RCSS Adaptive Version-Draft 1- August 2016 (Includes Success Criteria/ I Can Statements adapted from Johnston County) North Carolina Math 3 Collaborative Pacing Guide This pacing guide is the collaborative work of math teachers, coaches, and curriculum leaders from 38 NC public school districts. The teams worked through two face-to-face meetings and digitally to compile the information presented. NC Math 1, 2, and 3 standards were used to draft possible units of study for these courses. This is a first draft living document. Teams plan to meet throughout the year to continually tweak, update and refine these guides. Updates will be posted as available to this google document. Please reference the NC Math 1, 2, or 3 standards for any questions or discrepancies. This document should be used only after reading the NC Math 1, 2, and 3 standards and instructional guides provided by NC DPI. If you have suggestions or comments that you would like the collaborative writing team to consider for revisions, please email [email protected]or [email protected]. Units for NC Math 3 Number of Days (Block) Number of Days (Traditional) Unit 1: Functions and Their Inverses 10 20 Unit 2: Exponential and Logarithmic Functions 9 18 Unit 3: Polynomial Functions 9 18 Unit 4: Modeling with Geometry 7 14 Unit 5: Rational Functions 10 20 Unit 6: Reasoning with Geometry 15 30 Unit 7: Trigonometric Functions 10 20 Unit 8: Statistics 5 10 Total (allowing for flex days) 75 150
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RCSS Adaptive Version-Draft 1- August 2016 (Includes Success Criteria/ I Can Statements adapted from Johnston County)
North Carolina Math 3
Collaborative Pacing Guide
This pacing guide is the collaborative work of math teachers, coaches, and curriculum leaders from 38 NC public school districts. The teams worked through two
face-to-face meetings and digitally to compile the information presented. NC Math 1, 2, and 3 standards were used to draft possible units of study for these
courses. This is a first draft living document. Teams plan to meet throughout the year to continually tweak, update and refine these guides. Updates will be posted
as available to this google document.
Please reference the NC Math 1, 2, or 3 standards for any questions or discrepancies. This document should be used only after reading the NC Math 1, 2, and 3
standards and instructional guides provided by NC DPI.
If you have suggestions or comments that you would like the collaborative writing team to consider for revisions, please email [email protected] or
Understand and recognize piecewise defined relationships. Understand and solve absolute value equations and inequalities. Understand and solve systems of equations that include piecewise defined relationships (including absolute value) and
can use a graphing tool when necessary. Identify key parts in expressions and equations.
Learning Intentions: These are big ideas, understandings, important math that needs to be developed. They are not necessarily measurable
statements. Ideally a unit will have a handful of learning intentions.
Success Criteria: These are directly associated with a learning intention and articulate to students measurable, tangible, observable
demonstrations of the learning intention. Typically one learning intention has around 3 to 5 success criteria.
Unit 1: Functions and Their Inverses
Estimated Days: 10 Semester or 20 Year Long
Suggested Order: 1 of 8
Suggested Time: 10 days semester block (90-minute classes)
Rationale: The first unit builds upon students’ previous work with modeling functions in Math 1 and Math 2. This unit helps students
transition from modeling in the real world to more abstract mathematical concepts like polynomial and rational functions. It develops
the notion of the inverse function of quadratic, exponential, and linear functions and introduces piecewise-defined and absolute value
functions through multiple representations, i.e. graphing, equations, tables, verbal descriptions, etc. Since students in Math 1 and
Math 2 have already worked with linear, quadratic, and exponential functions, this allows teachers a chance to begin with content that
is familiar to students. It also assists teachers in identifying misconceptions, obstacles, and gaps in prior learning.
Students understand surface area and volume of geometric figures can be modeled by polynomial functions.
NC.M3.N-CN.9 Recognize parts of a polynomial, and apply the Fundamental Theorem of Algebra to determine the types and number
of solutions. NC.M3.A-SSE.1 NC.M3.A-APR.2 NC.M3.A-APR.3 NC.M3.A-CED.1 NC.M3.A-CED.2 NC.M3.F-BF.1a
Understand and apply the Remainder Theorem, the Factor Theorem, and the Division Algorithm. Create polynomial equations in one or two variables and use them to solve problems algebraically and graphically.
