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# Week 2: GeoGebra

This task has two parts, and each part has a question in it, to answer here in comments. It introduces GeoGebra - a dynamic geometry + algebra software, and the communities using it.

Video: Michael Borcherds, the creator of GeoGebra, talks about computer-based mathematics.

Part 1

Download GeoGebra and play around freely for a while, trying different things. You can draw objects, like points and lines, and make designs. You can move your objects around. You can also make objects stay put! For example, if you add a new point exactly at the intersection of two lines, however you move the lines, the point stays attached to that intersection.

If you need help, these sites have additional tutorials:

Make a square using GeoGebra. Now drag lines and points that form your square. Does the shape stay square? Make a square that stays exactly square (maybe changing in size or rotating) however you drag its points and lines around. If you are an advanced GeoGebra user or want to play harder, feel free to expand on this task by constructing other objects.

Find three ideas about geometry that students can learn if they make an exact square in GeoGebra, like you did. It can be something students will have to notice themselves, or something you will be able to point out for them once they had that experience.

Part 2

Put your square construction into GeoGebraTube, the community collection of materials. Include the link to your construction in your response to this task. While there, check out other people's materials!

Find a GeoGebra applet that models something and that you think students would like. What does the applet model? What would students learn about the properties of what is modeled if they built such an applet?

• Part 1 and Part 2:

I constructed a square by using the "Angle with a Given Size" tool and making all 4 angles 90 degrees. I then used the "Polygon" tool to close the figure. I then labeled the sides to be able to show the students how the sides change as they move the points on the square in the applet. Here is a link to my construction on GeoGebraTube: http://www.geogebratube.org/student/m28683. This is the first time I used this program, so I had to play around with it a lot. I could not figure out why I could not move point C and D in my figure. Any advanced GeoGebra users out there that can figure out why? I am really interested in learning more about this software and how it works.

Question on Part 1:

Three ideas that students can learn from making a perfect square in GeoGebra is that all the angles are 90 degrees, the sides are congruent, and that all the exterior angles are the same. You can also have the students construct the diagonals of the square and then they will also be able to see that the diagonals are congruent and intersect to form a 90 degree angle, which makes the bisectors perpendicular lines.

Question on Part 2:

I found an applet that involves perpendicular lines and their slopes. Here is a link to the applet: http://www.geogebratube.org/student/m23016. This would be great for my Algebra 1 students when they are learning about this property in Chapter 5. It is easier for them to see how parallel lines have the same slope, but the opposite reciprocal slope of the perpendicular lines is harder to remember. This applet allows students to move the points on the perpendicular lines around and discover the relationship between the slopes of the two lines. If the students constructed the perpendicular lines themselves in this applet, they would also learn what points, lines, or measurements are important to include in the applet to be able to show the opposite reciprocal slopes and that the lines are perpendicular (showing the right angle the lines form). I think when students are put into the “creating” role, they learn so much more about the concept because they have to take into account everything needed to make each figure and label the necessary information to prove their point. I think that the creation step would have to be after the students have already learned and mastered the concept. This is a great challenging to enhance their understanding of the material!

• The question you ask about points D and C is excellent. The reason I love it so much: it introduces an idea fundamental to all areas of mathematics: dependent and independent objects (see the picture). Your points A and B are independent, so you can move them as you wish. However, your points D and C are constructed by rotation, so they are dependent. If you construct things in different order, you can change that - for example, you can make C and D independent and A and B dependent. The same idea (dependent and independent) appears in algebra about variables, in calculus about derivatives and integrals and so on.

So this idea can be added to your list of what students can learn by building a square.

When you move things around and see the living changes, as you note, it's easier to notice patterns and discover relationships. This is a big reason to use software. Good point!

People argue about the order of creation vs. mastery - it's a big debate in education. However, most agree with your statement that Creating tasks require students to learn much more than lower-level tasks!

• Part 1

One idea that scholars could develop is the fact that sqaures are composed of two triangles when choosing a vertex of their choice and connecting to another vertex of their choice. They could extend that activity to see if they can discover how many triangles does it take to compost of polygons outside of quadrilaterals.

