- 1 Lower and Upper Sum
- 2 Reducing the Size of the GeoGebra Window
- 3 Upload to GeoGebra
- 4 Exporting a Dynamic Worksheet to a Webpage (for Advanced Users)
- 5 Visualizing Triangle Inequalities
- 6 Design Guidelines for Dynamic Worksheets
- 6.1 Design Guidelines 1: Layout of Dynamic Worksheets
- 6.2 Design Guidelines 2: Dynamic Figures
- 6.3 Design Guidelines 3: Explanations and Tasks
- 7 Creating a Tangram Puzzle
- 8 Challenge of the Day: Enhance Your Tangram Puzzle
Lower and Upper Sum
You will now learn how to create a dynamic worksheet that illustrates how lower and upper sums can be used to approximate the area between a function and the x-axis, which can be used to introduce the concept of integral to students.
- Open a new GeoGebra window.
- Switch to Perspectives – Algebra & Graphics.
|1||Enter the cubic polynomial f(x) = -0.5x3 + 2x2 – x + 1.|
|2||Create two points A and B on the x-axis. |
Hint: These points will determine the interval which restricts the area between the function and the x-axis.
|3||Create slider for the number n with Interval 1 to 50 and Increment 1.|
|4||Enter uppersum = UpperSum[f, x(A), x(B), n]. |
Hint: x(A) gives you the x-coordinate of point A. Number n determines the number of rectangles used in order to calculate the lower and upper sum.
|5||Enter lowersum = LowerSum[f, x(A), x(B), n].|
|6||Insert dynamic text Upper Sum = and select uppersum from Objects.|
|7||Insert dynamic text Lower Sum = and select lowersum from Objects.|
|8||Calculate the difference diff = uppersum – lowersum.|
|9||Insert dynamic text Difference = and select diff from Objects.|
|10||Enter integral = Integral[f, x(A), x(B)].|
|11||Insert dynamic text Integral = and select integral from Objects.|
|12||Fix slider and text using the Properties Dialog.|
Use slider n in order to modify the number of rectangles used to calculate the lower and upper sum. 1. Compare the values of the upper sum / lower sum to the value of the integral for different values of slider n. What do you notice? 2. What happens to the difference of the upper and lower sum (a) if n is small (b) if n is big?
Reducing the Size of the GeoGebra Window
GeoGebra will export the algebra and graphics view into the dynamic figure of the worksheet. In order to save space for explanations and tasks on the dynamic worksheet you need to make the GeoGebra window smaller prior to the export.
- If you don’t want to include the Algebra View you need to hide it prior to the export.
- Move your figure (or the relevant section) to the upper left corner of the Graphics View using the Move Graphics View Tool.
- Reduce the size of the GeoGebra window by dragging its lower right corner with the mouse (see right figure below). Hint: The pointer will change its shape when hovering above an edge or corner of the GeoGebra window.
Upload to GeoGebra
After adjusting the size of the GeoGebra window, you are now ready to export the figure as a dynamic worksheet using the File Menu.
- File – Share…
- The GeoGebra website opens automatically where you have to login (or register if you do not have an account yet) before you are able to continue your upload.
- Fill in the information for your students. If you want, you can also select to show the Toolbar, the Input Bar or the Menubar. Click Continue.
- Type a short explanation for other teachers, so that they are able to use your materials, too. This information is not shown on the student worksheet. Choose a target group and select tags that describe your material to help others with searching.
- Finish your Upload with the Save button.
Your worksheet is now saved on GeoGebra where people are able to like/dislike the material or write comments.
Exporting a Dynamic Worksheet to a Webpage (for Advanced Users)
Instead of uploading to GeoGebra it is possible to export your dynamic worksheet to a webpage.
- Export – Dynamic Worksheet as Webpage Hint: You could also use the key combination Ctrl + Shift + W.
- Fill in the text fields in the appearing window in the Export as Webpage Tab (title of the worksheet, name of the author, and date).
- Type a short explanation of the dynamic figure into the text field Text above the construction.
- Enter tasks and directions for students into the text field Text below the construction.
