Activity 3 - Transformation Matrices
Day 4
Objectives:
- Observe matrices as rotations as they act on shapes in the plane.
- Observe matrices as translations as they act on shapes in the plane,
- Observe matrices as reflections as they act on shapes in the plane.
- Observe how each entry of the transformation matrix affects the graph.
Materials:
- Pencil and Paper
- Technology: Geometer's Sketchpad
In this handout, answers are in orange and suggestions/directions are in red.
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Description of Activity:
In this activity, students use the interactive software Geometer's Sketchpad to explore how matrices can be used to transform the plane. Specifically, they view the effect of matrices on a triangle embedded in the plane. This activity uses three separate Geometer's Sketchpad files, linked above. In each of these, students will be able to interact with the matrices in a dynamic fashion and observe the result in real time. The first file has students entering an angle and observing how the corresponding matrix changes to yield that rotation. Students can then hypothesize the generic matrix for rotating using their knowledge of trigonometry. The second file has students dragging an arrow to translate the triangle and viewing the corresponding matrices for different translations. The last file allows the students to enter their own values into the transformation matrix and view the resulting transformation. This file is specifically focused on reflections but is generalized in the worksheet.
Rationale:
In Geometer's Sketchpad, students are able to interact with the transformation matrices in intuitive ways. The dynamic features and immediate feedback can allow students to reach conclusions about the transformation matrices at a quicker pace. Additionally, the files let students skip arduous graphing and drawing and instead focus the students on the affects of the transformation matrix. The multiple angles at which transformation matrices are viewed here allow students to gain a robust understanding of the affects matrices can have on the plane.
In this activity, students use the interactive software Geometer's Sketchpad to explore how matrices can be used to transform the plane. Specifically, they view the effect of matrices on a triangle embedded in the plane. This activity uses three separate Geometer's Sketchpad files, linked above. In each of these, students will be able to interact with the matrices in a dynamic fashion and observe the result in real time. The first file has students entering an angle and observing how the corresponding matrix changes to yield that rotation. Students can then hypothesize the generic matrix for rotating using their knowledge of trigonometry. The second file has students dragging an arrow to translate the triangle and viewing the corresponding matrices for different translations. The last file allows the students to enter their own values into the transformation matrix and view the resulting transformation. This file is specifically focused on reflections but is generalized in the worksheet.
Rationale:
In Geometer's Sketchpad, students are able to interact with the transformation matrices in intuitive ways. The dynamic features and immediate feedback can allow students to reach conclusions about the transformation matrices at a quicker pace. Additionally, the files let students skip arduous graphing and drawing and instead focus the students on the affects of the transformation matrix. The multiple angles at which transformation matrices are viewed here allow students to gain a robust understanding of the affects matrices can have on the plane.
Standards Addressed By This Activity:
Mathematical Practices Developed During Activity:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
- MCC9-12.N.VM.11 - Multiply a vector (one-column matrix) by a matrix of suitable dimensions to get another vector. Work with matrices as transformations of vectors.
- MCC9-12.N.VM.12 - Work with 2x2 matrices as transformations of the plane.
Mathematical Practices Developed During Activity:
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
5. Use appropriate tools strategically.
7. Look for and make use of structure.