Posted in Algebra

Making Sense of Power Function

The power function, ax^n, n = positive integral exponents is actually the ‘basic’ polynomial function.They are the first terms in the polynomial function.

With graphing utility, it is no longer as much fun to graph function. What has become more challenging is interpreting them. Here’s are a set of tasks you can ask your learners as review for function. You can give it as homework as well.

Consider the sets of power function in the diagrams below. Answer the following based on the diagram

  1. What are the coordinates of the points of intersection?
  2. Why do all the graphs intersect at those points?
  3. When is x^4 < x^2?
  4. When is x^7 > x^3?
  5. Why is it that as the degree or exponent of x that defines the function increases, the graph becomes flatter for the interval -1<x<1 and steeper for x > 1 or x >-1 ?
  6. Sketch the following in the graphs below: t(x) = x^{10}, l(x)=x^9
  7. Why is it that power function with even exponents are in Quadrants I and II while power function with odd exponents are in Quadrants I and III? Why are they not in Quadrant IV?
power function with even exponents
Power function with even exponents
power function with odd exponents
Power function with odd exponents

What other questions can you pose based on the graphs above? Kindly use the comment section to suggest more questions. Thanks.

My other posts about function

  1. Teaching the concept of function
  2. What is an algebraic function?
  3. How to find the equation of graphs of functions
  4. Evolution of the definition of function
  5. Strengths and limitations of each representation of function
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 🙂 )

[iframe https://math4teaching.com/wp-content/uploads/2012/12/PEACE.html 650 700]

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

Math knowledge for teaching tangent to a curve

I am creating a new category of posts about mathematical tasks aimed at developing teachers’ math knowledge for teaching. Most of the tasks I will present here have been used in studies about teaching and teacher learning. Mathematical knowledge for teaching was coined by J. Boaler based on what Shulman (1986) call pedagogical content knowledge (PCK) or subject-matter knowledge for teaching. I know this is a blog and not a discussion forum but with the comment section at the bottom of the post, there’s nothing that should prevent the readers from answering the questions and giving their thoughts about the task. Your thoughts and sharing will help enrich knowledge for teaching the math concepts involve in the task.

The following task was originally given to teachers to explore teachers beliefs to sufficiency of a visual argument.

The task:

Year 12 students, specializing in mathematics, were given the following question:
Examine whether the line y = 2 is tangent to the graph of the function f, where f(x) = x^3 + 2.

Two students responded as follows:

Student A: I will find the common point between the line and the graph and solving the system

math

The common point is A(0,2). The line is tangent of the graph at point A because they have only one common point (which is A).’

Student B: The line is not tangent to the graph because, even though they have one common

tangentpoint, the line cuts across the graph, as we can see in the figure.

Questions:

a. In your view what is the aim of the above exercise? (Why would a teacher give the problem to students?)

b. How do you interpret the choices made by each of the students in their responses above?

c. What feedback would you give to each of the students above with regard to their response to the exercise?

Source: Teacher Beliefs and the Didactic Contract on Visualisation by Irene Biza, Elena Nardi, Theodossios Zachariades.