CCGPS Coordinate Algebra Unpacked Standards Page 1 of 32 Bibb County School System May 2012
CCGPS Coordinate Algebra
This document is an instructional support tool. It is adapted from
Appendix A of the Common Core State Standards, the PARCC Model
Content Frameworks and from documents created by the Georgia
Department of Education, the Arizona Department of Education, and the
Ohio Department of Education.
Highlighted standards are transition standards for Georgia's
implementation of CCGPS in 2012-2013. The highlighted standards are
included in the curriculum for two grade levels during the initial year of
CCGPS implementation to ensure that students do not have gaps in their
knowledge base. In 2013-2014 and subsequent years, the highlighted
standards will not be taught at this grade level because students will
already have addressed these standards the previous year.
High school standards specify the mathematics that all students should
study in order to be college and career ready. Additional mathematics
that students should learn in fourth credit courses or advanced courses
such as calculus, advanced statistics, or discrete mathematics is indicated
by (+). All standards without a (+) symbol should be in the common
mathematics curriculum for all college and career ready students.
Standards with a (+) symbol may also appear in courses intended for all
students.
High school standards are organized in conceptual categories: Number
and Quantity, Algebra, Functions, Modeling, Geometry, and Statistics
and Probability. Modeling is best interpreted not as a collection of
isolated topics but in relation to other standards. Making mathematical
models is a Standard for Mathematical Practice, and specific modeling
standards appear throughout the high school standards indicated by a
star symbol (). A standard may apply to more than one course and
include content not addressed in this course. Parts of standards which
are struck through are not to be included in instruction for this course.
This document uses the units of the CCGPS Coordinate Algebra
curriculum map provided by the Georgia Department of Education as its
organizational tool. The curriculum map contains quick links to the
standards of each unit along with explanations, examples, and
corresponding mathematical practices.
What is in the document? The “unpacking” of the standards done in
this document is an effort to answer a simple question “What does this
standard mean that a student must know and be able to do?” and to
ensure the description is helpful, specific and comprehensive for
educators.
CCGPS Coordinate Algebra Unpacked Standards Page 2 of 32 Bibb County School System May 2012
Coordinate Algebra The fundamental purpose of Coordinate Algebra is to formalize and extend the mathematics that students learned in the middle grades. The critical areas, organized into units, deepen and extend understanding of linear relationships, in part by contrasting them with exponential phenomena, and in part by applying linear models to data that exhibit a linear trend. Coordinate Algebra uses algebra to deepen and extend understanding of geometric knowledge from prior grades. The final unit in the course ties together the algebraic and geometric ideas studied. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.
Coordinate Algebra Overview
Number and Quantity
Quantities (N-Q)
Algebra
Seeing Structure in Expressions (A-SSE)
Creating Equations (A-CED)
Reasoning with Equations and Inequalities (A-REI)
Functions
Interpreting Functions (F-IF)
Building Functions (F-BF)
Linear, Quadratic, and Exponential Models (F-LE)
Geometry
Congruence (G-CO)
Expressing Geometric Properties with Equations (G-GPE)
Modeling Statistics and Probability
Interpreting Categorical and Quantitative Data (S-ID)
Mathematical Practices (MP) 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of
others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.
Critical Areas of Focus (Units)
1. Relationships Between Quantities 2. Reasoning with Equations and Inequalities 3. Linear and Exponential Functions 4. Describing Data 5. Transformations in the Coordinate Plane 6. Connecting Algebra and Geometry Through Coordinates
CCGPS Coordinate Algebra Unpacked Standards Page 3 of 32 Bibb County School System May 2012
Key Advances from Grade K – 8 Discussion of Mathematical Practices in Relation to Course
Content
Students build on previous work with solving linear equations and systems of linear equations in two ways: (a) They extend to more formal solution methods, including attending to the structure of linear expressions, and (b) they solve linear inequalities.
Students formalize their understanding of the definition of a function, particularly their understanding of linear functions, emphasizing the structure of linear expressions. Students also begin to work on exponential functions, comparing them to linear functions.
Work with congruence and similarity motions that started in grades 6-8 progresses. Students also consider sufficient conditions for congruence of triangles.
Work with the bivariate data and scatter plots in grades 6-8 is extended to working with lines of best fit.
Modeling with mathematics (MP.4) should be a particular focus as students see the purpose and meaning for working with linear and exponential equations and functions.
Using appropriate tools strategically (MP.5) is also important as students explore those models in a variety of ways, including with technology. For example, students might be given a set of data points and experiment with graphing a line that fits the data.
As Coordinate Algebra continues to develop a foundation for more formal reasoning, students should engage in the practice of constructing viable arguments and critiquing the reasoning of others (MP.3).
Fluency Recommendations A/G High school students should become fluent in solving characteristic problems involving the analytic geometry of lines, such as finding the
equation of a line given a point and a slope. This fluency can support students in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two variables).
G High school students should become fluent in using geometric transformation to represent the relationships among geometric objects. This
fluency provides a powerful tool for visualizing relationships, as well as a foundation for exploring ideas both within geometry (e.g., symmetry) and outside of geometry (e.g., transformations of graphs).
S Students should be able to create a visual representation of a data set that is useful in understanding possible relationships among variables.
CCGPS Coordinate Algebra Unpacked Standards Page 4 of 32 Bibb County School System May 2012
Standards for Mathematical Practices
The Common Core Standards for Mathematical Practice are expected to be integrated into every mathematics lesson for all students Grades K – 12. Below are a few examples of how these Practices may be integrated into tasks that students complete.
Mathematical Practices Explanations and Examples 1. Make sense of
problems and persevere in solving them.
High school students start to examine problems by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. By high school, students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. They check their answers to problems using different methods and continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
2. Reason abstractly and quantitatively.
High school students seek to make sense of quantities and their relationships in problem situations. They abstract a given situation and represent it symbolically, manipulate the representing symbols, and pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Students use quantitative reasoning to create coherent representations of the problem at hand; consider the units involved; attend to the meaning of quantities, not just how to compute them; and know and flexibly use different properties of operations and objects.
3. Construct viable arguments and critique the reasoning of others.
High school students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. High school students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. High school students learn to determine domains to which an argument applies, listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
4. Model with mathematics.
High school students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. High school students making assumptions and approximations to simplify a complicated ituation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
5. Use appropriate tools strategically.
High school students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. High school students should be sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical
CCGPS Coordinate Algebra Unpacked Standards Page 5 of 32 Bibb County School System May 2012
models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. They are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
6. Attend to precision. High school students try to communicate precisely to others by using clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. By the time they reach high school they have learned to examine claims and make explicit use of definitions.
7. Look for and make use of structure.
By high school, students look closely to discern a pattern or structure. In the expression x2 + 9x + 14, older students can see the
14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. High school students use these patterns to create equivalent expressions, factor and solve equations, and compose functions, and transform figures.
8. Look for and express regularity in repeated reasoning.
High school students notice if calculations are repeated, and look both for general methods and for shortcuts. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, derive formulas or make generalizations, high school students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.
CCGPS Coordinate Algebra Unpacked Standards Page 6 of 32 Bibb County School System May 2012
Connecting the Standards for Mathematical Practice to the Standards for Mathematical Content The Standards for Mathematical Practice describe ways in which developing student practitioners of the discipline of mathematics increasingly ought to engage with the subject matter as they grow in mathematical maturity and expertise throughout the elementary, middle and high school years. Designers of curricula, assessments, and professional development should all attend to the need to connect the mathematical practices to mathematical content in mathematics instruction. The Standards for Mathematical Content are a balanced combination of procedure and understanding. Expectations that begin with the word “understand” are often especially good opportunities to connect the practices to the content. Students who lack understanding of a topic may rely on procedures too heavily. Without a flexible base from which to work, they may be less likely to consider analogous problems, represent problems coherently, justify conclusions, apply the mathematics to practical situations, use technology mindfully to work with the mathematics, explain the mathematics accurately to other students, step back for an overview, or deviate from a known procedure to find a shortcut. In short, a lack of understanding effectively prevents a student from engaging in the mathematical practices. In this respect, those content standards which set an expectation of understanding are potential “points of intersection” between the Standards for Mathematical Content and the Standards for Mathematical Practice. These points of intersection are intended to be weighted toward central and generative concepts in the school mathematics curriculum that most merit the time, resources, innovative energies, and focus necessary to qualitatively improve the curriculum, instruction, assessment, professional development, and student achievement in mathematics.
