Tag: teaching functions

Posted in Algebra, Assessment

What are the big ideas in function ?

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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, High school mathematics

Teaching the concept of function

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Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study. Function is the central underlying concept in calculus. It is also one of the key concepts of mathematics that can model many quantitative relationships.

Textbooks and teachers usually introduce function via a situation with the related quantities already identified. What is required of the  students is to learn how to set up and represent the relationships in tables, graphs, and equation and analyze the properties. In the real world, when function is used as a model, the first thing that needs to be done is to identify the varying quantities. So, it is important to let students identify the quantities and let them determine which of these quantities may be related. This way they get a sense of what function really is and what it is for. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties, not finding y  or f(x) given x and vice versa, not reading graphs,  and not translations among the representations. These are important knowledge and skills, yes, but only in the context for investigating or learning more about the relationships between the quantities, that is, the function. Thus, for an introductory lesson for function, I find it useful to use a situation where students themselves will:

  1. identify the changing and unchanging quantities;
  2. determine the effect of the change of one quantity over the others;
  3. describe the properties of the relationship; and,
  4. think of ways for describing and representing these relationships.

These are the ‘big ideas’ students should learn about function. Of course, there are others like looking or dealing with function as a mathematical object and not only as a process or procedure for generating or predicting values. However, for an introductory lesson on function, teachers need not focus on this yet.

Sample introductory activity:

What are the quantifiable attributes or quantities can you see in the figure below? Which of these quantities will change and remain unchanged if GC is increased or decreased? Click the figure and move point C. Are there ways of predicting the values of these changing quantities?

teaching function
Identifying related quantities

Click here or the image above to go to dynamic window for the worksheet.

I like this particular activity because it gives students the opportunity to link geometry/measurement concepts to algebra and learn mathematics through solving problems.

Click link for a synthesis of the evolution of the definition of function and What are the big ideas in function?

Posted in Algebra, Assessment

Assessing understanding of graphs of functions

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Problems about graphs of functions can be grouped into interpretation or construction tasks. The tasks may involve interpreting individual points, an interval, or the entire graph. The same may be said about construction tasks. It may involve point-plotting,  a part of the graph, or constructing the whole graph.

Tasks involving constructing graphs are considered more difficult than interpreting graphs tasks but with the available graphing technology, constructing graphs is now easy.  But not when you have to construct a relationship, not just graphs! In fact, I would consider it as an indicator of students deep understanding of graphs and functions when he or she can interpret and reason in terms of relationship shown in the graphs and from these be able to construct a new relationship, a new function. Here is a task you can use to assess this level of understanding. Note that in this task the graphs are not on grids to encourage holistic analysis of the graph rather than point-by-point. Interpreting graphs not on grids encourages algebraic thinking.

graphs
Relating graphs

Below is a a sample a Year 8 student solution to the task above. This answer indicates that the student understands graphs and the function it is representing but  he/she could still not reason in terms of relationship so resorted to interpreting individual points in x vs y and y vs z in order to relate x and z.

solutions by point-by-point analysis

The figure below shows a solution of a Year 10 student who could reason in terms of the relationships of the variables represented by the graphs.

reasoning in terms of relationship

A similar solution to this would be “x is directly proportional y but y is inversely proportional to z hence x would also be inversely proportional to z”.

Both solutions are correct and both solved the problem completely. Note that initially students will use the first solution just like the Year 8 student. The Year 10 however should be expected and encouraged to reason in terms of relationship.

A good assessment task not only assesses students’ mathematical knowledge and skills but also assesses the level of thinking and reasoning students are operating on. See posts on features of good problem solving tasks.

Posted in Algebra, Assessment, High school mathematics

Levels of understanding of function in equation form

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There are at least three representational systems used to study function: graphs, tables and equations. But unlike graphs and tables that are used to visually show the relationships between two varying quantities, students first experience with equation is not as a representation of function but a statement which state the condition on a single unknown quantity.  Also, the equal sign in the early grades is taught to mean “do the operation” and not to denote equality between the quantities of both sides of the sign. To complicate the matter, when equation is used as a representation of function, it takes an additional meaning, that of a representation of two varying quantities! For example, let y be number of t-shirts and x be the number of t-shirts to be printed. If the cost of printing a t-shirt is 2.50 then the function that defines y in terms of x is y = 2.50x. Understanding function equation form is not an easy concept for many students.

I believe that if mathematics teachers are aware of the differing level of abstraction in students’ thinking and reasoning  when they work with function in equation form then the teachers would be better equipped to design appropriate instruction to lead students towards a deeper understanding of this concept.Failure to do so would deprive students the opportunity to understand other advanced algebra and calculus topics.I would like to share a framework for assessing students’ developing understanding of function in equation form. This framework is research-based. You can download the full paper here or you can view the slides in my post Learning Research Study Module for Understanding Function.

The framework is in terms of levels of understanding. You can use it to design tasks or assess your students understanding of function in equation form. Each of these levels are “big ideas” or schemas in the understanding of function.
Level 1 – Equations are procedures for generating values.
Students at this level can find x given y or vice versa. Some can generate a number of pairs but not really see the equation as a rule for all pairs of values in a situation.
Level 2 – Equations are representations of relationships.
Students at this level understands domain and range, can generate pairs of values and graph it. They also know that that relationship is unique and true for the values in the domain and range.
Level 3 – Equations describe properties of relationships.
Students at this level can interpret the properties of the function like rate of change and intercepts form the equation but can do it by generating values.
Level 4 – Functions are objects that can be manipulated and transformed
This is the highest level. At this point students see the equation as a math object. They can do composition of function, can find its inverse by algebraic manipulation and can interpret the meaning or effect of the parameters to the graph of the function. 

You can reference the above framework:

Ronda, E. (2009). Growth points in students’ developing understanding of function in equation form. Mathematics Education Research Journal, 21, 31-53.