COURSE: ALGEBRA 1
TOPIC: Real Number System
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STAGE 1:
IDENTIFY DESIRED RESULTS |
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Content Standard(s) Generalizations about what students should know and be able to do |
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Established Goals: CT Frameworks: a)
The number system extends from natural numbers
to include integers, rational numbers, and real numbers. b)
Properties of number systems are used to
develop strategies for computation and estimation and judging the
reasonableness of results. c)
Multiplication, division and power properties
of exponents can simplify calculations with expressions and scientific notation. d)
Proportional reasoning can be used to make
predictions and describe relationships between variables. NCTM Standards: a)
Compare and contrast the properties of numbers
and number systems including the rational and real numbers. b)
Use number-theory arguments to justify
relationships involving whole numbers. c)
Develop fluency in operations with real
numbers. Using mental computation or
paper-and-pencil calculations for simple cases and technology for more
complicated cases. d)
Understands numbers, ways of representing
numbers and relationships among numbers. |
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Enduring Understandings Insights earned from exploring generalizations via the essential
questions (Students will understand THAT…) |
Essential Questions Inquiry used to explore generalizations |
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The student will understand that: 1. The real number system has subsystems. 2. The real number system is a mathematical field. 3. There is a specific order of operations in the real number system that must be followed for all computations. 4. There are laws of integral exponents that are used to simplify and evaluate algebraic expressions. 5. Different forms of numerical data are more appropriate to represent the magnitude of numbers and simplify calculations.
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1. What are the subsystems of the real number system? 2. What are the field properties of the real number system? 3. What is the order of operations for simplifying an algebraic or numerical expression? 4. How do you use the laws of integral exponents to evaluate algebraic expressions? 5. Why use scientific notation and exponents? |
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Knowledge and Skills What students are expected to know and be able to do |
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Students will know… 1. The real number system has subsystems that include the natural, whole, integers, and rational numbers. 2. The field properties of the real number system. 3. The order of operations (PEMDAS). 4. The laws of integral exponents. 5. How to perform conversions and operations with numbers in scientific notation and solve real life situations. |
Students will be able to… 1. Identify the subsystems of the real number system. 2. List, identify and use the field properties of the real number system. 3. Use the order of operations to simplify numerical and algebraic expressions. 4. Use the laws of integral exponents to simplify numerical and algebraic expressions. 5. Convert, multiply, and divide numbers in scientific notation and solve real world problems. |
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STAGE 2:
DETERMINE ACCEPTABLE EVIDENCE |
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Performance Task(s) Authentic application in
new context to evaluate student achievement of desired results designed
according to GRASPS (Goal, Role, Audience, Setting Performance, Standards) |
Other Evidence Application that is functional in a classroom context only to
evaluate student achievement of desired results |
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As a stockbroker, it is your job to advise your clients what stocks to invest in to make a profit. Your client wants to invest 100 shares at an opening value in three companies. You will select three companies you feel will be profitable and follow the stocks for one month. You will compute weekly profit and loss and weekly percent changes. Finally compute a monthly profit and loss and monthly percent change in value for each of the three stocks. You will present your results to your client. |
In addition to tests and quizzes, one or more of the following will be used:
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STAGE 3:
DEVELOP LEARNING PLAN |
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Learning Activities:
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COURSE: ALGEBRA 1
TOPIC: Equations,
Inequalities and Graphs
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STAGE 1:
IDENTIFY DESIRED RESULTS |
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Content Standard(s) Generalizations about what students should know and be able to do |
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Established Goals: CT Frameworks: a) Tables, graphs, and equations can be used to
analyze linear functions. b) Functions are used in a variety of
situations including to model data, to make predictions, and to find the rate of change. c)
A wide variety of functions can be used to
model real world situations. d)
Geometric relationships may be verified and
proved using synthetic and coordinate methods. e)
Exponential growth and decay models are based
on repeatedly multiplying by the same factor. NCTM Standards: a)
Understands patterns, relations, and
functions. b)
Represent and analyze mathematical situations
and structures using algebraic symbols. c)
Use mathematical models to represent and
understand quantitative relationships. d)
Analyze change in various contexts. e)
Apply and adapt a variety of appropriate
strategies to solve problems. f)
Use the language of mathematics to express
mathematical ideas precisely. g)
Recognize and use connections among
mathematical ideas. h)
Select,
apply and translate among mathematical representations to solve problems. |
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Enduring Understandings Insights earned from exploring generalizations via the essential
questions (Students will understand THAT…) |
Essential Questions Inquiry used to explore generalizations |
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The student will understand that: 1. Equations and inequalities can be represented by graphs that reveal patterns and enable predictions and solutions. 2. Graphs of linear equations, linear inequalities, and absolute value equations are constructed using data and algebraic rules. 3. Equations and inequalities are solved systematically by using inverse operations. 4. Real world situations can be modeled and solved by using equations and inequalities.
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1. How are equations and graphs related? 2. What strategies can you use to graph linear equations, linear inequalities, and absolute value equations? 3. How do you solve equations or inequalities algebraically? 4. How can you use equations, inequalities, and linear systems to solve real life problems?
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Knowledge and Skills What students are expected to know and be able to do |
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Students will know… 1. How to interpret the graphs of linear equations, linear inequalities, absolute value equations, and systems of equations and inequalities. 2. How to solve and graph linear equations, inequalities, absolute value equations, and systems of equations and inequalities. 3. How to determine the slope and intercepts of a line. 4. How to write the equation of a line. 5. How to write and use an equation, inequality or linear system as a real life model to solve problems and answer questions. 6. How to relate algebraic and geometric representations of equations, system of equations, and inequalities. |
Students will be able to… 1. Describe, analyze and generalize patterns using tables, rules, algebraic equations, and graphs. 2. Model, solve, and graph linear equations. 3. Use a table of values; intercepts, and slope intercept form to graph a linear equation. 4. Find the slope of a line. 5. Interpret the slope as a rate of change. 6. Find x and y intercepts. 7. Write the equation of a line given a graph, given the slope and one point, or given two points in slope intercept form, point slope form and standard form. 8. Model, solve, and graph inequalities, absolute value equations, and exponential growth and decay functions. 9. Model and solve systems of linear equations and inequalities by graphing, substitution, and linear combinations. 10. Translate real life problems into linear equations or inequalities and solve. 11. Interpret algebraic equations and inequalities geometrically and describe geometric relationships algebraically using the Cartesian coordinates. |
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STAGE 2:
DETERMINE ACCEPTABLE EVIDENCE |
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Performance Task(s) Authentic application in
new context to evaluate student achievement of desired results designed
according to GRASPS (Goal, Role, Audience, Setting Performance, Standards) |
Other Evidence Application that is functional in a classroom context only to
evaluate student achievement of desired results |
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Carnival Time: You are on a planning committee for a school carnival. You must decide what activities to include and what prices to charge. You have two options to consider. Option 1 charges $.25 a ticket and $1.00 for admission while option 2 charges $.50 a ticket and no admission charge. You will write a linear equation to model each option to find the profit y for selling x tickets. Profit equals income minus expenses. Assume 200 people will attend and use $ 500 for expenses. Graph and interpret the linear equations in slope intercept form to decide which option is better. Explain. |
In addition to tests and quizzes, one or more of the following will be used: 1. Cooperative learning activities. 2. Graphing calculator activities. 3. Informal and formal checks: homework checks, problem of the day and review worksheets. 5. Game review (Bingo and Decoder). 6. Student Tutor Software. |
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STAGE 3:
DEVELOP LEARNING PLAN |
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Learning Activities: 1. Cooperative learning activities. 2. Graphing calculator activities. 3. Homework and review worksheets. 4. Game review (Bingo and Decoder). 5. Student Tutor Software. |
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COURSE: ALGEBRA 1
TOPIC: Polynomials and
Factoring
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STAGE 1:
IDENTIFY DESIRED RESULTS |
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Content Standard(s) Generalizations about what students should know and be able to do |
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Established Goals: CT Frameworks: e)
Tables, graphs, and equations can be used to
analyze non-linear functions for which the rate of change is not constant. f)
Functions are used in a variety of situations
including to model data and to make predictions. g)
A wide variety of functions can be used to
model real world situations. h)
Functions can be viewed as objects on which
operations can be performed. NCTM Standards: a)
Understands patterns, relations and functions. b)
Represents and analyze mathematical situations
and structures using algebraic symbols. c)
Use mathematical models to represent and
understand quantitative relationships. d)
Analyze change in various contexts. e)
Apply and adapt a variety of appropriate
strategies to solve problems. f)
Use the language of mathematics to express
mathematical ideas precisely. g)
Recognize and use connections among
mathematical ideas. h)
Select, apply and translate among mathematical
representations to solve problems. |
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Enduring Understandings Insights earned from exploring generalizations via the essential
questions (Students will understand THAT…) |
Essential Questions Inquiry used to explore generalizations |
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The student will understand that: 1. Polynomials can be added, subtracted, and multiplied. 2. Polynomial equations can be factored and solved. 3. Factoring is used to solve quadratic equations and find x-intercepts. 4. The graph of a quadratic function is a parabola and the x-intercepts are solutions to the quadratic equation. 5. Quadratic equations can model real world problems. |
1. How do you add, subtract and multiply polynomials? 2. What techniques can you use to factor and solve polynomial equations? 3. How do you solve a quadratic equation? 4. How are quadratic functions and their graphs related? 5. When will you ever use quadratic equations in real life?
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Knowledge and Skills What students are expected to know and be able to do |
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Students will know… 1. How to perform operations on polynomials. 2. How to factor and solve polynomials. 3. How to use square roots or factoring to solve quadratic equations. 4. How to relate the factors and x intercepts of a quadratic equation. 5. That a parabola is the graph of a quadratic equation. 6. How to write quadratic models for real life situations and find the solution. |
Students will be able to… 1. Add, subtract, and multiply polynomials. Multiply polynomials using the distributive property, FOIL pattern and special product patterns (Sum and Difference Pattern, Square of a Binomial Pattern). 2. Factor a polynomial using the following techniques. a) GCF and the distributive property. (common monomial factor) b) Special product patterns (Difference of Two Squares, Perfect Square Trinomial). c) Quadratic trinomial where leading coefficient equals 1 or does not equal 1. 3. Solve a quadratic equation by factoring and the zero product property or by finding square roots. 4. Find x intercepts by factoring. 5. Recognize that quadratic equations are U shaped and can be solved by graphing. 6. Apply factoring skills to solve real life problems. |
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STAGE 2:
DETERMINE ACCEPTABLE EVIDENCE |
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|
Performance Task(s) Authentic application in
new context to evaluate student achievement of desired results designed
according to GRASPS (Goal, Role, Audience, Setting Performance, Standards) |
Other Evidence Application that is functional in a classroom context only to
evaluate student achievement of desired results |
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As director of the marching high school band, it is your duty to determine if the space allotted your group for the parade is adequate. The band is allowed 3200 square feet in the parade and must stay at least 5 feet from the curb. The width of the street is x feet and the length of the space allotted is x+30 feet. Rows should be 4 feet apart and each row will contain 8 band members. Write a polynomial expression representing the area the band is allotted. Find the dimensions for the area of the space allowed for the band by solving the quadratic equation. Find the number of rows your band will take up. Can all 200 members be in the band? If not, how many square feet are needed. |
In addition to tests and quizzes, one or more of the following will be used: 1. Cooperative learning activities. 2. Graphing calculator activities. 3. Informal and formal checks: homework checks, problem of the day and review worksheets. 4. Game review (Math bowl and Bingo) 5. Student Tutor Software. |
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STAGE 3:
DEVELOP LEARNING PLAN |
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Learning Activities: 1. Cooperative learning activities. 2. Graphing calculator activities. 3. Homework and review worksheets. 4. Game review (Math bowl and Bingo) 5. Student Tutor Software. |
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What’s the
Best Deal?
Name ______________________ Date _______
Directions:
Using the
information provided on the worksheet “What’s the Best Deal”, you are to
calculate both salary schemes and record your data results on the table
provided. Throughout your calculations, answer the following questions:
1. During your calculations, stop at Day 5 and
answer the question:
What
choice is the best at this point?
Why? Write a short paragraph
justifying your choice.
_______________________________________________________
_______________________________________________________
_______________________________________________________
2. Continue to compute, but stop at Day 10. Answer the following questions:
Which
choice is the best? Why? What happened?
_______________________________________________________
_______________________________________________________
_______________________________________________________
3. Using the data, plot the results on centimeter
graph paper and record your responses to the following questions:
For
what days does choice 1 yield better total earnings than choice 2?
_______________________________________________________
_______________________________________________________
_______________________________________________________
For
what days does choice 2 yield better total earnings than choice 1?
_______________________________________________________
_______________________________________________________
_______________________________________________________
Which
plan is the slowest in achieving the earnings objective? Why?
_______________________________________________________
_______________________________________________________
_______________________________________________________
Which
format, the table or the graph, would you use to communicate the results of
this investigation? Why?
_______________________________________________________
_______________________________________________________
_______________________________________________________
What
were the rules that generated the function for the daily salary for each
choice? Remember, you will need to support your choice by describing the graphs
for each situation with specifics.
_______________________________________________________
_______________________________________________________
_______________________________________________________
Standards and Criteria for Success
Scoring Rubric
The student:
3 Meets
or exceeds the objectives of the task.
Demonstrates a high level of understanding. Student demonstrates understanding of
the task. Written explanations are
thoughtful and complete. Calculations
are correct. The graph represents the
data results accurately.
2 Partially
meets the objectives of the task.
Demonstrates a high level of understanding. Student demonstrates some
understanding of task. Written
explanations are brief. Calculations are
made with some errors. The graph does not
quite represent the data results, but the explanations do represent
understanding.
1 Does
not meet the objectives of the task.
Demonstrates poor or incorrect understanding. Student
demonstrates little understanding of the task.
Written explanations are unclear.
Calculations are made with several errors. The graph does not represent the data results
and the explanations are incorrect.
0 Shows
no understanding of the problem or how to arrive at a solution. Student
demonstrates no understanding of the task.
There are no written explanations and many calculations errors. The graphs were not created.