http://www.MathEdPage.org/func-diag/index.html
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Function Diagramsby Henri Picciotto | |||
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http://www.MathEdPage.org/func-diag/index.html
Some rights reserved
| Visit Henri Picciotto's Math Education Page. | Send me e-mail . | ||
|
| |||
Function Diagramsby Henri Picciotto | |||
|
| |||
| |||
These changes are motivated in part by the availability of electronic graphers, and in part by sound pedagogical thinking: different representations are more accessible to different students, and the ability to switch representations both reflects and facilitates depth of understanding for all students. I very much support these changes. (See for example the article A New Algebra, and the graphing calculator lessons on this site.)
One underused representation of functions, which I have found educationally rich, is the one with parallel x and y axes. I heard about it in a talk by Martin Flashman (who got the idea from a calculus book by Sherman Stein, I believe). I was inspired to develop some lessons for the textbook I co-authored with Anita Wah (Algebra: Themes, Tools, Concepts -- ATTC), using the name "function diagrams" for that representation. I also read an interesting paper by Abraham Arcavi about the use of this representation in teacher training. Indeed, even if teachers never introduce this representation to their students, they gain some understanding from thinking about it themselves.
On this page, I introduce function diagrams, and provide a number of lessons and activities to use them at various levels in secondary school. Of course, this is meant to complement, not replace other approaches to these ideas. Indeed, function diagrams are essentially a pedagogical tool, while the other representations of functions have a power and reach that make them life tools.
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A function is a way to assign a single y value (an output) to each x value (input). It can be thought of as a set (perhaps infinite) of ordered pairs (x,y). In secondary school, we work mostly with functions on the real numbers. For example, the function y = 2x - 3 can be looked at in tabular, numerical form:
| x | y |
| 3 | 3 |
| 2 | 1 |
| 1 | -1 |
| 0 | -3 |
| -1 | -5 |
| -2 | -7 |
| -3 | -9 |
The (x, y) pairs listed on the table can also be represented as points on a Cartesian graph:
A function diagram is yet another representation of this, with parallel x and y axes. Each (x, y) pair is represented by a line segment connecting the input point on the x axis to the output point on the y axis. The line segment is called the input-output line. (In some cases, it is preferable to draw the whole line, instead of just a line segment. See "The Focus" below.)
Note that while on a cartesian graph one can show as many points as one wants, a function diagram becomes illegible if too many input-output lines are drawn.
In general, do not ask your students to spend too much time drawing function diagrams: it is tedious, and not all that enlightening. There may be exceptions to this, but it is usually more effective for students to analyze existing diagrams, to discuss diagrams you create on the overhead projector, or to use computers or calculators to create the diagrams.
Electronic Tools: A good discussion of function diagram basics can be had with this animation. See the function diagram applets page for more animations, and the function diagram software page for Cabri figures and TI calculator programs you can download.
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One way to motivate many of the rules for operations with signed numbers is to generalize from patterns that work for positive numbers. For example:
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3 + 4 = 7 3 + 3 = 6 3 + 2 = 5 3 + 1 = 4 3 + 0 = 3 + (-1) = 3 + (-2) = 3 + (-3) = 3 + (-4) = |
Another way to think about this pattern, is to recognize the function 3+x=y, which can be represented by a function diagram:
It is easy to figure out how the diagram can be extended, and thus to fill out the rest of the pattern.
Similar methods can be used to explore other patterns of this type, thus motivating the rules for signed number arithmetic, and providing students with a way to reconstruct them on their own.
| Connection: For a manipulatives-based approach to motivating signed arithmetic rules, see my book The Algebra Lab: Middle School (Where to get it.) . This approach can also be found in The Algebra Lab: High School, and in the back matter of the Teacher's Edition of ATTC. The manipulatives-based model is perhaps more accessible than the pattern-extension approach described above, which may itself be more accessible than models based on motion on the number line. All these approaches complement each other -- decide which ones you have time for, and how to articulate them. |
Download Nine Function Diagrams, a worksheet slightly edited from the ATTC Teachers' Resource Binder.
