Tutorial

Plane geometry

In this section, we will cover how you define geometrical constructions, specify properties of its elements, and define criteria for automatic evaluation.

The Geometry & Graphs Authoring Tool

Create a new exercise and choose Geometry & Graphs in the interaction. You will see the Geometry & Graphs authoring tool as in the image below.

The authoring tool consists of the following building blocks

• The Editor is the working area where you build the construction. You can use the tool buttons in the blue area to create elements such as points, circles, polygons, angles, bisectors, orthogonal projections, etc. We will describe each tool later in this section.
• Construction Details provides a list of all the elements you created and allows you to adjust properties, such as label, color, line type, etc.
• In Coordinate System you can enable and configure a coordinate grid and axes.
• The Settings block hosts general settings, such as which tool buttons are provided for the student and whether a background image should be inserted.
• In the Evaluation Criteria block, you list the rules that AlgebraKiT will use to automatically evaluate the student’s input.

An example

The following two examples show how these building blocks are used to create an interactive geometry question. If you are new to Geometry & Graphs then we suggest reading through these examples and performing the same tasks in your authoring environment.

Dynamic construction of the circumscribed circle of a triangle. You can drag the blue points

Interaction where the student has to create the circumscribed circle. Click to activate.

Example: Construct the circumscribed circle of a triangle

In this example, we construct the circumscribed circle of an arbitrary triangle. The result will be as in the live demo above (left).

Note that you can drag the vertices of the triangle around and that the circle automatically adapts.

Step 1: Create the triangle

• Choose the Polygon tool in the toolbar
• Click in the Editor area to create the three vertices of the triangle. Complete the polygon by clicking on the first vertex again.

You can run the exercise and see the triangle is visible. Note that the student will be able to drag the points around.

Step 2: Construct the circumscribed circle
As you probably know, the circumscribed circle can be constructed by intersecting the perpendicular bisectors.

• Choose the Perpendicular Bisector tool, which can be found from the constructions button, as shown in the image below.
• Click on one vertex of the triangle and then on another to create the perpendicular bisector.
• Repeat this for two other vertices to create another perpendicular bisector.

Step 3: Create the circle
The midpoint of the circumscribed circle is on the intersection of the two perpendicular bisectors and passes through the three vertices of the triangle.

• Use the Point tool to create a point on the intersection of the two perpendicular bisectors
• Select the Circle tool and click on the new point and then on one of the vertices of the triangle.

Note how the circle adapts when you modify the triangle.

Step 4: Configure the construction
Now the construction is done, we can configure some settings to make it look and behave as we want it. For this, we move our attention to the Construction Details panel, which lists the elements we created.

• Adjust the label of the midpoint. Click on the element in the list that corresponds to point D. Adjust the label to $M_c$. You can use the underscore ( _ ) to create the subscript ‘c’.
• Change the color and thickness of the circle. Follow the same process as the previous step.
• Optional: hide the perpendicular bisectors. You can easily hide elements by clicking on the ‘eye’ icon in the list.
• Optional: fix vertices of the triangle. You might not always want students to be able to move points around. Enable the ‘fixed’ checkbox in the properties of these points.

Extension: Turn this into an automatically evaluated question

Let’s now turn this construction into a question where the student has to create the circumscribed circle. To do this, we need to do the following:

• Hide the solution (bisectors and circle) from the construction
• Choose the tools the student is allowed to use
• Create evaluation criteria for automatic scoring.

The result will be the live demo above (right).

Step 1: Hide the solution
The solution of the exercise consists of the perpendicular bisector and the circle. Obviously, we need to hide them from the student. However, we do not want to simply delete them as we need these elements to check the solution.

In the Construction Details panel, there is a tab Evaluation. When this tab is active, any new elements will be invisible to students.

• Remove the circle and the perpendicular bisectors using the trash can in the Construction Details panel
• Activate the Evaluation tab
• Create the perpendicular bisectors and circle again.

Note that the new elements are shown with a different color in the left margin. If you activate the Template tab again, these new elements become invisible. This is what the student will see.

Step 2: Choose which tools the student can use
The Geometry & Graphs interaction supports a large number of tools to construct elements. To make the question easy to use, you select which tools should be available to the student.

• Click on the Settings tab to open it.
• Click on the tools you want to be available to the student.

Step 3: Create evaluation criteria
The student’s answer is correct if we find a circle that is equal to the circle we created ourselves. A mathematician would phrase this as follows: “There exists a circle z, such that z = g". Here, 'g' indicates the circle we created ourselves, as you can see in the elements list.

You can create such an evaluation criterion in the Evaluation Criteria panel.

• Click on the Evaluation Criteria panel to open it
• Select 'Existence' for the type.
• Type 'z' for the element label and select 'circle' for the Element type. This indicates we are searching for some circle in the construction and we refer to this circle as 'z'
• type 'z=g' for the condition.
• In the Description box, type the feedback you want to give to the student if the condition is not met. For instance: "You did not create the circumscribed circle correctly."

The question is ready. Run it to see how it works. Note that evaluation still works if you move the vertices!