Thursday, December 18, 2014

Math problems of the week: Common Core-inspired algebra problems

Extra Credit: Is it possible to have mastered algebra without knowing:

a. the meaning of the phrase "solve using elimination" in 12 (as opposed to ...? don't all algebraic strategies for solving systems of equations involve eliminating one of the variables?)
b. what is meant by "the variable that is undefined" in 13 (as opposed to "the variable that is defined"?)
c. what a "discriminant"is
d. what matrices are?

Addendum [with corrections]: The Common Core on Matrices

Included in the Common Core's supplementary material for the high school standards is a topic is a topic that is not covered in many algebra curricula, namely, matrices:

CCSS.Math.Content.HSN.VM.C.6 (+) Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

CCSS.Math.Content.HSN.VM.C.7 (+) Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

CCSS.Math.Content.HSN.VM.C.8 (+) Add, subtract, and multiply matrices of appropriate dimensions.

CCSS.Math.Content.HSN.VM.C.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.

CCSS.Math.Content.HSN.VM.C.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.

CCSS.Math.Content.HSN.VM.C.11 (+) Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

CCSS.Math.Content.HSN.VM.C.12 (+) Work with 2 × 2 matrices as a transformations of the plane, and interpret the absolute value of the determinant in terms of area.

While this is only supplementary material, it appears to be inspiring self-proclaimed "Common Core" tests that include problems involving matrices (see above). The latter would appear to be at odds with:

1. The Common Core's claim not to specify particular curricula (since matrices are absent from many algebra curricula).

2. The Common Core's advocacy of a narrower, deeper, focus (especially since the matrices problems one finds in Common-Core inspired material are generally trivial, cookbook style problems of the sort seen in problem 15 above).

Given this, Common Core authors and advocates will want to clarify the Common Core's position on whether or not, and in what ways, matrices should be included in Common Core-based tests.

Anonymous said...

Augmented matrices are covered in the Algebra II texts I've used with my kids (Lial and Foerster). The issue with the problem you showed is that you don't have to know how to do anything with matrices other than set the problem up.

Auntie Ann said...

I don't think I saw matrices until my differential equations class in college.

Deirdre Mundy said...

We had matrices starting in Algebra. Had to crank them out by hand until Calculus, when we were suddenly allowed to use graphing calculators.

They were presented as an easy way to solve systems of equations, but often seemed to take more work than NOT using a matrix!

Katharine Beals said...

I didn't learn matrices until college-level Linear Algebra, and I do not think it would have been useful to have learned about them any sooner than that.