ALGEBRA 100 GRADE: 8

APPROVED BY BOARD OF EDUCATION AUGUST 25, 2015 ELMWOOD PARK PUBLIC SCHOOLS

ALGEBRA 100

GRADE: 8

Prerequisite: Pre Algebra, Grade 7 or Pre Algebra Honors, Grade 7; teacher recommendation

Credits: 5

ABSTRACT This full year course prepares students for more rigorous study in higher mathematics courses by providing the necessary foundations of algebraic concepts and skills. In order to accomplish this, emphasis will be placed on the following: developing number sense and an ability to represent numbers in a variety of forms; understanding, selecting and applying various methods of performing numerical operations; understanding the symbolism of algebra, using exponents, solving linear equations and inequalities; applying absolute value; factoring expressions; exploring real numbers; understanding the basic concepts of coordinate geometry, developing spatial sense and an ability to use geometric properties and relationships; identifying quadratic functions; introducing basic trigonometric functions; developing the understanding of patterns, relationships and functions to analyze real world situations; using statistical analysis and probability to describe data; model situations and support appropriate inferences and arguments; applying the concepts of discrete mathematics to model and explore a variety of practical situations; using a variety of estimation strategies and recognizing situations in which estimation is appropriate. Students will use all available technology to enhance mathematical knowledge and application. Technology usage is recommended through the use of graphing calculators and computers. They will analyze relationships among variables and quantities and use these relationships to solve real-world problems, ranging from everyday applications to the

2 APPROVED BY BOARD OF EDUCATION AUGUST 25, 2015 ELMWOOD PARK PUBLIC SCHOOLS

sciences. They will develop reasoning ability and will become self-reliant, independent mathematical thinkers willing to take part in decision-making and risk-taking in order to arrive at reasonable conclusions. They will be able to pose and solve mathematical problems found in all disciplines. Technology usage is recommended through the use of graphing calculators and computers.

UNIT: (Title, Month(s), Number of Days)

1: Tools of Algebra September

14 days

2: Solving Equations & Inequalities

October-November 24 days

3: Solving & Applying Proportions November

10 days

STAGE 1: DESIRED RESULTS What will students understand as a result of the unit? What are the BIG ideas?

ESTABLISHED GOALS: (NJ CCCS and/or CCS)

Math CCSS EE.8.7, SP.8.1

A-REI.1

Technology CCCS 8.1, 8.2

21st Century Life and Careers CCCS

9.1, 9.2, 9.3

Math CCSS EE.8.7

A-CED.1, A-REI.3



9.1, 9.2, 9.3

Math CCSS EE.8.5

A-SSE.1



9.1, 9.2, 9.3 ENDURING UNDERSTANDINGS: (Students will Understand that . . .)

• will use variables to transform English phrases into mathematical expressions.

• will extend your ability to calculate with whole numbers, decimals, and factions to include integers.

• will use the order of operations and the distributive property to simplify expressions.

• will show the relationship between

• will solve equations, including equations with variables on both sides, using properties of equality.

• will develop the ability to solve problems by defining variables, relating them to one another, and writing an equation.

• will use measures of central tendency to describe a set of data.

• will learn how to graph inequalities.

• will find ratios and rates to model real-world situations.

• will use proportions to measure objects indirectly.

• will solve problems that involve discounts, taxes, and interest.




14 days




10 days two sets of real-world data, using a scatter plot.

• will solve inequalities, noting the differences from the methods used for solving equations.

• will write and solve compound inequalities by interpreting phrases that use and or or.

ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?)

• Why do you sometimes need variables when writing equations to represent real-world situations?

• Why are the rules for the order of operations necessary?

• How can you express an integer as a rational number?

• How would you explain the process for graphing a point on a coordinate plane to another student?

• When solving equations, what type of operations must you use to get the variable alone on one side of the equal sign?

• How can you clear an equation of fractions or decimals?

• How is solving an equation with 2 variable terms on the same side of the equal sign different than solving an equation with variable terms on both sides of the equal sign?

• What are the two ways to help set up a distance problem?