The second idea that scholars could develop is that they are able to create the other quadrilaterals by moving a specific point or points and a line or lines on the square. Leaving the line connected to the vertices will also prove that all quadrilaterals are composed of two triangles.

The third idea that scholars could develop is what happens to the perimeter and area of square when you double, triple, quadruple, etc... the length of a side.

Part 2

http://www.geogebratube.org/student/m28684 -> My Square Construction

Golf Parabola
The applet models the trajectory of a golf ball depending on the angle at which you hit the ball and its initial speed. The goal is for scholars to have their ball land where the golf flag is. They could complete their calculations on a piece of paper to determine the angel and intial velocity to get the golf ball to the flag. Once they figured out their calculations, they could enter it in the applet to see how close they came to reaching a hole in one. They would learn how the angle at which something is projected and the force used to project the item determines how far it would travel. They should also notice that is creates a parabolo in which they could determine its min or max.

• You are focusing on powerful, deep ideas students can learn - higher-order thinking related to the topic of construction. This makes for interesting classes and helps students appreciate the power of math. I can see how you can extend the project to support these ideas - say, invite different students to build different polygons out of triangles, and compare notes. You may need to plan good discussion questions to make sure students ponder the concepts deeply. Building + talking should get most people on board with your adventures!

I noticed your square stops being square if you drag points around. The challenge is: "Make a square that stays exactly square (maybe changing in size or rotating) however you drag its points and lines around." - Can you add this please? You can add another comment.

• Part 2:

First, let me apologize for saying 380 instead of 360 degrees in PART 1...I meant to hit PREVIEW, but mistakenly hit POST.

http://www.geogebratube.org/material/show/id/28655

HERE IS THE APPLET link I found would be a good example of something I could use in a middle school setting:
http://www.geogebratube.org/student/m10861

"Vertical Angles & Linear Pairs."
This applet is a great way to demonstrate the relationship to the degrees of circles and lines & angles.

-Students can take the points of the rays & move them & watch how no matter what the circle when you add all 4 angles together, will always add to your circle 360 degrees.

-This a a great example of what a linear pair is...meaning, no matter where the intersection lands, each correlating opposite angle with be equal in order for the lines to really truly be LINES. I love how this is shown where the student can manipulate this.

-The linear pair example does the same thing & its shows how when you have LINES vs line segments also, if you have vertical angles, you will also create linear pairs.

You can take this further because I always think of geometry as leading to building REAL life objects...like a shelf...or even a building, etc. Geometry can really begin to get students thinking of a future career as an engineer, or something like that. As they see these examples & play around with these online examples, maybe they can begin to imagine themselves building a skyscraper.

These geometric applets truly are the building blocks of drafting designs of many things.

• I see that you played a lot with line tools, for your own applet and for the one you found. It reminded me of unit studies, where you can focus on one idea (in this case, lines). Then you can use this idea to ponder deeper issues - for example, the point you made about the power of students manipulating lines to notice something about opposite angles. LINE is a fundamental building block in engineering drafting, as you explain - and also in art and now in computer design. So many kids now want to work in game design or other computer-related fields where they need tools of that sort...

There is an "edit" button at the top right corner of your comment, so you can correct typos or add something you forgot, etc.

• 1- you will find out the your angles are each 90 degrees

2-you will find out that your lines are equal in length because you've created 2 radiuses of the same circle

3-you will find that when you have a right angle that you've formed, you can create a circle around that vertex point of the right angle. Because you know that a circle is 380 degrees. If you have to points on each of your angle's rays, you CAN create a perfect circle because a circle is made up of 4 right angles.

• These are good math content items to notice. I am tempted to add items from your other comment here, as well - maybe students can discuss where these items (angles, lines) play important roles in life. Here is a cute little cartoon about it, one of my favorites!

"The dot and the line: a romance in lower mathematics"

• PART 1:Boy, it has been a REALLY long time since geometry 101 (which I'm taking this summer AGAIN!) This exercise was so helpful in remembering what makes a square a square & a circle a circle. At first, I was simply using the first graphing tool that pops up and playing around with that. With the numbers shown on the graph, you are making a square, but you CAN move the lines around....which means, you really havent proven that you've made an OFFICIAL square.

Once I use the tutorial, it made sense to start with the circle. If you don't have the luxury of the pre-filled numbers on a graph, you need to be able to prove that your object truly is a square. In order to know for sure that your line sements of your square are indeed equal as well as your angles are indeed right angles, you must use the radiuses of the circle. Knowing that every radius of each circle is the same length, this is how you are assuring that your square segments are exactly the same length (hence a square.)

Having my future students using this tutorial makes the most sense to me...starting with the circle & moving it from there into a square. Being able to have students recognize the relationship between a circle and a square is really an amazing correlation for anyone who is new to geometry...like they would be in middle school.

• Relationships - that's where it's at!

"It" being math.

"OFFICIAL square" made me chuckle. Love the phrase! When you work with kids, you can observe them using many different terms of that sort, making up some of their own, and so on. For example, they may call a proven construction "ubreakable" or "perfect." I try to pick up and repeat kids' own language, in addition to the traditional terms, because it helps them relate to math.

• PART 1

I constructed my square using point capture fixed to grid, so I could put my points at exact coordinates.  After I drew line segments to connect my 4 points in a square shape, I fixed my object so it could not be distorted (edit, select all, object properties, fix object).

I'm answering this question from the perspective of using GeoGebra to introduce my students to formal geometry.  My ideas that relate to drawing a square with GeoGebra are:

1.  The interior angle at each of a square's 4 vertices is 90 degrees.  I'm not sure if students would experiment with the angle tool on their own to discover this; they may need to be told to measure each angle (using the tool) and then discuss the findings.

2.  A regular polygon's sides are all the same length.  Students may or may not notice this by themselves via experimenting with the polygon drop-down menu.  Having students draw both a polygon and regular polygon (such as a quadrilateral and a square) and compare the two would hopefully prompt them to discover this for themselves.

3.  Students sometimes confuse line segments and lines.  GeoGebra will demonstrate the difference very nicely if students draw a square by connecting points with a "line through two points", and then draw another (preferably identical) polygon connecting points using "segment between two points" (both commands are on the 'Line Through Two Points' drop down menu).  I'm not sure if students would discover this by themselves via experimentation, but the teacher could assign the task and ask students to describe what they see, and explain why.

PART 2

My username is MsMockingbird and my square is at

The model that I chose is geared toward middle-school students, and has has an interdisciplinary approach.  http://www.geogebratube.org/student/m18487demonstrates how manipulating angles can greatly alter the appearance (specifically, perspective) of a tunnel/passageway.  I chose this applet because I've often heard students complain about not being able to relate curriculum to the real world; this provides an excellent, yet easily understood elementary example of how mathematics is present in the arts.  Many students are in awe of those who can transform 3D perspective into a 2D medium, while many young people who are gifted with the ability to intuitively do so cannot explain why their work is so aesthetically pleasing (such as my youngest sister many years ago).  A demonstration of how angles influence visual perspective would benefit all students, for they would be exposed to yet another way that mathematics influences the world as we know it.

• Hiya Sue -  1. Thank you for your perspectives about drawing a square with GeoGebra and about students confusing segments and lines. I have been thinking about just these two things this week- but as separate problems - and reading your thoughts helped me think with a different perspective.

2. And speaking of perspectives, I really liked Daniel's Tunnel Perspective that you found. I adapted it for devices (ipads, tablets and smartphones - albeit very slow on the latter) and posted it: http://www.geogebratube.org/student/m28417. I classified it as CCSS-6-g-4, but perspective is not really discussed at all and representation of 3d objects in 2d is barely mentioned at all in the CCSS.

3. Now I have been thinking about your comment that a regular polygon with 4 sides is always a square and realizing that the definition of a regular polygon must have 2 parts (I purposely am not looking it up). So (a) Why isn't a rhombus a regular polygon? and (b) Why isn't a rectangle a regular polygon? Also, it was interesting for me to realize that there are not analogous questions for triangles. That is, if a triangle has 3 equal angles it is equilateral and vice versa.

• The math term for your construction is "the coordinate method." You used particular coordinates to make a geometric construction. You can invite students to compare and contrast this with "coordinate-free" constructions such as using circles of the same radii to make a square. As you can imagine, the coordinate method is used a whole lot in computer graphics!

You ask a valid question about discovery. "Guided discovery" is when teachers gently lead students to notice various mathematical properties. This has to be done with good taste and with intellectual honesty. Otherwise, guided discovery turns into a mundane "guess my answer" game where students simply try to figure out what is in teacher's head, rather than observing the math itself.

What a neat tunnel applet! I played with it for good many minutes, because I draw a lot and this would help me understand things about art when I studied perspective. As you say, it's a strong power, almost something out of Marvel comics universe. A 3D artist superhero?

• Part 1: I’m definitely not an advanced GeoGebra user, but I’m taking Modern College Geometry this semester and have just started learning how to use Geometer’s Sketchpad. As a result, I was able to construct a square that stays a square pretty easily, so I played around with constructing some other shapes. My square construction is here.

For the square, the key to keeping the adjacent sides the same length is to use two radii of the same circle. This way, no matter how you stretch and drag the square, those sides are still radii of the same circle. I think that connecting circles to polygons is a really important concept, so that is definitely something I would want my students to see during this task. I hid my circle, but by keeping it visible, students could see that relationship as they move the square around. I think it’s easy to forget about circles when you’re looking at squares/triangles/other polygons, so I like the idea of my students seeing them together since there are important relationships, such as using circles to construct equilateral triangles.

I would also want my students to see that when two parallel lines are cut by a line that is perpendicular to one of the lines, it is also perpendicular to the other. This is a special case of the results of parallel lines cut by a transversal, and I think this task would illustrate that concept nicely.

I also think this task would be good for getting students to think about the minimum requirements for a square. It’s pretty easy for students to understand that squares have four equal sides, four equal angles, and both pairs of opposite sides are parallel to each other. But you can define a square with fewer criteria than that, and the actual construction of the square would give students the opportunity to really think about that concept.

I also constructed a quadrilateral, and then connected the midpoints of each side to make a parallelogram. My construction is here. I think this is one of those results that seems surprising at first – why is it always a parallelogram when quadrilaterals can be so different? I think that students constructing their own quadrilaterals and observing the parallelogram as they stretch the quadrilateral is a really good way to illustrate this result. Students can also construct the diagonals (which I did) to really visualize it as a combination of triangles. I really love the idea of students using GeoGebra or similar software to model geometric concepts like this because it really reinforces the concepts more than a static sketch on a whiteboard.

Part 2: One of the applets I found on GeoGebraTube was this model of planetary motion. This model illustrates Kepler’s Second Law of Planetary Motion, and I really like this because that’s one of those physics concepts that sounds confusing in words, but the model makes it very clear. I love the idea of using a model like this with students in physics class, and having them construct something like this could help them understand it conceptually, rather than it just being a law of physics that they memorize.

I was very pleased to see that there is a tag for physics, and I started thinking about all of the concepts that could be modeled with this kind of software. As a novice user, I still have a lot to learn about GeoGebra before I can create any really exciting applets, but just for fun, I created this very basic construction that models vector addition using the tip-to-tail method. My model is very basic, but this exercise really gave me a lot to think about as a future math and physics teacher.

• "Minimum requirement" is a strong overarching concept. It comes up in many areas of math, for example, what does it take for a function to be differentiable? What does it take for an equation to have solutions? Great choice of focus for a task! And this idea is easier and clearer with technology, because you can literally count how many steps are there in your construction, or how many commands are in your code.

"Confusing in words, clear in models" - YES! That's a strength of good software.

I played with your vector model a bit. Building these constructions does make you think! One suggestion I would make for when you play with it again - make point T dependent on points P and Q. This way, when you drag P and Q around, T will move automatically. There are other directions you can take it, of course. Physics is a great context for geometry. I am reading Newton's "Principia" at the moment and it's mostly geometry, to my surprise (I expected much more pure physics and calculus).

• And this idea is easier and clearer with technology, because you can literally count how many steps are there in your construction, or how many commands are in your code.

I keep saying that one of these days I am actually going to read Principia for myself. You have to give credit to someone who, when lacking the mathematical language needed for physics, went ahead and invented calculus. I'm also surprised to hear it's a lot of geometry, although I suppose it makes sense since calculus was a relatively new branch of mathematics and hadn't been given the rigorous treatment of years of study, while geometry was very well accepted and understood.

I spent all weekend reviewing for the Physics Praxis (which I am taking next week) and every time I turned the page, I thought, "this would be a great place to incorporate technology!" In high school physics, students have to learn the concepts without calculus, which can be really difficult when physics is basically the physical world described with calculus. There ends up being a lot of information that students have to take on faith because they don't have the calculus background to understand the derivation. I think that using modeling software would be incredibly helpful for so many topics, so that students can construct models that give them what they need to understand the ideas without the language of calculus.

One of the physics topics I came across this weekend that I thought could really benefit from the use of technology is reflection and refraction of light applied to ray tracing for mirrors and lenses. It's a lot of geometry and trigonometry, plus some algebra. The diagrams can get so messy and cluttered, and it can be hard to keep track of which line is which. I think that a model would be an excellent tool because students could get their hands in there and see why the image is real or virtual; upright or inverted. And what happens when the observer or object moves? What happens when the angle of incidence changes? Etc.

I have no idea how to create these kinds of models yet, but just these first few tasks have really shifted how I think about the different ways I can present material to my students so that they can truly understand it.

• Part 2

Here is the link to my construction, http://www.geogebratube.org/material/show/id/28163.

The applet that I came across that I found most interesting is A Tessellation. http://www.geogebratube.org/student/m27031  The author asks what the basic shape is, what the motions are and can you figure out how it was made. The tessellation shown is more complicated than ones I have created or had students create. Tessellations are a good way to teach rotation and reflection. I would give the students some specific parameters. Such as, no more than two different shapes and no more than two sizes of each shape to start. As they began to better understand the manipulation of the shapes they could add and expand on the original. I really like that the shapes can be adjusted so the students can see how by altering the original shape the tessellations change in the same way. This is hard to do with paper and pencil and even cut out objects.

• Tessellation is where computers shine. You know how students get frustrated when they can't cut shapes precisely and they don't fit? Any tessellation project with paper is just slow and tedious. Some people are okay with it, but it puts limits on ages and skills.

• Here is my Part 1. It took longer than I allowed time for. I did get distracted playing with the different tools though. I will do part 2 later but wanted to get this up so I do not forget!

The first square I made I used the points (0,0), (0,4), (4,4), (4,0). When I move the point (4,4) the square is no longer a square. However, the point on the origin I am unable to move and the points on the axises are only able to move along the same axis it was placed. I am unable to move any of the lines. I found this odd so I made a different square using points not on an axis or the origin. Now I am able to move any point and any line in any direction. This movement does make my square a square no more though!

This is how I made my exact square. I first created a circle with radius AB. I then created a perpendicular line by selecting the perpendicular line tool then clicking on segment AB and point A. I labeled the point where the perpendicular line intersects the circle point C. Next I created a parallel line to segment AC and passing through point B. I used the parallel line tool and selected segment AC and point B. Then I created my fourth side by making a perpendicular line selecting the line going through point B and point C. then I marked the point where these line intersect as point D.

No matter how I move points A or B (the only points I am able to move) the square remains a square. I made my picture less cluttered by hiding the lines and making them line segments.

Three ideas about geometry that students could discover by making an exact square could be…

After they make this square, measure all the sides and explain why they are all equal. The answer would include that the radius of a circle is the same everywhere, so the two segments interior to the circle will always be the same measurement.

The measure of the angles will all be 90 degrees because of properties of perpendicular lines.

When the angle using the center of the circle is measured GeoGebra gave the reflex angle. This could be a new topic to some students. They could try to explain why that measure was given and what it means. This might be some the teacher does as a closing for the day. It would be a good way to close the lesson and bring the group back together if this is done in a classroom setting.

• GeoGebra interprets the coordinate lines as constructed objects. When you put a point on a line, it stays on the line - and that includes coordinate lines. So the origin always stays put...

Measurement vs. deduction is a very interesting math area. You can measure segments to show they are equal, or you can use reasoning to prove them equal. Comparing and contrasting the two methods, as you suggest, can lead to interesting discussions.

And yes, this sort of work can take more time than we anticipate! I've seen people (including myself) getting lost for hours...