CCGPS Coordinate Algebra Unpacked Standards Page 7 of 32 Bibb County School System May 2012
Common Core Georgia Performance Standards
High School Mathematics
CCGPS Coordinate Algebra
Common Core Georgia Performance Standards: Curriculum Map
FIRST SEMESTER SECOND SEMESTER
Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6
Relationships Between
Quantities
Reasoning with
Equations and
Inequalities
Linear and
Exponential Functions Describing Data
Transformations in
the Coordinate Plane
Connecting Algebra
and Geometry
Through Coordinates
5 weeks 5 weeks 8 weeks 6 weeks 5 weeks 5 weeks MCC9-12.N.Q.1
MCC9-12.N.Q.2
MCC9-12.N.Q.3
MCC9-12.A.SSE.1a,b
MCC9-12.A.CED.1
MCC9-12.A.CED.2
MCC9-12.A.CED.3
MCC9-12.A.CED.4
MCC9-12.A.REI.1
MCC9-12.A.REI.3
MCC9-12.A.REI.5
MCC9-12.A.REI.6
MCC9-12.A.REI.12
MCC9-12.A.REI.10
MCC9-12.A.REI.11
MCC9-12.F.IF.1
MCC9-12.F.IF.2
MCC9-12.F.IF.3
MCC9-12.F.IF.4
MCC9-12.F.IF.5
MCC9-12.F.IF.6
MCC9-12.F.IF.7a,e
MCC9-12.F.IF.9
MCC9-12.F.BF.1a,b
MCC9-12.F.BF.2
MCC9-12.F.BF.3
MCC9-12.F.LE.1a,b,c
MCC9-12.F.LE.2
MCC9-12.F.LE.3
MCC9-12.F.LE.5
MCC9-12.S.ID.1
MCC9-12.S.ID.2
MCC9-12.S.ID.3
MCC9-12.S.ID.5
MCC9-12.S.ID.6a,b,c
MCC9-12.S.ID.7
MCC9-12.S.ID.8
MCC9-12.S.ID.9
MCC6.SP.5 .
MCC9-12.G.CO.1
MCC9-12.G.CO.2
MCC9-12.G.CO.3
MCC9-12.G.CO.4
MCC9-12.G.CO.5
MCC8.G.8 .
MCC9-12.G.GPE.4
MCC9-12.G.GPE.5
MCC9-12.G.GPE.6
MCC9-12.G.GPE.7
Considerations Considerations Considerations Considerations Considerations Considerations
All units will include the Standards for Mathematical Practice and indicate skills to maintain.
1. Make sense of problems and persevere in solving them.
2. Reason abstractly and quantitatively.
3. Construct viable arguments and critique the reasoning of others.
4. Model with mathematics.
5. Use appropriate tools strategically.
6. Attend to precision.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.
CCGPS Coordinate Algebra Unpacked Standards Page 8 of 32 Bibb County School System May 2012
Unit 1: Relationships Between Quantities
By the end of eighth grade students have had a variety of experiences working with expressions and creating equations. In this first unit, students continue this work by using
quantities to model and analyze situations, to interpret expressions, and by creating equations to describe situations.
Reinforce understanding of the properties of integer exponents. The initial experience with exponential expressions, equations, and functions involves integer exponents and builds
on this understanding.
BIG IDEA: The purpose of this unit is to stress reasoning and making sense of relationships and quantities by modeling in context.
Standards Mathematical
Practices
What students
should DO
What students
should KNOW
Examples/Explanations
Reason quantitatively and use units to solve problems.
MCC9-12.N.Q.1 Use
units as a way to understand
problems and to guide the
solution of multi-step
problems; choose and
interpret units consistently
in formulas; choose and
interpret the scale and the
origin in graphs and data
displays.
DOK 2
4. Model with
mathematics.
5. Use appropriate
tools strategically.
6. Attend to precision.
Use
Choose
Interpret
Units
Solution
Formulas
Scale
Origin
Graphs
Data displays
Working with quantities and the relationships between them provides
grounding for work with expressions, equations, and functions.
Include word problems where quantities are given in different units, which
must be converted to make sense of the problem. For example, a problem
might have an object moving 12 feet per second and another at 5 miles per
hour. To compare speeds, students convert 12 feet per second to miles per
hour:
hr 24
day 1
min 60
hr 1
sec 60
min 1 sec 24000 , which is more than 8 miles per hour.
Graphical representations and data displays include, but are not limited to:
line graphs, circle graphs, histograms, multi-line graphs, scatterplots, and
multi-bar graphs.
MCC9-12.N.Q.2 Define
appropriate quantities for the
purpose of descriptive
modeling.
DOK 2
4. Model with
mathematics.
6. Attend to precision.
Define
Appropriate
quantities
Descriptive
modeling
Examples:
What type of measurements would one use to determine their income
and expenses for one month?
How could one express the number of accidents in Georgia?
MCC9-12.N.Q.3 Choose
a level of accuracy
appropriate to limitations on
measurement when
reporting quantities.
This standard does not stand
alone – it must be integrated
into other standards in the
unit. DOK 2
5. Use appropriate
tools strategically.
6. Attend to precision.
Choose
Level of accuracy
Limitations
Measurement
Quantities
The margin of error and tolerance limit varies according to the measure,
tool used, and context.
Example:
Determining price of gas by estimating to the nearest cent is
appropriate because you will not pay in fractions of a cent but the cost
of gas is
..$
gallon
4793
CCGPS Coordinate Algebra Unpacked Standards Page 9 of 32 Bibb County School System May 2012
Interpret the structure of expressions.
MCC9-12.A.SSE.1
Interpret expressions that
represent a quantity in terms
of its context.
a. Interpret parts of an
expression, such as
terms, factors, and
coefficients.
b. Interpret complicated
expressions by viewing
one or more of their
parts as a single entity.
DOK 1
1. Make sense of
problems and
persevere in solving
them.
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
7. Look for and make
use of structure.
Interpret
Expressions
Quantity
Context
Terms
Factors
Coefficients
Single entity
Emphasis on linear expressions and exponential expressions with integer
exponents.
Students should understand the vocabulary for the parts that make up the
whole expression and be able to identify those parts and interpret their
meaning in terms of a context.
Create equations that describe numbers or relationships.
MCC9-12.A.CED.1
Create equations and
inequalities in one variable
and use them to solve
problems. Include equations
arising from linear and
quadratic functions, and
simple rational and
exponential functions.
DOK 2
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
Create
Use
Solve
Equations
Inequalities
One variable
Linear functions
Exponential
functions
Equations can represent real world and mathematical problems. Include
equations and inequalities that arise when comparing the values of two
different functions, such as one describing linear growth and one describing
exponential growth.
Examples:
Given that the following trapezoid has area 54 cm2, set up an equation
to find the length of the base, and solve the equation.
MCC9-12.A.CED.2
Create equations in two or
more variables to represent
relationships between
quantities; graph equations
on coordinate axes with
labels and scales.
DOK 2
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
Create
Represent
Graph
Equations
Two or more
variables
Relationships
Quantities
Coordinate axes
Labels
Scales
(Limit to linear and exponential equations, and, in the case of exponential
equations, limit to situations requiring evaluation of exponential functions
at integer inputs.)
CCGPS Coordinate Algebra Unpacked Standards Page 10 of 32 Bibb County School System May 2012
MCC9-12.A.CED.3
Represent constraints by
equations or inequalities,
and by systems of equations
and/or inequalities, and
interpret solutions as viable
or non-viable options in a
modeling context.
