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The solutions to the Test 3 problems are posted on the Test solutions page (warning: an earlier version was posted there which had not been fully edited. Please make sure you look at this version.) There were, as usual, two versions of the test. Look at the bottom left corner of the Cover Sheet to see which version yours was.

I have a few general comments about these problems also:

• As usual, please pay attention to the comments I wrote. Very common comment this time was “invalid canceling” or “incorrect canceling”: please make sure you understand why what you did was wrong.

• If I wrote any comment that said that you did not show enough work (however phrased) please understand that this a serious matter which you need to attend to. I will not be lenient on this in future tests.

• A lot of people did well on the first page and then very poorly on page two. One big cause of this was failure to read the instructions: students treated the expressions that they were to simplify as equations to be solved, which they were not. READ THE FRIENDLY INSTRUCTIONS!

• On the third page, where you had to solve equations, many people made the problems much harder than they needed to be, because of not CLEARING the denominators, or because of not using the LCM but rather using the product of all the denominators to clear the denominators. I will put some examples of how this makes things more difficult on this post later on.

• There will be a few problems like problems 2, 3, and 4 on the next Test as well, so make sure that you correct your erroneous methods. The WeBWorKs on Rational Expressions and on Fractional (really rational) Equations are available for you to practice and they offer hints and solutions if you use “show me another”

 

Test 4 is rescheduled to Monday 2 December, and will not be re-scheduled barring emergency situation.

 

Topics:

More rationalizing  denominators and complex numbers

• Trigonometry topic: special right triangles

The two important right triangles are:
• The isosceles right triangle,
• The half of an equilateral triangle – which I will call the half-equilateral triangle for short.
These are commonly called by other names which refer specifically to the degree measure of their angles, but it is much better to call them by these names for two reasons (at least):
• Later we will use the radian measure of the angles, so we do not want to tie ourselves too much to degree measure
• The names given above remind us of what these triangles really are, so that we can recreate them if we forget the relationships of their sides, for example.

Trigonomety in right triangles:

 

 

Homework:

• Review the isosceles right triangles and how we found the length of the hypotenuse using the Pythagorean Theorem, and also the half-equilateral right triangle and how we found its height using the Pythagorean Theorem. If you do this a few times, you will end up memorizing the triangle! You must learn it by heart as we will use it (and the other special right triangle) a lot in Trigonometry. Here is a good web source that shows how these triangles were developed. (Do not get too attached to the degree-measure names, though!)

Here is another good source on the half-equilateral triangle. (This one was also linked above in the notes.)

 

Here is the chapter from Prof. Africk’s textbook which discusses similar triangles: AfrickGeometryTexctbookSimilarTriangles

In addition to the WeBWorK, you should do problems 7 – 10 from p. 174 there. The answers will be posted here later.

 

Topics:

Simplifying radicals (including higher roots)

Adding and subtracting radicals

Multiplying radicals

Dividing radicals (simple case)

Here are the notes I handed out in class: OperationsWithRadicals

The problems we worked on in class were (some of) the even-numbered problems from the textbook, page 39

Homework homework yes you have homework

Please make sure to work on the WeBWorK assignments which are due tomorrow night, but start them now! There will be a quiz on adding/subtracting and multiplying radicals tomorrow.

Also, do the following problems on dividing radicals: assume all variables are $\ge 0$

(1) $\frac{\sqrt{57}}{\sqrt{19}}$

(2)  $\frac{\sqrt{2x^{3}}}{\sqrt{18x}}$

(3) $\frac{3\sqrt{2x}}{\sqrt{8x}}$

(4) $\frac{5\sqrt{30}}{10\sqrt{120}}$

 

 

Topic: Fractional (Rational) Exponents

Today’s active learning was on extending our work on exponents to include rational exponents. We started out by reviewing how to simplify square roots. Here are the problems:

Simplify, assuming that $x$ and $y$ are both $\ge 0$. [Why do we need to assume this? What difference does it make?]

