integrated math 1

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Integrated Math 1

2012-13 Curriculum Guide

Iredell-Statesville Schools

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Purpose and Use of the Documents The Curriculum Guide represents an articulation of what students should know and be able

to do. The Curriculum Guide supports teachers in knowing how to help students achieve

the goals of the new standards and understanding each standard conceptually. It should

be used as a tool to assist teachers in planning and implementing a high quality

instructional program.

• The “At-a-Glance” provides a snapshot of the recommended pacing of instruction

across a semester or year.

• Learning targets (“I can” statements) and Criteria for Success (“I will” statements)

have been created by ISS teachers and are embedded in the Curriculum Guide to

break down each standard and describe what a student should know and be able to

do to reach the goal of that standard.

• The academic vocabulary or content language is listed under each standard. There are

30-40 words in bold in each subject area that should be taught to mastery.

• The unpacking section of the Curriculum Guide contains rich information and

examples of what the standard means; this section is an essential component to help

both teachers and students understand the standards.

Teachers will be asked to give feedback throughout the year to continually

improve their Curriculum Guides.

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College and Career Readiness Anchor Standards for Reading

The K-12 standards on the following pages define what students should understand and be able to do by the end of each

grade. They correspond to the College and Career Readiness (CCR) anchor standards below by number. The CCR and grade-specific

standards are necessary complements – the former providing broad standards, the latter providing additional specificity – that together

define the skills and understandings that all students must demonstrate.

Key ideas and Details

1. Read closely to determine what the text says explicitly and to make logical inferences from it; cite specific textual evidence when

writing or speaking to support conclusions drawn from the text.

2. Determine central ideas or themes of a text and analyze their development; summarize the key supporting details and ideas.

3. Analyze how and why individuals, events, and ideas develop and interact over the course of a text.

Craft and Structure

4. Interpret words and phrases as they are used in a text, including determining technical, connotative, and figurative meanings, and

analyze how specific word choices shape meaning or tone.

5. Analyze the structure of texts, including how specific sentences, paragraphs, and larger portions of the text (e.g. a section,

chapter, scene, or stanza) relate to each other and the whole.

6. Assess how point of view or purpose shapes the content and style of a text.

Integration of Knowledge and Ideas

7. Integrate and evaluate content presented in diverse media and formats, including visually and quantitatively, as well as in

words.*

8. Delineate and evaluate the argument and specific claims in a text, including the validity of the reasoning as well as the relevance

and sufficiency of the evidence.

9. Analyze how two or more texts address similar themes or topics in order to build knowledge or to compare the approaches the

authors take.

Range of Reading and Level of Text Complexity

10. Read and comprehend complex literary and informational texts independently and proficiently.

* Please see “Research to Build and Present Knowledge” in writing and “Comprehension and Collaboration” in Speaking and Listening for additional standards

relevant to gathering, assessing, and applying information from print and digital sources.

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College and Career Readiness Anchor Standards for Writing

The K-12 standards on the following pages define what students should understand and be able to do by the end of each

grade. They correspond to the College and Career Readiness (CCR) anchor standards below by number. The CCR and grade-specific

standards are necessary complements – the former providing broad standards, the latter providing additional specificity – that together

define the skills and understandings that all students must demonstrate.

Text Types and Purposes*

1. Write arguments to support claims in an analysis of substantive topics or texts, using valid reasoning and relevant and sufficient

evidence.

2. Write informative/explanatory texts to examine and convey complex ideas and information clearly and accurately through the

effective selection, organization, and analysis of content.

3. Write narratives to develop real or imagined experiences or events using effective technique, well-chosen details, and well-

structured event sequences.

Production and Distribution of Writing

4. Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and

audience.

5. Develop and strengthen writing as needed by planning, revising, editing, rewriting, or trying a new approach.

6. Use technology, including the internet, to produce and publish writing and to interact and collaborate with others.

Research to Build and Present Knowledge

7. Conduct short as well as more sustained research projects based on focused questions, demonstrating understanding of the

subject under investigation.

8. Gaither relevant information from multiple print and digital sources, assess the credibility and accuracy of each source, and

integrate the information while avoiding plagiarism.

9. Draw evidence from literacy or informational texts to support analysis, reflection, and research

Range of Writing

10. Write routinely over extended time frames (time for research, reflection, and revision) and shorter time frames (a single sitting or

a day or two) for a range of tasks, purposes, and audiences.

* These broad types of writing include many subgenres. See Appendix A for definitions of key writing types.’

Taken from Common Core Standards (www.corestandards.org)

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Common Core: Standards for Mathematical Practice

(from: http://www.corestandards.org/the-standards/mathematics/introduction/standards-for-mathematical-practice/ )

The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These

practices rest on important “processes and proficiencies” with longstanding importance in mathematics education. The first of these are the NCTM process

standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency

specified in the National Research Council’s report Adding It Up: adaptive reasoning, strategic competence, conceptual understanding (comprehension of

mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately), and

productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).

1. Make sense of problems and persevere in solving them.

Mathematically proficient students start 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. Mathematically proficient 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. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient

students check their answers to problems using a different method, and they 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.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on

problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the

representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during

the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation

of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using

different properties of operations and objects.

3. Construct viable arguments and critique the reasoning of others.

Mathematically proficient 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. Mathematically proficient 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—

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explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can

make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an

argument applies. Students at all grades can 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.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this

might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or

analyze a problem in the community. 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. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to

simplify a complicated situation, 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.

Mathematically proficient 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. Proficient students are 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, mathematically proficient 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. Mathematically proficient

students at various grade levels 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.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning.

They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about 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. In the elementary grades, students give carefully formulated explanations to each other. 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.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same

amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well

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remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. 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.

8. Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might

notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to

the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y –

2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x + 1), and (x – 1)(x

3 + x2 + x + 1) might lead them to the

general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient 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.

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A Year at a Glance

1st Quarter

Unit 1: Simplifying and Rewriting Expressions

2nd Quarter

Unit 2: Function Notation and Behavior

Unit 3: Finding and Interpreting Solutions

3rd Quarter

Unit 4: Using Equations and Geometry to Solve Problems

4th Quarter

Unit 5: Understanding and Interpreting Data

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Integrated Math I- Middle School and High School

Testing Timeline 2012-13

Benchmarks followed by DASH will be scored by testing department. All Open-ended (OE) items will be scored at schools. All optional

benchmarks and optional open-ended items will have one copy provided to school and will be scored at schools.

