(Draft 12-17-13) Updates may be available as new resources become available.

2013-14 Preface to Algebra Instructional Guide Draft Semester 2

This Instructional Guide for Semester 2 has been created as a complement to the Algebra Bridging Document implemented in July, 2013. Between the beginning of this academic year and the present, landmark events have occurred that impact the content and instruction of mathematics within our school district. Most importantly, our Superintendent and Board - weighing the input of several diverse advisory groups - have furthered the implementation of the California Common Core within SUHSD by their approval of an integrated pathway for the three high school level courses comprising the A-G requirements for graduation. This pathway, as outlined in Appendix A of the CCSS, configures content to be taught within units in each of the three courses. With these broader descriptions of content defined, it then becomes possible to assign to a particular course discrete standards that specifically describe what a student should know and be able to do. These standards are significantly more rigorous than those that they replace in the old framework. Moreover, the selection of an integrated pathway supplants the “traditional” sequencing of topics characterized in our three current high school courses (Algebra, Geometry, and Intermediate Algebra). As a result, content and student expectations through the three courses are arranged in a significantly different, but cognitively appropriate, order from that in our experience of the recent past. All that being said, this Algebra Instructional Guide for Semester 2 of the current year must be viewed as serving two purposes, not necessarily obvious on first examination. First, it is a transitional piece that addresses content consistent with this level (first year of the integrated high school sequence) as presented in the new Framework. Second, it prepares students fairly for the materials they will be encountering in their next course (second year of the integrated high school sequence). The coursework partially addresses discrepancies between students’ background knowledge in certain key topics and their supposed entry point in these topics with subsequent success. From the teacher’s point of view, these two less obvious purposes of this instructional guide present challenges. On an immediate note, the question will be raised: where do I begin my instruction for the second semester? While an order of topics may be implied by this guide, how to sequence the semester for your learners is discretionary and should be the subject of PLC discussion. An examination of the suggested resources in this instructional guide may influence your decisions. In all Common Core courses standards are not taught in isolation, but are interwoven with related standards that promote deeper conceptual understanding. The suggested resources, far from exhaustive, have been selected from an array of credible institutions and reflect this interrelatedness. At the same time, these resources (and others recalled from your own experience), aside from their content merits, are notable for embedding one or more the Standards for Mathematical Practices. A simple segue into the second semester may be a reexamination of the properties of integer exponents for deep understanding. This will advance a later examination of geometric sequences and exponential functions. On one final note, please note that standards for Number and Quantity cited below are embedded within appropriate content throughout the semester and are not to be construed as stand-alone content at any one point in time. Reason quantitatively and use units to solve problems.

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.

2. Define appropriate quantities for the purpose of descriptive modeling.

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

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Algebra Instructional Guide Overview

The fundamental purpose of Mathematics I is to formalize and extend students’ understanding of linear functions and their applications. The critical topics of study deepen and extend understanding of linear relationships, in part by contrasting them with exponential phenomena, and in part by applying linear models to data that exhibit a linear trend.

Mathematics I uses properties and theorems involving congruent figures to deepen and extend understanding of geometric knowledge from prior grades. The courses in the Integrated Pathway follow the structure begun in the K-8 standards of presenting mathematics as a coherent subject, mixing standards from various conceptual categories.

The Mathematical Practice Standards are applied throughout the course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations.

The attached Instructional Guide is a recommendation and has organized the standards according to the units outlined in Appendix A of the CCSS. The ordering of the units does not necessarily imply a particular sequence of instruction, but rather allows the “clustering” of related concepts. The standards are not topics to be checked off a list during isolated units of instruction, but rather content that should be present throughout the school year through rich instructional experiences. Possible resources are not meant to be used in isolation, nor do they fully address each standard, but merely serve as a starting point for lesson planning.

Critical Areas of Study Student Outcomes

Relationships Between Quantities By the end of eighth grade students have had a variety of experiences working with expressions and creating equations. Students continue this work by using quantities to model and analyze situations to interpret expressions, and by creating equations to describe situations.

• Reason quantitatively and use units to solve problems.

