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
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
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
Is there more than one letter?
- Treat all letters EXCEPT the one you are solving for as if they were
numbers. |
Sec. 1.5 |
Is there an absolute value symbol in the
equation?
- Use the definition to separate into cases. |
Sec. 4.3 |
In the equation quadratic in form ?
- Rewrite using u (or other) letter for the variable expression that is
to be the
middle term then solve as an ordinary quadratic, then substitute back
into
the equation involving u and x to find x . |
Sec. 8.4 |
Is there a variable under a radical?
- Solve for the radical first, then square both sides of the equation
and
simplify . You MUST check the solutions because some may not check in
the original. |
Sec. 7.6 |
Are there one or more fractions in the equation ?
- Multiply both sides of the equation by the denominator (or LCD.)
Always
be sure that no denominator can be zero . |
Sec. 6.6 |
Is there an x3or higher power of x?
- The only way we could work this would be to gather all terms on one
side
of the equation and then factor.
- Use the Zero Product Principle to set each factor equal to zero, and
then
solve. There might be as many solutions as the highest power of x. |
Sec. 5.7 |
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.
a) Simplify the radical , if possible.
b) If the radical contains a perfect square, rewrite the solution twice,
once
with a + and once with a -, then simplify further.
c) If the radical contains a negative number there is no real solution |
Sec. 5.7
Sec 8.1
Sec. 8.2 |
Are there only x terms and constants?
1) Remove parentheses using the Distributive Law and simplify both
sides. If
you now find an x2 term, see instruction above.
2) Using the Addition Property of Equality, gather the x terms on one
side of
the equation.
3) Using the Addition Property of Equality, gather the constants on the
other
side.
4) Divide both sides of the equation by the coefficient of x.
5) If the x terms disappear and:
a) You get nonsense such as 5 = 2, then there is NO SOLUTION.
b) You get a true statement such as 6 = 6, then ANY NUMBER is a
solution. (This is called an identity). |
Sec. 1.4 |
| Is there a logarithmic expression? |
Sec. 9.5 |
Is this an inequality?
Is there an x2 or higher power?
Gather all terms on one side and factor.
Do a SIGN ANALYSIS to find the solutions. |
Sec. 8.5 |
Is there no higher power of x? (than the first
power)
Treat it as an equation except that when you multiply or divide by a
negative
you must REVERSE the inequality. Graph the solution. |
Sec. 4.1 |
Is this a compound of inequality? (It has 2
inequality symbols )
Perform operations on all 3 parts to isolate the x value.
Translate “and” and “or” into intersection and union of sets. |
Sec. 4.2 |
Absolute value inequality.
Translate into inequalities without absolute value sign and solve. |
Sec. 4.3 |
Rational inequality, boundary points.
Is there a variable in the numerator or denominator of a fraction ? Find
all
boundary points and test a point in each interval. |
Sec. 8.5 |
To solve 2 linear equations in 2 unknowns there
are 2 (equally good) methods.
Each eliminates one variable in the first step.
1) Substitution
2) Addition Method
(You can observe the approximate solution by graphing both equations on
the same
graph. The solution is the coordinates of the point where the lines
intersect.)
What can ‘go wrong?’
a) You lose BOTH variables in the first step and end up with nonsense
like
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
inconsistent
system.
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
solution. |
Sec. 3.1
Sec. 3.1
Sec. 3.1
Sec. 3.1
Sec. 3.1 |
To solve 2 equations in 2 unknowns with squares
or higher powers of one or both
variables.
Use addition or substitution – whichever allows you to eliminate a
variable in
the first step. Then use substitution to find the other part of the
ordered pair. |
Sec. 10.5 |
To solve 3 or more equations in 3 or more
unknowns:
Form a matrix and use elementary row operations.
Note carefully to find inconsistent or dependent systems. |
Sec. 3.3,
3.4 |
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 |
| |
Radicals: No negative under even index radical
(COMPLEX) |
Sec. 7.1 |
| 1 |
Is the expression under the radical a perfect
square? cube? Simplify. |
Sec. 71 |
| 2 |
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
positive
value, so if the expression is a variable, when it is negative the value
of the
RADICAL is its opposite. |
Sec. 7.3 |
| 3 |
Is there a fraction under the radical?
Simplify the expression into a single fraction and separate
into two separate radicals. |
Sec. 7.4 |
| 4 |
Is there a product or quotient of radicals?
Perform the operations. |
Sec. 7.3 |
| 5 |
Are two radicals in a sum alike (same index and
radicand?)
Add using the coefficients of the radicals. |
Sec. 7.4 |
| 6 |
Is there a radical in a denominator?
Rationalize it:
a) If it is a single radical, multiply numerator and denominator by that
radical.
b) If there is a sum or two terms where one or both are radicals
multiply
the numerator and denominator by the CONJUGATE of the
denominator. |
Sec. 7.5
Sec. 7.5 |
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
and let me know. Lastly, you can call and leave a message on my office phone.
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
tests. Your grade will be calculated with the following weights:
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
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
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
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
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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