Year-long Approach (this will include middle school students taking Int. Math I)

Baseline

(DASH)

Benchmark 1

(DASH) and

Open-ended

item

Open-ended

item from

BM2

Benchmark 2

(DASH)

During Finals

Open-ended

item from

BM3

Benchmark 3

(DASH)

Benchmark 3

(Optional)

Testing Windows

8/27/12-

9/6/12

10/18/12-

10/25/12

12/11/12-

12/13/12

1/10/13-

1/16/13

3/6/13-

3/8/13

3/25/13-

3/28/13

5/15/13-

5/17/13

Plus/Delta Feedback Due

9/20/12 11/8/12 N/A 1/30/13 N/A 4/11/13 N/A

Fall Semester Only (1st Semester) Spring Semester Only (2nd Semester)

Baseline

(DASH)

Benchmark 2

(DASH) and

Open-ended

item

Benchmark 3

(Optional)- 1

copy to school

8/27/12-9/6/12 10/18/12-

10/25/12

12/11/12-

12/13/12

Plus/Delta

Feedback Due

9/20

11/8/12 1/2/13

Please note: 10/18/12 – 10/25/12 – Year-long students take Benchmark 1

10/18/12 – 10/25/12 – Fall Semester students take Benchmark 2

3/25/13 – 3/28/13 – Year-long students take Benchmark 3

3/25/13 – 3/28/12 – Spring Semester students take Benchmark 2

Baseline

(DASH)

Benchmark 2

(DASH) and

Open-ended

item

Benchmark 3

(Optional)- 1

copy to school

1/22/13-

1/31/13

3/25/13-

3/28/13

5/15/13-

5/17/13

Plus/Delta

Feedback Due 4/12/13 5/31/13

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Correlation of ISS Integrated Math I Units to Core-Plus

ISS

Unit

Math I

Standard(s)

Concepts Core-Plus

Course

Core-Plus Unit

and Page #

Materials Needed

(Check Teaching

Resources for handouts)

1 Review Exponent Rules – Product of Powers

1 5 p.304 None

1 Review Exponent Rules – Quotient of Powers

1 5 p.332 None

1 A.SSE.2

A.SSE.3a

Rewriting quadratics into equivalent forms 1 7 p.495 None

2 F.BF.1b

F.LE.1

Creating a function from a context 1 1 p.48 Ruler

2 A.REI.10

F.IF.4

F.IF.7

Using the tables and graphs of functions 1 1 p.52 Graphing Calc.

2 F.IF.4

F.IF.7

Identifying the differences in linear, inverse,

quadratic and exponential functions

1 1 p.56 Graphing Calc.

3 A.REI.6 Solving systems of equations by elimination

2 1 p. 54 None

3 A.CED.3 Solving inequalities & equations with tables and

graphs

1 3 p. 188 Graphing Calc.

3 A.SSE.3 Factoring and expanding

2 5 p. 336 None

4 GGPE.4 -

GGPE.6

Understanding rays, lines, points, segments,

midpoint & distance on the coordinate planes.

2 3 p. 164 CPMP Tools (a free

download online)

4 FLE.1 Explain parts of exponential equation and

context.

1 5 p.294 None

4 G.GPE.5 Identifying parallel & perpendicular lines and

using their properties to solve problems.

2 3 p.170 None

5 SID.2 Shapes of Distribution

1 2 p. 76

5 SID.1 Variability and Box Plot

1 2 p. 108

5 SID.3 Identifying outliers

1 2 p. 113

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Integrated Mathematics I is divided into 5 units.

Hold the Ctrl button and click the unit to jump to that section. The Unit should be completed by:

1. Simplifying and Rewriting Expressions .................................................Quarter 1

2. Function Notation and Behavior ...........................................................Quarter 2

3. Finding and Interpreting Solutions........................................................Quarter 2

4. Using Equations and Geometry to Solve Problems ...............................Quarter 3

5. Understanding and Interpreting Data...................................................Quarter 4

Common Core - Integrated Mathematics I This course is based on the North Carolina Math I Standards.

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Unit 1: Simplifying and Transforming Expressions

Standards Covered in this Unit: N.RN.1, N.RN.2, A.SSE.1a, A.SSE.1b, A.SSE.2, A.SSE.3a, and A.APR.1

Students will become fluent in simplifying and identifying the parts of expressions. This includes using the order of operations, simplifying

polynomial expressions, and simplifying expressions with exponents and powers with rational exponents.

-Identify: terms, factors, coefficients, bases, and exponents

-Define quadratic expression and determine when an expression is or can be rewritten into quadratic form.

-Operations with polynomials (add, subtract, and multiply).

-Rational exponents should be limited to fractions with a numerator of one.

-Explain why the exponent rules work and demonstrate they still work with rational exponents.

-Rewrite and simplify expressions with radicals as expressions with rational exponents and vice versa.

Students will become fluent in factoring quadratic expressions.

-Limit factoring to the GFC method, difference of squares, and the trinomial method for when A≠1.

-Explain the Closure property relating to the addition, subtraction and multiplication of polynomials and demonstrate when polynomial

division is not closed, for example when the result is a rational expression.

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Common Core Unit: 1. Simplifying and Transforming Expressions

Common Core Standard(s): N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those

values, allowing for a notation for radicals in terms of rational exponents.

N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: Student may need to review the exponent rules prior to this section.

N-RN.1 In order to understand the meaning of rational exponents, students can initially investigate them by considering a pattern such as:

?2

22

42

162

2

1

1

2

4

=

=

=

=

What is the pattern for the exponents? They are reduced by a factor of ½ each time. What is the pattern of the simplified values? Each successive value is the

square root of the previous value. If we continue this pattern, then 22 2

1

=

Once the meaning of a rational exponent (with a numerator of 1) is established, students can verify that the properties of integer exponents hold for rational

exponents as well. For example,

3333312

1

2

1

2

1

2

1

===⋅+

since 393333 2

1

2

1

==⋅=⋅ 55551

33

13

3

1

===

⋅

since ( ) 5553

3

3

3

1

==

105

1

5

1

5

1

5

1

===−

xx

x

x

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Ex. Use an example to show why nm

n

m

x

x

x −= holds true for expressions involving rational exponents like

2

1or

5

1.

N-RN.2 Students should be able to use the properties of exponents to rewrite expressions involving radicals as expressions using rational exponents. At this

level, focus on fractional exponents with a numerator of 1.

Ex. Simplify the following.

a. 63

55 ⋅

b. 63

1

9xx ⋅

N-RN.2 Students should be able to use the properties of exponents to rewrite expressions involving rational exponents as expressions using radicals. At this

level, focus on fractional exponents with a numerator of 1.

Ex. Simplify the following.

a. 4

3

4

1

xx ⋅

b. 4

1

2

1

164 ⋅

Mathematical Language: exponent, base, rational exponent, radical expression, exponent laws

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can explain why a rational exponent can be written as a radical

expression.

I will demonstrate rewriting a rational exponent to a radical

expression using a pattern and vice versa.

I will demonstrate rewriting rational exponents to a radical

expression using exponent laws and vice versa.

2. I can rewrite a rational exponent into a radical expression and simplify

the expression and vice versa.

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Common Core Standard(s): 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.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: In this unit, limit to vocabulary and identification.

Mathematical Language: linear expression, quadratic expression, exponential expression, coefficient, constant, variable,

term, factors, base, exponent, quadratic term, linear term, constant term

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can identify an algebraic expression as linear, quadratic or

exponential.

I will explain why an expression is identified as a linear, quadratic or

exponential expression.

2. I can identify the parts of a linear, exponential, or quadratic

expression.

I will identify the coefficient, constant, variable, term, and factors in a

linear expression.

I will identify the base and exponent in an exponential expression.

I will identify the coefficients and factors in a quadratic expression.

3. I can breakdown complicated expressions into simple parts. I will identify quadratic term, linear term and constant term in a

quadratic expression.

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Common Core Standard(s):

A.SSE.2 Use the structure of an expression to identify ways to rewrite it.

A.APR.1 Understand that polynomials form a system analogous to the integers; namely, they are closed under the operations of addition,

subtraction, and multiplication; add, subtract, and multiply polynomials.

A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defines.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: For A.SSE.2, include identifying the GCF and simple factoring by grouping.

Note: For A-APR.1, EOC will be limited to addition and subtraction of quadratics and the multiplication of linear expressions.

Note: For A.SSE.3, factor expression up to quadratic trinomials. Do not solve in this unit.

Mathematical Language: like terms, simplify, closure property, polynomials, GCF, difference of squares, grouping,

trinomial method

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can rewrite algebraic expressions by adding, subtracting, multiplying

or factoring and identifying the new form as linear, quadratic, or

exponential.

I will add like terms to simplify expressions.

I will add, subtract, and multiply polynomials.

I will explain the meaning of the closure property as it pertains to the

addition, subtraction, multiplication, and division of polynomials.

I will find the greatest common factor and use it to rewrite the

expression.

I will use different methods of factoring to rewrite quadratic

expressions (difference of squares, grouping, and a trinomial

method).

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Unit 2: Function Notation and Behavior

Standards Covered in this Unit: N.Q.1, N.Q.3, A.SSE.1a, F.IF.1, F.IF.2, F.IF.4, F.IF.5, F.IF.6, F.IF.7a, F.IF.7e, F.IF.8a,

F.IF.8b, F.IF.9, F.BF.1b, F.BF.3, F.LE.1a, F.LE.1b, F.LE.1c, F.LE.3, F.LE.5, G.GMD.1, and G.GMD.3

Students will become fluent in identifying, writing, interpreting and evaluating functions. This includes understanding functions graphically,

numerically, symbolically and the limitations of a function.

-Identify domain and range, from and variety of circumstances and explain the reasons a relation is or is not a function.

-Evaluate linear and exponential functions and explain in the context of a situation.

-Define linear, quadratic, and exponential functions.

-Identify important features of a linear, quadratic, or exponential function by using a table or graph and explain their meaning in the

context of a situation: intercepts; intervals of increasing and decreasing output; positive and negative output; symmetries; and relative

maximums and minimums.

-Graph a function when given a verbal description of the relationship between quantities.

-Identify the domain of a function and the limitation of the domain based on the context of a situation.

-Describe the meaning of any point in context of a function and situation.

-Find the average rate of change of a function with a given interval if given a table, equation or graph.

-Compare functions represented in different forms. For example, find the function with the largest maximum when one function is

represented graphically while another function is represented algebraically.

-Identify a set of points as being linear or exponential.

-Describe and compare the rates of change of linear and exponential functions.

-Define and identify arithmetic and geometric sequences through sets of points and in the context of a situation.

Student will be able to explain how addition creates new functions and explain the relationship of the new function to the original function.

-Explain the relationship between f(x) and f(x+k) or f(x)+k

-Explain how to change a function from one situation to another.

Students will become fluent in giving an informal argument for the basic geometric formulas and using the formulas to solve problems.

-Describe the meaning of π in relation to both the diameter and radius and to the formula for circumference

-Give an informal argument for the area of a circle formula using dissection. (Using the sector to create a parallelogram)

-Give an informal argument for the volume of a cylinder using Cavalieri’s principal. (A cylinder is a stack of circles.)

-Give informal arguments for the volume of pyramids and cones.

-Use the volume formulas to solve problems in a given context. (Formulas will be provided.)

-Use the volume or dimensions of one 3-d object to find the volume or a dimension of another 3-d object. (For example, a cone inside a

cylinder.)

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Common Core Unit: 2. Function Notation and Behavior


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.

N.Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: This section should be used in conjunction with the study of functions and formulas and should be reinforced throughout the course.

Mathematical Language: origin, scale, units, accuracy

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can use units as a way to understand and guide the solution to a

problem.

I will choose and use units consistently in formulas.

I will use units to identify errors.

2. I can graph data and interpret a data display using the appropriate

scale and units.

I will identify the origin.

I will use the appropriate scale to fit the context.

I will identify and use the scale and units to interpret a graph.

3. I can state and use the appropriate level of accuracy of a

measurement.

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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).

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.

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.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: For A.SSE.1, identified parts in Unit 1. In this unit, interpret the meaning of the parts in context.

Mathematical Language: function, domain, range, function notation, input, output, independent, dependent, explanatory,

response, set, table, graph, coordinates

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can define and identify a function. I will explain the definition of a function.

I will identify a function in various forms: set, table, graph, and

coordinates

I will recognize and explain function notation.

2. I can define and identify domain and range. I will recognize the various terms for domain and range (input &

output, independent & dependent, explanatory & response).

3. I can evaluate functions and explain the input and output in context.

4. I can explain the parts of a function in context.

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G.GMD.1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.

Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: Emphasize the link between formulas and functions. This is the application of the functions section.

Note: For G.GMD.3, only use the formulas for evaluation. Solving for anything other then the volume will be covered in Unit 3.

Mathematical Language: circumference of a circle, area of a circle, volume (cylinder, pyramid, cone), sector, diameter,

Cavalieri’s principal

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can explain the formulas for the circumference of a circle, area of a

circle, and the volume of cylinder, pyramid, and cone.

I will use the relationship between the diameter and circumference

to explain the circumference formula.

I will use sectors arranged as a parallelogram to explain the area of a

circle formula.

I will use Cavalieri’s principal to explain the volume formula for a

cylinder.

2. I can find the volume of cylinders, pyramids, cones, and spheres using

a formula.

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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.

F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

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.

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 functions, showing intercepts and end behavior.

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

descriptions).

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: For this section, the focus is relating functions to their graphs and vice versa.

Note: For F.IF.4 - Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums

and minimums; symmetries; end behavior

Note: For F.IF.9 – Compare maximums, minimums and end behavior of various functions.

Mathematical Language: dependent, independent, variable, x-intercept, y-intercept, max, min, relative max, relative min,

symmetry, domain intervals, range intervals, increase, decrease, rate of change

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Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can identify the dependent and independent variables in the context

of a problem.

2. I can identify the important features of a graph and explain them in

the context of a problem.

I will find the x and y intercepts of functions.

I will find the max and min of a function.

I will recognize symmetries on the graph of function.

I will identify the shape of a graph behaving as linear, quadratic, or

exponential.

I will identify domain intervals as increasing and decreasing.

I will identify domain intervals where the range is positive or

negative.

I will sketch a graph given a verbal description.

I will explain the meaning of the features in context.

3. I can determine the appropriate domain based on the context of a

problem.

I will state the restrictions of the domain in the context of a problem.

I will explain the reasoning for the restrictions.

4. I can find the average rate of change. I will find the average rate of change given an equation, table or

graph.

I will interpret the average rate of change in the context of a

problem.

5. I can use technology to identify the important features of a graph.

6. I can compare functions represented in different forms (algebraically,

verbally, graphically, and with numerical tables).