DOK 3
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
Represent
Interpret
Constraints
Equations
Inequalities
Systems of
equations
Systems of
inequalities
Solutions
Viable options
Non-viable options
Modeling context
(Limit to linear equations and inequalities.)
Example:
A club is selling hats and jackets as a fundraiser. Their budget is $1500
and they want to order at least 250 items. They must buy at least as
many hats as they buy jackets. Each hat costs $5 and each jacket costs
$8.
o Write a system of inequalities to represent the situation.
o Graph the inequalities.
o If the club buys 150 hats and 100 jackets, will the conditions be
satisfied?
o What is the maximum number of jackets they can buy and still
meet the conditions?
MCC9-12.A.CED.4
Rearrange formulas to
highlight a quantity of
interest, using the same
reasoning as in solving
equations.
DOK 1
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
7. Look for and make
use of structure.
Rearrange
Formulas
Quantity of interest
(Limit to formulas with a linear focus.)
Examples:
The Pythagorean Theorem expresses the relation between the legs a
and b of a right triangle and its hypotenuse c with the equation
a2 + b2 = c2.
o Why might the theorem need to be solved for c?
o Solve the equation for c and write a problem situation where
this form of the equation might be useful.
Solve 34
3V r for radius r.
Motion can be described by the formula below, where t = time elapsed,
u = initial velocity, a = acceleration, and s = distance traveled.
s = ut+½at2
o Why might the equation need to be rewritten in terms of a?
o Rewrite the equation in terms of a.
CCGPS Coordinate Algebra Unpacked Standards Page 11 of 32 Bibb County School System May 2012
Teaching Considerations for Unit 1
The purpose of this unit, which consists entirely of Modeling () standards, is to stress reasoning and making sense of relationships and quantities by modeling in context.
MCC9-12.N.Q.3 does not stand alone; it must be integrated into other standards for the unit.
o Continuous or discrete (i.e., do you connect the dots for this domain?): For academic years 2012-13, 2013-14, and 2014-15, realize that students were taught this
in 6th grade. Beginning 2015-16, this will be a new concept for 9th grade students because it is not specifically addressed in CCGPS in middle school.
o Domain or range (should you connect your data points for range?): In 2012-13, students will come from 8th grade having been taught this concept. Beginning
with 2013-14, this will be a new concept for all students.
o Estimation or precision: Have students choose appropriate units and determine precision levels based on context.
o Use of appropriate units (including unit conversions, where appropriate): Early elementary GPS begins work with this, but all students need to focus on this
regardless of previous exposure.
MCC9-12.A.CED.1 – For this standard, you may use exponential models for contexts. Do not include problems that require students to use logarithms to solve them (no
variable exponents).
MCC9-12.A.CED.3 – This standard is setting the stage for Unit 2. Emphasis is not on finding the solution, but checking the reasonableness of a solution within the given
context. After 2012-13, systems of inequalities will be a new concept, and teachers will have to teach linear inequality graphing. It is suggested that teachers use colored
pencils, crosshatching, highlighters for shading systems of inequalities – “mustache graphs” where students just shade a small area do not show an understanding of the
inclusion of the entire half-plane. Teachers should be attentive to ensure that students understand the concept of “half-plane.”
MCC9-12.A.CED.1-2 should be taught before NQ and SSE standards. Students are creating and interpreting at the same time. Emphasize both equations and
inequalities.
MCC9-12.A.CED.3-4 should be taught at the end of the unit.
o Teachers need to include background on problem solving, and be tolerant of and encourage student use of multiple representations for problem solving
o Make sure all examples of interpreting complicated expressions are given in context, such as parts of specific equations, so that students are able to identify what
each part of the equation represents.
o Students are creating equations and expressions at the same time. The goal here is for students to be able to write equations fluently by first identifying the
unknowns, then writing a true equation or expression that represents that true statement. This standard is much more than just identifying “coefficient,”
“exponent,” “variable,” or “constant.” Student should be able to take statements like
Jane ordered two burritos and five tacos, and paid $8.75 (2B + 5T = 8.75);
Cassie is five years younger than Hal (C = H – 5 or H = C+5);
Mary, who had exactly $100, needed to buy a present for her two brothers and her three sisters. Write a mathematical sentence to represent this
situation so that she does not run out of money (2B + 3S ≤ 100); or
You have one penny and it doubles every day. Write a mathematical sentence that will show how much money, M, you would have after d days.
Return to Curriculum Map
CCGPS Coordinate Algebra Unpacked Standards Page 12 of 32 Bibb County School System May 2012
Unit 2: Reasoning with Equations and Inequalities
By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear
equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation and to justify the process used
in solving a system of equations. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and using them to solve
problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations.
Students explore systems of equations and inequalities, and they find and interpret their solutions. All of this work is grounded on understanding quantities and on relationships
between them.
BIG IDEA: Students should solve equations in different ways, especially in context. Emphasis should not be solely upon computation and
steps to solve; students need to understand why the “steps” to solving equations work, and what properties of equality each step represents
through proof. Students should graph two-variable systems of inequality, with focus upon the meaning of shading.
Standards Mathematical
Practices
What students
should DO
What students
should KNOW
Examples/Explanations
Understanding solving equations as a process of reasoning and explain the reasoning.
MCC9-12.A.REI.1 Explain
each step in solving a simple
equation as following from
the equality of numbers
asserted at the previous step,
starting from the assumption
that the original equation has
a solution. Construct a
viable argument to justify a
solution method.
DOK 3
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
7. Look for and make
use of structure.
Explain
Construct
Justify
Step
Simple equation
Equality of numbers
Solution
Viable argument
Solution method
Students should focus on and master linear equations and be able to extend
and apply their reasoning to other types of equations in future courses.
Students will solve exponential equations with logarithms in future courses.
Properties of operations can be used to change expressions on either side of
the equation to equivalent expressions. In addition, adding the same term to
both sides of an equation or multiplying both sides by a non-zero constant
produces an equation with the same solutions. Other operations, such as
squaring both sides, may produce equations that have extraneous solutions.
Example:
Explain why the equation x/2 + 7/3 = 5 has the same solutions as the
equation 3x + 14 = 30. Does this mean that x/2 + 7/3 is equal to 3x +
14?
Solve equations and inequalities in one variable.
MCC9-12.A.REI.3 Solve
linear equations and
inequalities in one variable,
including equations with
coefficients represented by
letters.
DOK 1
2. Reason abstractly
and quantitatively.
7. Look for and make
use of structure.
8. Look for and express
regularity in
repeated reasoning.
Solve
Linear equations
Linear inequalities
One variable
Coefficients
Extend earlier work with solving linear equations to solving linear
inequalities in one variable and to solving literal equations that are linear
in the variable being solved for. Include simple exponential equations that
rely only on application of the laws of exponents, such as 5x = 125 or 2x =
1/16.
Examples:
11183
7 y
3x > 9
ax + 7 = 12
4
9
7
3
xx
Solve for x: 2/3x + 9 < 18
CCGPS Coordinate Algebra Unpacked Standards Page 13 of 32 Bibb County School System May 2012
Solve systems of equations.
MCC9-12.A.REI.5 Prove
that, given a system of two
equations in two variables,
replacing one equation by
the sum of that equation and
a multiple of the other
produces a system with the
same solutions.
DOK 3
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
Prove
System of two
equations in two
variables
Sum
Multiple
Same solutions
Limit to linear systems.
Example:
Given that the sum of two numbers is 10 and their difference is 4,
what are the numbers? Explain how your answer can be deduced
from the fact that they two numbers, x and y, satisfy the equations
x + y = 10 and x – y = 4.
MCC9-12.A.REI.6 Solve
systems of linear equations
exactly and approximately
(e.g., with graphs), focusing
on pairs of linear equations
in two variables.
DOK 2
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
6. Attend to precision.
7. Look for and make
use of structure.
8. Look for and express
regularity in
repeated reasoning.
Solve
Systems of linear
equations
Exactly
Approximately
The system solution methods can include but are not limited to graphical,
elimination/linear combination, substitution, and modeling. Systems can be
written algebraically or can be represented in context. Students may use
graphing calculators, programs, or applets to model and find approximate
solutions for systems of equations.