• When should the mean be used to describe data and when should the median be used?

• What is the main difference between solving inequalities using multiplication and division and solving equations using multiplication and division?

• How can you use proportions to find a distance that is difficult to measure?

• What are two ways to write percent problems?

• What are the differences between theoretical probability and experimental probability?

• How does finding the probability of independent events differ from finding the probability of dependent events?




14 days




10 days

STAGE 2: ASSESSMENT EVIDENCE What evidence will be collected to determine whether or not the understandings have been developed, the knowledge and skills attained, and the

state standards met? [Anchor the work in performance tasks that involve application, supplemented as needed by prompted work, quizzes, observations, etc.]

PERFORMANCE TASKS: (Through what authentic performance tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?)

Assessments of each learning activity Mathematical Practices: 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.

Mathematical Practices: 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.


OTHER EVIDENCE: (Through what other evidence (e.g. quizzes, tests, academic prompts, observations, homework, journals) will students demonstrate achievement of the desired results?) (How will students self-assess their

Teacher observations Peer evaluations

Rubrics Bulletin Boards of exemplars

Tests Quizzes

Peer and self evaluations Presentations Daily notes



Tests Quizzes




Tests Quizzes





14 days




10 days learning?)

Homework Benchmark at the end of the unit



RESOURCES:

Notebooks/binder Agenda

NJ ASK Mathematics Reference Sheet, Grade 8

Differentiated Instruction Resources Assessment Resources

Solution Key Explore activities

iPads My.hrw.com

www.khanacademy.org Manipulatives

NJ ASK preparation problems www.funbrain.com





iPads My.hrw.com







iPads My.hrw.com



STAGE 3: LEARNING PLAN

What learning experiences and instruction will enable students to achieve the desired results? Utilize the WHERETO* acronym to consider key design elements.

SKILLS AND TOPICS: (What specific activities will students do and what skills will students know as a result of the unit?)

Lessons EE.8.7. Solve linear equations in one variable. • Give examples of linear equations

in one variable with one solution, infinitely many solutions, or no solutions. Show which of these

EE.8.7. Solve linear equations in one variable. • Give examples of linear equations

in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by

EE.8.5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two




14 days




10 days possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. SP.8.1. 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. 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.

successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

• Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

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-REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve

moving objects has greater speed. A-SSE.1. Interpret expressions that represent a quantity in terms of its context. • Interpret parts of an expression, such

as terms, factors, and coefficients. • 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.

8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration




14 days




10 days 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy.

problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and

and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.




14 days




10 days 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.

preparation in order to navigate the globally competitive work environment of the information age.

CROSS-CURRICULAR / DIFFERENTIATION: (What cross-curricular (e.g. writing, literacy, math, science, history, 21st century life and careers, technology) learning activities are included in this unit that will help achieve the desired results?) (What type of differentiated instruction will be used for ELL, SP.ED. and G&T students?)

Centers Tiered Lessons Open-Ended Responses

Differentiation: Special Education: • Modifications as dictated in the

student's IEP • Collaboration with resource teacher

and parent • Positive reinforcement. • Modify lesson as needed according

to ability. English Language Learners: • Use cooperative grouping • Provide written and oral

instructions. • Encourage support from native

language speakers who are more proficient in English.

• Extended time for completing













to ability. English Language Learners: • Use cooperative grouping • Provide written and oral instructions. • Encourage support from native


• Extended time for completing assessments.




14 days




10 days assessments.

• Go Math Spanish resources for assessments and assignments.

• *ELL Students- Instruction will be based on language proficiency.

Gifted and Talented: • Advanced Problem Solving • Above grade level math placement

option for qualified students, including Geometry Honors

• Higher order, critical and creative thinking skills.

• Cluster grouping • Flexible skill grouping within a

class or across grade level for rigor. • Teacher-selected instructional

strategies that are focused to provide challenge, engagement, and growth opportunities.

• Multi-disciplinary unit and/or project.

• Applied and integrated skills for the 21st Century learner.

