Download Clear Learning Targets Precalculus

Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-BF.1
Build a function that models
Essential Understanding
Academic Vocabulary/Language
a relationship between two
quantities
- Students should be able to
identify and describe
characteristics of parent
functions. Students should
develop understanding of
translations/transformations and
their applications to all functions.
-zeros, roots, transformation,
translation reflections, dilation,
parent square root, constant
identity, quadratic cubic, reciprocal
absolute value step, greatest
integer function, composition
Write a function that describes a relationship between two
quantities.
Compose functions. For example, if T(y) is the temperature in
the atmosphere as a function of height, and h(t) is the height of
a weather balloon as a function of time, then T(h(t)) is the
temperature at the location of the weather balloon as a function
of time.
I Can Statements
 I can use graphs of functions to estimate function values.
 I can identify, graph, and describe parent functions.
 I can identify and graph transformations of parent
functions.
 I can perform operations with functions.
 I can find compositions of functions.
Tier 2 Vocabulary
Extended Understanding
- describe, compose, transform
- Students can translate piecewise
functions or functions that they
have not yet seen.
- Students may find one function
when given the composition of the
functions and the other function.
Prior Knowledge
Future Learning
- transform/translate linear,
quadratic, and exponential
functions.
- Use compositions and
transformations to find inverses of
relations and functions both
algebraically and graphically.
- identify domain and range of
linear, quadratic, and exponential
functions.
- evaluate functions.
- operations on functions.
Columbus City Schools
Clear Learning Targets Precalculus 2015-2016
1
Instructional Strategies
Utilize graphing technology so that students can investigate behaviors of transformations on different families of functions.
Provide examples of when functions can be combined, such as determining a function describing the monthly cost for owning two vehicles
when a function for the cost of each (given the number of miles driven) is known. Using visual approaches (e.g., folding a piece of paper in half
multiple times), use the visual models to generate sequences of numbers that can be explored and described with both recursive and explicit
formulas. Emphasize that there are times when one form to describe the function is preferred over the other.
Common Misconceptions and Challenges
Students may not consider order when transforming functions.
Students may misunderstand function notation for composition of functions to represent multiplication (e.g., f(g(x)) means to multiply the f
and g function values).
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Lessons 1-2, 1-5, 1-6
Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/BF/B/4 Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career Connections
Students will research and evaluate several options when purchasing a vehicle (e.g., new versus used, lease versus own, down payment, and
interest rate). They will examine the differences in gas mileage consumption by selecting two vehicles to evaluate (e.g., SUV versus compact
hybrid). Once they choose a vehicle, they will use their evaluations to show why they chose the vehicle. Their research will include
interviewing automotive professionals, visiting dealerships, and navigating company websites.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
2
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-BF.4
Build new functions from
Essential Understanding
Academic Vocabulary/Language
existing functions
- Students should be able to find
inverse functions algebraically
and graphically.
- Students should be able to
evaluate whether two functions
are inverses by examining their
graphs.
- set notation, interval notation,
implied domain, relevant domain,
piecewise function, continuous,
limit, discontinuous, infinite, jump,
point, removable discontinuity,
nonremovable discontinuity, end
behavior, inverse relation, inverse
function, one-to-one
Find inverse functions. a. Solve an equation of the form f(x) = c
for a simple function f that has an inverse and write an
expression for the inverse. For example, f(x) = 2x3 for x>0 or
f(x) = (x+1)/(x-1) for x ≠1. b. (+) Verify by composition that one
function is the inverse of another. c. (+) Read values of an
inverse function from a graph or table, given that the function
has an inverse. d. (+) Produce an invertible function from a noninvertible function by restricting the domain.
I Can Statements
 I can describe subsets of real numbers.
 I can identify and evaluate functions and state their
domain.
 I can identify odd and even functions.
 I can use limits to determine the continuity of a function.
 I can use limits to describe the end behavior of functions.
 I can use the horizontal line test to determine whether a
function has an inverse.
 I can find inverse functions algebraically and graphically.
Columbus City Schools
Extended Understanding
- Students can evaluate whether
piecewise functions have inverses.
- Investigate inverses of even and
odd functions.
Tier 2 Vocabulary
- compose, construct, evaluate,
produce
Prior Knowledge
Future Learning
- Find domain and range of
functions.
- Analyze graphs of polynomial and
rational functions.
- Find composition of functions.
- Evaluate functions.
- use limits to determine function
continuity.
- use limits to define end behavior.
Clear Learning Targets Precalculus 2015--‐2016
3
Instructional Strategies
Provide examples of inverses that are not purely mathematical to introduce the idea. For example, given a function that names the capital of a
state, f(Ohio) = Columbus. The inverse would be to input the capital city and have the state be the output, such that f --1 (Denver) = Colorado.
Use real-world examples of functions and their inverses. For example, students might determine that folding a piece of paper in half 5 times
results in 32 layers of paper, but that if they are given that there are 32 layers of paper, they can solve to find how many times the paper would
have been folded in half.
Students should also recognize that not all functions have inverses. Again using a nonmathematical example, a function could assign a
continent to a given country’s input, such as g(Singapore) = Asia. However, g-1 (Asia) does not have to be Singapore (e.g., it could be China).
Exchange the x and y values in a symbolic functional equation and solve for y to determine the inverse function. Recognize that putting the
output from the original function into the input of the inverse results in the original input value.
Common Misconceptions and Challenges
Students may believe that all functions have inverses and need to see counter examples, as well as examples in which a non-invertible function
can be made into an invertible function by restricting the domain. For example, f(x) = x 2 has an inverse ( f -1 (x) = x ) provided that the domain
is restricted to x ≥ 0.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Lessons 1-1, 1-2, 1-3, 1-7
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSF/TF/A/4
Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career Connections
Aerospace engineers Perform a variety of engineering work in designing, constructing, and testing aircraft, missiles, and spacecraft. May
conduct basic and applied research to evaluate adaptability of materials and equipment to aircraft design and manufacture. May recommend
improvements in testing equipment and techniques.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
4
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-IF.7
Analyze functions using
Essential Understanding
Academic Vocabulary/Language
different representations
- Students should be able to solve
rational equations algebraically,
finding extraneous solutions.
- Students should be able to
analyze rational functions,
determining whether there are
any asymptotes.
- Students should be able to graph
rational functions, including
asymptotes.
- rational functions, asymptote,
vertical asymptote, horizontal
asymptote, oblique asymptote,
holes
Graph functions expressed symbolically and show key features
of the graph, by hand in simple cases and using technology for
more complicated cases. d. (+) Graph rational functions,
identifying zeros and asymptotes when suitable factorizations
are available, and showing end behavior.
Tier 2 Vocabulary
- analyze, construct, evaluate
Extended Understanding
- Students can construct a rational
function when given intercepts
and asymptotes or when given a
graph.
I Can Statements




I can analyze and graph rational functions.
I can find zeros of rational functions.
I can solve rational equations.
I can find asymptotes of rational functions and explain
discontinuities.
 I can describe end behavior of rational functions using
limit notation.
Columbus City Schools
Prior Knowledge
Future Learning
- Identify points of discontinuity
and end bahavior of graphs of
funtions using limits.
- Write partial fraction
decompositions of rational
expressions.
