Mathematics GRADE: 7

Transcription

1 Mathematics GRADE: 7 Prerequisite: Mathematics, Grade 6 Credits: 5 ABSTRACT The purpose of this full year, five credit course is to develop the students aptitude for solving problems by using a variety of strategies. Students 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. In Grade 7, instructional time should focus on four critical areas: (1) developing understanding of and applying proportional relationships; (2) developing understanding of operations with rational numbers and working with expressions and linear equations; (3) solving problems involving scale drawings and informal geometric constructions, and working with two- and three-dimensional shapes to solve problems involving area, surface area, and volume; and (4) drawing inferences about populations based on samples.

2 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October ESTABLISHED GOALS: (NJ CCCS and/or CCS) STAGE 1: DESIRED RESULTS What will students understand as a result of the unit? What are the BIG ideas? Math CCSS EE.7.4, EE.7.2, NS.7.1 Technology CCCS 8.1, 8.2 Math CCSS NS.7.1, NS.7.2, NS.7.3, EE.7.3 Technology CCCS 8.1, 8.2 Math CCSS NS.7.1, NS.7.2, NS.7.3, EE.7.3 Technology CCCS 8.1, st Century Life and Careers CCCS 9.1, 9.2, 9.3, st Century Life and Careers CCCS 9.1, 9.2, 9.3, st Century Life and Careers CCCS 9.1, 9.2, 9.3, 9.4 ENDURING UNDERSTANDINGS: (Students will Understand that...) Evaluating expressions and extending patterns. Evaluate numerical expressions involving powers and exponents. Evaluate algebraic expressions for given variable values. Use properties to evaluate expressions. Describe and extend sequences. Find square roots of perfect squares and estimate roots of numbers that are not perfect squares. Students read and write integers that correspond to positive and negative situations and find absolute values. They will use and justify the rules for adding, subtracting, multiplying, and dividing integers. These skills will build a foundation for students to apply integer rules to all rational numbers in order to solve algebraic equations and realworld problems. Students develop an understanding of rational numbers and their different forms. They express fractions as decimals and decimals as fractions. Rational numbers are compared and ordered. As with integers, students will learn how to add, subtract, multiply, and divide fractions. Students solve real-world problems with rational numbers. Students use exponents to multiply and divide monomials and to express numbers in scientific notation. 2

3 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) What is a mathematical expression, and how is it useful? How are expressions helpful in real-life? How can a sequence be used to solve a problem? How can you use the square roots of non-perfect squares in problemsolving? How can you use numbers to describe concepts such as sea level, losing yardage in a football game, or temperatures that drop below zero. What is the significance of absolute value? How is adding and subtracting integers like adding and subtracting whole numbers? How is it different? How do positive and negative numbers affect number operations? What are rational numbers? What is the relationship between rational numbers, integers, and whole numbers? How is adding unlike fractions similar to adding like fractions? How is it different? How is multiplying fractions different from adding fractions? How is it similar? What is the relationship between negative exponents and rational numbers? 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?) 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. 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. 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. 3

4 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October 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?) RESOURCES: 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 7 Graphic Organizers Manipulatives 4 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 7 Graphic Organizers Manipulatives 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 7 Graphic Organizers Manipulatives

5 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October 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. SKILLS AND TOPICS: (What specific activities will students do and what skills will students know as a result of the unit?) Use properties of operations to generate equivalent expressions. EE.7.2: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. EE.7.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to 5 Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. NS.7.1: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. NS.7.1: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by

6 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. NS.7.1: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. b. Understand p + q as the number located a distance q from p, in the 6 describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. NS.7.2: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. NS.7.2: Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as ( 1)( 1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with

7 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. 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, 7 (with non-zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q =p/( q). Interpret quotients of rational numbers by describing real-world context. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. NS.7.3: Solve real-world and mathematical problems involving the four operations with rational numbers. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. EE.7.3: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of non-zero divisor) is a rational number. If p and q are integers, then (p/q) = ( p)/q =p/( q). Interpret quotients of rational numbers by describing real-world context. c. Apply properties of operations as strategies to multiply and divide rational numbers. d. Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. NS.7.3: Solve real-world and mathematical problems involving the four operations with rational numbers. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. EE.7.3: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental

8 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October 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. 9.4: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, 8 answers using mental computation and estimation strategies. 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 computation and estimation strategies. 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 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October CROSS-CURRICULAR / DIFFERENTIATION: (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?) (What type of differentiated instruction will be used for ELL, SP.ED. and G&T students?) licenses, and/or degrees. 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. 9 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. 9.4: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 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. 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. 9.4: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 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.

