Geogebra/Desmos Functions Resource Page

I came across this resource page that I made for a presentation back in January.  I thought it was going to continue to get filled out, but I had to move on to other projects.  I wanted to share this in case it inspires any summer learning, and I hope the resources can help give an idea of how dynamic graphing software can enhance the student learning experience for functions.  A lot of these links are to pre-made Desmos or Geogebra pages by other authors, and some I created on my own.  I linked to blogposts for a few of the activities as well to give a better idea of how the activity was used in class.



You can also open this document in Google Drive and make your own copy at this link.  I've spent a lot of time over the past couple of years thinking about how to use Geogebra (and now Desmos) to improve student understanding of functions.  Most of the textbooks that I've used in Algebra and Algebra II in our district don't go very far beyond "graph this function" for practice problems in these sections.  Geogebra and Desmos offer us a chance to change the task and the types of questions that we ask when teaching functions.  

For the project above I included links to sliders, flashcards, and transformation activities when I could find/create them.  Students can play with sliders when first being introduced to a new function type to help them attend to the structure of the function with its parameters, and they can build on the background knowledge from previous function types.  Flashcards can be used to review vocabulary, and to check for understanding of the values of the parameters.  More on Geogebra Flashcards here.  Transformation activities or "other" activities mostly consist of Geogebra pages where students will find the equation for a given function.  

Though I haven't used all of these pages, the ones that I have used in Algebra II and Pre-Calc facilitated richer conversations, and overall better understanding of functions and transformations in my own class.

LAHS Geogebra Training 1-6-2015

Please sign in, then fill out this quick form to help us guide our time together.

Vision: Teachers and students will be proficient at using Geogebra to demonstrate thinking and to solve problems.

Goals:
-Learn about support available for integrating Geogebra into your class
-Learn how to locate and use pre-made Geogebra presentations
-Learn some basics for using Geogebra

Basics Tutorial
Screencasts
Functions
Geogebra Tools Page

I Spy: Introduction to Congruent Triangles

Below is an introduction to congruent triangles that makes use of student prior knowledge of transformations.  It also encourages students to deduce information from a diagram and to justify their reasoning. 

Show students the following diagram, and go through the steps below.  Some questions/prompts are shown in blue.

1.  What type of transformation is shown in this diagram?

2.  Reveal the measures of ABC.  Ask students, “Who thinks they know another measure in the diagram?  Be ready to explain your answer.  Show me ONE other measure that you know, and hold up your whiteboard when you are ready.”  Teacher can show answers from students that used notation correctly.  Reveal the measures of AFE.

3. Show students the statement below.

Ask them how we might fill in the blank, and why.

4.  Move point C to the horizontal line (you can do this in the Geogebra applet).  

Leaving the measurements showing, ask students to tell you what they “spy”.  They can write on their whiteboards, “I spy __________________.”  Answers must be defended.  Make sure to have the list of possible answers written on the whiteboard.  

The choices are angle bisector, midpoint, isosceles triangle, parallel lines, and right angles.

3.  Move point B to F. Move point A down the vertical line so your diagram looks like the one below.

Leaving the measurements showing, ask student to tell you what they “spy”.  They can write on their whiteboards, “I spy __________________.”  Answers must be defended.

I SPY CHOICES:  angle bisector, midpoint, isosceles triangle, parallel lines, and right angles.

4.  Turn the measurements off.  Move point A back up to the horizontal line, and make sure point B is on point F.  Ask student to tell you what they “spy”.  They can write on their whiteboards, “I spy __________________.”  Answers must be defended.

I SPY CHOICES:  angle bisector, midpoint, isosceles triangle, parallel lines, and right angles.

Extension if there is time:  Since we haven’t seen parallel lines yet in this dynamic diagram, ask students how we can modify the diagram to form parallel lines (and explain why).

There is a second page in the Geogebra Book that has a triangle rotated around a vertex by 180 degrees to form the second triangle.  

Part II will have the I Spy Class Activity if you have time for it.

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:

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'.

Modeling with Trig Functions

The activity below was used by some of the Algebra 2 teachers at LAHS last year, but it is also a great activity for trig/math analysis.  If you'd like to use this activity with a different context, please let me know.

I wanted to share this as blogpost to give an example of what we can do with the blog.  In this case, I am sharing parts of an activity that took place in a class, and some of the questions that a teacher might ask as they go through the activity.  So the blogpost can serve as a script if you find that to be helpful.

Towards the end of this unit in Algebra 2, students were asked to model real world situations using trigonometric functions.  We picked a few problems involving tides and ended with a problem from Illustrative Mathematics called Foxes and Rabbits 2.  We used a series of scaffolded Geogebra presentations to help students get started.  The first presentation is below, and is an embedded applet that you can play with.  Type a function into the f(x) input bar and press enter to see it on the graph.  You should try this right now. It is fun! The goal is to get the sine graph to pass through all of the data points.  



We introduced this to students by first talking about the data and how tides are measured.  You can talk about how the moon impacts tides and how the data is roughly periodic.  At this point students should know the amplitude and period for y=sin(x). Next, have a class conversation about how to transform this graph to pass through the given points.  The power of a premade interactive model is that you can pause during the conversation to allow students time to process.  I make use of the think-pair-share structure with the expectation that students may be called on to explain what they know or discussed with a partner.  Below is a list of questions that can be used with this presentation after the initial conversation about the data.  Feedback  on this list of questions would be great!

1. What is the midline for the Santa Cruz Tides data? (Discuss first, then reveal the midline by selecting the midline box.)

2.  How can I change (transform) the function f(x)=sin(x) so that it has this midline? (Discuss, then type in the correct function and reveal the change in the graph.)

3.  What is the maximum value of this function?  What does it represent with respect to the tides?  Find the minimum value as well.  Discuss and then reveal the max and min lines by selecting the appropriate box.

4.  What is the difference between the highest and lowest tide measurements?  Does this help me find the amplitude, period, or vertical shift of my function? 

5. Now that I know that the amplitude is 1.5, how can I change my function f(x)=sin(x)+2.5 to account for an amplitude that is not 1?

6.  Discuss with your partner what one period of y=sin(x) looks like.  My current graph has been shifted horizontally from the parent graph y=sin(x).  Can you find a new starting point?  How can we change our function y=1.5sin(x)+2.5 so that it has been shifted horizontally to your new starting point? (There are multiple answers here, which can be discussed now or later depending time.)

7.  What about the period?  Note: Most classes will have a formula to use to account for a change in period.  I haven't taught trig in years, so I just use horizontal stretch/shrink reasoning.  y=sin(x) has a period of 2pi, and Santa Cruz tides has a period of 12.  Since 12 is larger than 2pi, I will multiply x by a factor of (2pi)/12.  My new function is f(x)=1.5sin(2pi/12(x-9))+2.5.

One of the teachers from this team made a worksheet for students to use as we modeled the thinking.  I find this to be an important step so that students can refer back as they practice and study.  Please let me know if you are interested in having this worksheet, and I can get you a copy.  

Below is a list of all the tide problems that students worked on.  

Santa Cruz Tides
San Mateo Bridge Tides
Bay of Fundy Tides

Students also worked on Foxes and Rabbits from Illustrative Mathematics.

Illustrative Mathematics Foxes and Rabbits 2 Problem
Illustrative Mathematics Foxes and Rabbits Geogebra Tool