Mathematics Core Curriculum: Algebra Strand

fractions, decimals, and percents

IT’S A FAMILY AFFAIR!! Shelia M. Kelow School: Leland School Park Grade Level : 6th grade •Teaching objective(s) 6C1.f Explain the relationship among fractions, decimals, and percents with models and representation (DOK level 2) •(Note using non repeating decimals)

Instructional Activities

1. Review concept of converting fraction to decimal , decimal to fraction, and fractions to
percents using division and calculators to check the division.
•To convert fractions to decimals we simply divide the numerator by the denominator.

• Example
( instruct students to put a decimal with at least two zeros
following the numerator)

•To convert decimals to percent, simply move the decimal two places to
the right.
Example .75 = 75%
.40 = 40%
2. Probe for prior knowledge of fractions and decimals by having a question answer
conference with the students.
•Question #1 What fraction is equivalent to 0.50?

•Question #2 3/5is equivalent to what percent and decimal?

3. Provide guided practice on conversions & equivalent expressions. Example chart will be
completed with the teacher.

Questions for Students

1. How do we convert a fraction to a decimal? (Instructing students to write the
fraction by dividing. Example

2. How are decimals converted to percents? (Instructing students to move the
decimal two places to the right. Example .25 = 25%

3. How do we convert decimals to fractions? Instructing students to write the
fraction by putting what they say into numbers. Example

4. How do we convert percents to decimals? Instructing students to write the
percent and move the decimal now two places to the left. Example 55% = .55

5. What is a percent or the definition of percent? Amount per 100.

Student Activities

1. Observe, record and monitor notes provided by teacher
2. Complete and participate in matching game called “Where’s My Family?”
Given flash cards on Attachment 1 Families should be grouped together based on equivalence.

Sample:

Closing
1. Question and answer period on the practice
2. Have students give their interpretation of the questions and why they choose these answers
(Say Why)
3. Do a final wrap-up by clearing up any questions and misconceptions from the practice
activities.

Materials and Resources

=>Flash Cards (Attachment 1)
=>Teacher-made chart Attachment 2)
=>Overhead Projector used to provide visual execution of converting by teacher.

Assessment

Teacher-made quiz.

Attachment #1
Teacher will distribute flash cards at random instructing students to stand or find their family
members or equivalent parts

(THE FAMILY AFFAIR)

FLASH CARDS

Attachment # 2 NAME ___________________

MATH 096 COURSE SCHEDULE

Website for worksheets:

This schedule will be followed as closely as possible, but I reserve the right to make changes as they become necessary. Please read the given sections in the book before each lecture.

Notes on Polynomial Functions

Definition (polynomial function): Let n be a non-negative integer and let
with . Then the function
is called a polynomial function of x with
degree n.

Ex: Is 3x+5 a polynomial function?
Is a polynomial function?

Classification of polynomial functions

Constant Function:
Linear Function: f (x) = mx + b , with m,b∈R and m ≠ 0 (Ex: f (x) = 3x + 2 )
Quadratic Function: , with a,b, c∈R and a ≠ 0
-Quadratic formula (in abstract forms ) were first seen in 1800 BC
-Babylonian and Chinese Mathematicians used completing the square to solve
quadratic equations with positive roots between 600-400 BC. (doing this today)
-Euclid (inventor of geometry) came up with abstract method in 300 BC .
-Brahmagupta (Indian Mathematician) first man to write out quadratic formula for
positive roots
-Abraham bar ha-Nasi (jewish mathematician) was first man to write the
full solution to the general quadratic equation in 1136 AD

*Draw a graph of a quadratic function
*Quadratic equations are in the form of a parabola, have an axis of symmetry,
and contain a vertex
*If a>0, then parabola opens upward , and if a<0, it opens downward

Review of Transformations

y = f (x ±c)=>Horizontal Shift
y = f (x) ±c=>Vertical Shift
y = - f (x)=>Reflection in x-axis
y = f (-x)=>Reflection in y-axis

The Standard Form of a Quadratic Equation

Definition (Standard form of a quadratic function): The quadratic function
is said to be in standard form. The axis of symmetry is x=h, and
vertex is at (h,k). (h,k) is a maximum if a>0 and a minimum if a<0.

