curriculum activity packets correlation to daily lesson...
TRANSCRIPT
Curriculum Activity Packets Correlation to Daily Lesson Plans
The Georgia Department of Juvenile Justice
8th Grade Mathematics
Units of Instruction Resource Manual
Table of Contents
8th Grade Mathematics
Acknowledgments
Superintendents Letter
Mission and Vision Statements
Chapter 1: Introduction
Chapter 2: Teachers Guide
Chapter 3: Instructional Rotation
Chapter 4: Georgia Performance Standards
Chapter 5: Curriculum Map
Chapter 6: Essential Questions and Enduring Understandings
Chapter 7: Units of Instruction
Unit 1: Principles of Algebra & Rational Numbers
Task 1
Task 2
Task 3
Task 4 Focus CAPs
Unit 2: Graphs, Functions & Sequences
Task 1
Task 2
Task 3
Task 4
Task 5
Task 6 Focus CAPs
Unit 3: Exponents& Roots/ Ratios, Proportions & Similarity/ Percents
Task 1
Task 2
Task 3
Task 4
Task 5 Focus CAPs
Unit 4: Foundation of Geometry & Perimeter, Area & Volume
Task 1
Task 2
Task 3
Task 4 Focus CAPs
Unit 5: Data & Statistics
Task 1
Task 2
Task 3 Focus CAPs
Unit 6: Probability
Task 1
Task 2
Task 3
Task 4 Focus CAPs
Unit 7: Multi-Step Equations & Inequalities
Task 1
Task 2
Task 3
Task 4
Task 5 Focus CAPs
Unit 8: Graphing Lines, Sequences & Functions & Polynomials
Task 1
Task 2
Task 3
Task 4 Focus CAPs
Chapter 8: Task websites
Acknowledgements
The Georgia Department of Juvenile Justice Department of Education would like to thank the many educators who have helped to create this 8th Grade Math Units of Instruction Resource Manual. The educators have been particularly helpful in sharing their ideas and resources to ensure the completion and usefulness of this manual.
Students served by the DJJ require a special effort if they are to become contributing and participating members of their communities. Federal and state laws, regulations, and rules will mean nothing in the absence of professional commitment and dedication by every staff member.
The Georgia Department of Juvenile Justice is very proud of its school system. The school system is Georgias 181st and is accredited by the Southern Association of Colleges and Schools (SACS). The DJJ School System has been called exemplary by the US Department of Justice. This didnt just happen by chance; rather it was the hard work of many teachers, clerks, instructors and administrators that earned DJJ these accolades and accreditations. The DJJ education programs operate well because of the dedicated staff. These dedicated professionals are the heart of our system.
These Content Area Units of Instruction were designed to serve as a much needed tool for delivering meaningful whole group instruction. In addition, this resource will serve as a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs).
I would like to thank all the DJJ Teaching Staff, the Content Area Leadership Teams, Kimberly Harrison, DJJ Special Education/Curriculum Consultant and Martha Patton, Curriculum Director for initiating this project and seeing it through. Thank you all for your hard work and dedication to the youth we serve.
Sincerely yours,
James Jack Catrett, Ed.D.
Associate Superintendent
Mission
The mission of Department of Juvenile Justice Math Consortium (DJJMC) is to build a multiparty effort statewide to achieve continuous, systemic and sustainable improvements in the education system serving the Math students of the Department of Juvenile Justice (DJJ).
Vision
To achieve the mission of the DJJMC, members work collaboratively in examining the Georgia Performance Standards. These guidelines speak specifically to teachers being able to: deliver meaning content pertaining to the Characteristics of Math and its content standards across the Math Units of Instruction Resource Manual. The DJJMC will master and develop whole-group unit lessons built around Curriculum Activity Packets (CAPs), critique student work, and work as a team to solve the common challenges of teaching within DJJ. Additionally, the DJJMC jointly analyzes student test data in order to: develop strategies to eradicate common academic deficits among students, align curriculum, and create a coherent learning pathway across grade levels. The DJJMC also reviews research articles, attends workshops or courses, and invites consultants to assist in the acquisition of necessary knowledge and skills. Finally, DJJMC members observe one another in the classroom through focus walks.
Introduction
The 8th Grade Math Units of Instruction Resource Manual is a tool that has been created to serve as a much needed tool for delivering meaningful whole group instruction. This manual is a supplement to the skills and knowledge provided by the Georgia Department of Juvenile Justice Curriculum Activity Packets (CAPs). It is imperative that our students learn to reason mathematically, to evaluate mathematical arguments both formally and informally, to use the language of mathematics to communicate ideas and information precisely, and to work in cooperative learning groups. Best practices in education indicate that teachers should first model new skills for students. Next, teachers should provide opportunities for guided practice. Only then should teachers expect students to successfully complete an activity independently. The 8th Grade Math Units of Instruction meets that challenge.
The Georgia Department of Juvenile Justice
Office of Education
Direct Instruction Lesson Plan
Teacher:
Subject:______________________________
Date:_____________to__________________
Period
1st
2nd
3rd
4th
5th
6th
Students will engage in:
Independent activities pairing
Cooperative learning hands-on
Peer tutoring Visuals
technology integration Simulations
a project centers
lecture Other
Essential Question(s):
Standards:
CAPs Covered:
Grade Level:____ Unit:______
RTI Tier for data collection: 2 or 3
Tier 2 Students:
Tier 3 Students:
Time
Procedures Followed:
Material/Text
_______
Minutes
Review of Previously Learned Material/Lesson Connections:
Recommended Time: 2 Minutes
_______
Minutes
Display the Georgia Performance Standard(s) (project on
blackboard via units of instruction located at
http://thevillage411.weebly.com/units-of-instruction3.html, or print on blackboard) Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.
Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard). Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.
Recommended Time: 2 Minutes
_______
Minutes
Introduce task by stating the purpose of todays lesson.
Recommended Time: 2 Minutes
_______
Minutes
Engage students in conversation by asking open ended questions related to the essential question(s).
Recommended Time: 2 Minutes
_______
Minutes
Begin whole group instruction with corrective feedback:
Recommended Time: 10 Minutes
_______
Minutes
Lesson Review/Reteach:
Recommended Time: 2 Minutes
_______
Minutes
Independent Work CAPs:
Recommended Time: 30 Minutes
Teacher Reflections:
The Instructional Rotation Matrix has been designed to assist language arts teachers in providing a balanced approach to utilizing the Math Units of Instruction across all grade levels on a rotating schedule.
Monday
Tuesday
Wednesday
Thursday
6th Grade Content
Middle School
9th Grade Content
High School
7th Grade Content
Middle School
10th Grade Content
High School
8th Grade Content
Middle School
11th Grade Content
High School
6th Grade Content
Middle School
12th Grade Content
High School
7th Grade Content
Middle School
9th Grade Content
High School
8th Grade Content
Middle School
10th Grade Content
High School
6th Grade Content
Middle School
11th Grade Content
High School
7th Grade Content
Middle School
12th Grade Content
High School
Georgia Performance Standards
M8A1 Students will use algebra to represent, analyze, and solve problems.
a. Represent a given situation using algebraic expressions or equations in one variable.
b. Simplify and evaluate algebraic expressions.
c. Solve algebraic equations in one variable, including equations involving absolute values.
d.Solve equations involving several variables for one variable in terms of the others.
e.Interpret solutions in problem contexts.
