Category: Geogebra

Posted in Algebra, Geogebra, Geometry

The Pythagorean Theorem Puzzle

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Math puzzles are great activities for enjoying and learning mathematics. The following is an example of Tachiawase. Tachiawase is a popular puzzle in Japan which involve dissecting a geometrical figure into several parts and then recombining them to form another geometrical figure. The puzzle below is credited to Hikodate Nakane (1743). This was one of the puzzles distributed at the booth of Japan Society of Mathematical Education during the ICME 12 in Seoul this year.

Make a shape that is made from two different sizes of squares by dividing them into three parts  then recombine them into one square. [Reformulated version: Make two cuts in the shapes below to make shapes that can be recombined into a bigger square.]

two squares puzzle

Here’s how I figured out the puzzle: I know that it must have something to do with Pythagorean Theorem because it asks to make a bigger square from two smaller ones. But where should I make the cut? I was only able to figure it out after changing the condition of the puzzle to two squares with equal sizes. It reduced the difficulty significantly. This gave me the idea where I could make the cut for the side of the square I will form. The solution to this puzzle also gave me an idea on how to teach the Pythagorean theorem.

I made the following GeoGebra mathlet (a dynamic math applet) based on the solution of the puzzle. I think the two-square math puzzle is a little bit tough to start the lesson so my suggestion is to start the lesson with this mathlet and then give the puzzle later.  As always, the key to any lesson are the questions you ask. For the applet below, here’s my proposed sequence of questions:

  1. What are the areas of each of the square in the figure? Show at least two ways of finding the area.
  2. How are their areas related? Drag F to find out if your conjecture works for any size of the squares.
  3. Can you think of other ways of proving the relationships between the three squares without using the measures of the sides?
  4. If the two smaller squares BEDN and GFNH have sides p and q, how will you express the area of the biggest square LEJG in terms of the area of the smaller ones?
  5. Express the length of the sides LEJG in terms of the sides of BEDN and GFNH.

[iframe https://math4teaching.com/wp-content/uploads/2012/08/Pythagorean.html 550 450]

After this lesson on Pythagorean relation you can give the puzzle. Once they have the correct pieces, ask the students to move the pieces using transformation in the least possible moves. They should be able to do this in three moves using rotation. Click here to download the applet. Note: If you don’t see the applet, enable java in your browser

Use the comment sections to share your ideas for teaching the Pythagorean relation. If you like this post, share it to your network. Thank you.

Posted in Geogebra

Mathlets – dynamic math applets

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‘An applet is any small application that performs one specific task that runs within the scope of a larger program, often as a plug-in. An applet typically also refers to Java applets, i.e., programs written in theJava programming language that are included in a web page’ -Wikipedia. That settles it. It has nothing to do with Apple and small apples. What about mathlets? Yes, you guess it right that it is an applet about mathematics. Not, it’s not yet in the dictionary. But I find it cute and I intend to use it from now on to describe the math applets I have been creating since I started using GeoGebra to create dynamic worksheets for learning and discovering mathematics and not for demonstrating mathematics. Below is a list of mathlets which I posted in the new website AgIMat which contains resources in science and mathematics teaching.

GeoGebra mathlets are interactive web pages (html file) that consist of a dynamic figure (interactive applet) with corresponding explanations, questions and tasks for students. Students can use the dynamic worksheets both on local computers or via the Internet to work on the given tasks by modifying the dynamic figure.

Geometry

  1. Congruent segments
  2. Bisecting a segment
  3. Congruent angles
  4. Bisecting an angle
  5. SSS congruence
  6. SAS congruence
  7. ASA congruence

Graphs and Functions

  1. Coordinates system _1
  2. Coordinates system_2
  3. Coordinates system_3
  4. Introducing function
  5. Exponential function and its inverse
Posted in Geogebra

Pathways to mathematical understanding using GeoGebra

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You may want to check-out the first-ever book about the use of GeoGebra on the teaching and learning of mathematics: Model-Centered Learning: Pathways to Mathematical Understanding Using GeoGebra by Ligguo Bu and Robert Schoen.

Supported by new developments in model-centered learning and instruction, the chapters in this book move beyond the traditional views of mathematics and mathematics teaching, providing theoretical perspectives and examples of practice for enhancing students’ mathematical understanding through mathematical and didactical modeling.

Designed specifically for teaching mathematics, GeoGebra integrates dynamic multiple representations in a conceptually rich learning environment that supports the exploration, construction, and evaluation of mathematical models and simulations. The open source nature of GeoGebra has led to a growing international community of mathematicians, teacher educators, and classroom teachers who seek to tackle the challenges and complexity of mathematics education through a grassroots initiative using instructional innovations.

The chapters cover six themes: 1) the history, philosophy, and theory behind GeoGebra, 2) dynamic models and simulations, 3) problem solving and attitude change, 4) GeoGebra as a cognitive and didactical tool, 5) curricular challenges and initiatives, 6) equity and sustainability in technology use. This book should be of interest to mathematics educators, mathematicians, and graduate students in STEM education and instructional technologies.

STEM – Science, Technology, Engineering, Mathematics

Wikipedia on model-centered instruction:

The model-centered instruction was developed by Andre Gibbons. It is based on the assumption that the purpose of instruction is to help learners construct knowledge about objects and events in their environment. In the field of cognitive psychology, theorists assert that knowledge is represented and stored in human memory as dynamic, networked structures generally known as schema or mental models. This concept of mental models was incorporated by Gibbons into the theory of model-centered instruction. This theory is based on the assumption that learners construct mental models as they process information they have acquired through observations of or interactions with objects, events, and environments. Instructional designers can assist learners by (a) helping them focus attention on specific information about an object, event, or environment and (b) initiating events or activities designed to trigger learning processes.

I’m not sure if the book cites research cases that show how using Geogebra or interacting with applets help students build those mental models. It would be interesting if somebody will really do a study on this.

Posted in Geogebra, Math blogs

Math and Multimedia Carnival # 17 will be hosted here

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Hello bloggers. It’s carnival time again. The 17th edition of Math and Multimedia Carnival will be over here at Mathematics for Teaching. It will go live on 28th November 2011.

You are most welcome to share your posts about mathematics and the teaching and learning of it. You can submit your posts in Math and Multimedia Blog Carnival #17 .

Wondering what on earth a math carnival is? Check out these out: (No, not Zac Efron, the carnivals). But if you want to see Zac, check out the dvd 17  Again.

Please Like, Tweet, and Share so more bloggers will know. Thanks.