Posted in Conferences and seminars

Announcement: National Seminar-Workshop in Science and Mathematics Education

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The University of the Philippines National Institute for Science and Mathematics Education Development (UP NISMED), in cooperation with the Department of Education will hold the National Seminar-Workshop in Science and Mathematics Education with the theme Nurturing Young Minds to Build their Future: Opportunities for Science and Mathematics Education in the K+12 Curriculumon October 26-28, 2011 at UP NISMED, Diliman, Quezon City.

The seminar-workshop will feature plenary and parallel presentations and sharing of experiences that aims to:

  1. familiarize teachers, teacher educators, and school administrators with the K+12 curriculum framework  with focus on science and mathematics;
  2. elicit feedback on various features of the curriculum  in science and mathematics lessons in relation to content, instructional materials, teaching learning approaches, and assessment;
  3. provide a venue for participants to exchange effective/relevant strategies in implementing curricular reforms in their classes.
  4. launch the Philippine science and mathematics online community of learners to support continuing professional development, curriculum development, and research.

The participants to the seminar-workshop are elementary and high school science and mathematics teachers from public and private schools, teacher educators, and school administrators nationwide.

A registration fee of Php5,000.00 per participant is inclusive of seminar-workshop handouts (e.g. science and mathematics curriculum frameworks, sample activities spiraled across grade/year levels with performance indicators and concept maps that show connections) preloaded in a USB drive, seminar-workshop t-shirt and bag, three (3) lunches and six (6) snacks.  The fee may be charged against local funds, local school board funds and other sources subject to the usual accounting and auditing rules and regulations.

The participants will shoulder and make arrangements for their own board and lodging.

For registration details and further information/queries, please contact UP NISMED at (02) 9283545, 9281563 ext. 212, send email to nismed@up.edu.ph  or up_nismed@yahoo.com  or visitwww.nismed.upd.edu.ph.

 

Posted in Algebra, Geogebra, High school mathematics

Embedding the idea of functions in geometry lessons

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GeoGebra is a great tool to promote a way of thinking and reasoning about shapes. It provides an environment where students can observe and describe the relationships within and among geometric shapes, analyze what changes and what stays the same when shapes are transformed, and make generalizations.

When shapes or objects are transformed or moved, their properties such as location, length, angles, perimeters, and area changes. These properties are quantifiable and may vary with each other. It is therefore possible to design a lesson with GeoGebra which can be used to teach geometry concepts and the concepts of variables and functions. Noticing varying quantities is a pre-requisite skill towards understanding function and using it to model real life situations. Noticing varying quantities is as important as pattern recognition. Below is an example of such activity. I created this worksheet to model the movement of the structure of a collapsible chair which I describe in this Collapsible  chair model.

Show angle CFB then move C. Express angle CFB in terms of ?, the measure of FCB. Show the next angle EFB then move C. Express EFB in terms of ?. Do the same for angle FBG.
[iframe https://math4teaching.com/wp-content/uploads/2011/07/locus_and_function.html 700 400]
Because CFB depends on FCB, the measure of CFB is a function of ?. That is f(?) = 180-2?. Note that the triangle formed is isosceles. Likewise, the measure of angle EFB is a function of ?. We can write this as g(?) = 2?. Let h be the function that defines the relationship between FCB and FBG. So, h(?)=180-?. Of course you would want the students to graph the function. Don’t forget to talk about domain and range. You may also ask students to find a function that relates f and g.

For the geometry use of this worksheet, read the post Problems about Perpendicular Segments. Note that you can also use this to help the students learn about exterior angle theorem.

Posted in Algebra, Geogebra

Solving algebra problems – which one should be x?

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Every now and then I get an e-mail from a friend’s son asking for help in algebra problems. When it’s about solving word problems, the email will start with “How about just telling me which one is the x  and I’ll figure out the rest”. The follow-up email will open with “Done it. Thanks. All I need is the equation and I can solve the problem”. The third and final e-mail will be “Cool”. Of course I let this happen only when I’m very busy. Most times I try to explain to him how to represent the problem and set-up an equation. Here’s our latest exchange.

Josh: What is the measure of an angle if twice its supplement is 30 degrees wider than five times its complement? All I need is to know which one’s  the x.

Me: How about sending me a drawing of the angle with its complement and supplement?

Josh: Is this ok?

Me: Great. Let me use your drawing to make a dynamic version using GeoGebra. Explore the applet below by dragging the point in the slider. What do you notice about the values of the angles? Which angle depend on which angle for its measure? If one of the measure of one of the angles is represented by x, how will you represent the other angles? (Click here for the procedure of embedding applet]
[iframe https://math4teaching.com/wp-content/uploads/2011/07/angle_pairs1.html 650 435]

Josh: They are all changing. The blue angle depends on the green angle. Their sum is 90 degrees. The red angle also depends on the green angle.  Their sum is 180 degrees. The measure of the red angle also depends on blue angle.

Me: Excellent. Which of the three angles should be your x so that you can represent the others in terms of x also? Show it in the drawing.

Josh: I guess the green one should be x. The blue should be 90-x and the red angle should be 180-x.

Me: Good. The problem says that twice the measure of the supplement is 30 degrees wider than five times the complement. Which symbol >, <, or = goes to the blank and why, to describe the relationship between the representations of twice the supplement and five times the complement:

2(180-x) _____ 5(90-x)

Josh: > because it is 30 degrees more.

Me: Good. Now, what will you do so that they balance, that is make them equal?  Remember that  2(180-x) is “bigger” by 30 degrees? What would the equation look like?

Josh: I can take away 30 degrees from 180-x. My equation would be (180-x) -30 = 5(90-x)?

Me: Is that the only way of making them equal?

Josh: Of course I can add 30 to 5(90-x). I will have 180-x = 5(90-x)+30.

Me: You said  you can do the rest. Try it using both equations and tell me the value of your x and the measures of the three angles.

Josh: x = 40. That’s the angle. It’s complement measures 50 degrees and its supplement is 140 degrees. They’re the same for both equations.

Me: Does it makes sense? Do you think it satisfies the condition set in the problem?

Josh: 2(140) = 280. 5(50) = 250. 280 is 30 degrees wider than an angle of 250 degrees. Cool.

Me: What if you make A’DC your x? Do you think you will get the same answer?

No reply. I guess I’ll have to wait till the teacher give another homework to get another e-mail from him.

I don’t know if the questions I asked Josh will work with other students. Try it yourself. Please share or send this post to your co-teachers. Thanks. I will appreciate feedback.

Problem solving is the heart of mathematics yet it is one of the least emphasized activity. Solving problems are usually relegated at the end of the textbooks and chapters.

Posted in Geometry

Teaching triangle congruence

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In this post I propose a way of teaching the concept of triangle congruence. Like most of the lessons I share in this blog, the teaching strategy for this lesson is  Teaching through Problem Solving. In a TtPS lesson, the lesson starts with a situation that students will problematize. The problems either have many correct answers or have multiple solutions and can always be solved by previously learned concepts and skills. Problems like these help students to make connections among the concepts they already know and the new concept that they will be learning in the present lesson. The ensuing discourse among students and between teacher and students during the discussions of the different solutions and answers trains students to reason and communicate mathematically and thereby help them to appreciate the power of mathematics as a language and a way of thinking. In mathematics, language is precise and concise.

Here’s the sequence of my proposed lesson:

1. Setting the Problem:

Myra draw a triangle in a 1-cm grid paper. Without showing the triangle, she challenged her friends to draw exactly the same triangle with these properties:  QR is 4 cm long. The perpendicular line from P to QR is 3 cm. 

Pose this question: Can you draw Myra’s triangle?

Give students enough time to think. When each of them already have at least one triangle, encourage the class to discuss their solutions with their seat mates. Challenge the class to draw as many triangles satisfying the properties Myra gave.

2. Processing of solutions: Ask volunteers to show their solutions on the board. Questions for discussion: (1) Which of these satisfy the information that Myra gave? (2) What is the same among all the correct answers? [They all have the same area]. Possible solutions are shown below.

triangle congruence


3. Introducing the idea of congruence: Question:  If we are going to cut-out all the triangles, which of them can be made to coincide or would fit exactly? [When done, introduce the word congruence then give the definition.]

Tell the class that Myra only drew one triangle. Show the class Myra’s drawing. Question: In order to draw a triangle congruent to Myra’s triangle, what conditions or properties of the triangle Myra should have told us?

Myra’s triangle

Possible answers:

  1. QR is 4 cm long. The perpendicular line PQ  is 3 cm.
  2. QR is 4 cm long. PQ is 3 cm and forms a right angle with QR.
  3. PQR is a right triangle with right angle at Q. QR is 4 cm and PQ is 3 cm.

4. Extending the problem solving activity: Which of the following sets of conditions will always give triangles congruent to each other?

  1. In triangle ABC, AB and BC are each 5 cm long.
  2. ABC is a right triangle. Two of its shorter sides have lengths of 4 cm and 5 cm.

I would appreciate feedback so I can improve the lesson. You feedback will inform the sequel to this lesson.. Thank you.