• Interpret the structure of expressions. • Create equations that describe numbers or

relationships. Linear and Exponential Functions

Students learn function notation and develop the concepts of domain and range. They move beyond viewing functions as processes that take inputs and yield outputs and start viewing functions as objects in their own right. They explore many examples of functions, including sequences; they interpret functions displayed graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. They work with functions given by graphs and tables, keeping in mind that depending upon the context, these representations are likely to be approximate and incomplete. Their work includes functions that can be described or approximated by formulas as well as those that cannot. When functions describe relationships between quantities arising from a context, students reason with the units in which those quantities are measured. Students build on and informally extend their understanding of integer exponents to consider exponential functions. They compare and contrast linear and exponential functions, distinguishing between additive and multiplicative change. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions.

• Represent and solve equations and inequalities graphically.

• Understand the concept of a function and use function notation.

• Interpret functions that arise in applications in terms of a context.

• Analyze functions using different representations. • Build a function that models a relationship between

two quantities. • Build new functions from existing functions. • Construct and compare linear and exponential

models and solve problems. • Interpret expressions for functions in terms of the

situation they model.

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Reasoning with Equations Students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. This unit builds on these earlier experiences by asking students to analyze and explain the process of solving an equation and to justify the process used in solving a system of equations. Students develop fluency in writing, interpreting and translating between various forms of linear equations and inequalities, and using them to solve problems. They master the solution of linear equations and apply related solution techniques and the laws of exponents to the creation and solution of simple exponential equations. Students explore systems of equations and inequalities, and they find and interpret their solutions. All of this work is grounded on understanding quantities and on relationships between them.

• Understand solving equations as a process of reasoning and explain the reasoning.

• Solve equations and inequalities in one variable. • Solve systems of equations.

Statistics and Probability Students experience a more formal means of assessing how a model fits data appropriateness of linear models. They represent data with plots on the real number line. They use statistics appropriate to the shape of the data distribution to compare and spread of two or more different data sets. They interpret differences in shape, center and spread in the context of the data sets, accounting for possible effects of outliers.

• Summarize, represent, and interpret data on a single count or measurement variable.

• Summarize, represent, and interpret data on two categorical and quantitative variables.

• Interpret linear models.

Congruence, Proof, and Constructions Students experience rigid motions: translations, reflections, and rotations and use these to develop notions about what it means for two objects to be congruent. Students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work.

• Experiment with transformations in the plane. • Understand congruence in terms of rigid motions. • Make geometric constructions.

Connecting Algebra and Geometry through Coordinates Building on their work with the Pythagorean Theorem in 8th grade to find distances, students use a rectangular coordinate system to verify geometric relationships, including properties of special triangles and quadrilaterals and slopes of parallel and perpendicular lines.

• Use coordinates to prove simple geometric theorems algebraically.

Websites and Books Referenced in this Document

Websites o Illustrative Math o Inside Mathematics o LearnZillion o National Science Digital Library

(NSDL) o NCTM – Illuminations o NRich Mathematics o Math Open Reference o Mathematics Vision Project

(MVP)

o Mathematics Assessment Resource

Service/ Mathematics Assessment Project (MARS/MAP)

o Math Open Reference o Shmoop o Smarter Balanced Assessment

Consortium (SBAC)

Books: (see your site for availability) o Connected Math Series © 2002 o Algebra 1, Holt o Algebra 1, McDougal Littell o Geometry, Holt

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Essential Questions What significant differences distinguish a linear model from an exponential model?

Notes For F.LE.3, limit to comparison between exponential and linear models. Limit exponential functions to those of the form ( ) Limit to simple algebraic exponents and their properties. California Common Core Integrated Math 1 Functions Linear, Quadratic, and Exponential Models F-LE Construct and compare linear, quadratic and exponential models and solve problems.

1. Distinguish between situations that can be modeled with linear functions and with exponential functions.

a. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals.

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.

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).

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.

Interpret expressions for functions in terms of the situation they model.

