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Gloucester Township Public SchoolsMath Curriculum

8th Grade Algebra 1

OverviewMathematics is a universal language enmeshed in both the everyday experiences of human society and the natural world

around us. The Gloucester Township Public School District recognizes that mathematics is a fluid and intricately connected web of conceptual understandings, as opposed to segmented isolated skills and arbitrary units of study.

A nation that trains and prepares students to become mathematically literate problem solvers is an entity that sends citizens into the workforce ready to compete in a global economy laden with technology and problem solving opportunities. A school district that intends to have an accomplished field of mathematicians, engineers, medical professionals, scientists, and innovative entrepreneurs must plan and prepare standards-based curriculum that adheres to the Common Core Standards, includes 21st Century technology skills, and explores the variety of careers steeped in mathematics.

In consideration of the rigor and depth of mastery needed by students in our Nation's public school system, we have constructed the following curriculum guide and supporting documentation for Gloucester Township Public Schools through adoption of the New Jersey Department of Education Model Curriculum for Mathematics. Every student in our schools shall have the opportunity to become engaged in an enriching, real world approach to mathematics instruction that is based on solid educational research and data-driven instruction.

Benchmark and Cross Curricular Key

__Red: ELA

__ Blue: Math

__ Green: Science

__ Orange: Social Studies

__ Purple: Related Arts

__ Yellow: Benchmark Assessment

2

Math – Algebra 1Unit 1 – Relationships between Quantities and Reasoning with Equations

Standards Topics Activities Resources AssessmentsN.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

Conversions STEM ProjectsUnit Projects

Geometer’s SketchpadReal-World Math

Throughout text Benchmark Exams 1-4

Chapter Test: 2A, 2B, 2C, or 2D

All Chapter QuizzesN.Q.2 Define appropriate quantities for the purpose of descriptive modeling 8.G.2

Units of Measurement Extend 2.6 Benchmark Exams 1-4


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

PrecisionExtend 1.3 Benchmark Exams 1-4


All Chapter QuizzesA.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.

*Adding and Subtracting Polynomials

1.11.4

Benchmark Exams 1-4


All Chapter QuizzesA.SSE.1 Interpret expressions that represent a quantity in terms of its context.*b. Interpret complicated expressions by viewing one or more of their parts as single entity. For example, interpret P(1 + r)n as the product of P and a factor not depending on P.*A.SSE.1: Focus on linear, quadratic, and an introduction to exponential expressions.

*Dividing Monomials 1.21.39.7

Benchmark Exams 1-4


All Chapter Quizzes

A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear functions.**A.CED.1: Limit to linear or quadratic

*Using Equations to Solve Problems*Problem Solving Using Charts*Cost, Income, and Value Problems

1.5, 2.1, 2.2, 2.3, 2.4, 2.5, 2.9, 3.2, 5.1, 5.2, 5.3, 5.4, 5.5, 7.6, 8.5, 8.6, 8.7, 9.4, 9.5, 10.4, 11.8

Benchmark Exams 1-4


3

equations. *Rate-Time-Distance Problems*Area Problem

All Chapter Quizzes

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

*The Graphing Method

*Problem Solving with Systems of Equations

Extend 1.7 3.1, 3.4, 3.5, 3.6, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 6.1, 6.2, 6.3, 6.4, 6.5, 7.5, 7.5, 8.6, 8.7, 8.8, 9.1, 9.2, 9.4, 9.5, 10.1, 10.4, 11.2, 11.8

Benchmark Exams 1-4


All Chapter QuizzesA.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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

*Problems Without Solutions

*Solving Problems Involving Inequalities

*Inequalities in Two Variables

*Systems of Linear equations

*Linear Program

4.25.66.16.2

Benchmark Exams 1-4


All Chapter Quizzes

A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s laws V= IR to highlight resistance R.***A.CED.4: Exclude cases that require extraction of roots or inverse functions.

*Transforming Formulas 2.84.1

Benchmark Exams 1-4


All Chapter Quizzes

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.

*Transforming Equations: Addition and Subtraction*Multiplication and Division*Using Several Transformations*Proof in Algebra

1.5, 2.2, 2.3, 2.4, 2.5, 2.6, 2.9, 8.6, 8.7, 8.9

Benchmark Exams 1-4


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

*Transforming Equations: Addition and Subtraction

*Multiplication and Division

*Using SeveralTransformations

*Solving Inequalities

*Solving Problems Involving

Explore 2.2Explore 2.3Explore 5.2

1.5, 2.2, 2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 5.1, 5.2, 5.3, 5.4, 5.5

Benchmark Exams 1-4


All Chapter Quizzes

4

Inequalities

5

Math – Algebra 1Unit 2- Linear Relationships

Standards Topics Activities Resources AssessmentsN-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. For

example, we define 513 to be the cube root of 5

because we want (5 13)3

= 5( 1

3 )(3)to hold, so

(5 13)3

must equal 5.

*Fractional Exponents STEM ProjectsUnit Projects


7.3 Benchmark Exams 1-4


All Chapter Quizzes

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

*Adding and Subtracting Radicals

Extend 10.3 7.310.3

Benchmark Exams 1-4


All Chapter Quizzes8.EE.8. Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

*The graphing Method Benchmark Exams 1-4


All Chapter Quizzes

8.EE.8. Analyze and solve pairs of simultaneous linear equations.b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

*The graphing Method

* The Substitution Method

*The Addition-or Subtraction Method

*Multiplication with the Addition-or-Subtraction Method

Benchmark Exams 1-4


All Chapter Quizzes

8.EE.8. Analyze and solve pairs of simultaneous linear equationsc. Solve real-world and mathematical problems

*Solving Problems with Two Variables

Benchmark Exams 1-4

Chapter Test: 2A, 2B, 2C,

6

leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

*Multiplication with the Addition-or Subtraction Method

or 2D

All Chapter Quizzes

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.

*Multiplication with the Addition-or Subtraction Method



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

*The Graphing Method*The Substitution Method* Addition-or Subtraction Method*Multiplication with the Addition-or Subtraction Method

Extend 6.1Extend 6.5

6.16.26.36.46.5

Benchmark Exams 1-4


All Chapter Quizzes

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).

*Equations in Two Variable

*Points, Lines, and Their Graphs

1.6, 1.7, 3.1, 3.2, 3.4, 7.5, 9.1, 10.1

Benchmark Exams 1-4


All Chapter QuizzesA.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.

*The Graphing Method

*The Graph ofy=|ax+b|+c

Extend 6.1Extend 7.5Extend 9.3Extend 11.8

Benchmark Exams 1-4


All Chapter Quizzes

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.

*Inequalities in two Variables

*Systems of Linear Inequalities


5.66.6

Benchmark Exams 1-4


All Chapter Quizzes

7

8.F.1. Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

*Functions Defined by Tables and Graphs

*Functions Defined by Equations

Benchmark Exams 1-4


All Chapter Quizzes8.F.2. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

*Comparing Functions Benchmark Exams 1-4


All Chapter Quizzes

8.F.3. Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A=s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

Benchmark Exams 1-4


All Chapter Quizzes

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) demotes the output of f corresponding to the input x. The graph of f is graph of the y = f(x).

*Functions Defined by Tables and Graphs

*Functions Defined by Equations



All Chapter Quizzes

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.



All Chapter QuizzesF.IF.3 Recognize that sequences are functions, sometimes defined recursively,

3.57.7

Benchmark Exams 1-4

8

whose domain is 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.

7.8 Chapter Test: 2A, 2B, 2C, or 2D

All Chapter Quizzes

8.F.4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

*Slope of a Line

*Interpret Rate of Change

Benchmark Exams 1-4


All Chapter Quizzes

8.F.5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

Benchmark Exams 1-4


All Chapter Quizzes

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

Explore 3.1Extend 4.1

1.83.17.59.19.710.1

Benchmark Exams 1-4


All Chapter Quizzes

F.IF.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.*

1.77.57.69.110.1

Benchmark Exams 1-4


All Chapter Quizzes

9

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.