I will identify the important feature of functions represented in

different forms.

I will compare the properties of two functions represented in

different forms.

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Common Core Standard(s): 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 the graph using technology.

F.BF.1 Write a function that describes a relationship between two quantities.

b. Combine standard function types using arithmetic operations.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: At this level, limit to vertical and horizontal translations of linear and exponential functions. Relate the vertical translation of a linear

function to its y-intercept.

Note: For F.BF.1, this standard puts a context to the translation.

Mathematical Language: translation of a function, constant, addition of a constant, vertical translation, horizontal

translation, value of k (f(x) + k, k f(x), f(kx), and f(x + k))

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can translate a function. I will explain the effect of adding a constant to a function, f(x) + k ,as

a vertical translation.

I will explain the effect of adding a constant to x, f(x+k), as a

horizontal translation.

I will identify vertical and horizontal translations.

I will identify the value of k given a graph.

2. I can create and interpret a function in the context of a problem. I will recognize when a relationship exists.

I will write a function that describes a relationship.

I will create a function by adding or subtracting a constant to an

existing function and explain how the original function is effected in

the context of a problem.

I will create a function by adding a linear function to a linear or

quadratic function and explain how the original function is effected

in the context of a problem.

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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; exponential functions grow by equal factors over equal intervals.

b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

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.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: The focus in this unit is recognizing when situations represent linear and exponential behavior. Use tables and graphs. Introduce arithmetic

and geometric sequences and have the students identify them as linear and exponential. Creating functions that model data will occur in a later

unit (NOW-NEXT is used in Unit 4). (Quadratics and other polynomial functions for this standard are not addressed in this course.)

Mathematical Language: linear function, exponential function, arithmetic sequence, geometric sequence, constant rate

(common difference), constant ratio

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can distinguish between situations that can be modeled with linear

functions and with exponential functions

I will define an arithmetic and geometric sequence.

I will demonstrate that a linear function changes at a constant rate

(common difference).

I will demonstrate that an exponential function changes at a constant

ratio.

I will recognize an arithmetic sequence as a linear function in the

context of a problem (verbally, with a table, and graphically).

I will recognize a geometric sequence as an exponential function in

the context of a problem (verbally, with a table, and graphically).

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F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.

F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph,

and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: For F.LE.5, the focus of this is to interpret the meaning of linear and exponential functions. With this standard, set and solve simple linear or

exponential word problems using evaluation.

Note: For F.IF.8, the focus is on interpreting quadratic and exponential functions in context. Give the functions and describe the meaning of the

parts. Creating an exponential or quadratic function from data is part of unit 4.

Mathematical Language: linear function, slope, y-intercept, exponential function, rate, principal, exponent

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can interpret the parts of a linear function in context. I will explain the meaning of the slope in context.

I will explain the meaning of the y-intercept in context.

2. I can interpret the parts of an exponential function in context. I will explain the meaning of rate in context of a problem.

I will explain the meaning of the exponent in the context of a

problem.

I will explain the meaning of the principal in the context of a problem.

3. I can use a given function to solve a problem. I will identify the key values in a problem and substitute them

appropriately. (match the values to the variables)

I will explain my answer in context.

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Unit 3: Finding and Interpreting Solutions

Standards Covered in this Unit: A.SSE.3a, A.CED.1, A.CED.3, A.CED.4, A.REI.1, A.REI.3, A.REI.5, A.REI.6, A.REI.10,

A.REI.11, A.REI.12, and G.CO.1

Students will become fluent in the solving linear equations, inequalities, quadratic equations, and simple exponential equations. Students

will need to justify their solving process, recognize the limitations of their process, and interpret the solutions in context of the problem.

-Define solution.

-Solve linear equations and inequalities with justification (mathematical properties) for each step

-Compare the processes of solving equations to solving inequalities

-Explain the difference between solving an equation and simplifying an expression.

-Identify points as solutions or non-solutions in context

-Solve for an identified variable in a formula, limited to linear equations, or squared or cubed variables.

-Solve quadratic equations by factoring.

-Explain the roots/zeroes/solutions of a quadratic in context.

-Solve simple exponential equations using tables and explain the solution in context.

Students will become fluent in solving systems of equations and inequalities. Students will need to justify their solving process, recognize the

limitations of their process, and interpret the solutions in context of the problem.

-Solve systems of linear equations using the intersection of graphs, substitution and elimination.

-Explain why the process of elimination produces that same solution as the original system.

-Explain why the x-coordinate of intersection of y=f(x) and y=g(x) is the solution to f(x)=g(x), limited to linear and exponential equations.

-Find approximate solutions to exponential equations by creating a system then finding the intersection.

-Solve systems of linear inequalities using the intersection of graphs and determine if the boundaries are part of the solution.

-Use linear programming to solve problems.

-Identify points as solutions or non-solutions for linear systems and systems of linear inequalities.

Students will become fluent in setting up and solving equations based on the properties of angles.

-Define point, line, plane, ray, segment, midpoint, endpoint and angle

-Explain the relationships between angles, including vertical and adjacent angles, linear pairs, the angles of a triangle and the angles

formed by a transversal intersecting parallel lines and use that relationship to set up equations to solve problems.

-Use the segment addition postulate, angle addition postulate, and midpoints to set up equations and solve problems.

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Common Core Unit: 3. Finding and Interpreting Solutions


G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notations of point,

line, distance along a line, and distance around a circular arc.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: In this unit, focus on the definitions of point, line, plane, ray, segment, midpoint, endpoint and angle. Use the angle addition postulate,

segment addition postulate, midpoint, vertical angles, adjacent angles, linear pairs, the angles of a triangle and the angles formed by a

transversal through parallel lines to create and solve various linear equations and systems of equations. These should be taught (integrated) with

the appropriate section. (Finding midpoints, endpoints and the length of segments on a coordinate plane will be covered in Unit 4.)

Mathematical Language: point, line, plane, segment, ray, angle, midpoint, endpoint, transversal, linear pairs,

supplementary, complementary, alternate interior, vertical angles, linear, nonlinear, corresponding, adjacent, same-side

interior, parallel, perpendicular

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can describe a point, line, and plane. I will name a point, line and plane appropriately.

2. I can define a segment, ray, angle, midpoint, and endpoint. I will name geometric figures appropriately.

3. I can classify pairs of angles. I will identify vertical angles, linear pairs, supplementary, adjacent

angles (nonlinear), and transversal angle pairs (corresponding,

alternative interior, etc.)

4. I can use the measures of angles to set up and solve equations.

(through out the entire unit)

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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.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: Algebraic proof.

Mathematical Language: order of operations, proof, properties of equality (addition, multiplication), simplifying an

expression, solving an equation

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can explain my reasoning for each step in solving an equation. I will define an equation and explain its importance in the solving

process.

I will explain the relationship between the order of operations and

the solving process. (Compare simplifying to solving)

I will explain the use of the properties of equality in the solving

process.

I will explain when and how rewriting an expression is needed in the

solving process.

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Common Core Standard(s): A.CED.1 Create equations and inequalities in one variable and use them to solve problems.

