Download Lesson Plan 6.1

Lesson Plan 6.1
Teacher Candidate:
Chiara Shah
Grade Level/Subject
9th/Algebra I
Unit Title
Unit 4: Solving and Graphing Inequalities
Lesson Title
6.1 Solving One-Step Linear Inequalities
Duration
45 Minutes
Lesson Outcomes
Students will be able to:
1) Make a conjecture about what happens when multiplying or dividing
inequalities by a negative number (NJCCCS 4.1.A.1, 4.1.A.2, 4.1.A.3,
4.1.B.1, 4.3.B.1; NCTM 2.2, 9.1, 9.2; NJPTS 7.iii4, 7.iii.5)
2) Graph inequalities, as evidenced by their successful completion of
problems 23-29 odd on page 337 of their textbook. (NJCCCS 4.1.A.1,
4.1.A.2, 4.1.B.1, 4.B.3.1; NCTM 5.1, 10.4; NJPTS 7.ii.4, 7.iii.5)
3) Solve inequalities, as evidenced by their successful completion of
problems 15, 17, and 19 on page 337 of their textbook. (NJCCCS
4.1.B.1, 4.3.B.1; NCTM 9.2; NJPTS 7.iii.4, 7.iii5)
Standards NJCCCS
STANDARD 4.1 (NUMBER AND NUMERICAL OPERATIONS) ALL
STUDENTS WILL DEVELOP NUMBER SENSE AND WILL
PERFORM STANDARD NUMERICAL OPERATIONS AND
ESTIMATIONS ON ALL TYPES OF NUMBERS IN A VARIETY OF
WAYS.
A. Number Sense
1. Extend understanding of the number system to all real numbers.
2. Compare and order rational and irrational numbers.
3. Develop conjectures and informal proofs of properties of number systems
and sets of numbers.
B. Numerical Operations
1. Extend understanding and use of operations to real numbers and algebraic
procedures.
STANDARD 4.3 (PATTERNS AND ALGEBRA) ALL STUDENTS WILL
REPRESENT AND ANALYZE RELATIONSHIPS AMONG VARIABLE
QUANTITIES AND SOLVE PROBLEMS INVOLVING PATTERNS,
FUNCTIONS, AND ALGEBRAIC CONCEPTS AND PROCESSES.
B. Functions and Relationships
1. Understand relations and functions and select, convert flexibly among, and
use various representations for them, including equations or inequalities, tables,
and graphs.
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Lesson Plan 6.1
Standards NCTM
Standard 2: Knowledge of reasoning and Proof
2.2
Make and investigate mathematical conjectures
Standard 5: Knowledge of Mathematical Representation
5.1
Use representations to model and interpret physical, social, and
mathematical phenomena
Standard 9: Knowledge of Number and Operations
9.1
Analyze and explain the mathematics that underlies the procedures used
for operations involving integers, rational, real, and complex numbers
9.2
Use properties involving number and operations, mental computation,
and computational estimation
Standard 10: Knowledge of Different Perspectives on Algebra
10.1 Analyze patterns, relations, and functions of one and two variables
10.4 Use mathematical models to represent and understand quantitative
relationships
Standards NJPST
7. Standard Seven: Special Needs. Teachers shall adapt and modify
instruction to accommodate the special learning needs of all students.
iii. Teachers engage in activities to:
(4) Meet the needs of all learners by using a wide range of teaching
techniques to accommodate and modify strategies, services and resources,
including technology; and
(5) Make appropriate provisions, in terms of time and circumstances for work,
task assigned, communication and response modes, for individual students who
have particular learning differences or needs.
Modifications &
Accommodations
Materials and Use of
Instructional
Technology
1) Calculators will be provided to those students who need them. (NJPTS
7.iii.4, 7.iii.5)
Item
1) Textbook
2) Calculators
Description
1)
2)
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Lesson Plan 6.1
PROCEDURES
Attendance, Announcements, Homework
Today is the start of a new chapter, chapter 6, Solving and Graphing Inequalities
Anticipatory Set
This activity targets learner outcome #1: Make a conjecture about what happens when
multiplying or dividing inequalities by a negative number (NJCCCS 4.1.A.1, 4.1.A.2,
4.1.A.3, 4.1.B.1, 4.3.B.1; NCTM 2.2, 9.1, 9.2; NJPTS )
DO NOW – Investigate what happens to an inequality when it is multiplied or divided by positive
and negative numbers.
Write the following table on the board.