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

Possible Resources McDougal Littell, Algebra 1 (textbook) • Chapter 8 Exponents and Radicals (use) • Sections 8.1-8.3, 8.6*

*focus on simple equivalencies such as

(

√

√

) Holt, Algebra 1 (textbook) • Chapter 4 Functions, Section 4.5 • Chapter 7 Exponents and Polynomials, Sections 7.1 – 7.4, 7.5*

*focus on simple equivalencies such as

(

√

√

) • Chapter 11 Radical and Exponential Functions, Sections 11.6-

11.9 (omit quadratics) Connected Math Series (books) Growing, Growing, Growing • Teacher Guide • Recommended reading for teachers: p. 1a – 1l Thinking with Mathematical Models Investigation 2 Nonlinear Models p. 30 Problem 2.3 see student work in teacher guide p. 90 Investigation 3 More Nonlinear Models p. 37 3.1 Earning Interest p. 40 Problem 3.2 p. 44 Extensions #6 Investigation 4 A World of Patterns p. 55 #7 p. 58 #10

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Teacher’s Guide p. 63 #1-3 p. 69 Unit Test p. 67 Math Vision Project (web link) Module 3 – Arithmetic and Geometric Sequences Tasks 1-7- Complete List (F.LE.1) Module 4 – Linear and Exponential Functions Task 4.2 – Sorting out the Change (F.LE.1, F.LE.2) Task 4.3 – Where’s My Change (F.LE.1, F.LE.2) Task 4.4 – Linear, Exponential or Neither (F.LE.1, F.LE.2) Task 4.6 – Growing, Growing, Gone (F.LE.1, F.LE.2, F.LE.3)

These are a series of lessons that address multiple standards. MARS (web link) Comparing Investments (A.SSE.1, F.LE.1, F.LE.2, F.LE.3, F.LE.5 ) LearnZillion (video link) Lesson Set 1 (F.LE.1a) Lesson Set 2 (F.LE.1b) Lesson Set 3 (F.LE.1c) Examples: Students recognize that for successive whole number input values, and +1, a linear function , defined by ( ) = 𝑚 + , exhibits a constant rate of change: ( +1) −( ) = [𝑚( +1) + ]−(𝑚 + ) = 𝑚( +1 − ) = 𝑚. T-Table Example (F.LE.1)

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Essential Questions How can patterns be used in solving problems? In what ways are recursive and explicit rules alike? Unlike? How are linear and exponential sequences similar? Different? Notes Limit functions to linear and exponential only. F.BF.2 connects arithmetic sequences to linear functions and geometric sequences to linear functions and geometric sequences to exponential functions. Focus on vertical translations of graphs of linear and exponential functions. Relate the vertical translation of a linear function to its y-intercept.

California Common Core Integrated Math 1

Functions Building Functions F-BF Build a function that models a relationship between two quantities.

1. Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for

calculation from a context. b. Combine standard function types using arithmetic operations. For

example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Build new functions from existing functions. 3. Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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 effect on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expression for them.

Possible Resources Math Vision Project (web link) Module 3 – Arithmetic and Geometric Sequences Tasks 1-7- Complete List (F.BF.1, F.LE.1, F.LE.2,

F.LE.5) Module 4 – Linear and Exponential Functions Task 4.6 – Growing, Growing, Gone (F.BF.1, F.BF.2,

F.LE.1, F.LE.2, F.LE.3, F.IF.7) Module 7 – Connecting Algebra and Geometry Task 7.4 – Training Day (F.BF.1, F.BF.3, F.IF.9) Task 7.5 – Training Day Part II (F.BF.1, F.BF.3, F.IF.9) Task 7.6 – Shifting Functions (F.BF.1, F.BF.3, F.IF.9) LearnZillion (video link) Lesson Set 1 (F.BF.1a) Lesson Set 2 (F.BF1a) Lesson Set 3 (F.BF.1b) Lesson Set 4 (F.BF.3)

