Tag: irrational numbers

Posted in Number Sense

Test your understanding of irrational numbers

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The following is a set of tasks which I think are great questions for assessing understanding of irrational numbers. These tasks were from the study of Natasa Sirotic and Rina Zazkis. The responses were analysed in terms of algorithmic, formal, and intuitive knowledge described at the end of the post.

Set A

This set of tasks assesses the formal and intuitive knowledge about about the relative sizes of two infinite sets – rationals and irrationals.

  1. Which set do you think is “richer”, rationals or irrationals (i.e. which do we have more of)?
  2. Suppose you pick a number at random from [0,1] interval (on the real number line). What is the probability of getting a rational number?
Set B

This set assesses knowledge about how the rational and irrational numbers fit together in relation to the density of both sets.

  1. It is always possible to find a rational number between any two irrational numbers. Determine True or False and explain your thinking.
  2. It is always possible to find an irrational number between any two irrational numbers. Determine True or False and explain your thinking.
  3. It is always possible to find an irrational number between any two rational numbers. Determine True or False and explain your thinking. 
  4. It is always possible to find a rational number between any two rational numbers. Determine True or False and explain your thinking.
Set C

This set investigate knowledge of  the effects of operations between irrational numbers

  1. If you add two positive irrational numbers the result is always irrational. True or false? Explain your thinking.
  2. If you multiply two different irrational numbers the result is always irrational. True or false? Explain your thinking.

You may want to analyse the responses using Tirosh et al.’s (1998) dimensions of knowledge:

  • The algorithmic dimension is procedural in nature – it consists of the knowledge of rules and prescriptions with regard to a certain mathematical domain and it involves a learner’s capability to explain the successive steps involved in various standard operations.
  • The formal dimension is represented by definitions of concepts and structures relevant to a specific content domain, as well as by theorems and their proofs; it involves a learner’s capability to recall and implement definitions and theorems in a problem solving situation.
  • The intuitive dimension of knowledge (also referred to as intuitive knowledge) is composed of a learner’s intuitions, ideas and beliefs about mathematical entities, and it includes mental models used to represent number concepts and operations.

At the conclusion of the study, Sirotic and Zaskis reported this short exchange:

What do you think of the teacher’s answer?

You may want to share your responses to the questions in the comment section below.

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.


www.wombat.com

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, 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