PRE ALGEBRA GRADE: 7 & 8

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1 PRE ALGEBRA GRADE: 7 & 8 Credits: 5 ABSTRACT In grade 7-8 the goal for students is to apply their problem-solving, communication, and reasoning skills in an increasingly diverse set of situations as they develop a better understanding of the connections within mathematics and between mathematics and other disciplines and the real world. Instructional time for students should focus on three critical areas: (1) formulating and reasoning about expressions and equations including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.

2 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days ESTABLISHED GOALS: (NJ CCCS and/or CCS) ENDURING UNDERSTANDINGS: (Students will Understand that...) STAGE 1: DESIRED RESULTS What will students understand as a result of the unit? What are the BIG ideas? Math CCSS 7-8.NS.1 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 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. As with integers, students learn how to manipulate positive and negative fractions and mixed numbers using addition, subtraction, multiplication, and division to solve problems in everyday contexts. It is often useful to convert numbers to other representations in 2 Math CCSS 7-8.EE.1, 7-8.EE.2, 7-8.EE.3, 7-8.EE.4, 7-8.NS.1, 7-8.NS.2 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 Basic skills about exponents and powers apply to multiplying and dividing monomials and finding powers of monomials. Exponents are used to express very large and very small numbers in scientific notation. There is an inverse relationship between powers and roots. Math CCSS 7-8.EE.7 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 A useful tool in problem solving is writing one- or two-step equations or inequalities The solution of an equation differs fro the solution of an inequality. Properties of real numbers allow you to rewrite complex expressions involving parentheses and like terms. Using these properties helps solve multi-step equations and inequalities, including those with variables on each side.

3 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) order to solve problems. 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? What are some real-world situations involving money that use percents? How do you add, subtract, multiply, and divide very large and very small numbers? How are exponents helpful in realworld situations? What is the relationship between negative exponents and rational numbers? What is the relationship between the subsets of the real number system? Why are equations and inequalities important in mathematics? How do properties and number patterns assist in solving equations? How does a two-step equation differ from a one-step equation? How is solving an inequality similar to and different from solving an equation? How does the solution a two-step inequality differ from the solution of a two-step equation? What are some similarities and differences between algebraic expressions and equations? How are mathematical properties helpful in simplifying expressions? How are the Properties of Inequalities similar to the Properties of Equality? How are they different? 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.] 3

4 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days PERFORMANCE TASKS: (Through what authentic performance tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?) 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?) 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals 4

5 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days Papers showing students corrections of errors and misconceptions Portfolios Benchmark at the end of the unit Papers showing students corrections of errors and misconceptions Portfolios Benchmark at the end of the unit Papers showing students corrections of errors and misconceptions Portfolios Benchmark at the end of the unit RESOURCES: Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 8 Explore Activities Go Math Differentiated Workbooks Differentiated Grouping Go Math Assessment resources ipads PARCC Coach Workbooks IXL.com My.hrw.com Renaissance Graphic Organizers Manipulatives NJ ASK preparation problems Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 8 Explore Activities Go Math Differentiated Workbooks Differentiated Grouping Go Math Assessment resources ipads PARCC Coach Workbooks IXL.com My.hrw.com Renaissance Graphic Organizers Manipulatives NJ ASK preparation problems Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 8 Explore Activities Go Math Differentiated Workbooks Differentiated Grouping Go Math Assessment resources ipads PARCC Coach Workbooks IXL.com My.hrw.com Renaissance Graphic Organizers Manipulatives NJ ASK preparation problems STAGE 3: LEARNING PLAN What learning experiences and instruction will enable students to achieve the desired results? Utilize the WHERETO* acronym to 5

6 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days SKILLS AND TOPICS: (What specific activities will students do and what skills will students know as a result of the unit?) consider key design elements. Lessons Know that there are numbers that are not rational, and approximate them by rational numbers. 7-8.NS.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. 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. 6 Work with radicals and integer exponents. 7-8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, = 3 3 = 1/33 = 1/ EE.2. Use square root and cube root symbols to represent solutions to equations of the form x 2 = p and x 3 = 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. 7-8.EE.3. Use numbers expressed in the form of a single digit times a wholenumber 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 10 8 and the population of the world as 7 times 10 9, and determine that the world population is more than 20 times larger. Analyze and solve linear equations and pairs of simultaneous linear equations. 7-8.EE.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 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. 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.

7 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days 7-8.EE.4. Perform operations with numbers expressed in scientific 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. 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. Know that there are numbers that are not rational, and approximate them by rational numbers. 7-8.NS.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. 7-8.NS.2. Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them 7

8 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days approximately on a number line diagram, and estimate the value of expressions (e.g., π 2 ). 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. 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 / DIFFERENTIATION: (What cross-curricular (e.g. writing, literacy, math, science, history, 21 st century life and careers, technology) Centers Tiered Lessons Open-Ended Responses Differentiation: Special Education: 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. Centers Tiered Lessons Open-Ended Responses Differentiation: Special Education: Centers Tiered Lessons Open-Ended Responses Differentiation: Special Education: 8

9 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days 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?) 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 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 9 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 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 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 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

10 Rational Numbers and Percents 14 days Real Numbers and Monomials 14 days Equations and Inequalities 20 days 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 21 st Century learner. At-Risk Students Tiered groups and tiered assignments Small group instruction, one-to-one instruction and additional online 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. 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 21 st Century learner. At-Risk Students Tiered groups and tiered assignments Small group instruction, one-to-one instruction and additional online 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. 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 21 st Century learner. At-Risk Students Tiered groups and tiered assignments Small group instruction, one-to-one instruction and additional online 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. *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 10

11 (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: Expressions and Functions Linear Functions and Systems of Equations 16 days Two- and Three-dimensional Geometry 18 days ESTABLISHED GOALS: (NJ CCCS and/or CCS) ENDURING UNDERSTANDINGS: (Students will Understand that...) STAGE 1: DESIRED RESULTS What will students understand as a result of the unit? What are the BIG ideas? Math CCSS 7-8.F.1, 7-8.F.3, 7-8.F.4, 7-8.F.5, 7-8.SP.1, 7-8.SP.4 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 Patterns help in writing algebraic expressions and translate between verbal, tabular, graphical, and algebraic representations of linear functions. 11 Math CCSS 7-8.EE.5, 7-8.EE.6, 7-8.EE.8, 7-8.F.2, 7-8.F.3, 7-8.F.4, 7-8.F.5 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 The slope and x- and y-intercepts, when graphing a linear equation, represent real-world situations. Use of tables and graphs help represent, analyze, and solve real- Math CCSS 7-8.G.5, 7-8.G.6 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 Properties of lines and angles assist in determining measures of unknown angles. There are different classifications of triangles and quadrilaterals.

12 ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) Expressions and Functions Analysis extends this to non-linear functions. How can words, tables, graphs, and equations be used to show the relationship between two quantities? How can tables, graphs, and expressions demonstrate the relationship between two quantities? How are functions, formulas, equations, tables, and graphs related? What are some ways in which functions and relations can be represented? How are the graphs of linear functions and quadratic functions different? Linear Functions and Systems of Equations 16 days world problems related to linear equations and systems of linear equations. How can linear equations be used to represent real-world situations? How can slope, rate of change, and direct variation be used to represent real-world situations? When is it useful to represent an equation on a graph? What type of real-world situations can be solved by finding the solution of a system of equations? Two- and Three-dimensional Geometry 18 days Applying what is known of 2D figures helps construct and identify 3D figures. Why is the study of angles and lines important to understanding our environment? 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? How does knowledge of twodimensional figures help you when working with three-dimensional figures? 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 Mathematical Practices: 1. Make sense of problems and 12 Mathematical Practices: 1. Make sense of problems and Mathematical Practices: 1. Make sense of problems and persevere

13 tasks will students demonstrate the desired understandings?) (By what criteria will performances of understanding be judged?) 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?) Expressions and Functions 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals Papers showing students corrections of errors and misconceptions Portfolios Linear Functions and Systems of Equations 16 days 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals Papers showing students corrections of errors and misconceptions Portfolios Two- and Three-dimensional Geometry 18 days 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals Papers showing students corrections of errors and misconceptions Portfolios 13

14 RESOURCES: Expressions and Functions Linear Functions and Systems of Equations 16 days Two- and Three-dimensional Geometry 18 days Benchmark at the end of the unit Benchmark at the end of the unit Benchmark at the end of the unit Homework Notebooks/binder Notebooks/binder Notebooks/binder Agenda Agenda Agenda NJ ASK Mathematics Reference Sheet, NJ ASK Mathematics Reference Sheet, NJ ASK Mathematics Reference Sheet, Grade 8 Grade 8 Grade 8 Explore Activities Explore Activities Explore Activities Go Math Differentiated Workbooks Go Math Differentiated Workbooks Go Math Differentiated Workbooks Differentiated Grouping Differentiated Grouping Differentiated Grouping Go Math Assessment resources Go Math Assessment resources Go Math Assessment resources ipads ipads ipads PARCC Coach Workbooks PARCC Coach Workbooks PARCC Coach Workbooks IXL.com IXL.com IXL.com My.hrw.com My.hrw.com My.hrw.com Renaissance Renaissance Renaissance Graphic Organizers Graphic Organizers Graphic Organizers Manipulatives Manipulatives Manipulatives NJ ASK preparation problems NJ ASK preparation problems NJ ASK preparation problems 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. 14

15 SKILLS AND TOPICS: (What specific activities will students do and what skills will students know as a result of the unit?) Expressions and Functions Define, evaluate, and compare functions. 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 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. Use functions to model relationships between quantities. 7-8.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 Functions and Systems of Equations 16 days Understand the connections between proportional relationships, lines, and linear equations. 7-8.EE.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 moving objects has greater speed. 7-8.EE.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. Analyze and solve linear equations and pairs of simultaneous linear equations. 7-8.EE.8. Analyze and solve pairs of simultaneous linear equations. Understand that solutions to a system of two linear equations in Two- and Three-dimensional Geometry 18 days Understand congruence and similarity using physical models, transparencies, or geometry software. 7-8.G.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. Understand and apply the Pythagorean Theorem. 7-8.G.6. Explain a proof of the Pythagorean Theorem and its converse. 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 15

16 Expressions and Functions linear function in terms of the situation it models, and in terms of its graph or a table of values. 7-8.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. Investigate patterns of association in bivariate data. 7-8.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. Linear Functions and Systems of Equations 16 days 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. Two- and Three-dimensional Geometry 18 days technology, engineering, technological design and the designed world as they relate to the individual, global society, and the environment. 7-8.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 Define, evaluate, and compare functions. 7-8.F.2. Compare properties of two functions each represented in a different way (algebraically, graphically, 16

17 Expressions and Functions 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? 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. Linear Functions and Systems of Equations 16 days 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. Use functions to model relationships between quantities. 7-8.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. 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 Two- and Three-dimensional Geometry 18 days 17

18 Expressions and Functions Linear Functions and Systems of Equations 16 days features of a function that has been described verbally. Two- and Three-dimensional Geometry 18 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. CROSS-CURRICULAR: (What cross-curricular (e.g. writing, literacy, math, science, history, 21 st century life and careers, technology) learning activities are included in this unit that will help achieve the desired results?) Centers Tiered Lessons Open-Ended Responses Differentiation: Special Education: Modifications as dictated in the 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. 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 Centers Tiered Lessons Open-Ended Responses Differentiation: Special Education: Modifications as dictated in the Centers Tiered Lessons Open-Ended Responses Differentiation: Special Education: Modifications as dictated in the

19 Expressions and Functions 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 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. 19 Linear Functions and Systems of Equations 16 days 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 assessments. Go Math Spanish resources for assessments and assignments. 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. Two- and Three-dimensional Geometry 18 days 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 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

20 Expressions and Functions 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 21 st Century learner. At-Risk Students Tiered groups and tiered assignments Small group instruction, one-to-one instruction and additional online 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. Linear Functions and Systems of Equations 16 days 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 21 st Century learner. At-Risk Students Tiered groups and tiered assignments Small group instruction, one-to-one instruction and additional online 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. Two- and Three-dimensional Geometry 18 days 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 21 st Century learner. At-Risk Students Tiered groups and tiered assignments Small group instruction, one-to-one instruction and additional online 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. 20