M8A2 Students will understand and graph inequalities in one variable.
a. Represent a given situation using an inequality in one variable.
b. Use the properties of inequality to solve inequalities.
c. Graph the solution of an inequality on a number line.
d. Interpret solutions in problem contexts.
M8A3 Students will understand relations and linear functions.
a. Recognize a relation as a correspondence between varying quantities.
b. Recognize a function as a correspondence between inputs and outputs where the output for each input must be unique.
c. Distinguish between relations that are functions and those that are not functions.
d. Recognize functions in a variety of representations and a variety of contexts.
e. Use tables to describe sequences recursively and with a formula in closed form.
f. Understand and recognize arithmetic sequences as linear functions with whole-number input values.
h. Interpret the constant difference in an arithmetic sequence as the slope of the associated linear function.
i. Identify relations and functions as linear or nonlinear.
j. Translate among verbal, tabular, graphic, and algebraic representations of functions.
M8A4 Students will graph and analyze graphs of linear equations and inequalities.
a. Interpret slope as a rate of change.
b. Determine the meaning of the slope and y-intercept in a given situation.
c. Graph equations of the form y = mx + b.
d. Graph equations of the form ax + by = c.
e.Graph the solution set of a linear inequality, identifying whether the solution set is an open or a closed half-plane.
f.Determine the equation of a line given a graph, numerical information that defines the line, or a context involving a linear relationship.
g.Solve problems involving linear relationships.
M8A5 Students will understand systems of linear equations and inequalities and use them to solve problems.
a.Given a problem context, write an appropriate system of linear equations or inequalities.
b. Solve systems of equations graphically and algebraically, using technology as appropriate.
c.Graph the solution set of a system of linear inequalities in two variables.
d.Interpret solutions in problem contexts.
M8D1 Students will apply basic concepts of set theory.
a. Demonstrate relationships among sets through use of Venn diagrams.
b. Determine subsets, complements, intersection, and union of sets.
c. Use set notation to denote elements of a set.
M8D2 Students will determine the number of outcomes related to a given event.
a. Use tree diagrams to find the number of outcomes.
b. Apply the addition and multiplication principles of counting.
M8D3 Students will use the basic laws of probability.
a. Find the probability of simple independent events.
b. Find the probability of compound independent events.
M8D4 Students will organize, interpret, and make inferences from statistical data.
a. Gather data that can be modeled with a linear function.
b. Estimate and determine a line of best fit from a scatter plot.
M8G1 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.
a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.
b. Apply properties of angle pairs formed by parallel lines cut by a transversal.
c. Understand the properties of the ratio of segments of parallel lines cut by one or more transversals.
d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.
M8G2 Students will understand and use the Pythagorean theorem.
a. Apply properties of right triangles, including the Pythagorean theorem.
b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right triangle.
M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.
a. Find square roots of perfect squares.
b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.
c. Recognize square roots as points and as lengths on a number line.
d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in sign.
e. Recognize and use the radical symbol to denote the positive square root of a positive number.
f. Estimate square roots of positive numbers.
g. Simplify, add, subtract, multiply, and divide expressions containing square roots.
h. Distinguish between rational and irrational numbers.
i. Simplify expressions containing integer exponents.
j. Express and use numbers in scientific notation.
k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.
M8P1 Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M8P2 Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M8P3 Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
b. Communicate their mathematical thinking coherently and clearly to peers, teachers, and others.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M8P4 Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M8P5 Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
M8RC1 Students will enhance reading in all curriculum areas by:
a. Reading in All Curriculum Areas
Read a minimum of 25 grade-level appropriate books per year from a variety of subject disciplines and participate in discussions related to curricular learning in all areas.
Read both informational and fictional texts in a variety of genres and modes of discourse.
Read technical texts related to various subject areas.
b. Discussing books
Discuss messages and themes from books in all subject areas.
Respond to a variety of texts in multiple modes of discourse.
Relate messages and themes from one subject area to messages and themes in another area.
Evaluate the merit of texts in every subject discipline.
Examine authors purpose in writing.
Recognize the features of disciplinary texts.
c. Building vocabulary knowledge
Demonstrate an understanding of contextual vocabulary in various subjects.
Use content vocabulary in writing and speaking.
Explore understanding of new words found in subject area texts.
d. Establishing context
Explore life experiences related to subject area content.
Discuss in both writing and speaking how certain words are subject area related.
Determine strategies for finding content and contextual meaning for unknown words.
DJJ 8th Grade Mathematics
Georgia Performance Standards: Curriculum Map
1st Semester
2nd Semester
Principles of Algebra & Rational Numbers
Graphs, Function & Sequences
Exponents & Roots Ratios, Proportions
& Similarity
& Percents
Foundations of Geometry & Perimeter, Area & Volume
Data & Statistics
Probability
Multi-Step Equations & Inequalities
Graphing Lines, Sequences & Functions & Polynomials
Chapter
1
CAPs
1-6
Chapter
3
CAPs
12-15
Chapter
4
CAPs
16-20
Chapter
7
CAPs
31-36
Chapter
9
CAPs
43-47
Chapter
10
CAPs
48-52
Chapter
11
CAPs
54-57
Chapter
12
CAPs
58-62
2
7-11
5
21-25
8
37-42
13
63-67
6
26-30
14
68-71
GPS:
M8A1a,b,c,d
M8P1a,b,c,d
M8P4a,b,c
M8P3a,c
M8P5a,b,c
M8P2c
M8A2b,c
M8A1a,b
M8P3c
GPS:
M8A1b,d
M8P1a,b
M8P5a,b,c
M8P2c,d
M8P3a,c
M8A4c,e
M8P4a,b,c
M8A3b,c,d,e,i
GPS:
M8N1a,b,c,d,e,f,g,h,i,j,k
M8A1a,b,c,d
M8P1a,b,c,d
M8P2a,b,c,d
M8P3a,c,d
M8P4a,b,c
M8P5a,b,c
M8G2a,b
M8G1a,b,d
M8A4.b
M8N1i,k
GPS:
M8A1a,b,c,d
M8P3a,b,c,d
M8P1a,b,c
M8P2a,c,d
M8G1a,b,d
M8P4a,b,c
M8A1a,b,c,d
M8P5a,b,c
M8G2a
M8N1.k
GPS:
M8P2.c
M8P3a,c,d
M8P4a,b,c
M8P1a,b,c,d
M8P5a,b,c
M8D4.a
GPS:
M8D3a,b
M8P1a,b,c,d
M8P3a,c,d
M8P4a,b ,c
M8P2.c
M8P5a,b,c
M8A1a,c,d
M8D2a,b
GPS:
M8A1a,b,c,d
M8P1b,c,d
M8P5a,b,c
M8P3a,c
M8P4c
M8A2a,b,c,d
M8A5b,c
GPS:
M8A3h,d,e,f,
M8A4a,b,c,d,e,f
M8P1a,b,c,d
M8P3a,c
M8P4a,b,c
M8P5a,b,c
M8A3d,h,i
M8D4b
M8A1a,b,c,d
M8P2a,b,c,d
M8N1i,k
Focus CAPs:
Chapter 1
2 & 6
Chapter 2
7 & 11
Focus CAPs:
12 & 15
Focus CAPs:
Chapter 4
16 & 20
Chapter 5
21
Chapter 6
26
Focus CAPs:
Chapter 7
31 & 36
Chapter 8
37 & 42
Focus CAPs:
43
Focus CAPs:
48 & 52
Focus CAPs:
54 & 57
Focus CAPs:
Chapter 12
58
Chapter 13
63
Chapter 14
68
Enduring Understandings & Essential Question
Principles of Algebra & Rational Numbers
Enduring Understandings:
Algebraic expressions, equations and inequalities are used to represent relationships
between numbers.