Explore 3.3Extend 7.7Explore 9.1



All Chapter QuizzesF-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.

*Linear and Quadratic Functions

Extend 3.2Extend 4.1Explore 9.3Extend 9.3

3.13.23.44.19.19.3

Benchmark Exams 1-4


All Chapter Quizzes

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.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)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.


3.13.23.44.19.19.29.3

Benchmark Exams 1-4


All Chapter Quizzes

F.IF.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.

1.73.64.37.89.19.3

Benchmark Exams 1-4


All Chapter Quizzes

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.

1.7, 3.1, 3.4, 3.6, 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 4.7, 7.6, 7.8

Benchmark Exams 1-4


All Chapter QuizzesF.BF.1 Write a function that describes a relationship between two quantitites.*-b-Combine standard function types using arithmetic operations. For example, build a

4,27,69.3

Benchmark Exams 1-4


10

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.

All Chapter Quizzes

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.



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

Extend 4.1Explore 7.5Explore 9.3

Benchmark Exams 1-4


All Chapter Quizzes

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

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

3.33.57.79.6

Benchmark Exams 1-4


All Chapter Quizzes

F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

3.53.69.6

Benchmark Exams 1-4


All Chapter Quizzes

F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions.c. Recognize situations in which a quantity



11

grows or decays by a constant percent rate per unit interval relative to another. All Chapter Quizzes

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).

3.53.67.57.67.7

Benchmark Exams 1-4


All Chapter QuizzesF.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.



All Chapter QuizzesF.LE.5 Interpret the parameters in a linear or exponential function in terms of a context.

Extend 4-1 3.44.17.57.6

Benchmark Exams 1-4


All Chapter Quizzes

12

Math – Algebra 1Unit 3- Descriptive Statistics

Standards Topics Activities Resources AssessmentsS-ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

*Statistics STEM ProjectsUnit Projects


12.312.4

Benchmark Exams 1-4


All Chapter QuizzesS-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.

*Statistics Extend 12.8 12.312.4

Benchmark Exams 1-4


All Chapter QuizzesS-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).

12.312.4

Benchmark Exams 1-4


All Chapter Quizzes8.SP.1 Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

Benchmark Exams 1-4


All Chapter Quizzes

8.SP.2 Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

Benchmark Exams 1-4


All Chapter Quizzes

8.SP.3 Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and

Benchmark Exams 1-4

Chapter Test: 2A, 2B, 2C,

13

intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

or 2D

All Chapter Quizzes

8.SP.4 Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Benchmark Exams 1-4


All Chapter Quizzes

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.

Extend 12.7 Benchmark Exams 1-4


All Chapter Quizzes

S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or chooses a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Extend 9.6 4.54.6

Benchmark Exams 1-4


All Chapter Quizzes

S-ID.6. Represent data on two 4.6 Benchmark Exams 1-4

14

quantitative variables on a scatter plot, and describe how the variables are related. b. Informally assess the fit of a function by plotting and analyzing residuals.


All Chapter Quizzes

S-ID.6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.c. Fit a linear function for a scatter plot that suggests a linear association.

4.54.6

Benchmark Exams 1-4


All Chapter Quizzes

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

Extend 4.1 4.1 Benchmark Exams 1-4


All Chapter QuizzesS-ID.8. Compute (using technology) and interpret the correlation coefficient of a linear fit.



All Chapter QuizzesS-ID.9. Distinguish between correlation and causation.



All Chapter Quizzes

15

Math – Algebra 1Unit 4 – Expressions and Equations

Standards Topics Activities Resources AssessmentsA.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.

*Adding and Subtracting Polynomials

*Factoring Integers

STEM ProjectsUnit Projects


1.11.4

Benchmark Exams 1-4


All Chapter Quizzes

A.SSE.1 Interpret expressions that represent a quantity in terms of its context*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.