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.

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).

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.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: For A.CED.1 – In this unit, focus on writing equations and inequalities from word problems and use them to solve problems. Creating

equations from data is in a future unit.

Note: This section does not focus on systems.

Mathematical Language: dependent, independent, equation, inequality, solution, greater than/less than/equal to symbols,

linear, quadratic, exponential, boundary (solid vs. dotted, open vs. closed)

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can define a solution to one variable equations and inequalities. I will explain the relationship between a solution and an

equation/inequality.

I will determine if a number is a solution to an equation/inequality.

I will describe and rewrite a solution to an inequality verbally,

symbolically, and graphically.

I will describe the regions of a line graph that are not part of the

solution of an inequality.

2. I can solve linear equations and inequalities with one variable. I will solve linear equations and inequalities.

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I will graph the solutions to linear inequalities on a number line.

I will explain when and why the inequality symbols are “flipped” in

the solving process.

I will check the solving process used by others, identify errors, and

provide recommendations.

I will compare linear equations and inequalities.

I will write equations/inequalities from word problems and use them

to solve equations.

3. I can define solutions to two variable equations and inequalities. I will explain the relationship between tables, graphs and solutions.

I will determine if a point is a solution to an equation (linear,

quadratic or exponential).

I will determine if a point is a solution to an inequality, including

points located on the boundary.

4. I can solve and graph an equation or inequality with two variables. I will solve for the dependent or identified variable.

I will explain the steps in the solving process.



I will graph linear equations by using the slope and y-intercept, the

intercepts, points from a table, and the slope and one point.

I will graph linear inequalities; shading to identify the region of

solutions and using the appropriate symbol to describe the boundary

as being included or excluded in the solution region.

I will write equations/inequalities from word problems and use them

to solve equations.

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A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

G-GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

A.REI.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: Solve for an identified variable in literal equations and formulas. Review the geometry formulas and introduce formulas from physical

science.

Mathematical Language: subscript, coefficient, variable, formula equation (literal equation)

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can solve for a variable in a formula equation. I will understand the use of subscripts in formulas.

I will identify the variable to be solved (told explicitly or implied in a

word problem), solve for that variable, and use the new equation to

solve problems.

I will solve for a coefficient when an equation is generalized.

I will explain the steps in the solving process.



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A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

a. Factor a quadratic expression to reveal the zeros of the function it defines.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: Find the zeros through factoring and using technology.

Mathematical Language: roots, zeros, x-intercepts, solutions, quadratic, factors, factoring, Multiplicative Property of Zero,

rational solution, non-rational solution (quadratics that do not factor)

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can use factoring to solve quadratic equations. I will explain the relationship between the x-intercepts and the

solutions to quadratic equations.

I will use factoring to rewrite quadratic expressions.

I will explain how rewriting the quadratic expression into factors

leads to solution(s). (Multiplicative Property of Zero)

I will solve quadratic equations through factoring and use the

solution to solve problems.

I will explain the limitations of the process of factoring to solve

quadratic equations.

2. I will use technology to find the zeros of quadratic equations.

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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.

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.

A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

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.

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, rational, absolute value, exponential, and logarithmic functions.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: The focus of this group should be systems.

Note: For A.REI.5, prove why elimination works using the properties of equality.

Note: For linear systems, solve using elimination, substitution, graphing by hand and technology.

Note: For A.REI.11, use this process to find solutions for exponential equations. Look at unpacking document in reference to this standard.

Mathematical Language: systems of equations, systems of inequalities, solutions, solution types (one, infinite, no solution),

feasible region, optimal solution, elimination process, substitution process, intersection of a graph

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can find the solution to a system of linear equations. I will understand the meaning of a system of equations.

I will solve or approximate the solution of systems of equations

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through graphing by hand.

I will explain why the solution of a system is the intersection of the

equations’ corresponding graphs.

I will solve systems of equations with elimination.

I will explain why the process of elimination works. (Properties of

Equality)

I will solve systems of equations using substitution.

I will explain why the process of substitution works.

I will solve or approximate the solution of systems of equations using

technology.

I will compare word problems that produce one equation and those

that produce a system of equations.

I will write a system of equations from a word problem and use its

solution to solve problems. (perimeter, area, total, and break even

word problems)

2. I can graph the solution to a system of linear inequalities. I will explain why the solution to a system of inequalities is the

intersection of the inequalities’ corresponding graphs.

I will write a system of inequalities from a word problem and use its

feasible region to find the optimal solution.

3. I can explain how the solution to any single variable equation is

related to the solution of the system of equations created by setting

each side of the equation equal to y.

(Explain how the solution of f(x)=g(x) is related to the solution of y=f(x)

and y=g(x).) Teacher Note: Look at unpacking document in reference

to this standard (A.REI.11).

I will explain when it is appropriate to use this method.

I will use this method to find approximate solutions to linear and

exponential equations.

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Unit 4: Using Equations and Geometry to Solve Problems

Standards Covered in this Unit: A.CED.1, A.CED.2, F.IF.3, F.IF.8a, F.IF.8b, F.IF.9, F.BF.1a, F.BF.2, F.LE.1a, F.LE.2,

G.CO.1, G.GPE.4, G.GPE.5, G.GPE.6, and G.GPE.7

Students will become fluent in creating linear, quadratic and exponential equations, knowing the criteria and corresponding methods.

Students will explain their process and reasoning.

-Create a linear equation given two or more points and a point and slope.

-Create linear, quadratic or exponential equations (including half-life, growth and decay) in the context of a situation.

-Explain the choice of linear, quadratic and exponential in context of the information and problem.

-Explain the part of the equation in context of the situation.

-Use the created equation to solve problems.

-Describe the effect of adding a constant or linear expression to a created function in context to a situation.

Students will become fluent in identifying arithmetic and geometric sequence and creating their corresponding functions, both in explicit and

recursive forms.

-Write arithmetic and geometric sequences in both explicit and recursive forms in context to a situation. (Do not write in formal

recursive form. For example, use NEXT=NOW+5, starting at 3)

Students will become fluent in the common vocabulary of geometry.

-Write the definition of segment, circle, perpendicular lines, parallel lines, intersecting lines, concurrent lines, midpoint, types of

polygons, regular, and types of quadrilaterals. (This list is not exclusive).

-Recognize and use the basic symbols of geometry: equal, congruent, parallel, perpendicular, line, segment, ray,

Students will become fluent in identifying parallel and perpendicular lines and using their properties to solve problems.

-Identify parallel, perpendicular, intersecting and concurrent lines.

-Find the equation of a line that passes through a point and is parallel or perpendicular to another line.

-Explain why the slopes of perpendicular lines multiply to be -1, except when one line is horizontal or vertical.

Students will become fluent in identifying geometric shapes on the coordinate plane and using the properties of those geometric shapes to

solve problems.

-Use slope and distance to identify and verify geometric shapes (triangles, quadrilaterals, and regular polygons) on the coordinate plane.

-Find the perimeter and area of geometric figure on the coordinate plane.

-Use the properties of geometric figures to find missing coordinates, justifying the steps in the process.

-Prove that a point lies on a circle.

