Tag: GeoGebra activities

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 Algebra, Geogebra

Making connections: Square of a sum

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One of the ways to connect concepts in mathematics is to make use of the same representation to teach mathematics. Let me take for an example the visual representation of the square of a sum, (a+b)^2 = a^2+2ab+b^2 . This concept is usually ‘concretized’ using algebra tiles. However, if you have facility for computer technology I would recommend using the GeoGebra applet below: [iframe https://math4teaching.com/wp-content/uploads/2011/09/square_of_a_sum.html 650 550]

Suggested tasks:

1. You can show the grid (click view to do that) to teach Grade 3 pupils about area. You can change the dimensions of the rectangles and squares by dragging D.

2. For older students you can give this task (don’t show grid but you can show lengths then drag D): The square AGHC is dissected into rectangles and squares.If the sides of square AEFD is 2 units and that of square DIJC is 3 units,

a) calculate the area of the other rectangles and square?

b) write two numerical expressions representing two ways of getting the area of the big square.

4. Having worked with numerical expressions, students will be ready to work with variables: If AD is x and DC is y, find two expressions for the area of the square AGHC. This of course leads to the identity (x+y)^2 = x^2 + 2xy + y^2 . The popular FOIL method should only come after this activity. For the record, I’m not a fan of this method.

5. This representation can also be used to teach how to calculate expressions such as 0.75 x 0.75 +0.25 x 0.75 x 2 + 0.25 x 0.25. This is one way to help students appreciate an application of the square of a binomial.

6. You can also use this figure to teach the idea of function: If you drag D along AC, how will a change in the distance of AD affect the area of the rectangles and squares?

a) Create a table comparing the area of the quadrilaterals as the side AD increases from 0 to 5 units.

b) Do as in a) but this time compare the perimeters.

c) Express the area/perimeter of each quadrilateral as a function of the length of AD.

I will write about square of a difference in the next post.

Posted in Algebra, Geogebra, GeoGebra worksheets

Teaching mathematics with GeoGebra

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GeoGebra constructions are great and it’s fun ‘watching’ them especially if you know the mathematics they are demonstrating. If you don’t and most students don’t then we have a bit of a problem. Even if the applet demonstrates the mathematics to students I don’t think there’ll be much learning there. No one learns mathematics by watching. We know that ‘mathematics is not a spectator sport’. You have to play the game. In Geogebra  and Mathematics I proposed that if GeoGebra is to help students in learning mathematics with meaning and understanding, then students should know how to use it. But these GeoGebra tools will be most useful only to students if they know the mathematics behind the tools and why they work and behave like that.  And so we teach students the mathematics first? Where’s the fun in that?

I believe (Translation: I’ve yet to do a study if my theory works) that it is possible to learn mathematics and the tools of GeoGebra at the same time .  I will be sharing in this blog GeoGebra activities where students learn to use GeoGebra as they learn mathematics. The main objective is of course to learn mathematics. The learning of GeoGebra is secondary. I will start with the most basic of mathematics and the most basic of the tools in GeoGebra: points, lines, and the coordinates system.

The lesson includes four GeoGebra activities:

Activity 1 – What are coordinates of points? Read the introduction about coordinates system here.

Activity 2 – What are the coordinates of points under reflection in x and y axes?

Activity 3 – How to describe sets of points algebraically Part 1?

Activity 4 – How to describe sets of points algebraically Part 2? (under construction)

Posted in Algebra, GeoGebra worksheets

What is a coordinates system?

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This is the first in the series of posts about teaching mathematics and Geogebra tools at the same time. I’m starting with the most basic of the tools in GeoGebra, the point tool. What would be a better context for this than in learning about the coordinate system. Teacher can use the following introduction about geographic coordinates system and the idea of number line as introduction to this activity.

A coordinates system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric elements. An understanding of coordinate system is very important. For example, a geographic coordinate system enables every location on the Earth to be specified by a set of numbers. The coordinates are often chosen such that one of the numbers represent vertical position, and two or three of the numbers represent horizontal position. A common choice of coordinates is latitude and longitude. Sometimes, a third coordinate, the elevation is included. For example:

Philippine  Islands are located within the latitude and longitude of 13° 00 N, 122° 00 E. Manila, the capital city of Philippines is 14° 35′ N, 121º 00 E’.

In mathematics we study coordinates systems in order to describe location of points, lines and other geometric elements. The numberline is an example of a coordinates system which describe the location of a point using one number. The coordinates of a point on a numberline tells us the location of a point from zero. But what if the point is not on the line but above of below it? How can we describe exactly the location of that point? This is what this activity is about: how to describe the position of points on a plane.

You would need to familiarize your students first about the GeoGebra window shown below before asking them to work on the GeoGebra worksheet.

Click here to go the GeoGebra worksheet – What are coordinates of points?