Absolute value is used to represent distances between numbers.
Graphs can be used to represent all of the possible solutions to a given situation.
Many problems encountered in everyday life can be solved using equations or
inequalities.
Essential Questions:
How can I simplify and evaluate an algebraic expression?
How can I solve an equation or inequality?
How can I tell the difference between an expression, equation and an inequality?
How can I determine the absolute value of an expression?
How can I represent absolute value on a number line?
Graphs, Functions & Sequences
Enduring Understandings:
Collecting and examining data can sometimes help one discover patterns in the way in which two quantities vary.
Changes in varying quantities are often related by patterns which, once discovered, can be used to predict outcomes and solve problems.
Written descriptions, tables, graphs and equations are useful in representing and investigating relationships between varying quantities.
Different representations (written descriptions, tables, graphs and equations) of the relationships between varying quantities may have different strengths and weaknesses.
Linear functions may be used to represent and generalize real situations.
Slope and y-intercept are keys to solving real problems involving linear relationships.
Essential Questions:
What does the data tell me?
How does a change in one variable affect the other variable in a given situation?
Which tells me more about the relationship I am investigating a table, a graph or an equation? Why?
What strategies can I use to help me understand and represent real situations involving linear relationships?
How can the properties of lines help me to understand graphing linear functions?
How is a linear inequality like a linear equation? How are they different?
Exponents& Roots/ Ratios, Proportions & Similarity/ Percents
Enduring Understandings:
An irrational number is a real number that cannot be written as a ratio of two integers.
All real numbers can be plotted on a number line.
Exponents are useful for representing very large or very small numbers.
Square roots can be rational or irrational.
Some properties of real numbers hold for all irrational numbers.
There are many relationships between the lengths of the sides of a right triangle.
Essential Questions:
When are exponents used and why are they important?
Why is it useful for me to know the square root of a number?
How do I simplify and evaluate algebraic expressions involving integer exponents and square roots?
What is the Pythagorean Theorem and when does it hold?
Foundation of Geometry & Perimeter, Area & Volume
Enduring Understandings:
Parallel lines have the same slope and perpendicular lines have opposite, reciprocal
slopes.
When two lines intersect, vertical angles are congruent and adjacent angles are
supplementary.
When parallel lines are cut by a transversal, corresponding, alternate interior and alternate
exterior angles are congruent.
The length of segments formed by two non-parallel transversals cutting parallel lines is
proportional to the distances of the parallel lines from the intersection of the transversals.
Parallel lines can be constructed using the properties of parallel lines cut by a transversal.
In Euclidean Geometry, there is exactly one line through a given point parallel to a
second given line.
Essential Questions:
How can I be certain whether lines are parallel, perpendicular, or skew lines?
Why do I always get a special angle relationship when any two lines intersect?
When I draw a transversal through parallel lines, what are the special angle and segment
relationships that occur?
What information is necessary before I can conclude two figures are congruent?
How can my knowledge of constructing congruent triangles be used to construct
perpendicular and parallel lines?
Can I find parallel lines that intersect? Why or why not?
Data & Statistics
Enduring Understandings:
Relations show any correspondence between sets, while functions are special relations in which each input from a fixed set is associated with a single output.
Linear functions are defined by constant slope.
Arithmetic sequences are numerical representations of linear functions.
Venn diagrams are visual tools for organizing members of related sets.
Like functions and relations, visualizing sets in multiple representations often reveals unexpected patterns.
Essential Questions:
How can I identify a function?
How can I tell the difference between a relation and a function?
How can I relate arithmetic sequences to linear functions?
When working with sets, when do I use a union, and when do I use an intersection?
Probability
Enduring Understandings:
Tree diagrams are useful for describing relatively small sample spaces and computing
probabilities, as well as for visualizing why the number of outcomes can be extremely large.
Essential Questions:
How do I determine a sample space?
How can a tree diagram help me to find the number of possible outcomes related to a given event?
When and why do I use addition to determine sample space size?
When and why do I use multiplication to determine sample space size?
When and why do I use addition to determine probabilities?
When and why do I use multiplication to determine probabilities?
How can I use probability to determine if a game is fair or to figure my chances of winning the lottery?
Multi-Step Equations & Inequalities
Enduring Understandings:
There are situations that require two or more equations to be satisfied simultaneously.
There are several methods for solving systems of equations.
Solutions to systems can be interpreted algebraically, geometrically, and in terms of
problem contexts.
In some problem contexts, the constraints that must be satisfied are modeled by
inequalities rather than equations.
The number of solutions to a system of equations or inequalities can vary from no
solution to an infinite number of solutions.
Essential Questions:
How can I interpret the meaning of a system of equations algebraically and
geometrically?
How does mathematical notation indicate that equations are to be treated as a system?
What does it mean to solve a system of linear equations?
How can the solution to a system be interpreted geometrically?
How can I recognize how many solutions a system of equations has prior to solving?
How do I decide which method would be easier to use to solve a particular system of
equations?
Why is graphing a system of inequalities a good way to show the solution set?
How can I translate a problem situation into a system of equations or inequalities?
What does the solution to a system tell me about the answer to a problem situation?
Graphing Lines, Sequences & Functions & Polynomials
Enduring Understandings:
Relations show any correspondence between sets, while functions are special relations in which each input from a fixed set is associated with a single output.
Linear functions are defined by constant slope.
Arithmetic sequences are numerical representations of linear functions.
Venn diagrams are visual tools for organizing members of related sets.
Like functions and relations, visualizing sets in multiple representations often reveals unexpected patterns.
Essential Questions:
How can I identify a function?
How can I tell the difference between a relation and a function?
How can I relate arithmetic sequences to linear functions?
When working with sets, when do I use a union, and when do I use an intersection?
Unit: Principles of Algebra & Rational Numbers
Georgia Performance Standards:
M8A1 Students will use algebra to represent, analyze, and solve problems.
a. Represent a given situation using algebraic expressions or equations in one variable.
b. Simplify and evaluate algebraic expressions.
c. Solve algebraic equations in one variable, including equations involving absolute values.
d.Solve equations involving several variables for one variable in terms of the others.