Clear Learning Targets Precalculus 2015--‐2016
5
Instructional Strategies
Important features of rational functions are vertical asymptotes, or boundaries that the function gets closer and closer to, but never actually
reaches. Vertical asymptotes exist when the denominator of the function is 0 and can be found by computing the zeros of the denominator of
the function. If the highest-order x terms in the numerator and denominator have the same exponent, then there will also be a horizontal
asymptote when their coefficients are divided.
https://www.khanacademy.org/math/algebra2/rational-expressions/rational-function-graphing/e/graphs-of-rational-functions
http://betterlesson.com/common_core/browse/640/ccss-math-content-hsf-if-c-7d-graph-rational-functions-identifying-zeros-andasymptotes-when-suitable-factorizations-are-availab
Common Misconceptions and Challenges
Students may believe that skills such as factoring a trinomial or completing the square are isolated within a unit on polynomials, and that they
will come to understand the usefulness of these skills in the context of examining characteristics of functions.
Students may forget to factor the numerator when finding asymptotes.
When using graphing technology, students may not consider the window they are viewing the function through
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Lesson 2-2
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSF/IF/C/7
Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career Connections
Chemical engineers Design chemical plant equipment and devise processes for manufacturing chemicals and products, such as gasoline,
synthetic rubber, plastics, detergents, cement, paper, and pulp, by applying principles and technology of chemistry, physics, and engineering.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
6
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
CN.3
Perform arithmetic
Essential Understanding
Academic Vocabulary/Language
operations with complex
numbers
- It is not possible in the realm of
real numbers to find the square
root of a negative number.
- The square root of -1 is referred to
as i.
- Rational Zero Theorem,
Descartes’ Rule of Signs,
Fundamental Theorem of Algebra,
Linear Factorization Theorem,
complex conjugate
(+) Find the conjugate of a complex number; use conjugates to
find moduli and quotients of complex numbers.
Extended Understanding
- The modulus of a complex
number is always a real number
and in fact it will never be
negative.
I Can Statements
 I can find the conjugate of a complex number.
 I can find the moduli of complex numbers.
 I can find the quotient of complex numbers.
Columbus City Schools
Tier 2 Vocabulary
- perform, find, understand
Prior Knowledge
Future Learning
- Student learned in Integrated
Mathematics II that the square root
of -1 is called i.
- Students will learn to build new
functions from current functions.
- Students performed addition,
subtraction, multiplication, and
division on complex numbers.
Clear Learning Targets Precalculus 2015--‐2016
7
Instructional Strategies
Refer here for some notes on this topic: http://tutorial.math.lamar.edu/Extras/ComplexPrimer/ConjugateModulus.aspx
Remind students of what they learned about the Fundamental Theorem of Algebra: A polynomial function of degree n can have at most n real
zeroes.
Refer to section 2-1 in the book. Go over examples 1-4. Have students use synthetic division to determine roots.
Common Misconceptions and Challenges
Students may need a reminder as to how to do synthetic division. You can use this puzzle to review (page 54). Remind students that when they
set up their synthetic division steps, they must change the sign on the divisor.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Chapter 2-1
Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSN-N-CN#HSN-CN.A.3
Gizmo activity: http://www.explorelearning.com/index.cfm?method=cResource.dspDetail&ResourceID=147
Shmoop:
http://www.shmoop.com/common-core-standards/ccss-hs-n-cn-3.html
Career Connections
Many careers that deal with imaginary number revolve around electronics and engineering. Electrical engineers and electronic engineers use
complex numbers to model AC with dealing with electrical issues. Other fields that use complex numbers include physicists, astronomers,
audio technicians, and audiologists.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
8
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-BF.5
Build new functions from
Essential Understanding
Academic Vocabulary/Language
existing functions
-Students should identify domain,
range, and end behavior of
exponential and logarithmic
functions.
-Students should use the properties
of exponents and logarithms to
solve exponential and logarithmic
functions.
-Students should use function
models to predict and make
decisions and judgments.
- Algebraic function transcendental
function, exponential function,
natural base, continuous
compound interest, APY, APR,
principal, logarithmic function,
base, common logarithm, natural
logarithm, exponential growth,
exponential decay
Extended Understanding
- analyze, predict, judge
Understand the inverse relationship between exponents and
logarithms and use this relationship to solve problems
involving logarithms and exponents.
Tier 2 Vocabulary
- Analyze real world data that can
be modeled by exponential and
logarithmic functions.
I Can Statements
 I can evaluate, analyze, and graph exponential functions.
 I can solve problems involving exponential growth and
decay.
 I can evaluate expressions involving logarithms.
 I can sketch and analyze graphs of logarithmic functions.
 I can apply properties of logarithms.
 I can evaluate logarithms.
 I can use properties of exponential functions to solve
equations.
 I can use properties of logarithmic functions to solve
equations.
Columbus City Schools
Prior Knowledge
Future Learning
- Identify, graph, and describe
different parent functions.
- describe parent functions
symbolically and graphically.
- identify domains and ranges of
functions.
- determine the domain and range
of functions using graphs, tables,
and symbols.
- understand properties of
exponents.
- solve algebraic equations.
Clear Learning Targets Precalculus 2015--‐2016
- use regression to determine the
appropriateness of an exponentail,
logarithmic, logisteic, ubic, quartic,
or quadratic model.
9
Instructional Strategies
Students need to recognize that exponential and logarithmic functions are inverses of one another and use these functions to solve real-world
problems. Nonmathematical examples of functions and their inverses can help students to understand the concept of an inverse and
determining whether a function is invertible.
Provide applied examples of exponential and logarithmic functions, such as the use of a logarithm to determine pH or the strength of an
earthquake on the Richter Scale. Both pH and Richter Scale values are powers of 10 and are, therefore, logarithms. For example, the magnitude
of an earthquake, M, on the Richter Scale can be calculated using the formula M = log10A , where A represents the amplitude of measured
seismic waves.
Common Misconceptions and Challenges
Students may confuse exponential and polynomial functions. While both have an exponent, 𝑦 = 𝑥 ! is a polynomial function while 𝑦 − 2! is an
exponential function.
Students may confuse the exponents in logarithmic and exponential forms.
Students my confuse properties of logarithms with properties of exponents and incorrectly represent the logarithm
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 3
Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/BF/B/5 Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
http://www.ccsso.org/resources/digital_resources/common_core_implementation_video_series.htmhttps://www.ixl.com/standards/ohio/m
ath/high-school
Career Connections
Exponential and logarithmic functions are used for the Richter scale, the pH scale, to model populations, carbon date artifacts, determine time
of death, and compute investments. Some of the career fields in which exponential and logarithmic functions are used include Economists,
Bankers, Financial Advisors, Insurance Risk Assessors, Biologists, Engineers, Computer Programmers, Chemists, Physicists, Geographers,
Sound Engineers, Statisticians, Mathematicians, and Geologists.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
10
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-TF.3
Extend the domain of
Essential Understanding
Academic Vocabulary/Language
trigonometric functions using
the unit circle
- Students should be able to
evaluate and graph inverse trig
functions and find compositions of
trig functions.
-Students should be able to use
the Laws of Sines and Cosines to
solve and find area of oblique
triangles.
