Category: Algebra

Posted in Algebra

Teaching irrational numbers – break it to me gently

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Numbers generally emerged from the practical need to express measurement. From counting numbers to whole numbers, to the set of integers, and to the rational numbers, we have always been able to use numbers to express measures. Up to the set of rational numbers, mathematics is practical, numbers are useful and easy to make sense of. But what about the irrational numbers? You can tell by the name how it shook the rational mind of the early Greeks.


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Unlike rationals that emerged out of practical need, irrational numbers emerged out of theoretical need of mathematics for logical consistency. It could therefore be a little hard for students to make sense of and hard for teachers to teach. Surds, \pi, and e are not only difficult to work with, they are also difficult to understand conceptually.

It is not surprising that some textbooks, teaching guides, and lesson plans uses the following stunts to introduce irrational numbers:

After discussing how terminating decimal numbers and repeating decimal numbers are rational, you can then announce that the NON-repeating NON-terminating decimal numbers are exactly the IRRATIONAL NUMBERS.

What’s wrong with this? Nothing, except that it doesn’t make sense to students. It assumes that students understand the real number system and that the set of real numbers can be divided into two sets – rational and irrational. But, students have yet to learn these.

Some start with definitions:

Rational numbers are all numbers of the form  \frac{p}{q} where p and q are integers and q \neq 0. Irrational numbers are all the numbers that cannot be expressed in the form of \frac{p}{q} where p and q are integers.

How would we convince a student that there is indeed a number that cannot be expressed as a quotient of two integers or that there is a number that cannot be divided by another number not equal to zero? It’s not a very good idea but even if we tell them that \sqrt{2} is an irrational number, how do we show them that it fits the definition without resorting to indirect proof or proof of impossibility? What I am saying here is it is not pedagogically sound to start with definitions because definitions are already abstraction of the concept. I would say the same for all other mathematical concepts.

Before introducing irrational numbers, students should be given tasks that raises the possibility of the existence of a number other than rational numbers. Another way is to let them realize that the set of rational numbers cannot represent the measures of all line segments. Tasks that would help them get a sense of infinitude of numbers will also help. The idea is to prepare the garden well before planting. Read my post on why I think it is bad practice to teach a mathematical concept via its definition.

Posted in Algebra, Geogebra

Teaching with GeoGebra – Investigating coordinates of points

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The most basic mathematics students need to know to understand GeoGebra is the coordinate axes. Must you teach students how to plot points and interpret coordinates of points before they use GeoGebra or the other way around? I think, at the same time. Below is a sample activity on how I think this can be done.  The lesson is about investigating coordinates of points on a Cartesian plane. Its objective is to teach how to use GeoGebra’ s point tool, interpret coordinates of points and make generalizations.

1. Locate the reflections of the points A, B, C, D, E, F, and G if they will be reflected along the y-axis. Use the point button [.A] or the reflect button [.\.] to plot the points.

2. Hover the cursor along the points A to E. These pairs of numbers are called the coordinates of the point. What do you notice about the coodinates of these set of points (A through E)? Will this observation be true to the reflections of A, B, C, D, and E you just plotted?
3. Hover the cursor to the other points. How do the coordinates of the points relate to the values in the x and y axes?
4. In the input bar type P=(5,-2). Before hitting the Enter key, predict the location of the point. Experiment using other coordinates. Use the Move button to drag the grid to see the points you plotted, if they are not visible in the panel.
5. The x and y axes divide the plane into four quadrants. Describe the coordinates of the points located in each quadrant. What about the points along the x -axis and y – axis?

Click here to explore.

Of course, the teacher need to understand a little about GeoGebra first before giving this activity to his/her class.