-Find the midpoint of a segment, given the endpoints.

-Find the endpoint of a segment, given the other endpoint and the midpoint.

-Apply the coordinate geometry and the properties of the geometric figures to solve problems.

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Common Core Unit: 4. Using Equations and Geometry to Solve Problems


G.CO.1 Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notations of point,

line, distance along a line, and distance around a circular arc.

G.GPE.5 Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

G.GPE.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles.

G-GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio

G.GPE.4 Use coordinates to prove simple geometric theorems algebraically.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: For G.CO.1, focus on circle, parallel, perpendicular, intersecting, and concurrent/collinear lines. (Arcs are not in this course)

Note: For G.GPE.6, focus on finding the midpoint and the endpoint of a segment given the appropriate information.

Note: For G.GPE.4, prove that coordinates form special geometric figure such as circles, rectangles, and regular polygons.

Mathematical Language: circles, parallel lines, perpendicular line, intersecting lines, concurrent lines, categories of

triangles (equilateral, scalene, isosceles), categories of quadrilaterals (parallelogram, rectangles, rhombi, squares, trapezoids

and kites), regular polygons, slope, y-intercept, parallel, perpendicular, concurrent/collinear, equation of a line (standard,

slope-intercept, point-slope), midpoint, endpoint, length of a segment, Pythagorean Theorem, diagonal, vertex point,

coordinates

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can identify and explain the geometric definition of significant shapes

and lines.

I will explain the definitions of circles, parallel lines, perpendicular

line, intersecting lines, concurrent lines, the categories of triangles

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and the categories of quadrilaterals.

I will explain the definition of regular polygons.

I will find the slope and equation of a line given two points on that

line.

I will use slope and y-intercept to identify parallel, perpendicular,

intersecting and concurrent/collinear lines.

2. I can use the properties of parallel and perpendicular lines to find

other lines.

I will find the equation of a line given a point on the line and the

slope.

I will find the equation of a line given a point on the line and that it is

parallel or perpendicular to another line.

I will use parallel and perpendicular lines to solve problems.

3. I can use the properties of a midpoint to solve problems. I will find the midpoint.

I will find an endpoint given the other endpoint and the midpoint.

4. I can identify and solve problems using the properties of significant

geometric shapes on the coordinate plane.

I will find the length of line segments on the coordinate plane.

I will identify equilateral, isosceles, and scalene triangles given vertex

points on the coordinate plane.

I will identify parallelogram, rectangles, rhombi, squares, trapezoids

and kites given vertex points on the coordinate plane.

I will use the properties of the diagonals of the categories of

parallelograms for identification and to solve problems.

I will find a missing vertex point of an identified shape on the

coordinate plane.

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A.CED.1 Create equations and inequalities in one variable and use them to solve problems.

A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with

labels and scales.

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; exponential functions grow by equal factors over equal intervals.

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).

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: In this unit, the word problems contain data that students will use to create the equations/inequalities to solve problems.

Note: For F.LE.2, students should be able to create a linear equation from a linear relation by hand and with technology. Use technology with an

exponential relation. In this unit, there should be no deviations from the model and data.

Mathematical Language: sequence (arithmetic, geometric), linear model, exponential model

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can use basic patterns to solve problems. I will identify basic patterns in a set of numbers and explain in

context.

I will use the identified pattern to solve problems.

2. I can create a linear or exponential model and use the model to solve

problems.

I will identify a sequence as arithmetic, geometric or neither.

I will explain why a set of data should be modeled with a linear or

exponential model.

I will create a linear model, by hand and with technology, for

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sequences identified as arithmetic.

I will create an exponential model for sequences identified as

geometric.

I will express the created model with a written description, equation,

and graph.

I will use created linear and exponential models to solve problems.

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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.

F.IF.3 Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

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.

F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal

descriptions).

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: For F.BF.2, in a previous unit students identified arithmetic and geometric sequences. In this unit, students will create equations that

describe the data set in the explicit and recursive form.

Note: For F.IF.9 and F.LE.1a, focus on the NOW-NEXT.

Mathematical Language: arithmetic sequence, geometric sequence, explicitly defined, recursively defined, NOW-NEXT

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can describe an arithmetic or geometric sequence explicitly and

recursively using the NOW-NEXT.

2. I can rewrite a function written explicitly as recursive and vice versa.

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F-IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph,

and interpret these in terms of a context.

b. Use the properties of exponents to interpret expressions for exponential.

Unpacking: What does this standard(s) mean that a student will know and be able to do? Note: In Unit 2, the focus was interpreting parts of these functions in context. In this unit, the focus is on creating the function from information

provided and using the function to solve a problem or describe the behaviors in context. For quadratics, students should only solve by factoring

or using technology (this does not include completing the square).

Mathematical Language: linear model, quadratic model, exponential model, half life, growth, decay

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can use data to create a quadratic model to solve problems. I will recognize and explain when a set of data should be modeled

with a quadratic model.

I will use a quadratic model to solve problems in context.

2. I can compare a quadratic model to linear and exponential models.

3. I can classify examples of exponential growth or decay. I will write exponential equations representing half life, growth, and

decay.

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Unit 5: Understanding and Interpreting Data

Standards Covered in this Unit: N.Q.2, S.ID.1, S.ID.2, S.ID.3, S.ID.5, S.ID.6a, S.ID.6b, S.ID.6c, S.ID.7, S.ID.8, and S.ID.9

Student will become fluent in interpreting data and creating a mathematical model.

-Represent data in various forms: dot plots, histograms, and box plots

-Compare the shape (skewed, normal, uniform), center (median and mean) and spread (interquartile range, mean absolute deviation

and standard deviation) of two or more data sets, taking extreme data points into account.

-Define and identify outliers and explain their effect shape, center and spread

-Explain the shape of the distribution in context to a situation.

-Explain the relation between spread and variability.

-Create a two-way frequency table.

-Calculate joint, marginal and conditional relative frequencies and interpret in context.

-Describe associations and trends in data from two-way tables.

-Create a scatter plot

-Describe the form, strength, and direction of relationship in a scatter plot.

-Create a function that best models a data set, interpreting the part of the function in context.

-Calculate residuals and analyze a residual plot.

-Interpret slope and the intercept in terms of a linear model in the context of a set of data.

-Interpret the correlation coefficient of a linear relationship.

-Explain the difference between high correlation and causation and how to best determine causation.

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Common Core Unit: 5. Understanding and Interpreting Data


N-Q.2 Define appropriate quantities for the purpose of descriptive modeling

S.ID.1 Represent data with plots on the real number line (dot plots, histograms, and box plots).

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.

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).

Unpacking: What does this standard(s) mean that a student will know and be able to do? N-Q.2 Define the appropriate quantities to describe the characteristics of interest for a population. For example, if you want to describe how dangerous the

roads are, you may choose to report the number of accidents per year on a particular stretch of interstate. Generally speaking, it would not be appropriate to

report the number of exits on that stretch of interstate to describe the level of danger.

Ex. What quantities could you use to describe the best city in North Carolina?

Ex. What quantities could you use to describe how good a basketball player is?

