Tag: GeoGebra

Posted in Algebra, Geogebra

Solving systems of linear equations by elimination method

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This short investigation  about the graphs of the sum and difference of two or more linear equations may be used as an introductory activity to the lesson on solving systems of linear equations by elimination. It will provide a visual explanation why the method of elimination works, why it’s ok to add and subtract the equations.

The  investigation may be introduced using the GeoGebra applet below.

1. Check the box to show the graph when equations b and c are added.

2. Where do you think will the graph of b – c pass? Check box to verify prediction.

3. Check the box to show graphs of the sum or difference of two equations. What do you notice about the lines? Can you explain this?
[iframe https://math4teaching.com/wp-content/uploads/2011/07/solving_systems_by_elimination.html 700 400]

When equations b and c intersect at A. The graph of their sum will also intersect point A.

b:  x + 2y =1

c:  x – y =-5

a:  2x+y=-4

After this you can then ask the students to think of a pair of equation that intersect at a point and then investigate graph of the sum and difference of these equations. It would be great if they have a graphing calculator or better a computer where they can use GeoGebra or similar software. In this investigation, the students will discover that the graphs of the sum and difference of two linear equations intersecting at (p,r) also pass through (p,r). Challenge the students to prove it algebraically.

If ax+by=c and dx+ey=f intersect at (p,r),

show that (a+d)x+(b+e)y=f +c also intersect the two lines at (p,r).

The proof is straightforward so my advise is not to give in to the temptation of doing it for the students. After all they’re the ones who should be learning how to prove. Just make sure that they understand that if a line passes through a point, then the coordinates of that points satisfies the equation of the line. That is if ax+by=c passes through (p,r), then ap+br=c.


The investigation should be extended to see the effect of multiplying the linear equation by a constant to the graph of the equation or to start with systems of equations which have no solution. Don’t forget to relate the results of these investigations when you introduce the method of solving systems of equation by elimination. Of course the ideal scenario is for students to come up with the method of solving systems by elimination after doing the investigations.

You can give Adding Equations for assessment.

Posted in Curriculum Reform, Mathematics education

Knowledge of Teaching with ICT

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In the 80’s, Lee Shulman introduced the concept of pedagogical content knowledge to differentiate it from content knowledge (CK) and knowledge of general pedagogy (PK). Pedagogical content knowledge or popularly known as PCK  is teachers’ knowledge of how a particular subject-matter is best taught and learned. Since Shulman introduced this concept, many others have contributed towards defining and describing it, the most important elements of its description include (1) knowledge of interpreting the content, (2) knowledge of the different ways of representing the content to the learner,  and (3) knowledge of learners’ potential difficulties, misconceptions, and prior conceptions about the content and related concepts. Click here for an example of a pedagogical concept map for teaching integers.

With the increasing dependence of almost everything to ICT, it is no longer a question of whether schools should integrate these technology in its curriculum. In fact it’s been decades since courses on ICT have been offered as a subject in many schools. But how about the use of technology in teaching traditional subjects like mathematics? Does knowledge of technology equip teachers to use it to teach effectively?

Some mathematics teachers jumped to it right away, used technology in teaching. Some teachers are still in testing-the-water mode. Some, until now, are still totally in the dark, sticking to their old method despite the availability of technology, oblivious to the reality that in today’s ICT-driven world, it’s the students who are the natives and the teachers are the migrants (heard this at an APEC Conference in Tokyo). The way students learn are influenced by their experiences with many forms of technology and the way these tools think and do things.

When the pen and the printing press were invented, everybody thought that they will give an end  to illiteracy (I heard this from the same conference). It didn’t take long for us to realize that it didn’t and can’t. The same can be said with computers, internet, softwares for teaching. Experience with these tools tell us that it is not enough to know how to use ICT  just us it was not enough to know mathematics content to teach mathematics so that students learn it with meaning and understanding .   Teachers must now be equipped not only with PCK but with TPCK – Technological Pedagogical Content Knowledge.

Punya Mishra and Matthew Koehler introduced this theoretical framework known as Technological Pedagogical Content Knowledge (TPACK) in 2005. The basic premise of TPACK is that a teacher’s knowledge regarding technology is multifaceted and that the optimal mix for the classroom is a balanced combination of technology, pedagogy, and content.

technological pedagogical content knowledgeThe figure at the right is popularly known as TPACK Framework (click image for source). It shows the kinds of knowledge teachers should posses. It can be used as framework for designing learning experiences for teachers and for planning, analyzing and describing the integration of technology in teaching.

 

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