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MATH 101: BEGINNING ALGEBRA
COURSE KIT
August 2011
Learning Outcomes: The student satisfactorily completing Math 101 will:
1.
2.
3.
4.
5.
6.
Perform operations with real numbers and algebraic expressions.
Solve linear equations, inequalities, and formulas for specified variables.
Graph linear equations and determine the equation of a line.
Simplify, evaluate, and perform operations with polynomials.
Factor polynomials and solve quadratic equations by factoring.
Solve application problems involving the procedures and techniques detailed
above.
Original Course Kit Authors/Editors: Habib Aghdami, Mark Gollwitzer, Debbie Moran, Hala
Nestberg, William Parker, Judy Walden, and Bonnie Mullinix
The development of this Course Kit was facilitated through the Unlock Your Future Initiative and funded
under Title III grant [insert grant #]
Math 101: Beginning Algebra Course Kit
pg 2
Introduction to the Course Kit
This Course Kit has been developed for use by instructors teaching Beginning Algebra at
Greenville Technical College with the aim of helping students attain the learning outcomes
identified above. It contains all instructional materials needed to conduct active, applied and
practice-oriented lessons including:





lesson plans (for all primary instructional sessions and sample review lesson plans)
activities & task sheets and
supplemental materials (links to videos, presentations, websites)
course syllabus
course outline(s) (with alternate structures)
While originally developed independent of specific learning resources (texts, software…), this
Course Kit has been adapted to reference the materials selected through the course redesign
and pilot for use beginning Fall 2011:
Wright (2011). Developmental Mathematics. Hawkes Learning Systems;
Hawkes Learning Systems Software and the
Multiview TI30-XS Multiview calculator
This Course Kit represents the collaborative work of full-time and part-time Math instructors
working together to review effective teaching/learning practices in mathematics and develop,
pilot and document our best practices. The core team for this working on both Math 101 and
Math 102 Curricular revision from Fall 2009 – Fall 2011 included: William Parker (team lead),
Habib Aghdami, Mark Gollwitzer, Judy Walden (implementation lead), Hala Nestberg and
Debbie Moran. In addition, during the first year this team benefited from contributions from
Jean Steele, Toi Graham (lead 09), Mary Ann Billig, and Lucius Pinkney. This is the product of
their dedicated and committed work.
Contributing to the Course Kit – The instructional power of this course kit comes from the fact
that it is built on the wisdom of practice from many Greenville Tech math instructors combining
their decades of experience on what works with GTC students. This Kit is not finished. We
need your contributions. From these we will build a kit with multiple alternative and equivalent
activities that can be interchanged and adapted to meet the distinct needs of various classes of
students. If you have a favorite activity that you believe would fit within this curriculum, please
propose it by ______________________________
Math 101: Beginning Algebra Course Kit
pg 3
Table of Contents
Math 101 Course Outline with Lesson Plan/Activity Links
General Daily Lesson Plan with Activity/Teaching Ideas
Unit 1: Solving Linear Equations
Unit 1 Overview - Teaching Insights & Strategies
1a. Course Introduction & Basic Math Review
1b. Simplifying Algebraic Expressions
1c. Solving Linear Equations in One Variable
1d. Solving Absolute Value Equations
1e. Applications of Linear Equations
Unit 2: Linear Inequalities; Graphing Linear Equations in Two Variables
Unit 2 Overview - Teaching Insights & Strategies
2a. Solving Linear Inequalities
2b. Cartesian Coordinate System
2c. Graphing Linear Equations in Two Variables
2d. Graphing: Slope-Intercept Form
2e. Graphing: Point-Slope Form
2f. Applications of Two Variable Equations
Unit 3: Systems of Linear Equations; Exponents
Unit 3 Overview - Teaching Insights & Strategies
3a. Systems of Linear Equations: Solve by Graphing & Substitution
3b. Systems of Linear Equations: Solve by Addition
3c. Applications of Linear Systems
3d. Exponents
Math 101: Beginning Algebra Course Kit
Unit 4: Polynomials; Factoring
Unit 4 Overview - Teaching Insights & Strategies
4a. Introduction to Polynomials & Operations
4b. Operations on Polynomials – Multiplying
4c. Operations on Polynomials - Division
4d. Greatest Common Factor, Factor by Grouping
4e. Factoring Trinomials
Review Sessions
Team-Based Jeopardy Game Review Session for Final (or Test)
Review Session for Test – Option 1 (Standard – Online Test)
Review Session for Test – Option 2 (Standard – In-Class Test)
Appendices
Appendix A: Syllabus - Math 101 Beginning Algebra
Appendix B: Course Outlines
Appendix C: Activities
pg 4
Math 101: Beginning Algebra Course Kit
Course Outline
pg 5
Math 101 Beginning Algebra
Lesson Plans with Activities across 30 sessions
Session/
Meeting
Lesson
Plan
1
1a
Introduction
Basic Math Review
2
1b
Unit 1 – Solving Linear Equations
Simplifying and Evaluating Algebraic Expressions
Translating English Phrases and Algebraic Expressions
3
1c
4
1c cont.
1d
5
1c
6
7
8
1d
Review
9
2a
10
11
12
13
14
15
16
2b
2c
2d
2e
2f
Review
17
3a
18
3b
19
3c
20
3d
21
22
Review
23
4a
24
4b
25
26
4c
4d
4d
4e
27
28
29
30
Review
Focus
Solving Linear Equations in One Variable
Solving Linear Equations in One Variable
Solving Absolute Value Equations
Applications of Linear Equations:
Number Problems, Consecutive Integers
Working with Formulas (primarily solving formulas for different variables)
Applications of Linear Equations: Distance-Rate-Time, Interest, Average
Review for test on Unit 1
Unit 1 Test
Unit 2 – Linear Inequalities; Graphing Linear Equations in Two Variables
Solving Linear Inequalities in One variable
Solving Absolute Value Inequalities (simplest cases)
The Cartesian Coordinate System
Graphing Linear Equations in Two Variables
Graphing: Slope-Intercept Form y = mx + b
Graphing: Point-Slope Form y – y1 = m(x – x1)
Applications of Two Variable Equations
Review for test on Unit 2
Unit 2 Test
Unit 3 – Systems of Linear Equations; Exponents
Systems of Linear Equations: Solve by Graphing
Systems of Linear Equations: Solve by Substitution
Systems of Linear Equations: Solve by Addition
Applications of Linear Systems:
D=RT, Numbers, Amounts, Costs, Interest and Mixture
Exponents
& Scientific Notation
Review for test on Unit 3
Unit 3 Test
Unit 4 – Polynomials; Factoring
Introduction to Polynomials
Adding and Subtracting with Polynomials
Operations on Polynomials - Multiplying
Special Products of Binomials
Operations on Polynomials - Division
Greatest Common Factor, Factor by Grouping
continue - Greatest Common Factor, Factor by Grouping
Factoring Trinomials
Review for test on Unit 4
Unit 4 Test
Review for final exam
Final Exam
Text Section
Assignments / Notes
7.1 – 7.6 Review
7.7 Note: Continue review
of 7.1 – 7.6 concepts while
covering 7.7
7.8
8.1 : x + b = c and ax = c
8.2: ax + b = c
8.3: ax + b = cx + d
Appendix A.4:
8.4
8.5
8.6
Unit 1 Test
8.7
Appendix A.4:
9.1
9.2
9.3
9.4
Supplemental material
Unit 2 Test
10.1
10.2
10.3
10.4
10.5
11.1
11.2
Unit 3 Test
11.3
11.4
11.5
11.6
11.7
12.1
12.1 continued
12.2: x2 + bx + c
Unit 4 Test
Math 101: Beginning Algebra Course Kit
pg 6
Developmental Studies
Lesson Plan
General Daily Lesson Plan
Initial Lesson by: Judy Walden
(& Bonnie Mullinix & Team)
Course/Unit Focus: Math 101
Lesson: All
Primary Course Outcome(s): changes with specific lesson
Learning Objectives: By the end of the session, learners will:
1.
3.
Identify the topic and its relationship to previous and upcoming topics
2. Note: Additional Objectives change for each lesson
Practice applying the math skills associated with the lesson topic.
Materials: Smart board / white board (Computer lab, where needed and other materials as required)
Duration: Length of a class period (example: 75 min)
Time
5 min
Description/Activity
Prior to class, write on the board what section(s) will be covered today and what will be
covered in the next class period.
Suggested: Post an entry activity or problem that students can do as they enter the class.
5 min
Introduce the focus and topic of the lesson. Ask students what was done last time and
describe where this lesson falls in the course (relationship to previous topics &/or upcoming
topics). Give (/ask for) an example of why this topic is important (real-life applications).
Make any announcements that need to be made about upcoming quizzes, team/group
responsibilities, etc.
[Note: When the announcements are important, remember to update Blackboard and/or email the
students as well.]
5-10
min
Review of problems from the previous class.
Use varied structure/activities to involve and assess students (see below for ideas & vary what you
do from class to class, add your own)
50-55
min
5 min
Incorporate a combination of techniques and interactive activities to involve students in
actively developing math skills, solving problems, working in groups, experiencing concepts,
using a variety of learning approaches and styles. Intersperse with brief lecturettes or
discussions to highlight key points and/or correct misconceptions.
Sum up the important points of the lesson. Restate what will be covered in the next lesson and
what assignments students should be working on between classes. Announce any upcoming
tests, quizzes or projects.
Math 101: Beginning Algebra Course Kit
pg 7
Guide & Ideas for Effective Math Lessons:
Preparation & Board Use



At the beginning of each chapter/section, write the chapter number and name and/or section number
and name on the board.
Write each new term and definition on the board.
If the instructions for a problem are not obvious from the problem itself, then write the instructions on
the board as well.
Review of Last Lesson
Use varied structures/activities to involve and assess students’ knowledge and abilities (see below for ideas & vary
what you do from class to class, add your own)



List review problems on board prior to class, or
Give a quiz (individually, in groups; graded or ungraded)
Give a quiz online (for homework/in class (lab); identify and work with the problems that gave the most
trouble
After the students have had time to work the problems, review responses by:





Collecting or peer grading quiz & discuss, or
Working problems, showing answers on the board, or
Having students work problems on the board, or
Have students share/compare their results with classmates (such as in a group),
“Think/pair/share” – have them think or work the problem. After they have had time, have them pair up
and share their results with their partner, explaining their thinking & problem solving strategies
(instructor rotates among pairs listening/helping where needed)
Class Activities
Use a variety of activities in your class to strategically involve students in actively developing math skills, solving
problems, working in groups, experiencing concepts, using a variety of learning approaches and styles. Use a
combination of teaching and learning techniques and select activities that are well-matched to the topic and focus
of the lesson. Design your lessons to ensure that students become involved in learning rather than simply taking
notes on a lecture. Intersperse brief lecturettes or discussions to highlight key points and/or correct
misconceptions.
Selecting Activities
There are many activities that can be selected and techniques that can be used. Select activities that
relate to the lesson topic/focus:



Small Groups - Have students work a similar problem at their desk. Encourage them to work with
a partner
Teams - Assign working teams (of 3-4 students) to work together. Build teams based on varying
levels of expertise (using early diagnostics or other early quiz or placement information). Use
these teams regularly and/or shift teams part way through the semester
Jigsaw – Put students into groups where they first work a problem (master it) and then in
another group (with all problem types represented) where they share their problem and learn
how to solve the others.
Math 101: Beginning Algebra Course Kit


