Absolute value and degree questions

[swimmermom3]

Ds and I are continuing to work our way through AoPS Intermediate Algebra after pulling him from the Adv. Algebra 2 class at the high school. We are working slowly and stopping to review any area that he has questions on. Right now, I am supplementing from Foerster Algebra and Trig. along with the Introduction to Algebra book.

 

Two issues came up this afternoon that surprised me and I don't seem to be doing a very good job of explaining them. One reason that I am not the best math teacher is that I can tell him what to do on a particular problem, but not the why of it and Sailor Dude always wants to know the "why" of math. He definitely needs a STEM mother. :tongue_smilie:

 

1.) The first problem was a quick review on absolute value:

 

Ix-2I = 8

 

He knew to answer that x is equal to 10, but didn't know to look for a negative value for x. We talked about absolute value being the number of the distance from zero and that's why it's always positive, but he still looked a bit puzzled as to how x can be a negative number or how to solve for it.

 

Can someone point me to a better explanation?

 

2.) What is the degree of 5^3x^9 - y^4z^6?

 

Ds answered "9" using the definition of degree and the examples in Foerster. I am not sure that he understands that there is a difference between "degree" and the exponent.

 

Again, can someone please point me to a succinct explanation?

 

3.) My last question has to do with teaching math.  Ds is graphing functions for me today. Again, it should be a review. I get notebook paper with Cartesian graphs with no numbers, just the shape of the function. I am used to making a quick table with possible values for x and y and then making my graph with with numbers on it. Ds told me his method was acceptable at school. The graphs look correct, but he is using that darn graphing calculator. Should I make him do it by hand or is that busy work? 

 

Thanks again for being here. Math posts actually make my face flush while I write them. Maybe they will help someone else, if only to make another boardie think "Whew, and I thought I had problems teaching high school math!"

 

 

 

 

 

 

[regentrude]

1.) The first problem was a quick review on absolute value:

 

Ix-2I = 8

 

He knew to answer that x is equal to 10, but didn't know to look for a negative value for x. We talked about absolute value being the number of the distance from zero and that's why it's always positive, but he still looked a bit puzzled as to how x can be a negative number or how to solve for it.

 

Can someone point me to a better explanation?

 

 

|x-2|=8 should be translated into two equations:

it means that either x-2 = + 8 or that x-2 = - 8.

He always needs to examine both cases separately.

 

The first equation gives x=10, the second equation gives x= -6.

 

 

3.) My last question has to do with teaching math.  Ds is graphing functions for me today. Again, it should be a review. I get notebook paper with Cartesian graphs with no numbers, just the shape of the function. I am used to making a quick table with possible values for x and y and then making my graph with with numbers on it. Ds told me his method was acceptable at school. The graphs look correct, but he is using that darn graphing calculator. Should I make him do it by hand or is that busy work?

 

The value of graphing functions is in doing it by hand. In AoPS, he will NEVER be expected to use a calculator for something like this!!!

The student should learn: which values are good values to examine (often x=0, x=1, x = +/-infinity), in which areas it is necessary to go in smaller steps because interesting things will be happening,, how to sketch basic shapes of basic functions without resorting to a graphing calculator. Finding zeros by hand, understanding that there must be local minima/maxima between the zeros, seeing what the function does in asymptotic limits - these are all skills that can only be developed if he graphs by hand.

Putting it in the graphing calculator does not teach anything.

 

I can't answer your question about degree right now, since I have forgotten the precise definition.

.What ddoes your text say?

[Julie of KY]

1) if the absolute value of x-2 is 8, then this means it is eight units from zero, it can be either eight steps in the positive direction or eight steps in the negative direction. Therefore x-2 = 8 --> x=10; or x-2= -8 --> x= -6.

Not sure if that helps.

 

2) degree of function - I'm lazy today and don't feel like explaining it in English; if no one else does I'll come back to this thread.

 

3) All the graphing in AoPS should be done by hand with no help from a calculator. At the minimum, a few basic points should be labeled so you can see the scale of the graph. If graphing trig functions often pi or pi/2 is labeled. Oftentimes the x or y intercepts can be labeled. Any asymptotes should be labeled.

[kiana]

The shape is not enough, and if I saw this (the shape with nothing labeled) it would not receive credit. How much should be labeled depends on the specific function, but I would usually expect local maxima/minima (if they exist, and if the student has the tools to find them -- for example, the vertex of a parabola), asymptotes, and intercepts to be labeled at a bare minimum. Any monkey can put a function into a graphing calculator and copy it down, but this will not develop the ability to understand that, for example, if f(x) has a point at (0, 0), then f(x-5) + 3 has a point at (5, 3).

 

For the absolute value, does he understand problems such as |x| = 8? If he does not understand those, I would start there before moving to |x-2|=8, as the second type are quite difficult to understand if the first type are not understood.

 

For the degree, I would go back and briefly review how to tell the degree of monomials. I'd suspect that he only has problems with the ones with more than one variable. If you can do monomials, polynomials are easy (you can just write the degree of each term underneath it and look for the highest (including ties) number if you are a little shaky on monomials). VERY brief review here -- also a video lesson, but quite honestly I think he'll be able to grok it with a brief explanation and probably respond with 'Oh yeah duh, I knew that'. http://www.mathplanet.com/education/algebra-1/factoring-and-polynomials/monomials-and-polynomials

[Corraleno]

2.) What is the degree of 5^3x^9 - y^4z^6?

 

Ds answered "9" using the definition of degree and the examples in Foerster. I am not sure that he understands that there is a difference between "degree" and the exponent.

 

Again, can someone please point me to a succinct explanation?

 

To find the degree of a polynomial with one variable, you just take the largest exponent of the variable; this is what Sailor Dude was thinking when he answered 9. That would be correct if the expression had been something like 5^3x^9 - x^4 + x^6.

 

However, to find the degree of a polynomial with more than one variable, like 5^3x^9 - y^4z^6, you add the exponents of the variables in each term. So in this case the degree of the first term is 9 (because 5^3 is a constant, not a variable), and the degree of the second term is 6+4=10. So the degree of the polynomial is 10.

 

ETA: this site has a good explanation, and also covers finding the degree of fractional expressions and square roots