Category: Algebra

Posted in Algebra, Assessment, High school mathematics

Algebra test items – Graphs of rational functions

Posted on by

TIMSS (Trends in international Math and Science Study) classifies test items in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

In my earlier post about this topic on using the TIMSS Assessment Framework for constructing test items I presented a set of questions about zeros of cubic polynomial function. Here are three more test items about graphs of rational function based on the framework. Note that questions should be independent of each other, that is, an answer in one item should not serve as clue to the other items. I only used the same rational function here to highlight the differences among the cognitive domains – knowing, applying, reasoning.

Knowing

What may be the equation of the graph below?

 

Applying

The graph above the x-axis is function f and the graph below the x-axis is function g.  Which of the following equations describes the relationships between f and g?

a. g(x) = f(-x)              b. g(x) = f-1(x)                c. g(x) = f-1(-x)                d. g(x) = -f(x)              e. g(x) = /f(x)/

Reasoning

Carlo drew the figure below by graphing two functions on the same coordinate axes. The graph on the left is f(x) = 4/x2. Which of the following function is represented by the other graph on the right (the blue one)?

a. g(x)=\frac {4}{x^2}        b. g(x)=4+\frac {4}{x^2}        c. g(x)=\frac {4}{(x-2)^2}       d. g(x)=\frac {4}{(x-4)^2}                                   e. g(x)=\frac {4}{(x+4)^2}

All the graphs in these post were made using Geogebra graphing software. It’s a free graphing tool you can download here.

Posted in Algebra, Assessment

Algebra test items – Zeroes of function

Posted on by

I find the Trends in International Math and Science Study (TIMSS) Assessment Framework useful for constructing test items. TIMSS classified the questions in terms of cognitive domains namely, Knowing facts, procedures, concepts; Applying the facts, procedures and concepts usually in a routine problem solving task; and, Reasoning. Click here for detailed descriptions of each.

Here are three items assessing students’ learning about zeroes of function using each category in the framework. I used the same polynomial function to illustrate the differences among the three. In the real exam use different polynomial for each category so it won’t stand as hint to other questions.

Knowing

Which of the following is a zero of f(x) =6x3 – 17x2 – 5x + 6?

a. -6                b. -3               c. 0           d. 3        e. 6

There is no way apart from luck that students will choose the correct answer in this question if they don’t know what a zero of a function is. There are many ways of getting the correct answer of course (graphical, applying factor theorem, definition of zero of function).

Applying

What is the value of k if 3 is a zero of f(x) = 6x3 – kx2 – 5x + 6?

Questions about applying usually include standard textbook problem like the one shown above. It involves knowledge of a a fact/concept or procedure to complete the task. It does not only involve straightforward application of concepts unlike those under Knowing questions.

Reasoning

If 3 is a zero of a third degree polynomial function f, which of the following statements can never be true about this function?

a. f(0) = 3.

b. f(-3) = 0.

c. (0,0) is a point on the graph of f

d. (-3,3) is a point on the graph of f

e. (3,-3) is a point on the graph of f

Unlike questions under Applying which are standard or routine tasks, tasks under Reasoning category are usually non-routine and involves decision-making.

Click link to view another set of test items about graphs of rational functions.

Posted in Algebra, Curriculum Reform

Algebraic thinking in algebra

Posted on by

Algebraic thinking is an approach to thinking about quantitative situations in general and relational manner. This kind of thinking is optimized by a considerable understanding of the objects of algebra, a disposition to think in generality, and engagement in high-level tasks which provide contexts for applying and investigating mathematics and the real-world.

big ideas in algebra
Ingredients in Algebraic Thinking
Objects of Algebra

The objects are the content of algebra which I classify into three overlapping categories. The first category and the most basic are those for representing changing and unchanging quantities and relationships. These include the idea of variables, numbers, graphs, equations, matrices, etc. The second category are ideas for working with unknown quantities which involve solving equations and inequalities under which are linear equations and inequalities in one variable, systems of linear equations and inequalities, exponential equations, quadratic, trigonometric equations, etc. The third and last category involves the ideas for investigating relationships between changing quantities which include directly and inversely proportional relationships; relationships with constant rate of change; relationships with changing rate of change; relationships involving exponential growth and decay; periodic relationships, etc.

Thinking dispositions

Knowledge of algebraic content do not necessarily translate in algebraic thinking. Computational fluency in simplifying, transforming, and generating expression for example, while important, do not necessarily involve a person in algebraic thinking if one is doing it for its own sake. Thinking processes that contribute to the development of algebraic thinking are those that require purposeful representations of quantities and relationships, multiple interpretations of representations, finding structures, and generalization of patterns, operations and procedures. These should become part of students’ thinking disposition.

High level tasks

The higher-order tasks in mathematics  include problem solving, mathematical investigations (sometimes referred to also as open-ended problem solving tasks), and modeling.

Posted in Algebra, High school mathematics, Math investigations

Teaching combining algebraic expressions with conceptual understanding

Posted on by

In Math investigation about polygons and algebraic expressions I presented possible problems that students can explore. In this post, I will share some ideas on how the simple investigation of drawing polygons with the same area can be used as an introductory lesson to teach operations with algebraic expressions with meaning and understanding. Like the rest of the lessons in this blog, this lesson is not so just about learning the math but also making sense of the math and engaging students in problem solving.

The lesson consists of four problem solving tasks to scaffold  learning of adding, subtracting, multiplying and dividing algebraic expression with conceptual understanding.

Problem 1 – What are the different ways can you find the area of each polygons? Write an algebraic expression that would represent each of your method.

The diagram below are just some of the ways students can find the area of the polygons.

1. by counting each square
2. by dissecting the polygons into parts of a rectangle
3. by completing the polygon into a square or rectangle and take away parts included in the counting
4. by use of formula

The solutions can be represented by the algebraic expressions written below each polygon. Draw the students’ attention to the fact that each of these polygons have the same area of 5x^2 and that all the seven expressions are equal to5x^2 also.

Multiple representations of the same algebraic expressions

Problem 2 – (Ask students to draw polygons with a given area using algebraic expressions with two terms like in the above figure. For example a polygon with area 6x^2-x^2.

Problem 3 – (Ask students to do operations. For example 4.5x^2-x^2.)

Note: Whatever happens, do not give the rule.

Problem 4 – Extension: Draw polygons with area 6xy on an x by y unit grid.

These problem solving tasks not only links geometry and algebra but also concepts and procedures. The lesson also engages students in problem solving and in visualizing solutions and shapes. Visualization is basic to abstraction.

There’s nothing that should prevent you from extending the problem to 3-D. You may want to ask students to show the algebraic expression for calculating the surface area of  solids made of five cubes each with volume x^3. I used Google SketchUp to draw the 3-D models.

some possible shapes made of 5 cubes

Point for reflection

In what way does the lesson show that mathematics is a language?