(1) $\sqrt{9x^{2}}$

(2) $\sqrt{128}$

(3) *Sorry, I’ve forgotten exactly what I wrote for this problem. Can anyone tell us what it was?

(4) $\sqrt{x^{4}y^{9}}$

The active learning problems are here:

MAT1275-F15-Shaver.sshaver.RootsRadicalsShort

For these we used the definition of the fractional (rational) exponents, together with all the previous definitions and properties of exponents. All of the properties continue to work even with rational exponents, as long as we are careful about when the radicand needs to be $\ge 0$.

 

Homework homework yes you have homework!

Recommended order for the WeBWorK:

• First complete the assignments that are due tonight, obviously.

• Then complete the assignments HigherRootsAlgebraic, RationalExponents, and FrractionalEquations, if you have not already done so.

• At the same time you are working on those assignments, start working the Test3Review a few problems at a time. It’s best to schedule time on at least 3 of the 4 days before the Test (including today) and not try to do all the Test Review at once.

Here are some typed notes and a summary of the definitions and properties of exponents, which may be helpful:

MAT1275coRootsNotes

MAT1275coExponentsDefinitionsLaws-Condensed

A more comprehensive post on reviewing for Test 3 will go up soon, but until then, see the Test 2 Review post, which contains much of the same advice.

There are important elections happening even though it’s an “off-off” year. And that means that your vote can count more!

If you are registered to vote in NYC, there is an important election for Public Advocate and also there are 5 proposed changes to the City Charter which look very interesting. You can read about them here.

 

Find your polling place here, and congratulations on exercising your precious right to vote!

Homework for Thursday 31 October

Simplify each of the complex fractions by both methods:

1) $\frac{\frac{3}{2} + 3}{2}$

2) $\frac{27 – \frac{4}{3x}}{\frac{4}{27x} – 3}$

3 $\frac{\frac{6}{t} – \frac{5}{t-8}}{\frac{6}{t-8} – \frac{5}{t}}$

I will call on students at random to put these on the board.

We began with a diagnostic designed to check your understanding of basic fraction information and vocabulary, and tell you what you need to work on. You can see the diagnostic at this link  (it starts on the second page) and you can also read the reasoning behind the choice of answers, if you are interested. Please respect the (rather strict) copyright which the authors assert, and do not reproduce this.

Then on to the new topic, Rational Expressions: reducing them to lowest terms. We will be looking at the basic operations addition, subtraction, multiplication, and division. They work in the same way as the operations on fractions (rational numbers) with a few warnings we must pay attention to.

For review of how we build up the Real number system starting with the Natural Numbers, see the links in this post

It is very important that you become familiar with the various sets of numbers: their names, and what they are (and also where they come from). For example, why do I say that the number 0 (which is indeed a number!) is a very important invention – specifically, why is it an invention?

Rational Expressions are just algebraic expressions which are formed by ratios of polynomials, in the same way that rational numbers are formed by ratios of integers. (The links are to the Website Math Is Fun, which is a wonderful resource for reviewing basic concepts.)

The trouble we can have with rational expressions comes from the fact that, if a variable appears in the denominator, then it is possible for the denominator to become 0 for certain values of that variable. For example, the simple rational expression $\frac{1}{x}$ is undefined when $x = 0$. We must be careful to be aware of this potential problem.

Two warnings that come from this fact:

• When we reduce a rational expression by canceling a common factor which contains a variable, the reduced form is only valid as long as that factor is not equal to 0. Simple example: $\frac{3x}{x} = 3$ only if $x \neq 0$

• When solving an equation, it is important not to divide both sides of the equation by a variable or any expression containing a variable (because you might be dividing by 0.) For example, some people try to solve $x^{2} = 3x$ by dividing both sides by x, which gives $x=3$. But that is not the only solution of the equation! Find the other one by “guess and check” or by solving it properly.

