Goals of the Project
This post will show teachers how to implement the Matrix Project module on DeltaMath. The project allows students to build their own figure digitally using polygons, circles and lines. They can then define up to five matrices representing transformations that will be applied to their figure in a particular order. After defining their matrix transformations, students will attempt to calculate a single matrix that represents the composition of all of their matrices, and then the inverse of this composition matrix, taking the final image of their figure back to its original shape. All along the way, students can experiment and see animations of their matrices in action.
Step 1: How to Assign
You should assign the project just as you would assign any skill on DeltaMath, but there is a note that explains that the module will not contribute in any way to the automatically graded assignment. This is just a way to give the students access to the project.
Step 2: Students Create Their Figure
Students must create a unique figure which will be the subject of their animation. There are tools to create circles, lines and polygons. To fill a circle or polygon, students may choose any web color, using the name or the hex code, or typing “none”. Standard color words should all work. Polygons are created by typing a list of any number of points.
Notice how the order of the shape definitions matter in order to determine which shapes are “on top of” any other. The order can be changed by dragging the shape definition. Opacity is a value from 0 to 1 describing how opaque (or transparent) the background fill is.
Step 3: Defining the Transformation Matrices
Students can define up to five transformation matrices to apply to their figure. They can play the transformations in succession, experiment and reorder the matrices if they wish. Have the students experiment with changing the order and seeing that often the resulting figure is different.
It would probably be beneficial for students to do a writeup in which they explain what each of their individual transformation matrices is doing. For example, the following five matrices would (1) reflect over the x-axis, (2) rotate 90 degrees counterclockwise, (3) dilate by a factor of 1.2, (4) rotate 60 degrees counterclockwise and finally (5) shear parallel to the x-axis by a factor of -0.5 times the y-value:
You may require or optionally allow kids to challenge themselves in allowing 3 by 3 matrices using a dropdown box:
The main benefit here is that translations would now be possible:
The bottom row is always [0 0 1] and the vector representing the point always has a 1 on bottom. This is just a byproduct of having 3 by 3 square matrices. Using translations allows students to do cool things like rotate / dilate with respect to other points besides the origin. Students should translate their figures to the origin, perform the rotation / dilation, and then translate the figure back:
Step 4: Determine the Composition Matrix
At this point, students need to carefully multiply all of their matrices together (in the proper order!) Their goal is to come up with a single matrix that would complete the entire transformation. As they are doing this, they see a transparent figure representing their target (based on the previous matrices). If students get the matrix wrong, they will see their figure does not overlap, but they are also given textual feedback. When they finally get it right, they will be notified and they can move on to the final part of the project.
Step 5: Determine the Inverse Matrix
Students should understand than an inverse matrix is a transformation that takes the image created by their original composition matrix and transforms the figure back into its preimage. For a 2 by 2 matrix, an inverse can be calculated using the determinant:
Students can round their values to four decimal places if necessary. Now students see the transparent target is their original figure. They must keep trying inverse matrices until they succeed in getting their original figure back:
Step 6: Viewing Student Projects
Unlike most modules on DeltaMath with a simple answer and automatically graded questions, the Matrix Project will require teachers to use their judgment in grading the projects. To see student data, the teacher must click on the module “Matrix Transformation Project”. There will be a dropdown box where you can select which student data to look at:
When you are on the “Teacher Data” option you can make your own project that will save automatically for the purposes of classroom demonstrations. Otherwise, clicking on a student will allow you to see their figures, their transformations, and whether their composition and inverse matrices were calculated accurately:
This project is a tool to have students creatively make their own figure and to be able to quickly test out different transformation matrices and their effects on the figure. DeltaMath will animate the transformations and also let them know if the composition and inverse transformation matrices that they calculated are correct. The teacher is responsible for grading the projects as well as making sure it is properly differentiated for students. Some students may do two or three 2 by 2 matrices that involve no decimal values, while others may use decimals and 3 by 3 matrices. Remember that calculating an inverse of a 3 by 3 matrix can be particularly difficult with decimal values, so you may want to steer kids away from that. Of course there are lots of practice exercises for these skills that can be assigned independently on DeltaMath before completion of the project:
I hope this post was useful and I am open to questions or suggestions in the comments!