High School Math Teacher
Email
p>Mathematics Instructor
Hometown: Cedarburg WI
College: UW-Madison(1986)
Places I've lived/taught: Milwaukee WI, Kirtland NM, Delafield WI, Gillett WI and Belmont WI
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Classes Taught
Algebra
al·ge·bra
noun
the part of mathematics in which letters and other general symbols are used to represent numbers and quantities in formulae and equations
(ex)
In this case let’s notice that we can factor out a common factor (distributive) of 3x2 from all the terms so let’s do that first.
What is left is a quadratic that we can use the technique of complex trinomial factoring. Doing this gives us,
Geometry
ge·om·e·try
noun
the branch of mathematics concerned with the properties and relations of points, lines, surfaces, solids, and higher dimensional analogs.
(ex) In the figure below lines A'A" and C'C" are parallel. AB is the bisector of angle CAA" and BC is the bisector of angle ACC". Show that the size of angle ABC is equal to 90 degrees.
solution
Angles A'AC and angle ACC" are alternate interior angles and their sizes are equal.
which gives angle A'AC = angle ACC"
Angles A'AC and angle ACC" are alternate interior angles and their sizes are equal.
which gives angle A'AC = angle ACC"
Angles A'AC and angle A"AC are supplementary so that
which gives angle A"AC = 180 - angle A'AC = 180 - angle ACC"
Rearrange the above to obtain
which gives angle A"AC + angle ACC" = 180 0
Because AB and CB are bisectors(they divide the angle into two equal angles), angle ABC in triangle ABC is given by
which gives angle ABC = 180 - (angle A"AC + angle ACC") / 2
= 180 - 180 / 2 = 90 0
Algebra 2
al·ge·bra
noun
- A branch of mathematics in which symbols, usually letters of the alphabet, represent numbers or members of a specified set and are used to represent quantities and to express general relationships that hold for all members of the set.
(ex) Simplify the equation .
We are going to complete the square here.
The thing that we’ve got to remember here is that we must have a coefficient of 1 for the x2 term in order to complete the square. So, to get that we will first factor the coefficient of the x2 term out of the whole right side as follows.
Note that this will often put fractions into the problem that is just something that we’ll need to be able to deal with. Also note that if we’re lucky enough to have a coefficient of 1 on the x2 term we won’t have to do this step.
Now, this is where the process really starts differing from what we’ve seen to this point. We still take one-half the coefficient of x and square it. However, instead of adding this to both sides we do the following with it.
We add and subtract this quantity inside the parenthesis as shown. Note that all we are really doing here is adding in zero since 9-9=0! The order listed here is important. We MUST add first and then subtract.
The next step is to factor the first three terms and combine the last two as follows.
As a final step we multiply the 2 back through.
Pre-Calculus
pre·cal·cu·lus
noun
a course in mathematics that prepares a student for calculus.
(ex) Prove the identity
Working the left side of the problem:
Calculus
cal·cu·lus
noun
1.
the branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences. The two main types are differential calculus and integral calculus.
(ex) Differentiate .
At this point, we will continue to simplify the expression, leaving the final answer with no negative exponents.
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