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Wentzville School District
Algebra 1: Unit 10
Stage 1 – Desired Results
Unit 10 - Comparing Functions
Unit Title: Comparing Functions
Course: Algebra I
Brief Summary of Unit: In this unit students will review the features of various functions. Students will be able to
recognize and utilize the similarities between linear, quadratic and exponential to solve problems. Students will extend
these commonalities as they identify and graph step and absolute value functions. Finally, students will solve systems
of equations involving more than one type of function.
Textbook Correlation: Glencoe Algebra I Chapter 1: section 8, chapter 9: 6 & 7 (section 7, is only step & absolute value),
Lab 9.3
Time Frame: 2 weeks
WSD Overarching Essential Question
Students will consider…
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How do I use the language of math (i.e. symbols,
words) to make sense of/solve a problem?
How does the math I am learning in the classroom
relate to the real-world?
What does a good problem solver do?
What should I do if I get stuck solving a problem?
How do I effectively communicate about math
with others in verbal form? In written form?
How do I explain my thinking to others, in written
form? In verbal form?
How do I construct an effective (mathematical)
argument?
How reliable are predictions?
Why are patterns important to discover, use, and
generalize in math?
How do I create a mathematical model?
How do I decide which is the best mathematical
tool to use to solve a problem?
WSD Overarching Enduring Understandings
Students will understand that…
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Mathematical skills and understandings are used
to solve real-world problems.
Problem solvers examine and critique arguments
of others to determine validity.
Mathematical models can be used to interpret and
predict the behavior of real world phenomena.
Recognizing the predictable patterns in
mathematics allows the creation of functional
relationships.
Varieties of mathematical tools are used to
analyze and solve problems and explore concepts.
Estimating the answer to a problem helps predict
and evaluate the reasonableness of a solution.
Clear and precise notation and mathematical
vocabulary enables effective communication and
comprehension.
Level of accuracy is determined based on the
context/situation.
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How do I effectively represent quantities and
relationships through mathematical notation?
How accurate do I need to be?
When is estimating the best solution to a
problem?
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Using prior knowledge of mathematical ideas can
help discover more efficient problem solving
strategies.
Concrete understandings in math lead to more
abstract understanding of math.
Transfer
Students will be able to independently use their learning to…
know that a function can be written to model a real world situation.
Meaning
Essential Questions
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What do the key features of graphs of functions
represent in a real-world context?
When would technology be useful in comparing
functions?
How is a transformed function related to its parent
function?
Which key feature(s) will best help interpret a
problem?
How can you determine the practical domain of
the function as it relates to the numerical
relationship it describes?
How many solutions does a system with various
types of functions contain?
How can you determine if a function is linear,
quadratic or exponential from a graph, a table or
an equation?
What does the intersection of two graphs of
functions represent?
Why are transformations of graphs the same
regardless of the type of function?
What are similarities and differences between two
graphs?
Understandings
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Functions can be modeled using an algebraic
expression, table or graph and each model can be
used to analyze the function
Technology can be utilized to help investigate
functions.
Various representations of the same function
emphasize different characteristics of the given
function.
Knowing how vertical translations, horizontal
translations, reflections, and dilations effect
parent functions, makes writing equations of
functions and graphing functions more efficient.
Transformations of functions can be an effective
tool to graph and describe functions efficiently.
Contextual situations can have restricted
domains and/ranges.
When two functions are graphed simultaneously,
where they meet has meaning/context in an
equation.
Linear, quadratic and exponential functions
model different real world situations and they
could be represented graphically, as a table or in
equation format.
Functions written in different formats can be
compared to identify additional information
about the function for interpretation.
Acquisition
Key Knowledge
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Key Skills
Key features of graphs including: intercepts;
intervals where the function is increasing,
decreasing, positive, or negative; relative
maximums and minimums; symmetries; end
behavior
Average rate of change of a function
Domain & range
Linear Equations
Exponential Equations
Quadratic Equations
Absolute and relative maximums and minimums
successive differences
step function
absolute value function
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Identify intercepts
Determine if a function is increasing or decreasing
over a given interval
Identify relative extrema in the graph of a function
Identify symmetries in the graph of a function
Identify end behavior for the graph of a function
Solve real world problems using linear,
exponential, and quadratic equations
Graph linear and quadratic functions showing
intercepts, maxima, and minima.
Graph linear, exponential and quadratic equations
on the coordinate plane with appropriate labels
and scales
Determine if a given data set can be described
with a linear, an exponential, or a quadratic model
by:
○ graphing
○ using successive differences
○ using ratios
Given a set of data, write either a linear,
exponential, or quadratic equation that will model
the data
Use technology (graphing calculator, spreadsheet,
computer algebra system, etc.) to determine a
model (linear, exponential, quadratic) that will fit a
set of data
Demonstrate that transformations on linear,
quadratic, exponential, step and absolute value
functions follow the same patterns.
Identify and graph step functions
Identify and graph absolute value
Solve systems of linear and quadratic equations
using technology.
Solve real-world problems that can be modeled
with linear, exponential, quadratic, step, and
absolute value, functions using multiple strategies.
Standards Alignment
MISSOURI LEARNING STANDARDS
F.IF.4
For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of
the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features
include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and
minimums; symmetries; end behavior; and periodicity.★
F.IF.6
Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified
interval. Estimate the rate of change from a graph.★
F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using
technology for more complicated cases.★ b. Graph square root, cube root, and piecewise-defined functions, including
step functions and absolute value functions.
F.IF.8
Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of
the function.
a. Use the process of factoring and completing the square in a quadratic function to show zeroes, extreme values, and
symmetry of the graph, and interpret these in terms of a context.
F.IF.9
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in
tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for
another, say which has the larger maximum.
A.REI.7
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and
graphically. For example, find the points of intersection between the line y = –3x and the circle x2 + y2 = 3.
F.BF.3
Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive
and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on
the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for
them.
F.LE.1
Distinguish between situations that can be modeled with linear functions and with exponential functions.
a. Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by
equal factors over equal intervals.
b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to
another.
F.LE.2
Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description
of a relationship, or two input-output pairs (include reading these from a table).
1. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer
exponents to those values, allowing for a notation for radicals in terms of rational exponents. For example, we define
51/3 to be the cube root of 5 because we want (51/3)3 = 51/3*3 to hold, so (5)1/3*3must equal 5.
2. Rewrite expressions involving radicals and rational exponents using the properties of exponents
F.LE.3
Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing
linearly, quadratically, or (more generally) as a polynomial function.
F.LE.5
Interpret the parameters in a linear or exponential function in terms of a context.
A.CED.2
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
A.REI.11
Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the
solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions,
make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial,
rational, absolute value, exponential, and logarithmic functions.★
MP.1 Make sense of problems and persevere in solving them.
MP.2 Reason abstractly and quantitatively.
MP.3 Construct viable arguments and critique the reasoning of others.
MP.4 Model with mathematics.
MP.5 Use appropriate tools strategically.
MP.6 Attend to precision.
MP.7 Look for and make use of structure.
MP.8 Look for and express regularity in repeated reasoning.
SHOW-ME STANDARDS
Goals:
1.1, 1.4, 1.5, 1.6, 1.7, 1.8
2.2, 2.3, 2.7
3.1, 3.2, 3.3, 3.4, 3.5, 3.6, 3.7, 3.8
4.1, 4.4, 4.5, 4.6
Performance:
Math 1, 4, 5