1 Naugatuck High School Algebra I: Unit 1

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1 Subject(s) Grade/Course Unit of Study Unit Type(s) Pacing Algebra I 9 th grade #1: Patterns Skills-based 2 weeks plus 1 week re-teaching Overarching Standards: Understand the concept of a function and use function notation Build a function that models a relationship between two quantities Math Practices: (Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.) 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. Priority and Supporting Standards: (CCSS/Content Standards, ELL) F-IF 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers... Explanations and Examples: Recursive and explicit rules for sequences are first introduced in the context of chemistry. For example, the number of hydrogen atoms in a hydrocarbon is a function of the number of carbon atoms. This relationship may be defined by the recursive rule add two to the previous number of hydrogen atoms or explicitly as h = 2 + 2c. The function may also be represent in a table or a graph or with concrete models. F-BF 1. Write a function that describes a relationship between two quantities.* a. Determine an explicit expression, a recursive process, or steps for calculation from a context. F-BF 2. Write arithmetic and geometric sequences... recursively and [arithmetic sequences] with an explicit formula, use them to model situations, and translate between the two forms.* Example; You frequently go to the gym to work out lifting weights. You plan to gradually increase the size of the weights over the next month. You always put two plates that appear to be the same weight on each side of the bar. The plates are not labeled but you do know the bar weighs 20 kg. How can we express the total weight you lifted on any day? Students will assign a variable for the weight of a plate (say w) and derive an expression for the total weight lifted, 4w +20 or its equivalent. Arithmetic sequences may be introduced through geometric models. For example, the number of beams required to make a steel truss in the following pattern is an arithmetic sequence. 1

2 Priority and Supporting Standards: (CCSS/Content Standards, ELL) Essential Question(s): Explanations and Examples: Arithmetic sequences are also found in patterns for integers which can be used to justify and reinforce rules for operations. For example in completing this pattern 5 * 4 = 5 * 3 = 5 * 2 = 5 * 1 = 5 * 0 = 5 * - 1 = 5 * - 2 = students will find an arithmetic sequence with a common difference of -5. This pattern illustrates that the product of a positive integer and a negative integer is negative. Compound interest is a good example of a geometric sequence. For example, You just won first prize in a poetry writing contest. If you take the $500 you won and invest it in a mutual fund earning 8% interest per year, about how long will it take for your money to double? Fractals are another example of geometric sequences. For example in creating the Sierpinski triangle the number of unshaded triangles in each stage form a geometry sequence. In this case the explicit rule, T= 3 n, is readily determined. In general, however in this unit we focus on recursive rules for geometric sequences since they will be revisited in Unit 7 in conjunction with exponential functions, where the explicit rule will be given greater emphasis. Enduring Understanding(s): What is a sequence? How can patterns be represented? What are the advantages and disadvantages of a recursive rule compared to an explicit rule? Exact from State of Connecticut Curricula: Analyzing patterns and writing recursive and explicit algebraic rules provides a powerful way to extend patterns and make predictions. Concepts What Students Need to Know Recursive rule Explicit rule Arithmetic sequence Geometric sequence Bloom s Taxonomy Levels Skills What Students Need To Be Able To Do Find (specific term) Write (recursive rule) Write (explicit rule) Draw (next in sequence) Predict (nth term) Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills Algebra uses symbols to represent quantities that are unknown or vary. Mathematical phrases and real-world relationships can be represented using symbols and operations. Any real number can be added or subtracted using a number line or using rules involving absolute value. Level 1: Recall/Reproduction Level 1: Recall/Reproduction Level 1: Recall/Reproduction To write algebraic expressions (C1.1) Applying existing knowledge to generate new ideas, products, or processes To find sums and differences of real numbers (C1.4) Identifying trends and forecast possibilities To find products and quotients of real numbers To solve equations using tables and (C2.4) Using multiple processes and diverse perspectives to explore alternative solutions mental math 2

