Algebra

Fun with ‘definitions’ in algebra

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WARNING: use the following definitions with great caution.

A number phrase is a mathematical phrase which does not express a complete thought.
An arithmetic expression is any grammatically sensible expression made up of numbers and (possibly) arithmetic operations (like addition, division, taking the absolute value, etc). Notice that it only has to be grammatically sensible; an undefined expression like 5/0 is still an arithmetic expression, but something like ‘5)+/7?’ is just nonsense. You can always work out an arithmetic expression to a specific value, unless it’s undefined (in which case you can work that out).
An algebraic expression is any grammatically sensible expression made up of any or all of the following:

– specific numbers (called constants);
– letters (or other symbols) standing for numbers (called variables); and
– arithmetic operations.

By an algebraic expression in certain variables, we mean an expression that contains only those variables, and by a constant, we mean an algebraic expression that contains no variables at all.
A polynomial is an algebraic sum, in which no variables appear in denominators or under radical signs, and all variables that do appear are raised only to positive-integer powers.
A monomial is an algebraic expression made up only from any or all of these:

– Constants;
– Variables;
– Multiplication;
– Taking opposites (optional);
– Division by nonzero constants (optional);
– Raising to constant whole exponents (optional).

An algebraic expression is made up of the signs and symbols of algebra. These symbols include the Arabic numerals, literal numbers, the signs of operations, and so forth.

I got these definitions from where else, www. Of course we just want to simplify things for students but … . Anyway, just make sure that you don’t start your algebra lessons with definition of terms, be they legitimate or not legitimate.

Algebra, GeoGebra worksheets

What is a coordinates system?

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This is the first in the series of posts about teaching mathematics and Geogebra tools at the same time. I’m starting with the most basic of the tools in GeoGebra, the point tool. What would be a better context for this than in learning about the coordinate system. Teacher can use the following introduction about geographic coordinates system and the idea of number line as introduction to this activity.

A coordinates system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric elements. An understanding of coordinate system is very important. For example, a geographic coordinate system enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position. A common choice of coordinates is latitude and longitude. Sometimes, a third coordinate, the elevation is included. For example:

Philippine Islands are located within the latitude and longitude of 13° 00 N, 122° 00 E. Manila, the capital city of Philippines is 14° 35′ N, 121º 00 E’.

In mathematics we study coordinates systems in order to describe location of points, lines and other geometric elements. The numberline is an example of a coordinates system which describe the location of a point using one number. The coordinates of a point on a numberline tells us the location of a point from zero. But what if the point is not on the line but above of below it? How can we describe exactly the location of that point? This is what this activity is about: how to describe the position of points on a plane.

You would need to familiarize your students first about the GeoGebra window shown below before asking them to work on the GeoGebra worksheet.

Click here to go the GeoGebra worksheet – What are coordinates of points?

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 – x²D. 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.

Algebra, Assessment

Assessing understanding via constructing test items

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Assessing understanding of mathematics can also be done by asking students to write test items. Here’s my favorite assessment item. I gave this to a group of teachers.

Possible answers/ questions.

Year level: Third year (Year 9)

Question 1 – What is the distance of P from the origin?

Question 2 – What is the area of circle P with radius equal to its distance from the origin?

Question 3 – With P as one of the vertex, draw square with area 2 square units.

Year level: Second year (Year 8 )

Question 1 – Write the equation of the line that passes through P and the origin.

Question 2 – Write 3 equations of lines passing through (2,1).

Question 3 – Write the equation of the family of lines passing through (2,1).

Year level: First year (Year 7)

Question 1 – What is the ordinate of point P?

Question 2 – Locate (-2, 1). How far is it from P?

Question 3 – Draw a square PQRS with area 9 square units. What are the coordinates of that square?

How about using this exercise to assess your students? Ask them to construct test items instead of asking them to answer questions.

Here are a few more assessment items which I constructed based on the TIMSS Framework:

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