*Multiplying Monomials

*Dividing Monomials

1.21.39.7

Benchmark Exams 1-4


All Chapter Quizzes

A.SSE.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

*Monomial Factors of Polynomials

*Difference of Two Squares

*Squares of Binomials

Explore 8.5Explore 8.6

1.1, 1.2, 1.3, 1.4, 7.1, 7.2, 7.3, 7.4, 8.5, 8.6, 8.7, 8.8, 8.9

Benchmark Exams 1-4


All Chapter Quizzes

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.

*Solving Equations by Factoring

8.58.68.78.88.9

Benchmark Exams 1-4


All Chapter Quizzes

16

A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.


*Completing the Square

Extend 9.4 9.39.4

Benchmark Exams 1-4


All Chapter Quizzes

A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.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 approximate equivalent monthly interest rate if the annual rate is 15%



All Chapter Quizzes

17

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.

*Basic Assumptions Explore 8.1Explore 8.3

8.18.28.38.4

Benchmark Exams 1-4


All Chapter Quizzes

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

*A Problem Solving Plan

*Solving Linear Equations and Problem Solving with Linear

Equations

*Solve Problems Involving Inequalities

*Solving Problems Involving Quadratic Equations

1.5, 2.1, 2.2, 2.3, 2.4, 2.5, 2.9, 3.2, 5.1, 5.2, 5.3, 5.4, 5.5, 7.6, 8.5, 8.6, 8.7, 9.4, 9.5, 10.4, 11.8

Benchmark Exams 1-4


All Chapter Quizzes

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

*Problem Solving with Systems of Equations Extension 1.7 3.1, 3.4, 3.5, 3.6, 4.1, 4.2, 4.3,

4.4, 4.5, 4.6, 4.7, 6.1, 6.2, 6.3, 6.4, 6.5, 7.5, 7.6, 8.6, 8.7, 8.8, 9.1, 9.2, 9.4, 9.5, 10.1, 10.4, 11.2, 11.8

Benchmark Exams 1-4


All Chapter Quizzes

A.CED.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.

*Transforming Formulas2.84.1

Benchmark Exams 1-4


All Chapter Quizzes

A.REI.4 Solve quadratic equations in one variable.a. Use the method of completing the square to


*Methods of Solution9.410.2

Benchmark Exams 1-4


18

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 for this form.

All Chapter Quizzes

A.REI.4 Solve quadratic equations in one variable.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 andwrite them as a ± bi for real numbers a and b.

*Square Roots of Variable Expressions

*Quadratic Equations with Perfect Squares


*The Quadratic Formula

*Complex Numbers

Solving Equations by Factoring

*Methods of Solution

8.68.78.89.29.49.5

Benchmark Exams 1-4


All Chapter Quizzes

A.REI.7 Solve a simple system consisting of linear equation and 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.



All Chapter Quizzes

Math – Algebra 1Unit 5 – Quadratic Functions and Modeling

Standards Topics Activities Resources Assessments

19

N.RN.3 Explain why the sum or product 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 non-zero rational number and an irrational number is irrational.

STEM ProjectsUnit Projects


Extension 10.2

Benchmark Exams 1-4


All Chapter Quizzes

8.G.6 Explain a proof of the Pythagorean Theorem and its converse.

*The Pythagorean Theorem Benchmark Exams 1-4


All Chapter Quizzes

8.G.7Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.



All Chapter Quizzes

8.G.8Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.



All Chapter QuizzesF.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

Explore 3.1Extend 4.1

1.83.17.59.19.710.1

Benchmark Exams 1-4


All Chapter Quizzes

20

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.F.IF.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.*

1.77.57.69.110.1

Benchmark Exams 1-4


All Chapter Quizzes

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.

Explore 3.3Extend 7.7Extend 9.1



All Chapter Quizzes

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

*Points, Lines, and Their Graphs

*Slope of a Line

*Slope-Intercept Form of a Linear Equation

Linear and Quadratic Functions


3.13.23.44.19.19.29.3

Benchmark Exams 1-4


All Chapter Quizzes

F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple


9.710.1

Benchmark Exams 1-4


21

cases and using technology for more complicated cases.-b-Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. Exponential, growth or decay.

All Chapter Quizzes

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.

*Solving Equations by Factoring



Extend 9.4 9.29.4

Benchmark Exams 1-4


All Chapter Quizzes

F-IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.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)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.



All Chapter Quizzes

F.IF.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

1.73.64.37.89.19.3

Benchmark Exams 1-4


All Chapter Quizzes

22

quadrant function and an algebraic expression for another say which has the larger maximum.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.

1.7, 3.1, 3.4, 3.6, 4.1, 4.2, 4.3, 4.4, 4.5, 4.5, 4.7, 7.6, 7.8

Benchmark Exams 1-4


All Chapter Quizzes

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

4.27.69.3

Benchmark Exams 1-4


All Chapter Quizzes

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

Extend 4.1Explore 7.5Explore 9.3

Benchmark Exams 1-4


All Chapter Quizzes

F.BF.4 Find inverse functions.-a-solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for

Explore 10.1 4.7Benchmark Exams 1-4


23

the inverse. For example, f(x)=2x3 or f(x)=(x+1)/(x-1) for x≠1

All Chapter Quizzes

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.

9.6Benchmark Exams 1-4


All Chapter Quizzes

Appendix A Adaptations for Special Education Students, English Language Learners, and Gifted and Talented Students

Making Instructional Adaptations

Instructional Adaptations include both accommodations and modifications.

An accommodation is a change that helps a student overcome or work around a disability or removes a barrier to learning for any student.

Usually a modification means a change in what is being taught to or expected from a student.

-Adapted from the National Dissemination Center for Children with Disabilities

ACCOMMODATIONS MODIFICATIONSRequired when on an IEP or 504 plan, but can be implemented for any student to support their

Only when written in an IEP.

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learning.

Special Education Instructional Accommodations

Teachers will use Approaching Level Tier 2: Strategic Intervention in RtI Differentiated Instruction section of Glencoe lessons.

Teachers will use the Targeted Strategic Intervention from the Glencoe Online Support. Teachers shall implement any instructional adaptations written in student IEPs. Teachers will implement strategies for all Learning Styles (Appendix B) Teacher will implement appropriate UDL instructional adaptations (Appendix C )

Gifted and Talented Instructional Accommodations

Teachers will use Beyond Level in RtI Differentiated Instruction section of Glencoe lessons Teachers will use the Enrichment Masters from the Glencoe Online Support Teacher will implement Adaptations for Learning Styles (Appendix B) Teacher will implement appropriate UDL instructional adaptations (Appendix C)

English Language Learner Instructional Accommodations

Teachers will use the ELL Differentiated English Language Learner Support section of Glencoe lessons. Teachers will use the Differentiated ELL Support from the Glencoe Online Support. Teachers will implement the appropriate Teachers will implement the appropriate instructional adaptions for English Language Leaners (Appendix E)

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APPENDIX BLearning Styles

Aadapted from The Learning Combination Inventories (Johnson, 1997)and VAK (Fleming, 1987)

Accommodating Different Learning Styles in the Classroom:All learners have a unique blend of sequential, precise, technical, and confluent learning styles. Additionally, all learners

have a preferred mode of processing information- visual, audio, or kinesthetic.It is important to consider these differences when lesson planning, providing instruction, and when differentiating

learning activities. The following recommendations are accommodations for learning styles that can be utilized for all students in your class.

Since all learning styles may be represented in your class, it is effective to use multiple means of presenting information, allow students to interact with information in multiple ways, and allow multiple ways for students to show what they have learned when applicable.

Visual Utilize Charts, graphs, concept maps/webs, pictures, and cartoons

Watch videos to learn information and concepts

Encourage students to visualize events as they read math word problems

Use flash cards to practice basic math facts

Model by demonstrating tasks or showing a finished product

Have written directions available for student

Use power point presentations

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Color code and highlight operation symbols (+, -, x, ÷)

Color code and highlight key words in math word problems

Audio Allow students to give oral presentations or explain concepts verbally

Present information and directions verbally or encourage students to read directions aloud to themselves.

Allow students to work in pairs

Utilize songs and rhymes

Ask for choral responses in instruction, example have the entire class chant in unison multiples, evens/odds, or skip counting by 2s, 5,s or 10s

Repeat, clarify, or reword directions

Verbally guide students through task steps

Kinesthetic Act out concepts and dramatize events

Use flash cards

Use manipulatives

Allow students to deepen knowledge through hands on projects

Sequential: following a plan. The learner seeks to follow step-by-step directions, organize and plan work carefully, and complete the assignment from beginning to end without interruptions.Accommodations:Repeat/rephrase directionsProvide a checklist or step by step written directionsBreak assignments in to chunks

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Provide samples of desired productsHelp the sequential students overcome these challenges: over planning and not finishing a task, difficulty reassessing and improving a plan, spending too much time on directions and neatness and overlooking concepts

Precise: seeking and processing detailed information carefully and accurately. The learner takes detailed notes, asks questions to find out more information, seeks and responds with exact answers, and reads and writes in a highly specific manner.Accommodations:Provide detailed directions for assignmentsProvide checklistsProvide frequent feedback and encouragementHelp precise students overcome these challenges: overanalyzing information, asking too many questions, focusing on details only and not concepts

Technical: working autonomously, "hands-on," unencumbered by paper-and-pencil requirements. The learner uses technical reasoning to figure out how to do things, works alone without interference, displays knowledge by physically demonstrating skills, and learns from real-world experiencesAccommodations:Allow to work independently or as a leader of a groupGive opportunities to solve problems and not memorize informationPlan hands-on tasksExplain relevance and real world application of the learningWill be likely to respond to intrinsic motivators, and may not be motivated by gradesHelp technical students overcome these challenges: may not like reading or writing, difficulty remaining focused while seated, does not see the relevance of many assignments, difficulty paying attention to lengthy directions or lectures

Confluent: avoiding conventional approaches; seeking unique ways to complete any learning task. The learner often starts before all directions are given; takes a risk, fails, and starts again; uses imaginative ideas and unusual approaches; and improvises.Accommodations:Allow choice in assignmentsEncourage creative solutions to problemsAllow students to experiment or use trial and error approach

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Will likely be motivated by autonomy within a task and creative assignmentsHelp confluent students overcome these challenges: may not finish tasks, trouble proofreading or paying attention to detail

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APPENDIX CUniversal Design for Learning Adaptations

Adapted from Universal Design For Learning

Teachers will utilize the examples below as a menu of adaptation ideas.

Provide Multiple Means of Representation

Strategy #1: Options for perception

Goal/Purpose ExamplesTo present information through different modalities such as vision, hearing, or touch.

Use visual demonstrations, illustrations, and models

Present a power point presentation.

Use appropriate manipulatives, such as base 10 block, counters, or pattern blocks

Differentiate operation symbols by color coding

Draw pictures when possible

Use interactive websites and apps

Use modeling to help students solve problems

Provide examples of a correctly solved problem at the beginning of each lesson

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Have students work each step in a different color

Use songs and rhymes to help remember information

Use mnemonics like “Please Excuse My Dear Aunt Sally” (order of operations) to remember sequenced steps

Simplify and rephrase vocabulary in word problems

Strategy #2: Options for language, mathematical expressions and symbols

Goal/Purpose ExamplesTo make words, symbols, pictures, and mathematical notation clear for all students.

Use larger font size and/or magnifiers

Highlight important parts of problems, example: key words or operation signs

Use place value charts, number grids, and operation tables (addition/subtraction and multiplication/division tables)

Allow students to trace important visual patterns

Use graph paper to keep numbers aligned

Put boxes around each problem to visually separate

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them

Simplify and rephrase vocabulary in word problem

Turn lined paper vertically so the student has ready made columns

Color code and highlight keywords in math word problems

Strategy #3: Options for Comprehension

Purpose ExamplesTo provide scaffolding so students can access and understand information needed to construct useable knowledge.

