More 7th Grade Math

[jyl]

For all you wishing you were kids again, here - just for fun - are some "practice test" problems for part 2 of my daughter's final 7th grade algebra exam this week.

1. Rectangle has area 600 and the length of two adjacent sides sum to 64. What is the length of these sides.

2. Construct a polynomial function with 5 roots. How many vertices does it have? No use of graphing calculator permitted.

3. Create a system of equations having only two distinct solutions. Neither can be a horizontal or vertical line. State the solutions as two ordered pairs.

#1 is meant to test their abiilty to solve quadratic equations. #2 tests their intuitive understanding of polynomial roots and vertices. #3 is meant to test their familiarity with the graphs produced by linear and quadratic equations. (At least, this is what I assume the teacher is getting at.)

Unfortunately, I am out of town and helping her with this over the phone is rather trying.

Part 1, which she took Monday, was more of the nuts and bolts of factoring and solving equations, etc.

[Rick Lee]

#1 I can handle. The others, I'm totally lost.

[jyl]

#2 is easy, its just the terminology that is unfamiliar.

The "root" of y = f(x) means x for which y = 0.
E.g. y = (x+1) has root x = -1. That's one root.
To add another root, just add another term in the form (x+b)
E.g. y = (x+1)*(x-1). The second root is x = 1.
And so on. So, you end up w/ a polynomial of the form
y = (x+a)*(x+b)* . . . *(x+e) where a . . . e are constants of your choice.
And this polynomial has five roots. x = -a, -b, . . . -e. At each of these roots, y = 0, meaning the curve intersects the x-axis.
"Vertex", in this context, means local maxima or local minima.
As you move from one root to the adjacent root, the curve rises (or falls) from the x-axis, then falls (or rises) to intersect the x-axis again. So there is (at least) one vertex between each pair of adjacent roots. Five roots implies 4 vertices.

#3 as well.

The kids have learned only a few function types so far. The main ones are linear ones like y = a*x +b (which describe a straight line that may be sloped) and quadratic ones like y = a*x^2 + b*x + c (that describe a parabolic curve).
So, you can see that a sloping line and a parabola could easily intersect at two and only two points.
I told her to draw such a picture, then figure out the equations that would describe the curves, then solve for the intersection points.
For example, y = 2x +10 and y = x^2 will intersect at two points. To solve, she'll probably have to do more quadratic equation stuff.

I think her teacher is pretty good. The problems she sets for the kids are well thought out and quite instructive. Sometimes the kids don't fully grasp the material, but I don't think they need perfect comprehension at this point.

Next year, I would assume they move on to trig. Not sure what else - can't spend a whole school year on trigometry.

It occurs to me that, at this rate, she will likely graduate from high school (we have it picked out, will be a public one) with far more education than I had at that point. I graduated HS with some math but very little physics, zero chemistry or biology, self taught in English and history, and basically not much else. In fact, my HS education basically sucked, and it was supposed to be a good HS (El Segundo in L.A.).

We moan about education today, but there is good education to be had out there if the kids and parents make it a priority.

[Schumi]

No book I've ever used has used the term 'vertices' to describe what I've always seen referred to as local maxima and minima.

Other than that...cake. Although being a graduate student, my math skills are constantly being honed.

I was discussing with my mother a simulation program I was writing, when she asked my over the phone if it 'involved algebra'... I answered that everything I have done for the last 4 years has involved 'algebra' at every waking moment. It blows my mind that there can be (and were, when I was graduating high school) young people who could not solve simultaneous algebraic functions or take a derivative of a function. Or be able to sketch what y=x^2 looks like on a napkin.


Basic stuff. And knowing how to apply differential calculus to things in life can never be considered to NOT be useful. It sickens me that, at my (public) high school some years ago... 95% of my graduating class never took Calculus. Because they didn't have to. While I was in Calculus 1 class in high school (with 6 other people), 95% of everyone else was taking Home economics or Gym or something else... enough to just get by...

[jyl]

Yeah, this use of "vertex" threw me too.

[GH85Carrera]

It helps if you like math and it comes easy for you. How many times have you heard kids in school proclaim "I will never need algebra in real life." My co-worker did just enough to get through high school math. Now we both work in map making and photogrammetry. We use lots of simple algebra all the time just to get through the day.

I am glad to see that a school actually teaching!