10 Expressions and Patterns 11 Days September Integers 13 Days September-October Rational Numbers October 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. Gifted and Talented: Advanced Problem Solving Above grade level math placement option for qualified students, including Pre-Algebra, Algebra I, and Geometry 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 21 st Century learner. 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. Gifted and Talented: Advanced Problem Solving Above grade level math placement option for qualified students, including Pre-Algebra, Algebra I, and Geometry 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 21 st Century learner. 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. Gifted and Talented: Advanced Problem Solving Above grade level math placement option for qualified students, including Pre-Algebra, Algebra I, and Geometry 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 21 st Century learner. 10

11 *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: Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January ESTABLISHED GOALS: (NJ CCCS and/or CCS) STAGE 1: DESIRED RESULTS What will students understand as a result of the unit? What are the BIG ideas? Math EE.7.4 Technology 8.1, 8.2 Math RP.7.1, RP.7.2, RP.7.3, G.7.1 Technology 8.1, 8.2 Math RP.7.2, RP.7.3, EE.7.3 Technology 8.1, 8.2 ENDURING UNDERSTANDINGS: (Students will Understand that...) 21 st Century Life and Careers 9.1, 9.2, 9.3, 9.4 Students explore, investigate, and solve equations and inequalities using properties. 21 st Century Life and Careers 9.1, 9.2, 9.3, 9.4 Students will understand that if a scale factor describes how corresponding lengths in two 21 st Century Life and Careers 9.1, 9.2, 9.3, 9.4 Students will understand and apply proportionality, including similarity. Extend their work with ratios to 11

12 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) Students will understand operations on all rational numbers and solving linear equations. Students will apply properties of arithmetic and consider negative numbers in everyday contexts. Use the arithmetic of rational numbers as they formulate and solve linear equations in one variable and use equations to solve problems. Make strategic choices of procedures to solve linear equations in one variable and implement them efficiently. What do you know about the solution to an equation? An inequality? How can you use Properties of Equality to solve addition and subtraction equations? Why do both sides of an equation remain equal when both sides are multiplied or divided by the same number? 12 similar objects are related, the square of the scale factor describes how corresponding areas are related, and the cube of the scale factor describes how corresponding volumes are related. Students apply their work on proportionality to measurement in different contexts, including converting among different units of measurement to solve problems involving rates such as motion and constant speed. Students will apply proportionality when they work with the circumference, radius, and diameter of a circle; when they find the area of a sector of a circle; and when they make scale drawings. What are ratios and proportions and how do we use them in our daily lives? How can you determine if two quantities are proportional? How are ratios, proportions, and scale drawings used in the real world? How are similar figures scale drawings used in real-world develop an understanding of proportionality that they apply to solve single and multistep problems in numerous contexts. Use ratio and proportionality to solve a wide variety of percent problems, including problems involving discounts, interest, taxes, tips, and percent increase or decrease. Solve problems about similar objects by using scale factors that relate corresponding lengths of the objects. Graph proportional relationships (y/x = k, or y = kx) from other relationships, including inverse proportionality (xy = k, or y = k/x). What are percentages and how are they used in everyday life? How can equivalent fractions be helpful in estimating percentages? How can percent equations be applied to real-world situations? How is calculating sales tax similar to calculating tips?

13 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January Why is it important to combine like terms before completing number operations when solving two-step equations? How is solving an inequality similar to solving and equations? How is it different? situations? 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?) OTHER EVIDENCE: (Through what other evidence (e.g. 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 13 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 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

14 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January quizzes, tests, academic prompts, observations, homework, journals) will students demonstrate achievement of the desired results?) (How will students self-assess their learning?) RESOURCES: 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 7 Graphic Organizers Manipulatives NJ ASK preparation problems 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 7 Graphic Organizers Manipulatives NJ ASK preparation problems 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, Grade 7 Graphic Organizers Manipulatives NJ ASK preparation problems

15 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January 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?) Solve real-life and mathematical problems using numerical and algebraic expressions and equations. EE.7.4: Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. a. Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach b. Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. Analyze proportional relationships and use them to solve real-world and mathematical problems. RP.7.1: Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. RP.7.2: Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. Analyze proportional relationships and use them to solve real-world and mathematical problems. RP.7.2: Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. 15

16 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January 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 16 d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. RP.7.3: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Draw construct, and describe geometrical figures and describe the relationships between them. G.7.1: Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. 8.1: All students will use digital tools to access, manage, evaluate, and synthesize information in order to solve problems individually and RP.7.3: Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. Solve real-life and mathematical problems using numerical and algebraic expressions and equations. EE.7.3: Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. 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

17 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January 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. 9.4: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 17 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 preparation in order to navigate the globally competitive work environment 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. 9.4: All students who complete a career and technical education program will