Ex: Suppose . Identify the axis of symmetry and the vertex.
→ Answer: Axis of symmetry is x=-1 and vertex is (-1,2)

Ex: Describe the graph of and identify the vertex.
→ Convert it to standard form
→ Answer:

Ex: Describe the graph of and identify any x-intercepts
→ Convert it to standard form (for vertex)
→ We can continue solving (using completing the square), or factor original
equation to find x-intercepts
→ Answer: x-intercepts are (2,0) and (4,0), and vertex is (3,1)
→ Standard form is

Ex: Find the standard form of an equation that has a vertex at (1,2) and passes through
the point (3,-6).
→ Plug vertex in standard form:
→ Plug point (3,-6) in formula and solve for a
→ Answer:

Final Application

Suppose we have a quadratic function . If we convert this to standard
form (and complete the square), we obtain .
-Axis of symmetry:
-Vertex:

-Recall that the vertex is a maximum if a>0 and a minimum if a<0

For those that hate completing the square, this is an alternative. I don't recommend it!

Homework: Page 143-144 #1-59 odd

--Extension--

Cubic and Quartic Function :
-Not Addressed in book
-Solved in the 17^th Century by Italian mathematicians Cardano (a lawyer and
friend of Leonardo Da Vinci) and Tartaglia (a bookkeeper, who wrongfully
published some of Cardano's works)
-The solution is so difficult, that it is rarely mentioned in college courses
Quintic Functions+:
-There is no explicit formula for quintic equations and above.
-This was proved by Abel and Ruffini (called Abel–Ruffini theorem), and uses
applications of abstract algebra

Linear Algebra & Differential Equations

Text: Linear Algebra and Differential Equations, by Peterson and Sochacki

Prerequisites: You must have passed Math 116 (Calculus II) prior to enrolling.

Important Web Pages: You should check all of these sites regularly (at least once a week), as
there may be minor changes throughout the semester. All changes will also be announced in class.

(1)

Announcements, assignments, course documents, the discussion board, exam dates etc.
can be found on blackboard. You are expected to log in and check announcements at least
once a week.
Username: The first part of your Duquesne University email address.

Password: The first letter of your last name in lower case and your full social security
number without spaces or hyphens (you should change your password under 'student tools'
once you've logged on).

(2)

This is an up to date homework list which contains all of your assignments. You should
check this page regularly, as there may be slight changes throughout the semester.

(3)

This is an up to date schedule containing the dates of all quizzes and exams. You should
check this page regularly, as there may be slight changes throughout the semester.

Topics to be Covered: Systems of linear equations, vectors and matrices, determinants, vector
spaces, linear transformations, eigenvalues and eigenvectors, first order differential equations, higher
order and systems of linear differential equations, differential operators, Laplace transforms

Tentative Schedule: A tentative calendar can be found on the website listed on the previous
page. This calendar includes the dates of all exams and quizzes. There may be minor changes to
this schedule, so you should check this website on a regular basis.

Attendance: Each student is responsible for all of the material covered during class. This includes
lecture notes, homework assignment due dates, material for each quiz and exam, and any important
announcements. Attendance is not a factor in your final grade, however, missing class is never an
excuse for not knowing any of the above mentioned material. In the case that you are forced to
miss class due to a verifiable medical emergency, contact me as soon as possible and I will let you
know what we covered. Otherwise, you are solely responsible for obtaining the information that
you missed. Important announcements will often be posted on blackboard, but it's not guaranteed
that everything will be posted.

Homework: Students should read the relevant sections of the text prior to the lecture. Homework
problems will be assigned for each lecture. These represent the minimum number of problems you
should do in each section and serve together with lecture notes as a basis for questions on the
quizzes and exams. All homework assignments will be posted on the website listed on
the previous page.

Quizzes / In-Class Worksheets: There will be 10 quizzes. The questions will be based on
lecture notes and homeworks. There may be several occasions throughout the semester which I
will give unannouced in-class worksheets. The grades from these worksheets can be used to replace
your lowest quiz grades. There will be NO make-up quizzes or worksheets except in the case of a
verifiable medical emergency.

Exams: There will be 4 in class exams and a final exam. Exam questions will be based on lecture
notes and homeworks. There will be NO make-up exams except in the case of a verifiable medical
emergency.

Grades: You can earn up to 650 points which will be determined by the following criteria.