- arcsine function, arccosine
function, arctangent function,
oblique triangles, Law of Sines,
ambiguous case, Law of Cosines,
Heron’s Formula
Extended Understanding
- solve, evaluate, compare
Use special triangles to determine geometrically the values of
sine, cosine, tangent for π/3, π/4, and π/6, and use the unit
circle to express the values of sine, cosines, and tangent for x, π
+ x, and 2π – x in terms of their values for x, where x is any real
number
Tier 2 Vocabulary
-Students can compare
translations of inverse functions
-Students can find area of irregular
figures
I Can Statements
 I can evaluate and graph inverse trigonometric
functions.
 I can find compositions of trigonometric functions
 I can solve oblique triangles by using the Law of Sines or
the Law of Cosines.
 I can find areas of oblique triangles
Columbus City Schools
Prior Knowledge
Future Learning
- Find and graph inverses of
relations and functions
- Solve trig equations
- Solve right triangles using trig
functions
Clear Learning Targets Precalculus 2015--‐2016
- Verify trig identities
11
Instructional Strategies
Students can use what they know about 30-60-90 triangles and right isosceles triangles to determine the values for sine, cosine, and tangent for
π/3, π/4, and π/6. In turn, they can determine the relationships between, for example, the sine of π/6, 7π/6, and 11π/6, as all of these use
the same reference angle and knowledge of a 30-60-90 triangle.
http://betterlesson.com/common_core/browse/686/ccss-math-content-hsf-tf-a-3-use-special-triangles-to-determine-geometrically-the- valuesof-sine-cosine-tangent-for-3-4-and-6-an?from=domain_core
Common Misconceptions and Challenges
Students may believe that there is no need for radians if one already knows how to use degrees. Students need to be shown a rationale for how
radians are unique in terms of finding function values in trigonometry since the radius of the unit circle is 1. Students may also believe that all
angles having the same reference values have identical sine, cosine and tangent values. They will need to explore in which quadrants these
values are positive and negative.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Lesson 4-6, 4-7
Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/TF/A/3 Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career Connections
Drafters prepare the drawings used to build everything from spacecraft to bridges. Using rough sketches done by others, they produce detailed
technical drawings with specific information to create a finished product. Drafters use handbooks, tables, calculators, and computers to do
their work. Many specialize in architecture, electronics, or aeronautics.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
12
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-TF.4
Extend the domain of
Essential Understanding
Academic Vocabulary/Language
trigonometric functions using
the unit circle
- Students should find trig values
for any angle.
-Students should find values of trig
functions using the unit circle.
- quadrantal angle, reference angle,
unit circle, circular function,
periodic function, period
Use the unit circle to explain symmetry (odd and even) and
periodicity of trigonometric functions.
Extended Understanding
- Explore relationships between
equivalent values of sine and
cosine on the unit circle.
I Can Statements
 I can find the values of trigonometric functions for any
angle.
 I can construct the unit circle
 I can find values of trigonometric functions using the
unit circle.
Columbus City Schools
Tier 2 Vocabulary
- explore, construct
Prior Knowledge
Future Learning
- find values of trig functions for
acute angels uisng ratios in right
triangles
- graph transformations of sine and
cosine functions
Clear Learning Targets Precalculus 2015--‐2016
- graph tangent and reciprocal trig
functions
13
Instructional Strategies
Provide students with real-world examples of periodic functions. One good example is the average high (or low) Ohio Department of Education,
March 2015 Page 17 Mathematics Model Curriculum temperature in a city in Ohio for each of the 12 months. These values are easily located at
weather.com and can be graphed to show a periodic change that provides a context for exploration of these functions. Allow plenty of time for
students to draw – by hand and with technology – graphs of the three trigonometric functions to examine the curves and gain a graphical
understanding of why, for example, cos (π/2) = 0 and whether the function is even (e.g., cos(-x) = cos(x)) or odd (e.g., sin(-x) = -sin(x)). Similarly
students can generalize how function values repeat one another, as illustrated by the behavior of the curves.
http://betterlesson.com/common_core/browse/687/ccss-math-content-hsf-tf-a-4-use-the-unit-circle-to-explain-symmetry-odd-and-even-andperiodicity-of-trigonometric-functions?from=domain_core_container
Common Misconceptions and Challenges
Students may also believe that all angles having the same reference values have identical sine, cosine and tangent values. They will need to
explore in which quadrants these values are positive and negative.
Students may ignore negative values of trig functions in the unit circle. Students may confuse the coordinates of the ordered pair on the unit
circle.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Lesson 4-3
Illustrative Mathematics https://www.illustrativemathematics.org/contentstandards/HSF/TF/A/4 Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career Connections
Civil engineers perform engineering duties in planning, designing, and overseeing construction and maintenance of building structures and
facilities, such as roads, railroads, airports, bridges, harbors, channels, dams, irrigation projects, pipelines, power plants, water and sewage
systems, and waste disposal units. Includes architectural, structural, traffic, ocean, and geo-technical engineers.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
14
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-TF.7 & 9
Model periodic
Essential Understanding
Academic Vocabulary/Language
phenomena with
trigonometric functions
- Students should be able to graph
transformations of sine and cosine
functions.
- Students should be able to use
sine and cosine functions to model
data.
Students should be able to graph
the tangent and reciprocal trig
functions and damped trig
functions.
- sinusoid, amplitude, frequency,
phase shift, vertical shift, midline,
damped trigonometric function,
damping factor, damped
oscillation, damped wave, damped
harmonic motion
Prove and apply trigonometric identities
Use inverse functions to solve trigonometric equations that
arise in modeling contexts; evaluate the solutions using
technology, and interpret them in terms of context.
Prove the addition and subtraction formulas for sine, cosine,
and tangent and use them to solve problems.
I Can Statements
 I can graph tangent and reciprocal trigonometric
functions
 I can graph transformations of the sine and cosine
function.
 I can use a graphing calculator to graph the sine
functions and its inverse.
 I can use sinusoidal functions to solve problems.
 I can graph and examine the periods and sums and
differences of sinusoids.
 I can graph damped trigonometric functions.
Columbus City Schools
Tier 2 Vocabulary
- model, evaluate, prove
Extended Understanding
- Write functions to represent
graphs and when given
characteristics of functions.
Prior Knowledge
Future Learning
- Students should be aware that
sine is odd and cosine is even.
Students should know the relations
between sine, cosine and tangent
and should know the relation
between trigonometric values of
complementary angles.
- Evaluate and graph inverse trig
functions.
Clear Learning Targets Precalculus 2015--‐2016
15
Instructional Strategies
Students can explore the inverse trigonometric functions, recognizing that the periodic nature of the functions depends on restricting the
domain. These inverse functions can then be used to solve real-world problems involving trigonometry with the assistance of technology.
Students can explore other trigonometric identities, such as the half-angle, double-angle, and addition/subtraction formulas to extend on the
Pythagorean relationship. Formulas should be proven and then used to determine exact values when given an angle measure, to prove
identities, and to solve trigonometric equations. For example, by dividing the formula sin2 (θ) + cos2 (θ) = 1 by cos2 (θ), a new formula is
generated ( tan2 (θ) +1= sec 2 (θ) ).
Common Misconceptions and Challenges
Students may believe that sin-1 A = 1/sin A, thus confusing the ideas of inverse and reciprocal functions. Additionally, students may not
understand that when sin A = 0.4, the value of A represents an angle measure and that the function sin-1 (0.4) can be used to find the angle
measure.
Students may believe that sin(A +B) = sinA + sinB and need specific examples to disprove this assumption.
Students may have difficulty in remembering which of the trig functions pass through the origin.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Lessons 4-4, 4-5
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSF/TF/B
https://www.illustrativemathematics.org/content-standards/HSF/TF/C/9
Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career Connections
Civil engineers perform engineering duties in planning, designing, and overseeing construction and maintenance of building structures and
facilities, such as roads, railroads, airports, bridges, harbors, channels, dams, irrigation projects, pipelines, power plants, water and sewage
systems, and waste disposal units. Includes architectural, structural, traffic, ocean, and geo-technical engineers.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
16
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
F-TF.8
Prove and apply
Essential Understanding
Academic Vocabulary/Language
Trigonometric Functions
- Students should be able to use
trig identities to find trig values,
simplify expressions, and solve
equations.
-Students should verify trig
identities and determine whether
equations are trig identities.
- identity, trig identity, cofunction,
odd-even identities, Pythagorean
identities,
Prove the Pythagorean identity sin2 (θ) + cos2 (θ) = 1 and use it
to calculate trigonometric ratios.
Tier 2 Vocabulary
- verify, prove
Extended Understanding
- Students can solve trig
inequalities
I Can Statements





I can identify and use trig identities to find trig values.
I can use trig identities to simplify trig expressions.
I can verify trig identities.
I can determine whether equations are trig identities.
I can solve trig equations using trig identities.
Columbus City Schools
Prior Knowledge
Future Learning
- Students should be able to find
trig values using the unit circle.
- Students will use trig identities to
transform expresisons into forms
that will be used for integration
and differentiation.
- Students should be able to solve
right triangles.
- Students should be able to find
values of trig functions for any
angle.
Clear Learning Targets Precalculus 2015--‐2016
- Students will use trig substitution
for integration.
17
Instructional Strategies
In the unit circle, the cosine is the x-value, while the sine is the y-value. Since the hypotenuse is always 1, the Pythagorean relationship sin2 (θ) + cos2 (θ) = 1 is
always true. Students can make a connection between the Pythagorean Theorem in geometry and the study of trigonometry by proving this relationship. In
turn, the relationship can be used to find the cosine when the sine is known, and vice-versa. Provide a context in which students can practice and apply skills of
simplifying radicals.
Drawings of the unit circle can be useful in showing why the Pythagorean relationship must be true.
Dynamic geometry software, such as Geometer’s Sketchpad or Geogebra, can be used to demonstrate that, regardless of the angle measure, the Pythagorean
relationship always holds in the unit circle.
http://betterlesson.com/common_core/browse/694/ccss-math-content-hsf-tf-c-8-prove-the-pythagorean-identity-sin2-cos2-1-and-use-it-to-find-sin-cos-or-tan-given- sin-cos-or-tana?from=domain_core
Common Misconceptions and Challenges
Students may believe that there is no connection between the Pythagorean Theorem and the study of trigonometry.
Students may also believe that there is no relationship between the sine and cosine values for a particular angle. The fact that the sum of the
squares of these values always equals 1 provides a unique way to view trigonometry through the lens of geometry.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Lesson 5-1, 5-2, 5-3
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSF/TF/C/8
Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career connections
Physicists study the matter that makes up the universe. They also study forces of nature such as gravity and nuclear interaction. They use their
studies to design medical equipment, electronic devices, and lasers. Astronomers study the moon, sun, planets, galaxies, and stars. Their
knowledge is used in space flight and navigation. Many teach in colleges and universities.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
18
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
REI.8
Reasoning with Equations &
Essential Understanding
Academic Vocabulary/Language
Inequalities
- Gaussian elimination is a method
that is used to solve systems of
equations where row operations
are performed.
- multi-variable linear system, reduce
row-echelon, Gaussian elimination,
augmented matrix, coefficient
matrix, Gauss-Jordan elimination.
Extended Understanding
Tier 2 Vocabulary
- The solution of a system solved by
reduced row-echelon method is an
ordered triple labeled (r, s, t).
- represent
(+) Represent a system of linear equations as a single matrix
equation in a vector variable.
I Can Statements
 I can perform operations with matrices.
 I can use matrices to represent a system of linear
equations.
Columbus City Schools
Prior Knowledge
Future Learning
- Students have solved systems of
equations in two variables before.
They have also solved in three
variables but not by these methods.
- Students will perform various
operations on matrices.
Clear Learning Targets Precalculus 2015--‐2016
19
Instructional Strategies
Remind students how to solve a system of three equations with three unknowns by elimination. You can refer to this spreadsheet to help
instruct students on how elimination methods work.
Show the following Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSA-A-REI#HSA-REI.C.8
Common Misconceptions and Challenges
If students make a mistake early in a problem, chances are the rest of the problem will be incorrect. Teachers may wish to start with problems
that are integers and have students ask if the first variable they solve for is correct before moving on with the rest of the problem.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 6-1
Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-a-rei-8.html




Solve a system of equations using augmented matrices (Algebra 1)
Solve a system of equations using augmented matrices: word problems (Algebra 1)
Solve a system of equations using augmented matrices (Algebra 2)
Solve a system of equations using augmented matrices: word problems (Algebra 2)
Career Connections
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
20
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
N-VM.6
Perform operations on
Essential Understanding
Academic Vocabulary/Language
matrices and use matrices in
applications.
- Data can be organized into
matrices. When this is done, data
can be manipulated easily.
- transformations, translations,
reflections, rotations, dilations,
vertex matrix, translation matrix,
pre-image, image
(+) Use matrices to represent and manipulate data, e.g., to
represent payoffs or incidence relationships in a network.
Extended Understanding
- Students can see where matrices
can be used in real world
applications.
I Can Statements
 I can use matrices to represent data.
 I can use matrices to manipulate data.
 I can use matrices in real life applications.
Columbus City Schools
Tier 2 Vocabulary
- perform, operation, use
Prior Knowledge
Future Learning
- Students just used matrices in
Gaussian elimination to solve a
system of three equations with
three unknows.
- Student will perform scalar
operations on matrices.
Clear Learning Targets Precalculus 2015--‐2016
21
Instructional Strategies
Complete the Focus activity on page 384 of the textbook, having students draw a triangle and labeling the matrix formed by its vertices.
Show the Khan Academy video at: https://www.khanacademy.org/commoncore/grade-HSN-N-VM#HSN-VM.C.6
Common Misconceptions and Challenges
Remind students that when translating, right and up imply positive change where down and left imply negative change.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 6-1, 6-6
Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-6.html
Sophia: https://www.sophia.org/ccss-math-standard-9-12nvm6-pathway
http://www.ct4me.net/Common-Core/hsnumber/hsn-vector-matrix-quantities.htm
Career Connections
Actuaries design insurance plans that will help their company make a profit. They study statistics and social trends to decide how much money
an insurance company should charge for an insurance policy. They predict the amount of money an insurance company will have to pay to its
customers for claims. Some actuaries are self-employed and work as consultants.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
22
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
N-VM.7&8
Perform operations on
Essential Understanding
Academic Vocabulary/Language
matrices and use matrices
in applications
- Students will add, subtract, and
multiply matrices by a scalar and
should be able to interpret the
resulting data.
-Students will multiply matrices if
possible.
-Students will determine whether
the properties of real numbers
hold under operations on
matrices.
- matrix, element, dimensions,
spreadsheet, row, cell, column,
scalar, scalar multiplication
Multiply matrices by scalars to produce new matrices, e.g., as
when all of the payoffs in a game are doubled.
Add, subtract, and multiply matrices of appropriate dimensions.
Tier 2 Vocabulary
- evaluate, model
Extended Understanding
- Students can use
transformations on matrices to
model motion.
I Can Statements
 I can perform algebraic operations with matrices.
 I can multiply matrices.
 I can use the properties of matrix multiplication.
Columbus City Schools
Prior Knowledge
Future Learning
-Students should be able to
organize data into matrices.
Students will use matrices to solve
systems of equations.
Clear Learning Targets Precalculus 2015--‐2016
23
Instructional Strategies
Point out the existence of more than one type of matrix multiplication. Scalar multiplication refers to the multiplication of a matrix by a constant (a scalar)
to produce another matrix of the same size. This is similar to multiplying a number by a scale factor to increase or decrease its value in proportion to its
original value.
The scalar multiplication is performed by multiplying each element of a matrix by the same constant. To help students understand the operations with
matrices, review properties of real number operations (commutative, associative, identity and inverse for addition and multiplication) prior to introducing
operations with matrices. Matrix addition (subtraction) and multiplication are similar to real number addition (subtraction) and multiplication in many
instances, but there are some important differences. Begin with defining equality of matrices and emphasize the importance of the same size. Two matrices
are equal if they have the same size and their corresponding elements are equal. The sum of two matrices of the same size is a matrix with elements that are
the sums of the corresponding elements of the two given matrices. Addition is not defined for matrices of different sizes. Because two matrices are added by
adding their corresponding elements, it follows from the properties of real numbers that matrices of the same size are commutative and associative relative
to addition. When matrices are used in a context, they should not be added or subtracted unless their labels match. Provide students with contextual
applications to demonstrate the proper use of matrix addition (subtraction).
Common Misconceptions and Challenges
A side effect of treating matrices as an isolated area of study with many symbol manipulation rules similar to operations with real numbers is that students
may overestimate the similarity between operations with matrices and operations with real numbers. For example, some students may believe that: matrix
multiplication is commutative; a product of matrices with the same sizes is a product of their corresponding elements; the identity matrix has all elements
that are ones; and the inverse matrices have all entrees that are reciprocals of the elements of the original matrix. Teachers can remedy these
misconceptions by offering students a wide range of applications that show connections between matrices, transformation, vectors, and systems of linear
equations.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Lesson 6-4, 6-5
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSN/VM/C/7
https://www.illustrativemathematics.org/content-standards/HSN/VM/C/8
Achieve the Core Modules, Resources: http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio: https://www.ixl.com/standards/ohio/math/high-school
Career connections
Actuaries design insurance plans that will help their company make a profit. They study statistics and social trends to decide how much money
an insurance company should charge for an insurance policy. They predict the amount of money an insurance company will have to pay to its
customers for claims. Some actuaries are self-employed and work as consultants.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
24
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
N-VM.9-10
Perform operations on
matrices and use matrices in
applications.
VM.9: (+) Understand that, unlike multiplication of numbers,
matrix multiplication for square matrices is not a commutative
operation, but still satisfies the associative and distributive
properties.
VM.10: (+) Understand that the zero and identity matrices play a
role in matrix addition and multiplication similar to the role of 0
and 1 in the real numbers. The determinant of a square matrix is
nonzero if and only if the matrix has a multiplicative inverse.
I Can Statements
 I can understand that the multiplication of square matrices
is not commutative.
 I can understand that multiplication of square matrices
works with the associative property.
 I can understand that the multiplication of square matrices
works with the distributive property.
Columbus City Schools
Essential Understanding
Academic Vocabulary/Language
- Multiplication of square matrices
is not commutative.
- Multiplication of square matrices
does hold true for associative and
distributive properties.
- identity matrix, inverse matrix,
inverse, invertible, singular matrix,
deteminant
Extended Understanding
- perform, operation, use
Tier 2 Vocabulary
- The determinant of a square
matrix is nonzero only if the matrix
has a multiplicative inverse.
Prior Knowledge
Future Learning
- Students should already know how
to find the determinant of matrices
and how to multiply matrices.
- Students will study conic sections
next.
Clear Learning Targets Precalculus 2015--‐2016
25
Instructional Strategies
Show students how multiplying two numbers works either way; that is, 4 x 8 is the same as 8 x 4. Explain that this is always true for numbers.
Does it work for square matrices? Show the students an example of a 4x4 matrix being multiplied by another 4x4 matrix. Then, reverse the two
matrices and multiply again. Are the results the same? Are they as expected? Why do they think the products are different?
Multiply a matrix and its inverse together to obtain a zero matrix. Show students how the determinant is 0.
Common Misconceptions and Challenges
Students may think that given Matrix A and Matrix B, that A x B and B x A yield the same result. Use an example as described above to prove this
is not the case.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 6-2, 6-2 (Extend), 6-3




Matrix operation rules (Algebra 1)
Matrix operation rules (Algebra 2)
Properties of matrices (Algebra 2)
Matrix operation rules (Precalculus)
Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSN-NVM#HSN-VM.C.9











Determinant of a matrix (Algebra 2)
Is a matrix invertible? (Algebra 2)
Inverse of a matrix (Algebra 2)
Identify inverse matrices (Algebra 2)
Solve matrix equations using inverses (Algebra 2)
Determinant of a matrix (Precalculus)
Is a matrix invertible? (Precalculus)
Inverse of a 2 x 2 matrix (Precalculus)
Inverse of a 3 x 3 matrix (Precalculus)
Identify inverse matrices (Precalculus)
Solve matrix equations using inverses (Precalculus)
Career Connections
Actuaries design insurance plans that will help their company make a profit. They study statistics and social trends to decide how much money
an insurance company should charge for an insurance policy. They predict the amount of money an insurance company will have to pay to its
customers for claims. Some actuaries are self-employed and work as consultants.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
26
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
GPE.3
Translate between the
Essential Understanding
Academic Vocabulary/Language
geometric description and
the equation for a conic
section.
- Students will be able to derive
the equation of an ellipse.
- Students will be able to derive
the equation of a hyperbola.
- hyperbola, conic section, ellipse,
foci, constant, transverse axis,
conjugate axis
Tier 2 Vocabulary
(+) Derive the equations of ellipses and hyperbolas given the
foci, using the fact that the sum or difference of distances from
the foci is constant.
I Can Statements
 I can derive the equation of an ellipse given the foci.
 I can derive the equation of a hyperbola give the foci.
 I can understand that the sum or difference of distances
from the foci is constant.
Columbus City Schools
Extended Understanding
- translate, derive, using facts
- Students will realize that the
sum or difference of distances
from the foci is constant.
Prior Knowledge
Future Learning
- Students learned in GPE.2 how to
derive the equation of a parabola
given a focus and diretrix.
- Students will be able to explain
volume formulas.
Clear Learning Targets Precalculus 2015--‐2016
27
Instructional Strategies
You can remind students of how they derived the equation of a parabola in earlier lessons. Go over section 7-1 and 7-2 from the book, following
the examples. You can use the idea of roller coasters to spark interest: Why might elliptical-shaped loops be safer than circle-shaped loops on
roller coasters?
Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSG-G-GPE#HSG-GPE.A.3
Common Misconceptions and Challenges
Make sure students keep the terms ellipse and hyperbola straight. They often confuse the two. Review the form of the equation of each.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 7-1, 7-2
Shmoop: http://www.shmoop.com/common-corestandards/ccss-hs-g-gpe-3.html







Find properties of ellipses (Precalculus)
Find the eccentricity of an ellipse (Precalculus)
Write equations of ellipses in standard form (Precalculus)
Find properties of hyperbolas (Precalculus)
Find the eccentricity of a hyperbola (Precalculus)
Write equations of hyperbolas in standard form (Precalculus)
Convert equations of conic sections from general to standard form (Precalculus)
Career Connections
There are many careers that involve upper-level geometry. Many jobs can be found in the medicine field, transportation and construction
industries, arts and architecture fields, and engineering. For more information, read the article found on the link below:
http://work.chron.com/careers-require-geometry-10361.html
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
28
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
Explain volume formulas
GMD.2
and use them to solve
problems.
Essential Understanding
Academic Vocabulary/Language
- Students can use explain how
Cavalieri's principle can be used
to find the volume of a sphere.
- volume, Cavalieri’s principle, solid
figure
Tier 2 Vocabulary
(+) Give an informal argument using Cavalieri’s principle for the
formulas for the volume of a sphere and other solid figures.
Extended Understanding
- explain, use
- Student can apply Cavalieri’s
principle to find the volume of a
sphere.
I Can Statements
 I can I can give an informal argument using Cavalieri’s
principle for the formula for the volume of a sphere.
 I can I can give an informal argument using Cavalieri’s
principle for the formula for the volume of other solid
figures.
Columbus City Schools
Prior Knowledge
Future Learning
- Students should already know the
formula for finding the volume of a
sphere.
- Students will explore vectors.
Clear Learning Targets Precalculus 2015--‐2016
29
Instructional Strategies
The textbook does not specifically discuss Cavalieri’s principle. More information is coming out all the time, and this topic will be updated.
Here is one explanation: http://www.shmoop.com/common-core-standards/ccss-hs-g-gmd-2.html
https://en.wikipedia.org/wiki/Cavalieri's_principle
Video: https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=14&cad=rja&uact=8&ved=0CGoQtwIwDWoVChMI5ZnBoZDFxwIVCTMCh3jbALs&url=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DCYp824FeJ9s&ei=TNfcVeXDE4nm-AHj2YngDg&usg=AFQjCNGCY65n9Mvf1VMzovGakPZKhjqu0Q&sig2=3a9n1TYLy5Mp7nYI3o5gNA
Common Misconceptions and Challenges
None at this time.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 7-3
https://www.google.com/url?sa=t&rct=j&q=&esrc=s&source=web&cd=9&ved=0CFUQFjAIahUKEwjlmcGhkMXHAhUJMz4KHeNsAuw&url=http%3A%2F%2Fwww.mat h.was
hington.edu%2Fnwmi%2Fmaterials%2FCavalieri.pptx.pdf&ei=TNfcVeXDE4nmAHj2YngDg&usg=AFQjCNFszUSk67KEmQV1gzuUJLsymAuY_g&sig2=zrwV_VTBUYpSIWMMF6_zrw
Career Connections
The following list briefly describes work associated with some upper-level mathematics-related professions : actuary, mathematics teacher,
operations research analyst, statistician, physician, research scientist, computer scientist, inventory strategist, air traffic control, analyst,
attorney, economist, environmental mathematician, robotics engineer, geophysical mathematician, design ecologist, photogrammetrist, civil
engineer, geomatics engineer.
https://www.math.ucdavis.edu/~kouba/MathJobs.html
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
30
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
VM.1-2
Represent and model with
Essential Understanding
Academic Vocabulary/Language
vector quantities.
- Vectors have magnitude and
direction.
- Vector quantities can be
represented by directed line
segments.
- vector, magnitude, direction,
directed line segment, initial point,
terminal point.
Extended Understanding
- represent, model
VM.1: (+) Recognize vector quantities as having both magnitude
and direction. Represent vector quantities by directed line
segments, and use appropriate symbols for vectors and their
magnitudes (e.g., v, |v|, ||v||, v).
VM.2: (+) Find the components of a vector by subtracting the
coordinates of an initial point from the coordinates of a
terminal point.
I Can Statements
 I can understand that vector quantities have both
magnitude and direction.
 I can represent vector quantities by directed line
segments.
 I can use appropriate symbols for vectors and their
magnitude.
 I can find the components of a vector by subtraction.
Columbus City Schools
Tier 2 Vocabulary
- Students will find the
components of a vector by using
subtraction of coordinates.
Prior Knowledge
Future Learning
- Students are already
knowledgable about the
Coordinate plane. They should be
comfortable with positive
directions going up and right, while
negative directions go down and
left.
- Students will perform multiple
operation on vectors, including
several different ways to add them.
- Students know the distance
formula and Pythagorean Theorem.
Clear Learning Targets Precalculus 2015--‐2016
31
Instructional Strategies
Explore the links in the support section. There are several for Geometry as well as for Precalculus.
You may wish to review the Pythagorean Theorem and Distance Formulas.
Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-1.html
Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-1.html
OpenEd: https://www.opened.com/homework/n-vm-1-recognize-vector-quantities-as-having-both-magnitude-and/3689205 Khan
Academy: https://www.khanacademy.org/commoncore/grade-HSN-N-VM#HSN-VM.A.1
Common Misconceptions and Challenges
When students create a triangle on a graph and try to determine a missing side, students often mis-label the ‘long side’ (hypotenuse).
Sometimes students put an entire ordered pair in each ( ) in the magnitude formula. Instead of having (x2 – x1), for instance, they will do (x1y1). Remind students that you are finding the length of each side, and each side is represented by the same variable.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 8-1, 8-4, 8-6




Find the magnitude of a vector (Geometry)
Find the magnitude of a vector (Precalculus)
Find the direction angle of a vector (Precalculus)
Find the magnitude of a three-dimensional vector (Precalculus)
Career Connections
Management occupations
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Industrial production managers
Medical and health services managers
Property, real estate, and community association managers
Purchasing managers, buyers, and purchasing agents
Columbus City Schools





Find the component form of a vector (Geometry)
Find the component form of a vector given its magnitude and direction angle
(Geometry)
Find the component form of a vector (Precalculus)
Find the component form of a vector from its magnitude and direction angle
(Precalculus)
Find the component form of a three-dimensional vector (Precalculus)
Computer and mathematical occupations
Actuaries
Computer software engineers
Mathematicians
Statisticians
Engineers
Aerospace engineers
Chemical engineers
Clear Learning Targets Precalculus 2015--‐2016
32
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
Perform operations on vectors.
VM.4-5
VM.4a: Add vectors end-to-end, componentwise, and by the parallelogram rule. Understand
that the magnitude of a sum of two vectors is
typically not the sum of the magnitudes.
VM.4b: Given two vectors in magnitude and direction form, determine the
magnitude and direction of their sum.
VM.4c: Understand vector subtraction v - w as v + (-w), where -w is the additive
inverse of w, with the same magnitude as w and pointing in the opposite
direction. Represent vector subtraction graphically by connecting the tips in
the appropriate order, and perform vector subtraction component-wise.
VM.5a: Represent scalar multiplication graphically by scaling vectors and
possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(vx, vy) = (cvx, cvy).
Essential Understanding
- Students will add vectors in a
variety of ways including end-toend and the parallelogram rule.
- Students will perform scalar
multiplication.
Extended Understanding
Academic Vocabulary/Language
- component form, unit vector,
linear combination, cross product,
dot product, magnitude
Tier 2 Vocabulary
- perform, understand, determine,
represent
- Students can represent scalar
multiplication graphically.
VM.5b: Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v.
Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is
either along v (for c > 0) or against v (for c < 0).
I Can Statements









I can add vectors end-to-end.
I can add vectors component-wise.
I can add vectors by the parallelogram rule.
I can determine the magnitude and direction of the sum of
two vectors.
I can understand vector subtraction.
I can multiply a vector by a scalar.
I can represent scalar multiplication graphically.
I can perform scalar multiplication component-wise.
I can compute the magnitude of a scalar multiple.
Columbus City Schools
Prior Knowledge
Future Learning
- Student previously defined
vectors and learned that vectors
have magnitude and direction.
They performed subtraction on
vectors.
- Students will multiply a vector by
a matrix.
Clear Learning Targets Precalculus 2015--‐2016
33
Instructional Strategies
Use the links below to help you teach this topic.



Multiply a vector by a scalar (Precalculus)
Scalar multiples of three-dimensional vectors (Precalculus)
Find the magnitude or direction of a vector scalar multiple (Precalculus)
Common Misconceptions and Challenges
Student confuse the unit vector i with the imaginary number i. The unit vector is denoted by a bold, nonitalic letter i. The imaginary number is
denoted by a bold italic letter i.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 8-1, 8-2, 8-3, 8-4, 8-5




Add and subtract vectors (Precalculus)
Graph a resultant vector using the triangle method (Precalculus)
Graph a resultant vector using the parallelogram method
(Precalculus)
Add and subtract three-dimensional vectors (Precalculus)
Career Connections
Management occupations
Engineering and natural sciences managers
Farmers, ranchers, and agricultural managers
Industrial production managers
Medical and health services managers
Property, real estate, and community association managers
Purchasing managers, buyers, and purchasing agents
Columbus City Schools
 Find the magnitude and direction of a vector sum (Precalculus)


Add and subtract vectors (Precalculus)
Add and subtract three-dimensional vectors (Precalculus)
Computer and mathematical occupations
Actuaries
Computer software engineers
Mathematicians
Statisticians
Engineers
Aerospace engineers
Chemical engineers
Clear Learning Targets Precalculus 2015--‐2016
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Ohio’s Learning Standards – Clear Learning Targets
Precalculus
VM.11-12
Perform operations on
Essential Understanding
Academic Vocabulary/Language
matrices and use matrices in
applications.
- Vectors can be multiplied by
other matrices to create a new
matrix.
- cross product, torque,
parallelepiped, triple scalar
product
VM.11: (+) Multiply a vector (regarded as a matrix with one
column) by a matrix of suitable dimensions to produce another
vector. Work with matrices as transformations of vectors.
VM.12: (+) Work with 2 × 2 matrices as a transformations of the
plane, and interpret the absolute value of the determinant in
terms of area.
I Can Statements
 I can multiply a vector by a matrix to produce another
vector.
 I can work with 2x2 matrices as a transformation of the
plane.
 I can interpret the absolute value of the determinant in
terms of area.
Columbus City Schools
Extended Understanding
- Students will interpret the
absolute value of the determinant
in terms of area.
Tier 2 Vocabulary
- perform, use, produce, interpret
Prior Knowledge
Future Learning
- Students have already add vectors
in several ways and performed
scalar multiplication of vectors.
- Students will continue their study
of complex numbers.
Clear Learning Targets Precalculus 2015--‐2016
35
Instructional Strategies
Watch the Khan Academy video: https://www.khanacademy.org/commoncore/grade-HSN-N-VM#HSN-VM.C.11
OpenEd: https://www.opened.com/homework/n-vm-11-multiply-a-vector-regarded-as-a-matrix-with-one-column/3689775
Shmoop: http://www.shmoop.com/common-core-standards/ccss-hs-n-vm-11.html
Common Misconceptions and Challenges
Review the characteristics of triangles so students do not confuse them: A right triangle has a 90 degree angle and two sides that are
perpendicular to each other. An isosceles triangle has two sides that have the same length. And equilateral triangle has three sides that have
the same length. A scalene triangle has no sides that are the same length.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill: Chapter 8-3, 8-4 (Extend), 8-5
Coming soon: https://www.superteacherworksheets.com/common-core/n.vm.11.html
Career Connections
Physical scientists
Atmospheric scientists
Chemists and materials scientists
Environmental scientists and hydrologists
Physicists and astronomers
Metal workers and plastic workers
Computer control programmers and operators
Machinists
Columbus City Schools
Drafters and engineering technicians
Drafters
Engineering technicians
Life scientists
Biological scientists
Medical scientists
Electrical and electronic equipment mechanics,
installers, and repairers
Electronic home entertainment equipment
installers and repairers
Clear Learning Targets Precalculus 2015--‐2016
Engineers
Aerospace engineers
Chemical engineers
Civil engineers
Electrical engineers
Environmental engineers
Industrial engineers
Nuclear engineers
36
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
N-CN.4,5,&6
Summarize, represent,
Essential Understanding
Academic Vocabulary/Language
and interpret data on a
single count or
measurement variable
- Students should be able to graph
polar coordinates, converting
between polar and rectangular
forms.
-Students should be able to convert
between polar and rectangular
forms of complex numbers and
perform operations in polar
form.
- polar coordinate system, pole,
polar axis, polar coordinates, polar
equation, polar graph, limacon,
cardioid, rose, lemniscuses, spiral
of Archimedes, complex plane, real
axis, imaginary axis, Argand plane,
absolute value of a complex
number, polar form, trig form,
modulus, argument, pth roots of
unity
4. Represent complex numbers on the complex plane in rectangular
and polar form (including real and imaginary numbers), and explain
why the rectangular and polar forms of a given complex number
represent the same number.
5. Represent addition, subtraction, multiplication, and conjugation of
complex numbers geometrically on the complex plane; use properties
of this representation for computation. For example, (1 - √3i)3 = 8
because (1 - √3i) has modulus 2 and argument 120˚.
6. Calculate the distance between numbers in the complex plane as
the modulus of the difference, and the midpoint of a segment as the
average of the numbers at its endpoints.
I Can Statements







I can graph points with polar coordinates
I can graph polar equations
I can convert between polar and rectangular coordinates
I can convert between polar and rectangular equations
I can identify polar equations of conics
I can write and graph the polar equation of a conic
I can convert complex numbers from rectangular to
polar from and vice versa
 I can find products, quotients, powers and roots of
complex numbers in polar form.
Columbus City Schools
Extended Understanding
- The distance between numbers
in the complex plane is the
modulus of the difference.
Tier 2 Vocabulary
- calculate, summarize, represent,
interpret (data), use (properties).
Prior Knowledge
Future Learning
-Students should know how to find
distance between points on the
rectangular coordinate plane.
-Students will find the area of a
region ounded by a urve whose
equationis given in polar form.
-Students should be able to analyze and
draw the graph of conic equations in
the rectangular coordinate plane.
-Students will solve a system of
polar equations.
-Students should be able to define,
simplify, add, multiply, and solve
equations with imaginary numbers.
-Students should be abel to add and
subtract complex numbers and find
their conjugates.
Clear Learning Targets Precalculus 2015--‐2016
37
Instructional Strategies
A complex plane is obtained by associating the ordered pairs of real numbers with points in a rectangular coordinate system. Students should see a
correspondence between complex numbers written in rectangular form (a +bi) and a unique ordered pair of real numbers (a, b), where a represents a real
number and b represents an imaginary number. They also need to view a pair of conjugate complex numbers (a - bi) and (a + bi) as a reflection of the point
(a, b) to the point (a, -b) over the x- axis. Similar to a vector, a complex number is characterized by length and direction.
A complex number can be written in the alternative polar form, such that an arbitrary point P representing a number is associated with polar coordinates (r,
θ), where r is a modulus and θ is an argument.
Point out that, similar to a complex number in a coordinate plane associated with the ordered pair of real numbers, each geometric vector in the standard
position on a coordinate plane is associated with the ordered pair of real numbers that are coordinates of its terminal point. Students need to realize that
vectors and complex numbers as systems have some common properties and operations. For example, addition, subtraction or multiplication of complex
numbers by a real number is performed by the same rules as addition, subtraction or multiplication of vectors by a real number. Multiplication of two
complex numbers results in the cross product of two vectors.
Common Misconceptions and Challenges
Students may believe that complex numbers, as an area of study of mathematics, is completely isolated from other areas of study, such as vectors and
matrices, and only has a connection to quadratic equations with negative discriminants. The variety of approaches to connect complex numbers, vectors
and matrices can help students develop their understanding of important concepts of all three overlapping areas.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Chapter 9
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSN/CN/B
Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career connections
Computer control programmers and operators use special computer-controlled machines to cut and shape products such as car parts, machine
parts, and compressors. They use lathes, spindles, and milling machines following blueprints from engineers. While they work they must
constantly monitor readouts from the computer to make sure parts are being made properly. Because of the machinery they use, this can be a
dangerous job.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
38
Ohio’s Learning Standards – Clear Learning Targets
Precalculus
S-MD.1,2,3,4,5
Summarize,
represent, and
interpret data on
a single count or
measurement variable
1. Represent data with plots on the real number line (dot plots, histograms,
and box plots).
2. Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation)
of two or more different data sets.
3. Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).
4. Use the mean and standard deviation of a data set to fit it to a normal
distribution and to estimate population percentages. Recognize that there
are data sets for which such a procedure is not appropriate. Use calculators,
spreadsheets, and tables to estimate areas under the normal curve.
5. (+) Weigh the possible outcomes of a decision by assigning probabilities to
payoff values and finding expected values.
I Can Statements
 I can represent the sum of a series using sigma notation.
 I can find the nth term of arithmetic series and
sequences.
 I can find the nth term of geometric series and
sequences.
 I can use Pascal’s Triangle and/or the Binomial Theorem
to write binomial expansion.
 I can use a power series to represent a rational function.
 I can use approximate values of transcendental
functions.
Columbus City Schools
Essential Understanding
Academic Vocabulary/Language
- Students will be able to use sigma
notation to represent and calculate
sums of series, find nth terms of
arithmetic/geometric sequences and
series.
-Students will be able to use Pascal’s
Triangle or the Binomial Theorem to
write binomial expansions.
-Students will be able to use a power
series to represent a rational
function.
- principle of mathematical induction,
anchor step, inductive hypothesis,
inductive step, extended principle of
mathematical induction, binomial
coefficient, Pascal’s triangle, Binomial
Theorem, power series, exponential
series, Euler’s Formula
Tier 2 Vocabulary
- compare, interpret, observe
Extended Understanding
- Students should be able to assign
probabilities to payoff values.
Prior Knowledge
Future Learning
-Students should be able to use funcions
to generate ordered pairs and use
graphs to analyze the end behavior of
functions and write usning limit
notation.
-Students will find limits of
convergent functions
-Students should be able to perfom
operaitons ith imaginary and complex
number.
-Students will represent functions as
series for use in differrntial equations.
-Studetn wil proie interavl estimates
from sample proportions to estimate
population proportions.
-students should be able to graph and
analyze rational funtions.
Clear Learning Targets Precalculus 2015--‐2016
39
Instructional Strategies
Have students practice their understanding of the different types of graphs for categorical and numerical variables by constructing statistical posters. Note
that a bar graph for categorical data may have frequency on the vertical (student’s pizza preferences) or measurement on the vertical (radish root growth
over time - days). Measures of center and spread for data sets without outliers are the mean and standard deviation, whereas median and interquartile
ranges are better measures for data sets with outliers. Introduce the formula of standard deviation by reviewing the previously learned MAD (mean
absolute deviation). The MAD is very intuitive and gives a solid foundation for developing the more complicated standard deviation measure. Informally
observing the extent to which two boxplots or two dot plots overlap begins the discussion of drawing inferential conclusions. Don’t shortcut this
observation in comparing two data sets. As histograms for various data sets are drawn, common shapes appear. To characterize the shapes, curves are
sketched through the midpoints of the tops of the histogram’s rectangles. Of particular importance is a symmetric unimodal curve that has specific areas
within one, two, and three standard deviations of its mean. It is called the Normal distribution and students need to be able to find areas (probabilities) for
various events using tables or a graphing calculator.
Common Misconceptions and Challenges
That a bar graph and a histogram are the same. A bar graph is appropriate when the horizontal axis has categories and the vertical axis is labeled by either
frequency (e.g., book titles on the horizontal and number of students who like the respective books on the vertical) or measurement of some numerical
variable (e.g., days of the week on the horizontal and median length of root growth of radish seeds on the vertical). A histogram has units of measurement of
a numerical variable on the horizontal (e.g., ages with intervals of equal length). That the lengths of the intervals of a boxplot (min,Q1), (Q1,Q2), (Q2,Q3),
(Q3,max) are related to the number of subjects in each interval. Students should understand that each interval theoretically contains one-fourth of the total
number of subjects. Sketching an accompanying histogram and constructing a live boxplot may help in alleviating this misconception. That all bell-shaped curves
are normal distributions. For a bell-shaped curve to be Normal, there needs to be 68% of the distribution within one standard deviation of the mean, 95% within
two, and 99.7% within three standard deviations.
Common Core Support and Curriculum Resources
Integrated Math 4, McGraw Hill, Chapter 10 and 11
Illustrative Mathematics
https://www.illustrativemathematics.org/content-standards/HSS/ID/A
Achieve the Core Modules, Resources
http://achievethecore.org/dashboard/300/search/6/2/0/1/2/3/4/5/6/7/8/9/10/11/12
The Common Core in Ohio
https://www.ixl.com/standards/ohio/math/high-school
Career connections
Using national statistics, underwriters decide whether a person applying for insurance is a good risk. They also help the company decide how much to
charge. If they set the rates too low, the company will lose money. Too high, and the company will lose business to competitors. They answer questions
such as: How much should an insurance company charge for insurance?
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
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References
Many of these standards can be found on Shmoop.com: http://www.shmoop.com
Others can be found at OpenED: http://www.OpenEd.com
Some of the topics cite Illustrative Mathematics. However, others are still being added to their website.
Much of the material, especially in later sections of this course, is still being written. As they are found, they will be added to this Clear Learning
Target.
Click here for Santa Fe’s explanation of some of these standards.
Columbus City Schools
Clear Learning Targets Precalculus 2015--‐2016
41