Multiply by 2
Multiply by -2
Divide by 2
Divide by -2
4<8
5 > −2
3x < −7
Have students complete it as soon as they arrive in class.
Ask students for the answers for each cell.
(Do not switch direction of symbols yet, unless class appears to already know this rule or
unless they notice something is wrong.)
Multiply by 2
Multiply by -2
Divide by 2
Divide by -2
4<8
−4 < −8
−4 < −8
5 > −2
10 > −4
−10 > 4
5
> −1
2
5
− >1
2
1< 2
−1 < −2
3x < −7
6x < −14
−6x < 14
3x −7
<
2
2
3x 7
−
<
2 2
Go through each cell in the table above and ask: Is this true? Is this true? How about this one?
(Note: Tell the students, we are not sure if the ones with the variables are true because it
depends on what number we plug in for x. Right now, because we haven’t solved for
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Lesson Plan 6.1
x, x can be any value we wish. BUT, regardless, to make this inequality match the
original, the symbol needs to be switched.)
A: The inequalities in the cells circled above are not true.
Q: How can we make them all true?
A: Change the direction of the inequality sign.
Q: Can anyone spot a pattern?
A: We multiplied or divided by a negative number.
Q: Do you think that every time you multiply or divide by a negative number that you’ll need to
reverse the inequality symbol?
A: Yes.
Q: Can we make this a rule?
A: Yes.
Write rule on the board.
RULE: When you multiply or divide an inequality by a
negative number you need to change the direction of the
inequality symbol.
Let’s look at the third column of the table. We couldn’t tell earlier if the inequality was
true or false because of the variable. But now that we know this rule, we need to change
the direction of the inequality symbol. Do it now.
Multiply by 2
Multiply by -2
Divide by 2
Divide by -2
4<8
−4 < −8
−4 > −8
5 > −2
10 > −4
−10 < 4
5
> −1
2
5
− <1
2
1< 2
−1 > −2
4
3x < −7
6x < −14
−6x > 14
3x −7
<
2
2
3x 7
−
>
2 2
Lesson Plan 6.1
Practice problems: Students should work independently on the following problems. Page
333, problems 8-17. This is a Section/Chapter pre-work page that investigates the symbol
reversal issue.
NOTE: These problems assess if learner outcome #1 was met.
Lesson
PART I: Graphing Inequalities
The following activities targets learner outcome #2: Graph inequalities, as evidenced by their
successful completion of problems 23-29 odd on page 337 of their textbook. (NJCCCS
4.1.A.1, 4.1.A.2, 4.1.B.1, 4.B.3.1; NCTM 5.1, 10.4)
While students are working on the above problems independently, put a new table on the board.
Statement
Inequality
“All real numbers less than two”
“All real numbers greater than negative two”
“All real numbers less than or equal to one”
“All real numbers greater than or equal to
zero”
Graph
Work with students to fill in the table.
Explain how to graph using open (“less than” or “greater than”) and closed (“less than or equal to”
or “greater than or equal to”) dots.
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Lesson Plan 6.1
Statement
Inequality
“All real numbers less than two”
x<2
“All real numbers greater than negative two” x > −2
“All real numbers less than or equal to one”
x ≤1
“All real numbers greater than or equal to
x≥0
zero”
Graphing: Independent Practice problems: page 337, 23-45 odd.
These problems assess if learner outcome #2 was met.
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Graph
On board
On board
On board
On board
Lesson Plan 6.1
PART II: Solving Inequalities
Learner Outcome #3: Solve inequalities, as evidenced by their successful completion of problems
15, 17, and 19 on page 337 of their textbook. (NJCCCS 4.1.B.1, 4.3.B.1; NCTM 9.2)
Solving in equalities is the same as solving equalities. We need to isolate the variable.
Q: What does isolate mean?
A: Get the variable by itself.
The only thing that’s different is that we need to make sure we change direction of the symbol.
Q: When do we change direction of the symbol?
A: When multiplying or dividing by a negative number.
Q: How do we solve this?
On board:
x−3= 5
A: Add three to both sides.
Q: So how do you think we solve this?
On board:
x−3<5
A: Add three to both sides.
On board:
x−3<5
+3 +3
x<8
Right. Let’s try a few more. I’m going to right six problems on the board. I want you to work on
them at your desk. I’ll walk around to help if you need it.
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Lesson Plan 6.1
On board:
1. x + 6 ≥ 10
subtract six from both sides
2. x − 5 ≤ 3
add 5 to both sides
3. −2 > n − 4
add four to both sides
a
≤ 12
3
multiply both sides by 3
4.