Examples:
José had 4 times as many trading cards as Phillipe. After José gave
away 50 cards to his little brother and Phillipe gave 5 cards to his
friend for this birthday, they each had an equal amount of cards. Write
a system to describe the situation and solve the system.
Solve the system of equations: x+ y = 11 and 3x – y = 5.
Use a second method to check your answer.
Solve the system of equations:
x – 2y + 3z = 5, x + 3z = 11, 5y – 6z = 9.
The opera theater contains 1,200 seats, with three different prices. The
seats cost $45 dollars per seat, $50 per seat, and $60 per seat. The
opera needs to gross $63,750 on seat sales. There are twice as many
$60 seats as $45 seats. How many seats in each level need to be sold?
CCGPS Coordinate Algebra Unpacked Standards Page 14 of 32 Bibb County School System May 2012
Represent and solve equations and inequalities graphically.
MCC9-12.A.REI.12
Graph the solutions to a
linear inequality in two
variables as a half-plane
(excluding the boundary in
the case of a strict
inequality), and graph the
solution set to a system of
linear inequalities in two
variables as the intersection
of the corresponding half-
planes.
DOK 2
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
Graph
Solutions
Linear inequality in
two variables
Half-plane
Solution set
System of linear
inequalities in two
variables
Intersection
Students may use graphing calculators, programs, or applets to model and
find solutions for inequalities or systems of inequalities.
Examples:
Graph the solution: y < 2x + 3.
A publishing company publishes a total of no more than 100
magazines every year. At least 30 of these are women’s magazines, but
the company always publishes at least as many women’s magazines as
men’s magazines. Find a system of inequalities that describes the
possible number of men’s and women’s magazines that the company
can produce each year consistent with these policies. Graph the
solution set.
Graph the system of linear inequalities below and determine if (3, 2) is
a solution to the system.
33
2
03
yx
yx
yx
Solution:
(3, 2) is not an element of the solution set (graphically or by
substitution).
Teaching Considerations for Unit 2
All students should come in with background knowledge of transformations on a coordinate plane from 7th and 8th grade GPS math.
MCC9-12.A.REI.12 – Graph the solutions to a linear inequality in two variables as a half‐plane (excluding the boundary in the case of a strict inequality), and graph the
solution set to a system of linear inequalities in two variables as the intersection of the corresponding half‐planes.
Return to Curriculum Map
CCGPS Coordinate Algebra Unpacked Standards Page 15 of 32 Bibb County School System May 2012
Unit 3: Linear and Exponential Functions
In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this unit, students will learn function notation and
develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own
right. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between
representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that, depending upon the context,
these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot.
When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students build on and
informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between
additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.
BIG IDEA: The comparison of linear vs. exponential is a contrast between repeatedly adding (algebraic sequences) and repeatedly
multiplying (geometric sequencing). How do they differ graphically, in a table, long-term, and in context?
Standards Mathematical
Practices
What students
should DO
What students
should KNOW
Examples/Explanations
Represent and solve equations and inequalities in one variable.
MCC9-12.A.REI.10
Understand that the graph of
an equation in two variables
is the set of all its solutions
plotted in the coordinate
plane, often forming a curve
(which could be a line).
DOK 1
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
Understand
Graph of an
equation
Set of all its
solutions
Plotted in the
coordinate plane
Curve
Line
Focus on linear and exponential equations and be able to adapt and apply
that learning to other types of equations in future courses.
Example:
Which of the following points is on the circle with equation
521 22 )()( yx ?
a. (1, -2)
b. (2, 2)
c. (3, -1)
d. (3, 4)
MCC9-12.A.REI.11
Explain why the x-
coordinates of the points
where the graphs of the
equations y = f(x) and y =
g(x) intersect are the
solutions of the equation f(x)
= g(x); find the solutions
approximately, e.g., using
technology to graph the
functions, make tables of
values, or find successive
approximations. Include
cases where f(x) and/or g(x)
are linear, polynomial,
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
6. Attend to precision.
Explain
Find
Use
Make
x-coordinates
Points
Graphs of equations
Approximate
solutions
Technology
Table of values
Successive
approximations
Linear
Exponential
Students need to understand that numerical solution methods (data in a
table used to approximate an algebraic function) and graphical solution
methods may produce approximate solutions, and algebraic solution
methods produce precise solutions that can be represented graphically or
numerically. Students may use graphing calculators or programs to
generate tables of values, graph, or solve a variety of functions.
Example:
Given the following equations determine the x value that results in
an equal output for both functions.
15
23
xxg
xxf
)(
)(
CCGPS Coordinate Algebra Unpacked Standards Page 16 of 32 Bibb County School System May 2012
rational, absolute value,
exponential, and logarithmic
functions.
DOK 2
Understand the concept of a function and use function notation.
MCC9-12.F.IF.1
Understand that a function
from one set (called the
domain) to another set
(called the range) assigns to
each element of the domain
exactly one element of the
range. If f is a function and x
is an element of its domain,
then f(x) denotes the output
of f corresponding to the
input x. The graph of f is the
graph of the equation y =
f(x).
DOK 1
2. Reason abstractly
and quantitatively.
Understand
Function
Set
Domain
Range
Element
Output
Input
Graph
Equation
Draw examples from linear and exponential functions.
The domain of a function given by an algebraic expression, unless
otherwise specified, is the largest possible domain.
MCC9-12.F.IF.2 Use
function notation, evaluate
functions for inputs in their
domains, and interpret
statements that use function
notation in terms of a
context.
DOK 2
2. Reason abstractly
and quantitatively.
Use
Evaluate
Interpret
Function notation
Functions
Inputs
Domain
Statements
Context
Draw examples from linear and exponential functions.
The domain of a function given by an algebraic expression, unless
otherwise specified, is the largest possible domain.
Examples:
If 1242 xxxf )( , find ).(2f
Let )()( 32 xxf . Find )(3f , )(2
1f , and )(af .
If P(t) is the population of Tucson t years after 2000, interpret the
statements P(0) = 487,000 and P(10)-P(9) = 5,900.
MCC9-12.F.IF.3 Recognize
that sequences are functions,
sometimes defined
recursively, whose domain
is a subset of the integers.
DOK 2
8. Look for and express
regularity in
repeated reasoning.
Recognize Sequences
Functions
Defined recursively
Domain
Subset
Integers
Draw connection to F.BF.2, which requires students to write arithmetic
and geometric sequences.
CCGPS Coordinate Algebra Unpacked Standards Page 17 of 32 Bibb County School System May 2012
Interpret functions that arise in applications in terms of the context.
MCC9-12.F.IF.4 For a
function that models a
relationship between two
quantities, interpret key
features of graphs and tables
in terms of the quantities,
and sketch graphs showing
key features given a verbal
description of the
relationship. Key features
include: intercepts; intervals
where the function is
increasing, decreasing,
positive, or negative;
relative maximums and
minimums; symmetries; end
behavior; and periodicity.
DOK 2
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
6. Attend to precision.
Interpret
Sketch
Function
Relationship
between quantities
Key features
Graphs
Tables
Quantities
Verbal description
Intercepts
Intervals of increase
Intervals of decrease
Positive
Negative
Relative maximums
Relative minimums
Symmetries
End behavior
Focus on linear and exponential functions.
Flexibly move from examining a graph and describing its characteristics
(e.g., intercepts, relative maximums, etc.) to using a set of given
characteristics to sketch the graph of a function.
Examine a table of related quantities and identify features in the table, such
as intervals on which the function increases or decreases.
Recognize appropriate domains of functions in real-world settings. For
example, when determining a weekly salary based on hours worked, the
hours (input) could be a rational number, such as 25.5. However, if a
function relates the number of cans of soda sold in a machine to the money
generated, the domain must consist of whole numbers.
Given a table of values, such as the height of a plant over time, students can
estimate the rate of plant growth. Also, if the relationship between time and
height is expressed as a linear equation, students should explain the
meaning of the slope of the line. Finally, if the relationship is illustrated as
a linear or non-linear graph, the student should select points on the graph
and use them to estimate the growth rate over a given interval.
MCC9-12.F.IF.5 Relate
the domain of a function to
its graph and, where
applicable, to the
quantitative relationship it
describes.
DOK 2
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
6. Attend to precision.
Relate
Domain of a
function
Graph
Quantitative
relationship
Focus on linear and exponential functions.
Students may explain orally, or in written format, the existing relationships.
MCC9-12.F.IF.6
Calculate and interpret the
average rate of change of a
function (presented
symbolically or as a table)
over a specified interval.
Estimate the rate of change
from a graph. DOK 3
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
Calculate
Interpret
Estimate
Average rate of
change of a function
Symbolically
Table
Specified interval
Graph
Focus on linear functions and intervals for exponential functions whose
domain is a subset of the integers.
The average rate of change of a function y = f(x) over an interval [a, b] is
ab
afbf
x
y
)()(
. In addition to finding average rates of change from
functions given symbolically, graphically, or in a table, students may
collect data from experiments or simulations (e.g., falling ball, velocity of a
car, etc.) and find average rates of change for the function modeling the
CCGPS Coordinate Algebra Unpacked Standards Page 18 of 32 Bibb County School System May 2012
situation.
Examples:
Use the following table to find the average rate of change of g over
the intervals [-2, -1] and [0, 2]:
x g(x)
-2 2
-1 -1
0 -4
2 -10
Estimate the average rate of change of the function graphed below
over the intervals [-3, 0] and [0, 3].
Analyze functions using different representations.
MCC9-12.F.IF.7 Graph
functions expressed
symbolically and show key
features of the graph, by
hand in simple cases and
using technology for more
complicated cases.
a. Graph linear and
quadratic functions
and show
intercepts, maxima,
and minima.
e. Graph exponential
and logarithmic
functions, showing
intercepts and end
behavior, and
trigonometric
5. Use appropriate
tools strategically.
6. Attend to precision.
Graph
Show
Linear functions
Intercepts
Maxima
Minima
Exponential
functions
End behavior
Focus on linear and exponential functions. Include comparisons of two
functions presented algebraically.
Key characteristics include but are not limited to maxima, minima,
intercepts, symmetry, end behavior, and asymptotes. Students may use
graphing calculators or programs, spreadsheets, or computer algebra
systems to graph functions.
Examples:
Sketch the graph and identify the key characteristics of the
function described below.
0for 2
0for 2
x
xxxF
x)(
Graph the function f(x) = 2x by creating a table of values. Identify
the key characteristics of the graph.
2 4 6–2–4–6 x
2
4
6
–2
–4
–6
y
CCGPS Coordinate Algebra Unpacked Standards Page 19 of 32 Bibb County School System May 2012
functions, showing
period, midline,
and amplitude.
DOK 1
MCC9-12.F.IF.9 Compare
properties of two functions
each represented in a
different way (algebraically,
graphically, numerically in
tables, or by verbal
descriptions).
DOK 2
6. Attend to precision.
7. Look for and make
use of structure.
Compare
Properties of
functions
Algebraic
Graphic
Numeric
Tables
Verbal descriptions
Focus on linear and exponential functions. Include comparisons of two
functions presented algebraically.
Example:
Examine the functions below. Which function has the larger y-
intercept? How do you know?
o y = 5 – 2x and f(x) = 3x
o g(x) = 2x + 1 and
Build a function that models a relationship between two quantities.
MCC9-12.F.BF.1 Write a
function that describes a
relationship between two
quantities.
a. Determine an
explicit expression,
a recursive process,
or steps for
calculation from a
context.
b. Combine standard
function types
using arithmetic
operations.
DOK 2
1. Make sense of
problems and
persevere in solving
them.
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
6. Attend to precision.
7. Look for and make
use of structure.
8. Look for and express
regularity in
repeated reasoning.
Determine
Combine
Explicit expression
Recursive process
Steps for calculation
from a context
Standard function
types
Arithmetic
operations
Limit to linear and exponential functions.
Provide a real-world example (e.g., a table showing how far a car,
traveling at a uniform speed, has driven after a given number of
minutes) and examine the table by looking “down” the table to
describe a recursive relationship, as well as “across” the table to
determine an explicit formula to find the distance traveled if the
number of minutes is known.
Write out terms in a table in an expanded form to help students see
what is happening. For example, if the y-values are 2, 4, 8, 16,
they could be written as 2, 2(2), 2(2)(2), 2(2)(2)(2), etc., so that
students recognize that 2 is being used multiple times as a factor.
Focus on one representation and its related language – recursive or
explicit – at a time so that students are not confusing the formats.
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.
2 4–2–4 x
2
4
–2
–4
y
CCGPS Coordinate Algebra Unpacked Standards Page 20 of 32 Bibb County School System May 2012
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.
MCC9-12.F.BF.2 Write
arithmetic and geometric
sequences both recursively
and with an explicit formula,
use them to model
situations, and translate
between the two forms.
DOK 3
4. Model with
mathematics.
5. Use appropriate
tools strategically.
8. Look for and express
regularity in
repeated reasoning.
Write
Use
Translate
Arithmetic sequence
Geometric sequence
Recursive formula
Explicit formula
Model situations
An explicit rule for the nth term of a sequence gives an as an expression in
the term’s position n; a recursive rule gives the first term of a sequence, and
a recursive equation relates an to the preceding term(s). Both methods of
presenting a sequence describe an as a function of n.
Examples:
Generate the 5th-11th terms of a sequence if A1= 2 and
121 )()( nn AA
Use the formula: An= A1 + d(n - 1) where d is the common difference to
generate a sequence whose first three terms are: -7, -4, and -1.
There are 2,500 fish in a pond. Each year the population decreases by
25 percent, but 1,000 fish are added to the pond at the end of the year.
Find the population in five years. Also, find the long-term population.
Given the formula An= 2n - 1, find the 17th term of the sequence. What
is the 9th term in the sequence 3, 5, 7, 9, …?
Given a1 = 4 and an = an-1 + 3, write the explicit formula.
Students may believe that the best (or only) way to generalize a table of
data is by using a recursive formula. Students naturally tend to look “down”
a table to find the pattern but need to realize that finding the 100th term
requires knowing the 99th term unless an explicit formula is developed.
Students may also believe that arithmetic and geometric sequences are the
same. Students need experiences with both types of sequences to be able to
recognize the difference and more readily develop formulas to describe
them.
Build new functions from existing functions.
MCC9-12.F.BF.3 Identify
the effect on the graph of
replacing f(x) by f(x) + k, k
f(x), f(kx), and f(x + k) for
specific values of k (both
positive and negative); find
the value of k given the
graphs. Experiment with
cases and illustrate an
explanation of the effects on
4. Model with
mathematics.
5. Use appropriate
tools strategically.
7. Look for and make
use of structure.
Identify
Find
Experiment
Illustrate
Effects on the
graphs
Positive k values
Negative k values
Cases
Explanation of the
effects
Even functions
Focus on vertical translations of graphs of linear and exponential
functions. Relate the vertical translation of a linear function to its y-
intercept.
Examples:
Compare the shape and position of the graphs of each set of
functions. Explain the differences, orally or in written
format, in terms of the algebraic expressions for the functions.
CCGPS Coordinate Algebra Unpacked Standards Page 21 of 32 Bibb County School System May 2012
the graph using technology.
Include recognizing even
and odd functions from their
graphs and algebraic
expressions for them.
DOK 3
Odd functions
Graphs
Algebraic
expressions
o
13
23
xxg
xxf
)(
)(
o
32
2
x
x
xg
xf
)(
)(
Describe the effect of changing the values of a and b on the
position of the graphs of the functions listed below. What effect
do values between 0 and 1 have? What effect do negative values
have?
o baxxf )(
o baxf x )(
Construct and compare linear, quadratic, and exponential models and solve problems.