We will also see that the problem of zero denominators causes difficulties when we solve equation which contain rational expressions.

 

Important note: for many reasons, we usually prefer to see rational expressions in the form where their numerator and denominator are completely factored. So when reducing a rational expression, leave your answer in this factored form.

 

Information about the assignments and the order and timing of them:

If you feel that you could benefit from practice with operations on fractions (and it really does help), I recommend the Math Is Fun pages on reducing, addition and subtraction, multiplication and division, which are linked from the Fractions menu there. Each page has a number of questions at the bottom that you can use for practice or learning more, and the website checks your answers and gives more information as you work them!

There are several open assignments on WeBWorK. You should start with the assignment on Reducing Rational Expressions. I would recommend then starting the assignment on Adding Rational Expressions, which is not too hard if you remember how to add fractions that have the same denominator. I plan to work on the Multiplying and Diving Rational Expressions tomorrow in class, so hold off on that unless you are very ambitious and/or bored.

The related chapter in the textbook is Lesson 2 (because we are not going in the same order as the book does!), and there are also video resources.

 

Question to think about: Why is it that division by 0 is not allowed? After all,  when we could not take the square root of $-1$ in the Real Numbers, we just invented a new number for it. It turns out that there is no way to invent a number that would equal $\frac{3}{0}$, for example, but why?

To understand the reason, you will need to think mathematically about what division is. Resist the urge to use clever wordplay like, “well, you can’t divide by nothing” – I could convince you that anything at all is impossible by that kind of vague semantic trickery.

In mathematics, in the branch that is called algebra, there are two basic operations (addition and subtraction) and the other operations are defined in terms of those. This means that division is defined in terms of multiplication: $12 \divide 3 = 4$ because $4 \cdot 3 = 12$. (In the clumsy way I talk about square roots, I could similarly say that “12 divided by 3 is the number that, when you multiply it by 3, you get 12.”) That’s the definition of division.

When we say that division by 0 is undefined, what that means is that division by 0 cannot be defined in terms of multiplication the way I did it in the previous paragraph, and furthermore there is no way to define division by 0 so that it “works” inside our number system.

To see a little bit of why this is true, try to definen 3 divided by 0 using the definition in terms of multiplication. (This is anyway how you check a division problem.) With some thought you should be able to see what goes wrong.

It’s less easy to see why there is no alternative way to define 3 divided by 0, but I’ll give you a writeup or a link next time.

And it turns out that 0 divided by 0 is even worse! We say that $\frac{0}{0}$ is indeterminate. See if you can figure out why.

That’s the news. Check back later for more information.

Here are notes and the classwork from Thursday the 10th, including advice on the homework. Please read!

If you want to use Desmos as I did in class, here is the webpage. There is also an app which works very well. See the desmos.com homepage for links. Also see below!

Here is a page with some fun and informative stuff that you can enjoy reading or viewing (I hope) – including some pretty amazing graphs made using Desmos.

Here is a page with Quiz solutions (not quite up to date yet)

Here is a page with the (tentative) schedule of the Tests.

Here is a page with links to the tutoring schedules and also other resources

here is the classwork:

MAT1275coGraphingCircles

See Lesson 11 in the textbook for more notes on this.

Please complete the problems on finding the perpendicular bisector for next Wednesday. The answers are here:

There is an assignment on WebWorK about Nonlinear Systems, and you can take a look at it if you are ambitious: I would recommend looking at Problem 1 to start with. You can graph both of the equations in Desmos and see if you can figure out the solution or solutions that way! (In fact, you can try that with all of them.)

Please do work the CircleLab and try to get it correct 4 times, as directed. Since this problem uses Geogebra, you may want to work in the Computer Lab.

 

The nQuiz next time will be on putting the equations of circles into standard form by completing the square, and reading off the center and radius. (No graphing. But remember that you will have to draw the graphs on the Final Exam!)