2. Now, present theequation x2 - 2x - 6 = 0. Ask the
students how to solve an equation
like this for a positive root, since it does not factor.
Introduce New Material:
1. Introduce the concept of “completing the square.” The goal of completing the
is to manipulate an equation into one that factors nicely, like the examples
Begin with an easier example (x2 + 10x - 39 = 0), one that can be factored
nicely , so
that students can solve the problems both ways, to see that they get the same
2. First, give students a brief history of the “completing the square method,
formulated by al-Khwārizmī when algebra was invented (information found on pages
1-2 of the “Islamic Mathematics” information packet). Then, follow the steps
pages 3-5 of the “Islamic Mathematics” information packet. These teach students
complete the square using the method of al-Khwārizmī. It might be easiest for
students to use the second method (pp 4-5), but both should be presented.
3. Show the solution to the example problem (x2 + 10x - 39 = 0) by factoring so
students can recognize that factoring and completing the square methods both
the same solution.
1. Pass out the Completing the Square worksheet, and help students to complete
problem (x2 - 2x - 6 = 0) by completing it on the board.
Independent Practice: 1. Have students work in groups of 2-3 people to complete the rest of the
(Make sure the students are keeping their plus and minuses the same during
completing the square, depending on the sign in the original problem!) Walk
the room to help students who have trouble completing any of the problems.
Closing / Assessment:
1. For homework, assign problems similar to these (from a textbook or
asking students to solve the problems either by factoring or by completing the
Or, have students complete side 1 of the worksheet in class and side 2 for
Math 20 STUDY GUIDE
To the students:
When you study Algebra, the material is presented to you in a logical sequence.
Many ideas are developed, left, and then returned to when your knowledge is
Many different kinds of problems have similar instructions. This presents great
when trying to prepare for a final exam or keep up in the next Math class . You
mastered all the skills, but which one do you use in a specific problem? This
written to help you to re-organize your knowledge into a more usable form.
When you are faced with a problem that begins, “Solve for x.” What should you
do? As you will see, there are at least 9 different situations where you have
instruction. This guide will give you the key questions to ask yourself in order
what procedure to use. The main steps that are involved are included. The
asked in the ORDER that you should ask them. Each is referenced with a section
(or if only part of the section is involved, the specific page or problem number
To use this guide effectively, you should first read through the guide. Each
reference to a section should be examined carefully. Can you make up a problem
one being described? Would you know how to solve that problem without any clues?
Look at the problem or section referenced. Is it like yours? Can you work those
so, go on to the next topic. If not, highlight that line with a marker for
Perhaps you should put an example problem on a 3 by 5 card (include the page
for practice later . Now read the section again carefully. Work the examples and
few similar problems from the exercises (odd ones so you can check the answers)
practice. When you finish a whole type (i.e. Solve for x) mix your 3 by 5 cards
them like a test. Simply verify that you know how to start the problem. Any that
will direct you back to the sections where you need further study.
If you need further help, consider asking for a tutoring appointment in the Math
Lab. When you know what SPECIFIC topics present a problem for you, you can make
tutoring session much more effective and be of help so your tutor can know what
you need. See the Math Lab Coordinator early in the semester to fill out an
SOLVE FOR X
Is there more than one letter?
- Treat all letters EXCEPT the one you are solving for as if they were
Is there an x3or higher power of x?
- The only way we could work this would be to gather all terms on one
of the equation and then factor.
- Use the Zero Product Principle to set each factor equal to zero, and
solve. There might be as many solutions as the highest power of x.
Is x2 the highest power of x? Use any
of the following:
1) Try factoring, it sometimes works
2) Complete the square. WARNING: If a perfect square equals a negative
number, quit. There is no real solution.
3) Put the equation in standard form and apply the Quadratic Formula.
Rational inequality, boundary points.
Is there a variable in the numerator or denominator of a fraction ? Find
boundary points and test a point in each interval.
To solve 2 linear equations in 2 unknowns there
are 2 (equally good) methods.
Each eliminates one variable in the first step.
