Activity 1 - Beetle Survey
Days 1-2
Objectives:
- Examine matrices as an extension of tables.
- Explore addition, subtraction, and multiplication of matrices, via self-discovery where possible.
- Become comfortable producing matrices based on data.
- Encounter determinants and inverses of matrices.
- Encounter the 0 and identity matrices.
Materials:
- Pencil and Paper
- Website: Online Matrix Calculator
In this handout, answers are in orange and suggestions/directions are in red.
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Description of Activity:
This activity begins the exploration of matrices by relating them to tables, a construct familiar to students. It provides a scenario in which it is natural to define what a matrix is. Students discover matrix addition by making a matrix and seeing its relation to two previous matrices. The web applet is used so that students can examine matrix addition before they know the definition or process and come to their own conclusion. Another matrix is added using percents, which is then used for matrix multiplication. Again, students are asked to find a third matrix before knowing that it is the result of multiplying two together so that they appreciate the strangeness of matrix multiplication. After officially defining matrix multiplication, students examine the non-commutativity of matrix multiplication for general matrices and then square matrices. The web applet is used to allow students to test commutativity in the case of square matrices. At the end of the day, the handout can be collected by the teacher as a low-stakes assignment, to be returned the next day so that students can learn from their mistakes and be motivated to do the activity.
In the (suggested) second day, students create square matrices and record them, along with their determinant, inverse (if it exists) and the result of multiplying the matrix by its inverse. Here students should get vague intuition of the relationship between the determinant and inverse. They are also introduced to identity matrices for the first time. At the end of this exercise, the teacher discusses the parallels between 0 and 1 in the real numbers and the 0 and identity matrices.
Rationale:
A part of the standards for matrices is using technology to perform operations on matrices. While the standards which include this requirement are not covered in this activity, they are covered by the other two activities. Thus there is no reason not to let students familiarize themselves with the web applet. The inclusion of the web applet also allows a shift towards discovery-based learning in this unit, which is very different from how the unit is normally presented and should make students more receptive towards matrices. Finally, using a matrix calculator allows students to skip laborious calculations and observe the matter at hand, which can allow for accelerated learning even among lower-achieving students.
This activity begins the exploration of matrices by relating them to tables, a construct familiar to students. It provides a scenario in which it is natural to define what a matrix is. Students discover matrix addition by making a matrix and seeing its relation to two previous matrices. The web applet is used so that students can examine matrix addition before they know the definition or process and come to their own conclusion. Another matrix is added using percents, which is then used for matrix multiplication. Again, students are asked to find a third matrix before knowing that it is the result of multiplying two together so that they appreciate the strangeness of matrix multiplication. After officially defining matrix multiplication, students examine the non-commutativity of matrix multiplication for general matrices and then square matrices. The web applet is used to allow students to test commutativity in the case of square matrices. At the end of the day, the handout can be collected by the teacher as a low-stakes assignment, to be returned the next day so that students can learn from their mistakes and be motivated to do the activity.
In the (suggested) second day, students create square matrices and record them, along with their determinant, inverse (if it exists) and the result of multiplying the matrix by its inverse. Here students should get vague intuition of the relationship between the determinant and inverse. They are also introduced to identity matrices for the first time. At the end of this exercise, the teacher discusses the parallels between 0 and 1 in the real numbers and the 0 and identity matrices.
Rationale:
A part of the standards for matrices is using technology to perform operations on matrices. While the standards which include this requirement are not covered in this activity, they are covered by the other two activities. Thus there is no reason not to let students familiarize themselves with the web applet. The inclusion of the web applet also allows a shift towards discovery-based learning in this unit, which is very different from how the unit is normally presented and should make students more receptive towards matrices. Finally, using a matrix calculator allows students to skip laborious calculations and observe the matter at hand, which can allow for accelerated learning even among lower-achieving students.
Standards Addressed By This Activity:
Mathematical Practices Developed During Activity:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure.
- MCC9-12.N.VM.6 - Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.
- MCC9-12.N.VM.7 - Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.
- MCC9-12.N.VM.8 - Add, subtract, and multiply matrices of appropriate dimensions.
- MCC9-12.N.VM.9 - Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.
- MCC9-12.N.VM.10 - Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.
Mathematical Practices Developed During Activity:
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
7. Look for and make use of structure.