TECHNOLOGY Lesson Plans
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2.6 Properties of Equality (Day 2)
NJ Core Curriculum
Content Standards
4.2A Geometric Properties
1. Use reasoning and some form of proof to verify or refute conjectures and theorems.
4.3D Patterns and Algebra: Procedures
5.
Understand and apply the properties of operations,
numbers, equations, and inequalities.
Behavioral
Objectives
1. Level One: Students will be able to identify examples of the Addition, Subtraction, Multiplication, Division, and Substitution Properties of Equality.
2. Level Two: Students will be able to form their own examples of the Addition, Subtraction, Multiplication, Division, and Substitution Properties of Equality.
3. Level Three: Students will be able to use the Addition, Subtraction, Multiplication, Division, and Substitution Properties of Equality to verify or refute conjectures.
Materials
§ Colored Pencils (one per student)
§
Powerpoint Presentation –
Properties of Equality
§
Transparencies
§
Worksheets
§
Journals
Motivation
Previously, the students learned about the Reflexive, Symmetric, and Transitive Properties of Equality and Congruence. Today, they will learn the remaining Properties of Equality. After learning these properties, the students will have the skills to form logical arguments about segments and angles. Because most geometry students seem to have weak algebraic ability, these properties should also give them a set of guidelines to follow in solving equations in the future.
Procedure
1. The teacher will instruct the class to work on a warm-up activity in their journals, as they enter the room. This warm-up activity will require students to answer a few HSPA sample questions. Because there are many Juniors in this class who will be taking the HSPA, it is important to review this material as often as possible in this class. The students will be encouraged to work with partners if they want to, in order to complete these problems.
2.
The teacher will review these problems on the chalkboard
and/or orally. Students will be
encouraged to ask questions about these problems.
3. The teacher will inform students that they will be learning about the remaining Properties of Equality during today’s lesson. The teacher will provide a sample problem of each property before introducing the name of each property. This will be done on a PowerPoint presentation. Students will be instructed to record the notes and examples from the Powerpoint into their notebooks.
4. After all of the properties have been taught, the teacher will orally review Exercises 1-20 on pages 91 and 92 of the textbook. These exercises incorporate the properties learned during today’s lesson, as well as during the previous lesson. It provides a good cumulative review of all of the Properties of Equality and Congruence.
5. The students will then be instructed to complete a journal entry in which they must create their own examples of each Property of Equality.
6. While students are completing their journal entries, the teacher will pass out a worksheet for the students to complete for homework.
Questions
Before
§ Given this diagram on the board:
Ø What is an example of the Reflexive Property of Congruence?
Ø What is an example of the Symmetric Property of Equalilty?
Ø What is an example of the Transitive Property of Congruence?
During
§ After each example problem is shown to the students, ask:
What operation was performed to both sides of this equation?
Is the equation still true?
§ While the students are answering questions from the book, they will be
repeatedly asked:
Ø What property is demonstrated in this exercise?
After
§ Students will be instructed to work on a journal entry in which they must complete the following task:
Ø Create your own example of each Property of Equality.
Assessments and
feedback
The students will be assessed in several ways during this lesson. The teacher will review the responses to the warm-up activity, as well as student corrections, following today’s lesson. This will allow the teacher to determine which types of HSPA problems students are struggling with. There will be many opportunities for the teacher to assess students’ understanding of the material through oral responses to questions, and while exercises in the book are reviewed. The most informative assessment will come in the form of today’s journal entry. The teacher will be able to conclude whether or not students truly understand the differing properties, based on their own examples.
Homework
2.6 Practice A* worksheet
This worksheet requires the students to match statements with the properties illustrated, name properties illustrated by statements, and complete arguments with these properties, creating “informal proofs”. This worksheet assesses their knowledge of content in today’s lesson, along with their knowledge of content from the previous lesson.
Sources
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3.3 Angles Formed by Transversals & 3.4 Parallel
Lines and Transversals
NJ Core Curriculum
Content Standards
4.2A Geometric Properties (grade 8)
1.
Understand and apply concepts involving lines, angles,
and planes
4.2A Geometric Properties (grades 9-12)
3. Apply the properties of geometric shapes.
§ Parallel lines – transversal, alternate interior angles, corresponding angles
Objectives
Level One: Students will be able to identify and measure the eight angles formed by the intersection of parallel lines and a transversal.
NJCCCS:
4.2A (grade 8, grades 9-12)
Level Two: Students will be able to identify the relationships between pairs of angles formed by the intersection of parallel lines and a transversal.
NJCCCS:
4.2A (grade 8, grades 9-12)
Level Three: Students will be able to compare and contrast the relationships between pairs of angles formed by the intersection of parallel lines and a transversal.
NJCCCS:
4.2A (grades 9-12)
Materials
Motivation
Prior to this lesson, the students studied the properties of perpendicular lines. This lesson will allow the students to perform a discovery activity in which they make conjectures about the properties of parallel lines. The students will follow step-by-step instructions in order to construct a pair of parallel lines cut by a transversal. The students will then measure each angle formed by this intersection, and compare and contrast the relationships between specific pairs of angles. The students will be encouraged to connect prior knowledge about angles to parallel lines and transversals.
1.
The teacher will begin the lesson by reviewing the
homework from the previous lesson.
2. Next, the teacher will explain to the students that they will be investigating the properties of parallel lines through a Geometer’s Sketchpad activity, and lead the students in beginning the first task. Each student should then be given the instructions to complete the activity individually.
3. The teacher should circulate the room while students are working individually, to ensure that students are on task. If there are questions that are repeatedly asked, the teacher should lead the students in completing difficult steps in the activity. However, due to the fact that this is a discovery activity, students should be encouraged to work on their own or with a partner prior to asking the teacher for assistance.
4.
When
the students are finished creating their sketches, they should be instructed to
print out their results.
The teacher should then instruct the students to
complete the remainder of the activity based on their findings.
Again, they should be allowed to work with a partner
if they need assistance.
Differentiated
Instruction and Inclusion
It is expected that students will need much individualized assistance when completing this activity. The students will be guided in completing the beginning of the activity, but will be required to work individually to complete the remainder of the activity. During this time, the teacher should assist each student so that they have adequate time to complete the sketches. Some students have difficulty following series of instructions, and therefore it might be helpful to allow those students to work with partners or provide visual representations of the steps of the activity.
Assessments and
Feedback
During this lesson, the teacher will be able to assess students’ progress and results while circulating the room. Since students will all be working at various paces throughout this lesson, the teacher should observe each student and provide assistance when necessary. The teacher can monitor each student’s computer screen, and/or review the results when printed as additional forms of assessment. During the following lesson, the teacher can assess students’ understanding when the discussion of their findings occurs.
Reflections/Self-Evaluations
§
Students enjoyed working in computer lab; they
worked well and remained on task
§
In future – might be beneficial for students to
work in pairs in order to complete task quicker, allowing more time for
discussion
§
Some printouts were on two separate sheets – in
future, instruct students to make sketches small enough to fit on one sheet
Homework
Properties of
Parallel Lines* (if not finished)
Sources
Bennett, Dan. Exploring Geometry with The Geometer’s Sketchpad. Key Curriculum Press. Emeryville, 1999.
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Graphing Linear Equations
(Chapter
4 Graphing Calculator Activity)
Objectives / NJ
Core Curriculum Content Standards
Level One:
Students will be able to apply their knowledge of linear equations to
create graphs using the graphing calculator.
4.3 Patterns and Algebra
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.
Level Two:
Students will be able to compare and contrast representations of linear
equations on the graphing calculator through cooperative learning.
4.3 Patterns and Algebra
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.
2.
Analyze and explain the general
properties and behavior of functions of one
variable, using appropriate graphing technologies.
§ Slope of a line or curve
§ Domain and range
§ Intercepts
§ Rates of change
D. Procedures
2. Select and use appropriate methods to solve equations and inequalities.
§ All types of equations using graphing, computer, and graphing calculator techniques.
Level Three:
Students will be able to make and verify conjectures about linear
equations through cooperative learning and the use of the graphing calculator.
4.3 Patterns and Algebra
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.
3.
Analyze and explain the general properties and behavior of functions of
one
variable, using appropriate graphing technologies.
§ Slope of a line or curve
§ Domain and range
§ Intercepts
§ Rates of change
D. Procedures
2. Select and use appropriate methods to solve equations and inequalities.
§ All types of equations using graphing, computer, and graphing calculator techniques.