S-ID.1 In grades 6-8, students describe center and spread of a data distribution. Here they choose a summary statistic appropriate to the characteristics of the

data distribution, such as the shape (skewed vs. normal) or the presence of outliers.

Construct appropriate graphical displays (dot plots, histogram, and box plot) to describe sets of data values.

Ex. Make a dot plot of the number of siblings that members of your class have.

Ex. Create a frequency distribution table and histogram for the following set of data:

Age (in months) of First Steps

13 9 12 11

10 8.5 14 9

12.5 10 13.5 9.5

6 7.5 15 9

8 11.5 10 12

10.5 11 13 12.5

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Ex. Construct a box plot of the number of buttons each of your classmates has on their clothing today.

S-ID.2 Understand which measure of center and which measure of spread is most appropriate to describe a given data set. The mean and standard deviation

are most commonly used to describe sets of data. However, if the distribution is extremely skewed and/or has outliers, it is best to use the median and the

interquartile range to describe the distribution since these measures are not sensitive to outliers.

Ex. You are planning to take on a part time job as a waiter at a local restaurant. During your interview, the bosstold you that their best waitress, Jenni, made

an average of $70 a night in tips last week. However, when you asked Jenni about this, she said that she made an average of only $50 per night last week. She

provides you with a copy of her nightly tip amounts from last week (see below). Calculate the mean and the median tip amount.

a. Which value is Jenni’s boss using to describe the average tip? Why do you think he chose this value?

b. Which value is Jenni using? Why do you think she chose this value?

c. Which value best describes the typical amount of tips per night? Explain why.

Day Tip Amount

Sunday $50

Monday $45

Wednesday $48

Friday $125

Saturday $85

S-ID.2 Select the appropriate measures to describe and compare the center and spread of two or more data sets in context.

Ex. Delia wanted to find the best type of fertilizer for her tomato plants. She purchased three types of fertilizer and used each on a set of seedlings. After 10

days, she measured the heights (in cm) of each set of seedlings. The data she collected is shown below. Construct box plots to analyze the data. Write a brief

description comparing the three types of fertilizer. Which fertilizer do you recommend that Delia use?

Fertilizer A Fertilizer B Fertilizer C

7.6 6.3 1.0 11.0 9.2 5.6 10.5 11.8 15.5

5.0 4.5 5.2 8.4 7.2 12.1 14.7 11.0 10.8

3.2 4.6 2.4 10.5 14.0 15.3 13.9 12.7 9.9

5.5 3.8 1.5 6.3 8.7 11.3 10.3 10.1 15.8

6.2 6.9 2.6 17.0 13.5 14.2 9.5 13.2 9.7

S-ID.3 Understand and be able to use the context of the data to explain why its distribution takes on a particular shape (e.g. are there real-life limits to the

values of the data that force skewness? are there outliers?)

Ex. Why does the shape of the distribution of incomes for professional athletes tend to be skewed to the right?

Ex. Why does the shape of the distribution of test scores on a really easy test tend to be skewed to the left?

Ex. Why does the shape of the distribution of heights of the students at your school tend to be symmetrical?

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S-ID.3 Understand that the higher the value of a measure of variability, the more spread out the data set is.

Ex. On last week’s math test, Mrs. Smith’s class had an average of 83 points with a standard deviation of 8 points.

Mr. Tucker’s class had an average of 78 points with a standard deviation of 4 points. Which class was more consistent with their test scores? How do you

know?

S-ID.3 Explain the effect of any outliers on the shape, center, and spread of the data sets.

Ex. Explain the relationship between the mean and the median for a data set that has a few high outliers. What would most likely be the shape of its

distribution?

Ex. The heights of Washington High School’s basketball players are: 5 ft 9in, 5 ft 4in, 5 ft 7 in, 5ft 6 in, 5 ft 5 in, 5 ft 3 in, and 5 ft 7 in. A student transfers to

Washington High and joins the basketball team. Her height is 6 ft 10in.

a. What is the mean height of the team before the new player transfers in? What is the median height?

b. What is the mean height after the new player transfers? What is the median height?

c. What affect does her height have on the team’s height distribution and stats (center and spread)?

d. How many players are taller than the new mean team height?

e. Which measure of center most accurately describes the team’s average height? Explain.

Mathematical Language:

Quantitative, Standard deviation, Mean, Median, Mode, Interquartile Range, 5 number summary, Distribution, Outliers

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can take a set of data and plot it on the number line(P) • I will identify quantitative data.

• I will make dot plots from a data set.

• I will make a histogram from a data set or a dot plot.

• I will mark the five number summary on a number line.

• I will draw a box plot from the five number summary.

2. I can compare data sets using statistics that are appropriate to the

shapes of the distributions.(R)

• I will identify the differences between mean and median and

when they are used.

• I will identify the difference between standard deviation and

the five number summary and when they are used.

• I will identify the shape of a distribution.

• I will compare two data sets using mean, median, standard

deviation, and the five number summary.

3. I can draw conclusions from the shape and individual points of a data

set.(R)

• I will identify outliers/extreme points in a data set.

• I will match shapes to data sets without collecting data.

• I will determine the effects of outliers on a data set.

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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.

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.

b. Informally assess the fit of a function by plotting and analyzing residuals.

c. Fit a linear function for scatter plots that suggest a linear

Unpacking: What does this standard(s) mean that a student will know and be able to do? S-ID.5 Create a two-way frequency table from a set of data on two categorical variables.

Ex. Make a two-way frequency table for the following set of data. Use the following age groups: 3-5, 6-8, 9-11, 12-14, 15-17.

Youth Soccer League

Youth Soccer League

Gender Age Gender Age Gender Age Gender Age Gender Age

M 4 F 7 M 17 M 5 F 10

M 7 M 7 M 16 M 9 M 6

F 8 F 15 F 14 F 13 F 4

F 6 M 13 M 14 M 15 M 5

M 4 M 12 F 12 M 17 M 9

F 10 M 15 F 8 M 12 M 10

F 11 F 16 M 13 F 13 F 15

S-ID.5 Calculate joint, marginal, and conditional relative frequencies and interpret in context. Joint relative frequencies are compound probabilities of using

AND to combine one possible outcome of each categorical variable (P(A and B)). Marginal relative frequencies are the probabilities for the outcomes of one

of the two categorical variables in a two-way table, without considering the other variable. Conditional relative frequencies are the probabilities of one

particular outcome of a categorical variable occurring, given that one particular outcome of the other categorical variable has already occurred.

Ex. Use the frequency table to answer the following questions.

Youth Soccer League

Age Group

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Gender 3-5 years old 6-8 years old 9-11 years old 12-14 years old 15-17 years old Total

Male 4 3 3 5 5 20

Female 1 4 3 4 3 15

Total 5 7 6 9 8 35

a) What is the relative frequency of players who are male and 9-11 years old? (joint relative frequency)

b) What is percentage of female players that are 15-17 years old? (conditional relative frequency)

c) What percentage of league members are male? (marginal relative frequency)

S-ID.5 Recognize associations and trends in data from a two-way table.

Ex. Given the segmented bar graph below, describe any trends in the context of the data.

