Reflection: Rigor What is Algebra? - Section 2: Discussion: An Historic Problem - reasoning abstractly and quantitatively

 

A Common Mistake on Problem 1:

Students wrote x^2 + 21 = 10*sqrt(x) instead of x^2 + 21 = 10x.  I saw that the class was split about 50-50 on these two equations, so when one student asked me which one was correct, I responded by sending him to the board.  He wrote down both equations, and then explained to us that he wasn’t sure which was correct but that he was inclined toward the former.  He and some other students were helped when someone pointed out that since we used “x^2” for the square number, we should use “x” for its root. 

We also used some numerical examples to help students understand the relationship between the “square” and the “root of the square”.  They understood the concept but had trouble making use of the symbols correctly.  To do this, I built on the class's understanding of the term "square" by asking them for an example of a square number.  They suggested 25, so we tested that against the given criteria.  

First, we added 21 to our square and wrote down "25 + 21".  Next, we set this equal to "10 roots of the square".  Since our square was 25, the class recognized that the "root of the square" must be 5, so we wrote down "25 + 21 = 10(5)".  They immediately saw two things: first, that 25 was not the correct solution, and second that if we replaced our 25 with "x^2" we should also replace our 5 with "x", not "sqrt(x)". 

One student also noticed that our square number would have to end in 9 so that the sum could be a multiple of 10.  This led us to discover one of the solutions (x = 7), but we had to solve the equation explicitly to find the other.  You can see some of these things in the attached copy of student work

Additionally, very few students remembered how to solve a quadratic equation, but when one suggested using the Quadratic Formula they all recognized the formula and remembered how to apply it.  The changes being made to the Geometry curriculum should provide future students with more algebra practice, so I hope this won't be such a problem in the future.

  Rigor: A Common Mistake
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What is Algebra?

Unit 1: Modeling with Algebra
Lesson 1 of 15

Objective: Students will be able to explain why algebra needs definitions and axioms, what some of these first principles are, and what it means to "do algebra".

Big Idea: Algebra is built on axioms and definitions and relies on proofs just as much as geometry.

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