Teacher's Notes: This is a rich activity, which can generate discussion about signed numbers, inverse operations, opposites and reciprocals, the use of symbols... It deserves plenty of time, perhaps two class periods. Before answering the questions, students will need to figure out the formulas for each diagram -- many find it useful to first make tables of values. Diagram f is the most difficult, and you can allow students who have trouble with it to temporarily skip it.
The worksheet sets up a crucial review of fundamental concepts of arithmetic, and of basic algebraic notation, in a novel format. The novelty makes for a lesson that is accessible and interesting to a wide range of students. The concepts reviewed include the following:
These ideas are not explicitly addressed in the questions, but they are necessary to a full understanding of the answers. It is the teacher's responsibility to bring those out in class discussion.
Note that some students are surprised or even resentful that two consecutive questions could have the same answers as each other. That is a good opportunity to explain that what makes an answer correct is its mathematical validity, not such an incidental feature as whether it is the same answer as the previous one.
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Any interval in the domain is "magnified" by a continuous function into an interval in the range. The magnification is the factor by which we multiply the directed length of the x interval to get the directed length of the corresponding y interval.
For example, in the first diagram below, the magnification is 2 for any x interval. However in the second diagram, the magnification depends on the selected x interval. For example, for the x interval from 0 to 4, the magnification is 1/2, however for the x interval from 1 to 4, the magnification is 1/3.
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The magnification is constant only for linear functions. (We will discuss this in more detail below.)
Of course, magnification is just another word for the rate of change of the function. Slope in a cartesian graph and magnification in a function diagram are two representations of the same underlying idea (rate of change), and are calculated in the same way. But using different terminology in different contexts has some pedagogical value: it allows students to approach the concept afresh in a different context, and gives them a chance to make the connections themselves.
The composition of two functions is shown with linked function diagrams. For example:
Here is the resulting function:
Questions for discussion:
| Connection: When a function is composed with itself, it's called iteration. Function diagrams provide an excellent tool to help develop an understanding of what happens in the linear case. (See Iterating Linear Functions on this site.) |
Download a worksheet about composition and magnification, based on the figures above.
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The remaining activities are about the function diagrams of linear functions. In those diagrams, all the input-output lines meet in a single point callled the focus, or else they are all parallel, and the focus is a point at infinity.
Download the worksheet Focus on Function Diagrams and the figure Sixteen Function Diagrams. Together they constitute a lesson from ATTC, intended for use later on in an Algebra 1 course, or as review in a subsequent course. An alternative to students doing the whole lesson on paper is to have them work on #5 in small groups, then to have class discussions of the ideas in #1-4 and #6-20, and finally to have students do #22 individually, perhaps as homework. This could take two or three days, but it is time well spent, as it provides another way to think about familiar ideas that are sometimes remembered mechanically.
Note that the lesson described above assumes without proof the existence and uniqueness of the focus. Download the worksheet Geometry of Function Diagrams, which addresses this question with the help of geometry.
| Connection: An interesting result about the diagram of y=1/x, and its proof, using geometry. |
In the case of linear functions, the focus contains all the information about the function, while individual ordered pairs are represented by lines. This is exactly the dual of the situation in a Cartesian graph, where individual ordered pairs are represented by a point, and the whole function is represented by a line. This leads to an interesting application of function diagrams. A system of linear equations in two variables can be solved graphically by seeking the intersection of two lines. In a dual fashion, the same system can be solved graphically by joining the foci of the two functions. The resulting line (if it exists) represents the unique ordered pair solution of the system. The cases where there are no solutions or an infinite number also have a clear representations with function diagrams. Download Representing Linear Functions, a lesson from ATTC on this duality.
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Arcavi, Abraham and Rafi Nachmias (1989) "Re-exploring familiar concepts with a new representation." Proceedings of the 13th International Conference on the Psychology of Mathematics Education, 3, 174-181. (back)
Anita Wah and Henri Picciotto, Algebra: Themes, Tools, Concepts, Mountain View, CA: Creative Publications, 1994.
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| Visit Henri Picciotto's Math Education Page. | Send me e-mail . |
