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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.


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


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.


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


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.


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.


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.


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.


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.


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



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


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate


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.)


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate



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.


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


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.


Practices

What students

should DO

What students

should KNOW


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


and quantitatively.

3. Construct viable

arguments and

critique the

reasoning of others.


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


and quantitatively.


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


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


and quantitatively.

3. Construct viable

arguments and

critique the


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate




use of structure.


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?


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate


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).


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.



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?


Practices

What students

should DO

What students

should KNOW


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


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,


and quantitatively.

4. Model with

mathematics.

5. Use appropriate



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

)(

)(


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


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


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


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.


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate



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


and quantitatively.

4. Model with

mathematics.


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


and quantitatively.

4. Model with

mathematics.

5. Use appropriate


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


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



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


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



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


them.


and quantitatively.

4. Model with

mathematics.

5. Use appropriate




use of structure.


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


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



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



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.


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


4. Model with

mathematics.

5. Use appropriate



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


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.


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.


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


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


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.


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.



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.


Practices

What students

should DO

What students

should KNOW


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


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


and quantitatively.

3. Construct viable

arguments and

critique the


4. Model with

mathematics.


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?


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


and quantitatively.

3. Construct viable

arguments and

critique the


4. Model with

mathematics.

5. Use appropriate


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


them.


and quantitatively.

3. Construct viable

arguments and

critique the


4. Model with

mathematics.

5. Use appropriate



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.


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


and quantitatively.

3. Construct viable

arguments and

critique the


4. Model with

mathematics.

5. Use appropriate




use of structure.


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


them.


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


4. Model with

mathematics.

5. Use appropriate



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



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


4. Model with

mathematics.


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


and quantitatively.

3. Construct viable

arguments and

critique the


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.


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



Striking deviations

Reference to context


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.



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.


Practices

What students

should DO

What students

should KNOW


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


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


5. Use appropriate


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.


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



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


5. Use appropriate



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.


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?



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.


Practices

What students

should DO

What students

should KNOW


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


and quantitatively.


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


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.


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



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


and quantitatively.


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


and quantitatively.

5. Use appropriate



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.


Teach transition standard MCC8.G.8 during 2012-13 only. Return to Curriculum Map

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