M8P1 Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M8P4 Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M8P3 Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
c. Analyze and evaluate the mathematical thinking and strategies of others.
M8P5 Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
M8P2 Students will reason and evaluate mathematical arguments.
c. Develop and evaluate mathematical arguments and proofs.
M8A2 Students will understand and graph inequalities in one variable.
b. Use the properties of inequality to solve inequalities.
c. Graph the solution of an inequality on a number line.
M8A1 Students will use algebra to represent, analyze, and solve problems.
a. Represent a given situation using algebraic expressions or equations in one variable.
b. Simplify and evaluate algebraic expressions.
M8P3 Students will communicate mathematically.
c. Analyze and evaluate the mathematical thinking and strategies of others.
Selected Terms and Symbols:
Independent events: Events for which the occurrence of one has no impact on the occurrence of the other.
Relative frequency: The number of times an outcome occurs divided by the total number of trials.
Sample space: All possible outcomes of a given experiment.
Event: A subset of a sample space.
Simple Event: An event consisting of just one outcome. A simple event can be represented by a single branch of a tree diagram.
Compound Event: A sequence of simple events.
Complement: The complement of event E, sometimes denoted E (E prime), occurs when E doesnt. The probability of E equals 1 minus the probability of E: P(E) = 1 P(E).
Counting Principle: If an event A can occur in m ways and for each of these m ways, an event B can occur in n ways, then events A and B can occur in ways. This counting principle can be generalized to more than two events that happen in succession. So, if for each of the m and n ways A and B can occur respectively, there is also an event C that can occur in s ways, then events A, B, and C can occur in ways.
Tree diagram: A tree-shaped diagram that illustrates sequentially the possible outcomes of a given event.
Teachers Place:
Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.
1. Explain the activity (activity requirements)
2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at http://thevillage411.weebly.com/units-of-instruction3.html.
3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.
4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)
5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.
6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard.
7. Discuss answers with the students using the following questioning techniques as applicable:
Questioning Techniques:
Memory Questions
Signal words: who, what, when, where?
Cognitive operations: naming, defining, identifying, designating
Convergent Thinking Questions
Signal words: who, what, when, where?
Cognitive operations: explaining, stating relationships, comparing and
contrasting
Divergent Thinking Questions
Signal words: imagine, suppose, predict, if/then
Cognitive operations: predicting, hypothesizing, inferring, reconstructing
Evaluative Thinking Questions
Signal words: defend, judge, justify (what do you think)?
Cognitive operations: valuing, judging, defending, justifying
8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.
9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter)
10. At the end of the *whole group learning session, students will transition into independent CAP assignments.
*The phrase, whole group learning session is utilized rather than, the end of the activity because all of the activities may not be completed in one day.
Task: 1
Resources:
http://www.purplemath.com/modules/nextnumb2.htm
Activity
Part 1:
a) Do the following sequence of operations in order:
b) What did you get for your final number?
c) Check with your partner, what did that person get for their final number?
d) Everyone should have the same number. What number is that?
e) Why did everyone end with the same number?
f) How does this trick work?
Write down any number.
(This is your start number.)
Add to it the number that comes before it.
Add 11. Divide by 2.
Subtract your start number.
Part 2:
Now try this one:
Take the number of your birth month. Add 32.
Add the difference between your birth month number and 12.
Divide by 4.
Add 2.
This is your Lucky Number!
Do you feel lucky? Why or why not? Explain what made this trick work.
Part 3:
Work with a partner to write your own Number Trick. Be sure to list your instructions in the appropriate order. Swap Number Tricks with another group or share your Tricks one at a time with the entire class. See if your friends can describe how your Number Trick works.
Discussion, Suggestions, Possible Solutions
The challenge is to see if students can discover how the trick works. If students need a hint, suggest that instead of using an actual number, they use a box or a letter to begin with.
Part 1:
Write down any number. (This is your start number.)
Start with N. N
Add to it the number that comes before it.
The previous number is N-1 N + (N 1) = 2N 1
Add 11.
Add the 11 to the 2N 1 2N 1 + 11 = 2N + 10
Divide by 2.
Dividing 2N + 10 by 2 2N + 10 = N + 5
2
Subtract your start number.
The start number was N. N + 5 N = 5
This means that everyone should have ended with a 5 regardless of the start number.
Part 2:
Take the number of your birth month.
The birth month will be a number between 1 and 12.
If x is the number of a students birth, then 1 x 12.
Add 32.
Adding 32 gives x + 32.
Add the difference between your birth month number and 12.
When you add the difference between your birth month and 12 you get x + 32 + (12 x) which is equal to 44.
Divide by 4.
Dividing by 4 leaves 11.
Add 2.
Adding 2 yields 13; this is usually considered an unlucky number.
Part 3:
Work with a partner to write your own Number Trick. Be sure to list your instructions in the appropriate order. Swap Number Tricks with another group or share your Tricks one at a time with the entire class. See if your friends can describe how your Number Trick works.
Task: 2
Resources:
http://www.math.com/school/subject2/lessons/S2U4L1GL.html
Activity
The students at Eastman RYDC and Eastman YDC are participating in a game room survey. Eastman RYDC is located 5 miles from the game room and Eastman YDC 3 miles from the game room. The owner, of the game room wonders how far apart the centers are.
On grid paper, pick a point to represent the location of the game room.
Illustrate all of the possible points where Eastman RYDC could be located on the grid paper.
Using a different color, illustrate all of the possible points where Eastman YDC could be located.
What is the smallest distance, d, that could separate the centers?
How did you know?
What is the largest distance, d, that could separate the centers?
How did you know?
Write and graph an inequality in terms of d to show the owner of the game room all of the possible
distances that could separate the two centers.
Discussion, Suggestions, Possible Solutions
Students should understand that the centers could be located anywhere on the circle with the game room as the center and the radius as the distance that they are from the game room.
Eastman RYDC
Eastman YDC
Game Room
Therefore, the closest the centers could be would be 5 3 = 2 miles and the farthest apart that they could be would be 5 + 3 = 8 miles. This may be written as 5 3 = d where d represents the distance from Eastman
Task: 3
Resources:
http://www.math.com/school/subject2/lessons/S2U4L1GL.html
Activity
Part 1:
Your science teacher says the grades, g, in your class can be represented by the inequality |g 85| < 10. What is the lowest grade and what is the highest grade in the class? Explain your thinking.
Part 2:
Suppose the grades in your Language Arts class range from 68 to 94. Represent this information on a number line. Then write a compound inequality to represent the information. Finally, write an absolute value inequality to represent the same information.
Discussion, Suggestions, Possible Solutions
Part 1:
This absolute value inequality means that g 85 10 and g 85 -10.
Another way to write this is known as a compound inequality -10 g 85 10 and may be graphed on a number line.
Using the same thinking process as used when solving equations we have
-10 g 85 10
+85+85 +85
75 g 95
74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96
Part 2:
Suppose the grades in your Language Arts class range from 68 to 94. Represent this information on a number line. Then write a compound inequality to represent the information. Finally, write an absolute value inequality to represent the same information.