MCC9-12.F.LE.1
Distinguish between
situations that can be
modeled with linear
functions and with
exponential functions.
a. Prove that linear
functions grow by
equal differences
over equal intervals
and that
exponential
functions grow by
equal factors over
equal intervals.
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
7. Look for and make
use of structure.
Prove
Linear functions
Equal differences
Equal intervals
Exponential
functions
Equal factors
Students can investigate functions and graphs modeling different situations
involving simple and compound interest. Students can compare interest
rates with different periods of compounding (monthly, daily) and compare
them with the corresponding annual percentage rate. Spreadsheets and
applets can be used to explore and model different interest rates and loan
terms.
Students can use graphing calculators or programs, spreadsheets, or
computer algebra systems to construct linear and exponential functions.
Compare tabular representations of a variety of functions to show that
linear functions have a first common difference (i.e., equal differences over
equal intervals), while exponential functions do not (instead function values
grow by equal factors over equal x-intervals).
Apply linear and exponential functions to real-world situations. For
example, a person earning $10 per hour experiences a constant rate of
2 4 6–2–4–6 x
2
4
6
–2
–4
–6
y
2 4 6–2–4–6 x
2
4
6
–2
–4
–6
y
CCGPS Coordinate Algebra Unpacked Standards Page 22 of 32 Bibb County School System May 2012
DOK 3
b. Recognize
situations in which
one quantity
changes at a
constant rate per
unit interval
relative to another.
DOK 1
c. Recognize
situations in which
a quantity grows or
decays by a
constant percent
rate per unit
interval relative to
another. DOK 1
Recognize
recognize
Situations
Quantity
Constant rate per
unit interval
Situations
Grows
Decays
Constant percent
rate per unit interval
change in salary given the number of hours worked, while the number of
bacteria on a dish that doubles every hour will have equal factors over
equal intervals.
Examples:
A cell phone company has three plans. Graph the equation for
each plan, and analyze the change as the number of minutes used
increases. When is it beneficial to enroll in Plan 1? Plan 2? Plan 3?
1. $59.95/month for 700 minutes and $0.25 for each
additional minute,
2. $39.95/month for 400 minutes and $0.15 for each
additional minute, and
3. $89.95/month for 1,400 minutes and $0.05 for each
additional minute.
A computer store sells about 200 computers at the price of $1,000
per computer. For each $50 increase in price, about ten fewer
computers are sold. How much should the computer store charge
per computer in order to maximize their profit?
Sketch and analyze the graphs of the following two situations.
What information can you conclude about the types of growth
each type of interest has?
o Lee borrows $9,000 from his mother to buy a car. His mom
charges him 5% interest a year, but she does not compound
the interest.
o Lee borrows $9,000 from a bank to buy a car. The bank
charges 5% interest compounded annually.
MCC9-12.F.LE.2
Construct linear and
exponential functions,
including arithmetic and
geometric sequences, given
a graph, a description of a
relationship, or two input-
output pairs (include reading
these from a table).
DOK 2
4. Model with
mathematics.
8. Look for and express
regularity in
repeated reasoning.
Construct
Linear functions
Exponential
functions
Arithmetic
sequences
Geometric
sequences
Graph
Description of a
Provide examples of arithmetic and geometric sequences in graphic, verbal,
or tabular forms, and have students generate formulas and equations that
describe the patterns.
Students may use graphing calculators or programs, spreadsheets, or
computer algebra systems to construct linear and exponential functions.
Examples:
Determine an exponential function of the form f(x) = abx using
data points from the table. Graph the function and identify the key
characteristics of the graph.
CCGPS Coordinate Algebra Unpacked Standards Page 23 of 32 Bibb County School System May 2012
relationship
Input-output pairs
Table
x f(x)
0 1
1 3
3 27
Sara’s starting salary is $32,500. Each year she receives a $700
raise. Write a sequence in explicit form to describe the situation.
MCC9-12.F.LE.3
Observe using graphs and
tables that a quantity
increasing exponentially
eventually exceeds a
quantity increasing linearly,
quadratically, or (more
generally) as a polynomial
function.
DOK 1
2. Reason abstractly
and quantitatively.
Observe
Graphs
Tables
Increasing
exponentially
Exceeds
Increasing linearly
Example:
Contrast the growth of the functions f(x) = 3x and f(x) = 3x.
Interpret expressions for functions in terms of the situation they model.
MCC9-12.F.LE.5
Interpret the parameters in a
linear or exponential
function in terms of a
context.
DOK 3
2. Reason abstractly
and quantitatively.
4. Model with
mathematics.
Interpret
Parameters
Linear function
Exponential
function
Context
Limit exponential functions to those of the form f(x) = bx + k.
Students may use graphing calculators or programs, spreadsheets, or
computer algebra systems to model and interpret parameters in linear,
quadratic or exponential functions.
Example:
A function of the form f(n) = P(1 + r)n is used to model the
amount of money in a savings account that earns 5% interest,
compounded annually, where n is the number of years since the
initial deposit. What is the value of r? What is the meaning of the
constant P in terms of the savings account? Explain either orally
or in written format.
Teaching Considerations for Unit 3
Because of its scope, this unit will require approximately half of the semester. When planning instruction for first semester, this needs to be kept in mind so that adequate
time is allowed.
Return to Curriculum Map
CCGPS Coordinate Algebra Unpacked Standards Page 24 of 32 Bibb County School System May 2012
Unit 4: Describing Data
Experience with descriptive statistics began as early as Grade 6. Students were expected to display numerical data and summarize it using measures of center and variability. By
the end of middle school they were creating scatterplots and recognizing linear trends in data. This unit builds upon that prior experience, providing students with more formal
means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations
and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit.
Standards Mathematical
Practices
What students
should DO
What students
should KNOW
Examples/Explanations
Summarize, represent, and interpret data on a single count or measurement variable.
MCC9-12.S.ID.1
Represent data with plots on
the real number line (dot
plots, histograms, and box
plots).
DOK 1
4. Model with
mathematics.
5. Use appropriate
tools strategically.
Represent Data
Plots
Real number line
Dot plots
Histograms
Box plots
MCC9-12.S.ID.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.
DOK 2
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
7. Look for and make
use of structure.
Use
Compare
Statistics
Appropriate to the
shape
Data distribution
Center
Median
Mean
Spread
Interquartile range
Data sets
Standard deviation is left for Advanced Algebra; use MAD as a measure of
spread.
Students may use spreadsheets, graphing calculators and statistical software
for calculations, summaries, and comparisons of data sets.
Examples:
The two data sets below depict the housing prices sold in the King
River area and Toby Ranch areas of Pinal County, Arizona. Based
on the prices below which price range can be expected for a home
purchased in Toby Ranch? In the King River area? In Pinal
County?
o King River area {1.2 million, 242000, 265500, 140000,
281000, 265000, 211000}
o Toby Ranch homes {5 million, 154000, 250000, 250000,
200000, 160000, 190000}
Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68,
find the mean, median, and MAD. Explain how the values vary
about the mean and median. What information does this give the
teacher?
CCGPS Coordinate Algebra Unpacked Standards Page 25 of 32 Bibb County School System May 2012
MCC9-12.S.ID.3
Interpret differences in
shape, center, and spread in
the context of the data sets,
accounting for possible
effects of extreme data
points (outliers).
DOK 3
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
Interpret
Account for
Differences
Shape
Center
Spread
Data sets
Effects of extreme
data
Students may use spreadsheets, graphing calculators and statistical software
to statistically identify outliers and analyze data sets with and without
outliers as appropriate.
Summarize, represent, and interpret data on two categorical and quantitative variables.
MCC9-12.S.ID.5
Summarize categorical data
for two categories in two-
way frequency tables.
Interpret relative frequencies
in the context of the data
(including joint, marginal,
and conditional relative
frequencies). Recognize
possible associations and
trends in the data.
DOK 3
1. Make sense of
problems and
persevere in solving
them.
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
8. Look for and express
regularity in
repeated reasoning.
Summarize
Interpret
Recognize
Categorical data
Two-way frequency
tables
Relative frequencies
Context of the data
Joint relative
frequencies
Marginal relative
frequencies
Conditional relative
frequencies
Associations in data
Trends in data
Students may use spreadsheets, graphing calculators, and statistical
software to create frequency tables and determine associations or trends in
the data.
Examples:
Two-way Frequency Table
A two-way frequency table is shown below displaying the relationship
between age and baldness. We took a sample of 100 male subjects, and
determined who is or is not bald. We also recorded the age of the male
subjects by categories.
Two-way Frequency Table
Bald Age Total
Younger than 45 45 or older
No 35 11 46
Yes 24 30 54
Total 59 41 100
The total row and total column entries in the table above report the
marginal frequencies, while entries in the body of the table are the joint
frequencies.
Two-way Relative Frequency Table
The relative frequencies in the body of the table are called conditional
relative frequencies.
CCGPS Coordinate Algebra Unpacked Standards Page 26 of 32 Bibb County School System May 2012
Two-way Relative Frequency Table
Bald Age Total
Younger than 45 45 or older
No 0.35 0.11 0.46
Yes 0.24 0.30 0.54
Total 0.59 0.41 1.00
MCC9-12.S.ID.6
Represent data on two
quantitative variables on a
scatter plot, and describe
how the variables are
related.
a. Fit a function to the
data; use functions
fitted to data to solve
problems in the context
of the data. Use given
functions or choose a
function suggested by
the context. Emphasize
linear, quadratic, and
exponential models.
DOK 3
b. Informally assess the fit
of a function by plotting
and analyzing residuals.
DOK 2
c. Fit a linear function for
a scatter plot that
suggests a linear
association. DOK 2
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
5. Use appropriate
tools strategically.
6. Attend to precision.
7. Look for and make
use of structure.
8. Look for and express
regularity in
repeated reasoning.
Fit
Use
Solve
Use
Choose
Emphasize
Informally assess
Plot
Analyze
Fit
Function
Data
Context of the data
Given functions
Function suggested
by context
Linear models
Exponential models
Fit of a function
Residuals
Linear function
Scatterplot
Linear association
The residual in a regression model is the difference between the observed
and the predicted y for some x(y the dependent variable and x the
independent variable).
So if we have a model baxy and a data point ),( ii yx , the residual is
for this point is )( baxyr iii . Students may use spreadsheets,
graphing calculators, and statistical software to represent data, describe
how the variables are related, fit functions to data, perform regressions, and
calculate residuals.
Example:
Measure the wrist and neck size of each person in your class and
make a scatterplot. Find the least squares regression line. Calculate
and interpret the correlation coefficient for this linear regression
model. Graph the residuals and evaluate the fit of the linear
equations.
Interpret linear models.
MCC9-12.S.ID.7
Interpret the slope (rate of
change) and the intercept
(constant term) of a linear
model in the context of the
data. DOK 3
1. Make sense of
problems and
persevere in solving
them.
2. Reason abstractly
and quantitatively.
Interpret
Slope
Rate of change
Intercept
Constant term
Linear model
Students may use spreadsheets or graphing calculators to create
representations of data sets and create linear models.
Example:
Lisa lights a candle and records its height in inches every hour. The
results recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.6), (3,
14.9), (4, 13.2), (5, 11.5), (7, 8.1), (9, 4.7), and (10, 3). Express the
CCGPS Coordinate Algebra Unpacked Standards Page 27 of 32 Bibb County School System May 2012
4. Model with
mathematics.
5. Use appropriate
tools strategically.
6. Attend to precision.
Context of the data candle’s height (h) as a function of time (t) and state the meaning of the
slope and the intercept in terms of the burning candle.
Solution: h = -1.7t + 20
Slope: The candle’s height decreases by 1.7 inches for each hour
it is burning.
Intercept: Before the candle begins to burn, its height is 20 inches.
MCC9-12.S.ID.8
Compute (using technology)
and interpret the correlation
coefficient of a linear fit.
DOK 2
4. Model with
mathematics.
5. Use appropriate
tools strategically.
8. Look for and express
regularity in
repeated reasoning.
Compute using
technology
Interpret
Correlation
coefficient
Linear fit
Students may use spreadsheets, graphing calculators, and statistical
software to represent data, describe how the variables are related, fit
functions to data, perform regressions, and calculate residuals and
correlation coefficients.
Example:
Collect height, shoe-size, and wrist circumference data for each
student. Determine the best way to display the data. Answer the
following questions: Is there a correlation between any two of the three
indicators? Is there a correlation between all three indicators? What
patterns and trends are apparent in the data? What inferences can be
made from the data?
MCC9-12.S.ID.9
Distinguish between
correlation and causation.
DOK 1
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
6. Attend to precision.
Distinguish
Correlation
Causation
Some data leads observers to believe that there is a cause and effect
relationship when a strong relationship is observed. Students should be
careful not to assume that correlation implies causation. The determination
that one thing causes another requires a controlled randomized experiment.
Example:
Diane did a study for a health class about the effects of a student’s end-
of-year math test scores on height. Based on a graph of her data, she
found that there was a direct relationship between students’ math
scores and height. She concluded that “doing well on your end-of-
course math tests makes you tall.” Is this conclusion justified? Explain
any flaws in Diane’s reasoning.
MCC6.SP.5 Summarize
numerical data sets in
relation to their context,
such as by:
c. Giving quantitative
measures of center
(median and/or mean)
and variability
(interquartile range
and/or mean absolute
2. Reason abstractly
and quantitatively.
3. Construct viable
arguments and
critique the
reasoning of others.
4. Model with
mathematics.
5. Use appropriate
Summarize
Give
Describe
Numerical data sets
Relation to context
Quantitative
measures of
variability
Mean absolute
deviation
Overall pattern
Teach during academic years 2012-13, 2013-14, and 2014-15 only.
The Mean Absolute Deviation (MAD) describes the variability of the data
set by determining the absolute value deviation (the distance) of each data
piece from the mean and then finding the average of these deviations.
Higher MAD values represent a greater variability in the data.
Students should understand the mean distance between the pieces of data
and the mean of the data set expresses the spread of the data set. Students
can see that the larger the mean distance, the greater the variability.
Comparisons can be made between different data sets.
CCGPS Coordinate Algebra Unpacked Standards Page 28 of 32 Bibb County School System May 2012
deviation), as well as
describing any overall
pattern and any striking
deviations from the
overall pattern with
reference to the context
in which the data were
gathered. DOK 2
tools strategically.
6. Attend to precision.
Striking deviations
Reference to context
Teaching Considerations for Unit 4
MCC9-12.S.ID.1 – MCC9-12.S.ID.3 should be taught after MCC9-12.S.ID.5 – MCC9-12.S.ID.9, in order to build on the work from Unit 3. Line of best fit flows from Unit 3
with its emphasis on linear and exponential graphs. The line of best fit taught here is not a hand-computed linear regression; it is not median-median line. It is only done with
technology and informally. CCGPS Advanced Algebra will address regressions. After line of best fit, teach MAD standards.
Return to Curriculum Map
CCGPS Coordinate Algebra Unpacked Standards Page 29 of 32 Bibb County School System May 2012
Unit 5: Transformations in the Coordinate Plane
In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations
and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid
motions and formal constructions. They solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain
why they work.
Standards Mathematical
Practices
What students
should DO
What students
should KNOW
Examples/Explanations
Experiment with transformations in the plane.
MCC9-12.G.CO.1 Know
precise definitions of angle,
circle, perpendicular line,
parallel line, and line
segment, based on the
undefined notions of point,
line, distance along a line,
and distance around a
circular arc. DOK 1
6. Attend to precision. Know
Precise definitions
of angle, circle,
perpendicular line,
parallel line, line
segment
Undefined notions
of point, line,
distance along a
line, distance around
a circular arc
Review vocabulary associated with transformations (e.g. point, line,
segment, angle, circle, polygon, parallelogram, perpendicular, rotation
reflection, translation).