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Examples

Populations of cyanobacteria can double every 6 hours under ideal conditions, at least until the nutrients in its supporting culture are depleted. This means a population of 500 such bacteria would grow to 1000 in the first 6-hour period, 2000 in the second 6-hour period, 4000 in the third 6-hour period, etc. Evidently, if represents the number of 6-hour periods from the start, the population at that time ( ) satisfies ( ) = 2 ⋅( −1). This is a recursive formula for the sequence ( ), which gives the population at a given time period in terms of the population at time period −1. To find a closed, explicit, formula for 𝑃( ), students can reason that (0) = 500,(1) = 2 ⋅ 500,𝑃(2) = 2 ⋅ 2 ⋅ 500,𝑃(3) = 2 ⋅ 2 ⋅ 2 ⋅ 500,… A pattern emerges, that ( ) = 2 ⋅ 500. In general, if an initial population 𝑃0 grows by a factor 𝑟 > 1 over a fixed time period, then the population after time periods is given by 𝑃( ) = 𝑃0𝑟 Illustrative Math - Algae Problem (F.BF.1)

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Essential Questions Could I create various contexts from a given Equation? Conversely, could I create an equation that fits a specific context?

Notes Limit A.CED.1 and A.CED.2 to linear and exponential equations, and in the case of exponential equations, limit to situations requiring evaluation of exponential functions at integer inputs.

California Common Core Integrated Math 1 Creating Equations A-CED Create equations that describe numbers or relationships.

1. Create equations and inequalities in one variable including one with absolute value and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

2. Create equations in two or more variables to represent relationship between quantities; graph equations on coordinate axes with labels and scales.

Possible Resources

Math Vision Project (web link) Module 4 – Linear and Exponential Functions Task 4.1 – Connecting the Dots (F.IF.3) Task 4.7 – Making My Point (A.SSE.1, A.CED.2, F.LE.5) Task 4.8 – Efficiency Experts (A.SSE.1, A.SSE.3, A.CED.2, F.LE.5) Task 4.9 – Up a Little, Down a Little (A.SSE.1, A.CED.2, F.LE.5, F.IF.7)

LearnZillion (web link) Lesson Set 1 (A.CED.1) Lesson Set 2 (A.CED.2)

Examples

Suppose a certain bolt is to be mass-produced in a factory with the specification that its width should be 5mm with an error no larger than 0.01mm. If 𝑤 represents the width of a given bolt produced on the line, then we want 𝑤 to satisfy the inequality |𝑤 −5| ≤ 0.01, i.e., the difference between the actual width 𝑤 and the target width should be less than or equal to 0.01 Coffee Beans Intervals

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Essential Questions How does an expression differ from an equation? Notes Limit to linear expressions and to exponential expressions with integer exponents.

California Common Core Integrated Math 1 Algebra Seeing Structure in Expressions A-SSE Interpret the structure of expressions (exponential expressions with integer exponents)

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. For example, interpret 𝑃( 𝑟) as the product of P and a factor not depending on P.

Possible Resources Math Vision Project (web link) Module 4 – Linear and Exponential Functions Task 4.7 - Making My Point (A.SEE.1, A.CED.2, F.LE.5) Task 4.8 - Efficiency Experts (A.SSE.1, A.SSE.3, A.CED.2, F.LE.5) Tasks 4.9 - Up a Little, Down a Little (A.SSE. 1, A.CED.2, F.LE.5,

F.IF.7)

Examples

𝑝 +0.05𝑝 can be interpreted as the addition of a 5% tax to a price 𝑝. Rewriting 𝑝 + 0.05𝑝 as 1.05𝑝 shows that adding a tax is the same as multiplying the price by a constant factor Sand Truck

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Essential Questions How is technology useful in determining the solution of an equation in two variables? Notes REI.10, focus on linear and exponential equations and be able to adapt and apply that learning to other types of equations in future courses. REI.11, focus on cases where f(x) and g(x) are linear or exponential.

California Common Core Integrated Math 1 Algebra Reasoning with Equations and Inequalities A-REI Represent and sole equations and inequalities graphically.

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).

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.

Possible Resources Math Vision Project (web links) Module 5 – Features of Functions Task 5.4 - The Water Park (F.IF.2, F.IF.4, F.IF. 5, F.IF.7, A.REI.11,

A.CED.3) Task 5.5 - Pooling it Together (F.BF.1b, F.IF.2, F.IF.4, F.IF. 5, F.IF.7,

A.REI.11, A.CED.3) Task 5.6 - Interpreting Functions (F.BF.1b, F.IF.2, F.IF.4, F.IF. 5,

F.IF.7, A.REI.11, A.CED.3) Task 5.9 - Match that Function (F.IF.2, F.IF.4, F.IF. 5, F.IF.7, A.REI.11,

A.CED.3)

LearnZillion (video links) Lesson Set 9 (A.REI.10) Lesson Set 10 (A.REI.11)

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Essential Questions What characteristics describe functions? How does a function’s domain provide vital information regarding the context of a problem and its solutions? How can the same relationship be expressed by algebraic, graphic, tabular, and verbal representations? What are some examples and non-examples of a function? Notes F-IF 2 and 3, Learn as a general principle. Focus on exponential (integer domains) and on arithmetic and geometric sequences. Detailed analysis of any particular class of function at this stage is not advised. Draw example from linear and exponential functions. In F.IF.3 draw connection to F.BF.2, which requires student to write arithmetic and geometric sequences.

California Common Core Integrated Math 1 Functions Interpreting Functions F-IF Understand the concept of function and use function notation.

2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

3. Recognize that sequences are functions, sometimes defined recursively whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n-1) for n 1.).

Possible Resources

Math Vision Project (web links) Module 5 – Features of Functions Task 5.4 - The Water Park (F.IF.2, F.IF.4, F.IF. 5, F.IF.7, A.REI.11, A.CED.3) Task 5.5 - Pooling it Together (F.BF.1b, F.IF.2, F.IF.4, F.IF. 5, F.IF.7, A.REI.11,

A.CED.3) Task 5.6 - Interpreting Functions (F.BF.1b, F.IF.2, F.IF.4, F.IF. 5, F.IF.7, A.REI.11,

A.CED.3) Task 5.9 - Match that Function (F.IF.2, F.IF.4, F.IF. 5, F.IF.7, A.REI.11, A.CED.3) LearnZillion (video links) Lesson Set 2 (F.IF.2) Lesson Set 3 (F.IF.3)

Examples The equation ( ) = 2 by itself does not describe a function entirely. Similarly, though the expressions in the equations ( ) = 3 −4 and 𝑔( ) = 3 −4 look the same except for the variables used, may have as its domain all real numbers, while 𝑔 may have as its domain the natural numbers (i.e. 𝑔 defines a sequence). A sequence is a function whose inputs consist of a subset of the integers, such as: {0,1,2,3,4,5,…} Students can begin to study sequences in simple contexts, such as when calculating their total pay 𝑃38T when working for days at $6538T per day, obtaining a general expression 𝑃( ) = 65 ⋅ . Students investigate geometric sequences of the form 𝑔( ) = 𝑟 , ≥ 1, or 𝑔(1) = 𝑟, 𝑔( +1) = 𝑟 ⋅𝑔( ), for ≥ 2, when they study population growth or decay, as in the availability of a medical drug over time, or financial mathematics, such as when determining compound interest.

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Essential Questions What is the relationship between a function and its graphic representation? What are the characteristics of a graph? How do I use a graph to identify characteristics of a function and translate the information to the function notation?

Notes For F.IF.6, focus on (linear functions) intervals for exponential functions whose domain is a subset of the integers. Revisit N.RN.1 and N.RN.2 before discussing exponential models with continuous domains. For F.IF. 7e and 9 compare two functions algebraically. For example, compare the growth of two exponential functions such as .

California Common Core Integrated Math 1 Functions Interpreting Functions F-IF Interpret functions that arise in applications in terms of the context.

4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior, and periodicity.

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

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.

Analyze functions using different representations. 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 exponential functions, and logarithmic functions, showing intercepts and end behavior and trigonometric functions, showing period, midline, and amplitude.

