Math 099, Intermediate Algebra

Course Schedule & Homework Assignments

If you see the symbol below, it means that math class was videoed and you can get a link by e-mailing me.

*:*- Move-in Day

*:*- Orientation Day 1... but, alas, no math.

*:*- Orientation Day 2... but, alas, no math, again!

*:***Excursion:***Garden of the Gods and Pow Wow*

*:**Summary:*General ideas about the broad project of mathematics, motivated often by a desire for**beauty**and out of**curiousity**. Curiousity being the search for**truth**(which mathematicians believe exists, at least in the realm of pure mathematics, although applications to the real world are fraught with many assumptions and approximations), the way we go about this is by**simplifying**as much as possible, and only accepting statements and explanations which are absolutely crystal clear and have withstood careful examination by the whole mathematical community.*Detailed contents:*- We got curious about how tall were the rocks at Garden of the Gods.
- We had few tools to make measurements, but could at least use a friend of known height and could pace off some (approximate) measurements on the ground.
- We made as simplified a diagram as possible, whereupon doodling a bit, we noticed some things from basic geometry.
**Definition:**Two triangles are said to be**similar**if there is a single number $k$ with the property that each side of the second triangle is $k$ times as long as the correspond side of the first triangle.- The way to think of similar triangles as that they look very much the same, only one of them has been run through a zoom lens, so segment in it is $k$ times as long as it used to be.
- We can use similar triangles to learn something new as follows:
- Say $\Delta ABC$ is similar to $\Delta DEF$ and we know the lengths of the sides $\overline{AB}$ and $\overline{DE}$.
- The "zoom factor" must be $k=\frac{length(\overline{DE})}{length(\overline{AB})}$ — a number we can calculate from the lengths we know. So think of this $k$ as a known number from now on.
- The rations of the other side lengths must be equal to that same zoom factor $$\frac{length(\overline{EF})}{length(\overline{BC})}=k$$
- If we know $length(\overline{BC})$, that means that we can calculate $length(\overline{EF})$ by cross-multiplying that last equation to get $length(\overline{EF})=k\cdot length(\overline{BC})$, the right-hand side of which is the product of two known numbers!

**Theorem:**If all three angles of one triangle equal the corresponding angles of a second triangle, then the two triangles are similar.- A
**theorem**is a mathematical statement for which there is a careful, complete explanation of why it is true. This explanation is called a**proof**. **Theorem:**The sum of the three angles in any triangle equals $180^{\circ}$.- This theorem only applies to flat ["planar"] triangles, not, for
example, trangles written on the surface of a globe. It also tells
us that once we know two the values of two of the angles in a
triangle, the third one is forced to be whatever angle is left to
make the total equal $180^{\circ}$. This means, for example, that
we only need to match
**two**paris of corresponding angles in the above theorem about similar triangles, since the third triangle for both triangles will fixed with the same value in both triangles. - Another thing we noticed in our simple diagram of GotG rocks and assistant standing tall, was that the two triangles we drew both had a right angle in them.
**Definition:**An angle is called a**right angle**if its value is $90^{\circ}$. Equivalently, if the two rays which form the sides of the angle are on two perpendicular lines.**Definition:**A triangle is called a**right triangle**if one of its angles is a right angle. The two sides of a right triangle which meet at that right angle are called its**legs**, while the third side, the one opposite the right angle, is called the**hypothenuse**.- The most famous fact about right triangles, which was actually already known hundreds of years before Pythagoras was born, is:
**The Pythagorean Theorem:**Suppose the sides of a triangle have lengths $a$, $b$, and $c$. Suppose also that the angle between the sides of lengths $a$ and $b$ is a right angle — meaning that $a$ and $b$ are the lengths of the legs and $c$ is the length of the hypothenuse of this triangle. Then $$a^2+b^2=c^2$$- If you are every asked to state the Pythagorean Theorem, you have
state (almost) all of the above. If you just give the equation, as
many students do, the concerns are "what are those $a$, $b$, and
$c$?" and "you never mentioned that the triangle must have a
particular property [of being a
*right*triangle]." - Notice that if you are going to use the Pythagorean Theorem to solve a problem, you need to know the lengths of two of the sides of a triangle — which must be a right triangle — and then you can solve for the third side length.
- In particular, for our question "How tall are the GotG rocks?" we would need to know, maybe, the distance from the observer to the highest point of the rock — which would be the length of the hypothenuse of a right triangle in our diagram — and the distance from the observer to the base of the rock — being the length of one of the legs of the triangle. The hypothenuse length is hard to get without something like a high-tech "laser rangefinder."
- Dr. Montoya pointed out that there is a right triangle whose
sides have lengths 3, 4, and 5. (This nice right triangle was
actually known to the Babylonians.) Your math instructor then
proposed a conjecture (a
**conjecture**is a theorem that some people think is true, but no one yet has been able to prove) that this is the only right triangle whose sides are all whole numbers.