Use diagrams.

Use semantic maps and diagrams

Chunk pieces of information together, example: learn facts in sets of 3

Review previous lessons

Use a buddy system to clarify

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Use mnemonic aids to signal steps, example “Does McDonalds Sell Cheese Burgers” (long division: divide, multiply, subtract, check, bring down)

Provide students with a strategy to use for solving word problems

Use graph paper to keep numbers aligned

Use modeling to help students solve problems

Introduce concepts using real life examples whenever possible

Teach fact families and build fluency with games and understanding

When teaching number lines use tape or draw a number line on the floor for students to walk on

Provide Multiple Means of Action and Expression

Strategy #4: Options for physical action

Purpose Examples

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To provide materials that all learners can physically utilize

Use of computers when available

Preferential or alternate seating

Provide assistance with organization

Provide graph paper to organize place value

Provide appropriate manipulatives

Use flash cards

Provide highlighters for students when solving problems

Allow students to use desk top copies of fact sheets, multiplication/division tables etc.

Use individual dry-erase boards

Strategy #5: Options for expression and communication

Purpose Examples

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To allow the learner to express their knowledge in different ways

Allow oral responses or presentations

Students show their knowledge with charts and graphs

Give students extra time to respond to oral questions

Have students verbally or visually explain how to solve a math problem

Strategy #6: Options for executive function

Purpose ExamplesTo scaffold student ability to set goals, plan, and monitor progress

Provide clear learning goals, scales, and rubrics

Model skills

Utilize checklists

Give examples of desired finished product

Chunk longer assignments into manageable parts

Teach and practice organizational skills

Use a problem solving strategy checklist so that students can monitor their progress

Teach students to use self-questioning techniques

Reduce the number of practice or test problems on a

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page

Provide Multiple Means of Engagement

Strategy #7: Options for recruiting interest

Purpose ExamplesTo make learning relevant, authentic, interesting, and engaging to the student.

Provide choice and autonomy on assignments

Use colorful and interesting designs, layouts, and graphics

Use games, challenges, or other motivating activities

Provide positive reinforcement for effort

Use manipulatives

Provide learning aids such as calculators and/or operation tables (addition/subtraction and multiplication/division tables)

Introduce concepts using real life examples whenever possible

Use individual dry-erase boardsUse magnetic manipulatives examples: numbers, operation signs, ten frames, base ten blocks, etc.

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Strategy #8: Options for sustaining effort and persistence

Purpose ExamplesTo create extrinsic motivation for learners to stay focused and work hard on tasks.

Show real world applications of the lesson

Utilize collaborative learning

Assign a peer tutor

Incorporate student interests into lesson

Praise growth and effort

Recognition systems

Behavior plans

Repeat directions as needed

Provide immediate feedback

Strategy #9: Options for self-regulation

Purpose Examples

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To develop intrinsic motivation to control behaviors and to develop self-control.

Give prompts or reminders about self-control

Self-monitored behavior plans using logs, records, journals, or checklists

Ask students to reflect on behavior and effort

Post class rules using pictures and words

Post daily schedule using pictures and words

Circulate around the room

Develop a signal for when a break is needed

Provide consistent praise to elevate self-esteem

Model and role play problem solving

Desensitize students to anxiety causing events

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Appendix D Gifted and Talented Instructional Accommodations

How do the State of NJ regulations define gifted and talented students?

Those students who possess or demonstrate high levels of ability, in one or more content areas, when compared to their chronological peers in the local district and who require modification of their educational program if they are to achieve in accordance with their capabilities.

What types of instructional accommodations must be made for students identified as gifted and talented?

The State of NJ Department of Education regulations require that district boards of education provide appropriate K-12 services for gifted and talented students. This includes appropriate curricular and instructional modifications for gifted and talented students indicating content, process, products, and learning environment. District boards of education must also take into consideration the PreK-Grade 12 National Gifted Program Standards of the National Association for Gifted Children in developing programs..