67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95
This means that the lowest score included 68 and the highest score included 94. That is why the points are shaded along with all points in between.
One possible answer could be determined by 68 x 94
-81-81-81
-13 x 81 13
Course Title: Course 3 8th grade Ga DJJ
State Code: 27.0230000 CAP: 2
Georgia Performance Standard(s):
M8P1.b Solve problems that arise in mathematics and in other contexts.
M8P4.c Recognize and apply mathematics in contexts outside of mathematics.
M8P5.b Select, apply, and translate among mathematical representations to solve
problems.
M8P5.c Use representations to model and interpret physical, social, and mathematical
phenomena.
M8P2.c Develop and evaluate mathematical arguments and proofs.
M8P3.a Organize and consolidate their mathematical thinking through communication.
M8A1.b Simplify and evaluate algebraic expressions.
M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.
Objective(s):
The student compares and orders integers and evaluates expressions containing absolute values.
The student will be able to add integers.
Instructional Resources:
Holt Mathematics 8th grade Course 3 Pgs. 14-21
Chapter 1 Resource Book (CRB)
One-Stop Planner
Activities:
Read textbook pgs. 14-17.
Complete Think and Discuss, pg. 15.in textbook
Complete Practice and Problem Solving, Problems 1-5, 15-19, and 29-39 on pg. 16.in textbook
Complete Practice A 1-3 CRB, pg. 19.
Complete Reading Strategies 1-3 CRB, pg. 25.
Read textbook pgs. 18-21.
Complete Think and Discuss, pg. 19.in textbook
Complete Practice and Problem Solving, Problems 1-8, 13-20, 31-38, and 49-56 on pgs. 20-21.in textbook
Complete Practice A 1-4 CRB, pg. 27.
Complete Reading Strategies 1-4 CRB, pg. 33.
Evaluation:
Complete Power Presentations Lesson Quiz 1-3, pg. 17 and 1-4, pg. 21.
Modifications:
Performance Tasks: IDEA works CD
Course Title: Course 3 8th grade Ga DJJ
State Code: 27.0230000 CAP: 6
Georgia Performance Standard(s):
M8A2.b Use the properties of inequality to solve inequalities.
M8A2.c Graph the solution of an inequality on a number line.
M8P5.b Select, apply, and translate among mathematical representations to solve
problems
M8P5.c Use representations to model and interpret physical, social, and
mathematical
phenomena.
M8P3.a Organize and consolidate their mathematical thinking through
communication
Objective(s):
The student organizes and reviews key concepts and skills presented in Chapter One.
The student assesses mastery of concepts and skills in Chapter One
Instructional Resources:
Holt Mathematics Course Two Textbook
Chapter 1 Resource Book (CRB)
One Stop Planner
Activities:
Complete the odd number problems on the Study Guide, pgs. 53-54 in textbook.
Have teacher check your work on review. If score is at least 80%, then go on to
Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in
CRB to cover concepts not understood.
Complete Chapter Test pg.55 in textbook.
Have teacher check your work on test. If score is at least 80%, then go on to next CAP. If score is less than 80%, then teacher will give worksheets in CRB covering concepts still not understood.
Evaluation:
Complete Chapter Test pg.55 in textbook with 80% accuracy
Modifications: IDEA Works CD
Course Title: Course 3 8th grade Ga DJJ
State Code: 27.0230000 CAP: 7
Georgia Performance Standard(s):
M8P1.b Solve problems that arise in mathematics and in other contexts.
M8P4.c Recognize and apply mathematics in contexts outside of mathematics.
M8P5.c Use representations to model and interpret physical, social, and mathematical
phenomena.
M8P2.c Develop and evaluate mathematical arguments and proofs.
M8P3.a Organize and consolidate their mathematical thinking through communication.
M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.
M8P4.a Recognize and use connections among mathematical ideas.
M8P4.b Understand how mathematical ideas interconnect and build on one another to
produce a coherent whole.
M8P5.b Select, apply, and translate among mathematical representations to solve
problems.
Objective(s):
The student writes rational numbers in equivalent forms. The student will be able to
compare and order positive and negative rational numbers written as fractions, decimals,
and integers.
Instructional Resources:
Holt Mathematics 8th grade Course 3, Pgs. 60-71
Chapter 2 Resource Book (CRB)
One-Stop Planner
Activities:
Complete Are You Ready in textbook pg., pg. 61.
Read textbook pgs. 60-67
Complete Think and Discuss, pg. 65.in textbook
Complete Practice and Problem Solving, Problems 1-10, 29-38, and 67-79 on pgs. 66-
67.in textbook
Complete Practice A 2-1 CRB, pg. 3.
Complete Reading Strategies 2-1 CRB, pg. 10.
Read textbook pgs. 68-71.
Complete Think and Discuss on pg 69.in textbook
Complete Practice and Problem Solving, Problems 1-4, 10-17, 27, and 42-53 on pgs. 70-
71.in textbook
Complete Practice A 2-2 CRB, pg. 12.
Complete Reading Strategies 2-2 CRB, pg. 18.
Evaluation:
Complete Power Presentations Lesson Quiz 2-1, pg. 67 and 2-2, pg. 71.
Modifications:
Performance Tasks: IDEA works CD
Course Title: Course 3 8th grade Ga DJJ
State Code: 27.0230000 CAP: 11
Georgia Performance Standard(s):
M8A1.a Represent a given situation using algebraic expressions or equations in one
variable.
M8A1.c Solve algebraic equations in one variable, including equations involving absolute values.
M8A1.d Interpret solutions in problem contexts.
M8P4.c Recognize and apply mathematics in contexts outside of mathematics.
M8P5.b Select, apply, and translate among mathematical representations to solve
problems.
M8P3.a Organize and consolidate their mathematical thinking through communication.
M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.
M8P5.c Use representations to model and interpret physical, social, and mathematical
phenomena.
M8P1.a Build new mathematical knowledge through problem solving.
M8P1.b Solve problems that arise in mathematics and in other contexts
Objective(s):
The student organizes and reviews key concepts and skills presented in Chapter Two.
The student assesses mastery of concepts and skills in Chapter Two
Instructional Resources:
Holt Mathematics Course Two Textbook
Chapter 2 Resource Book (CRB)
One Stop Planner
Activities:
Complete the odd number problems on the Study Guide, pgs. 53-54 in textbook.
Have teacher check your work on review. If score is at least 80%, then go on to
Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in
CRB to cover concepts not understood.
Complete Chapter Test pg.55 in textbook.
Have teacher check your work on test. If score is at least 80%, then go on to next CAP.
If score is less than 80%, then teacher will give worksheets in CRB covering concepts
still not understood.
Evaluation:
Complete Chapter Test pg.55 in textbook with 80% accuracy.
Modifications: IDEA Works CD
Unit: Graphs, Functions & Sequences
Georgia Performance Standards:
M8A1 Students will use algebra to represent, analyze, and solve problems.
b. Simplify and evaluate algebraic expressions.
d. Solve equations involving several variables for one variable in terms of the others.
