Mathematics Curriculum Guide Algebra

Math_Curriculum Guide Algebra_07.28.11_v1 Page | 1 This Curriculum Guide is to be used in conjunction with the Pacing Guide and Instructional Design Lesson Plan. The Instructional Design will help with lesson planning using Launch-Explore-Summarize-Apply.

Mathematics Curriculum Guide Algebra

MILWAUKEE

PUBLIC SCHOOLS


Mathematics – High School Algebra Course: Introduction

The fundamental purpose of this course is to formalize and extend the mathematics that students learned in the middle grades. Because it is built on the middle grades standards, this is a more ambitious version of Algebra than has generally been offered. The critical areas deepen and extend understanding of linear and exponential relationships by contrasting them with each other and by applying linear models to data that exhibit a linear trend, and students engage in methods for analyzing, solving, and using quadratic functions. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. Critical Area 1: By the end of eighth grade, students have learned to solve linear equations in one variable and have applied graphical and algebraic methods to analyze and solve systems of linear equations in two variables. Now, students analyze and explain the process of solving an equation. Students develop fluency writing, interpreting, and translating between various forms of linear equations and inequalities, and use 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. Critical Area 2: In earlier grades, students define, evaluate, and compare functions, and use them to model relationships between quantities. In this area, students will learn function notation and develop the concepts of domain and range. They explore many examples of functions, including sequences; they interpret functions given graphically, numerically, symbolically, and verbally, translate between representations, and understand the limitations of various representations. 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. Students explore systems of equations and inequalities, and they find and interpret their solutions. They interpret arithmetic sequences as linear functions and geometric sequences as exponential functions. Critical Area 3: This area builds upon prior students’ prior experiences with data, providing students with more formal means of assessing how a model fits data. Students use regression techniques to describe approximately linear relationships between quantities. They use graphical representations and knowledge of the context to make judgments about the appropriateness of linear models. With linear models, they look at residuals to analyze the goodness of fit. Critical Area 4: In this area, students build on their knowledge from critical area 2, where they extended the laws of exponents to rational exponents. Students apply this new understanding of number and strengthen their ability to see structure in and create quadratic and exponential expressions. They create and solve equations, inequalities, and systems of equations involving quadratic expressions. Critical Area 5: In this area, students consider quadratic functions, comparing the key characteristics of quadratic functions to those of linear and exponential functions. They select from among these functions to model phenomena. Students learn to anticipate the graph of a quadratic function by interpreting various

Standards for Mathematical Practice

1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning


forms of quadratic expressions. In particular, they identify the real solutions of a quadratic equation as the zeros of a related quadratic function. Students expand their experience with functions to include more specialized functions – absolute value, step, and those that are piecewise-defined. The assessments and strategies for learning listed on this page are not meant to be an “end all”. Additionally, these are repeated for every grade level curriculum guide with the intent that our students experience some of the same strategies year to year, teacher to teacher. On the following pages, you will find specific usage for some of these.


NOTE: The misconceptions listed within the template are only samples. You will encounter additional misconceptions as you work with students.

Classroom Assessment For Learning Literacy Connections Assessment for learning provides students with insight to improve achievement and helps teachers diagnose and respond to student needs. (Stiggins, Rick, Judith Arter, Jan Chappuis, Steve Chappuis. Classroom Assessment for Student Learning. Educational Testing Service, 2006.) Formative Assessment tools include: Constructed response items Descriptive Feedback (oral/written) Effective use of questioning Exit slips Milwaukee Math Partnership (MMP) CABS Observational checklists and anecdotal notes Portfolio items Projects (PBL) Student journals Student self-assessment Students analyze strong and weak work samples Use of Learning Intentions Use of rubrics with students Use of talk formats and talk moves, p. 55 of the CMSP Use of Success Criteria Summative Assessment: Unit Tests WKCE/WAA Progress Monitoring Tool Measure of Academic Progress (MAP)

Comprehensive literacy is the ability to use reading, writing, speaking, listening, viewing and technological skills and strategies to access and communicate information effectively inside and outside of the classroom and across content areas. (CLP, p.11) Literacy strategies can help students learn mathematics: Concept mapping Graphic organizers Journaling K-W-L Literature RAFT Reciprocal teaching Talk moves Think Aloud strategy Think, Pair, Share strategy Three-minute pause Two column note taking Vocabulary strategies (e.g. Marzano’s 6 steps; Frayer model)


Mathematics – High School Algebra Course

Modeling

Modeling Standards Modeling is best interpreted in relation to other standards, not as a collection of isolated topics. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards indicated by a star symbol (*).

The basic modeling cycle is summarized in the diagram. It involves (1) identifying variables in the situation and selecting those that represent essential features, (2) formulating a model by creating and selecting geometric, graphical, tabular, algebraic, or statistical representations that describe relationships between the variables, (3) analyzing and performing operations on the relationships to draw conclusions, (4) interpreting the results of the mathematics in terms of the original situation, (5) validating the conclusions by comparing them with the situations, and then either improving the model or, it is acceptable, (6) reporting on the conclusions and the reasoning behind them. Choices, assumptions, and approximations are present throughout this cycle.





Number and Quantity The Real Number System

N-RN Quantities*

N-Q Extend the properties of exponents to rational exponents 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. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1.3)3 to hold, so (51/3)3 must equal 5. 2. Rewrite expressions involving radicals and rational exponents using the properties of exponents Use properties of rational and irrational numbers 3. Explain why the sum of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

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.





Number and Quantity Essential/Enduring Understandings Assessments A real-world quantity is described as using a number and an appropriate

unit. Expanding understanding of real numbers as rational numbers together

with irrational numbers and using them to make sense of real-world and abstract situations

Have students do a Quick Write to answer the following prompt: “What makes some ways of representing a relationship better or more appropriate than others?”

Have students do a Think-Pair-Share to describe their thinking on the following prompt: “Describe a function that relates the area of a square to the length of its side.”

CABS (Classroom Assessment Based on Standards)

Fractals and Fractions Toothpick Hexagons Jacki’s Homework

CR (Constructed Response)

Free Throws

Common Misconceptions/Challenges A real-world quantity is described as using a number and an appropriate unit.

Students have difficulty in understanding the attributes of measurement. One way to surface this misconception is to ask students to match physical objects to appropriate units. For example, would a human be measured in feet, kilometers, meters, kilograms, liters, or inches? In order to help students with this misconception have students estimate different attributes of an object stating both the value and an appropriate unit. This would be an on-going practice.

Expanding understanding of real numbers as rational numbers together with irrational numbers and using them to make sense of real-world and abstract situations.

Students have difficulty in understanding exact vs. approximate values of irrational numbers. One way to surface this misconception is to have students estimate and graph rational and irrational numbers on a number line. Listen for students explanations of where they would graph an irrational number on the number line. Challenge the students’ approximate number (for example √2) by using a calculator to see the result.


Instructional Practices N-RN: The Real Number System Have students apply the properties of exponents to include fractional exponents by utilizing a variety of expressions. N-Q: Quantities* Have students calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the context of the problem, as well

as unit.

Example: Jeremy has a job at the movie theater. His hourly wage is $7.38. Suppose 15% of his income is withheld for taxes and Social Security. What is his hourly take-home wage? While the numerical answer is 6.273, students need to see this as a money situation where only two decimal places make since in terms of realistic hourly wage.

Create opportunities for students to explore numbers, both geometrically and algebraically.

Example: Suppose the original large triangle has an area of 24 square units. Have students explain the shaded area at least two different ways. See diagram below.

Differentiation


A real-world quantity is described as using a number and an appropriate unit. When asking students to estimate, allow students to use physical models to experience benchmark measurements. For example, use a dollar bill to have a

visual reference of approximately six inches, or when working with liquid measurements have a 20 ounce bottle and a 12 ounce can. Expanding understanding of real numbers as rational numbers together with irrational numbers and using them to make sense of real-world and abstract situations. Allow students to use calculators to explore rational and irrational numbers. For example, ask students to find the √2 with out using the square root key on

a calculator.

Literacy Connections Academic Vocabulary Students address the writing standard by the Quick Write. Students address the speaking and listening standards by the Think-Pair-

Share.

Exponent Irrational Radical Rational Roots Square Roots

Resources Use square tiles to build patterns Use graphing calculators to build functions to represent real-world problems Use calculators to explore irrational numbers Keypress.com (log in is required)



Algebra Seeing Structure in Expressions

A-SSE

Arithmetic with polynomials

A-APR

Creating Equations*

A-CED

Reasoning with Equations and Inequalities

A-REI

Interpret the structure of expressions 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(1+r)n as the product of P and a factor not depending on P. 2. Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4

Perform arithmetic operations on polynomials 1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the opertions of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Create equations that describe numbers or relationships 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a

Understand solving equations as a process of reasoning and explain the reasoning 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. 2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. Solve equations and inequalities in one variable 3. Solve linear equations and inequalities in one variable, including




as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2). Write expressions in equivalent forms to solve problems 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. b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approxmate equivalent monthly interest rate if the annual rate is 15%.

modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. 4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

equations with coefficients represented by letters. 4. Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. Solve systems of equations 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. 6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.


7. Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x2 + y2 =3 Represent and solve 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.* 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.



Algebra Essential/Enduring Understandings Assessments Solving equations is a process of reasoning and explaining the reasoning

Certain operations will produce a simpler equivalent equation

Reason, communicate, and compute with numeric and algebraic

expressions and equations

Students will be able to describe a real world situation through multiple representations: Tables Graphs Symbols Written and verbal explanations Models

Students will be able to connect the work done in number and

operations with algebraic properties through prompts like “Toothpick Hexagons”

CABS Umbrella Equations

CR

The Scale Experiment Paxolai

Common Misconceptions/Challenges Solving equations is a process of reasoning and explaining the reasoning Certain operations will produce a simpler equivalent equation Students have a difficult time understanding the meaning of the equal sign between two expressions and how they can use algebraic properties to create

equivalent expressions. For example, ask students to complete this equation: 7 + 5 = __ + 8. Continually ask students to explain why they are making their next step and why the operation produces an equivalent equation.

Reason, communicate, and compute with numeric and algebraic expressions and equations Students are challenged when needing to see through the symbols to the underlying quantities they represent through the context of the problem. To

address this challenge, encourage students to pause as needed during a manipulation process in order to probe into the referents for the symbols involved.


Instructional Practices A-CED: Creating Equations When solving equations, the student is involved in creating expressions and equations from context, this includes students making critical decisions about

the variables.

Example: The local bagel store sells a baker’s dozen of bagels for $6.49, while the grocery store down the street sells a bag of 6 bagels for $2.50. Allow students to make sense of the context and create an algebraic expression for each store. Have students explain the terms in the context of the problem.

A-REI: Reasoning with Equations and Inequalities When students explore equations, students need to interpret and understand the different forms of equations that can represent the same pattern.

Students need to justify this using algebraic relationships or the context of the situation, to show the relationships among the various forms of the equations.

Example: When given a situation of a line that passes through the points (-2, -1) and (5, 13). Allow students to explore these two equations in point-slope, y = -1 + 2(x + 2) and y = 13 + 2(x – 5). Students should be able to explore the equivalence of these equations by graphing, tables, or symbolic manipulations. It is critical to allow students to make their own decisions on how to explore the equivalence of the equations.

A-SSE: Seeing Structure in Expressions Connect symbolic manipulation with the laws of arithmetic.

Example: Many students learn the area model for multiplication. This model can be used when exploring the distributive property (see diagram).

Differentiation Solving equations is a process of reasoning and explaining the reasoning Have students solve linear equations and inequalities numerically, graphically, including the use of appropriate technology, and symbolically to see

multiple representations of equivalent equations.


Certain operations will produce a simpler equivalent equation Allow students to explore equivalent equations in a table by using different input values in the expressions to see the equivalency.

Reason, communicate, and compute with numeric and algebraic expressions and equations When asking students reason with algebraic expressions and equations in a context, surface students’ different back ground knowledge before asking

them to explain. Literacy Connections Academic Vocabulary Have students create multiple and visual representations through

algebraic models Use Think Aloud strategies to help students make sense of the context.

Associative Property Additive Property Commutative Property Distributive Property Equations Expressions Inequalities Integers Inverse Property Patterns Polynomials Variables

Resources Graphing Calculators to represent algebraic models Use manipulative for students to see the visual algebraic models Keypress.com (log in is required)



Functions Interpreting Functions

F-IF Building Functions

F-BF Linear, Quadratic, and Exponential Models*

F-LE Understand the concept of a function and use

function notation 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).

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) +

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

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




f(n – 1) for n>1. 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. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function.*

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.*

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. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

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.


a. Graph linear and quadratic functions and show intercepts, maxima, and minima.

b. Graph square root, cube root, and piecewise-

defined functions, including step functions and absolute value functions.

c. Graph polynomial functions, identifying zeros

when suitable factorizations are available, and showing end behavior.

d. (+) Graph rational functions, identifying zeros

and asymptotes when suitable factorizations are available, and showing end behavior.

e. Graph exponential and logarithmic functions,

showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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 functions. For example, identify percent rate of change in functions such as y = (1.02)2, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as


representing exponential growth or decay. 9. Compare properties of two functions each

represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.



Functions Essential/Enduring Understandings On-going Assessments Interpret expressions for functions in terms of the situation they model

Interpret and create algebraic models to make predictions and evaluate

decisions

Functions can be represented in multiple ways, including algebraic, graphical, verbal, and tabular representations

Given data from a problem situation, students should be able to: Choose a family of functions that sensibly models the situation Find parameters for a function to approximately match the data Use the resulting function to solve the problem Interpret and reflect on their solution in the context of the

problem CABS

Toothpick Hexagons Unidel Mary’s Dots Sally’s Doorbell

CR Canned Hams

Common Misconceptions/Challenges Interpret expressions for functions in terms of the situation they model Students have a difficult time understanding what the word problem means. When asking students to create an expression from a word problem, engage

students in the Think Aloud strategy to tell and retell situations in the word problem before students write an equation.

Interpret and create algebraic models to make predictions and evaluate decisions Students will often see relationships in terms of a linear function. For example when students analyze collected data, they only see a linear relationship.

Have students create a scatter plot of the collected data prior to writing an equation that models the situation.

Functions can be represented in multiple ways, including algebraic, graphical, verbal, and tabular representations Students are challenged in their fluency with multiple representations. For example, students may be able to graph from a table, but do not connect that

to making a table from a graph (or doesn’t necessarily mean they can make a table from a graph). To address this challenge, create questions and opportunities for students to be able to connect a table to a graph and a graph to a table.

Instructional Practices F-BF: Building Functions


Students should experience a variety of types of situations modeled by functions. Draw examples from linear and exponential functions and make the connection to recursive sequences.

Example: Each of the 30 people at a party shook hands with everyone else. How many handshakes were there altogether? Have students collect the data by modeling the situation and create a table to find a pattern. Have students develop a function rule from the situation to see how many handshakes were at the party.

F-LE: Linear, Quadratic, and Exponential Models* Students should explore a variety of situations that can be modeled with linear and exponential functions.

Example: Grace manages a local charity. A wealthy benefactor has offered two options for making a donation over the next year. One option is to give $50 now and $25 each month after that. The second option is to give $1 now and twice that amount next moth; each month afterward the benefactor would give twice the amount given the month before. Have students create a table, graph, and an equation that models both situations. Ask students to decide what option would be best and justify their reasoning using the mathematics they’ve developed.

F-IF: Interpreting Functions Emphasis on the connection of functions to real world phenomena

Example: Have students sketch a graph that reflects all the information given in this story. It was a dark and stormy night. Before the torrents of rain came, the bucket was empty. The rain subsided at daybreak. The bucket remained untouched through the morning until Old Dog Trey arrived as thirsty as a dog. The sun shone brightly through the afternoon. Then Billy, the kid next door, arrived. He noticed two plugs in the side of the bucket. One of them was about a quarter of the way up, and the second one was near the bottom. As fast as you could blink an eye, he pulled out the plugs and ran away.

Differentiation Interpret expressions for functions in terms of the situation they model When given a contextual situation, have students use literacy strategies, such as highlighting/circling key mathematical words, that will help students sort

the important information in the problem to see the connections among the situation, table, graph, and equation.

Interpret and create algebraic models to make predictions and evaluate decisions Have students use a graphing calculator to be able to change the window, table frequency, and equation so that students can quickly recognize the

patterns in the information to make a better mathematical decision. Functions can be represented in multiple ways, including algebraic, graphical, verbal, and tabular representations Using a variety of representations can help make functions more understandable to a wider range of students than can be accomplished by working with


symbolic representations alone.

Example: Students need to understand numbers in their raw form. Otherwise, they will not have the flexibility to present them accurately in other arrangements. Further, some students are adept at interpreting numbers and noticing patterns among them, and for these students it is important that they have the opportunity to view numbers in their most basic state. With regard to the Rule of Four, using a numerical representation refers to displaying data in an organized fashion, possibly as an ordered list or in a table. The table below, for instance, could be used to verify that x2 + 6x + 8 = (x + 2)(x + 4) for every value of x, the values of x2 + 6x + 8 and (x + 2)(x + 4) are equal.

x (x + 2) (x + 4) (x + 2)(x + 4) x2 + 6x + 8

-8 -6 -4 24 24

-7 -5 -3 15 15

-6 -4 -2 8 8

-5 -3 -1 3 3

-4 -2 0 0 0

-3 -1 1 -1 -1

-2 0 2 0 0

-1 1 3 3 3

0 2 4 8 8

1 3 5 15 15

2 4 6 24 24


Literacy Connections Academic Vocabulary Students will have to write their relationships from the real-world

phenomena that occurs in a function Students can highlight key terms from a contextual situation

Continuous Discrete Domain Exponential Model Family of Functions Linear Models Rate of Change Slope Table of Values Quadratic Models Function Ordered Pair Parameters Range

Resources Graphing Calculators to make connections between multiple representations of functions Keypress.com (log in is required)



Geometry Congruence

G-CO Similarity, Right Triangles, and Trigonometry

G-SRT Expressing Geometric Properties with Equations

G-GPE Experiment with transformations in the plane 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). 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Define trigonometric ratios and solve problems involving right triangles 6. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. 8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

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. *





Geometry Essential/Enduring Understandings Assessments Both quantitative and qualitative attributes of a geometric object can

impact measurements associated with that object.

The transformations that are fundamental to determining the geometric relationships between shapes apply equally to algebraic situations

Create an image and perform all transformations and have students explain the meaning and relationship of the transformations to the original image

CABS My Initials

CR

Triangle PBJ

Common Misconceptions/Challenges Both quantitative and qualitative attributes of a geometric object can impact measurements associated with that object. Students may know the formulas, but don’t know how to derive and connect the formulas to geometric properties. For example, students have a difficult

time remembering and applying the slope formula when given two coordinates. To address this, have students use a slope triangle to develop the formula for slope and refer back to the slope triangle when students are challenged. Continue to ask questions, “does your slope make sense in the context of the problem?”

The transformations that are fundamental to determining the geometric relationships between shapes apply equally to algebraic situations Students often don’t understand the relationship between the parent function and the transformation. For example, students get confused by the rules of

transformation and are too quick to find the rule instead of understanding and visualizing how the transformation moves on a coordinate grid. To address this, have students explore transformations on graph paper, patty paper, and geometric software to see the relationship between the parent function and the transformation before giving them the rules.

Instructional Practices G-GPE: Expressing Geometric Properties with Equations Have students articulate the use of coordinate geometry to justify geometric properties as an important merging of geometry and algebra.

Example: Have students explore the property of slope by using the slope triangle on the coordinate plane. Remind students the direction of the line will indicate a positive vs. a negative slope. It is important to include various examples where the line passes through different quadrants. See diagram below for the slope triangle explanation.


G-CO: Congruence Have students use visual and spatial reasoning, and geometric modeling to solve problems involving transformations.

Example: Consider the graph of y = x2 and its image after a horizontal stretch by a factor of 2. Write an equation for the image. Describe the image in terms of a vertical stretch or shrink. Write an equation that shows this transformation. Is this equation equivalent to the one that shows a horizontal

stretch? When you vertically stretch or shrink the graph of y = f(x) by a factor of a, you get a graph of y = a • f(x). If you horizontally stretch or shrink the graph of y = f(x) by a factor of b, you will get the graph of what equation?


G-SRT: Similarity, Right Triangles, and Trigonometry Have students represent geometric situations algebraically and algebraic situations geometrically and use them to make connections in solving problems.

Example: A baseball diamond is a square with 90 ft between first and second base. What is the distance from home plate to second base? The distance between home plane and second base is the hypotenuse of a right triangle. Call it c. This means that the area of a square on c equals the sum of the areas of the squares on each leg. The legs a and b are equal in this case. See diagram.

Differentiation Both quantitative and qualitative attributes of a geometric object can impact measurements associated with that object. To help students create and find the values of the slope triangle, use Geoboards or Geometric Software for students to construct the slope triangle, and

find the slope of the hypotenuse. The transformations that are fundamental to determining the geometric relationships between shapes apply equally to algebraic situations To help students experience transformation, have students physically model various transformations. For example, one student would physically

represent the parent function while other students physically move around the coordinates to represent a transformation of the parent function.


Literacy Connections Academic Vocabulary Students will create a graphic organizer to compare and contrast:

Slope of parallel and perpendicular lines Area and volume Various transformations

Area Dilation Reflection Rotation Parent Function Transformation Translation Volume

Resources Geometer’s Sketchpad to explore the effects of transformations on a particular image Keypress.com (log in is required)



Statistics and Probability

Interpreting Categorical and Quantitative Data S-ID

Conditional Probability and the Rules of Probability S-CP

Summarize, represent, and interpret data on a single count or measurement variable 1. Represent data with plots on the real number line (dot 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). Summarize, represent, and interpret data on two categorical and quantitative variables 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 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Understand independence and conditional probability and use them to interpret data 1. Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”). 2. Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. 3. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. 5. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of being unemployed if you are female with the chance of being female if you are unemployed.




a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. c. Fit a linear function for a scatter plot that suggests a linear association. Interpret linear models 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. 9. Distinguish between correlation and causation.

Use the rules of probability to compute probabilities of compound events in a uniform probability model 6. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.



Statistics and Probability Essential/Enduring Understandings Assessments Organize, display, and compare both quantitative and qualitative data to

analyze situations and predict outcomes

Probability whether obtained through theoretical analysis or experimentation, are useful for predicting what should happen over the long run

Create an authentic project originating from a research question Students can articulate a question, a plan to address the question,

a process to collect the data, and communicate a summary of the data that defends an answer to the question

CABS

Bob’s Blocks

Common Misconceptions/Challenges Organize, display, and compare both quantitative and qualitative data to analyze situations and predict outcomes Students have difficulties choosing appropriate variables and graphs to represent the relationship to the data and the story behind the data. For example,

after students collect data from an experiment, they are challenged with finding the best graph that represents the collected data (i.e. scatter plot, histogram, bar graph, number line, box plot, circle graph, etc). To address this, have students explore news papers with various data and graphs to generate conversation as to why the graph was chosen for that particular set of data.

Probability whether obtained through theoretical analysis or experimentation, are useful for predicting what should happen over the long run Students have a difficult time understanding random and the process that creates a random sample through probability. For example, when students are

asked to conduct a survey, they don’t know who to ask that would create a random sample. To address this, use the random number generator on the calculator to have students explore the idea of randomness. Ask students if any of the numbers repeat, what were the parameters placed on the numbers, and how would you apply the idea of the random number generator to conduct a survey?

Instructional Practices S-ID: Interpreting Categorical and Quantitative Data By incorporating randomness into data collection, probability provides a way to make sense of the variation in sample results from one sample to another.

The sampling distribution of a sample statistic summarizes the long-run behavior of the statistic from repeated random sampling. The sampling distribution provides a mechanism for describing the variation expected in a sample statistic and for determining whether an observed value might be reasonable from chance variation or whether it is more likely due to some other factor.

Example: Students should discuss outliers in relation to the data as to why they may have occurred. There may have been an error in collecting data or other factors.


S-CP: Conditional Probability and the Rules of Probability Fostering students’ understanding of random variables and probability requires engaging them in the development of both simulation and mathematical

probability models. The design of a simulation model requires that students set up a logical sequence of steps for describing the possible outcomes of a random variable. The implementation of a simulation model allows students to both experience and make sense of the notion of long-run behavior of a random variable and to determine experimental probabilities.

Example: If you tossed a coin many times, you would expect the ratio of the number of heads to the number of tosses to be close to ½. The more times you toss the coin, the closer the ratio of heads to total tosses will be to ½. With random events, patterns often emerge in the long rum, but these patterns do not help predict a particular outcome. When flipping a coin, you know what the theoretical probabilities are for heads and tails. However, in some situations you cannot calculate the theoretical probability of an outcome. After performing many trials, you can determine an experimental probability based on your experimental results.

Differentiation Organize, display, and compare both quantitative and qualitative data to analyze situations and predict outcomes To help students who are struggling with creating the authentic project in the assessment, provide them with more specific questions to help them connect

the process of the data collection to the conclusion. Continue to connect the data, graphs, and conclusions to real-world examples from the news paper.

Probability whether obtained through theoretical analysis or experimentation, are useful for predicting what should happen over the long run To help students understand how probabilities are determined, have them create and use tables and counting trees to illustrate the number of outcomes in

a random event.

Literacy Connections Academic Vocabulary In the authentic project students will need to read, write, speak, and listen Use Marzano’s steps to help build students academic vocabulary

Box Plots Conditional Probability Experimental Probability Frequency Tables Histograms Mean Median Mode Population Probability Random


Random Sample Sample Scatter plots Theoretical Probability Variable

Resources The activity “Sampling Rectangles” in Navigating through Data Analysis in Grades 9-12 (Burill et al 2003) illustrates important ideas related to the sampling

distribution of the sample mean. Graphing Calculator to create random number generator, graphs, and tables Manipulative to model probability experiments Keypress.com (log in is required)

Mathematics Curriculum Guide Algebra

Documents

Transcript of Mathematics Curriculum Guide Algebra