Posted in Algebra, Geogebra, High school mathematics

Teaching simplifying and adding radicals

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The square root of a number is usually introduced via an activity that involves getting the side of a square with the given area. For example the side of a square with area 25 sq unit is 5 unit because 5 x 5 = 25. To introduce the existence of \sqrt{5}, a square of area 5 sq units is shown. The task is to find the length of its side. The student measures it then square the measure to check if it will equal to 5. Of course it won’t so they will keep on adjusting it. The teacher then introduces the concept of getting the root and the symbol used. This is a little boring.  A more challenging task is to start with this problem: Construct a square which is double the area of a given square. In my post GeoGebra and Mathematics: Squares and Square Roots I described a teaching sequence for introducing the idea of square root using this problem. There are 4 activities in the sequence. The construction below can be an extension of Activity 4. This extension can be used to teach simplifying radicals and addition of radicals. The investigation still uses the regular polygon tool  and introduces the text tool of GeoGebra.  Click links for the tutorial on how to use these tools. You will find the procedure for constructing the figure here.

radicalsThe construction shows the following equivalence:

1. 2\sqrt{5} = \sqrt{5}+\sqrt{5} since EA = EF+FH

2. 4\sqrt{5} = 2\sqrt{5}+2\sqrt{5} since AK = AB+BJ

3. 2\sqrt{10} =\sqrt{10}+\sqrt{10}

4. 4\sqrt{10} = 2\sqrt{10}+2\sqrt{10}

5.7\sqrt{5} = \sqrt{5}+2\sqrt{5}+4\sqrt{5}

6. 2\sqrt{5} = \sqrt{20} because they are both lengths of the sides of square EHBA or poly3 whose area is 20 (see algebra panel)

7. 2\sqrt{10} = \sqrt{40} because they are both lengths of the sides of square AHJI or poly4 whose area is 40.

8. 4\sqrt{5} = \sqrt{80} because they are both sides of square AJLK or poly5 whose area is 80.

Posted in Algebra, Geogebra, Geometry, High school mathematics

Teaching with GeoGebra: Squares and Square Roots

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This post outlines a teaching sequence for introducing the concept of square roots in a GeoGebra environment. Of course you can do the same activity using grid papers, ruler and calculator. However, if the students have access to computers then I highly recommend that you use GeoGebra to do this. In my post GeoGebra and Mathematics, I argued that the more the students understand the mathematics behind GeoGebra, the more confident they could become in using this tool. The earlier the exposure to this environment, the better. The way to do this is to integrate the learning of the tool in learning mathematics.

The figure below is the result of the final activity in my proposed teaching sequence for teaching square roots of numbers and some surds or irrational numbers. The GeoGebra tool that the students is expected to learn is the tool for constructing general polygons and regular polygons (the one in the middle of the toolbar).

Squares and Square Roots

The teaching sequence is composed of four activities.

Activity 1 involves exploration of the two polygon tools: polygons and regular polygons. To draw a polygon using the polygon tool is the same as drawing polygons using a ruler. You draw two pints then you use the ruler/straight edge to draw a side. But with Geogebra you click the points to determine the corners of the polygon and Geogebra will draw the lines for you. In the algebra window you will see the length of the segment and the area of the polygon. Click here to explore.

GeoGebra shows further its intelligence and economy of steps in Activity 2 which involves drawing regular polygons. Using the regular polygon tool and then clicking two points in the drawing pad, GeoGebra will ask for the number of sides of the polygon. All the students need to do is to type the number of sides of their choice and presto they will have a regular polygon. Click here to explore.

Activity 3 is the main activity which involves solving the problem Draw a square which is double the area of another square. Click here to take you to the task.

Activity 4 consolidates ideas in Activity 3. Ask the students to click File then New to get a new window from the previous activity’s applet then ask them to draw the figure above – Squares and Square Roots.  You can also use the figure to compare geometrically the values of \sqrt{2} and 2 or  show that \sqrt{8} = 2\sqrt{2}. This activity can be extended to teach addition of radicals.

Like the rest of the activities I post here, the learning of mathematics, in this case the square roots of numbers, is in the context of solving a problem. The activities link number, algebra, geometry and technology. Click here for the sequel of this post.

This is the second in the series of posts about integrating the teaching of GeoGebra and  Mathematics in lower secondary school. The first post was about teaching the point tool and investigating coordinates of points in a Cartesian plane.

GeoGebra book:

Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra