|
|
Integrated Algebra & Geometry, Unit 2:
Linear Functions
(@
18 days)
Overview
The overarching theme of the unit is usefulness of
mathematics in the real world. In this unit you will use linear
equations to graph and model real-life situations. You will learn to
write and graph linear equations given the starting value/y-intercept
and the rate / slope of the line. Using the properties of linear
functions, you will apply these properties to real - world problems. You
will be able to distinguish between linear and non-linear functions as
well as identify a direct variation as a specific case of a linear
function. You will learn the effects of changes in the slopes and
y-intercepts of linear functions. Each lesson is introduced by an
essential question that serves as the foundation for the teaching
objective of a particular lesson.
|
|
|
|
Enduring Understandings
are important ideas that students should carry with them years beyond the
instruction received this year.
-
Algebra is a language.
-
Numbers can be represented, ordered, and communicated in many different
forms.
-
Solutions must be reasonable.
-
Compu tation is a vital mathematical tool.
- Computation and
reasoning are a vital mathematical tools.
Essential Questions
are the most important “big picture” questions students should be able to
answer after completing learning activities.
-
How do you demonstrate equivalent numbers using various forms?
-
Why do we need to represent, order, and communicate numbers?
-
How do you select and apply computational methods to solve a problem?
-
How do you justify your solution as reasonable?
Standards:
 Highest
Frequency
 High
Frequency
 Other
Standards and E-Skills
Standard
2: (Algebra and Functions) Students use algebraic methods to explore, model
and describe patterns and functions involving numbers, shapes, data, and
graphs in problem-solving situations and communicate the reasoning used in
solving these problems.
2.1
Recognize, extend and use geometric, numeric, linear, or visual patterns to
solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
2.5a Solve simple equations
2.5b Translate English expressions and algebraic expressions
2.2
Recognizing, describing, and extending a pattern and function using tables
and graphs in a problem solving situation. Using a table to find a
constant/unit rate.
2.5
Substituting in a formula to compute a value. Solving a simple linear
equation in a problem solving situation. Using a linear function given in a
context to solve a problem.
Standard 6: (Computation) Students link concepts and procedures as they
develop and use computational techniques, including estimation, mental
arithmetic, paper-and-pencil, calculators, and computers, in problem-solving
situations and communicate the reasoning involve
Compute with whole numbers.
Add and subtract decimals in problem solving situations
Apply appropriate computational methods to solve a problem.
Apply
computational skills to find reasonable answers
|
Lessons
Introduction: Introduction
to Linear Functions
Duration: @ 1-2 class periods
Standard information 2:
2.1
Recognize, extend and use geometric, numeric, linear, or visual patterns to
solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Questions: What real-world situations can be modeled by
linear functions? Why would we need a linear function to model a real-world
situation?
Activities
-
Introduction
- This lesson represents a transition between the current and previous
units. You already know the definition of a function as well as they
distinguish real-world situations that represent functions. In this
lesson, you will learn what makes functions become linear. You already
know how to distinguish from a graph if it represents a linear or
non-linear function. Now, you will learn how to recognize
real-world situations that can be modeled by linear functions. We
introduce these by pointing at the pattern of repeated
addition/subtraction from a starting value. You will learn to use a
“shortcut” for repeated subtraction, which ends with writing an equation
that models Adam’s account balance after x days. The key question that
points out the usefulness of using patterns (that represent linear
relationships) is question #20. The following question might lead to an
interesting discussion. Then, we can summarize properties of linear
patterns/functions.
-
Practice - Student Worksheet:
My Introductory Lesson
Differentiation
Extension:
Support:
Lesson 1: Using Linear
Functions
Duration: @ 1-2 class periods
Standard information 2:
2.1
Recognize, extend and use geometric, numeric, linear, or visual patterns to
solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language.
Essential Questions:
How can we use linear functions to solve real-world
problems? How does the starting value affect the graph?
Activities
-
Introduction -
In
the previous unit you have learned how to determine whether the set of
data is a function or not as well as properties that functions can
possess. You know that a function describes the way two quantities are
related. You will use the knowledge in this unit to solve real-world
situations related to linear functions. For example: On Memorial day
Hayden bought a new computer at Best Buy. The computer was not on sale,
it cost $672, but Hayden was offered an 18-month interest- free
financing. After considering the deal, Hayden decided to buy it. He
determined that he would pay off the computer within the free-interest
period. How high should Hayden’s monthly payments be in order to pay off
the credit card balance in 18 months? In 12 months? How would the answer
change if his previous balance on the credit card was $134?
-
Practice - Student Worksheet:
Student Worksheet Lesson 1
(Writing linear
equations, completing tables, graphing, and interpreting intercepts.)
Differentiation
Extension:
Support:
Lesson 2:
Elements of Linear Functions
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Questions:
What are two important
elements that define a linear function?
Activities
-
Introduction -
Lesson 2
-
Practice –
Student Worksheet 2
Writing linear equations, completing tables, graphing, and interpreting
intercepts, interpreting effects of changing rates and starting values
on graphs.)
Differentiation
Extension:
Support:
Lesson 3:
Finding Rates From Tables
Duration: @ 1 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Questions:
Do you need to
know the starting value in order to find the rate? What must you know in
order to find the rate?
Activities
-
Introduction –
Lesson
3
-
Practice
-
Student Worksheet 3
(Finding rates from tables, finding unit rates.)
-
Test Yourself
-
Differentiation
Extension: Watch the
Car
Speed video (3:13) and the
Bungee
Jumper video (1:43) to see how time distance graphs are used to display
and analyze speed.
Support:
Lesson 4:
Using Rates
Duration: @ 2 class period
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language.
Essential Question:
How can we use rates to make decisions?
Activities
-
Introduction -
Lesson 4
- Practice -
Student Worksheet 4. 1
(Writing and interpreting equations, finding real-world meanings of
rates.)
Student Worksheet 4.2
(Comparing rates, making decisions based on better rates.)
Differentiation
Extension:
Support:
Lesson 5:
Graphs
Duration: @ 2 class periods
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding: Algebra is a language.
Essential Question: How
do we find rates from graphs?
Activities
-
Introduction
-
Lesson
5
-
Practice -
Student Worksheet 5
(Reading and interpreting graphs, finding rates from graphs, converting
between units.)
Differentiation
Extension:
Support:
Lesson 6:
Using Points to Find Slope
Duration: @ 3 class periods
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Question:
How can points be used
to find the slope?
Assessment:
Formative Assessment #1
Activities
-
Introduction -
Lesson
6
-
Practice -
Student Worksheet 6.1
(Writing equations from graphs.)
Student Worksheet 6.2
(Graphing equations.)
Student Worksheet 6.3
(Finding rates/slopes from tables, graphs, equations, slope formula.)
-
Test
Yourself -
Formative Assessment #1
Differentiation
Extension:
Support:
Lesson 7: Direct
Variations
Duration: @ 2 class periods
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Questions:
Knowing how the
y-intercept (starting value) affects the graph of a linear function, what
will all graphs whose y-intercept is 0 have in common? How can you tell from
the context of a problem if it’s a direct variation?
Activities
-
Introduction
-
Lesson
7
-
Practice
-
Student Worksheet 7.1
(Writing direct variation equations, relationship between direct
variations and linear functions.)
Student Worksheet 7.2
(More practice on direct variations, tables, graphs, interpreting
tables.)
Differentiation
Extension:
Support:
Lesson 8:
Linear Equations
Duration: @ 3-4 class periods
Standard information
2.1 Recognize, extend and use geometric, numeric, linear, or visual patterns
to solve a problem. Using the rule for a pattern to represent it in a table,
graph, and problem solve. Recognize an equation that models a given
situation.
Enduring Understanding:
Algebra is a language.
Essential Questions: What
is a reasonable mathematical model, in symbolic form, for relationships that
have an initial value and a constant rate of change? How can we efficiently
solve real-life problems using linear equations?
Activities
-
Introduction
-
Lesson 8
-
Practice
-
Student Worksheet 8.1
(Evaluating and solving linear equations, interpreting linear
equations.)
Student
Worksheet 8.2
(Additional practice on solving real-life problems that can be modeled
by linear functions.)
Student Worksheet 8.3
(Solving linear equations without a context.)
Student Worksheet 8.4 (A
huge variety of problems on solving linear equations. The problems were
provided by Wasson High School.)
-
Test Yourself
-
Formative Assessment #3
Differentiation
Extension:
Support:
|
|