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pg 8
Peer teaching – have students (in pairs or groups) responsible for preparing for and introducing
new problems to the class (rotating responsibility). Allow students to select which topic they will
be responsible for. Provide some class time and/or support for preparing to present.
Projects – Groups/Teams are given a project to complete that demonstrates their knowledge of
key skills and concepts
Online Software – Individuals work on self-paced activities using online learning software.
Many More – insert your ideas for activities here (and share with others!)
After the students have completed an activity, make sure to “process” and discuss it to ensure that:



Answers and solutions to problems are clear
Strategies for working through and solving challenging problems are understood by everyone
Questions get answered.
Use “Lecturettes” Effectively
A lecturette is a short lecture of no more than 15-20 minutes that are targeted to a specific topic or
concept. They can be as short as 2-5 minutes and remain particularly effective in guiding students from
one step to the next. Lecturettes can be used to introduce a topic, clarify a common/shared
misconception (while working on problems), or to consolidate, highlight and underscore concepts or key
points following an activity. Whether they come before, during or after activities, use them strategically
and pointedly to support the other activities. As you deliver a lecturette use visuals, the board, text and
examples to help students gather complete and relevant notes that will guide them on their next step.
(Note: Students have different learning styles and not all students will remember things that are
communicated only verbally). Provide a clear and organized presentation that helps students see the
connection between ideas shared and the lesson topics/focus.
Review for Test
Include multiple ways for students to review for tests. Just as above, activities including lecturettes Use
several and/or change what you do:
 Use Review Games (e.g. Jeopardy game)
 Have students work sample problems (individually and/or in groups), asking questions of each
other or the instructor
o Write a list of problem numbers from the chapter review.
o Invite students to write & submit test problems (and answers)
 Provide incentives:
o Students may leave early if they finish all problems (individually or in groups) and show
them to the instructor to confirm accuracy.
o Agree to use selections from the best problems submitted by students in the test
o Provide participation points
Conclusion
Use your last few minutes of class to sum up the important points of the lesson and remind students of what
they’ve done, where they are, and where they are headed. Gather any feedback about the class from students
(“muddiest points”, how to clarify, additional needs). Mention what will be covered in the next lesson and what
assignments students should be working on between classes. Announce any upcoming tests, quizzes or projects.
Math 101: Beginning Algebra Course Kit
pg 9
See Unit 4 for example
Developmental Studies
Unit Overview - Teaching Insights & Strategies
Unit Overview - Teaching Insights & Strategies
Unit 1: Teaching Linear Equations
Submitted by:
Judy Walden &
Mark Gollwitzer
Purpose of Unit: This unit reviews simple arithmetic with signed numbers and solving linear equations
and its applications.
Where the Students are: Many of the students will be familiar with the concepts in this unit but don’t
let them get complacent. The average student will still be having problems with fractions, and many of
them don’t like doing math with the alphabet. This is a good time for them to perfect their skills.
Where to Begin: Begin with a review of operations with real numbers, then bring in the initial concepts
of linear equations.
Connect to Previous Work: This unit is a review of skills they should have learned if they took MAT 031,
MAT 032, or tested into MAT 101. Any review you can provide will be helpful, but it would be good to
remind the students that these beginning sections may not be covered in detail. Students will be
required to certify in the sections relevant to MAT 101.
Challenges: Many students will want to work the problems in their head. We can usually cure them of
this if we remind them that it is easy to get lost in problem solving. If they work the problems in their
head the problems will eventually get so complicated they won’t be able to solve them. When this
happens they will want to write something down but they won’t know where to begin. If on the other
hand they start solving the simple problems on paper and writing out each step of the problem they will
be building a strong foundation for their problem-solving practice and won’t notice as the problems get
harder.
Closure: Remind the students that math builds upon itself. Each time they learn new concepts, these
concepts are building blocks for the next section and next chapter. They need to learn each set of
concepts well so as to have a solid foundation to building upon.
Math 101: Beginning Algebra Course Kit
pg 10
Developmental Studies
Lesson Plan
Lesson 1a: Course Introduction & Basic Math Review
Initial Lesson by: Bonnie Mullinix
Course/Unit Focus: Beginning Algebra (Math 101), Solving Linear Equations
Primary Course Outcome(s): 1. Perform arithmetic operations with real numbers and algebraic
expressions.
Learning Objectives: By the end of the session, learners will:
1.1.
1.2.
Review the overview, structure and requirements of the course;
Define and identify key terms: variable, term, constant, coefficient, and an algebraic
expression;
Practice simplifying expressions by combining like terms.
1.3.
Materials: Whiteboard or interactive board notebook and pencil worksheet (provided) Activity Sheets
& grid paper
Duration: 1 class (75 min)
Key Concepts & Skills:



basic operations with real numbers
order of operations
operations with real numbers
Time
Description/Activity
10-20 min
Introduction: Instructor introduces the students to the course, referencing the one-page
summary and flowchart “Graphic Syllabus” for Math 101 to focus them on the math path they
will be pursuing.
Instructor points students to Blackboard, Text and software they will be using and demonstrates
where to find the full syllabus and encourages them to print and review the syllabus in detail.
5-10 min
Quick Kwiz Pt I: Instructor displays and distributes a quick quiz on real number operations (see
attached), explaining that the purpose of this “kwiz” is to check and develop their
understanding of basic math. So they will complete this alone and then share/compare their
answers with students and self-assess their work and understanding.
Students complete the quick kwiz individually.
Math 101: Beginning Algebra Course Kit
15-25 min
pg 11
Quick Kwiz Pt II: Students form groups of 3-4, introduce themselves and share and compare
their answers, agreeing on the “right answer” within their group.
Groups then ‘uncover’ the right answer using the IF-AT scratch-off answer sheets (or instructorled iClicker use). Instructor rotates among the groups and observes progress and patterns of
understanding, answering questions where appropriate.
Instructor asks students to score their kwiz in the following way:
1 pt for each individual correct answer
½ pt for each group correct answer
next to each answer they now understand (from group work)
? next to each answer they don’t understand
5-10 min
Kwiz Check: Instructor asks groups to identify all questions with a question mark next to them.
Instructor uses these responses to focus on specific trouble problems, clarify steps, provide
additional problems to try out, and demonstrate solutions.
If needed for review: Instructor introduces acronyms to underscore appropriate order of
operations – asking students to help identify what each stands for and ways to remember these
orders:



Complete the following acronyms:
PEMDAS _______ _______ _______ _______ _______ _______
BEDMAS _______ _______ _______ _______ _______ _______
Hint: As they relate to the Order of Operations
Write down how are they different? [2 min]
Share/compare your thinking with a partner [2 min]
Instructor invites students (or a student) to write their answers in the spaces as projected on
the SmartBoard, and then projects the answers on top of what they have written and invites
them to note the differences.
5-10 min
5-10 min
If time allows, instructor provides additional problems (from text or software and/or on board)
and students work in groups to solve them (to demonstrate and solidify their understanding of
real number operations). Otherwise, problems may be assigned for homework.
Summary – If necessary, instructor provides a brief lecturette to reinforce the key points of the
lesson, address objectives/topics covered, assign homework and note focus of the next lesson.
Homework Assignment: Log into Blackboard, Review syllabus (identify questions), Get text and
complete the basic diagnostic in the online math software.
[Back to Table of Contents]
Assessment Strategies/Comments: Quick Quiz covering relevant problems.
attached]

Complete the following acronyms:
[Need sample
Math 101: Beginning Algebra Course Kit
pg 12
PEMDAS
Parentheses
Exponents
Multiply Divide
Add
Subtract
BEDMAS
Brackets
Exponents
Divide
Add
Subtract
Remember by:

Please Excuse My Dear Aunt Sally

Big Elephants Destroy Mice And Snails

Pink Elephants Destroy Mice And Snails
Multiply
Math 101: Beginning Algebra Course Kit
pg 13
Review of Operations on Real Numbers
Order of operations: When you have multiple operations, use the following priority:
P – parentheses – Within grouping symbols, use order of operations.
(i.e., grouping symbols also including brackets, absolute value, fraction bars)
E – exponents
MD – multiplication and division – Perform these left to right.
AS – addition and subtraction – Perform these left to right.
Many people remember this as:
Please Excuse My Dear Aunt Sally.
Example:
2 + 3(4 − 1 ∗ 7)
2 + 3(4 − 7)
2 + 3(−3)
2 + (−9)
−7
Types of Numbers
Natural numbers
Whole numbers
Integers
Rational numbers
Also called counting numbers
Natural numbers plus zero
Whole numbers plus negatives
(typically, all tick marks on a number line)
Numbers that can be written as a fraction of
integers
Note: “ratio” in rational numbers
1, 2, 3, 4, 5, …
0, 1, 2, 3, 4, 5, …
…, -3, -2, -1, 0, 1, 2, 3, …
All integers, plus fractions,
terminating and repeating decimals
2
Irrational numbers
Real numbers
Numbers that cannot be written as a fraction of
integers
If written as a decimal, the digits never repeat.
All numbers that are either rational or irrational.
Ex: −4, 3, 1, 0, − , −.5,
3
.333333…
Ex: √6, 𝜋
You can find all of these numbers on a real
number line.
Operations with Real
Numbers
Exponent
Absolute value
The number of times a number (base) is
multiplied by itself
Distance of a number from zero.
43 = 4 ∙ 4 ∙ 4 = 64
|2| = 2
|−4| = 4
−|−3| = −3
Adding Real Numbers
With same signs
With different signs
Just add. Result has the same sign.
Subtract the absolute values (ignore
signs). Result has same sign as number
−3 + (−9) = −12
−5 + 2 = −3
Math 101: Beginning Algebra Course Kit
pg 14
−5 + 7 = 2
with largest absolute value.
−(−𝑎) = 𝑎
−(−6) = 6
Same thing as adding the negative of the
number.
−5 − 9 = −5 + (−9) = −14
Double negative rule
Subtracting Real Numbers
−1 − (−2) = −1 + [−(−2)]
= −1 + 2 = 1
Multiplying Real Numbers
Times zero
Same signs
Opposite signs
Dividing Real Numbers
Zero
−4 ∙ 0 = 0
−2 ∙ (−4) = 8
−3 ∙ 3 = −9
Any number times 0 = 0
Result is positive
Result is negative
Divide by 0: Result is “undefined”
Divide 0 by non-zero: Result is zero.
Helpful hint to remember which is which:
𝑁 𝑂
,
𝑂
−4
0
undefined
0
=0
−2
𝐾
Same signs
Result is positive
Opposite signs
Result is negative
−10
=5
−2
−12
= −3
4
25
= −5
−5
Properties of Real
Numbers
Commutative Property
(addition, multiplication)
Order doesn’t matter
(−2) + 9 = 9 + (−2)
(−2) ∙ 9 = 9 ∙ (−2)
Associative Property
(addition, multiplication)
Identity Property
(addition, multiplication)
Inverse Property
(addition, multiplication)
Grouping doesn’t matter
If you perform an operation using
an identity, nothing changes
−5 + 0 = −5
0 is additive identity
1 is multiplicative identity
−9 ∙ 1 = −9
If you perform an operations using
inverses (or opposites), you get
the identity.
−9 + 9 = 0
The multiplicative inverse has another
name: reciprocal.
Distributive Property
[(−2) + 9] + (−1) = −2 + [9 + (−1)]
When you have a number times a
sum or difference, you can
distribute.
1
−9 ∙ (− ) = 1
9
9(𝑎 + 5) = 9𝑎 + 45
−2(𝑏 − 7) = −2𝑏 + 14
−(𝑎 − 2) = −𝑎 + 2
Math 101: Beginning Algebra Course Kit
pg 15
[Back to Table of Contents]
Developmental Studies
Lesson Plan
Lesson 1b: Simplifying Algebraic Expressions
Initial Lesson by: Habib Aghdami
Course/Unit Focus: Beginning Algebra (Math 101), Solving Linear Equations
Primary Course Outcome(s): 1. Perform arithmetic operations with real numbers and algebraic
expressions.
Learning Objectives: By the end of the session, learners will:
1.4.
1.5.
1.6.
1.7.
Identify a variable, term, constant, coefficient, and an algebraic expression.
Practice simplifying expressions by combining like terms.
evaluate algebraic expressions
translate phrases from words to algebraic expressions
Emphasize:





terminology
how to identify like terms
use of operations on negative numbers
use of properties of real numbers
how to use Hawkes Learning System and importance of homework
Materials: Whiteboard or interactive board notebook and pencil worksheet (provided) Activity Sheets &
grid paper
Duration: 1 class (75 min)
Time
5 min
5 min
5-10 min
Description/Activity
Intro – Instructor introduces session noting: This session will use Properties of Real
Numbers and Operations with Real Numbers. It will also lead to Solving Linear
Equations in the following sections. He will also note that today’s class will be taught
using Hawkes in order to get the students proficient in the software. Future classes
may also be taught using Hawkes.
Instructor will open up the Hawkes lesson corresponding to Variable and Expressions.
Then he will use the Instruct button to define each concept and show examples.
Instructor should ask for one volunteer to do a couple of problems using the Practice
button on Hawkes. The rest of the students should help come up with the answer
Math 101: Beginning Algebra Course Kit
Time
5 min
5-10 min
5 min
5-10 min
5 min
5-10 min
pg 16
Description/Activity
and the volunteer will key it in. The instructor should take the time to show the
Tutor button and how it works.
Instructor will open up the Hawkes lesson corresponding to Simplifying Expressions.
Then he will use the Instruct button to define each concept and show examples.
Instructor should ask for one volunteer to do a couple of problems using the Practice
button on Hawkes (at least one problem with a squared variable and another with
Distributive Property). The rest of the students should help come up with the answer
and the volunteer will key it in.
Instructor will open up the Hawkes lesson corresponding to Evaluating Expressions.
Then he will use the Instruct button to define each concept and show examples.
Instructor should ask for one volunteer to do a couple of problems using the Practice
button on Hawkes (preferably problems with negative values for variables). Have
the rest of the students work in small groups to come up with the answer and the
volunteer will key it in.
Instructor will open up the Hawkes lesson corresponding to Translating Phrases to
Expressions. Then he will use the Instruct button to define each concept and show
examples.
Instructor should ask for one volunteer to do a couple of problems using the Practice
button on Hawkes (preferably problems with negative values for variables). Have
the rest of the students work in small groups to come up with the answer and the
volunteer will key it in.
Math 101: Beginning Algebra Course Kit
pg 17
Simplifying Algebraic Expressions
Activity Sheet
Here are vocabulary terms used in algebra that are important to learn.
Term: An algebraic term is either a number or a number multiplied by one or more variables.
52x2 - 9x + 36 = 7m + 82 Each green item is a separate term.
Expression: An algebraic expression is made up of one or more algebraic terms. Expressions do
not have equal signs.
52x2 - 9x + 36 = 7m + 82 Each blue item is a separate expression.
Coefficient: A coefficient is the number part of a term with variables.
52x2 - 9x + 36 = 7m + 82 Each red item is a separate coefficient.
Variable: In algebra, letters represent variables.
52x2 - 9x + 36 = 7m + 82 Each grey item is a separate variable.
Constant: Terms with no variables are called constants. They are constants because their value
is constant - it never changes.
52x2 - 9x + 36 = 7m + 82 Each purple item is a separate constant.
Equation: An equation is a statement that two expressions are equal. An equation always has
an equals sign ( = ).
52x2 - 9x + 36 = 7m + 82 3 + 5y = 8 Examples of equations
Math 101: Beginning Algebra Course Kit
pg 18
Simplify algebraic expressions by combining like terms.
Like terms are terms with the same variable or variables.
Like terms
Unlike terms
All constants are like terms.
Constants and variables are not like
terms.
2, -34, 0.59, ¼
18, x
All these terms have one variable, y.
Terms with different variables are
not like terms.
3y, -10y, y, 6y
7x, 7y
All these terms have the same
variable, t3.
Terms with the same variable but
different exponents are not like
terms.
t3, 9t3, ½t3, -14t3
10y, 3y2
All these terms have the same
variable combination, xy.
Terms with different variable
combinations are not like terms.
29xy, 0.5xy, xy, -xy
x, 8xy
Example 1: One variable
-x - 6 + 5x Simplified:
-x and 5x are like terms, so combine them.
-x - 6 + 5x = 4x - 6
The answer is 4x - 6
Example 2: Two variables
3y + 4 - y + 4x - 6 Simplified:
Combine the x and y terms and the constants.
3y + 4 - y + 4x - 6 =
4x + (3y - y) + (4 - 6) =
4x + 2y + -2 =
Math 101: Beginning Algebra Course Kit
pg 19
This simplifies to 4x + 2y - 2
The answer is 4x + 2y - 2
Example 3:
Simplify the expression and order your answer based on alphabetical letter and magnitude.
4x2 - 3y + x - 2x2 - 2 - 3y + 7 Simplified:
Combine the terms and simplify.
4x2 - 3y + x - 2x2 - 2 - 3y + 7 =
(4x2 - 2x2) + x + (-3y - 3y) + (-2 + 7) =
2x2 + x + (-6y) + 5 =
This simplifies to 2x2 + x - 6y + 5
The answer is 2x2+ x - 6y + 5
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 20
Developmental Studies
Lesson Plan
Lesson 1c: Solving Linear Equations in One Variable
Initial Lesson by: Judy Walden
Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities
Primary Course Outcome(s): 2. Solve linear equations, inequalities, and formulas for specified variables.
Learning Objectives: By the end of the lessons, learners will:
1.8.
1.9.
1.10.
1.11.
1.12.
Identify linear equations in one variable.
Use the addition and multiplication properties of equality to solve linear equations
Solve more complicated linear equations that require simplication of expressions and both
addition and multiplication properties of equality
Solve linear equations that contain fractions by clearing fractions.
Identify contradictions, identities, and conditional linear equations and state the solution set
for each.
Emphasize:






The difference between solving an equation and simplifying an expression.
What the solution of a linear equation means.
The importance of writing down each step (i.e., not skipping steps, not solving equations in your
head)
The importance of checking your answer.
Do not divide each side of an equation by a variable.
The difference between how many solutions there are versus what the solutions are.
Duration: 1.5 classes (115 min)
Time
5 min
10 min
5-10 min
Description/Activity
Intro: Homework challenges discussed and/or collected. Instructor introduces lesson
describing its importance, relationship to algebra and key concepts and skills to be
addressed.
Instructor opens up the Hawkes lesson corresponding to Solving Linear Equations:
x+b=c and ax=c. Then uses the Instruct button to define each concept and show
examples.
Instructor should ask for one volunteer to do a couple of problems using the Practice
button on Hawkes The rest of the students should work in small groups to help come
up with the answer and the volunteer will key it in. The instructor will show the work
on the board.
Math 101: Beginning Algebra Course Kit
Time
10 min
10-15
min
15 min
20-25
min
5-10 min
pg 21
Description/Activity
Instructor opens up the Hawkes lesson corresponding to Solving Linear Equations:
ax+b=c. Then uses the Instruct button to define each concept and show examples.
Instructor asks for one volunteer to do a couple of problems using the Practice button
on Hawkes The rest of the students work in small groups to help come up with the
answer and the volunteer will key it in. The instructor shows the work on the board.
Instructor opens up the Hawkes lesson corresponding to Solving Linear Equations:
ax+b=cx+d. Then uses the Instruct button to define each concept and show examples.
Instructor asks for one volunteer to do a couple of problems using the Practice button
on Hawkes The rest of the students should work in small groups to help come up with
the answer and the volunteer will key it in. The instructor will show the work on the
board.
Summary - Instructor provides a brief lecturette to reinforce the key points of the lesson
and foreshadow the next lesson and assigns homework (as needed)
Assessment Strategies/Comments: Homework to be assigned in Tutorial software
Activity: Algebra Balance Scales at the following link:
http://nlvm.usu.edu/en/nav/category_g_4_t_2.html
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 22
Developmental Studies
Lesson Plan
Initial Lesson by: Hala Nestberg
Lesson 1d: Solving Absolute Value Equations
Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities
Primary Course Outcome(s): 2. Solve linear equations, inequalities, and formulas for specified variables.
Learning Objectives: By the end of the lessons, learners will:
1.13.
1.14.
1.15.
1.16.
1.17.
1.18.
1.19.
Identify linear equations in one variable.
Use the addition and multiplication properties of equality to solve linear equations
Solve more complicated linear equations that require simplication of expressions and both
addition and multiplication properties of equality
Solve linear equations that contain fractions by clearing fractions.
Identify contradictions, identities, and conditional linear equations and state the solution set
for each.
Solve absolute value linear equations.
Graph the solution set for absolute value equations.
Emphasize:








The difference between solving an equation and simplifying an expression.
What the solution of a linear equation means.
The importance of writing down each step (i.e., not skipping steps, not solving equations in your
head)
The importance of checking your answer.
Do not divide each side of an equation by a variable.
The difference between how many solutions there are versus what the solutions are.
The interpretation of the absolute value of a number as a distance from zero on a number line.
Graphing on a number line the solution set for an absolute value equation.
Duration: .5 classes (35 min)
Time
5 min
10 min
10 min
Description/Activity
Transition: Instructor reviews what absolute value of a number means, finding the
absolute value of a number, and evaluating absolute value expressions for a given
value of a number
Instructor opens up the Hawkes lesson corresponding to Solving Absolute Value (A.4)
and uses the Instruct button to define each concept and show examples.
Have students work on at least two practice examples.
Math 101: Beginning Algebra Course Kit
Time
5 min
5 min
pg 23
Description/Activity
Ask each student to team up with one other student. Have them check their answers
with each other and explain how they came up with the solutions. Ask students to
explain to each other the rationale.
Summary - Instructor reiterates steps and rationale and that the � symbol and concept
are useful in cases where how far values are from each other regardless of which is
smaller than the other.
Assessment Strategies/Comments: Homework to be assigned in Tutorial software
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 24
Developmental Studies
Lesson Plan
Lesson 1e: Applications of Linear Equations
Initial Lesson by: Judy Walden
Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities
Primary Course Outcome(s): 2. Solve linear equations, inequalities, and formulas for specified variables.
Learning Objectives: By the end of the lessons, learners will:
1.20.
1.21.
Read and solve different kinds of applications using algebraic equations.
Solve formulas for different variables
Emphasize:



Defining what the variable represents in each application problem.
Using algebra to solve an application problem, not arithmetic.
Solving formulas for different variables uses the same principles as solving linear equations.
Duration: 2 class periods (2.5 hours)
Time
5 min
15 min
20 min
15 min
20 min
Description/Activity
Intro: Homework challenges discussed and/or collected. Instructor introduces lesson
describing its importance, relationship to algebra and key concepts and skills to be
addressed.
Instructor describes the steps to solving an application problem. Then selects a sample
problem and goes thru these steps.
Instructor goes to Practice button on Hawkes application section for number and
consecutive integer problems, picks one sample of each type of problem, and works
step by step to get the answer.
Instructor finds another of one of these types of problems in Hawkes and has students
work in groups to try to solve using algebra. After appropriate time, the instructor will
review the problem.
Instructor shows how to solve a formula for a different variable. Then goes to Hawkes
Practice and have the students work in groups to solve. After appropriate time, the
instructor will review the problem.
Math 101: Beginning Algebra Course Kit
Time
20 min
15 min
5-10 min
pg 25
Description/Activity
Instructor goes to Practice button on Hawkes application section for distance, interest,
and average, picks one sample of each type of problem, and works step by step to get
the answer.
Instructor finds another of one of these types of problems in Hawkes and has students
work in groups to try to solve using algebra. After appropriate time, the instructor will
review the problem.
Summary - Instructor provides a brief lecturette to reinforce the key points of the
lesson and foreshadow the next lesson and assigns homework (as needed)
Assessment Strategies/Comments: Homework to be assigned in Tutorial software
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 26
Developmental Studies
Unit Overview - Teaching Insights & Strategies
Submitted by:
Mark Gollwitzer
Unit 2: Teaching Linear Inequalities & Graphing Linear Equations in Two Variables
Purpose of Unit: In this unit we will be introducing the students to algebraic visualization, also known as
graphing. This is probably the single most important unit in the development of mathematicians. Our
goal is to explore the relationships between the equations, and their visual representations. If we do this
properly the student should begin to understand how, as mathematicians we look at real world
problems and find suitable equations that can produce solutions to those problems. If we give them a
descent gimps into algebraic visualizations, they should understand the need to analyze and graph
functions in future classes.
Where the Students are: When the students get to this chapter they will be in relatively good spirits as
it is still early in the semester, and they most of them will have done well on the first test. Unfortunately
many of them will have passed the first test without putting in much time or effort. Although this unit is
not that complicated as far as each individual skills they must master, we have found that this unit
produces the lowest grades. What makes this unit hard is the fact that up until now the student has
been given a skill or two they must master then tested on their ability to perform a specific task. Many
of the students are solving problems by memorizing what to do, with little care for what they are
actually looking for. This unit is different, as for every problem there may be several ways it can be
worked and they will have to decide for themselves, how they want to attack the problem.
Where to Begin: we need to spend some time introducing the concept of algebraic visualization to the
students. This is simply done by having the students imagine, or actually drawing a picture as you
describe it in words. It won’t take long for them to realize how challenging this is. If we then have them
fold their paper in half both length and widthwise, it will be easy for them to see the grid, or axis they
can measure from. We can then plot points then connect the dots. Next we take a simple linier equation
say X + Y = 6 and have them find pairs of numbers that satisfy the equation. QED.
Connect to Previous Work: when finding soultuions to linier equations we are using the problem solving
skills they found in the previous chapters.
Challenges: The greatest challenge in this unit is getting the students to understand before it is too late,
that being able to perform the required skills, will not be sufficient they must actually make deeper
connections.
Math 101: Beginning Algebra Course Kit
pg 27
Developmental Studies
Lesson Plan
Initial Lesson by: Hala Nestberg
Lesson 2a: Solving Absolute Value Inequalities
Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations & Inequalities
Primary Course Outcome(s):
Learning Objectives: By the end of the lessons, learners will:
1. Solve simple linear inequalities involving absolute value.
2. Graph the solution set to absolute value inequalities using the real number line.
Emphasize:




The interpretation of the absolute value of a number as a distance from zero on a number line.
Representing the solution set to an absolute value inequality by graphing the solution set on the
real number line.
Representing the solution set to an absolute value inequality by writing an appropriate
inequality (union, intersection)
Graphing on a number line the solution set for an absolute value equation.
Duration: .5 classes (40 min)
Time
5 min
15 min
15 min
5 min
5 min
Description/Activity
Transition: Instructor reviews what absolute value of a number means and what it
means to solve inequalities.
Instructor opens up the Hawkes lesson corresponding to Solving Absolute Value (A.4)
and uses the Instruct button to define each concept and show examples.
Have students work on at least four practice examples.
Ask each student to team up with one other student. Have them use the loose paper
activity to illustrate the solutions to absolute value inequalities and explain it to each
other.
Summary - Instructor reiterates steps and rationale.
Assessment Strategies/Comments: Homework to be assigned in Tutorial software
Activity: loose white paper; draw a number line and illustrate the less than a value versus the greater
than the value. Activity: Algebra Balance Scales at the following link:
http://nlvm.usu.edu/en/nav/category_g_4_t_2.html
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 28
Developmental Studies
Lesson Plan
Initial Lesson by: Habib Aghdami
Lesson 2b: Cartesian Coordinate System
Course/Unit Focus: Beginning Algebra (MAT 101) / Linear Equations in Two Variables
Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line.
Learning Objectives: By the end of the session, learners will:
2.1.
2.2.
2.3.
2.4.
2.5.
2.6.
Plot points on a rectangular coordinate system.
Identify what quadrant or axis a point lies on.
Tell if an ordered pair is a solution of an equation in two variables or not.
Complete an ordered pair that has one missing value.
Read a bar graph.
Read a line graph
Materials: Rectangular Coordinate System and Reading Graphs, Activity/Discussion Sheets, Whiteboard,
Computer & projector (or Overhead)
Duration: 1 class (75 min)
Time
20 min
Description/Activity
Intro: Instructor introduces lesson noting: this lesson contains properties of “Solving Linear
Equations” (topics related to the previous chapter) and contains topics which will “Leadin” to
the “Graphing Linear Equations in Two Variables”.
Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant
examples:
15-20 min
-
Sketches the Cartesian Plan on the board
-
Draws points in all quadrants (points out the characteristics of the points)
-
Draws point on axis (points out the characteristics of the points)
-
Evaluates a linear equation at several values for x and y
-
Goes over a small bar graph problem
-
Goes over a small line graph problem
Instructor forms students in groups of 4-6 and provides each student a “Rectangular Coordinate
Math 101: Beginning Algebra Course Kit
Time
pg 29
Description/Activity
System and Reading Graphs Activity/Discussion Sheet”
Students work individually to solve the problem.
Students compare answers within their groups and discuss their challenges and findings
10-20 min
5-10 min
Small Groups report out their solutions to the class. Instructor facilitates the discussion, posing
questions, challenging groups/students to explain and clarify. Alternatively, instructor displays
the Answer to Rectangular Coordinate System and Reading Graphs Activity/Discussion Sheet via
a computer or an overhead.
Summary - Instructor provides a brief lecturette to reinforce the key points of the lesson and
foreshadow the next lesson.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 30
Graphing Linear Equations in Two Variables Activity/Discussion Sheet
Name______________________________
Practice Problems 1a - 1b:
Determine whether the equation is linear or not.
1a. y = 2x - 1
1b.
Practice Problems 2a - 2b:
Graph the linear equation.
2a. y = 2x - 1
2b.
Math 101: Beginning Algebra Course Kit
Answers to Graphing Linear Equations in Two Variables Activity/Discussion Sheet
Answer/Discussion to 1a
y = 2x - 1
If we subtract 2x from both sides, then we can write the given equation as -2x + y = -1.
Since we can write it in the standard form, Ax + By = C, then we have a linear
equation.
Answer/Discussion to 1b
If we add x squared to both sides we would end up with
. Is this a linear
equation? Note how we have an x squared as opposed to x to the one power.
It looks like we cannot write it in the form Ax + By = C, because the x has to be to the
one power, not squared. So this is not a linear equation.
Answer/Discussion to 2a
y = 2x - 1
Step 1: Find three ordered pair solutions.
The three x values I'm going to use are -1, 0, and 1. (Note that you can pick ANY
three x values that you want. You do not have to use the values that I picked.)
You want to keep it as simple as possible. The following is the chart I ended up with
after plugging in the values I mentioned for x.
pg 31
Math 101: Beginning Algebra Course Kit
pg 32
x
y = 2x - 1
(x, y)
-1
y = 2(-1) - 1 = -3
(-1, -3)
0
y = 2(0) - 1 = -1
(0, -1)
1
y = 2(1) - 1 = 1
(1, 1)
Step 2: Plot the points found in step 1.
Step 3: Draw the graph.
Answer/Discussion to 2b
Math 101: Beginning Algebra Course Kit
pg 33
Step 1: Find three ordered pair solutions.
The three x values I'm going to use are -1, 0, and 1. (Note that you can pick ANY
three x values that you want. You do not have to use the values that I picked.)
You want to keep it as simple as possible. The following is the chart I ended up with
after plugging in the values I mentioned for x.
x
y = -1/2x
(x, y)
-1
y = -1/2(-1) = 1/2
y = -1/2(0) = 0
y = -1/2(1) = -1/2
(-1, 1/2)
0
1
Step 2: Plot the points found in step 1.
Step 3: Draw the graph.
(0, 0)
(1, -1/2)
Math 101: Beginning Algebra Course Kit
pg 34
Math 101: Beginning Algebra Course Kit
pg 35
Developmental Studies
Lesson Plan
Lesson 3b: Exploring Linear Data
Initial Lesson by: Habib Aghdami
Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing
Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line.
Learning Objectives: By the end of the session, learners will:
2.7.
2.8.
2.9.
2.10.
Explore relationships between symbolic expressions and graphs of lines, paying particular
attention to the meaning of intercept and slope.
Use graphs to analyze the nature of changes in quantities in linear relationships.
Describe the importance of scatterplots and use them to display data.
Estimate and write equations of lines.
Materials: Grid Paper (several sheets per student or group) Bike Weights and Jump Heights Activity
Sheet Weights and Drug Doses Activity Sheet, an overhead or a computer
Duration: 2 class periods (2.5 hrs)
Time
5 min
15-20 min
Description/Activity
Intro – Instructor introduces lesson noting: this lesson contains properties of “Solving Linear
Equations” (topics related to the previous chapter) and contains topics which will “Leadin” to
the “Solving the System of Linear Equations” (topics related to the next chapter).
Instructor invites student to choose which problem they wish to work on (Bike Jump
or Medicine Doses). Students then form groups of 4-6 for each of the two problems.
Students work individually to solve the problem
Students compare answers within their groups and discuss their findings.
20-30 min
Groups report out their solutions to the class. Instructor facilitates the discussion,
posing questions, challenging groups/students to explain and clarify.
20-30 min
Students find a partner who did the other problem and meet and share results; each
talking the other through their problems.
5-10 min
Instructor provides a brief lecturette to reinforce the key points of the lesson and
foreshadow the next lesson.
Math 101: Beginning Algebra Course Kit
pg 36
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 37
Developmental Studies
Lesson Plan
Initial Lesson by: Habib Aghdami
Lesson 2c: Graphing Linear Equations in Two Variables
Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing
Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line.
Learning Objectives: By the end of the session, learners will:
2.11.
2.12.
2.13.
2.14.
2.15.
Recognize when an equation in two variables is a linear equation (or not);
Graph a linear equation;
Find the x- and y-intercepts of a linear function;
Graph a linear function using the x- and y-intercepts;
Graph vertical and horizontal lines.
Materials: Graphing Linear Equations in Two Variables Activity/Discussion Sheet, a Computer or an Overhead
Duration: 1 class period (75 min)
Time
15 min
Description/Activity
Intro: Instructor introduces lesson noting: this lesson contains properties of “Rectangular
Coordinate System and Reading Graphs” (topics related to the previous section) and contains
topics which will “Leadin” to the “Slope of a Line” (topics related to the next section).
Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over
relevant examples:
15 -20 min
-
Characteristics of a linear equations vs. a non linear equation
-
Evaluating a linear equations in two variables
-
Find the x- and y-intercepts of a linear function.
-
Graph a linear function using the x- and y-intercepts.
-
Graph vertical and horizontal lines.
Instructor forms students in of 4-6 and provides each student a “Graphing Linear Equations in
Two Variables Activity/Discussion Sheet” and “Intercept Activity/Discussion Sheet”
Students work individually to solve the problem
Math 101: Beginning Algebra Course Kit
Time
pg 38
Description/Activity
Students compare answers within their groups and discuss their challenges and findings
10-20 min
5-10 min
Groups report out their solutions to the class. Instructor facilitates the discussion, posing questions,
challenging groups/students to explain and clarify. Alternatively, instructor displays the Answer to
Graphing Linear Equations in Two Variables Activity/Discussion Sheet and “Intercept Activity/Discussion
Sheet via a computer or an overhead
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the next
lesson.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 39
Developmental Studies
Lesson Plan
Lesson 2d: Graphing: Slope-Intercept Form
Initial Lesson by: Habib Aghdami
Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing
Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line.
Learning Objectives: By the end of the session, learners will:
2.16.
2.17.
2.18.
Find the slope given a graph, two points or an equation.
Write a linear equation in slope/intercept form.
Determine if two lines are parallel, perpendicular, or neither.
Materials: Slope Activity/Discussion Sheet, a Computer or an Overhead
Duration: 1 class period (75 min)
Time
20 min
Description/Activity
Intro: Instructor introduces lesson noting: this session contains properties of “Graphing Linear
Equations in Two Variables” (topics related to the previous section) and contains topics which will
“Leadin” to the “Equations of Lines” (topics related to the next section).
Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant
examples:
15-20 min
-
Rise/Run Formula
-
Slope – Intercept Formula
-
Point – Slope Formula
-
Characteristics of “X = c” Equation
-
Characteristics of “Y = c” Equation
Instructor forms students in of 4-6 and provides each student a “Slope Activity/Discussion Sheet”
Students work individually to solve the problem
Students compare answers within their groups and discuss their challenges and findings
Math 101: Beginning Algebra Course Kit
10-20 min
5-10 min
pg 40
Groups report out their solutions to the class. Instructor facilitates the discussion, posing
questions, challenging groups/students to explain and clarify. Alternatively, instructor displays the
Answer to Slope Activity/Discussion Sheet via a computer or an overhead
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the
next lesson.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 41
Developmental Studies
Lesson Plan
Lesson 2e: Graphing: Point-Slope Form
Initial Lesson by: Habib Aghdami
Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing
Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line.
Learning Objectives: By the end of the session, learners will:
2.19.
2.20.
2.21.
Use the slope/intercept form to write a linear equation given the slope and y-intercept.
Use the slope/intercept form to graph a linear equation.
Use the point/slope equation to set up an equation given:
a. any point on the line and the slope.
b. two points on the line.
c. a point on the line and a parallel line.
d. a point on the line and a perpendicular line.
Materials: Slope Activity/Discussion Sheet, a Computer or an Overhead
Duration: 1 class period (75 min)
Time
20 min
Description/Activity
Intro: Instructor introduces lesson noting: this lesson contains properties of “Slope of Lines” (topics
related to the previous section) and contains topics which will “Leadin” to the “Applications of Two
Variable.
Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant
examples:
15-00 min
Rise/Run Formula
Slope – Intercept Formula
Point – Slope Formula
Characteristics of “X = c” Equation
Characteristics of “Y = c” Equation
Characteristics of parallel lines
Characteristics of perpendicular lines
Instructor forms students in of 4-6 and provides each student a “Equations of Lines
Activity/Discussion Sheet”
Students work individually to solve the problem
Students compare answers within their groups and discuss their challenges and findings
Math 101: Beginning Algebra Course Kit
Time
10-20 min
5-10 min
pg 42
Description/Activity
Groups report out their solutions to the class. Instructor facilitates the discussion, posing
questions, challenging groups/students to explain and clarify. Alternatively, instructor displays the
Answer to Equations of Lines Activity/Discussion Sheet via a computer or an overhead
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the
next lesson.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 43
Developmental Studies
Lesson Plan
Lesson 2f: Applications of Two Variable Equations
Initial Lesson by: Habib Aghdami
Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing
Primary Course Outcome(s): 3. Graph linear equations and determine the equation of a line.
Learning Objectives: By the end of the session, learners will:
2.22. …
2.23. …
2.24. …
Materials:
Duration: 1 class period (75 min)
Time
20 min
Description/Activity
Intro: Instructor introduces lesson noting: this lesson contains properties of “Slope of Lines” (topics
related to the previous section) and contains topics which will “Leadin” to the “Applications of Two
Variable.
Lecturette: Instructor provides a brief lecturette key points of the lesson and goes over relevant
examples:
5-10 min
15-25 min
10-20 min
5-10 min
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the
next lesson.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 44
Developmental Studies
Unit Overview - Teaching Insights & Strategies
Submitted by:
Mark Gollwitzer
Unit 3: Teaching Systems of Linear Equations
Purpose of Unit: in this unit we will be teaching students the substitution, and addition methods for solving
problems with more than one variable.
Where the Students are: Many of the students just received their first bad test grade in math since high
school. Because of this many of them will be questioning whether they can succeed in math at all. This
chapter separates those that are willing to work hard for a grade from the rest. If handled correctly this
is a wonderful chapter for helping the students develop mathematical maturity.
Where to Begin: We need to convince the students that the material in this chapter is within their
grasp. We can do this by explaining to them that they will only be asked to learn two new skills, i.e. the
substitution, and addition methods. We then start with the substitution method as they have already
been doing this when solving word problems. This method should not be too far out of their reach,
therefore they should gain some confidence. When we show them the addition method they will grasp
it enthusiastically, for after working with the substitution method for a couple of days the addition
method will seem like childes play. Then all we need do is motivate them to practice, practice, and
practice.
Connect to Previous Work: The chapter starts out solving systems by graphing. This is a review of the
last chapter. Teaching substitution is easily done by reminding them of how they replaced unknowns,
with expressions when solving word problems. The addition method can easily be taught by reminding
the students about the addition property of equality.
Challenges: Remember, the students are usually more than a little frustrated with their performance on
the last test, and beginning to doubt themselves. Because of this we need to expose the students to as
many different levels of difficulty as possible, so that when they see the problems on their homework
and tests they don’t panic.
Closure: This is a good chapter for pushing students to work hard. The best thing about this chapter is,
the problems are complicated and time consuming enough, so that when their test grades improve they
will see the value in homework, and begin to gain some of the confidence they will need in future
chapters.
Math 101: Beginning Algebra Course Kit
pg 45
Developmental Studies
Lesson Plan
Lesson 3a: Solving Systems of Equations by Graphing
Initial Session by: W. Parker
Course/Unit Focus: MAT 101/Unit 3: Systems of Linear Equations
Primary Course Outcome(s): 4. Solve systems of two linear equations and interpret solutions;
Learning Objectives: By the end of the lesson, learners will:
3.1. Solve a system of equations by using appropriate graphing techniques.
3.2. Identify the solution of the system
3.3. Interpret the significance of the solution
Materials: Graph paper and straight edge/ruler
Duration: 75 minutes
Time
Description/Activity
5 min.
Review objectives. Introduce topic and uses in mathematics.
10 min
Define system of equations, the possible solutions, and how to interpret solutions.
10 min
20-25min
20 min
Review and demonstrate graphing of linear equations by different methods:
 Table of values;
 slope/intercept.
Demonstrate solving systems of equations by graphing. Show examples with:
 one solution,
 infinite solutions, and
 no solutions.
Review how a solution is checked.
Activity: Small group-solving practice problems. Students work individually and in groups to
solve 5 problems (either even or odd by group). Groups check their answers with each other
(10 min).
One person from the even group pairs with a person from the odd group and shares answers
(5-10 min).
5-10 min
Review solutions and interpretation of solutions. Instructor encourages students to try and
complete any problems that they did not attempt in class for homework in addition to
completing certification for Hawkes 10.1.
Assessment Strategies/Comments: Hawkes Learning Systems 10.1
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 46
Math 101: Beginning Algebra Course Kit
pg 47
Math 101: Beginning Algebra Course Kit
pg 48
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 49
Developmental Studies
Lesson Plan
Lesson 3b-1: Solving System of Equations by Substitution
Initial Session by: W. Parker
Course/Unit Focus: MAT 101/ Systems of Linear Equations
Primary Course Outcome(s): 4. Solve systems of equations and interpret solutions;
Learning Objectives: By the end of the session, learners will:
3.4. Use the substitution method to find the solution of the system of equations.
3.5. Identify the solution of the system
3.6. Interpret the significance of the solution
Materials: Text
Duration: 40 minutes
Time
5 min.
Description/Activity
Review objectives. Introduce topic and uses in mathematics.
5 min
Review and demonstrate solving a system of equations by graphing
10 min
Demonstrate solving systems of equations by substitution. Review how a solution is
checked and interpreted.
15min
Activity: Small group-solving practice problems. Students work cooperatively to solve 3
problems from worksheet.
5min
Review solutions by substitution and interpretation of solutions. Student volunteer will
explain solution from the group. Additional problems from worksheet and certification for
Hawkes 10.2 assigned for homework.
Assessment Strategies/Comments: Hawkes Learning Systems 10.2
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 50
Math 101: Beginning Algebra Course Kit
pg 51
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 52
Math 101: Beginning Algebra Course Kit
pg 53
Developmental Studies
Lesson Plan
Lesson 3b-2: System of Equations by Addition
Initial Session by: W. Parker
Course/Unit Focus: MAT 101/Unit 3: Systems of Linear Equations
Primary Course Outcome(s): 4. Solve systems of equations and interpret solutions;
Learning Objectives: By the end of the session, learners will:
3.7. Use the addition method to find the solution of the system of equations.
3.8. Identify the solution of the system
3.9. Interpret the significance of the solution
Materials: Text
Duration: 35 minutes
Time
2 min
Description/Activity
Review objectives. Introduce topic and compare addition method substitution method.
5 min
Review substitution to solve a system of equations, the possible solutions, and how solutions
are interpreted.
5min
Demonstrate solving the same system of equations by addition method. Review how the
solution is checked.
5min
Demonstrate solving different systems of equations by addition method. Show examples
with one solution, infinite solutions, and no solutions. Review how a solution is checked and
interpretation of the solution.
15min
Activity: Small group-solving practice problems. Students work in groups to solve 2-3
problems from worksheet. Groups will report/discuss using substitution and addition
methods for solving systems of equations.
3 min
Review solutions and interpretation of solutions. Instructor encourages students to complete
problems from the worksheet in addition to completing certification for Hawkes 10.3
Assessment Strategies/Comments: Hawkes Learning System 10.3
Math 101: Beginning Algebra Course Kit
pg 54
Math 101: Beginning Algebra Course Kit
pg 55
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 56
Developmental Studies
Lesson Plan
Lesson 3c: Applications of Linear Systems
Initial Session by: W. Parker
Course/Unit Focus: MAT 101/Unit 3: Systems of Linear Equations
Primary Course Outcome(s): 4. Solve systems of equations and interpret solutions;
Learning Objectives: By the end of the session, learners will:
3.10.
3.11.
3.12.
3.13.
Convert word problems to systems of equations
Use an appropriate method to find the solution of the system of equations.
Identify the solution of the system
Interpret the significance of the solution
Materials: Text
Duration: 75 minutes
Time
2 min
25 min
Description/Activity
Review objectives.
Activity: Small group-solving practice problems. Students work in groups to solve 2-3 problems from worksheet.
Groups will identify:
1.
2.
3.
4.
5.
Variables/unknowns
Two equations
Method used to find solution
Solution
Interpretation of solution
Groups will report/discuss the methods used for solving the problem by systems of equations.
25min
Demonstrate converting and solving word problems using systems of equations. Explain why a system of
equations might be necessary or more appropriate than an equation with a single variable. Review how the
solution is checked in the original word problem.
20min
Activity: Small group-solving practice problems. Students work in different groups to solve 2-3 problems from
worksheet. Groups will report/discuss using methods for solving systems of equations. Instructor will assist
groups as necessary.
3 min
Review solutions and interpretation of solutions. Instructor encourages students to complete problems from the
worksheet in addition to completing certification for Hawkes 10.4
Assessment Strategies/Comments: Hawkes Learning System 10.4
Math 101: Beginning Algebra Course Kit
pg 57
Math 101: Beginning Algebra Course Kit
pg 58
Math 101: Beginning Algebra Course Kit
[Back to Table of Contents]
pg 59
Math 101: Beginning Algebra Course Kit
pg 60
Developmental Studies
Lesson Plan
Lesson 3d: Exponents
Initial Lesson by:
Mark Gollwitzer
Course/Unit Focus: Beginning Algebra (MAT 101) / Systems of Linear Equations - Exponents
Primary Course Outcome(s): Use the rules of exponents to simplify algebraic expressions.
Learning Objectives: By the end of the session, learners will:
3.14.
Build and use the rules for exponents;
3.15.
Practice and apply rules for exponents;
3.16.
Structure and solve problems with scientific notation.
Materials: Smartboard/Whiteboard; preselected problems, quiz.
Duration: 2 class sessions/2.5 hours
Key Concepts & Skills:

Time
15 min
Description/Activity
Explain the definition of exponent: 23 = 2 ∙ 2 ∙ 2 , 𝑎𝑚 = 𝑎 ∙ 𝑎 ∙ 𝑎 … ∙ 𝑎.
Show the students how to expand and discover product and quotient rule for 𝑎2 ∙ 𝑎3 , and
30-45
min
𝑎5
𝑎3
𝑎 3
𝑎2
𝑏
𝑏
.
2
Have the students break up into groups and develop the rules for (𝑎𝑏)3 , (𝑎2 𝑏 3 )2 , ( ) , ( 3) ,
𝑎5
𝑎7
, and
𝑎2
𝑎2
50-80
min
Have students use exponent rules in class to simplify selected problems making students aware of
the potentially confusing ways the problem could be presented.
15-40
min
Explain what a polynomial is, then show the students how to divide a polynomial by a monomial.
Two or three examples are enough.
10-15
min
Give the students a short 3-5 question quiz on division of a polynomial by a monomial.
Note: Assessment.
The quiz should contain 1 monomial/monomial, 2 binomial/monomial, and 3 trinomial/monomial.
[Back to Table of Contents]
Assessment Strategies/Comments: Practice problems & quiz
Math 101: Beginning Algebra Course Kit
pg 61
to be integrated w/ Lesson 3d Exponents above
Initial Lesson by: Mark Gollwitzer
Lesson 3d-2: Application of Exponents - Scientific Notation Original Lesson 4d
Course/Unit Focus: Beginning Algebra (MAT 101) / Graphing
Primary Course Outcome(s): 4. Simplify, evaluate, and perform arithmetic operations with polynomials.
& 5.Use the rules of exponents to simplify algebraic expressions.
Learning Objectives: By the end of the session, learners will:
3.17.
3.18.
3.19.
Convert numbers to scientific notation.
Convert numbers from scientific notation.
Multiply and divide numbers in scientific notation.
Materials: Smart board /white board (Computer lab, where needed and other materials as required)
Duration: 30 min
Time
5 min
3 min
Description/Activity
Ask the students if they know what the national debt is. Talk about the national debt clock by
showing or explaining that the smaller numbers are changing so fast they aren’t worth mentioning.
Explain that as this is a large number it would be nice if we could express it in a more compact way.
Of to the side explain 5x10= 50, 5x100 or 10^=500, and 5x1000 or 10^3 =5000. Use this new
understanding to convert national debt to 13.5x10^12
Help the student see the relationship between the number of zero’s and the exponent.
2min
Explain that while it is easier to understand $13.5 Trillion in S.I. we would have moved the decimal
one more place, making it 1.35x10^13 to fit the true form of S.I...
Note: This helps them see that there is a specific form for scientific notation.
5 min
Discuss the number of people in the United States. Turn this number into S.I.
5 min
Ask the students how much every American citizen owes on the national debt. Show them how to
use the rules of exponents to do the division.
5 min
Ask the student how much an eyelash weighs (0.000304grams). Show the student how to convert
extremely small numbers to S.I.. Have students convert this weight to pounds. Have them express
this number in standard form.
5 min
Ask the students what the total weight of all eyelashes in America. Assume 100-150 eyelashes per
person.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 62
Developmental Studies
Unit Overview - Teaching Insights & Strategies
Unit 4: Teaching Polynomials and Factoring
Submitted by: Mark Gollwitzer &
Debbie Moran
Purpose of Unit: To understand the problems inherent in teaching exponents and polynomials to the
developmental student.
Where the Students are: The material in this section is relatively easy and accessible for most students.
When the students get this far many of them have struggled through, and in many cases been
disappointed with their grades on the chapters dealing with the graphing of linear equations. The last
day to withdraw has just passed so some students will have withdrawn from the class while, those that
remain may have grades that are borderline. In the following lesson plans we are trying to teach in a
way that gives the student some confidence back.
Where to Begin: Chapter 11 is already in progress and working with exponents has been a good
experience. They find this topic to be a bit easier than the graphing and systems they have been
studying previously. Those students that need to pick their grades up will get motivated now.
Connect to Previous Work: At this point students have been working with polynomials, but just the
simplest type while learning to solve equations. Once this idea is pointed out, the student will pick up
the idea of simplifying polynomials quickly. Function notation was introduced in chapter 9 and will be
reinforced here to evaluate polynomial expressions.
Challenges: Although long division of polynomials may look difficult at first glance, if we carefully
compare it to long division of numbers, they usually can get it as their confidence by this point is coming
back.
Closure: Students seem to thrive more on totally algebraic manipulation and this chapter provides just
that. Their grades should recover nicely and they will be in a better situation for the sections on
factoring.
Math 101: Beginning Algebra Course Kit
pg 63
Developmental Studies
Lesson Plan
Lesson 4a: Introduction to Polynomials & Operations
Initial Lesson by: Debbie Moran
Course/Unit Focus: Beginning Algebra (MAT 101) / Polynomials
Primary Course Outcome(s): 4. Simplify, evaluate, and perform operations with polynomials and 5.
Factor polynomials and solve quadratic equations by factoring.
Learning Objectives: By the end of the session, learners will:
4.1. Define a polynomial, and learn how to classify polynomials.
4.2. Evaluate a polynomial for given values of the variable.
4.3. Be able to add and subtract polynomials.
4.4. Simplify algebraic expressions by removing grouping symbols and combining like terms.
Materials:
Duration: 1 class period (75 min)
Time
10-15 min
Description/Activity
Entry Activity: Instructor projects entry problem(s) (or posts on whiteboard) for learners to work
on as they enter.
[from 9.5] 𝑓(𝑥) = 𝑥 2 − 3𝑥 + 2, find 𝑓(4), 𝑓(0), 𝑓(−3)
Simplify: [from 7.7] 4(𝑦 + 3) − 5(𝑦 − 2)
Simplify: [from 7.7] 2𝑥 2 − 2𝑦 + 5𝑥 2 + 6𝑥 2
Answer questions from previous class period.
40 min
Section 11.3 Give students the definition for a Monomial, Polynomial found on page 875.
Introduce the special names for 1st -, 2nd - and 3rd -degree polynomials and give examples.
Evaluate polynomials using the function notation.
Section 11.4 This should be an easy section as the student already have experienced adding and
subtracting like terms as well as removing parenthesis with the distributive property. Emphasis
that subtraction of two polynomials must remove the parenthesis by distributing a -1 factor to all
terms within the parenthesis. Subtracting in a vertical format is important for long division later in
the chapter.
5-10 min
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the
next lesson. Current assignments/homework is mentioned.
[Back to Table of Contents]
Assessment Strategies/Comments: Hawkes Learning software certification.
Math 101: Beginning Algebra Course Kit
pg 64
Developmental Studies
Lesson Plan
Lesson 4b: Operations on Polynomials – Multiplying
Initial Lesson by: Mark Gollwitzer &
Debbie Moran
Course/Unit Focus: Beginning Algebra (MAT 101) / Polynomials
Primary Course Outcome(s): 4. Simplify, evaluate, and perform arithmetic operations with polynomials.
Learning Objectives: By the end of the session, learners will:
4.5 Multiply monomials by polynomials
4.6 Multiply two polynomials
4.7 Multiply polynomials using the FOIL method
4.8 Multiply binomials find products that are the difference of squares
4.9 Multiply binomials find products that are perfect square trinomials.
Materials: Whiteboard/Chalk board
Duration: 1.5 hours
Time
5-15 min
Description/Activity
Multiplying Polynomials - Remind the students how the distributive property works. Expand
their understanding by showing examples of progressively more complicated monomials
distributed into longer and more complicated polynomials and demonstrating the use of rules
for exponents to multiply polynomials.
10-20 min
Multiplying Binomials & Trinomials - Show the students how to use the distributive property to
multiply binomials by trinomials and trinomials by trinomials. Notes: Clarify how distributive
property compares to FOIL; Highlight how when collecting like terms we add coefficients not
exponents.
20-30 min
Special product rules - Introduce the mnemonic FOIL to perform the operation of multiplying
two binomials. This mnemonic will remind students that there are 4 products that must be
found when multiplying two binomials. Expose the students to the special product rules and
short cuts for multiplying polynomials.
5 min
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow
the next lesson. Current assignments/homework is mentioned.
[Back to Table of Contents]
Assessment Strategies/Comments: Hawkes Learning software certification.
Math 101: Beginning Algebra Course Kit
pg 65
Developmental Studies
Lesson Plan
Lesson 4c:
Opera
tions on
Polynomia
ls Division
Initial Lesson
by: Mark
Gollwitzer &
Debbie Moran
Course/Unit Focus: Beginning Algebra (MAT 101) / Polynomials
Primary Course Outcome(s): 4. Simplify, evaluate, and perform arithmetic operations with polynomials.
Learning Objectives: By the end of the session, learners will:
4.10
4.11
4.12
4.13
Divide a polynomial by polynomials by a monomial.
Divide polynomials using the long division algorithm.
Check a long division using by multiplying quotient by divisor and adding the remainder.
Use place holders for missing terms.
Materials: Smart board/ white board / PowerPoint (Computer lab, where needed and other materials as required)
Duration: 70 min
Time
10min
Description/Activity
Show the students how we divide a polynomial by a monomial,
18𝑥 5 −6𝑥 4 +9𝑥
3𝑥
=
18𝑥 5
3𝑥
−
6𝑥 4
3𝑥
+
9𝑥
3𝑥
and then reduce each fraction.
25 min
Show the students another method using an example of a polynomial divided by a larger
polynomial and explain the need for a new method i.e. can’t cancel terms only factors. (Remind
the student how not to cancel.) example:
𝑥 2 −2𝑥−20
𝑥+4
Review long division of numbers by hand. 695/31. Explain in detail and show the math for all the
steps. Note when thinking about how many times 31 goes into 69 we can ignore the 1 in 31
Note: Use of long division; also, many Hispanic students didn’t learn the galley method of long
division.
Show example of long division. Take
𝑥 2 +5𝑥+6
𝑥+3
and as you are doing the long division note we can
Math 101: Beginning Algebra Course Kit
pg 66
Time
Description/Activity
ignore the +3 in x+3 just like we ignored the 1 in 31. Note that we draw the line and change the
signs in order to do the subtraction. Use distributive property to check.
Have students do one.
5 min
Show a new example. Take
𝑥 2 +7𝑥−12
𝑥−3
and as you are doing the long division note we can ignore
the -3 in x-3 just like we ignored the 1 in 31. Note that we draw the line and change the signs in
order to do the subtraction. This time make sure they understand when we subtract a negative
we actually add. Hence draw the line and change the signs. Have students do one.
5 min
Using this example
4𝑥 2 +12𝑥+9
2𝑥+3
note we can take
4𝑥 2
2𝑥
to get started. Then have students do one.
Note: Many students have trouble figuring out what they need to multiply by.
20 min
Give the students problems with 3rd and 4th degree polynomials, some that have remainders,
and some with missing terms. While students are working on them explain the need for place
holders.
Note: help the students make the final connections.
[Back to Table of Contents]
Assessment Strategies/Comments: Hawkes Learning software certification.
Math 101: Beginning Algebra Course Kit
pg 67
Developmental Studies
Lesson Plan
Initial Lesson by: Debbie Moran
Lesson 4d: The Greatest Common Factor, Factor by Grouping
Course/Unit Focus: Beginning Algebra (MAT 101) / Factoring
Primary Course Outcome(s): 5. Factor polynomials and solve quadratic equations by factoring.
Learning Objectives: By the end of the session, learners will:
4.10 Find the GCF of a set of terms
4.11 Factor polynomials by factoring out the monomial GCF.
4.12 Factor polynomials by grouping
Duration: 90 min
Time
10 min
30 min
Description/Activity
Begin by pointing out the difference between “factors” and “product”. (15)(3) = 45
Factors are 15 and 3 and the product is 45.
Talk about factorization of numbers. Find the GCF of two larger numbers, 30 and 54 using
the same method as with polynomials. Continue with an example of monomials,
150𝑥𝑦, 250𝑦 2 𝑥 3 , 100𝑥 2 𝑦 2
Practice finding and factoring out the GCF with more than one term. Start by using the
same terms as in the previous example: 150𝑥𝑦 + 250𝑦 2 𝑥 3 − 100𝑥 2 𝑦 2. Use the GCF
and rewrite as a product of the GCF and the remaining factors: 50𝑥𝑦(3 + 5𝑦𝑥 2 − 2𝑥𝑦)
Be sure to point that when a whole term is factored out, a 1 (or -1) must be used to retain
the “place” in the polynomial. This way when the GCF is re-distributed to each term, the
original polynomial will be the result.
10 min
30 min
Allow students to practice with several problems.
Factoring by Grouping – a development idea:
ax + ay = a(x+y)
$x + $y= $(x+y)
(Trash)x + (Trash)y = (Trash)(x+y)
“Taking out the Trash”
(a + b) x + (a+b)y = (a+b)(x+y)
Now proceed using polynomial examples.
Be sure to include examples when a -1 is the GCF for the 2nd group of terms.
Also show students that the terms can be reordered and the same resulting factorization.
Math 101: Beginning Algebra Course Kit
Time
pg 68
Description/Activity
Do not allow students to group by using parenthesis. The expression 𝑥𝑦 + 5𝑥 + 3𝑦 +
15 usually becomes (𝑥𝑦 + 5𝑥)(+3𝑦 + 15) showing these terms now as a product. It
might be best to use a squiggly underline for each pair of terms or a brace .
Allow students to practice several of these grouping problems.
5-10 min
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the
next lesson.
[Back to Table of Contents]
Assessment Strategies/Comments: Hawkes Learning software certification.
Math 101: Beginning Algebra Course Kit
pg 69
Developmental Studies
Lesson Plan
Initial Lesson by:
Debbie Moran
Lesson 4e: Factoring Trinomials
Course/Unit Focus: Beginning Algebra (MAT 101) /Factoring
Primary Course Outcome(s): 5. Factor polynomials
Learning Objectives: By the end of the session, learners will:
4.14 Factor trinomials with a coefficient of 1
4.15 Factor trinomials with a coefficient of 1 after factoring out the GCF.
Duration: 50 min.
Time
15 min
15 min
Description/Activity
Introduce Factoring an equation of the form 𝒙𝟐 + 𝒃𝒙 + 𝒄 by showing the product of two binomials
and how the numerical portions are related.
(𝒙 + 𝟐)(𝒙 + 𝟓) = 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟎 where 7 = 2+5 and 10 = (2)(5)
Now, show the reverse of this problem looking for factors of 10 that add to be 7.
Include other examples with values for b and c being different combinations of positive and
negative values.
Give students about 10 minutes to practice a few of these. Small groups would be beneficial is
allowing them to “teach each other”.
Now, give examples of a trinomial that has a GCF and when removed the remaining trinomial has a
= 1. Then factor the trinomial. Provide students with several examples.
Once again allow a few minutes to practice in their small groups.
Follow up with these 2 tips:
 there cannot exist a common factor within the binomial if there is not a common
factor in the trinomial
 the signs on the terms of the trinomial can assist in placement of the signs in the
binomial factors
5-10 min
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the
next lesson.
Assessment Strategies/Comments: Hawkes Learning software certification.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 70
Developmental Studies
Lesson Plan
Review Session – Factoring Trinomials
Initial Lesson by: Debbie Moran
Course/Unit Focus: Beginning Algebra (MAT 101) /Factoring
Primary Course Outcome(s): 5. Factor polynomials
Learning Objectives: By the end of the session, learners will:
4.16 Factor trinomials with a coefficient of 1
4.17 Factor trinomials with a coefficient of 1 after factoring out the GCF.
Duration: 50 min.
Time
15 min
15 min
Description/Activity
Introduce Factoring an equation of the form 𝒙𝟐 + 𝒃𝒙 + 𝒄 by showing the product of two binomials
and how the numerical portions are related.
(𝒙 + 𝟐)(𝒙 + 𝟓) = 𝒙𝟐 + 𝟕𝒙 + 𝟏𝟎 where 7 = 2+5 and 10 = (2)(5)
Now, show the reverse of this problem looking for factors of 10 that add to be 7.
Include other examples with values for b and c being different combinations of positive and
negative values.
Give students about 10 minutes to practice a few of these. Small groups would be beneficial is
allowing them to “teach each other”.
Now, give examples of a trinomial that has a GCF and when removed the remaining trinomial has a
= 1. Then factor the trinomial. Provide students with several examples.
Once again allow a few minutes to practice in their small groups.
Follow up with these 2 tips:
 there cannot exist a common factor within the binomial if there is not a common
factor in the trinomial
 the signs on the terms of the trinomial can assist in placement of the signs in the
binomial factors
5-10 min
Instructor provides a brief lecturette to reinforce the key points of the lesson and foreshadow the
next lesson.
Assessment Strategies/Comments: Hawkes Learning software certification.
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 71
Developmental Studies
Lesson Plan
Review Session – Final Jeopardy
Initial Lesson by:
Debbie Moran
Course/Unit Focus: Beginning Algebra (MAT 101) / Any
Primary Course Outcome(s): Any/all related to the test/final.
Learning Objectives: By the end of the session, learners will:
1. Practice a range of problems on factoring (in preparation for the chapter test);
2. Work together in teams to quickly identify answers to representative problems from each
category.
Activity: Use the Jeopardy activity to review the factoring, solving equations and applications.
Assessment Strategies/Comments:

Give extra credit on the test (1 to 3 points) for each group. (?)
Emphasize:

Materials:
Duration: 1 class (75 min)
Time
Description/Activity
5 min
Intro: Homework challenges discussed and/or collected. Instructor introduces lesson describing
its importance, relationship to algebra and key concepts and skills to be addressed.
Encourage Students to break into pairs
5-10 min
Summary - Instructor provides a brief lecturette to reinforce the key points of the lesson and
foreshadow the next lesson and assigns homework (as needed)
Assessment Strategies/Comments: Homework may be assigned in Tutorial software and/or
[Back to Table of Contents]
Math 101: Beginning Algebra Course Kit
pg 72
Appendix A: Syllabus - Math 101 Beginning Algebra
Arts and Sciences Division
Developmental Studies
MAT 101
Beginning Algebra
Syllabus ◊ Fall 2011
Course Description: This course includes the following topics: operations with signed
numbers; addition, subtraction, multiplication, and division with algebraic expressions; factoring;
techniques for solving linear and fractional equations; and an introduction to graphing.
In addition this course covers systems of linear equations, exponents and operations with
polynomials and serves as an introduction to Algebra and its applications.
Purpose and Prerequisites: MAT 101 provides a basic foundation in Algebra and serves as preparation
for MAT 102. Students may also elect to take MAT 101 as a refresher course. Registration for MAT 101
requires passing MAT 032 with a grade of C (70%) or higher or by satisfactory placement test (Compass/Asset).
Semester Credit Hours: 3.0 (non-credit)
Learning Outcomes: The student satisfactorily completing MAT 101 will have demonstrated
with a minimum of 70% accuracy on all coursework the ability to:
1.
2.
3.
4.
5.
6.
7.
Perform operations with real numbers and algebraic expressions;
Solve linear equations, inequalities, and formulas for specified variables;
Graph linear equations and determine the equation of a line;
Solve systems of two linear equations and interpret solutions;
Solve application problems involving the procedures and techniques above;
Simplify, evaluate, and perform operations with polynomials;
Factor polynomials.
Required Learning Resources (Textbooks/Materials/Online):
Text:
Wright, D. Franklin. Developmental Mathematics. Charleston, SC: Hawkes
Learning Systems, 2011.
Software: Hawkes Learning Systems
Other:
TI-30XS (Multiview) scientific calculator
Blackboard & GTC gmail: Blackboard will be used to access online documents (syllabus, course
outline), resources and announce assignments and tests in coordination with the Hawkes Learning
System. GTC gmail will be used to communicate important course information. Students should
regularly access both Blackboard and GTC gmail to keep up to date with course announcements and
assignments.
Math 101: Beginning Algebra Course Kit
pg 73
Course Requirements and Evaluation:
Course Outline/Schedule: A course outline/schedule will be provided that identifies specific topics
covered and assignments/assessments (test/quizzes) due dates across the semester.
Grading Scheme: This course will have the following types of assignment/assessments and final
grades will be weighted as listed.
Homework (Hawkes certification)
20%
Unit tests
(4 @ 11.25%)
45%
Final Exam
30%
Activities
5%
Grades:
A: 90-100%
B: 80-89%
C: 70-79%
D: 60-69%
F: 0-59%
There will be no extra credit given in this class and no curving of grades. Your final grade will be
the grade that you earned in the class and should reflect your knowledge of the material.
If this course is required as a prerequisite to another math course, you must receive a final grade
of C or higher to proceed to the next course.
Homework & Certification:
Homework will be done online using the Hawkes Learning System software. In this software you will
study, practice and “certify” on each topic covered in this class. To complete each assignment you must
demonstrate your mastery by passing the certification quiz at 80%. This will earn you 100% on your
homework assignment.
Each homework assignment for a class will be due at the beginning of the next class period. If you
submit assignments late, you may still receive partial credit for the first 5 days after each due date, but
there will be a late penalty of 10% per day subtracted from your score. After the late period is over, you
will receive a zero for a missed homework assignment. Because of the time needed to install and learn
to use the Hawkes software, your first week’s assignments will be due at the beginning of the first class
period of the second week.
Your 3 lowest homework assignment grades will be dropped.
Unit Tests: You cannot make up a missed test, but one missed test score will be replaced by the
final exam score. Other missed tests will receive a grade of zero. If you take all unit tests, your lowest
score will be replaced by the final exam score (if it is higher).
Final Exam: You must take the final exam to pass this course. The final exam will be comprehensive
and cannot be exempted.
Activities: Consistent participation in class activities, problem-solving practice, group work, in-class
quizzes, online discussions (in Blackboard), and application experiences is a central part of your learning
experience.
Partial Credit Rubric (A Learning Guide & Grading Tool) Where partial credit is available on a test or
exam, the following rubric will be used to award points for solutions.
0 (0%)
Non-responsive
Solution
contains no
correct
information.
1 (25%)
Preliminary
Solution contains
some correct
information/elem
ents, but problem
is unsolved/
unstarted.
2 (50%)
Beginning
Solution contains
evidence of
understanding the
key concept, and a
solution has been
attempted/started.
3 (75%)
Developing
Solution clearly
exhibits use of key
concept(s) to structure
answer, and a solution
has been proposed,
there is limited nonconceptual/careless
calculation errors.
4 (100%)
Exemplary
Solution effectively
uses key concept(s)
and appropriate
steps/methods to
structure answer and
provides a correct
solution with no errors.
Math 101: Beginning Algebra Course Kit
pg 74
How to Succeed in this Class: A Checklist
 Read your emails and check in on Blackboard regularly.
o
o
Your instructor will send out communications to the class via your GTC gmail account.
Announcements, assignments and grades will be posted to Blackboard periodically.
 Attend every class.
o If you miss a class, YOU are responsible for learning the material you missed.
Read the book, complete the assignments and learn the material before coming back
for the next class.
o Note: If you miss more than 3 classes AND have less than a 70 average, you may
be administratively withdrawn from the class (note: administrative withdrawals occur about
a week before the last day to withdraw).
 Be on time and don’t leave early.
o If you must leave early, inform your instructor before class.
o You must be in class for at least half the class to be counted present for the day.
 Bring your book and calculator to class.
 Participate in class activities, problem-solving and discussions.
 Stay focused on the class from beginning to end
o Do not pack up early.
o Turn your cell phone and ipod off and put any unneeded distractions away. It is
important for you to be focused on math (and nothing else) while in the classroom
– It will help you learn.
 Read your textbook and study the problems and examples it provides.
 Do Your Homework.
o As with all college classes, plan on doing at least 2 hours of study outside class
for every hour spent in class.
o Use Hawkes Learning System Help features and practice opportunities to master
each topic.
o Homework is a required part of your learning and you need your homework scores
to pass this class – so make this a regular part of your study plan.
 Ask for help.
o As soon as you have problems - Don’t wait until it is too late to recover from these
problems as you might miss your chance for doing well in (or passing) the course.
o See your instructor before/after class or come by his/her office.
o Go to the Aspire Learning Zone (104-357, Barton Campus)
o Go to the Math Center (see locations at each campus and schedules at:
http://gvltec.edu/instructional_support/).
o Get a private tutor - There is free tutoring at GTC both through the ALZ on the
Barton campus and Instructional Support Program (ISP).
o Contact Your Academic Coach – An Academic Coach will be associated with this
course. The Academic Coach is available to assist students with learning success
strategies such as: study skills, time management, and accessing campus
resources. Students may connect with their Academic Coach through the
ASPIRE Learning Zone (104-357). The ALZ serves as the learning and support
center for all Developmental Studies Students.
Additional information on these programs with schedules and locations can be found at
the GTC website (www.gvltec.edu) under Academic and Instructional Support/Tutoring
Programs at: http://gvltec.edu/tutoring/.
Math 101: Beginning Algebra Course Kit
pg 75
Greenville Tech Policies and Learning Resources
Greenville Tech has policies and learning resources that have been developed and designed to
help learners succeed. The following documents include information and guidelines for how to
access resources and complete information on time. It is important that you read through them,
understand your opportunities and your responsibilities and make the most of the supportive
learning environment that has been designed with your success in mind.
Developmental Studies Department Policies
(linked through Blackboard Course Content)
Arts & Sciences Division Policies
(linked through Blackboard Course Content)
Important Dates Fall 2011:
August 15
August 15-19
September 5
October 10-11
October 26
November 23-25
December 5
December 6-12
Classes begin
Add/Drop Week
Labor Day Holiday (M) No Classes
Fall Break (M-T) No Classes
Last Day to Withdraw from 15 Week Classes (W)
Thanksgiving Holiday (W-F) No Classes
Last Day of Classes (M)
Final Exams (T-M)
Math 101: Beginning Algebra Course Kit
pg 76
Appendix C: Activities
Developmental Studies
Activity
Title: Intercepts
Submitted by: Habib Aghdami
Duration: 1 class period
Type: Individually or in groups
Subject/Unit Focus: This activity is designed for Beginning Algebra (Math 101); the section on “Graphing Linear
Equations in Two Variables”
Purpose of Activity: This session is related to the concept of using intercepts to graph a linear equations in two
variables, as well as vertical and horizontal lines.
Objective(s) addressed: By the end of the session, learners will:
1.
Find the x- and y-intercepts of a linear function.
2.
Graph a linear function using the x- and y-intercepts.
3.
Graph vertical and horizontal lines.
Description of Activity:
This activity builds skills necessary for identifying equations of diagonal, horizontal and vertical lines, also graphing
linear equations by choosing the x- and y-intercepts.
Materials Needed: Intercepts Activity/Discussion Sheet
References: This activity is designed for Beginning Algebra (Math 101); the section on “Graphing Linear
Equations in Two Variables”
Support Materials: Overhead or a Computer, Answers to Intercepts Activity/Discussion Sheet
Math 101: Beginning Algebra Course Kit
pg 77
Intercept Activity/Discussion Sheet
Name______________________________
Practice Problems 1a - 1b:
Graph each linear function by finding x- and y-intercepts.
1a. 2x - 3y = -6
1b. x = 3y
Practice Problems 2a - 2b:
Graph each linear equation.
2a. x = 4
2b. y + 5 = 0
Math 101: Beginning Algebra Course Kit
Answers Intercept Activity/Discussion Sheet
Answer/Discussion to 1a
2x - 3y = -6
Step 1: Find the x- and y- intercepts.
Let's first find the x-intercept.
What value are we going to use for y?
You are correct if you said y = 0.
*Find x-int. by replacing y with 0
*Inverse of mult. by 2 is div. by 2
The x-intercept is (-3, 0).
Next we will find the y- intercept.
What value are we going to plug in for x?
If you said x = 0, you are right.
*Find y-int. by replacing x with 0
*Inverse of mult. by -3 is div. by -3
The y-intercept is (0, 2)
pg 78
Math 101: Beginning Algebra Course Kit
pg 79
Step 2: Find at least one more point.
We can plug in any x value we want as long as we get the right corresponding y value
and the function exists there.
Let's put in an easy number x = 1:
*Replace x with 1
*Inverse of add 2 is sub. 2
*Inverse of mult. by -3 is div. by -3
So the ordered pair (1, 8/3) is another solution to our function.
Note that we could have plugged in any value for x: 5, 10, -25, ..., but it is best to keep
it as simple as possible.
The solutions that we found are:
x
y
(x, y)
-3
0
(-3, 0)
0
2
(0, 2)
1
8/3
(1, 8/3)
Math 101: Beginning Algebra Course Kit
Step 3: Plot the intercepts and point(s) found in steps 1 and 2.
Step 4: Draw the graph.
Answer/Discussion to 1b
pg 80
Math 101: Beginning Algebra Course Kit
x = 3y
Step 1: Find the x- and y- intercepts.
Let's first find the x-intercept.
What value are we going to use for y?
You are correct if you said y = 0.
*Find x-int. by replacing y with 0
The x-intercept is (0, 0).
Next we will find the y- intercept.
What value are we going to plug in for x?
If you said, x = 0 you are right.
*Find y-int. by replacing x with 0
The y-intercept is (0, 0)
Step 2: Find at least one more point.
Since we really have found only one point this time, we better find two additional
solutions so we have a total of three points.
We can plug in any x value we want as long as we get the right corresponding y value
and the function exists there.
Let's put in an easy number x = 1:
pg 81
Math 101: Beginning Algebra Course Kit
pg 82
*Replace x with 1
*Inverse of mult. by 3 is div. by 3
So the ordered pair (1, 1/3) is another solution to our function.
Let's put in another easy number x = -1:
So the ordered pair (-1, -1/3) is another solution to our function.
The solutions that we found are:
x
y
(x, y)
0
0
(-3, 0)
1
1/3
(1, 1/3)
-1
-1/3
(-1, -1/3)
Step 3: Plot the intercepts and point(s) found in steps 1 and 2.
Math 101: Beginning Algebra Course Kit
Step 4: Draw the graph.
Answer/Discussion to 2a
x=4
pg 83
Math 101: Beginning Algebra Course Kit
pg 84
This is in the form x = c.
So, what type of line are we going to end up with?
Vertical.
Step 1: Find the x- and y- intercepts.
AND
Step 2: Find at least one more point.
Since this is a special type of line, I thought I would talk about steps 1 and 2 together.
It does not matter what y is, as long as x is 4.
Note that the x-intercept is at (4, 0).
Do we have a y-intercept? The answer is no. Since x can never equal 0, then there
will be no y-intercept for this equation.
Some points that would be solutions are (4, 0), (4, 1), and (4, 2).
Again, I could have picked an infinite number of solutions.
The solutions that we found are:
x
y
(x, y)
4
0
(4, 0)
4
1
(4, 1)
4
2
(4, 2 )
Step 3: Plot the intercepts and point(s) found in steps 1 and 2.
Math 101: Beginning Algebra Course Kit
Step 4: Draw the graph.
Answer/Discussion to 2b
y+5=0
If you subtract 5 from both sides, you will have y = -5. It looks like it fits the form y =
c.
With that in mind, what kind of line are we going to end up with?
Horizontal.
pg 85
Math 101: Beginning Algebra Course Kit
pg 86
Step 1: Find the x- and y- intercepts.
AND
Step 2: Find at least one more point.
Since this is a special type of line, I thought I would talk about steps 1 and 2 together.
It doesn't matter what x is, y is always -5. So for our solutions we just need three
ordered pairs such that y = -5.
Note that the y-intercept (where x = 0) is at (0, -5).
Do we have a x-intercept? The answer is no. Since y has to be -5, then it can never
equal 0, which is the criteria of an x-intercept.
So some points that we can use are (0, -5), (1, -5) and (2, -5). These are all ordered
pairs that fit the criteria of y having to be -5.
Of course, we could have used other solutions, there are an infinite number of them.
The solutions that we found are:
x
y
(x, y)
0
-5
(0, -5)
1
-5
(1, -5)
-1
-5
(1, -5)
Step 3: Plot the intercepts and point(s) found in steps 1 and 2.
Math 101: Beginning Algebra Course Kit
Step 4: Draw the graph.
pg 87