14 October 2009

Baddie of the Month: October 2009

Filed under: Uncategorized — admin @ 8:57 am

No wonder everyone hates mathematics and considers it useless and doesn’t want to hear about creating good national standards.

I was just reading this part: The Numbers Gap That Matters of the article in the New York Times Blog

In the article as an important mathematics question shown as an example of what our students must know:


From what I read:  It seems that the reason our children cannot do mathematics is because such questions are not in the standards. Now I am not sure that  (a)  I know the correct answer to this question and (b) knowing the answer to this question does anything to improve my understanding of mathematics.

OK,  I know that (d) is NOT an equation and (e) is definitely an equation, but for many (a) is a function so maybe not an equation, (b) has 2 solutions so maybe not an equation, (c) is a tautology (always true) so maybe not an equation.

Who the heck cares?  And – if you do care –  is x^2+3x+4=0 an equation? Oh wait, it has an equal signs. Oh no, it has no solutions. Oh wait, it has complex solutions. Oh no, we can’t graph them.  Let’s go on and on about semantics instead of doing some real mathematics.

== A REAL PROBLEM with Equations and Expressions in MATHEMATICS EDUCATION ==

It is for sure more important that students differentiate between a problem that requires the simplification of an expression and a problem that requires the solution of an equation.

(A) to “simplify an expression” uses exactly the word “simplify” and not the word “solve” and that the answer looks like a snake (expressions connected with equal signs) and that the pupil understands that he is doing “re-organizational work” and not solving and

(B) to “solve an equation” uses exactly the word “solve” and not the word “simplify” and that the answer is a list of “expression=expression” until the student gets to a solution of the form “x=3”.

I am sick to death of students who have graduated from high school and entering an engineering program (and let us assume actually understand order of operations)  that will tell me that the following is absolutely correct.

Given:  Solve 2(x+1)+x-2=6 . They will simply add the green to get

Solve 2(x+1)+x-2=6=2x+2+x-2=6=3x=6=2

Please – this is important.


BTW: Presumably -  given the designers’ “definition”:  An equation is a statement that two expressions are equal., the “correct” answer would be: Only d is NOT an equation. See Common Core Standards Initiative

27 March 2009

Goodie of the Month – Tangents to Quadratics

Filed under: Uncategorized — admin @ 2:17 pm

Goodie: “A technique/question that can be applied in many places and teaches thinking.”

Baddie: “A technique/question that is a waste of good teaching and learning-to-think-and-do-math time.”

March 2009 Goodie of the MonthTangents with Quadratics

Well now that we have saved all that time not factoring and completing the square and considering complex roots, we might want to study tangents to quadratics.

Now you might ask why? Answer: it reinforces so many topics from Algebra 1 and ties them all together.

How to study tangents to quadratics?

Tell the student that the slope of the tangent line to the quadratic function y(x)=ax²+bx+c is always s(x)=2ax+b.

Now ask them "Given y(x)=x²+x-3, find and graph the equation of the tangent line to this quadratic at x=-2?

1. First they must graph the quadratic by finding the y-intercept at c=-3, roots at -2.3 and 1.3 and vertex at (-0.5,-3.25).

2. Then, they must find the point on the quadratic by substituting y(-2) and check that this point (-2,-1) is on the quadratic.

3. Then, they must find the equation of the slope at this point: s(x)=2x+1.

4. Then they must find the slope of the tangent by substituting s(-2) to get m=-3.

5. Then they must find the equation of the tangent line using the point-slope formula: y= s(-2) (x+2)+y(-1).

6. Finally, they must graph this line y=-3x-7 and see that it is indeed tangent to the quadratic at the point.

So why is this a “Goodie of the Month”?

Just look at all the skills it reinforces from Algebra 1! And everything is visible and checkable!

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Related topics:

Past: March 2009 Baddie of the month – Teaching Complex Numbers with the Quadratic Formula

Past: February 2009 Goodie of the month – Good questions for Quadratic Equations/Functions

Past: February 2009 Baddie of the month – Teaching completing the square to graph a quadratic.

Past: January 2009 Baddie of the Month – Hand-factoring a quadratic with a≠1.

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