B1. I can divide polynomials by using long division, or synthetic
division.
B2. I can understand and apply the remainder theorem.
B3. I can understand and apply the factor theorem.
B4. I can create and solve one and two variable polynomial equations
algebraically and graphically..
NC.M3.F-IF.9 Compare key features of two
functions using different
representations by comparing
properties of two different
functions, each with a different
representation (symbolically,
graphically, numerically in
tables, or by verbal
descriptions).
NC.M3.F-IF.7 Analyze piecewise, absolute
value, polynomials, exponential,
rational, and trigonometric
functions (sine and cosine)
using different representations
to show key features of the
graph, by hand in simple cases
and using technology for more
complicated cases, including:
domain and range; intercepts;
intervals where the function is
increasing, decreasing, positive,
or negative; rate of change;
relative maximums and
minimums; symmetries; end
behavior; period; and
discontinuities.
NC.M3.F-LE.3 Compare the end behavior of
functions using their rates of
change over intervals of the
same length to show that a
quantity increasing exponentially
eventually exceeds a quantity
increasing as a polynomial
function.
NC.M3.F-BF.3 Extend an understanding of the
effects on the graphical and
tabular representations of a
function when replacing �(�)
with �∙�(�), �(�) + �, �(� + �) to
include �(�∙�) for specific values
of ɑ (both positive and
negative).
C. Analyze a polynomial function
and compare two or more
functions by using their key
features. Given solutions or a
graph write the equation of
polynomial function. Graph
transformations. Compare the
relative rates of growth of
exponential and polynomial
functions.
C1. I can identify the key features of a polynomial function.
C2. I can compare key features of two given functions in a variety of
representations.
C3. I can create a polynomial function given the zeros or a graph.
C4. I can determine whether an exponential or polynomial function
increases more rapidly.
C5. I can use rules of transformations on polynomial functions.
NC.M3.F-BF.1a Build polynomial and exponential functions with real solution(s) given a graph, a description of a relationship, or ordered pairs (include reading these from a table).
NC. M3. F-BF.1b.
Build a new function, in terms of a context, by combining standard function types using arithmetic operations.
Vocabulary: Polynomial, leading coefficient, degree, term, factor, Fundamental Theorem of Algebra, domain, range, x-intercept, y-intercept, increasing/decreasing intervals, relative vs. absolute maximum(s) & minimum(s), end behavior, zero, root, synthetic division, long division, Remainder Theorem, Factor Theorem, complex number, imaginary number, multiplicity
Task: Popcorn task
Unit 4: Modeling with Geometry
Estimated Days: 7 Semester or 14 Year Long
Rationale: This unit transitions from polynomial work to geometric concepts that require the use of algebra. It is intentionally placed after the
polynomials unit because the polynomials unit is suggested to begin with geometric modeling that results in a polynomial. Teaching this unit right
after the conclusion of polynomials, allows you to circle back to the geometric modeling concept and study it to its full depth. The placement of
this unit also gives students a break from the heavy algebra work of polynomials prior to beginning rational functions.
NC.M3.G-GMD.3 Use the volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems. NC.M3.G-GMD.4 Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects NC.M3.G-MG.1 Apply geometric concepts in modeling situations:
● Use geometric and algebraic
Implement surface area and
volume of geometric figures and
model using polynomial functions.
Furthermore, relating cross
sections with two-dimensional and
three-dimensional figures
A1: I can use volume formulas to solve problems.
A2: I can identify the two-dimensional shape formed by the cross-
section of a three dimensional figure.
A3. I can identify the three-dimensional shape formed by the rotation of
a two-dimensional figure.
A4: I can use geometric concepts to model and solve real-world
situations.
concepts to solve problems in modeling situations:
● Use geometric shapes, their measures, and their properties, to model real-life objects.
● Use geometric formulas and algebraic functions to model relationships.
● Apply concepts of density based on area and volume.
● Apply geometric concepts to solve design and optimization problems.
NC.M3.G-CO.14 Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems.
NC.M3.A-APR.7 Understand the similarities between arithmetic with rational expressions and arithmetic with rational numbers.
a. Add and subtract two rational expressions, (�) and (�), where the denominators of both (�) and (�) are linear expressions.
b. Multiply and divide two rational expressions.
NC.M3.A-SSE.1a Identify and interpret parts of a piecewise, absolute value, polynomial, exponential and rational expressions including terms, factors, coefficients, and exponents. NC.M3.A-CED.2 Create and graph equations in two variables to represent absolute value, polynomial, exponential and rational relationships between quantities.
NC.M3.A-CED.1 Create equations and inequalities in one variable that represent absolute value, polynomial, exponential, and rational relationships and use them to solve problems algebraically and graphically.
A. Recognize rational expressions
as the division of two polynomials
and use properties of simple
fractions to analyze, perform
arithmetic operations, create and
solve equations that model real
world phenomena.
A1. I can rewrite a rational expression in 2 ways by factoring and
simplifying or write as a quotient and remainder.
A2. I can explain how operations on rational expressions are the same as
simple fractions.
A3. I can multiply and divide rational expressions.
A4. I can find LCD in order to add and subtract rational expression when
the denominators are linear.
A5. I can recognize the difference between adding rational expressions
and solving rational expressions.
A6. I can solve a one variable rational equation algebraically or using a
graph.
A7. I can give examples showing how extraneous solutions may arise
when solving rational equation.
A8. I can create and solve a rational equation to solve an application
problem.
A9. I can interpret the terms, factors, and coefficients of rational
expressions.
NC.M3.A-REI.2 Solve and interpret one variable rational equations arising from a context, and explain how extraneous solutions may be produced.
NC.M3.F-IF.9 Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).
NC.M3.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities. NC.M3.F-IF.7 Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.
B. Understand and interpret the
key features, uses and limitations
of multiple representations of a
rational function.
B1. I can determine the domain and discontinuities of a rational
expression by finding vertical asymptotes and holes of the function.
B2. I can determine the range of a rational expression by finding the
horizontal asymptotes and holes of a function.
B3. I can find the end behaviors of a rational function by looking at a
graph or table.
B4. I can find x and y intercepts from a graph, while recognizing that they
may not exist.
Vocabulary: Rational function, common denominator, complex fraction, proportion, extraneous solution, domain restriction, domain, range, x-intercept, y-intercept, increasing/decreasing intervals, end behavior, asymptote, point of discontinuity (hole)
Unit 6: Reasoning with Geometry
Estimated Days: 15 Semester or 30 Year Long
Rationale: This unit transitions into geometric concepts with an emphasis on reasoning, justification, and formalizing proof. Students will extend
upon their work with proof in Math 2 (NC.M2.G.CO.9 and NC.M2.G.CO.10) focusing on both paragraph and flow proofs. Students are familiar
with the properties of parallelograms from middle school and have categorized parallelograms and informally verified parallelogram properties
through coordinate geometry in Math 1. Students will prove more theorems about triangles including the centers of triangles. This concept can be
used as a transition into reasoning with circles. The Reasoning with Geometry Unit purposefully concludes with circles. In students’ work with
circles, they will develop their understanding of radian measure through proportions in circles. This sets up a connection of circular motion to
NC.M3.G-GPE.1 Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.
A. Write the equation of a circle
given radius, and center or a
graph of a circle in standard and
general form. Determine radius
and center of a circle in general
form by completing the square.
A1. I can write the equation of a circle given center and radius in standard
and general form.
A2. I can determine the radius and center of a circle given an equation in
standard or general form , or the graph itself.
NC.M3.G-CO.14 Apply properties, definitions, and theorems of two-dimensional figures to prove geometric theorems and solve problems. NC.M3.G-C.2 Understand and apply theorems about circles.
● Understand and apply theorems about relationships with angles and circles, including central, inscribed and circumscribed angles.
● Understand and apply theorems about relationships with line segments and circles including, radii, diameters, secants, tangents and chords
NC.M3.G-C.5 Using similarity, demonstrate that the length of an arc, , for a given central angle is proportional to the radius, , of the circle. Define radian measure of the central angle as the ratio of the length of the arc to the radius of the
circle,
. Find arc lengths
and areas of sectors of circles.
B. Quantify the relationships
between angles and arcs of a
circle, whether those angles are
inscribed, circumscribed, or
central angles. Use theorems on
right angles inscribed in circles,
tangent lines, secants and chords
in circles to solve problems and
prove theorems. Use similarity to
verify arc-length formula. Use arc-
length and area formulas to solve
problems generally and in context.
B1. I can use theorems on angles and arcs of circles to solve problems
and prove theorems.
B2. I can use theorems on inscribed right angles, tangent lines, chords
and secant lines in circles to solve problems and prove theorems.
B3. I can determine the arc-length and area of a sector.
B4. I can use the similarity of circles to measure an angle by the ratio of
the radius and the arc length.
NC.M3.G-MG.1 Apply geometric concepts in modeling situations:
● Use geometric and algebraic concepts to solve problems in modeling situations:
● Use geometric shapes, their measures, and their properties, to model real-life objects.
● Use geometric formulas and algebraic functions to model relationships
NC.M3.F-TF.1 Understand radian measure of an angle as:
● The ratio of the length of an arc on a circle subtended by the angle to its radius.
● A dimensionless measure of length defined by the quotient of arc length and radius that is a real number.
● The domain for trigonometric functions.
NC.M3.F-IF.1 Extend the concept of a function by recognizing that trigonometric ratios are functions of angle measure. NC.M3.F-TF.2 Build an understanding of trigonometric functions by using tables, graphs and technology to represent the cosine and sine functions.
a. Interpret the sine function as the relationship between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its y coordinate.
b. Interpret the cosine function as the relationship
A. Understand that (right)
triangular trigonometric
functions are related to
circular trigonometric
functions in the coordinate
plane. Develop the sine
graph using the unit circle.
A1. I can define a radian measure of an angle as the length of the arc on the unit
circle subtended by the angle.
A2. I can explain how a ratio represents a value of a trig function for an angle.
A3. I can work with angles in standard position to find coterminal and reference
angles.
A4. I can sketch a sine graph from the values of the unit circle.
between the radian measure of an angle formed by the horizontal axis and a terminal ray on the unit circle and its x coordinate.
NC.M3.F-IF.9 Compare key features of two functions using different representations by comparing properties of two different functions, each with a different representation (symbolically, graphically, numerically in tables, or by verbal descriptions).
NC.M3.F-IF.4 Interpret key features of graphs, tables, and verbal descriptions in context to describe functions that arise in applications relating two quantities to include periodicity and discontinuities. NC.M3.F-IF.7 Analyze piecewise, absolute value, polynomials, exponential, rational, and trigonometric functions (sine and cosine) using different representations to show key features of the graph, by hand in simple cases and using technology for more complicated cases, including: domain and range; intercepts; intervals where the function is increasing, decreasing, positive, or negative; rate of change; relative maximums and minimums; symmetries; end behavior; period; and discontinuities.
B. Understand and interpret
the key features, uses and
limitations of multiple
representations of
trigonometric functions that
model real world periodic
behavior.
B1. I can use key features to construct the graph of sine function.
B2. I can state the amplitude, period, and midline of the sine function.
B3. I can use technology, graphs, and tables to compare sine graphs.
B4. I can describe the effect of a transformation on the graph of the sine function.
B6. I can use technology to interpret the key features of the sine graph in a real
world situation.
NC.M3.F-TF.5 Use technology to investigate the parameters, �, �, and ℎ of a sine function, (�) = �∙ sin (�∙�) + ℎ , to represent periodic phenomena and interpret key features in terms of a context. NC.M3.F-BF.3 Extend an understanding of
the effects on the graphical
and tabular representations
of a function when
replacing �(�) with �∙�(�),
�(�) + �, �(� + �) to include
�(�∙�) for specific values of
ɑ (both positive and
negative).
Vocabulary: Initial side of an angle, terminal side of an angle, ray, coterminal angles, sine, cosine, tangent, radian (angle measure), unit circle, domain, range, period, midline, amplitude, frequency, cycle, phase shift.
Unit 8: Statistics
Estimated Days: 5 Semester or 10 Year Long
Rationale: This unit, Statistics, is more flexible in the pacing, than the other suggested units. Statistics can be taught as a stand-alone unit, since
there is less integration and connection between standards. However, it is suggested that you do not break up the coherency of units that have
intentionally been suggested to be taught in a certain order. (i.e - do not teach this unit between Reasoning with Geometry and Trigonometric
Functions)
Standards Learning Intentions
NC.M3.S-IC.1 NC.M3.S-IC.3
Understand statistics as a process of making inferences about a population (parameter) based on results from a
random sample (statistic). Acknowledge the role of randomization in using sample surveys, experiments, and observational studies to collect
data and understand the limitations of generalizing results to populations (related to randomization).
Unit 8: Statistics
Estimated Days: 5 Semester or 10 Year Long
Suggested Order: 8 of 8
Suggested Time: 5 days on a semester schedule (90-minute classes)
Rationale: This unit, Statistics, is more flexible in the pacing, than the other suggested units. Statistics can be taught as a stand-alone
unit, since there is less integration and connection between standards. However, it is suggested that you do not break up the
coherency of units that have intentionally been suggested to be taught in a certain order. (i.e - do not teach this unit between Circles
NC.M3.S-IC.1 Understand the process of making inferences about a population based on a random sample from that population.
Supporting Standard
NC.M3.S-IC.3 Recognize the
A. Understand statistics as a
process of making inferences
about a population (parameter)
based on results from a random
sample (statistic).
Acknowledge the role of
randomization in using sample
A1. I can distinguish between a sample (statistic) and a population
(parameter).
A2. I can describe how to select a random sample from a given
population.
A3. I can explain the purposes and the differences of sample surveys,
observational studies, and experiments, including how randomization
applies to each.
A4. I can distinguish between sample surveys, observational studies, and
experiments.
NC.M3.S-IC.4 NC.M3.S-IC.5
Understand simulation is useful for using data to make decisions. Understand that samples can differ by chance.
NC.M3.S-IC.6 Understand not all data that is reported is valid. Reports should be evaluated based on source, design of the study, and
data displays.
purposes of and differences between sample surveys, experiments, and observational studies and understand how randomization should be used in each.
surveys, experiments, and
observational studies to collect
data and understand the
limitations of generalizing results
to populations (related to
randomization).
A5. I can determine how results of a statistical study can be generalized to
make conclusions about a population based on the sample.
Priority Standard
NC.M3.S-IC.4 Use simulation to understand how samples can be used to estimate a population mean or proportion and how to determine a margin of error for the estimate. Supporting Standard
NC.M3.S-IC.5 Use simulation to determine whether observed differences between samples from two distinct populations indicate that the two populations are actually different in terms of a parameter of interest.
B. Understand simulation is useful
for using data to make decisions.
Know how to carry out a
simulation with data for the
purposes of: estimating
population means or proportions,
determining the margin of error for
those estimates, and determining
statistical significance.
Understand that samples can
differ by chance.
B1. I can use data from a sample survey to estimate a population mean or
proportion with a margin of error.
B2. I can determine and justify if results from an experiment are
statistically significant.
● I can identify the parameter of interest in an experiment.
● I can select and calculate sample statistics.
● I can calculate the difference between the sample statistics.
● I can set up and complete a simulation re-randomizing the groups.
● I can compare the actual difference to the simulated differences to
determine statistical significance.
B3. I can state a conclusion about the effectiveness or accuracy of a claim
based on a sample.
Priority Standard
NC.MC.S-IC.6 Evaluate articles and websites that report data by identifying the source of the data, the design of the study, and the way the data are graphically displayed.
C. Understand not all data that is
reported is valid. Reports should
be evaluated based on source,
design of the study, and data
displays.
C1. I can evaluate and make sense of a statistical article or website.
Vocabulary: Sample, population, margin of error, standard deviation, simple random sample, systematic sample, stratified random sample, cluster