- Click Export and save your dynamic worksheet. Hint: GeoGebra will create several files which always need to stay together in order to maintain the functionality of the dynamic worksheet. We recommend creation of a new folder (e.g. Dynamic_Worksheets) within the GeoGebra_Introduction folder prior to saving your dynamic worksheet.
Tips and Tricks for Creating Dynamic Worksheets
- After saving the dynamic worksheet it will be automatically opened up in your web browser. Check the text you inserted as well as the functionality of the interactive applet. If you want to change your dynamic worksheet go back to the GeoGebra file and make your changes to the figure. Export the figure again (you can use the same file name to overwrite the old worksheet) in order to apply your changes. Hint: You can change the text of the dynamic worksheet in the same way.
- GeoGebra automatically saves your entries in the export window for dynamic worksheets. If you want to make changes to your figure while filling in the export dialog you can just close it and continue later on.
- Make sure your applet is not too big. Your students shouldn’t have to scroll between the tasks and the figure because this makes learning more difficult.
- Your dynamic worksheet should fit on one screen. If you want to include more than 3 tasks you should consider creation of another worksheet that includes the same dynamic figure but different tasks.
Enhancing Dynamic Worksheets
The export dialog window for Export as Webpage consists of two tabs: General and Advanced. In the last activity you used tab General in order to add explanations, tasks and directions to the dynamic figure prior to the export. You will now learn how to enhance your dynamic worksheet by including different features in the interactive figure using the tab Advanced.
Visualizing Triangle Inequalities
You will now create a dynamic worksheet that illustrates the construction steps for a triangle whose three side lengths a, b and c are given. Additionally, this worksheet will allow your students to discover triangle inequalities.
- Open a new GeoGebra window.
- Switch to Perspectives – Geometry.
|9||Create a point D on circle e.|
|10||Create segment g between the points A and D.|
|11||Construct the midpoint E of segment g.|
|12||Enter text1: b and attach it to point E.|
|13||Create a point F on circle f.|
|14||Create segment h between points B and F.|
|15||Construct the midpoint G of segment h.|
|16||Enter text2: a and attach it to point G.|
|17||Match colors of corresponding objects.|
|18||Show the Navigation bar for Construction Steps (View Menu).|
|19||Show the Button to open construction protocol (menu View – Navigation bar for Construction Steps).|
|20||Open the Construction Protocol.|
|21||Show the column Breakpoint.|
|22||Change the order of construction steps so that the radius of the circles and the attached text show up at the same time. |
Hint: You might also set some other breakpoints (e.g. show all sliders at the same time).
|23||Now check Show Only Breakpoints.|
(a) Export your triangle construction as a dynamic worksheet.
(b) Come up with explanations and tasks for your students that guide them through the construction process of the triangle and help them explore the triangle inequalities by modifying the given side lengths using the sliders.
Design Guidelines for Dynamic Worksheets
The following design guidelines for dynamic worksheets are the result of a formative evaluation of dynamic worksheets created by teachers in our NSF MSP classes during fall 2006 and spring 2007. The guidelines are based on design principles for multimedia learning stated by Clark and Mayer. These guidelines were summarized to address and avoid common mistakes during the creation process of dynamic worksheets as well as to increase their quality with the hope that they will foster more effective learning. Although some of these guidelines may seem obvious, we have found it very important in our work with teachers to discuss and explain them in detail. The following figure shows an entire dynamic worksheet created with GeoGebra that allows students to explore properties of the orthocenter of a triangle. By modifying the dynamic construction students can examine the orthocenter of a great variety of triangles instead of just one special case. Several key words within the explanation and tasks match the color of the corresponding objects in order to facilitate finding them within the construction. Furthermore, the tasks are placed next to the dynamic construction in order to fit all information on one screen and avoid additional cognitive load through scrolling.
Design Guidelines 1: Layout of Dynamic Worksheets
Your entire worksheet should fit on one screen. Students should not have to scroll between the tasks and the interactive figure. We consider 1024x768 or 1280x1024 pixels as today's usual screen size which constrains the size of the dynamic worksheet. Using an HTML editor like NVU you can use tables to arrange text, images and interactive figures so they fit on one screen. If this is not possible, consider breaking the dynamic worksheet into several pages.