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

Possible Resources

Math Vision Project (web links) Module 5 – Features of Functions Task 5.1 - Getting Ready for a Pool Party (F.IF. 4) Task 5.2 - Floating Down the River (F.IF. 4, F.IF. 5) Task 5.3 - Features of Functions (F.IF. 4, F.IF. 5) Task 5.4 - The Water Park (F.IF.2, F.IF.4, F.IF. 5, F.IF.7,

A.REI.11, A.CED.3) Task 5.5 - Pooling it Together (F.BF.1b, F.IF.2, F.IF.4, F.IF. 5,

F.IF.7, A.REI.11, A.CED.3) Task 5.6 - Interpreting Functions (F.BF.1b, F.IF.2, F.IF.4, F.IF.

5, F.IF.7, A.REI.11, A.CED.3) Task 5.9 - Match that Function (F.IF.2, F.IF.4, F.IF. 5, F.IF.7,

A.REI.11, A.CED.3)

MARS (web link) Functions and Everyday Situations (F.IF.4, F.IF.6, F.IF.9)

NCTM (web link) Domain Representations (F.IF.5)

Get the Math (web links) Math in Special Effects (A.SSE.1, A.CED.2, A.REI.10, F.IF.1,2,4,5,6,7, F.BF.1) Math in Basketball (A.SSE.1, A. CED.1, A.CED.2, A.REI.10, F.IF.1,2,4,5,6,7, F.BF.1)

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Rates – Annenberg Learner (F.IF.7, F.IF.9) Exponential Graphing using Technology - CPalms (F.IF.1, 4,5,7) Functions - NOYCE (F.IF.7, F.LE.3, A.CED.2)

Interactive Manipulatives (web links) Slope Slider Graphing Lines LearnZillion (video links) Lesson Set 4 (F.IF.5) Lesson Set 5 (F.IF.7a) Lesson Set 6 (F.IF.7a)

Examples Make the connection between the graph of the equation = ( )and the function itself—namely, that the coordinates of any point on the graph represent an input and output, expressed as ( , ( ))—and understand that the graph is a representation of a function Note that there is neither an exploration of the notion of relation vs. function nor the vertical line test in the CA CCSSM. This is by design.

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Essential Questions How do geometric constructions help me understand properties of plane figures? How can I describe the process of transforming one figure onto another? How can I define and differentiate among transformations (translations, rotations, reflections) in terms of essential concepts? Notes Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. Rigid motions are at the foundation of the definition of congruence. Students reason from the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof. Rigid motions and their assumed properties can be used to establish the usual triangle congruence criteria, which can then be used to prove other theorems. *All of the standards listed below are from the CA Framework, but have live links that are embedded throughout this section. Please use them as necessary.

California Common Core Integrated Math 1 Geometry Congruence G-CO Experiment with transformations in the plane.

1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.

See also: distance formula, equidistant points, collinear points, rays, opposite rays, vertex ,intersecting lines, horizontal lines, vertical lines, perpendicular bisector.

2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch.).

3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines,parallel lines, and line segments.

5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry

Possible Resources

Math Vision Project (web links) Module 6 – Congruence. Constructions and Proof Task 6.1 - Leaping Lizards! (G.CO.1, G.CO.4, G.CO.5) Task 6.2- Is It Right? (G.CO.1, G.GPE.5) Task 6.3 - Leap Frog (G.CO.4, G.CO.5) Task 6.4 - Leap Year (G.CO.1, G.CO.2, G.CO.4,

G.GPE.5) Task 6.5 - Symmetries of Quadrilaterals (G.CO.3,

G.CO.6) Task 6.6 - Symmetries of Regular Polygons (G.CO.3,

G.CO.6) Task 6.7 - Quadrilaterals‐Beyond Definition (G.CO.3,

G.CO.4, G.CO.6) Task 6.8 - Can You Get There From Here? (G.CO.5) Task 6.9 - Congruent Triangles (G.CO.6, G.CO.7,

G.CO.8) Connected Math Series (book) Kaleidoscopes, Hubcaps and Mirrors

Algebra Instructional Guide Semester 2 (Draft 12-18-13) Updates may be available as new resources become available. Page 15

software. Specify a sequence of transformations that will carry a given figure onto another.

