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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

Equations, Equations, Equations

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Students deal with a different ‘types’ of equations: equations in one unknown, equations in two unknowns, and the equation representations of function. There are others like the parametric equations but let’s talk about the first three I just enumerated. Is there a connection among all these three apart from being equations?

Let’s take for example the equation 4x – 1 = 3x + 2. To solve the equation, students are taught to use the properties of equality. When the topic gets to equation in two unknowns, this equation is learned independently of the equation in one unknown especially in  finding the solutions. When the topic gets to solving systems of equation say 3x+y = 4 and x–y = 5, the methods for solving the system of linear equation – substitution,  elimination, graphing – are also learned without making the connection to the methods of solving equation they already know. Then, function comes in the scene; the y‘s disappeared and out of nowhere comes f(x). Most times we assume the students will make the connection themselves.

How can we help students make connection among these three? To solve equation in one unknown, I think we should not rush to teaching them how to solve it using the properties of equality. There are other ways of solving these equations one of which is generating values which I’m sure you use in introducing equations in two variables. Using the example earlier, students can generate the values of 4x – 1 and then 3x+2. This way, the question “What is x so that 4x-1 = 3x+2 is true?” will make sense to students. They will have to find the value of x that belongs to the group of numbers generated by 4x-1 as well as to the group of numbers generated by 3x+2.

equation in one unknown

Now, why go through all these? Two reasons: 1) to reinforce the notion that algebraic expressions is a generalized expression representing a group of numbers/values and 2) to plant the seed of  the notion of function and equations in two unknowns which students will meet later. Of course this does not mean we should not teach how to solve equation using the properties of equality. I just mean we should teach them other solutions that will help students make the connection when they meet the other types of equations.

function machineAnother popular tool is the input-output machine which is the same really as the table of values. For some reason they are used mostly to introduce equations in two unknowns or to introduce function. Why not introduce it early with equations in one unknown? Of course you need a second machine for the other expression. The challenge for the students is to find what they need to input in both machine so they will have the same output. The outputs can be represented by the expressions on each side of the equal sign but later you get to the study of function you may introduce y provided that y = 4x-1. Students need to see that this equation does not just mean equality but that it also means the value of y depends on x according to the rule 4x-1. Since every x value generates a unique y value, y is said to be a function of x, in symbol, y=f(x). Since y = f(x), we can also write f(x) = 4x – 1.

In most curricula, the formal study of function comes after systems of linear equation so there’s no hurry with the f(x) thing. The use of the form y = 4x-1 would be enough. If students understand equations this way I think they can figure out the substitution method for solving systems of linear equations by themselves. Graphing would therefore also be a natural solution students can think of. Equation Solver is a simple GeoGebra applet I made to help students make the connection.

Wouldn’t it be nice if students see 4x-1=3x+2 not just a simple equation in one unknown where they need to find x but also as two functions who might share the same (x,y) pair? This will really come in handy later. Solutions #2 and #3 of solving problems by equations and graphs are examples of problems where this knowledge will be needed.

I recommend that you also read my post What Makes Algebra Difficult is the Equal Sign.

To understand is to make connections. This has become a mantra in this blog. Students will not make the connection unless you make it explicit in the design and implementation of the lesson.

Posted in Geometry

Regular Polygons Problems

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One  of my favourite lesson design is a sequence of problem solving tasks that requires repetition of same reasoning and analysis by varying the ‘mathematical context’ of the problem in increasing complexity. However the variation in the context of the problem should be such that they still share some properties. In the examples below, the number of sides of the polygons is varying but they are all regular polygons.It is also important that the problems can be solved/ explained in different ways – algebraically, geometrically, arithmetically or a combination of these.

Here is a sample sequence of problems. This lesson is good from Grade 5 up. If you are handling different grade levels and they all reason in the same way as your fifth graders reason, you have a big problem.

Problem 1

The segments in the figure below form equilateral triangles with the dotted line segment. Compare the total lengths of the red segments to the total lengths of the blue segments. You must be able to explain how you arrive at your conclusion or give a justification to it.

equal perimeter

Problem 2

What if the segments form squares instead of equilateral triangles with the dotted line segment? Compare the total lengths of the red segments to the blue segments. Which is longer?

perimeter problem

Problem 3

What if it the line segments form regular pentagons instead of squares? Do you think your conclusion will hold for any regular polygon? Prove.

Problem 4

What if instead of regular polygons, you have a semicircle? Click link to see the problem and solution.

Encourage students to use algebra and geometric constructions to justify their answers. This lesson is not about getting the correct conclusion. That’s the easy part. It is about explaining/ proving it.

You may want to view another similar lesson on quadrilaterals.