What is differentiation?

Curriculum Differentiation is a process teachers use to increase achievement by improving the match between the learner’s unique characteristics:

Prior knowledge Cognitive LevelLearning Rate Learning StyleMotivation Strength or Interest

And various curriculum components:Nature of the Objective Teaching ActivitiesLearning Activities ResourcesProducts

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Differentiation involves changes in the depth or breadth of student learning. Differentiation is enhanced with the use of appropriate classroom management, retesting, flexible small groups, access to support personal, and the availability of appropriate resources, and necessary for gifted learners and students who exhibit gifted behaviors (NRC/GT, University of Connecticut).

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Gifted & Talented Accommodations Chart

Adapted from Association for Supervision and Curriculum Development

Teachers will utilize the examples below as a menu of adaptation ideas.

Strategy Description Suggestions for AccommodationHigh Level Questions

Discussions and tests, ensure the highly able learner is presented with questions that draw on advanced level of information, deeper understanding, and challenging thinking.

Require students to defend answers Use open ended questions Use divergent thinking questions Ask student to extrapolate answers when given

incomplete informationTiered assignments

In a heterogeneous class, teacher uses varied levels of activities to build on prior knowledge and prompt continued growth. Students use varied approaches to exploration of essential ideas.

Use advanced materials Complex activities Transform ideas, not merely reproduce them Open ended activity

Flexible Skills Grouping

Students are matched to skills work by virtue of readiness, not with assumption that all need same spelling task, computation drill, writing assignment, etc. Movement among groups is common, based on readiness on a given skill and growth in that skill.

Exempt gifted learners from basic skills work in areas in which they demonstrate a high level of performance

Gifted learners develop advanced knowledge and skills in areas of talent

Independent Projects

Student and teacher identify problems or topics of interest to student. Both plan method of investigating topic/problem and identifying type of product student will develop. This product should address the problem and demonstrate the student’s ability to apply skills and knowledge to the problem or topic

Primary Interest Inventory Allow student maximum freedom to plan, based

on student readiness for freedom Use preset timelines to zap procrastination Use process logs to document the process

involved throughout the study

Learning Centers

Centers are “Stations” or collections of materials students can use to explore, extend, or practice skills and content. For gifted students, centers should

Develop above level centers as part of classroom instruction

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move beyond basic exploration of topics and practice of basic skills. Instead it should provide greater breadth and depth on interesting and important topics.

Interest Centers or Interest Groups

Interest Centers provide enrichment for students who can demonstrate mastery/competence with required work/content. Interest Centers can be used to provide students with meaningful learning when basic assignments are completed.

Plan interest based centers for use after students have mastered content

Contracts and Management Plans

Contracts are an agreement between the student and teacher where the teacher grants specific freedoms and choices about how a student will complete tasks. The student agrees to use the freedoms appropriately in designing and completing work according to specifications.

Allow gifted students to work independently using a contract for goal setting and accountability

Compacting A 3-step process that (1) assesses what a student knows about material “to be” studied and what the student still needs to master, (2) plans for learning what is not known and excuses student from what is known, and (3) plans for freed-up time to be spent in enriched or accelerated study.

Use pretesting and formative assessments Allow students who complete work or have

mastered skills to complete enrichment activities

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Appendix E English Language Learner Instructional Accommodations

Adapted from World-class Instructional Design and Assessment guidelines (2014), Teachers to English Speakers of Other Languages guidelines, State of NJ Department of Education Bilingual

Math

Instruction: Provide bilingual dictionaries. Simplify language, clarify or explain directions. Build background (discuss, allow for questions, and use visuals if applicable) prior to giving assessment make the text meaningful. Pre-teach difficult vocabulary. Highlight key word or phrases. Allow ELL students to hear word problems twice and have a second opportunity to check their answers. Allow ELL students extended time for word problems. Provide specific seating arrangement (close proximity for direct instruction, teacher assistance, and buddy).

Response: Allow for oral explanations Allow the use of word walls and vocabulary banks.

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