3 Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills Subtracting a real number is equivalent to adding its opposite. The rules for multiplying real numbers are related to the properties of real numbers and the definitions of operations. The product or quotient of two real numbers with different signs is negative. The product or quotient of two real numbers with the same sign is positive. The relationship between the quantities can be represented in different ways, including tables, equations and graphs. Some sequences have function rules that can be used to find any term of the sequence. When the pattern in a sequence is identified, the sequence can be extended. A geometric sequence has an initial value and a subsequent sequence of values based on a common ratio. A geometric sequence can be modeled with an exponential function. Level 2: Skills/Concepts To use tables, equations and graphs to describe relationships Level 3: Strategic Thinking/Reasoning Level 3: Strategic Thinking/Reasoning Assured Assessments: (Performance Based/Curriculum Based) To identify and extend patterns and sequences To represent arithmetic sequences using function notation To write and use recursive formulas for geometric sequences Assessment Evidence Other Evidence: (C3/4.9) Assuming shared responsibility for collaborative and ethical work, and valuing the individual contributions made by each team member (C5.2) Participating actively, as well as being reliable and punctual Unit 1 Test o Advanced o Academic o Algebra 1-A PBA Fractal Geometry Score Using the NHS School-Wide Critical Thinking & Problem Solving Rubric o Algebra 1-A Entrance slips Exit slips Observations 3

4 Learning Plan Days CCSS DOK Instructional Practices Ensured Assignments # of Days per Section A.SSE.1a Level 1 Variables and Expressions CCSS 1.1 CCSS: Pg. 7 #9-25, 27-38, PH 1.1 PH: Pg. 6 #1-20, Differentiation/ Resources D: Teacher Notes 2 N.RN.3 Adding and Subtracting Real Numbers CCSS 1.5 PH n/a CCSS: Pg. 33 # 1 D: 3 N.RN.3 Multiplying and Dividing Real Numbers CCSS 1.6 PH n/a CCSS: Pg. 42 #11-43 odd, #44-65, D: 4 A.CED.1 Introduction to Equations CCSS 1.8 CCSS: Pg. 56 #29-47 odd, # D:

5 PH n/a 5 6 A.REI.10 A.CED.2 Patterns, Equations and Graphs CCSS 1.9 PH n/a CCSS: Pg. 56 #49, 51-63, CCSS: Pg.64 #9-25 odd, 27-33, D: D: 7 A.SSE.1a A.SSE.1b F.IF.3 F.BF.2 F.BF.1a Arithmetic Sequences /Describing Number Patterns CCSS 4.7 PH 5.7 CCSS: Pg. 279 #9-35 odd PH: Pg. 294 #1-12, 34-44, D: CCSS: Pg. 279 #37-53 odd, #76-87 D: 8 PH: Pg. 294 #13-33, 45-48, A.SSE.1a F.IF.3 F.BF.2 F.BF.1a Geometric Sequences CCSS 7.8 CCSS: Pg. 470 #9-37 odd, #38-52, D:

6 PH 8.6 PH: Pg. 463 #1-35 odd Common Misunderstandings Interdisciplinary Connection Unit Vocabulary Technology Resources 6

7 Absolute Value Additive Inverses Algebraic Expression Arithmetic Sequence Common Difference Equation Explicit Formula Geometric Sequence Inductive Reasoning Multiplicative Inverses Numerical Expression Open Sentence Opposites Quantity Reciprocals Recursive Formula Sequence Solution Solution of an Equation Term of a Sequence Variable Math behind the scenes video Activity labs Lessons Chapter reviews Editable worksheets (remediate/enriched; ELL support) Portable study center Math videos Kuta Infinite Software Pre-Algebra Algebra I Geometry Algebra II 7

8 Subject(s) Grade/Course Unit of Study Unit Type(s) Pacing Algebra I 9 th grade #2: Linear Equations and Inequalities Skills-based 4 weeks plus 1 week re-teaching Overarching Standards: Reason quantitatively and use units to solve problems. Interpret the structure of expressions Create equations that describe numbers or relationships Understand solving equations as a process of reasoning and explain the reasoning Solve equations and inequalities in one variable Math Practices: (Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.) 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. Priority and Supporting Standards: (CCSS/Content Standards, ELL) N-Q 1. Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. N-Q 2. Define appropriate quantities for the purpose of descriptive modeling. Explanations and Examples: In all problem situations the answer should be reported with appropriate units. In situations involving money, answers should be rounded to the nearest cent. When data sets involve large numbers (e.g. tables in which quantities are reported in the millions) the degree of precision in any calculation is limited by the degree of precision in the data. These ideas are introduced in the solution to contextual problems in Unit 2 and reinforced throughout the remainder of the course. N-Q 3. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. A-SSE 1. Interpret expressions that represent a quantity in terms of its context.* a. Interpret parts of an expression, such as terms, factors, and coefficients. Understanding the order of operations is essential to unpacking the meaning of a complex algebraic expression and to develop a strategy for solving an equation. b. Interpret complicated expressions by viewing one or more of their parts as a single entity... Using the commutative, associative and distributive properties enables students to find equivalent expressions, which are helpful in solving equations. 8

9 Priority and Supporting Standards: (CCSS/Content Standards, ELL) A-CED 1. (part) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear... functions Explanations and Examples: Here are some examples where students can create equations and inequalities. a. (Two step equation) The bank charges a monthly fee of $2.25 for your Dad s checking account and an additional $1.25 for each transaction with his debit card, whether used at an ATM machine or by using the card to make a purchase. He noticed a transaction charge of $13.50 on this month s statement. He is trying to remember how many times he used the debit card. Can you use the information on the statement help him figure out how many transactions he made? b. (Equations with variables on both sides) Willie and Malia have been hired by two different neighbors to pick-up mail and newspapers while they are on vacation. Willie will be paid $7 plus $3 per day. Malia will be paid $10 plus $3 per day. By which day will they have earned the same amount of money? c. (Equations which require using the distributive property) Jessica wanted to buy 7 small pizzas but she only had four $2 off coupons. So, she bought four with the discount and paid full price for the other three, and the bill came to $ How much was each small pizza? d. (Inequality) The student council has set aside $6,000 to purchase the shirts. (They plan to sell them later at double the price.) How many shirts can they buy at the price they found online if the shipping costs are $14? 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. Begin with a familiar formula such as one for the perimeter of a rectangle: p = 2l + 2w. Consider this progression of problems: (a) Values for the variables l and w are given. We can find p by substituting for l and w and evaluating the expression on the right side. (b) Values for the variables l and p are given. We can find w by substituting for l and p and solving the equation for w. (c) We can find a formula for w in terms of l and p, by following the same steps as in (b) above to solve for w. This gives us a general method for finding w when the other variables are known. (Check this new formula by showing that it gives the correct value for w when the values of l and p from (b) are substituted. ) 9

10 Priority and Supporting Standards: (CCSS/Content Standards, ELL) A-REI 1. Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. Explanations and Examples: For two-step equations, flow charts may be used to help students undo the order of operations to find the value of a variable. For example, this flow chart may be used to solve the equation 4x 2 = 30. Then students learn to solve equations by performing the same operation (except for division by zero) on both sides of the equal sign. 4x 2 = x = x = 8 In solving multiple-step equations students should realize that there may be several valid solution paths. For example here are two approaches to the equation 3x + 10 = 7x 6. 3x + 10 = 7x 6 3x + 10 = 7x 6 3x 3x 7x 7x 10 = 4x 6 4x + 10 = = 4x 4x =