M8P1 Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
M8P5 Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
M8P2 Students will reason and evaluate mathematical arguments.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M8A4 Students will graph and analyze graphs of linear equations and inequalities.
c. Graph equations of the form y = mx + b.
e.Graph the solution set of a linear inequality, identifying whether the solution set is an open or a closed half-plane.
M8P4 Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M8A3 Students will understand relations and linear functions.
b. Recognize a function as a correspondence between inputs and outputs where the output for each input must be unique.
c. Distinguish between relations that are functions and those that are not functions.
d. Recognize functions in a variety of representations and a variety of contexts.
e. Use tables to describe sequences recursively and with a formula in closed form.
i. Identify relations and functions as linear or nonlinear.
Selected Terms and Symbols:
Additive Inverse: The sum of a number and its additive inverse is zero. Also called the opposite of a number. Example: 5 and -5 are additive inverses of each other.
Exponent: The number of times a base is used as a factor of repeated multiplication.
Exponential Notation: See Scientific Notation below.
Hypotenuse: The side opposite to the right angle in a right triangle.
Irrational: A real number whose decimal form is non-terminating and non-repeating that cannot be written as the ratio of two integers.
Leg: Either of the two shorter sides of a right triangle. The two legs form the right angle of the triangle.
Pythagorean Theorem: A theorem that relates the lengths of the sides of a right triangle: The sum of the squares of the lengths of the legs of a right triangle equals the square of the length of the hypotenuse.
Radical: A symbol that is used to indicate square roots.
Rational: A number that can be written as the ratio of two integers with a nonzero denominator.
Scientific Notation: A representation of real numbers as the product of a number between 1 and 10 and a power of 10, used primarily for very large or very small numbers.
Significant Digits: A way of describing how precisely a number is written.
Square root: One of two equal factors of a nonnegative number. For example, 5 is a square root of 25 because 55 = 25. Another square root of 25 is -5 because (-5)(-5) = 25. The +5 is called the principle square root of 25 and is always assumed when the radical symbol is used.
Teachers Place:
Prior to beginning the performance activity, the teacher should implement the following steps using teaching techniques you have found to be effective for your students.
1. Explain the activity (activity requirements)
2. Display the Georgia Performance Standard(s) (project on blackboard via units of instruction located at http://thevillage411.weebly.com/units-of-instruction3.html.
3. Read the Georgia Performance Standard(s) aloud and explain it to your students. You can rephrase the Georgia Performance Standard to make sure your students understand it.
4. Display the Essential Question(s) (project on blackboard via units of instruction, or print on blackboard)
5. Read the Essential Question (s) aloud and explain it to your students. You can rephrase the Essential Question (s) to make sure your students understand it.
6. Engage students in conversation by asking open ended questions related to the Essential Question (s) display answers on the blackboard.
7. Discuss answers with the students using the following questioning techniques as applicable:
Questioning Techniques:
Memory Questions
Signal words: who, what, when, where?
Cognitive operations: naming, defining, identifying, designating
Convergent Thinking Questions
Signal words: who, what, when, where?
Cognitive operations: explaining, stating relationships, comparing and
contrasting
Divergent Thinking Questions
Signal words: imagine, suppose, predict, if/then
Cognitive operations: predicting, hypothesizing, inferring, reconstructing
Evaluative Thinking Questions
Signal words: defend, judge, justify (what do you think)?
Cognitive operations: valuing, judging, defending, justifying
8. Guide students into the activity utilizing the web-based activities listed under the resource section below the task number.
9. Complete the activity with the students (some tasks may require students to work independently, peer to peer, learning circles [2-3 students] or as a whole group [the entire class]. Therefore the teacher may serve as activity leader and or facilitator. When an activity calls for students to work in learning circles you should assign roles to students individually i.e. recorder, discussion leader or presenter)
10. At the end of the *whole group learning session, students will transition into independent CAP assignments.
*The phrase, whole group learning session is utilized rather than, the end of the activity because all of the activities may not be completed in one day.
Task: 1
Resources:
Stopwatches or clocks with minute hands
Large grid paper or large grid transparencies
Activity
Part 1:
In the sixth and seventh grade, you studied proportional relationships. Do you think that the number of heartbeats you can count is proportional to the number of seconds that you check your pulse? Explain why or why not.
Part 2:
Work with your partner to take measurements and test your conjecture. One of you will be the timer and the other will count their own number of heart beats per period of time.
First, count and record the number of beats in 10 seconds, and repeat the experiment counting the number of beats in 20 seconds and 40 seconds.
Heartbeat Data for ___________________________
Number of Seconds
Number of Heartbeats
10
20
40
Predict how many times your heart would beat in 25 seconds, in 60 seconds, and in 120 seconds. Explain how you made your predictions.
Part 3:
After gathering this data, change jobs. The person who kept time now checks his/her pulse rate for 10 seconds, 20 seconds, and 40 seconds.
Heartbeat Data for ___________________________
Number of Seconds
Number of Heartbeats
10
20
40
Predict how many times your heart would beat in 25 seconds, in 60 seconds, and in 120 seconds. Explain how you made your predictions.
Part 4:
Develop a function rule (equation) to represent your pulse rate. How does your function rule compare with your partners function rule? Explain to your partner why your function rule is valid.
Draw a graph (scatter plot) to represent your pulse rate. How does your graph compare with your partners graph? Explain to your partner why your graph is valid.
Task: 2
Resources:
http://www.math.com/students/calculators/calculators.html
Graphing calculator
Graphing calculator presentation software
Activity
Part I:
Two students at Sumter YDC Matt and Tyler have been studying graphing linear equations. Matt challenged Tyler to a race to see who could graph y = 3x + 5 in the least amount of time. Matt is going to graph the equation by hand, and Tyler is going to use the graphing calculator. Matt says he can graph the equation in less than ten seconds, in less time than Tyler can enter the equation in the calculator and press the Graph key. Explain how you think Matt intends to graph the equation. Illustrate his method on two more linear equations that you make up to give to Matt and Tyler to use for their race.
Part II:
Who do you think will win the race if Matt and Tyler are given the equation 3x + 4y = 12 to graph? Why? How do you predict Matt will graph the equation?
Discussion, Suggestions, Possible Solutions
Regardless of whether students have used graphing calculators to graph linear equations, they should be able to explain how an equation in slope-intercept form may be graphed quickly by first locating the y-intercept and then using the rise and run to locate another point on the line. In the example given, the student should explain that Matt will locate the y-intercept (0, 5), and then go to the right one unit and up three units or up three units and over to the right one unit to locate another point on the line. These two points are then used to sketch the line.
Students might also respond that Matt could quickly get a table of values. Using x = 0, Matt has the y-intercept of 5. Using x = 1, Matt obtains the point (1, 8).
Task: 3
Resources:
Graph paper
Stopwatch or classroom timer
Uncooked spaghetti
Activity
Part : Data Collection
Work with your partner to take measurements and test your conjecture. One of you will be the timer and the other will count their own number of heart beats per period of time.
First, count and record the number of beats in 10 seconds, and repeat the experiment counting the number of beats in 20 seconds, 30 seconds, 40 seconds, 50 seconds, and 60 seconds.
