Montclair Public Schools CCSS Algebra 2 Honors Unit 3: Marshall ...

Montclair Public Schools
CCSS Algebra 2 Honors Unit 3: Marshall A.b
Subject Algebra 2 Grade 11 Unit # 3 Pacing Marking Period 3
Unit Name
2013-2014
Modeling with Exponential, Logarithmic, Trigonometric Functions
Overview
Students will begin by understanding the building blocks of geometry. Basic terms, theorems, and proofs will be introduced. Students will explore these
ideas by learning how to create proofs (formal and informal) and by generating constructions (using compass and straightedge, or geometric software
and other methods).
Standard # Standard MC,
SC,
or
AC
N.RN.1
N.RN.2
A.SSE.3
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
5^(1/3) to be the cube root of 5 because we
want (5^(1/3))^3 = 5*(3/3) to hold, so
(5^(1/3))^3 must equal 5.
Rewrite expressions involving radicals and
rational exponents using the properties of
exponents.
Use properties of exponents to transform
expressions for exponential functions. For
example, the expression 1.15^t can be
rewritten as (1.15^(1/12))^(12t) = 1.012^(12t)
to reveal the approximate equivalent monthly
SLO
#
MC 1
2
MC 3
4
MC 5
6
Student Learning Objectives
Use properties of integer exponents to explain and
convert between expressions involving radicals and
rational exponents, using correct notation. For
example, we define 5^(1/3) to be the cube root of 5
because we want (5^(1/3))^3 = 5*(3/3) to hold, so
(5^(1/3))^3 must equal 5.
Determine if a value raised to a rational exponent will
result in a real number rather than an imaginary value.
(e.g., 4^(1/2) = 2, (-27)^(1/3) = -3, (-4)^(1/4) is not real,
etc.)
Write equivalent expressions for exponential functions
using the properties of exponents.
Reinforcement of laws of exponents. Understand how
x^(1/4) * x^(3/4) is equivalent to 1. Rewrite 4^(1/2) *
16^(1/4)=2^2
Write equivalent expressions for exponential functions
using the properties of exponents.
Students to analyze examples of exponential growth
and decay models (relating to biology and chemistry
Depth of
Knowledge
4
3
4
3
4
4

F.BF.1
interest rate if the annual rate is 15%.
Write a function that describes a relationship
between two quantities. 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.
7
8
courses) as well as compounding interest models
(relating to finance and economics courses).
Compare exponential and linear growth rates and
determine during which intervals exponential growth
may be slower than that of linear growth.
Construct a function that combines standard function
types using arithmetic operations to model a
relationship between two quantities.
3
4
F.LE.4
For exponential models, express as a logarithm
the solution to ab^c = d, where a, c, and d are
numbers and the base b is 2, 10, or e, evaluate
the logarithm using technology.
9
SC 10
Students will construct functional models of real-world
situations.
Identify the inverse relationship that exists between
logarithms and exponential functions. Students
should apply this to problem solving involving
exponential and logarithmic functions.
3
3
11
Express as a logarithm, the solution to ab^c = d, where
a, c, and d are numbers and the base is 2, 10 , or e.
3
12
Evaluate the logarithm using either graphing or
scientific technology.
3
13
Apply laws of logarithms to either expand or contract
a given logarithmic expression.
4
F.IF.7e
Graph exponential and logarithmic functions
showing intercepts, and end behavior, and
trigonometric functions, showing period,
midline, and amplitude.
SC 14
Graph functions expressed symbolically and show key
features of the graph (including intercepts, intervals
where the function is increasing, decreasing, positive
or negative; relative maxima and minima; symmetries,
and end behavior) by hand in simple cases and using
technology in more complicated cases.
(Students encouraged to use graphing technologies)
2
15
Interpret the parameters in an exponential function in
2
2013-2014

terms of its context.
F.TF.1
Understand the radian measure of an angle as
the length of the arc on the unit circle
subtended by the angle.
16
AC 17
Contrast how exponential and logarithmic functions
are different from linear and quadratic functions.
Use the radian measure of an angle to find the length
of the arc in the unit circle subtended by the angle and
find the measure of the angle given the length of the
arc.
3
2
F.TF.2
Explain how the unit circle in the coordinate
plane enables the extension of trigonometric
functions to all real numbers, interpreted as
radian measures of angles traversed
counterclockwise around the unit circle.
18
AC 19
Reinforce calculation of arc length from prior
Geometry content.
Explain how the unit circle in the coordinate plane
enables the extension of trigonometric functions to all
real numbers (interpreted as radian measures of all
angles traversed counterclockwise around the unit
circle).
2
1
20
Relate prior content from Geometry to unit circle,
specifically 30-60-90 and 45-45-90 triangles.
2
F.TF.8
Prove Pythagorean Identity (sin x)^2 + (cos x)^2
= 1 to find sin x, cos x, tan x, given sin x, cos x,
or tan x, and the quadrant of the angle.
21
AC 22
Facilitate student discovery of the relationship of
quadrant 1 to the other quadrants of the unit circle
and relate to the ASTC mnemonic device.
Reinforce Pythagorean theorem from Geometry
content. Construct angle relationships using
fundamentals in right triangle trigonometry and
identify SOHCAHTOA mnemonic device
1
2
F.TF.5
Choose trigonometric functions to model
periodic phenomena with specified amplitude,
frequency and midline.
23 Use the Pythagorean Identity (sin x)^2 + (cos x)^2 = 1
to find sin x, cos x, tan x, given sin x, cos x, or tan x,
and the quadrant of the angle.
AC 24 Choose trigonometric functions to model periodic
phenomena with specified amplitude, frequency and
midline.
1
1
2013-2014

F.IF.7e
Mathematical
Practice #
Graph exponential and logarithmic functions
showing intercepts, and end behavior, and
trigonometric functions, showing period,
midline, and amplitude.
AC 25
Graph functions expressed symbolically and show key
features of the graph, by hand in simple cases and
using technology for more complicated cases.
26 Graph trigonometric functions showing period,
midline, and amplitude.
Selected Opportunities for Connections to Mathematical Practices
1
1
1
2
3
4
5
6
7
8
2013-2014
SLOs 6, 7, 10, 15, 16, 24
SLOs 1, 2, 7, 12
SLOs 6, 7, 8, 9, 24, 25, 26
SLOs 6, 7, 8, 9, 14, 15, 24, 26
SLOs 7, 11, 12, 13, 14, 18, 25
SLOs 1, 2, 5, 11, 13, 15, 17, 18
SLOs 10, 13, 14, 15, 16, 21
SLOs 1, 2, 3, 4, 19, 20
Big Ideas
Radical expressions can be written in terms of rational (fractional) exponents, which are governed by the laws of exponents in a manner
consistent with expressions containing integer exponents.
Exponential and logarithmic functions are inverses.
Exponential functions model real world situations modeling growth and decay.
Angles can be measured in degrees and radians, which unlike degrees relate to a distance measurement (i.e., arc length).
The unit circle is a useful tool in showing the relations between trigonometric functions and the repetitive values in angles.
Basic trigonometric functions have periodicity (repetitive behavior), a frequency, and may have amplitude.
Essential Questions
Can one conclude that exponential and logarithmic functions are inverses by analyzing algebraic and graphical details?
What are exponential functions and how can they be used to model growth and decay situations in the sciences and finance?
How can logarithms be used to solve exponential equations?
Why must students learn radian measure in addition to degree measure?
How can trigonometric functions be used to model periodic event?
Assessments
Tests, quizzes, and homework aligned to the curriculum content. Algebra 2 Academic Unit 3 Model Assessment is a 44-minute, single-class-period
assessment that should be administered at the end of the third marking period.

Key Vocabulary
Rational exponents, Radicals
Exponential modeling, Logarithms
Trigonometry – radians, unit circle, period, midline, amplitude
Function properties – increasing/decreasing, end behavior
Suggested Resources (list specific chapters and or page numbers from existing text that correspond to the SLOs and Standards)
SLO Sections and page numbers from Prentice Hall New Jersey Algebra 2, Pearson Education, Inc. 2009
1 7-4 on pp. 385-390
2 7-1 on pp. 370-373
3, 4, 5 7-1 on p. 368 and 7-4 on p. 387
6 to 9 8-1, 8-2 on pp. 430-445, 479
10 to 13 8-3 on pp. 439-452
14 to 16 8-3 on pp. 439-452 addressing graphical features of exponential and logarithmic graphs.
17 13-3 on pp. 726-730
18 13-3 on p. 728, 725 (classroom activity/demonstration)
19 13-2 on pp. 720-723.
20 Geometry Review on p. 717
21 Exercise #50 on p. 723 (All Students Take Calculus) Quadrant 1 ALL trig function values are positive, Q2 Only Sine is positive etc.
22 14-3 on pp. 792-794
23 14-1 on p. 779
24 to 26 Period 13-1 on pp. 710-716, Graphing sine 13-4 on pp. 734-742, Graphing Cosine 13-5 on pp. 743-748.
Common
Core
http://www.parcconline.org/samples/mathematics/high-school-mathematics
http://www.state.nj.us/education/modelcurriculum/math/
http://ccss.lexington4.net/unpacking-the-standards
http://www.engageny.org/
http://www.illustrativemathematics.org/standards/hs
http://www.ccsstoolbox.com/parcc/PARCCPrototype_main.html
http://www.smarterbalanced.org/sample-items-and-performance-tasks/
2013-2014