Engaging students: Adding, subtracting, and multiplying matrices

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Joe Wood. His topic, from Algebra: adding, subtracting, and multiplying matrices.

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A1. What interesting (i.e., uncontrived) word problems using this topic can your students do now?

One interesting real world problem for matrix operations can be found in Chapter 4.1.3 at http://spacemath.gsfc.nasa.gov/algebra2.html. The problem deals with astronomical photography. It starts by explaining the process by which NASA gets its images and relates the process of taking the pictures from blurry to clear using matrices. The problem goes as follows:

spacemath

For a way to engage students who are not interested in astronomy, and to allow students to learn more on their own time of the uses, a homework assignment could be for them to find places other than NASA that this process could be used.

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D1. What interesting things can you say about the people who contributed to the discovery and/or the development of this topic? (You might want to consult Math Through The Ages.)

“Nine Chapters of the Mathematical Art”, an ancient book that dates between 300 BC and AD 200, gives the first documented use of matrices. Even though matrices were used as early as 300 BC, the term “matrix” was not used until 1850 by James Joseph Sylvester. The term matrix actually comes from a Latin word meaning “womb”.

Below is a list published on the Harvard website of important matrix concepts and the years they were introduced.

200 BC: Han dynasty, coefficients are written on a counting board [6]
1545 Cardan: Cramer rule for 2×2 matrices. [6]
1683 Seki and Leibnitz independently first appearance of Determinants [6]
1750 Cramer (1704-1752) rule for solving systems of linear equations using determinants [8]
1764 Bezout rule to determine determinants
1772 Laplace expansion of determinants
1801 Gauss first introduces determinants [6]
1812 Cauchy multiplication formula of determinant. Independent of Binet
1812 Binet (1796-1856) discovered the rule det(AB) = det(A) det(B) [1]
1826 Cauchy Uses term “tableau” for a matrix [6]
1844 Grassman, geometry in n dimensions [14], (50 years ahead of its epoch [14 p. 204-205]
1850 Sylvester first use of term “matrix” (matrice=pregnant animal in old french or matrix=womb in latin as it generates determinants)
1858 Cayley matrix algebra [7] but still in 3 dimensions [14]
1888 Giuseppe Peano (1858-1932) axioms of abstract vector space [12]

 

 

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B1. How can this topic be used in your students’ future courses in mathematics or science?

Matrices and matrix operations are used in many math classes from Algebra and Calculus, to Linear Algebra and beyond. So any student interested in studying any discipline of Engineering or mathematics should become very familiar with matrices since they are used in a wide variety of ways (one way is seen above). Matrices are also useful in other courses as well. In Chemistry, matrices can be used for balancing chemical equations. In Physics, matrices can be used to decompose forces. Even in ecology or biology classes, matrices can be crucial. A great example would be studying animal populations under given conditions.
One hope in giving so many brief examples is that a student who cares nothing about the topic of matrices would here about a topic they are interested in (say animals) and that would spark questions into how or why matrices are useful. And of course, when dealing with matrices, addition subtraction, and multiplication of matrices follows closely behind.

 

 

 

 

References:

“Common Topics Covered in Standard Algebra II Textbooks.” Space Math @ NASA. NASA, n.d. Web. 18 Sept. 2015.

Knill, Oliver. “When Was Matrix Multiplication Invented?” When Was Matrix Multiplication Invented?  Harvard, 24 July 2014. Web. 18 Sept. 2015.

Smoller, Laura. “The History of Matrices.” The History of Matrices. University of Arkansas at Little Rock, Apr. 2001. Web. 18 Sept. 2015.

 

 

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