21 Triangles and Transformations 21 days Units of Measure 10 days Data Analysis and Statistics ESTABLISHED GOALS: (NJ CCCS and/or CCS) ENDURING UNDERSTANDINGS: (Students will Understand that...) STAGE 1: DESIRED RESULTS What will students understand as a result of the unit? What are the BIG ideas? Math CCSS 7-8.EE.2, 7-8.EE.6, 7-8.G.1, 7-8.G.2, 7-8.G.3, 7-8.G.4, 7-8.G.5, 7-8.G.7, 7-8.G F.4 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 Similar triangles help solve problems including height and distances. The Pythagorean Theorem solves problems with right triangles as well as finding the distance between two points on the coordinate plane. The effects a transformation has on congruency and location on the coordinate plane. Math CCSS 7-8.EE.8 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 They may solve equations for any variable in common formulas. They use unit ratios to convert larger units to smaller units. Use of unit ratios and dimensional analysis converts length, weight/mass, capacity, time, temperature, area, and volume between units of the same system and between units of different systems. Math CCSS 7-8.G.7, 7-8.SP.1, 7-8.SP.2, 7-8.SP.3 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3 Displays of data convey information and make conjectures about relationships among these data sets. Changes in data values affect summary measures. 21

22 Triangles and Transformations 21 days Units of Measure 10 days Data Analysis and Statistics ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) How are similar triangles, right triangle relationships, and transformations used in the real world? How can proportional reasoning be applied to problem-solving situations involving similar and right triangles? How is the Distance Formula related to the Pythagorean Theorem? How are the results of a transformation different than the original figure? How are they similar? Why is it important to know how to convert between units of measure? Why is it important for scientists who collaborate on a project to use the same temperature scale? What is the role of unit ratios when converting between customary and metric systems of measurement? How can you summarize a set of data using only a few numbers and graphical displays? Why are the measures of central tendency important for interpreting data? How are measures of variation and box-and-whisker plots helpful in interpreting data? How do scatter plots and lines of best fit enable you to make predictions about data? 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. 22 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.

23 Triangles and Transformations 21 days Units of Measure 10 days Data Analysis and Statistics 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?) 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals Papers showing students corrections of errors and misconceptions Portfolios Benchmark at the end of the unit 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals Papers showing students corrections of errors and misconceptions Portfolios Benchmark at the end of the unit 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. Teacher observations Rubrics Bulletin Boards of exemplars Tests Quizzes Peer and Self evaluations Presentations Daily notes Long-Term projects Interviews between student and peer Interviews between student and teacher Group projects Homework Self-generated problems and solutions Journals Papers showing students corrections of errors and misconceptions Portfolios Benchmark at the end of the unit RESOURCES: Notebooks/binder Agenda Notebooks/binder Agenda Notebooks/binder Agenda 23

24 Triangles and Transformations 21 days Units of Measure 10 days Data Analysis and Statistics NJ ASK Mathematics Reference Sheet, Grade 8 Explore Activities Go Math Differentiated Workbooks Differentiated Grouping Go Math Assessment resources ipads PARCC Coach Workbooks IXL.com My.hrw.com Renaissance Graphic Organizers Manipulatives NJ ASK preparation problems NJ ASK Mathematics Reference Sheet, Grade 8 Explore Activities Go Math Differentiated Workbooks Differentiated Grouping Go Math Assessment resources ipads PARCC Coach Workbooks IXL.com My.hrw.com Renaissance Graphic Organizers Manipulatives NJ ASK preparation problems NJ ASK Mathematics Reference Sheet, Grade 8 Explore Activities Go Math Differentiated Workbooks Differentiated Grouping Go Math Assessment resources ipads PARCC Coach Workbooks IXL.com My.hrw.com Renaissance Graphic Organizers Manipulatives NJ ASK preparation problems 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?) Work with radicals and integer exponents. 7-8.EE.2. Use square root and cube root symbols to represent solutions to 24 Analyze and solve linear equations and pairs of simultaneous linear equations. 7-8.EE.8. Analyze and solve pairs of simultaneous linear equations. Understand and apply the Pythagorean Theorem. 7-8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in

25 Triangles and Transformations 21 days Units of Measure 10 days Data Analysis and Statistics equations of the form x 2 = p and x 3 = 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. Understand the connections between proportional relationships, lines, and linear equations. 7-8.EE.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. Understand congruence and similarity using physical models, transparencies, or geometry software. 7-8.G.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 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. 8.1 All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve right triangles in real-world and mathematical problems in two and three dimensions. Investigate patterns of association in bivariate data. 7-8.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. 7-8.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. 7-8.SP.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, 25

26 Triangles and Transformations 21 days Units of Measure 10 days Data Analysis and Statistics the same measure. c. Parallel lines are taken to parallel lines. 7-8.G.2. Understand that a twodimensional 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. 7-8.G.3. Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates. 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. 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. 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. 7-8.G.4. Understand that a twodimensional 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. 7-8.G.5. Use informal arguments to 26

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