Grades will be assigned as follows:

average
points

Finding Zeros

Answer the following questions by creating sample polynomial equations with the specified characteristics.
Solution Methods for Polynomial Equations of Degree Two (Quadratic Polynomials)

1. Take the Square-Root of Both Sides; give a sample equation in which this method would be appropriate: x² = 9

2. Factor a Trinomial by Grouping; give a sample equation in which this method would be appropriate:

3. Factor the Difference of Two Squares; give a sample equation in which this method would be appropriate:

4. Use the Quadratic Formula; give a sample equation in which this method would be appropriate:

Solution Methods for Polynomial Equations of Degree Three (Cubic Polynomials)

5. Factor by Grouping; give a sample equation in which this method would be appropriate:

6. Factor the Difference of Two Cubes; give a sample equation in which this method would be appropriate:

7. Factor the Sum of Two Cubes; give a sample equation in which this method would be appropriate:

8. Use the Cubic Formula – only as a last resort!

Solution Methods for Polynomial Equations of Degree Four (Quartic Polynomial)

9. Factor by Grouping; give a sample equation in which this method would be appropriate:

10. Factor the Difference of Two Squares; give a sample equation in which this method would be appropriate:

11. Factor by Quadratic Form; give a sample equation in which this method would be appropriate:

12. Use the Quartic Formula – only as a last resort!

General Guidelines for Solving Higher Degree (n > 2) Polynomials

The theorems below allow us to determine the number & type of zeros. Knowing the number & type of zeros helps us narrow our search for zeros. Write a different sample problem to illustrate each of the theorems below.

1. Fundamental Theorem of Algebra or its corollary on
The equation 4x³ + 5x² – 3 = 0 is of degree 3 and has exactly 3 zeros.

2. Odd-Degree Theorem

3. Descartes Rule of Signs

4. Intermediate Value Theorem (aka Graphing Theorem)

5. (Optional) Bounds on Zeros

Next, we use information gained from the theorems above with the techniques listed below to determine the exact value of each zero.

6. Factor if possible and then use the Factor Theorem

Let p(x) = x³ + 2x² – x – 2. By factoring, p(x) = (x + 2)(x + 1)(x – 1). So p(x) has 3 zeros: x = -2, -1, 1.

7. List and test possible zeros from Rational Zeros Theorem

8. Locate and approximate zeros on a graph

9. Use long division to create a depressed equation & solve the depressed equation to locate additional zeros

10. Use the Conjugate Roots Theorem

From Descartes' revolutionary work, La Geometrie, (1638) on the discussion of roots of polynomial equations, we find, without hint of a proof, the rule of signs:

On connoift auffy de cecy combien il peut y auoir de vrayes racines, & combien de fauffes en chafque Equation. A fçauoir il y en peut auoir autant de vrayes, que les fignes + & -- s'y trouuent de fois eftre changés; & autant de fauffes qu'il s'y trouue de fois deux fignes +, ou deux fignes -- quie s'entrefuiuent.

"We can determine also the number of true and false roots that any equation can have, as follows: An equation can have as many true roots as it contains changes of sign, from + to - or from - to +; and as many false roots as the number of times two + signs or two - signs are found in succession."

Two Interesting Finding Zeros Problems:

(1) 4x³ + 4x² – 7x + 2 = 0
Here, Descartes’ Sign Rule suggests 2 positive zeros, but the 2 positives are not distinct, that is, x = 0.5 with multiplicity of 2.
 

2) – 6x³ – x² + x + 10 = 0
Here, there is one positive irrational zero. This irrational zero does not have a conjugate pair. Most irrational zeros in your textbook occur as conjugate pairs.

Generalized Conjugate Roots Theorem (not in your textbook)

For a polynomial with rational coefficients and even degree n, the irrational roots, if any, have even multiplicity or occur in conjugate pairs of the form c ± d where c and/or d are irrational, and the complex roots, if any, occur in conjugate pairs of the form a ± bi.

Example: Determine the roots of the polynomial

By the Rational Zeros Theorem, we find three roots x = -2, -1, and 1.
As a consequence of the Fundamental Theorem of Algebra, there must be a fourth root. Is this fourth root a rational, irrational number, or complex number?
By the Generalized Conjugate Roots Theorem (above), complex roots and irrational roots must occur in pairs. Thus, the single remaining root cannot be complex or irrational. The fourth root must be a rational number, that is, one of the three rational roots must have a multiplicity of 2.

Early History of Algebra: a Sketch

Algebra has its roots in the theory of quadratic equations which obtained its original and quite
full development in ancient Akkad (Mesopotamia) at least 3800 years ago. In Antiquity, this
earliest Algebra greatly influenced Greeks and, later, Hindus. Its name, however, is of Arabic
origin. It attests to the popularity in Europe of HighMiddle Ages of Liber algebre et almuchabole
- the Latin translation of the short treatise on the subject of solving quadratic equations:
(A
summary of the calculus of gebr and muqabala).