Heartbeat Data for ___________________________
Time (sec)
Number of Heartbeats
10
20
30
40
50
60
70
80
90
100
110
120
Part : Data Representation
Graph your data.
Part : Fitting a Line to Data
Method 1: Estimation
Use a piece of raw spaghetti to visualize and estimate a line of best fit.
Choose two points on the line.
Use your two points to write an equation for a line of best fit.
Method 2: Lower and Upper Quartiles
Find the five-number summary for your x-values (time).
Find the five-number summary for your y-values (number of heartbeats).
Record your lower and upper quartiles for your x-values and your y-values.
Time (sec)
Number of Heartbeats
Lower Quartile
Upper Quartile
Draw a horizontal box-and-whisker plot using the five-number summary for your x-values (time.) Plot your box-and whisker plot under the x-axis on your graph.
Draw a vertical box-and-whisker plot using the five-number summary for your y-values (number of heartbeats). Plot your box-and whisker plot next to the y-axis on your graph.
Draw vertical lines from the lower and upper quartile values on the x-axis box-and whisker plot.
Draw horizontal lines from the lower and upper quartile values on the y-axis box-and whisker plot.
Find the coordinates of the vertices of the rectangle formed by the intersection of the vertical and horizontal lines. We will refer to these vertices as quartile points. Note: Use only the quartile points (vertices) that follow the direction of the data.
Do the quartile points have to be actual data points? Why or why not?
Draw a line connecting the two quartile points.
Write an equation for this line. This is a line of best fit for your data.
Use one of your equations to predict how many times your heart would beat in 25 seconds, in 240 seconds, and in 3 minutes. Explain how you made your predictions.
Example
(No. of heartbeats) (Time (sec))
Part : Data Collection
After gathering this data, change jobs. The person who kept time now checks his/her pulse rate and repeat part : Fitting a Line to Data.
Part : Analysis How does your line of best fit compare with your partners line of best fit? Explain to your partner why your line of best fit is valid.
Discussion, Suggestions, Possible Solutions
Students could come up with possible explanations for variations in heart-rate, such as the effect of exercise, health conditions (thyroid problems, e.g.), etc.
Students should recognize that the variable representing time represents seconds and students will need to convert three minutes to seconds before using substitution.
Task: 4
Resources:
http://www.mathsisfun.com/sets/function.html
Activity
Last summer one of your classmate Brian went to the mountains and panned for gold. Although they didnt find any gold, they did find some pyrite (fools gold) and many other kinds of minerals. Brians friend, who happens to be a geologist, took several of the samples and grouped them together. The he told Brian that all of those minerals were the same. Brian had a hard time believing him, because they are many different colors. He suggested Brian analyze some data about the specimens. Brian carefully weighed each specimen in grams (g) and found the volume of each specimen in milliliters (ml).
Brian has asked his math class at Savannah RYDC to help him analyze the data. Write your analysis of his data given below:
Specimen Number
Mass or weight (g)
Volume (ml)
1
17
7
2
10
4
3
13
5
4
16
6
5
7
3
6
24
10
7
5
2
Discussion, Suggestions, Possible Solutions
Students could graph the volume as the independent variable (x) and mass as the dependent variable (y). Their graphs could be produced either by hand on graph paper or as a STAT PLOT on a graphing calculator. Students could choose two points that are on their line of best fit that they determined by eye-balling the data, using the lower and upper quartiles, or they could use the linear regression feature of the calculator to obtain the regression equation y = 2.41x + .40.
Students might divide the mass by the volume by hand for each specimen and then find the average of this value. They could also have the calculator divide the values and find the average. The average value (2.49) is close to the coefficient of the x term in the regression equation. As students explore the meaning of this slope in this problem context, they should come to understand that it means every time the volume goes up by one ml the mass goes up by approximately 2.4 or 2.5 grams. This rate of mass in grams per milliliter of volume is the density of the mineral.
Research could be done to find lists of densities for particular minerals. While earth science references will list the density (also called specific gravity) of quartz as 2.6, the samples used for the data above were quartz. This could lead to a discussion of the precision and accuracy of measurements, as well as to a discussion of impurities and other factors that could influence the results.
The relationship between the mass and volume of the specimens might be described by some students as a proportional relationship. This could lead to the conclusion that a theoretical model for this relationship might be written as y = 2.6x. In other words, a hypothetical sample with a volume of zero would have a mass of zero, so the y-intercept should be zero.
As an extension, ask students where data points would be for samples of minerals that have a density greater than 2.6. These would be in the half-plane above the regression line for the quartz samples. This can provide a transition to graphing inequalities.
Task: 5
Resources:
http://www.coolmath.com/algebra/23-graphing-rational-functions/index.html
Graph paper
Graphing calculator
Colored pencils
Activity
In math class one day Mrs. Smith conducted an experiment. In the first part of the experiment, her students wrote Ts on a sheet of paper and counted how many they were able to make in one minute. Then Mrs. Smith told them to change their pencils to their other hands, and they repeated the experiment. Mrs. Smith then gathered from each student the information about how many Ts were made with his/her right hand and how many were made with his/her left hand in one minute. Using the data for the right hand as the x values and the data for the left hand as the y values, explain what you expect a graph of this data would look like.
In this context, a point on the line y = x would represent data for an ambidextrous person, a person that works equally as well with their left hand as they do their right hand. Left-handed persons would be able to make more Ts with their left hands than with their right hands in one minute, so their data would lie above the line y = x. Since most students are right-handed, most of the points would be graphed below the line y = x.
Questions to ask students:
What is an ambidextrous person?
Can you think of an equation that represents data for an ambidextrous person?
Why does y = x represent data for an ambidextrous person?
Ambidextrous People
(y)
(Left Hand)
(x)
(Right Hand)
Right Handed People Left Handed People
(y) (y)
(x) (x) (Left Hand)
(Right Hand) (Right Hand)
Discussion, Suggestions, Possible Solutions
Using a graphing calculator is an efficient way to capture the picture of the actual data. Enter the x values in List 1 (L1), the y values in List 2 (L2), and enter y1 = x. Turn the statplot on to view the data in Lists 1 and 2, select Zoom Stat to set the viewing window appropriately, and select y1 to see the boundary line y = x along with the individual data points.
Course Title: Course 3 8th grade Ga DJJ
State Code: 27.0230000 CAP: 12
Georgia Performance Standard(s):
M8A1.b Simplify and evaluate algebraic expressions.
M8A1.d Interpret solutions in problem contexts.
M8P1.a Build new mathematical knowledge through problem solving.
M8P1.b Solve problems that arise in mathematics and in other contexts.
M8P5.a Create and use representations to organize, record, and communicate
mathematical ideas.
M8P2.c Develop and evaluate mathematical arguments and proofs.
M8P2.d Select and use various types of reasoning and methods of proof.
M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.
M8A4.c Graph equations of the form y = mx + b.
M8P4.a Recognize and use connections among mathematical ideas.
M8P4.b Understand how mathematical ideas interconnect and build on one another to
produce a coherent whole.
M8P5.b Select, apply, and translate among mathematical representations to solve
problems.
M8P3.a Organize and consolidate their mathematical thinking through communication.
Objective(s):
The student writes solutions of equations in two variables as ordered pairs. The student will be able to graph points and lines on the coordinate plane.
Instructional Resources:
Holt Mathematics 8th grade Course 3, Pgs. 115-126
Chapter 3 Resource Book (CRB)
One-Stop Planner
Activities:
Complete Are You Ready in textbook pg., pg. 115.
Read textbook pgs. 115-121.
Complete Think and Discuss, pg. 119 and 123.
Complete Practice and Problem Solving, Problems 1-4, 8-11, 23, 24, and 36-44 on pgs. 120-121.in textbook
Complete Practice A 3-1 CRB, pg. 3.
Complete Reading Strategies 3-1 CRB, pg. 9.
Read textbook pgs. 122-125.
Complete Think and Discuss, pg. 126.in textbook
Complete Practice and Problem Solving, Problems 1-6, 17-22, and 38-47 on pgs. 124-
125.in textbook
Complete Practice A 3-2 CRB, pg. 11.
Complete Reading Strategies 3-2 CRB, pg. 17.
Evaluation:
Complete Power Presentations Lesson Quiz 3-1, pg. 121 and 3-2, pg. 125.
Modifications:
Performance Tasks: IDEA works CD
Course Title: Course 3 8th grade Ga DJJ
State Code: 27.0230000 CAP: 15
Georgia Performance Standard(s):
M8A1.b Simplify and evaluate algebraic expressions.
M8A1.d Interpret solutions in problem contexts.
M8A3.i Translate among verbal, tabular, graphic, and algebraic representations of
functions.
M8A4.c Graph equations of the form y = mx + b.
M8A4.e Determine the equation of a line given a graph, numerical information that defines
the line, or a context involving a linear relationship.
M8P2.d Select and use various types of reasoning and methods of proof.
M8P5.a Create and use representations to organize, record, and communicate
mathematical ideas.
M8P5.b Select, apply, and translate among mathematical representations to solve
problems.
M8A3.e Use tables to describe sequences recursively and with a formula in closed form.
M8P3.c Analyze and evaluate the mathematical thinking and strategies of others.
M8P4.c Recognize and apply mathematics in contexts outside of mathematics
Objective(s):
The student organizes and reviews key concepts and skills presented in Chapter Three
The student assesses mastery of concepts and skills in Chapter Three.
Instructional Resources:
Holt Mathematics Course Two Textbook
Chapter 3 Resource Book (CRB)
One Stop Planner
Activities:
Complete the odd number problems on the Study Guide, pgs. 150-151 in textbook.
Have teacher check your work on review. If score is at least 80%, then go on to
Chapter Test. If score is less than 80%, then teacher will give Reteach worksheets in
CRB to cover concepts not understood.
Complete Chapter Test pg.153 in textbook.
Have teacher check your work on test. If score is at least 80%, then go on to next CAP.
If score is less than 80%, then teacher will give worksheets in CRB covering concepts
still not understood.
Evaluation:
Complete Chapter Test pg.153 in textbook with 80% accuracy.
Modifications:
IDEA Works CD
Unit: Exponents& Roots/ Ratios, Proportions & Similarity/ Percents
Georgia Performance Standards:
M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.
a. Find square roots of perfect squares.
b. Recognize the (positive) square root of a number as a length of a side of a square with a given area.
c. Recognize square roots as points and as lengths on a number line.
d. Understand that the square root of 0 is 0 and that every positive number has two square roots that are opposite in sign.
e. Recognize and use the radical symbol to denote the positive square root of a positive number.
f. Estimate square roots of positive numbers.
g. Simplify, add, subtract, multiply, and divide expressions containing square roots.
h. Distinguish between rational and irrational numbers.
i. Simplify expressions containing integer exponents.
j. Express and use numbers in scientific notation.
k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.
M8A1 Students will use algebra to represent, analyze, and solve problems.
a. Represent a given situation using algebraic expressions or equations in one variable.
b. Simplify and evaluate algebraic expressions.
c. Solve algebraic equations in one variable, including equations involving absolute values.
d.Solve equations involving several variables for one variable in terms of the others.
M8P1 Students will solve problems (using appropriate technology).
a. Build new mathematical knowledge through problem solving.
b. Solve problems that arise in mathematics and in other contexts.
c. Apply and adapt a variety of appropriate strategies to solve problems.
d. Monitor and reflect on the process of mathematical problem solving.
M8P2 Students will reason and evaluate mathematical arguments.
a. Recognize reasoning and proof as fundamental aspects of mathematics.
b. Make and investigate mathematical conjectures.
c. Develop and evaluate mathematical arguments and proofs.
d. Select and use various types of reasoning and methods of proof.
M8P3 Students will communicate mathematically.
a. Organize and consolidate their mathematical thinking through communication.
c. Analyze and evaluate the mathematical thinking and strategies of others.
d. Use the language of mathematics to express mathematical ideas precisely.
M8P4 Students will make connections among mathematical ideas and to other disciplines.
a. Recognize and use connections among mathematical ideas.
b. Understand how mathematical ideas interconnect and build on one another to produce a coherent whole.
c. Recognize and apply mathematics in contexts outside of mathematics.
M8P5 Students will represent mathematics in multiple ways.
a. Create and use representations to organize, record, and communicate mathematical ideas.
b. Select, apply, and translate among mathematical representations to solve problems.
c. Use representations to model and interpret physical, social, and mathematical phenomena.
M8G2 Students will understand and use the Pythagorean theorem.
a. Apply properties of right triangles, including the Pythagorean theorem.
b. Recognize and interpret the Pythagorean theorem as a statement about areas of squares on the sides of a right triangle.
M8G1 Students will understand and apply the properties of parallel and perpendicular lines and understand the meaning of congruence.
a. Investigate characteristics of parallel and perpendicular lines both algebraically and geometrically.
b. Apply properties of angle pairs formed by parallel lines cut by a transversal.
d. Understand the meaning of congruence: that all corresponding angles are congruent and all corresponding sides are congruent.
M8A4 Students will graph and analyze graphs of linear equations and inequalities.
b. Determine the meaning of the slope and y-intercept in a given situation.
M8N1 Students will understand different representations of numbers including square roots, exponents, and scientific notation.
i. Simplify expressions containing integer exponents.
k. Use appropriate technologies to solve problems involving square roots, exponents, and scientific notation.
Selected Terms and Symbols:
Absolute Value: The distance a number is from zero on the number line. Examples: |-4| = 4 and |3| = 3
Addition Property of Equality: For real numbers a, b, and c, if a = b, then a + c = b + c. In other words, adding the same number to each side of an equation produces an equivalent equation.
Additive Inverse: Two numbers that when added together equal 0.
Example, 3.2 and -3.2
AlgebraicExpression: A mathematical phrase involving at least one variable. Expressions can contain numbers and operation symbols.
Equation: A mathematical sentence that contains an equals sign.
Evaluate an Algebraic Expression: To perform operations to obtain a single number or value.
Inequality: A mathematical sentence that contains the symbols >,