5. −5m > 10
divide both sides by -5 … don’t forget to
switch signs
1
6. − x ≤ 3
2
multiply both sides by -2… don’t forget to
switch signs
Solving Inequalities. Independent Practice. Page 337, Problems 15,17, 19.
These problems assess if learner outcome #3 was met.
Closure
Question for Students
1) When do we need to change
direction of an inequality symbol?
2) How to we solve inequalities?
3) When do we use an open dot and
when do we use a closed dot when
graphing inequalities?
Answer
1) When we multiply or divide by a negative number
2) The same way as equalities, except need to switch
the inequality symbol if we multiply or divide by a
negative number
3) Open dot is used for “less than” and “greater than”
and a closed dot is used for “greater than or equal to”
and “less than or equal to”
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Lesson Plan 6.1
Homework
Page 337, 22-44 even.
Reflection
What Went Well (Strengths of lesson)
Students in both periods noticed immediately that something was wrong when they started
multiplying the inequalities by negative numbers. This is constructivism at its greatest!
As soon as a student raised his hand, I asked, “Does anyone else notice something strange
going on?” I let a few more students nod or raise their hands before asking, “What do you
think is happening? WHY is this happening?”
What Didn’t Go Well (Weaknesses of lesson)
Because this is not my class and I have not taught them the way I intend to teach this when
it’s my own class, they didn’t get the answer to this question, or they didn’t really understand
it. The answer is that negative means opposite, or a flip through the origin on the number
line. Because of the negative everything in the problem gets flipped across the number line.
Therefore, what was smaller is now larger and what was larger is now smaller.
Instead, I told them that “Negative means opposite, right? So we need to do the opposite.
The opposite of “greater than” is “less than.” The opposite of less than” is “greater than.”
Also, I said we need to do this because it’s obvious from the numbers on the board that the
sign needs to be switched.
Even still, some students didn’t understand why you need to change direction of inequality
symbol when there are variables in the inequality. It took a few more examples of
multiplying regular, non-variable inequalities, by negative numbers for them to be
comfortable with the idea that it’s got to happen with all inequalities that are
multiplied/divided by negatives.
Notes to Self/Notes for Next Time
When its my own class this changing directions of inequality symbol would be a GREAT
writing assignment. Don’t tell them why it happens in class. Assign them to complete a
reflection asking “Why do you think we need to change the inequality symbol? What is
happening when you multiply or divide by a negative?”
Actually, this little chart and activity would be a great independent problem for students to
do after they finish the chapter 5 test. Then, Icould assign the reflection writing assignment
as homework the night of the test, and they can come in today, on the first lesson of the
chapter, prepared to answer “WHY?” and begin solving inequalities. I could probably
combine sections 6.1 (solving one-step inequalities) with section 6.2 (solving multistep
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Lesson Plan 6.1
inequalities) into one day’s instruction.
Professional Development Recommendations
By changing this lesson to include a reflection writing assignment, I would be targeting the
following NJPTS:
1. Standard One: Subject Matter Knowledge. Teachers shall understand the central concepts, tools of
inquiry, structures of the discipline, especially as they relate to the New Jersey Core Curriculum Content
Standards (CCCS), and design developmentally appropriate learning experiences making the subject
matter accessible and meaningful to all students.
iii. Teachers engage in activities to:
(1) Promote the development of critical and creative thinking, problem solving and decision making skills by
engaging students in formulating and testing hypotheses according to the methods of inquiry and standards of
evidence within the discipline;
AND
4. Standard Four: Instructional Planning and Strategies. Teachers shall understand instructional
planning, design long and short term plans based upon knowledge of subject matter, students,
community, and curriculum goals, and shall employ a variety of developmentally appropriate strategies
in order to promote critical thinking, problem solving and the performance skills of all learners.
ii. Teachers value and are committed to the development of students’ critical thinking, independent problem
solving and performance capabilities.
AND
1. Standard One: Subject Matter Knowledge. Teachers shall understand the central concepts, tools of
inquiry, structures of the discipline, especially as they relate to the New Jersey Core Curriculum Content
Standards (CCCS), and design developmentally appropriate learning experiences making the subject
matter accessible and meaningful to all students.
i. Teachers know and understand:
(3) That literacy skills and processes are applicable in all content areas and help students to develop the
knowledge, skills and dispositions that enable them to construct meaning and make sense of the world through
reading, writing, listening, speaking and viewing; and
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