a value (often a fraction) | Slope in this problem is given as a rate (verbally) |
| Y-intercept in beginning algebra given as an ordered pair (or y-value) | Y-intercept in this problem is stated as an initial value |
| X-values to use … -10 to +10 (domain) | Domain stated verbally |
| Y-values to use … -10 to +10 | Y-values determined by actual outputs |
| Scale for x-axis is provided (normally -10 to +10) | Scale for x-axis must be determined (like 0, 10, 20, etc) |
| Scale for y-axis is provided (normally -10 to +10) | Scale for y-axis must be determined (like 0, 20, 40, 60, etc) |
| Find two points, draw line | Use table of values for points, draw line |
This is not a scientific analysis of the knowledge needed for this problem; there are details at a finer grain of analysis that would show more gaps.
Essentially, this is a problem caused by “Bumper Mathing” (see an earlier post on that). We constrain the graphing environment to the extent that the resulting knowledge is not applicable in any realistic situation. We can do better than this.
“Graphing”, as a collection of related concepts and procedures, is fairly complex yet very useful … and is worth doing well. We can certainly make more room in the algebra course so that students leave with good mathematics and knowledge that transfers.
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Content of developmental math courses, math reasoning and applications | Jack Rotman | 
May 9, 2012 4:42 pm
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Jack Rotman
NOTE: This blog will become 'inactive' on January 1, 2020.