Category: High school mathematics

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

Posted in Algebra, High school mathematics

Properties of equality – do you need them to solve equations?

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Should students learn the properties of equality before we ask them to solve equations in one variable? Would it be too hard for them to solve the equation 2(x + 7) = 4x without knowing the properties of equality?

In the grades, pupils learn to find equivalent ways of expressing a number. For example 8 can be written as 4 + 4, 3 + 5, 4 x 2, 10 – 2. Now, what has the pupils previous experience of expressing numbers in different ways got to do with solving equations in one variable?

Let us take this problem. What value of x will make the statement 2(x-5) = 20 true?. The strategy is to express the terms in equivalent forms.

2(x-5) = 20 can be expressed as 2(x-5) = 2(10).

2(x-5) = 2(10) implies (x – 5) = 10

x-5 = 10 can be expresses as x-5 = 15 – 5. Thus x = 15.

This way of thinking can be used to solve the equation 2(x + 7) = 4x.

2(x+7) = 2(2x)

=>    (x+7) = 2x

=>    x + 7 = x + x

=>    x = 7.

Of course not all equations can be solved by this method efficiently.   So you may asked ‘why not teach them the properties of equality first before asking them to solve equations like these?’  Here are some benefits of asking students to solve equations first before teaching the properties of equality:

1.  It makes students focus on the structure of the equation. Noticing equivalent structure is very useful in doing mathematics.

2.  It makes the equations like 3x = 18, x + 15 = 5, which are used to introduce how the properties are applied, problems for babies.

3. It is easier to do mentally. Try solving equations using the properties of equality mentally so you’ll know what I mean.

4. I hope you also notice that the technique has similarities for proving identities.

So when do we teach the properties of equality? In my opinion, after the students have been exposed to this way of solving and thinking.

Here’s on how to introduce the properties of equality via problem solving.

Posted in Algebra, High school mathematics

Teaching the concept of function

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Mathematics is not just about the study of numbers and shapes but also about the study of patterns and relationships. Function, which can define some of these relationships, is an indispensable tool in its study. Function is the central underlying concept in calculus. It is also one of the key concepts of mathematics that can model many quantitative relationships.

Textbooks and teachers usually introduce function via a situation with the related quantities already identified. What is required of the  students is to learn how to set up and represent the relationships in tables, graphs, and equation and analyze the properties. In the real world, when function is used as a model, the first thing that needs to be done is to identify the varying quantities. So, it is important to let students identify the quantities and let them determine which of these quantities may be related. This way they get a sense of what function really is and what it is for. The function is not the graph, not the table of values, and not the equation. The function is the relationship between the variables represented by these. The study of function is the study of these relationships and their properties, not finding y  or f(x) given x and vice versa, not reading graphs,  and not translations among the representations. These are important knowledge and skills, yes, but only in the context for investigating or learning more about the relationships between the quantities, that is, the function. Thus, for an introductory lesson for function, I find it useful to use a situation where students themselves will:

  1. identify the changing and unchanging quantities;
  2. determine the effect of the change of one quantity over the others;
  3. describe the properties of the relationship; and,
  4. think of ways for describing and representing these relationships.

These are the ‘big ideas’ students should learn about function. Of course, there are others like looking or dealing with function as a mathematical object and not only as a process or procedure for generating or predicting values. However, for an introductory lesson on function, teachers need not focus on this yet.

Sample introductory activity:

What are the quantifiable attributes or quantities can you see in the figure below? Which of these quantities will change and remain unchanged if GC is increased or decreased? Click the figure and move point C. Are there ways of predicting the values of these changing quantities?

teaching function
Identifying related quantities

Click here or the image above to go to dynamic window for the worksheet.

I like this particular activity because it gives students the opportunity to link geometry/measurement concepts to algebra and learn mathematics through solving problems.

Click link for a synthesis of the evolution of the definition of function and What are the big ideas in function?

Posted in High school mathematics, Lesson Study

Pedagogical Content Knowledge Map for Integers

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I’m working with a group of Year 7 mathematics teachers doing Lesson Study for the first time. The teachers chose to do a lesson study for what they believe to be the most difficult topic in this year level – integers. Part of my preparation as facilitator is to draw a map of what I know about teaching the topic. The map is more than a concept map because it includes not just related big ideas or concepts but also how  these are taught and learned. Hence, I call this pedagogical content knowledge map (PCK map).

The PCK map I present here is a product of my own readings and my own experiences of teaching the topic. This means that it may not be the same as other teachers especially the ‘teaching part’ of the map, the ones in orange colors. For example, experience and research results back my claim that the number line is a very good way of representing the set of integers but not in teaching operations. Click here for my post about this. Notice that I gave emphasis on students knowing when a negative, a positive or a zero result rather than the rules for operation. I believe that without this, a conceptual understanding of the operation involving integers will be weak. Also, experience has taught me that although integers are numbers, the teaching of it must be algebraic. The instructions should be so designed so that students are learning algebraic thinking as well. I have noted this in the PCK map.

The map is not yet complete. I intend to include descriptions of effective activities and students’ learning trajectory of the concept after my research with the teachers. Please feel free to give your comments and share experiences for teaching integers that I could look into in my study.

pedagogical content knowledge
PCK Map for Integers

Please click the link to see my PCK map for Algebraic Expressions.