MCC9-12.G.CO.2
Represent transformations in
the plane using, e.g.,
transparencies and geometry
software; describe
transformations as functions
that take points in the plane
as inputs and give other
points as outputs. Compare
transformations that
preserve distance and angle
to those that do not (e.g.,
translation versus horizontal
stretch). DOK 2
6. Attend to precision.
Represent
Describe
Compare
Transformations in
the plane
Functions
Points in the plane
Inputs
Outputs
Preserve distance
Preserve angle
Translation
Horizontal strech
Students may use geometry software and/or manipulatives to model and
compare transformations.
Provide both individual and small-group activities, allowing adequate time
for students to explore and verify conjectures about transformations and
develop precise definitions of rotations, reflections and translations.
Provide real-world examples of rigid motions (e.g. Ferris wheels for
rotation; mirrors for reflection; moving vehicles for translation).
MCC9-12.G.CO.3 Given a
rectangle, parallelogram,
trapezoid, or regular
polygon, describe the
rotations and reflections that
carry it onto itself. DOK 1
3. Construct viable
arguments and
critique the
reasoning of others.
5. Use appropriate
tools strategically.
Describe when
given a rectangle,
parallelogram,
trapezoid , or
regular polygon
Rotations
Reflections
Onto itself
Students may use geometry software and/or manipulatives to model
transformations.
Analyze various figures (e.g. regular polygons, folk art designs or product
logos) to determine which rotations and reflections carry (map) the figure
onto itself. These transformations are the “symmetries” of the figure.
CCGPS Coordinate Algebra Unpacked Standards Page 30 of 32 Bibb County School System May 2012
MCC9-12.G.CO.4 Develop
definitions of rotations,
reflections, and translations
in terms of angles, circles,
perpendicular lines, parallel
lines, and line segments.
DOK 3
6. Attend to precision.
7. Look for and make
use of structure.
Develop
Definitions of
rotations,
reflections,
translations
Angles
Circles
Perpendicular lines
Parallel lines
Line segments
Students may use geometry software and/or manipulatives to model
transformations. Students may observe patterns and develop definitions of
rotations, reflections, and translations.
Focus attention on the attributes (e.g. distances or angle measures) of a
geometric figure that remain constant under various transformations.
Make the transition from transformations as physical motions to functions
that take points in the plane as inputs and give other points as outputs. The
correspondence between the initial and final points determines the
transformation.
Emphasize understanding of a transformation as the correspondence
between initial and final points, rather than the physical motion.
MCC9-12.G.CO.5 Given a
geometric figure and a
rotation, reflection, or
translation, draw the
transformed figure using,
e.g., graph paper, tracing
paper, or geometry software.
Specify a sequence of
transformations that will
carry a given figure onto
another. DOK 3
3. Construct viable
arguments and
critique the
reasoning of others.
5. Use appropriate
tools strategically.
7. Look for and make
use of structure.
Draw when given
a geometric figure
and a rotation,
reflection, or
translation
Specify
Transformed figure
Sequence of
transformations
Onto itself
Use graph paper, transparencies, tracing paper or dynamic geometry
software to model transformations and demonstrate a sequence of
transformations that will carry a given figure onto another.
Provide students with a pre-image and a final, transformed image, and ask
them to describe the steps required to generate the final image. Show
examples with more than one answer (e.g., a reflection might result in the
same image as a translation).
Work backwards to determine a sequence of transformations that will carry
(map) one figure onto another of the same size and shape.
Teaching Considerations for Unit 5
Clarifications are needed on the following:
MCC9-12.G.CO.1 – What is undefined notions of point, line, distance along a line, and distance around a circular arc? Will distance formula be used here?
MCC9-12.G.CO.2 – Is a horizontal stretch related to dilation?
Return to Curriculum Map
CCGPS Coordinate Algebra Unpacked Standards Page 31 of 32 Bibb County School System May 2012
Unit 6: Connecting Algebra and Geometry through Coordinates
Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including
properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines.
Standards Mathematical
Practices
What students
should DO
What students
should KNOW
Examples/Explanations
Use coordinates to prove simple geometric theorems algebraically.
MCC8.G.8 Apply the
Pythagorean Theorem to
find the distance between
two points in a coordinate
system. DOK 1
2. Reason abstractly
and quantitatively.
8. Look for and express
regularity in
repeated reasoning.
Apply
Find
Pythagorean
Theorem
Distance between
two points in a
coordinate system
Students will create a right triangle from the two points given (as shown in
the diagram below) and then use the Pythagorean Theorem to find the
distance between the two given points. Generalize to the distance formula.
MCC9-12.G.GPE.4 Use
coordinates to prove simple
geometric theorems
algebraically.
DOK 3
3. Construct viable
arguments and
critique the
reasoning of others.
Use
Prove
Coordinates
Simple geometric
theorems
Algebraic
Restrict contexts that use distance and slope.
Students may use geometric simulation software to model figures and
prove simple geometric theorems.
Example:
Use slope and distance formula to verify the polygon formed by
connecting the points (-3, -2), (5, 3), (9, 9), (1, 4) is a parallelogram.
Use slopes and the Euclidean distance formula to solve problems about
figures in the coordinate plane such as:
o Given three points, are they vertices of an isosceles, equilateral, or
right triangle?
o Given four points, are they vertices of a parallelogram, a rectangle,
a rhombus, or a square?
o Given the equation of a circle and a point, does the point lie
outside, inside, or on the circle?
o Given the equation of a circle and a point on it, find an equation of
the line tangent to the circle at that point.
o Given a line and a point not on it, find an equation of the line
through the point that is parallel to the given line.
o Given a line and a point not on it, find an equation of the line
through the point that is perpendicular to the given line.
o Given the equations of two non-parallel lines, find their point of
intersection.
CCGPS Coordinate Algebra Unpacked Standards Page 32 of 32 Bibb County School System May 2012
MCC9-12.G.GPE.5 Prove
the slope criteria for parallel
and perpendicular lines and
use them to solve geometric
problems (e.g., find the
equation of a line parallel or
perpendicular to a given line
that passes through a given
point). DOK 3
3. Construct viable
arguments and
critique the
reasoning of others.
8. Look for and express
regularity in
repeated reasoning.
Prove
Use
Solve
Slope criteria for
parallel lines
Slope criteria for
perpendicular lines
Geometric problems
Lines can be horizontal, vertical, or neither.
Students may use a variety of different methods to construct a parallel or
perpendicular line to a given line and calculate the slopes to compare the
relationships.
MCC9-12.G.GPE.6 Find
the point on a directed line
segment between two given
points that partitions the
segment in a given ratio.
DOK 2
2. Reason abstractly
and quantitatively.
8. Look for and express
regularity in
repeated reasoning.
Find
Point
Directed line
segment
Partitions
Given ratio
Students may use geometric simulation software to model figures or line
segments.
Examples:
Given A(3, 2) and B(6, 11),
o Find the point that divides the line segment AB two-thirds of
the way from A to B.
The point two-thirds of the way from A to B has x-coordinate
two-thirds of the way from 3 to 6 and y coordinate two-thirds
of the way from 2 to 11. So, (5, 8) is the point that is two-
thirds from point A to point B.
o Find the midpoint of line segment AB.
Given two points, use the distance formula to find the coordinates
of the point halfway between them. Generalize this for two
arbitrary points to derive the midpoint formula.
Use linear interpolation to generalize the midpoint formula and
find the point that partitions a line segment in any specified ratio.
MCC9-12.G.GPE.7 Use
coordinates to compute
perimeters of polygons and
areas of triangles and
rectangles, e.g., using the
distance formula.
DOK 1
2. Reason abstractly
and quantitatively.
5. Use appropriate
tools strategically.
6. Attend to precision.
Use
Compute
Coordinates
Perimeter
Area
Distance formula
Example:
The vertices of ABC are located at A(-3,2), B(0, 6), and C(1, 2).
Find the perimeter and area of ABC.
Teaching Considerations for Unit 6
Teach transition standard MCC8.G.8 during 2012-13 only. Return to Curriculum Map