Download Understanding By Design Unit Template

Understanding By Design Unit Template
Title of Unit
Curriculum Area
Developed By
Unit 6: Exponents and Exponential Functions
Algebra 1
Anne Heyt
Grade Level
Time Frame
9th
3 weeks
Identify Desired Results (Stage 1)
Content Standards
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N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.
A.SSE.1 Interpret expressions that represent a quantity in terms of its context.
A.SEE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented
by the expression.★
A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate
axes with labels and scales.
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.★
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 zeros, extreme values,
and symmetry of the graph, and interpret these in terms of a context.
b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent
rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing
exponential growth or decay.
F.BF.1Write a function that describes a relationship between two quantities. ★
Determine an explicit expression, a recursive process, or steps for calculation from a context.
Combine standard function types using arithmetic operations. For example, build a function that models the
temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.
F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations,
and translate between the two forms.★
F.LE.1 Distinguish between situations that can be modeled with linear functions and with exponential functions.
Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by
equal factors over equal intervals.
Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.
Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to
another.
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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).
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.
S.ID.6 Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.
a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given
functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.
b. Informally assess the fit of a function by plotting and analyzing residuals.
c. Fit a linear function for a scatter plot that suggests a linear association.
Understandings
Essential Questions
Overarching Understanding
Overarching
Topical
Students must understand the multiplication properties of exponents.
Students must understand the division property of exponents.
Students must understand how to combine exponential expressions involving
both multiplication and division.
Students must understand how formulas for growth and decay relate to
geometric sequences.
Related Misconceptions
Multiplying and Dividing exponents is like multiplying and dividing any other
number.
Linear equations are the same as exponential equations.
The graph of an exponential equation is a line.
How can data sets and their graphs
be interpreted as linear or
exponential?
How are multiplication and division
related to addition and subtraction
in the laws of exponents?
How can exponential functions be
used to model applications that
include growth and decay in
different contexts?
Knowledge
Skills
The following terms:
Coefficient
Term
Factors
Exponent
Raised to a power
Exponential Growth
Exponential Decay
Constant Multiplier
I can apply the properties of exponents to simplify expressions with
integer and rational exponents. (N.RN.2)
I can write radical expressions as expressions with rational exponents
(N.RN.2)
I can write expressions with rational exponents as radical expressions.
(N.RN.2)
I can identify the terms, factors, and coefficients of an equation. (SSE.1)
I can define an exponential equation, y= a*b ^x (SSE.3)
I can rewrite and exponential equation using the properties of exponents.
Students will know…
Students will be able to…
Growth rate
Rate of Decay
When values change through multiplying by a constant, that constant is
called a constant multiplier. The rate of change increases as the amount
increases.
If a constant multiplier is greater then 1, we have exponential growth, as
represented by the equation y= A(1+r) ^X. In this case, r is called the
growth rate.
If a constant multiplier is less then 1, we have exponential decay, as
represented by the equation y= A(1-r)^X. In this case, r is called the rate of
decay.
When you multiply like bases, add the exponents when raising a power to a
power.
When dividing powers with like bases we subtract the exponents, that is you
take the exponent of the numerator and subtract the exponent of the
denominator.
A negative exponent can be changed to a positive exponent by taking the
reciprocal of the base, making the exponent positive.
Exponenital functions are in the form f(x) = ab^x. The graph of the
exponential function y=b^x decreases at a decreasing rate when b is
between 1 and 0, and increases at an increasing rate when b is greater then
1.
How to find the common ratio given a sequence of numbers.
How to take a sequence of numbers and write the recursive and explicit
formulas for a geometric sequence.
How to determine the common ratio given a graph of an equation.
(SSE.3)
I can determine the best model for a real world problem (for example,
growth or decay models). (CED.2)
I can determine if an equation is expressing growth or decay by
examining the equation. (F.IF.7)
I can create a table or graph of an exponential function. (F.IF.7)
I can explain how a simple geometric transformation changes a growth
graph to a decay graph. ( F.IF.7)
I can define a geometric sequence, and transform it into a recursive or
explicit function. (F.BF.1)
I can write a recursive or explicit expression to describe a real world
problem. (F.BF.1)
I can define a geometric sequence of numbers and understand that the
ratio of consecutive terms is called the common ratio. (F.BF.2)
I can determine the common ratio between two terms in a series.
(F.BF.2)
I can explain why the recursive formula for a geometric sequence uses
multiplication and why the explicit formula uses exponentiation. (F.BF.2)
I can translate between the recursive and explicit forms of geometric
sequences (F.BF.2)
I can define the exponential function y=ab^x. (F.LE.1)
I can identify situations the display equal ratios of change over equal
intervals, and understand that these can be modeled with exponential
functions. (F.LE.1)
I can tell if a real world situation should be modeled with a linear function
or an exponential function. (F.LE.1)
I can determine if a function is linear or exponential given a sequence,
graph, description, or a table. (F.LE.2)
I can describe the process used to construct an exponential function that
passes through two given points. (F.LE.2)
I can explain why exponential functions eventually have greater output
values the linear equations. (F.LE.3)
I can explain the meaning of the constant’s a and b of an exponential
function when the exponential function models a real world relationship.
I can determine when linear or exponential models should be used to
represent a data set. (S.ID.6)
I can determine whether linear and exponential models are increasing or
decreasing. (S.ID.6)
Assessment Evidence (Stage 2)
Performance Task Description
Goal
Role
Audience
Situation
Product/Performance
Standards
Other Evidence – Selected Response Assessments, Extended Response Assessments,
Exit Cards, work samples, observations
Student Self–Assessment and Reflection
Learning Plan (Stage 3)
Where are your students headed? Where have
they been? How will you make sure the
students know where they are going?
How will you hook students at the beginning of
the unit?
What events will help students experience and
explore the big idea and questions in the unit?
How will you equip them with needed skills and
knowledge?
How will you cause students to reflect and
rethink? How will you guide them in rehearsing,
revising, and refining their work?
How will you help students to exhibit and selfevaluate their growing skills, knowledge, and
understanding throughout the unit?
How will you tailor and otherwise personalize
the learning plan to optimize the engagement
and effectiveness of ALL students, without
compromising the goals of the unit?
How will you organize and sequence the
learning activities to optimize the engagement
and achievement of ALL students?
From: Wiggins, Grant and J. Mc Tighe. (1998). Understanding by Design, Association for Supervision and Curriculum Development
ISBN # 0-87120-313-8 (ppk)