Materials
§
Worksheets
-
Graphing
Calculator Exploration (Sheets 1 and 2)
§
Transparency
- Graphing Calculator Warm-Up
-
“Ticket to Leave”
Activity
Motivation
Prior to this lesson, the students have learned how to graph linear equations using a variety of methods. During today’s lesson, the students will manipulate linear equations and create representations of problems on the graphing calculators. The students will compare and contrast their results with the results of their classmates. The teacher will begin this lesson with a real-world problem that can be solved through the representation of a linear equation. The students will be able to relate this idea to their lives. This will motivate the students to manipulate similar equations and create graphs of such equations in order to solve real-world problems.
Procedure
1. The teacher will instruct students to work on two warm-up problems, presented on the overhead projector, when they enter the room. After the students have been given some time to complete these problems, the teacher will encourage students to share their ideas with the class. The techniques used by the students in solving these problems will be techniques used during today’s graphing calculator activity.
Ø
How can
we represent this problem using an equation?
Ø
What
would be the x and y intercepts of this line?
2. The teacher will then introduce today’s lesson by telling the students that they will be manipulating various linear equations using ideas from chapter four, and creating graphical representations of linear equations using the graphing calculators. The teacher should also explain that this activity requires cooperative learning, and the students must work with partners in order to complete each task.
3.
The teacher will distribute a
graphing calculator and a
Graphing Calculator Exploration
worksheet to each student.
If students are not sitting in pairs, the teacher
should rearrange seats to organize the students in this manner.
One student in the pair will receive the
Graphing Calculator
Exploration – Sheet 1 and the other student in
the pair will receive the
Graphing Calculator Exploration – Sheet 2.
4. The teacher will guide the students in completing the first problem on their worksheets. Because the problems are slightly different on Sheets 1 and 2, the teacher will model one of these problems and instruct the other students to modify their equations as needed. The teacher will use a graphing calculator that will be projected onto a screen. Since these students do not have much experience using the graphing calculator, the teacher will lead them through each step. The teacher will also use a large diagram of a graphing calculator to show the students where each button is located for the various tasks they must complete.
Questions (after
students have completed Problems 1 and 2):
Ø
What are
the x and y intercepts of this line?
Ø
What is
the equation of the line, written in slope-intercept form?
Ø
How is
your line similar to your partner’s line?
Ø
How is
your line different from your partner’s line?
Questions:
Ø
What is
the constant of variation for this equation?
Ø
What is
the equation of the line?
Ø
By
looking at your graphs, how do you know that this graph represents a direct
variation equation?
Ø
If the
constant of variation is changed, how does the graph change?
Ø
What
would be some examples of equations that are not direct variation equations?
How would the graphs of these equations be different from the graphs of
direct variation equations?
6. The students will complete a “Ticket to Leave” activity. This activity will require the students to complete two tasks related to the ideas from today’s lesson:
y = -5x + 7
Ø Given this table, explain why this information represents a direct variation equation. Write the direct variation equation.
Differentiated
Instruction and Inclusion
Because many of the students do not have much familiarity with graphing calculators, it is imperative that the teacher circulates the room and assists individual students when necessary. The teacher will have the opportunity to do this while students are completing tasks and working with their partners. Students should also be encouraged to ask their partners for assistance, if possible. The teacher should lead the class in completing as much of the Graphing Calculator Exploration worksheets as possible, based on the general pace of the students. As a result, the students might be instructed to complete the remainder of the activity for homework.
Assessments and
feedback
The students will be assessed in multiple ways throughout the lesson.
Throughout the lesson, the teacher should circulate the room to observe
the students’ progress while working on the graphing calculators and listen to
discussion between partners. After
each problem, the teacher will lead the class in discussion about their results
through use of the questions on the worksheets.
The teacher will randomly call on students to share their findings, in
order to gain a fair assessment of general understanding.
At the end of the period, the teacher will collect and review the “Ticket
to Leave” activity responses in order to evaluate conceptual understanding
Homework
If the students do not finish the Graphing Calculator Exploration worksheets, they will be required to complete these problems for homework.
Sources
Larson, Ron, Laurie Boswell, Timothy D. Kanold, and Lee
Stiff.
Algebra 1.
McDougal Littell.