11 4 = x x = 4 A-REI 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. Essential Question(s): Enduring Understanding(s): What is an equation? What does equality mean? What is an inequality? How can we use linear equations and linear inequalities to solve real world problems? What is a solution set for a linear equation or linear inequality? How can models and technology aid in the solving of linear equations and linear inequalities? Exact from State of Connecticut Curricula: To obtain a solution to an equation, no matter how complex, always involves the process of undoing the operations. Concepts What Students Need to Know Bloom s Taxonomy Levels Skills What Students Need To Be Able To Do Order of operations Expression vs. equation Inequality Associative property Commutative property Distributive property Inverse operation Model (with linear equation or inequality) Solve (linear equation or inequality) Simplify (expression) Use (algebraic properties) Select (appropriate units, degree of precision) Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills 11

12 When simplifying an expression operations must be performed in the correct order (Order of operations). Expression vs. equation. Important properties include the commutative, associative and identity properties, the zero property of multiplication and the multiplication property of -1. The distributive property can be used to simplify the product of a number and a sum or difference. A literal equation is an equation that involves two or more variables. An inequality is a mathematical sentence that uses an inequality symbol to compare the values of two expressions. Inequalities can be represented with symbols. The solution of an inequality can be represented on a number line. Level 1: Recall/Reproduction To simplify expressions involving exponents To use the order of operations to evaluate expressions Level 2: Skills/Concepts To identify and use properties of real numbers Level 1: Recall/Reproduction To use the distributive property to simplify expressions Level 2: Skills/Concepts To solve one-step, two-step and multi-step equations in one variable To solve equations with variables on both sides To identify equations that are identities or no solution (C1.1) Applying existing knowledge to generate new ideas, products, or processes; (C1.4) Identifying trends and forecast possibilities (C2.3) Collecting, analyzing, and evaluation data and/or sources to identify solutions and/or make informed decisions (C5.2) Participating actively, as well as being reliable and punctual Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills The addition, subtraction, multiplication and division properties of inequality can be used to solve inequalities. Level 1: Recall/Reproduction To rewrite and use literal equations and formulas Level 2: Skills/Concepts To write, graph and identify solutions of inequalities Assured Assessments: (Performance Based/Curriculum Based) Assessment Evidence Other Evidence: Unit 2 Test o Advanced o Academic o Algebra 1-A Entrance slips Exit slips Observations PBA- The Big Dig Use School-Wide Critical Thinking and Problem Solving Rubric 12

13 o Advanced o Academic o Algebra 1-A Learning Plan Days CCSS DOK Instructional Practices Ensured Assignments # of Days per Section Differentiation/ Resources A-SSE 1 Order of Operations and 1 D: Evaluating Expressions CCSS 1.2 CCSS: Pg. 13 #9-35 odd, PH 1.2 PH: Pg. 12 # s 1-47 odd, odd Teacher Notes 11 A-SSE 1 Properties of Real Numbers CCSS 1.4 PH 2.5 : CCSS: Pg. 26 #7-35 odd, PH:Pg.88 # s 1-47odd, D: 12 A-SSE 1 The Distributive Property CCSS 1.7 : CCSS: Pg. 50 #9-63 odd, D: PH 2.4 PH: Pg. 83 # s 1-45 odd, even 13

14 13 A.CED.1 A.REI.1 A.REI.3 : Solving One-Step Equations CCSS 2.1 CCSS: Pg. 85 #11-51 odd,52-76 PH pg. 118 PH: Pg. 118 # s D: 14 A.CED.1 A.REI.1 A.REI.3 Solving Two-Step Equations CCSS 2.2 PH 3.1 : CCSS: Page 91 #11-37 odd #38-60, PH: Pg. 1-53odd, D: 15 A.CED.1 A.REI.1 A.REI.3 Solving Multi-Step Equations CCSS 2.3 PH 3.2 : CCSS: Page 98 #11-29 odd 54, 61-63,65, PH: Pg. 129 # s 1-11,52-56even, D: 16 : CCSS: Page 98 #31-43 odd 45-53,55-60,64 D: 14