S-ID.6 Create a scatter plot from two quantitative variables.

Ex. Connie works for a telephone company. She calls existing customers to sell them additional services for their account. The table below shows how much

Connie earns for selling selected numbers of additional services.

Create a scatter plot of the number of services sold and the daily pay she received.

# of services sold 10 20 30 40 50

Daily Pay (in dollars) 60 80 100 120 140

S-ID.6 Describe the form, strength, and direction of the relationship between the two variables in context.

Ex. Describe, in context, the form, strength, and direction of the scatterplot from the problem above.

S-ID.6a Determine which type of function best models a set of data. Fit this type of function to the data and interpret constants and coefficients in the context

of the data (e.g. slope and y-intercept of linear models, base/growth or decay rate and y-intercept of exponential models). Use the fitted function to make

predictions and solve problems in the context of the data.

Ex. What type of function models the data found in the scatterplot above?

Find the function that best describes the data. What is the meaning of the slope and y-intercept in the context of the problem? Use the model to predict

Connie’s earnings for selling 100 services.

S-ID.6b Calculate the residuals for the data points fitted to a function. A residual is the difference between the actual y-value and the predicted y-value (! ! !),

which is a measure of the error in prediction. (Note: ! is the symbol for the predicted y-value for a given x-value.) A residual is represented on the graph of the

data by the vertical distance between a data point and the graph of the function.

Ex. Calculate the residuals from the plot above. What do they represent? Are the points with negative residuals located above or below the regression line?

S-ID.6b Create and analyze a residual plot. A residual plot is a graph of the x-values vs. their corresponding residuals. (Note that some computer software

programs plot ! vs. residual instead of x vs. residual. However, the interpretation of the residual plot remains the same.) If the residual plot shows a balance

between positive and negative residuals and a lack of a pattern, this indicates that the model is a good fit. For more accurate predictions, the size of the

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residuals should be small relative to the data. At this level, for part b, focus on linear models.

Ex. What is the sum of the squared residuals of the linear model that represents the situation described above? Can you find a different line that gives a

smaller sum? Explain.

S-ID.6c For data sets that appear to be linear, use algebraic methods and technology to fit a linear function to the data. To develop the concept of LSRL, begin

by finding the centroid (!! !! and selecting another point to fit a line through the center of the data. (Note: When describing a set of one-variable data, the

mean is the most common predictor of a value in that data set. Therefore, the centroid is a logical choice for a point on the line of best fit because it uses the

average of the x-values and the average of the y-values.) Find the sum of the squared errors of this line and compare to lines fitted to the same set of data

(but a different second point) by others. The Least squares Regression Line is a line that goes through the centroid and minimizes the sum of these squared

errors.

Ex. Below is the data for the 1919 season and World Series batting averages for nine White Sox players.

Season

Batting

Average

.319 .279 .275 .290 .351 .302 .256 .282 .296

World Series

Batting

Average

.226 .250 .192 .233 .375 .056 .080 .304 .324

a. Create a scatter plot for the data provided. Is there a linear association? Explain.

b. What is the Least squares Regression Line that models this data?

c. How do you know this equation is the line of best fit to model the data?


Categorical, two way table, marginal distribution, conditional distribution, explanatory variable, response variable, joint

probability, conditional probability, marginal probability, residual, scatter plot, fitted function, association, correlation

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can summarize two categorical variables using two way tables.(R) • I will identify categorical variables.

• I will make two way tables using two categorical variables.

• I will find marginal, conditional distributions using two way

tables.

• I will find marginal, conditional, and joint probabilities using

two way tables.

2. I can create and use scatter plots to describe how two quantitative

variables are related.(P)

• I will identify the explanatory variable and the response

variable for a scatter plot.

• I will make a scatter plot using two quantitative variables.

• I will identify the form, association, and strength of a

relationship on a scatter plot.

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• I will identify what type of function can best describe a scatter

plot (linear and exponential).

• I will create residuals for a scatter plot with a fitted function

(line of best fit etc.)

• I will interpret a residual in the context of a problem.

• I will use residuals and fitted functions to solve problems

from a scatter plot.

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S.ID.7 Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.8 Compute (using technology) and interpret the correlation coefficient of a linear fit.

S.ID.9 Distinguish between correlation and causation.

Unpacking: What does this standard(s) mean that a student will know and be able to do?

S-ID.7 Understand that the key feature of a linear function is a constant rate of change. Interpret in the context of the data, i.e. as x increases (or decreases)

by one unit, y increases (or decreases) by a fixed amount.

S-ID.7 Interpret the y-intercept in the context of the data, i.e. an initial value or a one-time fixed amount.

Ex. The equation _________represents a pay plan offered to employees who collect credit card applications.

What do the numbers in the rule tell you about the relationship between daily pay and the number of credit card

applications collected?

S-ID.8 Understand that the correlation coefficient, r, is a measure of the strength and direction of a linear relationship between two quantities in a set of data.

The magnitude (absolute value) of r indicates how closely the data points fit a linear pattern. If r = 1, the points all fall on a line. The closer ! is to 1, the

stronger the correlation. The closer ! is to zero, the weaker the correlation. The sign of r indicates the direction of the relationship – positive or negative.

Ex. A couple of friends decided to measure their compatibility by ranking their favorite activities.

Watching TV Listening to

Music

Reading Talking on the

Phone

Hanging out with

friends

Shopping Exercise

Mary 4 5 2 7 6 1 3

Maria 7 2 4 3 5 6 1

a. Using technology, make a scatterplot for the two rankings.

b. Predict what the rs value is. Use the scatterplot to help explain your answer.

c. Find the Least squares Regression Line that models this set of data.

d. Using technology identify what the correlation coefficient is and interpret what it means in the context of the data.

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S-ID.9 Understand that because two quantities have a strong correlation, we cannot assume that the explanatory (independent) variable causes a change in

the response (dependent) variable. The best method for establishing causation is to conduct an experiment that carefully controls for the effects of lurking

variables. If this is not feasible or ethical, causation can be established by a body of evidence collected over time (e.g. smoking causes cancer).

Ex. When you have an association between two variables, how can you determine if the association is a result of a cause-and-effect relationship?


Slope, intercept, correlation coefficient, explanatory(independent) and response(dependent) variable, linear model,

causation

Learning Targets: “I Can” Criteria For Success: “I Will” 1. I can interpret the slope and intercept for a linear model.(R) • I will identify the slope and intercept of a linear model.

• I will relate the slope of a linear model to a data set.

• I will relate the intercept of a linear model to a data set.

• I will use the explanatory and response variables to

determine what the intercept and slope of a linear model

mean.

2. I can find and explain the correlation coefficient for a linear model.(R) • I will use the calculator (or software) to find the correlation

for a linear model.

• I will describe what the correlation coefficient says about a

linear model.

3. I can describe the difference between correlation and causation.(R) • I will identify high, medium, and low correlations.

• I will identify when high correlations identifies strong

relationship.

• I will use deductive reasoning to determine if a high

correlation describes a possible causation.

The ISS Curriculum Guide is adapted from NC DPI

http://www.ncpublicschools.org/acre/standards/common-core-tools/