Posted in Humor

Christmas GeoGebra Applet

candlesHere’s wishing you the choicest blessings of the season. The applet was adapted from the work of Wengler published in GeoGebra Tube. Being an incurable math teacher and blogger I can’t help not to turn this unsuspecting christmas geogebra applet into a mathematical task.

Observe the candles.

1. When is the next candle lighted?

2. On the same coordinate axes, sketch the time vs height graph of each of the four candles?

3. What kind of function does each graph represents?  Write the equation of each function?

3. If the candles burns at the rate of 2 cm per second and all the candles are completely burned after 20 seconds, what is the height of each candle? (Note: These are fast burning candles 🙂 )

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Posted in Algebra

Strengths and limitations of each representation of function

Function is defined in many textbooks as a correspondence between two sets x  and y such that for every x there corresponds a unique y. Of course there are other definition. You can check my post on the evolution of the definition of function. Knowing the definition of a concept however does not guarantee understanding the concept. As Kaput argued, “There are no absolute meanings for the mathematical word function, but rather a whole web of meanings woven out of the many physical and mental representations of functions and correspondences among representations” (Kaput 1989, p. 168). Understanding of function therefore may be done in terms of understanding of its representations. Of course it doesn’t follow that facility with the representation implies an understanding of the concept it represents. There are at least three representational systems used to study function in secondary schools. Kaput described the strengths and limitations of each of these representational systems. This is summarised below:

Tables: displays discrete, finite samples; displays information in more specific quantitative terms; changes in the values of variables are relatively explicitly available by reading horizontally or vertically when terms are arranged in order (this is not easily inferred from graph and formula).

Graphs: can display both discrete, finite samples as well as continuous infinite samples; quantities involved are automatically ordered compared to tables; condenses pairs of numbers into single points; consolidates a functional relationship into a single visual entity (while the formula also expresses the relationship into a single set of symbols, individual pair of values are not easily available for considerations unlike in the graph).

Formulas/ Equations: a shorthand rule, which can generate pairs of values (this is not easily inferred from tables and graphs); has a feature (the coefficient of x) that conveys conceptual knowledge about the constancy of the relationship across allowable values of x and y — a constancy inferable from table only if the terms are ordered and includes a full interval of integers in the x column; parameters in equation aid the modelling process since it provides explicit conceptual entities to reason with (e.g. in y = mx, m represents rate).

It is obvious that the strength of one representation is the limitation of another. A sound understanding of function therefore should include the ability to work with the different representations confidently. Furthermore, because these representations can signify the same concept, understanding of function requires being able to see the connections between the different representations since “the cognitive linking of representations creates a whole that is more than the sum of its parts” (Kaput, 1989, p. 179). Below is a sample task for assessing understanding of the link between graphs and tables. Click solutions to view sample students responses.

tables and graphs

How do you teach function? Which representation do you present first and why?

Reference

Ronda, E. (2005). A Framework of Growth Points in Students Developing Understanding of Function. Unpublished doctoral dissertation. Australian Catholic University, Melbourne, Australia.

Posted in Algebra, Assessment

What are the big ideas in function ?

Function is defined in many textbooks as a correspondence relationship from set X to Y such that for every x (element of X), there is one and only one y value in Y. Definitions are important to know but in the case of function, the only time students will ever use the definition of function as correspondence is when the question is “Which of the following represents a function?”. I think it would be more useful for students to understand function as a dependence/covariational relationship  first than for them to understand function as a correspondence relationship. The latter can come much later. In dependence/co-variational relationship “a quantity should be called a function only if it depends on another quantity in such a way that if the latter is changed the former undergoes change itself” (Sfard, 1991, p. 15)

The concept of change and describing change is a fundamental idea students should learn about functions. Change, properties, and representations. These are the big ‘ideas’ or components we should emphasize when we teach functions of any kind – polynomial, exponential, logarithmic, etc. Answer the following questions to get a sense of what I mean.

1. Which equation shows the fastest change in y when x takes values from 1 to 5?

A.     y = 4x2               B.     y = -2x2                C.     y = x2 + 10              D.     y = 6x2 – 5

2. Point P moves along the graph of y = 5x2, at which point will it cross the line y = 5?

A. (5, 0) and (0,5)      B. (-5, 0) and (0,-5)     C.  (1, 5) and (-1, 5)    D. (5, 1) and (5,-1)

3. Which of the following can be the equation corresponding to the graph of h(x)? 

A.  h(x) = x3 + 1           B.   h(x) = x3 – 1

C. h(x) = 2x3 + 1          D. h(x) = 2x3 + 4

4. The zeros of the cubic function P are 0, 1, 2. Which of the following may be the equation of the function P(x)?

    A.  P(x) = x(x+1)(x+2)       B. P(x) = x(x-1)(x-2)        C. P(x) = x3 – x2         D. P(x) = 2x3 – x2 – 1

5.  Cubes are made from unit cubes. The outer faces of the bigger cube are then painted. The cube grows to up to side 10 units.

The length of the side of the cube vs the number of unit cubes painted on one face only can be described by which polynomial function?

A. Constant    B.  Linear       C. Quadratic      D.  Cubic function

Item #1 requires understanding of change and item #5 requires understanding of the varying quantities and of course the family of polynomial functions.

Of course we cannot learn a math idea unless we can represent them. Functions can be represented by a graph, an equation, a table of values or ordered pair, mapping diagram, etc. An understanding of function requires an understanding of this concept in these different representations and how a change in one representation is reflected in other representations. Items #2 and #3 are examples of questions assessing understanding of the link between graphs and equations.

Another fundamental idea about function or any mathematical concept for that matter are the properties of the concept. In teaching the zeroes of a function for example, students are taught to find the zeroes given the equation or graph. One way to assess that they really understand it is to do it the other way around. Given the zeroes, find the equation. An example of an assessment item is item #4.

You may also want to read  How to assess understanding of function in equation form and Teaching the concept of function.

Posted in Algebra, Geogebra, High school mathematics

Embedding the idea of functions in geometry lessons

GeoGebra is a great tool to promote a way of thinking and reasoning about shapes. It provides an environment where students can observe and describe the relationships within and among geometric shapes, analyze what changes and what stays the same when shapes are transformed, and make generalizations.

When shapes or objects are transformed or moved, their properties such as location, length, angles, perimeters, and area changes. These properties are quantifiable and may vary with each other. It is therefore possible to design a lesson with GeoGebra which can be used to teach geometry concepts and the concepts of variables and functions. Noticing varying quantities is a pre-requisite skill towards understanding function and using it to model real life situations. Noticing varying quantities is as important as pattern recognition. Below is an example of such activity. I created this worksheet to model the movement of the structure of a collapsible chair which I describe in this Collapsible  chair model.

Show angle CFB then move C. Express angle CFB in terms of ?, the measure of FCB. Show the next angle EFB then move C. Express EFB in terms of ?. Do the same for angle FBG.
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Because CFB depends on FCB, the measure of CFB is a function of ?. That is f(?) = 180-2?. Note that the triangle formed is isosceles. Likewise, the measure of angle EFB is a function of ?. We can write this as g(?) = 2?. Let h be the function that defines the relationship between FCB and FBG. So, h(?)=180-?. Of course you would want the students to graph the function. Don’t forget to talk about domain and range. You may also ask students to find a function that relates f and g.

For the geometry use of this worksheet, read the post Problems about Perpendicular Segments. Note that you can also use this to help the students learn about exterior angle theorem.