15 PH: Pg. 129 # s odd, odd 17 A.CED.1 A.REI.1 A.REI.3 Equations With Variables on Both Sides CCSS 2.4 PH 3.3 : CCSS: Page 105 #11-19 odd PH: Pg.136 # s 1-16, D: 18 : CCSS: Page 106 #21-31 odd 33-42,47-49,57-66 PH: Pg. 136 # s 17-31,39-40, odds D: 19 N.Q.1 A.CED.1 A.CED.4 A.REI.1 A.REI.3 Literal Equations and Formulas CCSS 2.5 PH pg. 140 : CCSS: Page 112 #11-35 odd 36-48, PH: Pg. 141 # 1-3 *H 1 D: 20 A.REI.3 : Inequalities and Their Graphs CCSS 3.1 CCSS: Pg. 168 # 9-39 odd, 1 D: 15

16 PH 4.1 PH: Pg. 202 # s 1-49 odd, 75-93odd. A.CED.1 A.REI.3 Solving Inequalities Using Addition and Subtraction CCSS 3.2 : CCSS: Pg. 174 #9-43 odd, #45-75, D: 21 PH 4.2 PH: Pg. 209 #23-41 odd, odd, odd 22 N.Q.2 A.CED.1 A.REI.3 Solving Inequalities Using Multiplication and Division CCSS 3.3 PH 4.3 : CCSS: Pg. 181 #7-31 odd, #33-62, PH: Pg. 215 #1-29 odd, #57-77 odd 1 D: 23 A.CED.1 A.REI.3 Solving Multi-Step Inequalities CCSS 3.4 : CCSS: Pg. 190 #9-21 odd, #44, D: PH 4.4 PH: Pg. 222, #1-12,

17 24 : CCSS: Pg. 190 #22-33 odd, #35-43, 45-46, 49-54, PH: Pg. 222 # s odd, even : Common Misunderstandings D: Interdisciplinary Connection Unit Vocabulary Technology Resources 17

18 Addition/Subtraction Property of Equality Base Coefficient Constant Counterexample Deductive Reasoning Distributive Property Equivalent Equations Equivalent Expressions Equivalent Inequalities Evaluate Exponent Formula Identify Inverse Operations Isolate Like Terms Literal Equation Multiplication/Division Property of Equality Power Simplify Solution of Inequality Solving Inequalities Solving Two-Step Equations Term Math behind the scenes video Activity labs Lessons Chapter reviews Editable worksheets (remediate/enriched; ELL support) Portable study center Math videos Kuta Infinite Software Pre-Algebra Algebra I Geometry Algebra II 18

19 Subject(s) Grade/Course Unit of Study Unit Type(s) Pacing Algebra I 9 th grade #3: Functions Skills-based 2 weeks plus 1 week re-teaching Overarching Standards: Create equations that describe numbers or relationships Represent and solve equations and inequalities graphically Understand the concept of a function and use function notation Interpret functions that arise in applications in terms of the context Analyze functions using different representations Priority and Supporting Standards: (CCSS/Content Standards, ELL) A-CED 2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales A-REI 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). Math Practices: (Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.) 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. Explanations and Examples: Students may collect data from water that is cooling using two thermometers, one measuring Celsius, the other Fahrenheit. From this they can create of the relationship and show that it can be modeled with a linear function. The graph below shows the height of a hot air balloon as a function of time. Explain what the point (50, 300) on this graph represents. F-IF 1. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). The domain of a function given by an algebraic expression, unless otherwise specified, is the largest possible domain. Mapping diagrams may be used to introduce the concepts of domain and range. The vertical line test may be used to determine whether a graph represents a function. 19

20 Priority and Supporting Standards: (CCSS/Content Standards, ELL) F-IF 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. Explanations and Examples: Examples: If f(x) = x 2 + 4x 12, find f(2) Let f(x) = 10x 5; find f(1/2), f(-6), f(a) If P(t) is the population of Tucson t years after 2000, interpret the statements P(0) = 487,000 and P(10)-P(9) = 5,900. F-IF 4. For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negative...* Students may be given graphs to interpret or produce graphs given an expression or table for the function, by hand or using technology. Examples: A rocket is launched from 180 feet above the ground at time t = 0. The function that models this situation is given by h = 16t t + 180, where t is measured in seconds and h is height above the ground measured in feet. What is a reasonable domain restriction for t in this context? Determine the height of the rocket 2 seconds after it was launched. Determine the maximum height obtained by the rocket. Determine the time when the rocket is 100 feet above the ground. Determine the time at which the rocket hits the ground. How would you refine your answer to the first question based on your response to the fifth question? It started raining lightly at 5am, then the rainfall became heavier at 7am. By 10am the storm was over, with a total rainfall of 3 inches. It didn t rain for the rest of the day. Sketch a possible graph for the number of inches of rain as a function of time, from midnight to midday. F-IF 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in factory, then the positive integers would be an appropriate domain for the function.* Students may explain orally, or in written format, the existing relationships. 20

21 Priority and Supporting Standards: (CCSS/Content Standards, ELL) F-IF 7b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. F-IF 9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Essential Question(s): Explanations and Examples: Emphasis in this unit is distinguishing the graphs of linear and non-linear functions; linear functions are emphasized in Unit 4; non-linear functions are studied in more detail in later units. Students may be asked to match graphs with tables or equations with which they may represent, and to explain their reasoning. Enduring Understanding(s): What is a function? What are the different ways that functions may be represented? How can functions be used to model real world situations, make predictions, and solve problems. Functions are a mathematical way to describe relationships between two quantities that vary. Exact from State of Connecticut Curricula: Concepts What Students Need to Know Independent variable Dependent variable Ordered pair Mapping Diagram Table Graph Equation for a function Function notation Domain Range Vertical Line test Bloom s Taxonomy Levels Skills What Students Need To Be Able To Do Determine (whether or not a relation is a function) Determine (range and domain of a function) Model (a real world situation with a function) Evaluate (a function) Represent a function (with table, graph, equation, mapping diagram) Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills Graphs can be used visually to represent the relationship between two variable quantities as they each change. Tables and graphs can show relationships between variables. Values of one variable may be uniquely determined by values of another (Independent/Dependent). Such relationships may be represented using words, tables, equations, sets of ordered pairs, and graphs. Level 1: Recall/Reproduction Level 1: Recall/Reproduction Level 2: Skills/Concepts To represent mathematical relationships using graphs To identify and represent patterns that describe linear and nonlinear functions To graph equations that represent functions (C1.1) Applying existing knowledge to generate new ideas, products, or processes (C1.4) Identifying trends and forecast possibilities (C2.3) Collecting, analyzing, and evaluation data and/or sources to identify solutions and/or make informed decisions (C2.7) Synthesizing and making connections between information and arguments 21

22 Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills A nonlinear function is a function whose graph is not a straight line. The set of all solutions of an equation forms its graph. A graph may include solutions that do not appear in a table. A real-world graph should show only points that make sense in a given solution. A function is a special type of relation where each value in the domain is paired with one value in the range. The vertical line test shows whether a relation is a function Level 2: Skills/Concepts To write equations that represent functions Level 1: To determine whether a relation is Recall/Reproduction function Level 2: Skills/Concepts To find domain and range and use function notation. (C2.8) Interpreting information and drawing conclusions based on the best analysis (C3/4.1) Interacting, collaborating, and publishing with peers, experts, or others (C5.2) Participating actively, as well as being reliable and punctual Assured Assessments: (Performance Based/Curriculum Based) Assessment Evidence Other Evidence: Unit 3 Test o Advanced o Academic o Algebra 1-A Entrance slips Exit slips Observations Learning Plan Days CCSS DOK Instructional Practices Ensured Assignments # of Days per Section F.IF.4 A.REI Using Graphs to relate two quantities CCSS 4.1 PH 5.1 CCSS: Pg. 237 #5-13 odd, PH: Pg. 254 #1-16, Differentiation/ Resources D: Teacher Notes 26 F.IF.4 A.REI.10 Patterns and Linear Functions D: 22