2) Addition Method
(You can observe the approximate solution by graphing both equations on
graph. The solution is the coordinates of the point where the lines
What can ‘go wrong?’
a) You lose BOTH variables in the first step and end up with nonsense
0 = 7. There is NO SOLUTION. (In this case the lines on the graph
would be parallel, so they don’t meet at all.) We call this an
b) You lose both variables in the first step and end up with truth like
0 = 0.
The answer is that there are MANY SOLUTIONS. (In this case if you
graphed the lines, one would be superimposed over the other.) Both
equations describe the same line so any point on the line represents a
To solve 2 equations in 2 unknowns with squares
or higher powers of one or both
Use addition or substitution – whichever allows you to eliminate a
the first step. Then use substitution to find the other part of the
To solve 3 or more equations in 3 or more
Form a matrix and use elementary row operations.
Note carefully to find inconsistent or dependent systems.
COMPUTE OR EVALUATE
Order of Operation
1) Work from the innermost grouping out.
a) The numerator and denominator of a fraction are each groupings.
b) An absolute value symbol is a grouping
c) If the fraction is a complex, find SOME part that can be simplified
and start there.
1) Write the expression in terms of the logs of single numbers.
2) Write each number in scientific notation using base 10 for ordinary
numbers. Use natural logs if the problem involves power of e.
3) Look up the log of each number using the table of logs in the book
(Mantissa) and write the log of each power of 10 by inspection
4) Simplify the expression into one with a positive mantissa and a
5) Write the answer in scientific notation using the body of the table
A Logarithmic Expression
Use the definition of logarithm to write in exponential form , and then
the missing number.
To graph ANY equation involving x and y:
1) Make a table for x and y.
2) Pick at least 5 values, some negative, for x. (Occasionally, it may
convenient to pick some values for y.)
3) Using the formula given to you, complete the table. (Substitute each
into the formula then compute the remaining value.) It is particularly
to substitute 0 for x to find the y-intercept (s) and 0 for y to find
4) Plot the points from your table on the graph.
5) Connect the points smoothly moving from left to right
To graph an equation like x = 4 (or any number.)
All x values are 4; pick anything
at all for y. The result will be a vertical line.
To graph an equation like y = 7 (or any number.)
All y values are 7, pick anything
at all for x. The result will be a horizontal line.
2) Pick any point well away from the dotted edge. (If the origin
qualifies, it is an
3) Substitute the coordinates of your point into the inequality.
a) If the test point makes the inequality true, shade in that side of
b) If the test point does NOT make the inequality true, shade in the
4) If the inequality allows =, (either ≥) fill in the edge of the graph
To graph a system of inequalities:
1) Graph the first inequality as above.
2) Using a different color, graph the second inequality on the same
3) The answer is the region that is shaded with BOTH colors.
The x-intercept of (a line) or curve is where it
crosses the x-axis. To find its value,
substitute 0 for y and then solve for x.
The y-intercept is where the line or curve
crosses the y-axis. To find its value,
substitute 0 for x and then solve for y.
If an equation can be put into the form y = mx +
b, then it is a straight line.
If an equation involves the second power of x or
y or both, it may be a conic
To find the distance between two points, ( x1
,y1) and ( x2 , y2 )
The Pythagorean Theorem gives us this formula:
The midpoint of the line segment between two
points, ( x1 ,y1) and ( x2 , y2
The slope of a line can be determined in two
1) If you know the equation of the line, solve it for y. The slope is
coefficient of x.
2) If you know the coordinates of two points,
( x1 ,y1) and ( x2 , y2
) use the formula:
Parallel lines have the same slope. Perpendicular
lines have slopes with product –1
Fractions: If there are no variables, see
To add or subtract:
a) Find the Lowest Common Denominator.
b) Change each fraction to an equivalent fraction by multiplying
numerator and denominator by the same value.
c) Add the numerators and use the common denominator. If there is a
“—“ in front of a fraction be sure to distribute it to EVERY TERM in
To multiply, factor numerators and denominators
reducing where possible.
Leave the answer in factored form unless it is part of a larger problem.
(i.e. must be added to other terms.)
To divide, FIRST invert the divisor, and then
proceed as in multiplication.
If there is a fraction within a numerator or
a) Multiply numerator and denominator of the largest fraction by the LCD
for all fractions
b) Treat numerator and denominator as a grouping and simplify, then
divide as indicated by the larger fraction.