The original was composed circa AD 830 in Arabic at the House of Wisdom - a kind of academy
in Baghdad where in IX-th century a number of books were compiled or translated into Arabic
chiefly from Greek and Syriac sources - by some Al-Khwarizmi whose name simply means that
he was a native of the ancient city of Khorezm (modern Uzbekistan).

Three Latin translations of his work are known: by Robert of Chester (executed in Segovia in
1140), by Gherardo da Cremona (Toledo, ca. 1170) and by Guglielmo de Lunis (ca. 1250).

Al-Khwarizmi's name lives today in the word algorithm as a monument to the popularity of his
other work, on Indian Arithmetic, which circulated in Europe in several Latin versions dating
from before 1143, and spawned a number of so called algorismus treatises in XIII-th and XIV-th
Centuries.

During the Middle Ages, Algebra was essentially limited to solving polynomial equations of degree
≤3. The aforementioned treatise of Al-Khwarizmi deals only with quadratic equations. A
native of Nishapur in Persia, Omar Khayyam (1048 -131), in the end of the XI-th Century employed
geometrical methods to solving cubic equations. In Europe, Omar Khayyam is primarily
known as the famous author of the collection of poems, Rubayyat. One of the earliest efforts to
free this early Algebra from relying on geometric methods was due to Al-Karaji (953 -ca. 1029),
whose family seems to have come to Baghdad from the Persian city of Karaj.

A major breakthrough occurred in 1515 when a native of Bologna, Scipione dal Ferro (1465-1526),
solved the cubic equation algebraically. Another Bolognese, Lodovico Ferrari (1522-1565), in 1540
discovered the solution of the quartic equation.

Apart from isolated attempts, like the use of letters to denote numbers by Jordanus de Nemore
(1225-1260), or the introduction of symbols + and − by JohannesWidman (1462-1498) in 1489,
no systematic “algebraic” notation was used by early algebraists before the XVI-th Century. Instead
of symbols and equations they have been using descriptive sentences. What we call today
“algebraic” notation makes its entry intoMathematics in the treatise In artem analyticamisagoge
published in 1591 in Tours by (1540-1603), a native of Vandèe (Western France).

Modern Algebra commences with the publication in 1830 of Treatise on Algebra by George Peacock
(1791-1858). Within the next hundred years Algebra becomes a theory of mathematical
structures.

Mathematics & Economics: Connections for Life

9:00 Introductions/ Making the Case for Mathematics and Economics

9:30 Lesson 10: Powerball Economics

Math Prerequisites : compute basic probabilities, calculate combinations
Math & Econ Terms: fair game, expected returns

10:30 Break

10:45 Breakout Sessions

Group 1
Lesson 1: The Nature of Demand Curve
Math Prerequisites : make a graph from a table of values, equation of a line
Math & Econ Terms: demand curve, inverse relationship, law of demand

Lesson 4: Understanding the Mathematics of Changes in Supply & Demand
Math Prerequisites: linear equations, equation of a parallel line
Math & Econ Terms: equilibrium, demand, change in supply, translation

Group 2
Lesson 4: Understanding the Mathematics of Changes in Supply & Demand
Math Prerequisites: linear equations, equation of a parallel line
Math & Econ Terms: equilibrium, demand, change in supply, translation

Lesson 8: Mathematics of Nonlinear Economic Shapes: The Cubic Cost Function
Math Prerequisites: Evaluate and graph cubic and quadratic functions
Math & Econ Terms: law of diminishing marginal returns, marginal cost,
variable cost, fixed cost, total cost

12:15 Lunch

12:45 Breakout Sessions

Group 1
Lesson 12: Autonomics
Math Prerequisites: calculate percentage changes, proportions, and ratios
Math & Econ Terms: fixed cost, implicit cost, opportunity cost

Lesson 14: The Mathematics of Savings
Math Prerequisites: simple and compound interest formulas
Math & Econ Terms: future value, present value, interest compounding

Lesson 13: Tax Math
Math Prerequisites: calculate rates, percentages, and proportions
Math & Econ Terms: proportional, regressive, and progressive taxes

Group 2
Lesson 7: Math of Nonlinear Economic Shapes: Production Possibilities Curve
Math Prerequisites: first derivative of a second-degree equation, concavity
Math & Econ Terms: scarcity, opportunity cost, increasing costs

Lesson 9: Profit Mathematics
Math Prerequisites: quadratic equations, first and second derivatives
Math & Econ Terms: marginal revenue, marginal cost, profit

2:45 Wrap-up/Evaluation

3:00 Adjourn

Workshop is made possible by a grant from State Farm Insurance to the National Council on Economic Education.