At the beginning of a dynamic worksheet, you should give an explanation of its content. Keep the text short (no more than one or two sentences) and write it in a personal style.
You will usually add questions or tasks to make sure that your students use the worksheet actively. Place these tasks close to the interactive applet (e.g. directly below it). Don't use more than three or four questions / tasks to avoid scrolling. If you have more tasks, consider breaking your worksheet into several pages.
Make sure that your dynamic worksheet just contains objects that are relevant for the objectives. Neither use unnecessary background or purely decorative images, nor background music on the web page in order not to distract your students from reaching the objectives.
Design Guidelines 2: Dynamic Figures
Allow as much interactivity as possible in your dynamic figure. As a rule of thumb, all visible objects should be movable or changeable in some way. Your dynamic figure should provide plenty of freedom to explore the relations of its mathematical objects and discover mathematical concepts.
Try to make your dynamic figure as easy to use as possible. If an object can be moved or changed, try to make this obvious, e.g. all movable points could be red or larger in size. If you don't want objects to be changed, fix them (e.g. text, functions or slider positions) so they cannot be moved accidentally.
Your dynamic figure should be large enough to allow all intended manipulations, but small enough to fit on one screen and still leave sufficient space for explanations and questions on the surrounding web page.
Use dynamic text
Dynamic text, like the length of a changeable segment, should be placed close to the corresponding object in your applet.
Avoid static text
Too much text can easily clutter your interactive applet. Instead, place static text like explanations or questions on the web page that includes your dynamic figure.
When a dynamic worksheet is opened you should be able to read all labels and important information. For example, a point label should not be crossed by a line.
Design Guidelines 3: Explanations and Tasks
Short, clear and personal style
Try to write your explanations and questions in a short, clear and conversational style. Use the term ‘you' within the text and try to address the students directly.
Small number of questions
Limit your number of questions or tasks per worksheet to three or four to avoid scrolling. If you want to ask more questions, create a new worksheet.
Use specific questions
Avoid general questions like ‘What is always true about X?' and make clear what the students should do, e.g. `What happens to X when you move Y?'. We recommend that your students should take notes while they work with a dynamic worksheet. If you want them to write down their answers on paper, say so on the worksheet.
Refer to your applet
Your text should support the use of your interactive applet. For example, try to explain a new term by referring to your applet instead of using an isolated textual definition. Additionally, you can color certain keywords to match the formatting style of the object they refer to. This makes the text easier to read and helps your students to find corresponding representations of the same object.
Your audience are learners
If you want to provide information for other educators (e.g. lesson plan, solutions) do so in a separate document (e.g. web page, pdf-document). Your students should not be distracted or confused by such information.
If your interactive figure is meant for presentation only it might be better to have no tasks or questions on the web page. If you include text, it should be understandable for students as well.
Creating a Tangram Puzzle
In this activity you will create the "Tangram" puzzle. It consists of 7 geometric shapes which can all be constructed using the side length a (see Tangram_puzzle.html).
Task 1: Figure out the side lengths of each partIn order to construct the parts of the Tangram puzzle you need to figure out the individual side lengths of the seven geometric figures first. They all depend on the side length a of the main square.
Task 2: Construct the individual parts of the Tangram
1. Enter the number a = 6. It will provide a basis for the construction of all triangles and quadrilaterals necessary for a "Tangram" puzzle.2. Try to figure out the side lengths of the geometric shapes.
3. Begin each of the geometric figures using a segment with given length. This will allow you to drag and rotate the figure later on.
4. Construction hints:
a. If the height of a right triangle is half the length of the hypotenuse you might want to use the theorem of Thales for the construction (see practice block 1).
b. If you know the legs of a right triangle you might want to construct it similar to a square construction.
c. For constructing a square using its diagonals, it is helpful to know that they are perpendicular and bisect each other.
d. For constructing the parallelogram it is helpful to know the size of the acute angle.
5. Check your construction by trying out if you can manage to create a square with side length a using all figures.
Challenge of the Day: Enhance Your Tangram Puzzle
With these geometric shapes other figures than a square can be created as well. Search the Internet for a "Tangram" figure other than a square and import this figure into the Graphics View. Export the GeoGebra construction again using a different name and different instructions.