Remember, no denominator of any fraction may ever
Always reduce final answers where possible by
dividing common factors from
numerator and denominator.
Radicals: No negative under even index radical
Is the expression under the radical a perfect
square? cube? Simplify.
Is there a factor of the expression under the
radical that is a perfect square, cube
etc? Factor it out and simplify. Remember, the radical always has a
value, so if the expression is a variable, when it is negative the value
RADICAL is its opposite.
Is there a fraction under the radical?
Simplify the expression into a single fraction and separate
into two separate radicals.
Is there a product or quotient of radicals?
Perform the operations.
Are two radicals in a sum alike (same index and
Add using the coefficients of the radicals.
Is there a radical in a denominator?
a) If it is a single radical, multiply numerator and denominator by that
b) If there is a sum or two terms where one or both are radicals
the numerator and denominator by the CONJUGATE of the
1) FIRST, review the rules of exponents.
2) You may apply any appropriate rule to the expression, but the
strategies may be useful:
a) Are there powers of other expressions? Use the Power (of a Product)
rule to remove parentheses.
b) Are there powers of exponential expressions? Use Power (of a
Power) rule where appropriate.
c) Are there like bases in numerator or denominator? Use the product
rule to simplify (add exponents.)
d) Are there like bases in both numerator and denominator? Divide (by
e) Are there negative exponents? Use the negative exponent rule to
write the reciprocal.
f) Write as a single fraction.
g) Are you finished?
Each exponent should apply to a single base. Each base should
appear only once. There should be no negative exponents. Powers of
numbers should be calculated. The fraction should be in lowest
First, review the properties of Logarithms.
Apply the properties – one at a time – until the goal is achieved.
a) We define that i 2=−1
b) Complex numbers are written as a + bi, where a and b are real
c) To remove a complex number from the denominator of an expression,
multiply by its conjugate.
d) For power of 1, substitute (-1) for i squared as many times as
substitute 1 for i to the fourth.
Place the decimal point after the first non-zero digit and multiply by
appropriate power of 10. Simplify by using exponent rules on powers of
To factor a number means to write it as a product of primes (numbers that cannot
factored further.) Begin with any product and then break each number down until
can be factored further.
To factor a polynomial:
1) Is there a factor common to all terms? Factor out the greatest common
2) Are there 4 terms? Try factoring by grouping.
3) Is there a common pattern?
a) Is this a difference of 2 squares ?
b) Is this a perfect square trinomial?
c) Is this the sum or difference of two cubes?
When all else fails on a trinomial:
4) Perform a structures search. (This is an organized version of the
from the text.)
a) List all the possible ways to factor the first (squared) term. These
the column headings.
b) In each column, list all the possible arrangements of the factors for
the last (constant) term. (These form the rows.)
c) Test each entry in your table using FOIL to see if this makes the
middle term possible. (If there are no candidates, report that it
DOES NOT FACTOR.)
d) If you have a candidate, insert signs to try to match original.
i. If the last sign (constant) is negative, the signs are different.
ii. If the last sign is positive, the two signs are alike, Use the sign
the middle term.
iii. If none of the above works, go on searching for new candidates.
iv. If you exhaust the list and none work, report that it DOES NOT
e) Check your solution.
Check to be sure that none of the factors can be factored further
“How to Solve Word Problems in Algebra” By
Mildred Johnson is an excellent and
inexpensive resource. It is available in the bookstore.
1) Read through the problem to determine type.
2) Draw a picture, if possible.
3) Write “Let x be …”
4) Pick out the basic unknown and finish the above sentence.
5) Write as many other quantities as possible in terms of x and label
6) Is there STILL another unknown? If so, write, “Let y be …” and
the sentence. Write all other quantities in terms of ‘x and y’. You may
one or more of the formulas below to complete this.
Note: Tables are useful in many of these problems. Make one like the
the text where appropriate.
7) Write any formula(s) that apply to this type of problem.
a) d = rt (distance, time and speed)
b) In wind or stream, when moving with the current, the speed is the
sum of the speed of the craft and the current.
c) i = Pr (interest for 1 year)
d) Concentration of a solution
(% target) (amount mixture) = amount target ingredient.
e) (cost per item) (number of items) = value
f) (denomination of a bill) (# of bills) = value
g) consecutive numbers x, x + 1, x + 2 , etc.
h) consecutive ODD or EVEN numbers (The value of the first
determines which) n, n + 2, n + 4 etc.
i) In age problems, when they say “in 5 years,” write each age + 5
j) Work rate problems convert the time to do a job into the work done
per time period by taking the reciprocal. THESE quantities can be
added or subtracted.
k) Geometric formulas are found on the back cover of the text. Ask
your instructor which you are responsible for knowing.
l) Fulcrum: Use weight x distance for each force.
m) Cost Analysis
n) Direct and Inverse Variation.
8) Use the formula or the words from the problem to write an equation.
9) Solve the equation for x (or x and y.)
10) REREAD the question. Write all the quantities from the original
using the value for x as a key.
11) Answer the question asked.
12) Check the answer with the problem’s original words.
Discard any answers that don’t fit.
Functions and composites
Sec 2.1, 2.2,
Inverse of a function
The Binomial Theorem
Intersection and union of intervals
MATH 060 ONLINE SYLLABUS
COMMUNICATING WITH THE INSTRUCTOR:
The best way to communicate with me is by email. Another great way to
with me live is by chat room . If you would like me to host a chat room, please
and let me know. Lastly, you can call and leave a message on my office phone.
This class will meet on campus for an orientation on Monday February 9, 2009 in
305 at 7:30 PM. You are required to attend the orientation if you haven’t
completed the online orientation by Saturday , February 7 at 11:59 PM. The class will also meet
14, April 4, May 2, and May 23 for exams in CCC 401 from 8:30 AM – 12:00 PM and
final exam will be given on May 30 in CCC 401 from 8:30 AM – 12:00 PM.
The online homework assignments will have due dates. The suggested due dates
be displayed online in course compass, but will be considered late if not done
on campus exam. The homework is not timed, so you may redo them until you get a
100%. No work is accepted after the due date has passed. The homework
will be made available after the suggested due dates so that you can always go
and redo/ review the problems before the exam.
Every week we don’t have an exam will have either an online quiz or online test.
dates that these are available as well as due dates are indicated in the
quizzes and tests will be timed. You will have multiple attempts on these. Only
highest score will count toward your grade.
TEXTBOOK HOMEWORK :
The textbook homework assignments will be posted online in course compass on the
announcements page. The homework must be neatly done following the sample
homework guidelines to receive full credit. You will have four homework
Each assignment is due on the day of the exam. NO LATE HOMEWORK WILL BE
ON CAMPUS EXAMS:
The exam days are given above and online in the schedule. If you miss an exam, I
replace your missed exam with the final grade. If you miss two tests, you will
dropped. If you know in advance that you will not be able to attend a scheduled
you may reschedule one test provided you contact me 1 week before the test is
Once you are enrolled in the online section with admissions and records you will
to complete the online orientation or attend the on campus orientation. Once you
complete the online orientation you will receive the course ID. Then you will be
register for the course online at course compass. Before you register online,
you have everything you need including a student access code (comes with the
can be purchased online), CourseCompass Course ID provided by your instructor,
a valid email address.
• You will get the student access code when you purchase the textbook on
campus or online
• The CourseCompass Course ID is provided once you successfully complete the
• You will need a valid email address (if you don’t have one, get one at hotmail
Once you have enrolled in the course and are able to login
to course compass, you
MUST run the installation wizard and load the plug-ins.
*** If you do not get registered with course compass by
Friday, February 20, you
will be dropped from the class.
*** If you are inactive in course compass for more than a
week, you will be
dropped. Inactive means that you are not completing the assignments, not
participating in discussion threads, and not participating in chat rooms.
IS ONLINE MATH FOR YOU??
There are obvious benefits for taking a Math class online, but there can be
disadvantages. There is no “real” teacher. Your “real” teacher is being replaced
online video clips. This means that you will need to be extremely self-motivated
online. You will also need to pace yourself and make sure you stay up with the
To do well in this class you should have about 24 hours a week set aside to
this class. If you feel that you are someone who needs more structure from the
traditional professor, you should take a regular Math class.
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