At-Risk Students • Tiered groups and tiered

assignments

assessments. • Go Math Spanish resources for

assessments and assignments. • *ELL Students- Instruction will

be based on language proficiency.










assignments






• Cluster grouping • Flexible skill grouping within a class

or across grade level for rigor. • Teacher-selected instructional




At-Risk Students • Tiered groups and tiered assignments • Small group instruction, one-to-one

instruction and additional online resources.




14 days




10 days • Small group instruction, one-to-one


• Tiered assignments will be provided by the Go Math series and differ based on number and/or complexity of problems and degree of scaffolding.

• NJDOE Resources

• Small group instruction, one-to-one instruction and additional online resources.


• NJDOE Resources


• NJDOE Resources

*WHERETO W = Help the students know WHERE the unit is going and WHAT is expected. Help the teacher know WHERE the students are coming from (prior knowledge, interests). H = HOOK all students and HOLD their interest. E = EQUIP students, help them EXPERIENCE the key ideas and EXPLORE the issue. R = Provide opportunities to RETHINK and REVISE their understandings and work. E = Allow students to EVALUATE their work and its implications. T = TAILORED to the different needs, interests, and abilities of learners. O = ORGANIZE to maximize initial and sustained engagement as well as effective learning. UNIT: (Title, Month(s), Number of Days)

4: Graphs & Functions November-December

10 days

5: Linear Equations & Their Graphs

December-January 13 days

6: Systems of Equations & Inequalities

January-February 12 days




10 days





STAGE 1: DESIRED RESULTS

What will students understand as a result of the unit? What are the BIG ideas? ESTABLISHED GOALS: (NJ CCCS and/or CCS)

Math CCSS F.8.1, F.8.2

F-IF.1, F-IF.2, F.BF.2, F-LE.2



9.1, 9.2, 9.3

Math CCSS EE.8.6, F.8.3, F.8.4, F.8.5, SP.8.2 A-REI.10, F-IF.5, F-IF.6, F-IF.7a,

F-BF.3, F-LE.5



9.1, 9.2, 9.3

Math CCSS EE.8.8

A-CED.2, A-REI.5, A-REI.6, A-REI.11



9.1, 9.2, 9.3

ENDURING UNDERSTANDINGS: (Students will Understand that they will. . .)

• move from the specific case of equations in one variable to the study of functions in two variables.

• learn about the function rules, and model data using equations, tables, and graphs.

• learn how to use inductive reasoning for recognizing number patterns called sequences

• learn how to write linear equations and recognize their different forms.

• by working with rate of change, interpret the slope in real-world situations.

• determine whether the graphs of two linear equations are parallel or perpendicular.

• extend your ability to solve equations to include solving a system of two equations with two variables.

• learn methods of solving a linear system and how to determine which method is best for a given situation.




10 days






• What are the key differences between relations and functions?

• What are the advantages and disadvantages of representing functions using rules, tables and graphs?

• What are the two ways to find the slope of a line?

• How does changing the value of the slope affect the graph of the line?

• How does changing the value of the y-intercept affect the graph of the line?

• How do you find the x- and y-intercepts of a linear equation?

• What is the difference between a trend line and a line of best fit?

• Why is it sometimes easier to solve equations using substitution rather than graphing?

• When is it best to solve a linear system by elimination and by using substitution?

• Compare graphing linear inequalities with graphing linear equations.

STAGE 2: ASSESSMENT EVIDENCE

What evidence will be collected to determine whether or not the understandings have been developed, the knowledge and skills attained, and the state standards met? [Anchor the work in performance tasks that involve application, supplemented as needed by prompted work, quizzes,

observations, etc.] PERFORMANCE TASKS: (Through what authentic performance tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?)

Assessments of each learning activity Mathematical Practices: 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.

Mathematical Practices: 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

Mathematical Practices: 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




10 days





8. Look for and express regularity in repeated reasoning.

repeated reasoning. repeated reasoning.

OTHER EVIDENCE: (Through what other evidence (e.g. quizzes, tests, academic prompts, observations, homework, journals) will students demonstrate achievement of the desired results?) (How will students self-assess their learning?)



Tests Quizzes

Peer and self evaluations Presentations Daily notes Homework

Benchmark at the end of the unit



Tests Quizzes





Tests Quizzes


Benchmark at the end of the unit RESOURCES:





iPads My.hrw.com







iPads My.hrw.com







iPads My.hrw.com






10 days





STAGE 3: LEARNING PLAN What learning experiences and instruction will enable students to achieve the desired results? Utilize the WHERETO* acronym to

consider key design elements. SKILLS AND TOPICS: (What specific activities will students do and what skills will students know as a result of the unit?)

F.8.1. 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.1 F.8.2. 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. F-IF.1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph

EE.8.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. F.8.3. 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. F.8.4. 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

EE.8.8. Analyze and solve pairs of simultaneous linear equations. • 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.

• 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.

• Solve real-world and mathematical problems 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.




10 days





of f is the graph of the equation y = f(x). F-IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-BF.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F-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). 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an

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. F.8.5. 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. SP.8.2. 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. A-REI.10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

A-CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A-REI.5. Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. A-REI.6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. A-REI.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.




10 days





understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.

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. F-IF.6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. F-IF.7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.

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

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

8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge




10 days





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. F-LE.5. Interpret the parameters in a linear or exponential function in terms of a context. 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration

about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.




10 days





and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.

CROSS-CURRICULAR: (What cross-curricular (e.g. writing, literacy, math, science, history, 21st century life and careers, technology) learning activities are included in this unit that will help achieve the desired results?)




and parent




and parent




and parent




10 days





• Positive reinforcement. • Modify lesson as needed according











class or across grade level for rigor.












class or across grade level for rigor.












strategies that are focused to provide




10 days





• Teacher-selected instructional strategies that are focused to provide challenge, engagement, and growth opportunities.




assignments • Small group instruction, one-to-one



• NJDOE Resources

• Teacher-selected instructional strategies that are focused to provide challenge, engagement, and growth opportunities.







• NJDOE Resources

challenge, engagement, and growth opportunities.






• NJDOE Resources



7: Exponents & Exponential Functions

February-March 14 days

8: Polynomials & Factoring March 15 days

9: Quadratic Equations & Functions

March-April 16 days



Math CCSS EE.8.1, EE.8.3, EE.8.4

F-IF.7e, F-IF.8b, F.BF.2, F-LE.2



9.1, 9.2, 9.3

Math CCSS A-SSE.2,

A-APR.1, A-APR.7



9.1, 9.2, 9.3

Math CCSS NS.8.2

A-SSE.3, A-APR.3, A-REI.4, A-REI.10, F-IF.7, F.IF.8, F-LE.1

Technology CCCS

8.1, 8.2

21st Century Life and Careers CCCS 9.1, 9.2, 9.3

ENDURING UNDERSTANDINGS: (Students will Understand that they will. . .)

• extend their knowledge about exponents to include zero and negative exponents.

• learn the properties of exponents, and how exponents are used to write a geometric sequence.

• graph exponential functions by making a table of values.

• categorize polynomials by their degree and number of terms.

• learn to add, subtract, and multiply polynomials.

• examine quadratic graphs and their equations.

• solve quadratic equations by various techniques.

• determine an appropriate linear, quadratic, or exponential model for real-world data.







March-April 16 days


• What is the meaning of a zero exponent?

• What is the meaning of a negative exponent?

• How is dividing powers with the same base different from multiplying powers with the same base?

• How is raising a quotient to a power different than raising a power to a power?

• How do you square a binomial? • How do you determine what

numbers are used in the binomial factors when factoring expressions of the type x2 + bx + c.?

• What is the first thing you should look for when factoring a trinomial?

• How do a and c affect a quadratic graph?

• How many solutions does a quadratic equation have?

• How can you determine the number of solutions from the graph?

• How do you complete the square? • When is the quadratic formula

useful? • What is the discriminant and how is

it useful? • How do you choose between a linear,

quadratic, or exponential model for data?



observations, etc.]

PERFORMANCE TASKS: (Through what authentic performance tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?)

Assessments of each learning activity Mathematical Practices: 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.

Mathematical Practices: 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.

Mathematical Practices: 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.







March-April 16 days

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.

6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.

6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning.




Tests Quizzes





Tests Quizzes





Tests Quizzes



RESOURCES:





iPads My.hrw.com

www.khanacademy.org





iPads My.hrw.com

www.khanacademy.org





iPads My.hrw.com

www.khanacademy.org







March-April 16 days

Manipulatives NJ ASK preparation problems

www.funbrain.com


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EE.8.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3–5 = 3–3 = 1/33 = 1/27. EE.8.3. Use numbers expressed in the form of a single digit times a whole-number power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger. EE.8.4. Perform operations with numbers expressed in scientific

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). A-APR.1. Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. A-APR.7. (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational

NS.8.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. 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. b. Complete the square in a







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notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. 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.

e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

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

expression; add, subtract, multiply, and divide rational expressions. 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment,

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 approximate equivalent monthly interest rate if the annual rate is 15%.

A-APR.3. Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. A-REI.4. Solve quadratic equations in one variable. • 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.

• Solve quadratic equations by







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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.

F-BF.2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F-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). 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge.

and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.

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.

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). 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. b. Graph square root and cube root functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.







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8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.

d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential functions, showing intercepts and end behavior.

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

a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential 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







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exponential growth or decay. F-LE.1. Distinguish between situations that can be modeled with linear functions and with exponential functions. Prove that linear functions grow by equal differences over equal intervals, and that exponential functions grow by equal factors over equal intervals. 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and







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organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.

CROSS-CURRICULAR / DIFFERENTIATION: (What cross-curricular (e.g. writing, literacy, math, science, history, 21st century life and careers, technology) learning activities are included in this unit that will help achieve the desired results?) (What type of differentiated instruction will be used for ELL, SP.ED. and G&T students?)





to ability.





to ability.





to ability.







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English Language Learners: • Use cooperative grouping • Provide written and oral











strategies that are focused to provide challenge, engagement, and

English Language Learners: • Use cooperative grouping • Provide written and oral











strategies that are focused to provide challenge, engagement, and

English Language Learners: • Use cooperative grouping • Provide written and oral instructions. • Encourage support from native











• Multi-disciplinary unit and/or







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growth opportunities. • Multi-disciplinary unit and/or

project. • Applied and integrated skills for the

21st Century learner. At-Risk Students • Tiered groups and tiered




• NJDOE Resources

growth opportunities. • Multi-disciplinary unit and/or


21st Century learner. At-Risk Students • Tiered groups and tiered




• NJDOE Resources


21st Century learner. At-Risk Students • Tiered groups and tiered assignments • Small group instruction, one-to-one



• NJDOE Resources

*WHERETO W = Help the students know WHERE the unit is going and WHAT is expected. Help the teacher know WHERE the students are coming from (prior knowledge, interests). H = HOOK all students and HOLD their interest. E = EQUIP students, help them EXPERIENCE the key ideas and EXPLORE the issue. R = Provide opportunities to RETHINK and REVISE their understandings and work. E = Allow students to EVALUATE their work and its implications. T = TAILORED to the different needs, interests, and abilities of learners.


O = ORGANIZE to maximize initial and sustained engagement as well as effective learning. UNIT: (Title, Month(s), Number of Days)

10: NJ ASK Preparation April

10 days

11: Radical Expressions & Equations

May 14 days

12: Rational Expressions & Functions May-June 18 days



Math CCSS NS.8.1, EE.8.2

G.8.1, G.8.2, G.8.3, G.8.4 G.8.5, G.8.9,

SP.8.3, SP.8.4

Technology CCCS 8.1,

Math CCSS G.8.6, G.8.7, G.8.8,

F-IF.7b



9.1, 9.2, 9.3

Math CCSS A-APR.2, A-APR.6,

A-REI.2



9.1, 9.2, 9.3 ENDURING UNDERSTANDINGS: (Students will Understand that they will. . .)

• Fractions can be expressed as terminating or repeating decimals and terminating decimals can be expressed as fractions.

• Rational numbers are compared and ordered in either fractional or decimal form.

• It is often useful to convert numbers to other representations in order to solve problems.

• Properties of lines and angles assist in determining measures of

• simplify expressions containing radicals.

• use the Pythagorean Theorem to find the lengths of sides of right triangles.

• solve radical equations.

• study inverse variation and learn to distinguish between direct and inverse variation.

• graph and solve equations involving rational expressions, utilizing knowledge of polynomials and domains.

use permutations and combinations to find the number of outcomes of real-world situations.




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May 14 days


unknown angles. • There are different classifications

of triangles and quadrilaterals. • Applying what is known of 2D

figures helps construct and identify 3D figures.


• How are fractions, decimals, and percents related?

• How do different forms of rational numbers help solve problems?

• How do different representations of rational numbers help you compare and order them?

• How are angle measurements used in real-world situations?

• Describe the relationship between two parallel lines cut by a transversal and the angles that are formed.

• Why is it important to understand the properties of two-dimensional figures, such as triangles and quadrilaterals?

• What are the uses of the Pythagorean Theorem?

• What is an extraneous solution and how do you find one?

• How is a direct variation different from and inverse variation?

• How do you add and subtract rational expressions?

How do you solve problems that require distinguishing between permutations and combinations?






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observations, etc.] PERFORMANCE TASKS: (Through what authentic performance tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?)

Assessments of each learning activity Mathematical Practices: 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.






Tests Quizzes





Tests Quizzes





Tests Quizzes


Benchmark at the end of the unit RESOURCES:


NJ ASK Mathematics Reference Sheet,








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Grade 8 Differentiated Instruction Resources

Assessment Resources Solution Key

Explore activities iPads

My.hrw.com www.khanacademy.org


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NS.8.1. Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. EE.8.2. Use square root and cube root symbols to represent solutions to

G.8.6. Explain a proof of the Pythagorean Theorem and its converse. G.8.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. G.8.8. Apply the Pythagorean Theorem to find the distance between two points

A-APR.2. Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x – a is p(a), so p(a) = 0 if and only if (x – a) is a factor of p(x). A-APR.6. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less




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equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. G.8.1. Verify experimentally the properties of rotations, reflections, and translations:

a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines.

G.8.2. Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them. G.8.3. Describe the effect of dilations, translations, rotations, and reflections

in a coordinate system. 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. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate

than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system. A-REI.2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge. 8.2 All students will develop an understanding of the nature and impact of technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 9.1: All students will demonstrate creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and




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on two-dimensional figures using coordinates. G.8.4. Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them. G.8.5. Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so. G.8.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

creative, critical thinking, collaboration and problem solving skills to function successfully as global citizens and workers in diverse ethnic and organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.

organizational cultures. 9.2: All students will develop skills and strategies that promote personal and financial responsibility related to financial planning, savings, investment, and charitable giving in the global economy. 9.3: All students will apply knowledge about and engage in the process of career awareness, exploration, and preparation in order to navigate the globally competitive work environment of the information age.




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May 14 days


SP.8.3. Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and 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. SP.8.4. 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




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also tend to have chores? 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and collaboratively to create and communicate knowledge.

CROSS-CURRICULAR: (What cross-curricular (e.g. writing, literacy, math, science, history, 21st century life and careers, technology) learning activities are included in this unit that will help achieve the desired results?)

























10 days


May 14 days


























assessments. • Go Math Spanish resources for

assessments and assignments. • *ELL Students- Instruction will

be based on language proficiency.










instruction and additional online




10 days


May 14 days





• NJDOE Resources




• NJDOE Resources

resources. • Tiered assignments will be provided

by the Go Math series and differ based on number and/or complexity of problems and degree of scaffolding.

• NJDOE Resources

Documents

ALGEBRA 100 GRADE: 8