18 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January 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 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. 18 of the information age. 9.4: All students who complete a career and technical education program will acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 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. acquire academic and technical skills for careers in emerging and established professions that lead to technical skill proficiency, credentials, certificates, licenses, and/or degrees. 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

19 Equations and Inequalities 17 Days November-December Proportions and Similarity 12 Days December Pecent January Extended time for completing assessments. Gifted and Talented: Advanced Problem Solving Above grade level math placement option for qualified students, including Pre-Algebra, Algebra I, and Geometry 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 21 st Century learner. Extended time for completing assessments. Gifted and Talented: Advanced Problem Solving Above grade level math placement option for qualified students, including Pre-Algebra, Algebra I, and Geometry 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 21 st Century learner. assessments. Gifted and Talented: Advanced Problem Solving Above grade level math placement option for qualified students, including Pre-Algebra, Algebra I, and Geometry 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 21 st Century learner. UNIT: Linear Functions January-February Probability and Predictions February-March Statistical Displays March 19

20 Linear Functions January-February Probability and Predictions February-March Statistical Displays March ESTABLISHED GOALS: (NJ CCCS and/or CCS) STAGE 1: DESIRED RESULTS What will students understand as a result of the unit? What are the BIG ideas? Math CCSS RP.7.2 Technology CCCS 8.1, 8.2 Math CCSS SP.7.1, SP.7.2, SP.7.5, SP.7.6, SP.7.8 Technology CCCS 8.1, 8.2 Math CCSS SP.7.3, SP.7.4 Technology CCCS 8.1, 8.2 ENDURING UNDERSTANDINGS: (Students will Understand that...) 21 st Century Life and Careers CCCS 9.1, 9.2, 9.3, 9.4 Students will understand how to apply proportionality, including similarity. Apply proportionality to solve single and multistep problems in numerous contexts. Use ratio and proportionality to solve a variety of percent problems, including discount, interest, taxes, tips, and percent increase or decrease. Solve problems about similar objects by using a scale factors that relate corresponding lengths of the objects. Graph proportional relationships and identify the unit rate as the 21 st Century Life and Careers CCCS 9.1, 9.2, 9.3, 9.4 Students will understand that when all outcomes of an experiment are equally likely, the theoretical probability of an event is the fraction of outcomes in which the event occurs. Use theoretical probability and proportions to make approximate predictions. Understand the distinction between independent and dependent events. Be able to use sample data to make predictions and determine if samples are valid. 21 st Century Life and Careers CCCS 9.1, 9.2, 9.3, 9.4 Students will understand how to use proportions to make estimates relating to a population on the basis of a sample. Apply percentages to make and interpret histograms and circle graphs. Explore how changes in data values and extreme values affect measures of central tendency. Students will understand why a specific measure of central tendency provides the most useful information in a given context. Draw conclusions and make predictions based on trends and relationships among data. 20

21 Linear Functions January-February Probability and Predictions February-March Statistical Displays March ESSENTIAL QUESTIONS: (What provocative questions will foster inquiry, understanding, and transfer of learning?) slope of the related line. Distinguish proportional relationships from other relationships, including inverse proportionality. What kinds of relationships might exist between two quantities and what do the graphs of these relationships look like? Why is it useful to be able to represent functions in various ways? How are slope, rate of change, and constant rate of change related? When is a linear function a direct variation? How is a direct variation different from inverse variation? What are some real-world situations in which you can use probability to make predictions? How is the sample space related to the probability of an event? How are compound events related to simple events? How can experimental probability be used to make predictions? Understand how to select and construct appropriate data displays for different kinds of data. How are statistical displays helpful to us in our everyday lives? Why are measures of central tendency important for interpreting data? How do you select an appropriate graphic display for a set of data? Why is it important to understand statistics and different types of graphs? Why is it important to select an appropriate display when graphing 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 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 21

22 Linear Functions January-February Probability and Predictions February-March Statistical Displays March 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?) RESOURCES: 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet, 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 Notebooks/binder Agenda NJ ASK Mathematics Reference Sheet,

23 Linear Functions January-February Probability and Predictions February-March Statistical Displays March Grade 7 Graphic Organizers Manipulatives NJ ASK preparation problems Grade 7 Graphic Organizers Manipulatives NJ ASK preparation problems Grade 7 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?) Analyze proportional relationships and use them to solve real-world and mathematical problems. RP.7.2: Recognize and represent proportional relationships between quantities. 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. 23 Use random sampling to draw inferences about a population. SP.7.1: Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. SP.7.2: Use data from a random sample Draw informal comparative inferences about two populations. SP.7.3: Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. SP.7.4: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations.