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PPS Mathematics Curriculum Algebra I Plainfield Public Schools Mathematics Unit Planning Organizer Grade/Course Unit of Study Pacing Dates MP1. MP2. MP3. MP4. MP5. MP6. MP7. MP8. Algebra 1 Unit 3 Expressions and Equations 4 weeks; 3 instructional week , 1 weeks for reteaching and enrichment January 5 – January 29.2016 CCSSM Mathematical Practices Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I I. Unit Standards A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* A.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. “Unwrapped” Skills “Unwrapped” Concepts DOK Levels (students need to be able to do) (students need to know) FOCUS STANDARD: A.CED.1 Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. Create equations and inequalities 3 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts DOK Levels (students need to know) FOCUS STANDARD: A.CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales Create equations 3 graph 3 “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts DOK Levels (students need to know) FOCUS STANDARD: A.CED.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. Rearrange formulas 3 “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts DOK Levels (students need to know) FOCUS STANDARD: A.REI.A.1 Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Explain equations 2 Construct 3 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts DOK Levels (students need to know) FOCUS STANDARD: A.REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Solve Equations 2 Inequalities “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts (students need to know) DOK Levels FOCUS STANDARD: A.REI.D.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.* Explain Equations 1 “Unwrapped” Skills (students need to be able to do) “Unwrapped” Concepts DOK Levels (students need to know) FOCUS STANDARD: A.REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding halfplanes. Graph Linear inequality 2 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I II Mathematical Practices………………………………………………… Explanation & Examples Algebra: Creating Equations (A-CED) Create equations that describe numbers or relationships. Standards Mathematical Explanations and Examples Students are expected to: Practices HS.A-CED.A.1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. HS.A-CED.A.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. HS.MP.2. Reason abstractly and quantitatively. Equations can represent real world and mathematical problems. Include equations and inequalities that arise when comparing the values of two different functions, such as one describing linear growth and one describing exponential growth. HS.MP.4. Model with mathematics. Examples: 2 ● Given that the following trapezoid has area 54 cm , set up an equation to find the length of the base, and solve the equation. ● Lava coming from the eruption of a volcano follows a parabolic path. The height h in feet of a piece of lava t seconds after it is ejected from the volcano is given by After how many seconds does the lava reach its maximum height of 1000 feet? HS.MP.5. Use appropriate tools strategically. HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I HS.A-CED.A.4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R. HS.MP.2. Reason abstractly and quantitatively. Examples: ● HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.7. Look for and make use of structure. ● The Pythagorean Theorem expresses the relation between the legs a and b of a right 2 2 2 triangle and its hypotenuse c with the equation a + b = c . o Why might the theorem need to be solved for c? o Solve the equation for c and write a problem situation where this form of the equation might be useful. o Solve for radius r. Motion can be described by the formula below, where t = time elapsed, u=initial velocity, a = acceleration, and s = distance traveled 2 s = ut+½at o o Why might the equation need to be rewritten in terms of a? Rewrite the equation in terms of a. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Algebra: Reasoning with Equations and Inequalities (A-REI) Understand solving equations as a process of reasoning and explain the reasoning. Standards Mathematical Practices Explanations and Examples Students are expected to: HS.A-REI.A.1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. HS.MP.2. Reason abstractly and quantitatively. HS.MP.3. Construct viable arguments and critique the reasoning of others. HS.MP.7. Look for and make use of structure. Properties of operations can be used to change expressions on either side of the equation to equivalent expressions. In addition, adding the same term to both sides of an equation or multiplying both sides by a non-zero constant produces an equation with the same solutions. Other operations, such as squaring both sides, may produce equations that have extraneous solutions. Examples: Explain why the equation x/2 + 7/3 = 5 has the same solutions as the equation 3x + 14 = 30. Does this mean that x/2 + 7/3 is equal to 3x + 14? Show that x = 2 and x = -3 are solutions to the equation Write the equation in a form that shows these are the only solutions, explaining each step in your reasoning. Algebra: Reasoning with Equations and Inequalities (A-REI) Solve equations and inequalities in one variable. Standards Mathematical Practices Explanations and Examples Students are expected to: HS.A-REI.B.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. HS.MP.2. Reason abstractly and quantitatively. HS.MP.7. Look for and make use of structure. HS.MP.8. Look for and express regularity in repeated reasoning. Examples: 7 y 8 111 3 3x > 9 ax + 7 = 12 3 x x 9 7 4 Solve for x: 2/3x + 9 < 18 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Algebra: Reasoning with Equations and Inequalities (A-REI) Represent and solve equations and inequalities graphically. HS.A-REI.D.11. Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. Connection: ETHS-S6C2-03 HS.MP.2. Reason abstractly and quantitatively. HS.MP.4. Model with mathematics. HS.MP.5. Use appropriate tools strategically. HS.MP.6. Attend to precision. Students need to understand that numerical solution methods (data in a table used to approximate an algebraic function) and graphical solution methods may produce approximate solutions, and algebraic solution methods produce precise solutions that can be represented graphically or numerically. Students may use graphing calculators or programs to generate tables of values, graph, or solve a variety of functions. Example: Given the following equations determine the x value that results in an equal output for both functions. f ( x ) 3x 2 g ( x ) ( x 3)2 1 Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Algebra: Reasoning with Equations and Inequalities (A-REI) Represent and solve equations and inequalities graphically. Standards Mathematical Practices Students are expected to: HS.A-REI.D.12. Graph the solutions HS.MP.4. Model with to a linear inequality in two mathematics. variables as a half-plane (excluding HS.MP.5. Use appropriate the boundary in the case of a strict tools strategically. inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. Explanations and Examples Students may use graphing calculators, programs, or applets to model and find solutions for inequalities or systems of inequalities. Examples: Graph the solution: y < 2x + 3. A publishing company publishes a total of no more than 100 magazines every year. At least 30 of these are women’s magazines, but the company always publishes at least as many women’s magazines as men’s magazines. Find a system of inequalities that describes the possible number of men’s and women’s magazines that the company can produce each year consistent with these policies. Graph the solution set. Graph the system of linear inequalities below and determine if (3, 2) is a solution to the system. x 3y 0 x y 2 x 3 y 3 Solution: (3, 2) is not an element of the solution set (graphically or by substitution). Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Ill .Essential Questions ……………………………………………Corresponding Big ideas Essential Questions Corresponding Big Ideas What can you learn about a function from a table, graph, or equation that represents the function? For functions that map real numbers to real numbers, certain patterns of covariation, or patterns in how two variables change together, indicate membership in a particular family of functions and determine the type of formula that the function has. How can you determine whether the graph of an equation is linear? A rate of change describes how one variable quantity changes with respect to another – in other words, a rate of change describes the covariation between two variables. Members of a family of functions share the same type of rate of change. This characteristic rate of change determines the kinds of real-world phenomena that the functions in the family can model. How do the solution of inequalities relate to the solution of equations? How can you use an equations to determine a function and the situation it represents ? How can you use graphs to find the solutions of an inequality ? How does applying the same operation to each side on an inequality change ( or not) the relationship of two quantities being . Changing the way that a function is represented does not change the function, although different representations highlight different characteristics, and some may only show part of the function. Some representations of a function may be more useful than others, depending on the context Links between algebraic and graphical representations of functions are especially important in studying relationships and change. Real numbers, composing a function with “shifting” or “scaling” functions changes the formula and graph of the function in readily predictable ways. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I compared ? Solving an equation is the process of rewriting the equation to make what it says about its variable(s) as simple as possible. Properties of number and equality can be used transform an equation ( or inequality) into equivalent ,simpler equation to find solution . Solving equation will include factoring as a means to acquire the zeroes of a function and on completing the square as a means to determine the maximum or minimum of the function. Quadratic equations can be solved by a variety of methods, finding the square root , using the Quadratic formula completing the square, factoring or Zero Product Property Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I IV. Student Learning Objective Student Learning Objective Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, simple rational and exponential functions and highlighting a quantity of interest in a formula. A.CED.1, A.CED.4 Create linear relationship between two or more variables. Graph equations on the coordinate axes with labels and scale. A.CED.2, Graph equations and inequalities, explain that the solution to an equation is all points along the curve A.REI.12 Describe constraints with linear equations and inequalities to determine if solutions are viable or non-viable A.REI.11 PARCC Instructional Clarification Mathematics Assessment Test Specifications Tasks have a real-world context. The quantity of interest is linear in nature. Mathematical Practices MP.2 MP.6 MP.7 MP.2 MP.4 • Solve multi-step contextual word problems with degree of difficulty appropriate to the course, requiring application of course-level knowledge and skills articulated in A-CED, N-Q.2, A-SSE.3,A-REI.6, A-REI.12, A-REI.11-1, limited to linear and quadratic equations CED is the primary content; other listed content elements may be involved in tasks as well. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I V. Unit Vocabulary Unit Vocabulary Terms Inequality Equivalent Expression Equivalent inequalities Commutative Property of addition Commutative Property of Multiplication Distributive Property Combining Expressions Solution of an inequalities Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I V . Differentiating Instruction Research Based Effective Teaching Strategies Task /Activities that solidifies mathematical concepts Use questioning techniques to facilitate learning Reinforcing Effort, Providing Recognition Practice , reinforce and connect to other ideas within mathematics Modifications ( how do I differentiate instruction?) Modifications Before or after school tutorial program Leveled rubrics Increased intervention Small groups Change in pace Calculators Extended time Alternative assessments Tiered activities/products Color coded notes Use of movements Use any form of technology Promotes linguistic and nonlinguistic representations Cooperative Learning Setting Objectives, Providing Feedback Varied opportunities for students to communicate mathematically Use technological and /or physical tools Extension Research sound frequencies that can be heard by various animals. Students present ranges as inequalities. Student can identify animals with the best and worst hearing, those that hear low best or high sound best. Student describe in written form how animals may use the range to their advantage Special Education Strategies for English Language Learners Change in pace Calculators Alternative assessments Accommodations as per IEP Modifications as per IEP Use graphic organizer to clarify mathematical functions for students with processing and organizing difficulties’. Whiteboards Small Group / Triads Word Walls Partially Completed Solution Gestures Native Language Supports Pictures / Photos Partner Work Work Banks Teacher Modeling Math Journals Constant review of math concepts to strengthen understanding of prior concepts for difficulties recalling facts. Use self-regulations strategies’ for student to monitor and assess their thinking and performance for difficultly attending to task Cooperative learning (small group, teaming, peer assisted tutoring) to foster communication and strengthen confidence. Use technology and/or hands on devices to: clarify abstract concepts and process for : 1. Difficulty interpreting pictures Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. . PPS Mathematics Curriculum Algebra I 21st Century Learning Skills : Teamwork and Collaboration Initiative and Leadership Curiosity and Imagination Innovation and Creativity Critical thinking and Problem Solving Flexibility and Adaptability Effective Oral and Written Communication Accessing and Analyzing Information and diagram. 2.difficulties with oral communications 3. Difficulty correctly identifying symbols of numeral 4.Difficulty maintaining attentions Simplify and reduces strategies / Goal structure to enhance motivation , foster independence and self-direction for: 1.difficulty attending to task 2. difficulty with following a sequence of steps to solution. 3.difficulty processing and organizing Scaffolding math idea/concepts guided practice and questioning strategies’ to clarify and enhance understanding of math big ideas for : 1.Difficulty with process and organization 2.difficulty with oral and written Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Enrichment : Use any form of technology (graphing calculators) Tiered activities/products Use of movement Use of realia, graphic organizers, visuals. communication Teacher models strategies’ and think out aloud strategies to specify step by step process for 1.Difficulties processing and organization 2. Difficulty attending to tasks. Use bold numbers and/or words to draw students’ attention to important information. **Use a straw or straight pasta to with coordinate grids to deepen students’ understanding of slope, y-intercept, and point of intersection and / or systems of equations. Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Instructional Resources Instructional Resources and Materials Formative Assessment Short constructed responses Extended responses Checks for Understanding Exit tickets Teacher observation Projects Timed Practice Test – Multiple Choice & Open-Ended Questions Performance Tasks: Illustrative Mathematics .Basketball Summative Assessment: End of Unit Assessment for Algebra 1 Unit 3 Equations and Inequalities Print Pearson Algebra 1 with Foundation © 2011 ( middle school) : Chapter 2 and Chapter 3 Holt McDougal Larson Algebra 1 ©2011 (high School) : Chapter 3 and Chapter 6 Technology Resources for teachers Annenberg Learning : Insight into Algebra 1 Mathematics Assessment Projects Get the Math Achieve the Core Webmath.com sosmath.com Mathplanet.com Interactive Mathematics.com Illustrative Mathematics Inside Mathmatics.org Asia Pacific Economic Cooperation : :Lesson Study Videos Genderchip.org Interactive Geometry Mathematical Association of America National Council of Teachers of Mathematics learner.org Math Forum : Teacher Place Shmoop /common core math Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. Resources for Students Khan Academy Math world : Wolfram.com Webmath.com sosmath.com Mathplanet.com Interactive Mathematics.com Mathexpression.com.algebra Math Words for Advance Algebra & Pre-Calculus Math TV Virtual Nerd : Algebra 1 PPS Mathematics Curriculum Algebra I Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved. PPS Mathematics Curriculum Algebra I Copyright 2011, The Leadership and Learning Center. 866.399.6019. All rights reserved.