Transformations with Geogebra

Use this Geogebra Book to help demonstrate translations and reflections.

Use this Geogebra Book by Jed Butler to demonstrate rotations.

An organizer used during a Geogebra presentation can help students organize their thinking and study later on.  Follow this Dropbox link to download one that can be used for section 4-8.  Below is a sample of what is on the organizer:

Geometry: Slogan Project

Below are two examples from MVHS Geometry Slogan Project.  This project almost speaks for itself, but I will post a link to the directions soon to clarify directions/scoring.  Student posters were presented in class at the end of the project.

Math Problems from Exeter

If you ever want to do math problems in your spare time, or are looking for inspiration, check out the math curriculum from Phillips Exeter Academy.  The math program is integrated, and the problems spiral like CPM.  They do a great job of developing concepts through problem sets.

Some things I wish I knew when I first started using Geogebra

Using Geogebra as a demonstration tool for teaching concepts such as graphing and transformations can be amazing, but there is so much that can go wrong during the lesson.  After some pretty intensive play sessions with Geogebra and reading through a few tutorials such as this one by Gerrit Stols I was ready to try a demo in my class.

One of the first concepts I showed to students on Geogebra was how the factors of a polynomial function are related to its roots.  I typed in the function f(x)=x(x+2)(x-7) into the input bar at the bottom of the screen, hit enter, and below is what we see.

Not so great.  If I want my students to understand the nature of cubic functions, this is definitely not  what I want them to see.  Select "Move Graphics View" from the toolbar near the top of the screen, and then drag on either axis to rescale.  Drag in any quadrant to recenter.  When you are done, select the "Move" tool from the toolbar so you can select or move object.

The graph below looks better, but the students in the back still can't see the numbers on the axes.  The function will be hard to see on worksheets, so changing the line thickness can help as well.

To change the font size, select "Options" from the top of the page, then font size, then 24.

To change the line thickness right click on the function and select "object properties".  Select the "style" tab to change the line thickness.

The modified graph is much easier to see (and definitely appeals more to my inner interior designer).  There is also a tab for color under the object properties, which is especially nice if you want to build a graph matching activity.

One final must-have tip is to use a text box to display the equation for the function.  Select the "text" tool from the toolbar, click in the graphics view, type f(x)= and then from the objects menu select your function f.

Select the "move" tool and then drag the text box to a good location.

I made the screencasts below for a presentation last year with the Mountain View High School math team.  You can check out the screencasts, though I'll give a fair warning that they were made at about 5am, pre-caffeine.

GeoGebra Tutorial: General Tools

Modify the axes and grid

Modify a point

Modify a function

Translations with Geogebra

Unit planning for transformations is almost complete.  We have a set of notes/practice worksheets from the county office of ed (really great!), an FAL (Transforming 2d figures), and a common test.  The next step is thinking about how to use Geogebra to help with demonstrating the concepts and with practice.

There are a few different ways you can use Geogebra to look at translations.  The first way is to use a vector to translate a point or a figure.  The diagram below has a point and a polygon, so my next step will be to add a translation vector anywhere on the screen.  It doesn't matter where you put the vector.  You can find the vector tool under the dropdown menu for lines.

 For this example I'll use a vector to translate my point/figure 3 units right and 2 units down.  You can see the vector below, at the origin.

It might make more sense for students to see the vector starting at the point that will be translated, but then you need to construct a new vector each time if you want to stay consistent.  If I put the vector away from my diagram, I can use it each time I want to translate an object, and explain to students that the translate tool works by selecting the object first, then the vector that describes its translation (see pic below, located in a dropdown menu).

The diagram below shows my point and my polygon after translation.  Notice how the program automatically names the points in the image using the prime notation.

If you want students to focus on using the coordinate rule (x,y) -> (x+3,y-2), then you can do the transformations without using a vector by making use of the coordinates of your pre-image point(s). Before you give this a try, it is best to open a new window and add point A.  The way Geogebra names the coordinates of point A is (x(A),y(A)).  To translate point A 3 units right and 2 units down, create a point with coordinates (x(A)+3,y(A)-2).  Create this point by typing into the input bar at the bottom of the screen.  Better yet, give this new point the name A'.

Select the "move" tool, then drag point A around and watch how point A' moves.  Another helpful strategy is to turn the trace feature on for both points.  Do this by right clicking on the point and selecting "trace on".

Make sure the "move" tool is selected, then drag point A and watch as point A' follows along and traces out a figure that is congruent but translated 3 units right and 2 units down.  To clear the traces from the screen, type ZoomIn[1] into the input bar at the bottom of the screen.  (This command zooms in your screen, making it 1 times as large as it was before.  Clever trick to get the traces off the screen).

Geometers Sketchpad has lots of presentations on transformations that can be viewed on the Dynamic Number Project website.  My understanding is that students can access these presentations with the Sketchpad Viewer on an iPad.  I haven't thought too much yet about how I can use these in my class.  We don't have Sketchpad, and we also don't have ipads for each student.  For now I am trying some of their ideas on Geogebra, though I have to admit the experience isn't as smooth.  One strategy that Sketchpad uses is to attach a point to the perimeter of an object.  Geogebra has a similar tool, which is an option in the dropdown menu under polygons.  Make sure you have the object and a point created first.  Then when you select the "attach/detach point" tool select the point first, then the object to attach it too.  I found out the hard way to select the interior of the object.  By selecting the perimeter of a polygon, the point was confined to the line segment that created that side only.  

Once I had point A attached to the polygon below, I dragged point A to the edge and around the perimeter.  This created a congruent the congruent shape in orange traced out by point A'.

Growth Mindset Posters

We have GOT to do this at our school!! #NavasotaInt @KellyKastner @bradfordkevin66 pic.twitter.com/s8QyVVD06Y
— Todd Nesloney (@TechNinjaTodd) August 17, 2014

QR Code Scavenger Hunt for parallel lines cut by a transversal

You can use this activity to help students review the vocabulary and relationships amongst alternate interior angles, corresponding angles, etc.  

I've set up this activity so that you can start at any problem number.  The first step is to go to any of these problems, and scan the code to get your first clue.  If you are doing this activity with a class, the following ten papers will be displayed on the walls around the classroom.  Below the pictures in this blogpost is an organizer that students can use to record their work.  If you want to try the scavenger hunt first, scan any of the codes below and get your first clue.  The clue will give you a description for a pair of angles, and then you are to find the picture that meets your description.  When you have located the correct diagram, scan the code below to get your second clue.  Repeat this process until finished.  You can check your answers below when you are finished (or skip ahead if you aren't ready to play with QR codes yet.  An easy to use QR code reader is Inigma).

First things first.  The answer key is as follows:

QR code 1 sends you to problem 10, QR code 10 sends you to problem 7, then to 6, 2, 5, 3, 9, 4, 8, and QR code 8 sends you back to problem 1.  This way you can have students start at any QR code and they can make their way around the room back to where they started.  

The student record keeping sheet consists of a table, with the first row below:

This entry is to be used with the clue that matches diagram 1.  Students will label the diagram with the correct angles.  Next they will write their clue as a statement.  I chose this as a next step because students would be able to refer back to this activity later if needed, and the writing won't take that long.  The third column is a possible extension.  I was thinking to give students a word bank of angle vocabulary, and they can pick from the word bank when choosing the relationship for angles f and g.  A few teachers have already decided not to include the third column in the activity, or to leave it as a follow up activity for those that finish early.  I have added an extra record keeping sheet below in case you do not want the third column.

Since this activity focuses on vocabulary and is at the lower level in terms of depth of knowledge, this can be used as an opportunity to incorporate math practice standards.  Set the expectation that students will explain their thinking to a partner as they go through the problems, and count this as part of the activity score. 

I'm sure there are some things that can go wrong during this activity.  One thing I wonder about is whether the diagrams will be too small to have posted on the wall.  This made me wonder if it would be better to have students browse the diagrams as we have in this blogpost.  I also wonder if the conversations will be better if students aren't wandering the classroom.  Students also need to know that the QR code reader needs to be facing the right way.  Sometimes it is the simple details that make the difference!  

Below are the resources if you'd like to try this activity.  And if you try it, please let us know how it goes.

Problem Cards and Clues

Record Keeping Sheet

Record Keeping Sheet for a shorter version of the activity (no third column)

Make your own QR Code Scavenger Hunt! by iLearn Technology