We then noticed that we could multiply all the sides by the same number (making a new triangle*similar*to the old one!) and get other right triangles with sides of whole-number lengths, such as 6, 8, 10 or 9, 12, 15. So a more modest conjecture might be that all right triangles with all sides having whole number lengths must be multiples of (similar to) the 3, 4, 5 example.

Unfortunately, this is not true, either. In fact, the famous philosopher Plato (who did a little math; in fact, he spent a little time in a Pythagorean community) described a way to generate an infinite number of whole-number-sided right triangles, non of which is simlar to any other one. - The French mathematician (and lawyer, actually) Pierre de
Fermat conjectured in 1637 that there are no whole number
solutions if we use cubes instead of squares in a Pythagorean
Theore-type equation. That is, he claimed that there are no
triples of whole numbers $a$, $b$, and $c$ for which
$a^3+b^3=c^3$ — or, in fact, for any power higher than 2!
— but he claimed not to have room for the proof in the
margin of the book where he wrote the claim. This is sometimes
called
**Fermat's Last Theorem**, although it really should have been called a*conjecture*for the next 357 years, at the end of which the British mathematician Andrew Wiles (who was then a professor at Princeton University) was able to find a proof [building upon the work of dozens of other mathematicians].

**Worksheet 1**

*:**Summary:*A little review of yesterday, and more discussion of mathematics as the search for beauty and truth (of a purely mental kind). Starting to talk about**lines**,**Cartesian coordinates**,**equations of lines**, and therefore**slope**and**$y$-intercept**.*Detailed contents:*- The
**hypothesis**is the assumption you must make in a theorem in order for the conclusion to be true. Many theorems have the form "*If $A$, then $B$,*" in which case $A$ is the hypothesis and $B$ is the conclusion. **Definition:**The points in the plane can be described by**Cartesian coordinates**in the form $(x,y)$, where the $x$ describes how many units to move along the horizontal ($x$-) axis and the $y$ tells how many units to move parallel to the vertical ($y$-) axis. The**origin**is the point with Cartesian coordinates $(0,0)$ (so: right where the axes cross).- Starting with an equation which has the variables $x$ and $y$,
we say that the
**graph**of that equation is the set of all points in the plane whose Cartesian coordinates $(x,y)$ make the equation true when plugged in. **Definition:**An equation of the form $y=mx+b$ is an equation for a line. The $m$ is the**slope**of the line and the $b$ is the $y$-intercetp.- If a line has slope $m$, that means that anywhere along the line, if you move 1 unit to the right (increase $x$ by 1), the line moves $m$ units up (increases the $y$ by $m$). Or, if you move (run) any distance, call it $run$, in the $x$ direction, the line goes up (rises) by a distance $rise$, where $$m=\frac{rise}{run}$$ In practice, this is often how we compute the $m$.
- If a line has $y$-intercept $b$, that means that it passes through the point $(0,b)$. In other words, if passes through the $y$-axis $b$ units above (or below, if $b$ is negative) the origin.
- When you want to find out where two lines intersect, you are looking for a point $(x,y)$ which satisfies both of the equations. That means you write them both down, and solve them together for first one variable and then the other.
- Did an example of linear equations for cost of internet service which had a basic price (which became the $y$-intercept) and price per MB (which became the slope), for two different providers. The intersection point told us how many MB would make the two providers cost the same amount, while fewere MB would make one provider cheaper and more MB would make the other cheaper.
- Another example was of finding the height of the rocks at Garden of the Gods (again!), by having choosing a coordinate system in which to work -- meaning: choosing where the $x$- and $y$-axes were, and where the origin was. Then, using these coordinates, we wrote the equations of two different sight lines from the over the top of the head of an assistant and to the top of the rocks, with the assistant standing in two different places. The point of intersection of those lines was the top of the rocks, written in the coordinate system we had chosen from which we could read off the hight of that point immediately.

- The
**Worksheet 2**

*:*- No AM math, instead an extra joyful (and long)
**Worksheet 3**

- No AM math, instead an extra joyful (and long)
*:**Summary:*Going over some issues in recent worksheets. Then discussing**exponential growth**,**exponential decay**, and some of the arithmetic of exponentiation.*Detailed contents:*- Going over
*Worksheets 1-3*. - Writing up your solution to a mathematical problem is about
*telling a story*, explaining to your reader what you are doing at every step, why you want to do that, and why you know it is OK to do that. - Exponential models as very like linear models, only with multiplication by the base of the exponenet taking the role of addition of the slope.
- Introduction to arithmetic of exponentiation, such as by repeated doubling — the function is $y=2^x$ — or compounding interest — the function is $y=1.05^x$, if the interest rate is 5%.

- Going over
**Worksheet 4**

*:***Math Quiz 1**, on the material from this week, immediately before lunch; here is a review sheet for the quiz.- No math worksheet in the PM.

*:***Excursion:***Royal Gorge*

*:**Summary:*Going over*Quiz 1*. Redos, worth up to half of the points you missed, are due no later than**tomorrow**.

*:**Summary:*Really getting into the algebra of exponentiation.*Detailed contents:*- Basic arithemetic of exponentiation:
- meaning of $a^n$, for $n$ a natural number
- $(a^n)^m=a^{nm}$, for $n$ and $m$ natural numbers
- generalizing this rule to $(a^b)^c=a^{bc}$ for all real numbers $b$ and $c$, which then gives us the rules
- $a^0=1$
- $a^{-n}=\frac{1}{a^n}$ for $n$ an integer
- $a^\frac{1}{q}=\sqrt[q]{a}$ for $n$ a non-zero integer

- Definition and some discussion of the
**natural numbers**$\NN$, the**integers**$\ZZ$, the**rational numbers**$\QQ$,**irrational numbers**, and the**real numbers**$\RR$. - applied exponential models, such as bacterial growth and radioactive decay

- Basic arithemetic of exponentiation:
**Worksheet 5**

*:*- No AM math.
*Summary:*More exponential models.

*:**Summary:*Arithmetic with complex numbers, more exponential models.**Here**are some practice problems to get ready for the quiz tomorrow.

*:***Math Quiz 2**, on the material from this week, here is a review sheet for the quiz.- No math worksheet in the PM.

*:***Excursion:***Ludlow Massacre Site*

*:**:**Summary:*Going over*Quiz 2*. Redos, worth up to half of the points you missed, are due no later than**tomorrow**.

*:**Summary:*Solving quadratic equations by**completing the square**.**Worksheet 7**

*:**Summary:*Some terminology, then applied problems with quadratic equations.*Detailed contents:*- The
**conjugate**of the complext number $a+bi$ is $a-bi$. - A
**function**is a formula $f(x)$ which has an input and an output. You can graph a function $f(x)$ by plugging in lots of numbers for $x$, getting output numbers $y=f(x)$, and putting a point on the Cartesian plane at location $(x,y)$. - The graph of a quadratic function $y=f(x)=ax^2+bx+c$ is shaped
like a bowl (technically, it's a
*parabola*) which points up if $a>0$ and down if $a<0$. - The maximum or minimum of a parabola is at the $x$-coordinate halfway between any two $x$ values whose corresponding $y$ is the same.
- Examples of solving and maximizing quadratic functions from economics, art, and farming.

- The
**Here**are some practice problems to get ready for the quiz tomorrow.

*:***Math Quiz 3**, on the material from this week, here is a review sheet for the quiz.- No math in the PM.

*:***Excursion:***Pueblo and Surrounding Area*

*:**:**Summary:**Detailed contents:***Worksheet 13**

*:*- Review and discussion of ACCUPLACER in AM math.

*:*- No math class today AM.
**ACCUPLACER**in the afternoon.

*:*- Non-ACCUPLACER component of math
**Final Exam**during AM math class. *Symposium, 5-7pm, LARC Lobby and 109*

- Non-ACCUPLACER component of math
*:*- Wrap-up