Download Patterns and Inductive Reasoning

Updated August, 2009
Pre-test and Patterns (3 weeks)
Essential Questions:
What is a sequence?
How can patterns be represented?
Why do we begin an Algebra I course with an exploration of patterns?
Can all real world situations be represented with a linear pattern?
What are the advantages and disadvantages of a recursive rule compared to an explicit rule?
How does technology aid in the recognition and construction of patterns?
Students should…
1. explore an application of
patterns
Effective Teaching Strategies and
Suggested Activities
Investigation 1 – Exploring Patterns with
Hydrocarbons
2. analyze and create different
representations of
patterns—from tables to
graphs to symbolic rules.
1. define a variable.
2. evaluate an expression
Middletown Public Schools
Investigation 2 – Modeling Algebraic
Expressions
Course-Level Expectations
1.1.1
Express and analyze
sequential patterns (including
arithmetic and geometric
sequences) drawn from real
world contexts using tables,
graphs, words and symbolic
rules; translate any
representation of a pattern
into another representation.
1.1.10 Make and justify predictions
based on patterns.
2.1.1 Compare, locate, label and
order integers, rational
numbers and real numbers on
number lines, scales and
coordinate grids.
1.1.1 Express and analyze sequential
patterns (including arithmetic
and geometric sequences)
drawn from real world contexts
using tables, graphs, words and
symbolic rules; translate any
representation of a pattern into
another representation.
August 2009
CAPT correlations
(1) Construct, describe, extend, and
analyze a variety of numerical,
geometric, and statistical patterns.
(2) Describe, analyze, and
generalize patterns using tables,
rules, algebraic expressions and
equations, and graphs.
(3) Make and justify predictions
based on patterns.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Make and justify predictions
based on patterns.
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Updated August, 2009
1. recognize, describe, and
extend sequences and
create an algebraic
expression to model the
rule.
Investigation 3 – Patterns with Fractals
2. create a fractal, find
geometric patterns and
write a rule to represent a
geometric sequence.
1. write both the recursive rule Investigation 4 – Arithmetic Sequences
and the explicit rule for the
nth term of an arithmetic
Investigation 5 – Building Bridges
sequence
2. use the rules to extend
patterns and find the
specific element in an
arithmetic sequence.
1.1.2 Determine the nth term of a
sequence by both hand and
with a graphing calculator.
1.1.10 Make and justify predictions
based on patterns.
1.1.3 Derive and apply a rule for
the nth term of an arithmetic
sequence.
1.1.4 Write both a recursive rule
and an explicit rule for a
sequence.
1.1.10 Make and justify predictions
based on patterns.
2.1.1 Use models to make, test and
describe conjectures
involving properties of twoand three-dimensional
figures.
1.1.3 Derive and apply a rule for
the nth term of an arithmetic
sequence.
1.1.4 Write both a recursive rule
and an explicit rule for a
sequence.
1.1.10 Make and justify predictions
based on patterns.
(3) Use recursive processes,
including iteration, to solve
problems.
(1) Solve problems using geometric
models.
(2) Construct, describe, extend, and
analyze a variety of numerical,
geometric, and statistical patterns.
(3) Make and justify predictions
based on patterns.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Make and justify predictions
based on patterns.
(3) Use recursive processes,
including iteration, to solve
problems.
Middletown Public Schools
August 2009
2
Updated August, 2009
1. Find a recursive rule to
calculate terms of
geometric sequences.
Note: students are not
necessarily expected to find
the explicit rule for
geometric sequences
2. Explain the difference
between an arithmetic and a
geometric sequence.
Investigation 6 – Geometric Sequences
1.1.1 Express and analyze sequential
patterns (including arithmetic
and geometric sequences)
drawn from real world contexts
using tables, graphs, words and
symbolic rules; translate any
representation of a pattern into
another representation.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Make and justify predictions
based on patterns.
1.1.4 Write both a recursive rule and
an explicit rule for a sequence.
Assessments:
Warm-ups, discussion/questioning, mid-unit assessment, performance task, end-unit assessment
Middletown Public Schools
August 2009
3
Updated August, 2009
Unit 2: Linear Equations and Inequalities (5 weeks)
Essential Questions:
What is an equation?
What does equality mean?
What is an inequality?
How can we use linear equations and linear inequalities to solve real world problems?
What is a solution set for a linear equation or linear inequality?
How can models and technology aid in the solving of linear equations and linear inequalities?
Students should…
1. represent real world
situations with algebraic
expressions
Effective Teaching Strategies
and Suggested Activities
Investigation 1 – Undoing Operations
2. apply real number properties
to simplify algebraic
expressions
Course-Level Expectations
2.2.1
Use algebraic properties,
including associative,
commutative and
distributive, inverse and
order of operations to
simplify computations with
real numbers and simplify
expressions.
(1) Identify appropriate operations
(including addition, subtraction,
multiplication, division,
exponentiation, and square roots)
and use these operations in a
variety of contexts.
1.2.1 Write linear equations and
inequalities that model real
world situations.
1.3.1 Simplify and solve linear
equations and inequalities.
2.1.1 Compare, locate, label and
order integers, rational
numbers and real numbers on
number lines, scales and
coordinate grids.
(1) Identify appropriate operations
and use these operations in a
variety of contexts.
3. describe reasonable values
that the variable and/or
expression may represent
1. perform integer operations
2. combine like terms
3. evaluate expressions
4. use the distributive property
5. solve one-step linear equations
6. solve one-step linear
inequalities
7. solve one-step linear equations
and inequalities in context
Middletown Public Schools
Investigation 2 – Unit Pre-test and
Review
CAPT correlations
August 2009
(2) Select appropriate methods for
computing (including mental
mathematics, estimation, paperand-pencil, and calculator
methods).
(2) Select appropriate methods for
computing.
(3) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
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Updated August, 2009
1. write a linear equation that
models a real world scenario
Investigation 3 – Two-Step Linear
Equations in Context
2. solve two-step linear equations
and justify their steps.
1. write a linear inequality that
models a real world scenario
2. solve two-step linear
inequalities and justify their
steps
Middletown Public Schools
Investigation 4 – Two-Step Linear
Inequalities
2.2.1 Use algebraic properties,
including associative,
commutative and distributive,
inverse and order of operations
to simplify computations with
real numbers and simplify
expressions.
2.2.3 Choose from among a variety
of strategies to estimate and
find values of formulas,
functions and roots.
2.2.4 Judge the reasonableness of
estimations and computations.
3.3.1 Select and use appropriate
units, scales, degree of
precision to measure length,
angle, and area of geometric
models.
1.2.1 Write linear equations and
inequalities that model real
world situations.
1.3.1 Simplify and solve linear
equations and inequalities.
1.2.1 Write linear equations and
inequalities that model real
world situations.
1.3.1 Simply and solve linear
equations and inequalities.
August 2009
(1) Select appropriate methods for
computing.
(2) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
(1) Select appropriate methods for
computing.
(2) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
5
Updated August, 2009
1. solve multi-step linear
equations and linear inequalities
and justify their steps
Investigation 5 – Multi-Step Linear
Equations and Linear Inequalities
1.2.1 Write linear equations and
inequalities that model real
world situations.
1.3.1 Simplify and solve linear
equations and inequalities.
(1) Assess the reasonableness of
answers to problems.
(2) Use integers, fractions,
decimals, percents, and scientific
notation in real-world situations to
count, measure, compare, order,
scale, locate, and label.
(3) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
Assessments:
Warm-ups, discussion/questioning, mid-unit assessment, performance task, end-unit assessment
Middletown Public Schools
August 2009
6
Updated August, 2009
Unit 3: Functions (3 weeks)
Essential Questions:
What is a function?
How can functions be used to model real world situations, make predictions, and solve problems
Students should…
1. explore and define relations,
functions, domain, and range.
Middletown Public Schools
Effective Teaching Strategies
and Suggested Activities
Investigation 1 – Relations and
Functions
Grade-Level Expectations
CAPT correlations
1.1.5 Identify the role of independent and
dependent variables in a function;
determine the domain and range of a
function describing a real world
problem.
1.1.9 Explore patterns and functions using
a variety of technologies (i.e.
graphing calculators, spreadsheets,
and on-line resources).
1.2.2. Represent functions (including linear
and nonlinear functions such as
square, square root, and piecewise
defined) with tables, graphs, words
and symbolic rules; translate any
representation of a function into
another representation.
1.2.3 Decide on appropriate axis labels and
scales for the graph of a function
representing a real-world situation.
4.1.1 Collect real data and create
meaningful graphical representations
of the data both by hand and with
graphing technologies.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
August 2009
(2) Use a variety of
representations (including
graphs, tables, words, number
lines, pictures, etc.) to present,
interpret and communicate
various kinds of numerical
information.
(3) Use the coordinate plane to
represent functions.
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Updated August, 2009
1. study functions from data in
tables and graph these
functions; look for trends.
Investigation 2 – What is a function? 1.1.5
2. examine by table and graph
the relationship of functions
and non-function.
3. collect data and investigate
the relationship.
4. apply the vertical line test
and identify the domain and
range.
1. evaluate a function in
function notation given a
value of the domain
Investigation 3 – Function Notation
and Evaluating Functions
Identify the role of independent and
dependent variables in a function;
determine the domain and range of a
function describing a real world
problem.
1.1.9 Explore patterns and functions using
a variety of technologies (i.e.
graphing calculators, spreadsheets,
and on-line resources).
1.2.2 Represent functions (including
linear and nonlinear functions such
as square, square root, and piecewise
defined) with tables, graphs, words
and symbolic rules; translate any
representation of a function into
another representation.
1.2.3 Decide on appropriate axis labels
and scales for the graph of a
function representing a real world
situation.
4.1.1 Collect real data and create
meaningful graphical representations
of the data both by hand and with
graphing technologies.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
1.3.2
(1) Use a variety of
representations to present,
interpret and communicate
various kinds of numerical
information.
1.3.3
2. evaluate a function in
function notation given a
value of the range
Use functional notation to evaluate
a function for a specified value of its
domain.
For a function y = f(x), find y for a
given x and find x for a given y.
(3) Construct and use linear
functions to model and solve
real-world situations.
(4) Use the coordinate plane to
represent functions.
(2) Construct and use linear
functions to model and solve
real-world situations.
3. explain the meaning of a
solution in the context of the
problem.
Middletown Public Schools
(2) Use a variety of
representations (including
graphs, tables, words, number
lines, pictures, etc.) to present,
interpret and communicate
various kinds of numerical
information.
August 2009
(3) Use the coordinate plane to
represent functions.
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Updated August, 2009
1. construct a function given a
function’s verbal description
Investigation 4 – Multiple
Representations and Applications of
Functions
1.2.2
2. identify the domain and range
of a function
3. graph functions by hand
1.2.3
4. determine whether a relation
is a function by examining its
table or graph (use vertical
line test)
1.3.2
5. use function notation
1.3.3
6. evaluate a function given
values of the independent
variable
2.1.1
7. evaluate a function given
values of the dependent
variable
4.1.1
8. construct a graph and table
for a function
Represent functions (including
linear and nonlinear functions such
as square, square root, and piecewise
defined) with tables, graphs, words
and symbolic rules; translate any
representation of a function into
another representation.
Decide on appropriate axis labels
and scales for the graph of a
function representing a real-world
situation.
Use functional notation to evaluate a
function for a specified value of its
domain.
For a function y = f(x), find y for a
given x and find x for a given y.
Compare, locate, label and order
integers, rational numbers and real
numbers on number lines, scales and
coordinate grids.
Collect real data and create
meaningful graphical representations
of the data both by hand and with
graphing technologies.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Use a variety of
representations (including
graphs, tables, words, number
lines, pictures, etc.) to present,
interpret and communicate
various kinds of numerical
information.
(3) Use the coordinate plane to
represent functions.
9. identify the domain and range
of a function in a real world
context
Assessments:
Warm-ups, discussion/questioning, mid-unit assessment, performance task, end-unit assessment
Middletown Public Schools
August 2009
9
Updated August, 2009
Unit 4: Linear Functions (6 weeks)
Essential Questions:
What is a linear function?
What are the different ways that linear functions may be represented?
What is the significance of a linear function’s slope and y-intercept?
How many linear functions model real world applications?
How many linear functions help us analyze real world situations and solve practical problems?
Students should…
Effective Teaching Strategies
and Suggested Activities
Grade-Level Expectations
1. Interpret distance-time
Investigation 1 – What makes a
graphs and tables in terms Function Linear?
of the motion of an
object.
1.1.5
2. Write a verbal description
of a distance-time
function, sketch its graph,
and construct a table of
values.
1.1.6
3. Distinguish between
linear and non-linear
functions by recognizing
functions with a constant
rate of change whether
the function is given
verbally, graphically, or
in table form. (Note:
Calculation of slopes will
be developed more fully
in a later investigation.)
Middletown Public Schools
1.1.9
1.2.2
Identify the role of independent
and dependent variables in a
function; determine the domain
and range of a function
describing a real world problem
Understand that linear functions,
which can be written
symbolically as
y = m x + b, represent constant
additive change; a unit increase
in the independent variable (x)
causes the value of the
dependent variable (y) to change
by m units; arithmetic sequences
are special cases of linear
functions.
Explore patterns and functions
using a variety of technologies.
Represent functions with tables,
graphs, words and symbolic
rules; translate any
representation of a function into
another representation.
August 2009
CAPT correlations
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Use a variety of representations
(including graphs, tables, words,
number lines, pictures, etc.) to
present, interpret and
communicate various kinds of
numerical information.
(3) Construct and use linear
functions to model and solve realworld situations.
(4) Use the coordinate plane to
represent functions.
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Updated August, 2009
4. Identify distance-time
functions with slopes of
different magnitudes from
the verbal description, the
graph, and the table.
1.2.3
5. Distinguish between
distance-time functions
with positive slopes
(increasing functions) and
functions with negative
slopes (decreasing
functions) given a verbal,
graphical or tabular
representation of the
function.
6. State the average velocity
of an object between two
points.
1. distinguish between a
linear and non-linear function
from a table of values and
from a graph
2. transform a function from
a verbal description, and
algebraic, tabular and
graphical forms
3. identify from a table of
values the linear function’s
constant average rate of
change
Middletown Public Schools
Investigation 2 – Recognizing Linear
Functions from Words, Tables and
Graphs
Decide on appropriate axis labels
and scales for the graph of a
function representing a real
world situation.
1.2.5 Recognize and explain the
meaning and practical
significance of the slope and the
x- and y-intercepts as they relate
to a context, graph, table or
equation.
1.2.6 Create a linear function from
two non-vertical ordered pairs or
from a single ordered pair and a
rate of change.
3.1.1 Develop and apply formulas to
solve measurement problems
indirectly.
4.1.1 Collect real data and create
meaningful graphical
representations of the data both
by hand and with graphing
technologies.
1.1.5 Identify the role of independent
and dependent variables in a
function; determine the domain
and range of a function
describing a real-world problem
1.1.6 Understand that linear
functions, which can be written
symbolically as y = m x + b,
represent constant additive
change; a unit increase in the
independent variable (x) causes
the value of the dependent
variable (y) to change by m
units; arithmetic sequences are
August 2009
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Use a variety of representations
(including graphs, tables, words,
number lines, pictures, etc.) to
present, interpret and
communicate various kinds of
numerical information.
(3) Construct and use linear
11
Updated August, 2009
4. identify from a table of
values the linear function’s y
intercept
1.2.1
1.2.3
5. use an equation or a graph
of a function that models a
real world situation to produce
a particular ordered pair and
to give an appropriate
interpretation of its meaning
in context
1.2.4
1.2.5
6. choose appropriate
increments and scales to
construct tables and fourquadrant graphs by hand
1.2.6
7. demonstrate ability to
select appropriate table set up
and windows to construct
tables and graphs using
technology
8. demonstrate how to create
a split screen (table/graph)
and use the trace feature to
demonstrate the relationship
between an ordered pair and a
point on the graph.
Middletown Public Schools
2.2.2
4.1.1
special cases of linear functions. functions to model and solve realworld situations.
Write linear equations and
inequalities that model real
(4) Use the coordinate plane to
world situations.
represent functions.
Decide on appropriate axis
labels and scales for the graph
of a function representing a
real-world situation.
Explain how changes in the
parameters m and b affect the
graph of a linear function.
Recognize and explain the
meaning and practical
significance of the slope and the
x- and y-intercepts as they
relate to a context, graph, table
or equation.
Create a linear function from
two non-vertical ordered pairs
or from a single ordered pair
and a rate of change.
Use technological tools such as
spreadsheets, probes, algebra
systems and graphing utilities to
organize, analyze and evaluate
large amounts of numerical
information.
Collect real data and create
meaningful graphical
representations of the data both
by hand and with graphing
technologies.
August 2009
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Updated August, 2009
1. determine run, rise, and
slope given two points in
the coordinate plane
Investigation 3 – Calculating and
Interpreting the Slope
1.1.6
2. identify the slope given the
verbal description, graphic
or tabular model of a linear
function
3. recognize the slope as the
coefficient of x given a
linear function in the form
f(x) = mx + b
4. graph a linear function in
slope-intercept form by
plotting an ordered pair and
using slope to find other
points on the line
1.2.3
1.2.4
1.2.5
5. recognize rates in the form
of units of the dependent
variable per units of
independent variable
4.1.1
6. interpret the rate of change
of the linear function in a
real world context
Understand that linear
functions, which can be written
symbolically as y = m x + b,
represent constant additive
change; a unit increase in the
independent variable (x) causes
the value of the dependent
variable (y) to change by m
units; arithmetic sequences are
special cases of linear
functions..
Decide on appropriate axis
labels and scales for the graph
of a function representing a
real-world situation.
Explain how changes in the
parameters m and b affect the
graph of a linear function.
Recognize and explain the
meaning and practical
significance of the slope and the
x- and y-intercepts as they
relate to a context, graph, table
or equation.
Collect real data and create
meaningful graphical
representations of the data both
by hand and with graphing
technologies.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Use a variety of representations
(including graphs, tables, words,
number lines, pictures, etc.) to
present, interpret and
communicate various kinds of
numerical information.
(3) Construct and use linear
functions to model and solve realworld situations.
(4) Use the coordinate plane to
represent functions.
(5) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
7. identify and graph
horizontal and vertical lines
8. determine whether lines
are parallel or
perpendicular.
Middletown Public Schools
August 2009
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Updated August, 2009
1. describe the changes in a
line that occur when the yintercept increases or
decreases
Investigation 4 – Effects of Changing
Parameters
1.2.3
1.2.4
2. describe the changes in a
line that occur when the
slope increases or decreases
1.2.5
3. explain the meaning of a
change in slope or a change
in y-intercept in the context
of a real world problem
Decide on appropriate axis
labels and scales for the graph
of a function representing a
real-world situation.
Explain how changes in the
parameters m and b affect the
graph of a linear function.
Recognize and explain the
meaning and practical
significance of the slope and the
x- and y-intercepts as they
relate to a context, graph, table
or equation.
(1) Make and justify predications
based on patterns.
(2) Construct, read, and interpret
tables, charts, and graphs of realworld data.
(3) Construct and use linear
functions to model and solve realworld situations.
4. identify the slope and yintercept of the line from
the graph of a linear
function
5. find the slope-intercept
form of the equation of a
line given its graph with the
y-intercept and an indicated
point.
1. recognize three forms of a Investigation 5 – Forms of Linear
linear equation; slopeEquations
intercept, standard and
point-slope
2. model a real world
situation with an
appropriate form of a linear
equation
3. find x- and y- intercepts
and slope of a linear
function given any form of
Middletown Public Schools
1.1.6
Understand that linear
functions, which can be written
symbolically as y = m x + b,
represent constant additive
change; a unit increase in the
independent variable (x) causes
the value of the dependent
variable (y) to change by m
units; arithmetic sequences are
special cases of linear functions.
1.2.1 Write linear equations and
inequalities that model real
world situations.
August 2009
(1) Construct and use linear
functions to model and solve realworld situations.
(2) Select appropriate methods for
computing.
(3) Construct, read, and interpret
tables, charts, and graphs of realworld data.
(4) Make and evaluate inferences
from tables, charts, graphs, and
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Updated August, 2009
the equation
4. draw the graph given the xand y- intercepts, slope and
y-intercept, or the slope and
a point on the graph
other representations of data.
(5) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
5. explain what the x- and yintercepts represent in the
context of a graph of a real
world problem
6. transform linear equations
from standard form to
slope-intercept form and
vice versa
7. transform linear equations
from point-slope form to
standard and to slopeintercept form.
1. Write an equation of a line
in the context of a real
world problem.
2. Write the equation of a line
in slope-intercept form,
point-slope form, or
standard form given (1) the
slope and y-intercept, (2)
the slope and one ordered
pair on the line, (3) two
ordered pairs or (4) an
ordered pair and the
Middletown Public Schools
Investigation 6 – Find the Equation
of a Line
1.1.5
Identify the role of independent (1) Construct and use linear
and dependent variables in a
functions to model and solve realfunction; determine the domain world situations.
and range of a function
describing a real-world problem
1.1.6 Understand that linear
functions, which can be written
symbolically as y = m x + b,
represent constant additive
change; a unit increase in the
independent variable (x) causes
the value of the dependent
variable (y) to change by m
units; arithmetic sequences are
special cases of linear functions.
1.2.1 Write linear equations and
August 2009
15
Updated August, 2009
equation of a parallel or
perpendicular line.
3. Identify when sufficient
information is given to write
an equation of a linear
function that models a real
world situation.
4. Make predictions based on
the meaning of the function.
1.2.2
1.2.3
1.2.4
1.2.5
1.2.6
1.3.1
2.2.1
Middletown Public Schools
inequalities that model realworld situations.
Represent functions (including
linear and nonlinear functions
such as square, square root, and
piecewise defined) with tables,
graphs, words and symbolic
rules; translate any
representation of a function into
another representation.
Decide on appropriate axis
labels and scales for the graph
of a function representing a
real-world situation.
Explain how changes in the
parameters m and b affect the
graph of a linear function.
Recognize and explain the
meaning and practical
significance of the slope and the
x- and y-intercepts as they relate
to a context, graph, table or
equation.
Create a linear function from
two non-vertical ordered pairs
or from a single ordered pair
and a rate of change.
Simply and solve linear
equations and inequalities.
Use algebraic properties,
including associative,
commutative and distributive,
inverse and order of operations
to simplify computations with
real numbers and simplify
expressions.
August 2009
16
Updated August, 2009
2.2.2
2.2.3
2.2.4
Use technological tools such as
spreadsheets, probes, algebra
systems and graphing utilities to
organize, analyze and evaluate
large amounts of numerical
information.
Choose from among a variety of
strategies to estimate and find
values of formulas, functions
and roots.
Judge the reasonableness of
estimations and computations.
Assessments:
Warm-ups, discussion/questioning, mid-unit assessment, performance task, end-unit assessment
Middletown Public Schools
August 2009
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Updated August, 2009
Unit 5: Scatter Plots and Trend Lines (3.5 weeks)
Essential Questions:
How do we make predictions and informed decisions based on current numerical information?
What are the advantages and disadvantages of analyzing data by hand versus using technology?
What is the potential impact of making a decision from data that contains one or more outliers?
Students should …
1. explore ways to fit a trend line
to data in a scatter plot and use
the trend line to make
predictions.
Effective Teaching Strategies
and Suggested Activities
Investigation 1 – Sea Level Rise
2. create the appropriate visual
and graphical representation of
real data.
.
1. identify the strength and
direction of the trend line
2. identify and explain the slope
and intercepts in the context of
the problem
3. explain what a coordinate pair
means in the context of the
situation
4. identify causal relationships
and explain the difference
between correlation and
Middletown Public Schools
Investigation 2 – Explorations of Data
Sets
Course-Level Expectations
5. Write linear equations that model real
world situations.
10. Decide on appropriate axes labels
and scales for the graph of a function
representing a real world situation.
18. Explore patterns and functions using
a variety of technologies (i.e.,
graphing calculators, spreadsheets,
on-line resources).
20. Collect real data and create
meaningful graphical representations
of the data both by hand and with
graphing technologies.
10. Decide on appropriate axes labels
and scales for the graph of a function
representing a real world situation.
13. Explore patterns and functions using
a variety of technologies (i.e.,
graphing calculators, spreadsheets,
on-line resources).
16. Recognize and explain the practical
significance of the slope and the xand y-intercepts of a linear function
in a real world problem.
18. Collect real data and create
meaningful graphical representations
of the data both by hand and with
August 2009
CAPT Correlations
(1) Use a variety of
representations (including
graphs, tables, words, number
lines, pictures, etc.) to present,
interpret and communicate
various kinds of numerical
information.
(2) Construct and use linear
functions to model and solve
real-world situations.
(1) Use a variety of
representations (including
graphs, tables, words, number
lines, pictures, etc.) to present,
interpret and communicate
various kinds of numerical
information.
(2) Construct and use linear
functions to model and solve
real-world situations.
(3) Use the coordinate plane to
represent functions.
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causation.
1. Students will be able to answer
a question about the world that
can be analyzed with bivariate
data.
2. For given bivariate data,
student will use a “guess and
check” strategy to manipulate
the slope and y intercept of a
trend line on a calculator in order
to find their best estimate for the
trend line.
3. For given or student-generated
bivariate data, students will be
able to use technology to graph a
scatter plot, calculate the
regression equation and
correlation coefficient, tell the
strength and direction of a
correlation, solve the equation
for y given x, interpolate and
extrapolate, explain the meaning
of slope and intercepts in
context, identify a reasonable
domain and distinguish between
data that is correlated compared
to causal.
Middletown Public Schools
graphing technologies.
Investigation 3 – Forensic
Anthropology: Technology and Linear
Regression
3. Make and justify predictions based on
patterns.
5. Write linear equations that model real
world situations.
9. Represent linear functions with tables,
graphs, words and symbolic rules.
Translate any representation of a
function into another representation.
10. Decide on appropriate axes labels
and scales for the graph of a function
representing a real world situation.
13. Explore patterns and functions using
a variety of technologies (i.e.,
graphing calculators, spreadsheets,
on-line resources).
16. Recognize and explain the practical
significance of the slope and the xand y-intercepts of a linear function
in a real world problem.
18. Collect real data and create
meaningful graphical representations
of the data both by hand and with
graphing technologies.
19. Use best-fit models on graphing
calculators to describe and analyze
collections of ordered pairs.
20. Predict an unknown value between
data points on a graph (interpolation)
and by extending the graph
(extrapolation). Interpolate and
extrapolate geometrically and
algebraically.
August 2009
(1) Use a variety of
representations (including
graphs, tables, words, number
lines, pictures, etc.) to present,
interpret and communicate
various kinds of numerical
information.
(2) Construct and use linear
functions to model and solve
real-world situations.
(3) Use the coordinate plane to
represent functions.
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1. define an outlier
Investigation 4 – Exploring the
Influence of Outliers
3. Make and justify predictions based on (1) Construct, read, and
patterns.
interpret tables, charts, and
13. Explore patterns and functions using graphs of real-world data.
a variety of technologies (i.e.,
graphing calculators, spreadsheets,
on-line resources).
19. Use best-fit models on graphing
calculators to describe and analyze
collections of ordered pairs.
Investigation 5 – Piecewise Functions
5. Write linear equations that model real
world situations.
9. Represent linear functions with tables,
graphs, words and symbolic rules.
Translate any representation of a
function into another representation.
2. identify if a potential outlier is
present on a scatter plot and
name the coordinates of the
outlier
3. draw trend lines and provide a
general description of the
influence that outliers have on
the slope as well as the direction
and strength of the relationship
between two variables
4. describe the impact that outliers
have on linear regression
equations, their related
components (i.e., slope, yintercept, correlation
coefficient), and the conclusions
drawn from an analysis of a
data set in which they are
included.
1. utilize multiple lists to input the
data and calculate linear
regression models
2. identify two points on each line
segment and use them to
calculate the equation of the line
Middletown Public Schools
August 2009
(1) Use the coordinate plane to
represent functions.
(2) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
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Updated August, 2009
that contains that segment
translate among representations.
3. identify the domain for which
the line segment exists
4. write the piecewise function
given the graph
5. write a story that describes the
piecewise graph.
Assessments:
Warm-ups, discussion/questioning, mid-unit assessment, performance task, end-unit assessment
Middletown Public Schools
August 2009
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Updated August, 2009
Unit 6: Systems of Linear Equations (3 weeks)
Essential Questions:
What does the number of solutions (none, one or infinite) of a system of linear equations represent?
What are the advantages and disadvantages of solving a system of linear equations graphically versus algebraically?
Students should…
Effective Teaching Strategies
and Suggested Activities
1. write the equations to model
Investigation 1 – Solving Systems of
the situation, graph the two
Linear Equations
equations (both by hand and
using the graphing calculator),
find the point of intersection
and interpret the solution in
the context of the problem
2. will solve a system of linear
equations that represents a
real world situation
graphically, algebraically, and
numerically
3. explain what the solution to a
system of linear equations
means in the context of the
problem
1. solve a system of linear
equations using the
substitution method.
Investigation 2 – Solving Systems of
Linear Equations Using Substitution
Course-Level Expectations
CAPT correlations
1.1.9 Explore patterns and functions
using a variety of technologies.
1.2.1 Write linear equations and
inequalities that model realworld situations.
1.2.3 Decide on appropriate axis labels
and scales for the graph of a
function representing a realworld situation.
1.2.7 Solve systems of linear equations
that model real world situations
using both graphical and
algebraic methods.
4.1.1 Collect real data and create
meaningful graphical
representations of the data both
by hand and with graphing
technologies.
(1) Represent and analyze situations
involving variable quantities with
tables, graphs, verbal rules, and
equations, and translate among
representations.
1.2.1
(1) Construct and use linear
functions to model and solve realworld situations.
1.2.7
2. explain what the solution to a
system of linear equations
means in the context of
practical problem.
Middletown Public Schools
Write linear equations and
inequalities that model realworld situations.
Solve systems of linear
equations that model real world
situations using both graphical
and algebraic methods.
August 2009
(2) Use a variety of representations
(including graphs, tables, words,
number lines, pictures, etc.) to
present, interpret and communicate
various kinds of numerical
information.
(3) Construct and use linear
functions to model and solve realworld situations.
(4) Use the coordinate plane to
represent functions.
(2) Assess the reasonableness of
answers to problems.
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1. Use the elimination method
to solve a system of equations
2. Explain the algebraic
properties upon which the
elimination method is based
Investigation 3 – Solving Systems of
Linear Equations Using Elimination
1.2.1
Write linear equations and
inequalities that model realworld situations.
1.2.7 Solve systems of linear equations
that model real world situations
using both graphical and
algebraic methods.
(1) Construct and use linear
functions to model and solve realworld situations.
(2) Assess the reasonableness of
answers to problems.
3. Explain the relationship
between the number of solutions
to a system of equations and the
relationship between the slopes
and y-intercepts of the equations
within a system
4. Identify the characteristics of
systems of equations that lend
themselves to the substitution
and elimination methods
Assessments:
Warm-ups, discussion/questioning, mid-unit assessment, performance task, end-unit assessment
Middletown Public Schools
August 2009
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Updated August, 2009
Unit 7: An Introduction to Exponential Functions (5 weeks)
Essential Questions:
What does the number of solutions (none, one or infinite) of a system of linear equations represent?
What characterizes exponential growth and decay?
What are real world models of exponential growth and decay?
What is the real world significance of the parameters in an exponential model?
How can we manipulate an exponential graph to model real data?
What are the limitations of exponential growth models?
How can one differentiate an exponential model from a linear model given a real world data set?
How can technology help model and analyze real world data sets?
What are the advantages and disadvantages of solving a system of linear equations graphically versus algebraically?
Students should …
Effective Teaching Strategies
and Suggested Activities
Course – Level Expectations
1. Recognize linear and nonlinear growth by examining a
table of values or a graph
Investigation 1 – A New Function
Family – World Population Growth
1.1.7 Recognize that exponential
functions represent constant
multiplicative change, written
symbolically as y = a (b to the x); a
unit increase in the independent
variable (x) causes the value of the
dependent variable (y) to be
multiplied by b; geometric
sequences are special cases of
exponential functions.
1.1.9 Explore patterns and functions
using a variety of technologies
1.2.1 Write linear equations and
inequalities that model real world
situations.
1.3.4 Use exponential functions to model
real world situations and solve
problems.
2. Make a scatter plot by hand
with appropriate scaling and
labels
3. Recognize that for nonlinear
growth the average rates of
change will not be constant
4. Define a linear equation
from a linear growth pattern
5. Use the home screen
recursive feature of a graphing
calculator to model
Middletown Public Schools
August 2009
CAPT Correlations
(1) Use ratios, proportions, and
percents to solve problems.
(2) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(3) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
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Updated August, 2009
exponential growth
6. Recognize that a
multiplicative pattern of
growth over Δ x = 1
represents exponential growth
2.2.4 Judge the reasonableness of
estimations and computations.
4.1.1 Collect real data and create
meaningful graphical
representations of the data both by
hand and with graphing
technologies.
7. Write very small and large
numbers in scientific notation
1. Recognize and describe the
parameters of an exponential
function
2. Explain the difference
between linear and
exponential growth.
Investigation 2 – Linear Growth and
Exponential Decay
Investigation 3 - Exploring the
Graph of y = a bx and Modeling
Exponential Growth
1.1.7 Recognize that exponential
functions represent constant
multiplicative change, written
symbolically as y = a (b to the x); a
unit increase in the independent
variable (x) causes the value of the
dependent variable (y) to be
multiplied by b; geometric
sequences are special cases of
exponential functions.
1.1.8 Explain the difference between
linear and exponential growth.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
1. Interpret and make
predictions from the graph of
an exponential function.
Investigation 4 – Collecting Data
and Modeling an Exponential
Relationship
1.1.10 Make and justify predictions based
on patterns.
(1) Make and justify predictions
based on patterns.
1. Use successive ratios to
approximate an exponential
pattern of growth or decay.
Investigation 5 – Using Data Tables
to Model Exponential Functions
4.2.1
(1) Make and justify predictions
based on patterns.
4.2.2
Middletown Public Schools
Analyze linear and exponential
models using trend lines and the
graphing calculator.
Estimate an unknown value
between data points on a graph or
list (interpolation) and make
predictions by extending the graph
or list (extrapolation).
August 2009
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Updated August, 2009
1. Identify and use the
percentage change of an
exponential function.
Investigation 6 – Exploring Percents
and Percent Change
1.1.7
1.3.4
1. Determine the doubling time
or half-life of an exponential
function
Investigation 7 – Compound Interest
and Exponential Decay
1.1.7
1.3.4
1. Create a table of carbon
dioxide (CO2) concentration
data from a graph
2. Explore a plot of data and
determine which function
family (linear or exponential)
to use as a model
Middletown Public Schools
Investigation 8 – The Mathematics
of Global Warming
Recognize that exponential
functions represent constant
multiplicative change, written
symbolically as y = a (b to the x);
a unit increase in the independent
variable (x) causes the value of the
dependent variable (y) to be
multiplied by b; geometric
sequences are special cases of
exponential functions.
Use exponential functions to
model real world situations and
solve problems.
(1) Use ratios, proportions, and
percents to solve problems.
Recognize that exponential
functions represent constant
multiplicative change, written
symbolically as y = a (b to the x);
a unit increase in the independent
variable (x) causes the value of the
dependent variable (y) to be
multiplied by b; geometric
sequences are special cases of
exponential functions.
Use exponential functions to
model real world situations and
solve problems.
(1) Use ratios, proportions, and
percents to solve problems.
1.1.5 Identify the role of independent and
dependent variables in a function;
determine the domain and range of a
function describing a real world
problem
1.1.6 Understand that linear functions,
which can be written symbolically
as y = m x + b, represent constant
August 2009
(2) Use integers, fractions,
decimals, percents, and scientific
notation in real-world situations to
count, measure, compare, order,
scale, locate, and label.
(2) Use integers, fractions,
decimals, percents, and scientific
notation in real-world situations to
count, measure, compare, order,
scale, locate, and label.
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
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Updated August, 2009
3. Determine an exponential
model for the CO2 vs. time
data
4. Determine a linear model for
the global temperature vs.
time data
5. Explain the significance of
the parameters a and b in the
context of the CO2
concentration vs. time model
6. Explain the significance of
the parameters m and b in the
context of the global
temperature vs. time model
7. Make projections for future
CO2 concentrations and global
temperature
additive change; a unit increase in
the independent variable (x) causes
the value of the dependent variable
(y) to change by m units; arithmetic
sequences are special cases of linear
functions.
1.1.8 Explain the difference between
linear and exponential growth.
1.1.9 Explore patterns and functions using
a variety of technologies (i.e.
graphing calculators, spreadsheets,
and on-line resources).
1.2.1 Write linear equations and
inequalities that model real world
situations.
1.2.7 Solve systems of linear equations
that model real world situations
using both graphical and algebraic
methods.
1.3.4 Use exponential functions to model
real world situations and solve
problems.
2.1.1 Compare, locate, label and order
integers, rational numbers and real
numbers on number lines, scales and
coordinate grids.
2.2.4 Judge the reasonableness of
estimations and computations.
4.1.1 Collect real data and create
meaningful graphical representations
of the data both by hand and with
graphing technologies.
4.2.1 Analyze linear and exponential
models using trend lines and the
graphing calculator.
Middletown Public Schools
August 2009
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Updated August, 2009
Assessments:
Warm-ups, discussion/questioning, mid-unit assessment, performance task, end-unit assessment
Middletown Public Schools
August 2009
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Updated August, 2009
Unit 8: Linear Programming Capstone (15 days)
Essential Questions:
How can linear programming be used to manipulate real world data?
Students should …
1. use linear programming to
identify optimal solutions to
practical problems.
Middletown Public Schools
Effective Teaching Strategies
and Suggested Activities
Unit 8 worksheets and activities
Course – Level Expectations
1.1.9 Explore patterns and functions
using a variety of technologies (i.e.
graphing calculators, spreadsheets,
and on-line resources).
1.2.1 Write linear equations and
inequalities that model real world
situations.
1.2.2 Represent functions (including
linear and nonlinear functions such
as square, square root, and
piecewise defined) with tables,
graphs, words and symbolic rules;
translate any representation of a
function into another
representation.
1.2.5 Recognize and explain the meaning
and practical significance of the
slope and the x- and y-intercepts as
they relate to a context, graph, table
or equation.
1.3.1 Simplify and solve linear equations
and inequalities.
2.1.1 Compare, locate, label and order
integers, rational numbers and real
numbers on number lines, scales
and coordinate grids.
August 2009
CAPT Correlations
(1) Represent and analyze
situations involving variable
quantities with tables, graphs,
verbal rules, and equations, and
translate among representations.
(2) Use variables, expressions,
equations, and inequalities,
including formulas, to model
situations and solve problems.
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Updated August, 2009
2.2.2 Use technological tools such as
spreadsheets, probes, algebra
systems and graphing utilities to
organize, analyze and evaluate large
amounts of numerical information.
2.2.3 Choose from among a variety of
strategies to estimate and find
values of formulas, functions and
roots.
2.2.4 Judge the reasonableness of
estimations and computations.
4.1.1 Collect real data and create
meaningful graphical
representations of the data both by
hand and with graphing
technologies.
Assessments:
Homework, research questions, student presentations and a differentiated Linear Programming Capstone Problem.
Reference
This document adopted all activities, assessments, objectives, and course-level expectations from a draft of the CT state algebra 1 curriculum.
Middletown Public Schools
August 2009
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