[masraum]

Originally Posted by Schumi

And knowing how to apply differential calculus to things in life can never be considered to NOT be useful. It sickens me that, at my (public) high school some years ago... 95% of my graduating class never took Calculus. Because they didn't have to. While I was in Calculus 1 class in high school (with 6 other people), 95% of everyone else was taking Home economics or Gym or something else... enough to just get by...

I'm a math geek, I took Calc in HS (AP tested out of Calc 1 in college, but had to take it again anyway). I've taken Calc 1, 2 & 3 and DE (twice, once 20 years ago and again 2 years ago). As much as I find math/calc really interesting, especially the practical application, I don't know that 99% of the population needs or would benefit from it.

[vash]

i SUCKED in math. got a cheerleader to do my homework all thru highschool. duh. plan backfired as i found myself in an engineering program. turns out i was just LAZY. i had decent skills and once i got the ball rolling, i rocked.

as mentioned above, you need a math dictionary to get some of those problems started. honestly, we dont use much math at work (construction) but when it comes up, i am surprised at how much basic stuff my coworkers have forgotten. i think ironically, because i was so late in the game with my math education, i had to scramble to catch up. in doing so, i have the best grasp of solving some of our problems. i love coming up with 3 different solutions to the same prob. 95% is basic alg, 5% cal I.

[sammyg2]

I've worked with my daughter on her math for years, one thing has become incresingly obvious.
It depends on who wrote the math book.

The vocabulary and the terminology is up to the author's discression in too many cases.
Guess it makes them feel smarter to take common math and regurgitate it with a twist so that it looks like they have come up with something new.
Now days doing math is less about understanding math concepts and more about being able to look things up in a book. It isn't supposed to be a puzzle.

[jyl]

Hmm, I guess we are the math illiterates.

http://en.wikipedia.org/wiki/Vertex_(curve)

From Wikipedia, the free encyclopedia
Jump to: navigation, search
For other uses, see Vertex.

An ellipse (red) and its evolute (blue). The dots are the vertices of the curve, each corresponding to a cusp on the evolute.In the geometry of curves a vertex is a point of where the first derivative of curvature is zero. This is typically a local maximum or minimum of curvature. Other special cases may occur, for instance when the second derivative is also zero, or when the curvature is constant. For a circle which has constant curvature, every point is a vertex.

[jyl]

I want my daughter to have a reasonable bit of math done by the time she graduates high school. At least through integral calculus. I also want her to have a reasonable bit of physics. Functional fluency in spoken/written Chinese is another goal.

[ChemMan]

I'll do #1 just for the "fun" of it.

Let one side=y
Let the other side =x

xy=600
x+y=64
x=64-y

Therefore,

(64-y)y=600
-y2+64y-600=0

throw this into the quadratic equation and you get 11.41 and 52.59. Units were specified in the problem

[GH85Carrera]

I still blame "new math" for my dislike of math. Math classes in grade school in the 60's was the pits.

[TerryH]

In 1988 at age 32 I went back to night school for technical math classes including calculus. Two years later with a straight 4.0 GPA, I had to quit for work and travel related reasons for a business start up.

Because I have never had to apply what I learned, my retention is close to zero. Perhaps a few refresher courses would bring some math theorem and formula back to life, but for now I feel like a big nil.

[Rick Lee]

I still use algebra regularly, but just about never geometry and certainly never calculus. I was so happy to place out of all that with AP's before college. But I think the real benefit to going through those courses is to teach problem solving and logic. I don't need to use the second derivative while shopping at the grocery store or doing my checkbook. I'm sure that knowledge has made my life easier in many other ways, though.

[Schumi]

I never necessarily 'liked' math. I was the one scribbling down the homework at the last minute to turn in. I was fortunate though that I understood easily even then and that has translated now later in life to be able to approach problems much easier.

For instance, a page from a paper I referenced when writing a MATLAB program last semester to solve for stress distributions in carbon fiber composite layups:




Now without a fundamental knowledge of matrices, which I learned in the 9th grade, I would have had a pretty rough time understanding how to implement that into my code (well, not so much the code, but the hand calculations to go with it). It's amazing how those little things from early in school come back to play grabass ten years later.

Of course not everyone needs to know the complex stuff but understanding the fundamentals makes whatever you do later in life much easier.

I'd feel much better knowing that the 24 year old nurse at the hospital passed high school algebra when she's doing the math on how much of a certain drug to inject into my ass..... :-)

[sammyg2]