Understand congruence in terms of rigid motions. 6. Use geometric descriptions of rigid motions to transform figures and to

predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

8. Explain how the criteria for triangle congruence ( ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Make geometric constructions. 12. Make formal geometric constructions with a variety of tools and methods

(compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

MARS (web link) Representing and Combining Transformations (8.G) Transforming 2D Figures (G.CO) LearnZillion (video link) Lesson Set 3 (G.CO.5) Math Open Reference (interactive web links) List of Links for Constructions

Examples

Two shapes are congruent if there is a sequence of rigid motions in the plane that takes one shape exactly onto the other. Defining Rotations

Algebra Instructional Guide Semester 2 (Draft 12-18-13) Updates may be available as new resources become available. Page 16

Essential Questions In what way is congruence revealed through transformation and rigid motions? How can I effectively prove theorems using coordinates? Notes Limit to reasoning about right triangles; e.g., derive the equation for a line through two points using similar right triangles. Relate work on parallel lines in G.GPE.5 to work on A.REI.5 involving systems of equations having no solution or infinitely many solutions. GPE.7 provides practice with the distance formula and its connection with the Pythagorean theorem.

California Common Core Integrated Math 1 Geometry Expressing Geometric Properties with Equations G-GPE Use coordinates to prove simple geometric theorems algebraically.

4. Use coordinates to prove simple geometric theorems algebraically. 5. Prove the slope criteria for parallel and perpendicular lines and use them

to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

7. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Possible Resources Math Vision Project (web link) Module 7 –Connecting Algebra and Geometry Task 7.1 – Go the Distance (G.GPE.7) Task 7.2 – Slippery Slopes (G.GPE.5) Task 7.3 – Prove It! (G.GPE.4)

Examples

Proving a figure defined by four points is a rectangle by proving that lines containing opposite sides of the figure are parallel and lines containing adjacent sides are perpendicular. Students must be fluent in finding slopes and equations of lines (where necessary) and understand the relationships between the slopes of parallel and perpendicular lines in order to do so. Students must be fluent in finding slopes and equations of lines (where necessary) and understand the relationships between the slopes of parallel and perpendicular lines in order to do so. Use relationships between slopes of parallel and perpendicular lines to solve problems, but they justify why they work. Proof

Algebra Instructional Guide Semester 2 (Draft 12-18-13) Updates may be available as new resources become available. Page 17

Essential Questions How do data influence real-world decisions? Do various displays allow us to interpret the same data in different ways?

Notes

California Common Core Integrated Math 1 Statistics and Probability Interpreting Categorical and Quantitative Data S-ID Summarize, represent and interpret data on a single count or measurement variable.

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

2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets.

3. Interpret differences in shape, center and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Possible Resources

Math Vision Project (web links) Module 8 – Modeling Data Task 8.1 – Texting by the Numbers (S.ID.1, S.ID.3) Task 8.2 – Data Distribution (S.ID.1, S.ID.3) Connected Math (book) Samples and Populations LearnZillion (video links) Lesson Set 1 (S.ID.1) Lesson Set 2 (S.ID.2) Lesson Set 3 (S.ID.3)

Examples Comparing Height Data

Appendix of Examples

Algebra Instructional Guide Semester 2 (Draft 12-18-13) Appendix Page 1

Linear and Exponential Models (cont. from pg 4)

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Appendix of Examples

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Functions (cont. from pg 6)

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Appendix of Examples

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Creating Equations (cont. from pg. 4)

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Seeing Structure in Expressions (cont. from pg. 5)

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Appendix of Examples

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Congruence (cont. from pg 13)

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Appendix of Examples

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Expressing Geometric Properties with Equations (cont. from pg )

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Appendix of Examples

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Statistics and Probability (cont. from pg. )

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Appendix of Examples

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Websites for Reference Illustrative Math Inside Mathematics LearnZillion National Science Digital Library (NSDL) NCTM – Illuminations NRich Mathematics Math Open Reference Mathematics Vision Project (MVP) Mathematics Assessment Resource Service/ Mathematics Assessment Project (MARS/MAP) Shmoop Smarter Balanced Assessment Consortium (SBAC)