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# Lesson Plan for Completing the Square

Overview:
Students will learn about the rich history of Islamic Mathematics, and will gain an
understanding of the relationship of Islamic mathematics to modern mathematics . This
unit requires some basic knowledge of algebra, and so is geared towards high school
students or middle-school students learning algebra . Students will learn some of the
methods used by Muslim mathematicians in “completing the square.” Using the history
intertwined with the mathematics lesson , students will be more interested in the lesson
and have a better understanding of completing the square.

Objectives:
Students will be able to:
1. Recount a brief history of Islamic mathematics and the expansion of Greek
mathematics in the Arab world
2. Solve for x by completing the square

Activity:
Opening / Hook:
1. Students should be familiar with algebraic methods for solving quadratic equations ,
such as factoring. First, review these methods , with examples such as the following:

2. Now, present the equation x 2 - 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 square
is to manipulate an equation into one that factors nicely, like the examples above.
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 answer.
2. First, give students a brief history of the “completing the square method, which was
formulated by al-Khwārizmī when algebra was invented (information found on pages
1-2 of the “Islamic Mathematics” information packet). Then, follow the steps from
pages 3-5 of the “Islamic Mathematics” information packet. These teach students to
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 that
students
can recognize that factoring and completing the square methods both give
the same solution.

Guided Practice:
1. Pass out the Completing the Square worksheet, and help students to complete the first
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 worksheet.
(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 around
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 worksheet ),
asking students to solve the problems either by factoring or by completing the square.
Or, have students complete side 1 of the worksheet in class and side 2 for homework.

# 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 broader.
Many different kinds of problems have similar instructions. This presents great difficulty
when trying to prepare for a final exam or keep up in the next Math class . You have
mastered all the skills, but which one do you use in a specific problem? This guide was

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 seen that
instruction. This guide will give you the key questions to ask yourself in order to decide
what procedure to use. The main steps that are involved are included. The questions are
asked in the ORDER that you should ask them. Each is referenced with a section number
(or if only part of the section is involved, the specific page or problem number is given.)

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 like the
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 easily? If
so, go on to the next topic. If not, highlight that line with a marker for further study.
Perhaps you should put an example problem on a 3 by 5 card (include the page number)
for practice later . Now read the section again carefully. Work the examples and select a
few similar problems from the exercises (odd ones so you can check the answers) for
practice. When you finish a whole type (i.e. Solve for x) mix your 3 by 5 cards and treat
them like a test. Simply verify that you know how to start the problem. Any that you miss
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 a
tutoring session much more effective and be of help so your tutor can know what help
you need. See the Math Lab Coordinator early in the semester to fill out an application
for tutoring.

SOLVE FOR X

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. 2) Within a group: a) Exponentiate first . (SEE Exponential Expressions for more detail .) i. Only the closest possible base is raised to the power. To raise a negative base or a fractional base to a power REQUIRES parentheses. ii. Any base to the zero power is 1. Any base to a negative power indicates that you take the reciprocal. iii. A fractional exponent can be broken into an integer power and a root. b) Multiply and divide, moving from left to right. c) Simplify signs, if necessary. d) Algebraic addition is last Sec. 1.1 Sec. 1.2 Sec. 1.6 Sec. 7.2 Problems using Logarithms 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 (Characteristic.) 4) Simplify the expression into one with a positive mantissa and a single characteristic. 5) Write the answer in scientific notation using the body of the table for the mantissa. A Logarithmic Expression Use the definition of logarithm to write in exponential form , and then fill in the missing number. Sec. 9.3

GRAPH

 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 be convenient to pick some values for y.) 3) Using the formula given to you, complete the table. (Substitute each value into the formula then compute the remaining value.) It is particularly useful to substitute 0 for x to find the y-intercept (s) and 0 for y to find the x- intercept (s). 4) Plot the points from your table on the graph. 5) Connect the points smoothly moving from left to right Sec. 1.3 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. Sec. 2.3 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. Sec. 2.3 To graph an INEQUALITY with absolute value on a number line (only one variable). Find critical values and test intervals. Sec. 4.3 To graph an INEQUALITY: 1) Graph as above but dot in line or curve. 2) Pick any point well away from the dotted edge. (If the origin qualifies, it is an easy choice.) 3) Substitute the coordinates of your point into the inequality. a) If the test point makes the inequality true, shade in that side of your graph. b) If the test point does NOT make the inequality true, shade in the other side. 4) If the inequality allows =, (either ≥) fill in the edge of the graph border solidly. Sec. 4.4 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 graph. 3) The answer is the region that is shaded with BOTH colors. Sec. 4.1 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. Sec. 2.3 If an equation involves the second power of x or y or both, it may be a conic section. Sec. 10.1- 10.4 To find the distance between two points, ( x1 ,y1) and ( x2 , y2 ) The Pythagorean Theorem gives us this formula: Sec. 10.1 The midpoint of the line segment between two points, ( x1 ,y1) and  ( x2 , y2 )  is Sec. 10.1

SLOPE

 The slope of a line can be determined in two ways. 1) If you know the equation of the line, solve it for y. The slope is the coefficient of x. 2) If you know the coordinates of two points, ( x1 ,y1) and  ( x2 , y2 )  use the formula: Sec. 2.3 Sec. 2.3 Parallel lines have the same slope. Perpendicular lines have slopes with product –1 Sec. 2.4

SIMPLIFY

 Fractions: If there are no variables, see compute. Sec. 1.1 1 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 the numerator. Sec. 6.2 2 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.) Sec. 6.1 3 To divide, FIRST invert the divisor, and then proceed as in multiplication. Sec. 6.1 4 If there is a fraction within a numerator or denominator, either: a) Multiply numerator and denominator of the largest fraction by the LCD for all fractions OR b) Treat numerator and denominator as a grouping and simplify, then divide as indicated by the larger fraction. Sec. 6.3 5 Remember, no denominator of any fraction may ever be zero. Sec. 6.1 6 Always reduce final answers where possible by dividing common factors from numerator and denominator. Sec. 6.1

 Exponential expressions 1) FIRST, review the rules of exponents. 2) You may apply any appropriate rule to the expression, but the following 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 subtracting exponents.) 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 terms Sec 1.6 Logarithmic Expressions First, review the properties of Logarithms. Apply the properties – one at a time – until the goal is achieved. Sec. 9.3, 9.4 Complex Numbers a) We define that i 2=−1 b) Complex numbers are written as a + bi, where a and b are real numbers. 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 possible or substitute 1 for i to the fourth. Sec 7.7 Scientific Notation Place the decimal point after the first non-zero digit and multiply by the appropriate power of 10. Simplify by using exponent rules on powers of 10. Sec. 1.7

FACTOR

To factor a number means to write it as a product of primes (numbers that cannot be
factored further.) Begin with any product and then break each number down until none
can be factored further.

 To factor a polynomial: 1) Is there a factor common to all terms? Factor out the greatest common factor (term.) 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 trial factors from the text.) a) List all the possible ways to factor the first (squared) term. These are 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 of 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 FACTOR. e) Check your solution. Check to be sure that none of the factors can be factored further Sec. 5.6 Sec. 5.3 Sec. 5.3 Sec. 5.5 Sec. 5.4

WORD PROBLEMS

 “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 them. 6) Is there STILL another unknown? If so, write, “Let y be …” and complete the sentence. Write all other quantities in terms of ‘x and y’. You may need one or more of the formulas below to complete this. Note: Tables are useful in many of these problems. Make one like the models in 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 problem 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. Sec. 1.5, 3.2, 5.7, 6.7, 10.5 Sec. 6.8

OTHER TOPICS

 Functions and composites Sec 2.1, 2.2, 9.2 Inverse of a function Sec. 9.2 Arithmetic Sequences Sec 11.2 Geometric Sequences Sec 11.3 The Binomial Theorem Sec 11.4 Intersection and union of intervals Sec 4.2

# MATH 060 ONLINE SYLLABUS

COMMUNICATING WITH THE INSTRUCTOR:
The best way to communicate with me is by email. Another great way to communicate
with me live is by chat room . If you would like me to host a chat room, please email me

INFORMATION
This class will meet on campus for an orientation on Monday February 9, 2009 in CCC
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 March
14, April 4, May 2, and May 23 for exams in CCC 401 from 8:30 AM – 12:00 PM and a
final exam will be given on May 30 in CCC 401 from 8:30 AM – 12:00 PM.

The class will consist of on campus exams (4), online homework , online quizzes , online
tests, textbook homework , and an on campus final. You will also turn in a homework
notebook (see the sample homework guidelines link on the website ) on the day of the
5% online homework
10% online quizzes /tests
5% homework
50% on campus exams
30% final

All students will need a PC based computer to access the online course. See the
orientation letter for details.

STUDENT LEARNING OUTCOMES
• Recognize and interpret equations of lines
• Solve systems of linear equations
• Factor Polynomials

ONLINE HOMEWORK:
The online homework assignments will have due dates. The suggested due dates will
be displayed online in course compass, but will be considered late if not done before the
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 assignments
will be made available after the suggested due dates so that you can always go back
and redo/ review the problems before the exam.

ONLINE QUIZZES/TESTS:
Every week we don’t have an exam will have either an online quiz or online test. The
dates that these are available as well as due dates are indicated in the schedule. The
quizzes and tests will be timed. You will have multiple attempts on these. Only your

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 assignments.
Each assignment is due on the day of the exam. NO LATE HOMEWORK WILL BE
ACCEPTED.

ON CAMPUS EXAMS:
The exam days are given above and online in the schedule. If you miss an exam, I will
replace your missed exam with the final grade. If you miss two tests, you will be
dropped. If you know in advance that you will not be able to attend a scheduled test,
you may reschedule one test provided you contact me 1 week before the test is given.

FINAL:
The final is cumulative. I will post a review for the final around 1 or 2 weeks before we
take it.

GETTING STARTED
Once you are enrolled in the online section with admissions and records you will need
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 able to
register for the course online at course compass. Before you register online, make sure
you have everything you need including a student access code (comes with the book or
can be purchased online), CourseCompass Course ID provided by your instructor, and
• 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
online orientation.
• You will need a valid email address (if you don’t have one, get one at hotmail or
yahoo)

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 with
online video clips. This means that you will need to be extremely self-motivated to learn
online. You will also need to pace yourself and make sure you stay up with the material.
To do well in this class you should have about 24 hours a week set aside to devote 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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