23 CCSS 4.2 CCSS: Pg. 243 #7-13 odd, PH 1.4 PH: Pg. 29 #1-15, F.IF.4 A.REI.10 Patterns and Nonlinear Functions CCSS 4.3 CCSS: Pg. 250 #7-13 odd, 14-18, D: 27 PH 5.2 PH: Pg. 259 #1-41 odd, A.REI.11 Graphing Functions and Solving Equations CCSS CB 4.4 PH n/a CCSS: Pg. 260 # D: 29 N.Q.1 F.IF.5 A.REI.10 Graphing a Function Rule CCSS 4.4 PH 5.3 CCSS: Pg. 257 #9-31odd, 33-39, PH: Pg. 266 #1-39 odd, odd 1 D: 30 N.Q.2 Writing a Function Rule D: 23

24 A.SSE.1a A.CED.2 CCSS 4.5 CCSS: Pg. 265 #9-21 odd, 22-30, PH 5.4 PH: Pg. 272 #1-31 F.IF.1 F.IF.2 Formalizing Relations and Functions CCSS 4.6 CCSS: Pg. 271 #9-23 odd, D: 31 PH n/a Common Misunderstandings Interdisciplinary Connection 24

25 Dependent Variable Independent Variable Input Output Function Linear Function Continuous Graph Unit Vocabulary Discrete Graph Relation Domain Range Vertical Line Test Function Notation Technology Resources Math behind the scenes video Activity labs Lessons Chapter reviews Editable worksheets (remediate/enriched; ELL support) Portable study center Math videos Kuta Infinite Software Pre-Algebra Algebra I Geometry Algebra II 25

26 Subject(s) Grade/Course Unit of Study Unit Type(s) Pacing Algebra I 9 th grade #4: Linear Functions Skills-based 5 weeks plus 1 week re-teaching Overarching Standards: Interpret functions that arise in applications in terms of the context Analyze functions using different representations Construct and compare linear and exponential models and solve problems Interpret expressions for functions in terms of the situation they model Priority and Supporting Standards: (CCSS/Content Standards, ELL) F-IF 6. Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.* Math Practices: (Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.) 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. Explanations and Examples: For linear functions, the rate of change is also the slope of the line. Example: A cell phone company uses the function y = 0.03x to determine the monthly charge, y, in dollars, for a customer using the phone fo x minutes. Interpret the slope of this function in the context of this problem and indicate the appropriate units. F-IF 7. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.* a. Graph linear...functions and show intercepts.. Example: Find the x-intercept and the y-intercept for each of these lines. y = 3x + 18 y 7 = 2/3(x + 4) 6x + 4y = 96 F-IF 8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. For linear functions, the slope intercept form easily reveals the slope and the y-intercept. From the standard form one can easily determine both intercepts. The point-slope form focuses attention on a particular point on the graph and is directly related to the definition of slope. 26

27 Priority and Supporting Standards: (CCSS/Content Standards, ELL) F-LE 1. Distinguish between situations that can be modeled with linear functions [and with exponential functions]. a. Prove that linear functions grow by equal differences over equal intervals... over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another... Explanations and Examples: Given several tables of values, determine which represent linear functions, and explain why. Input Output Input Output x y F-LE 2. Construct linear... functions, including arithmetic... sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). Essential Question(s): Example: Sara s starting salary is $32,500. Each year she receives a $700 raise. Write a sequence in explicit form to describe the situation. Draw a graph representing this situation. Is this function discrete or continuous? Enduring Understanding(s): What is a linear function? What are the different ways that linear functions may be represented? What is the significance of linear function s slope and y-intercept? How may linear functions model real world situations? How may linear functions help us analyze real world situations and solve practical problems? Exact from State of Connecticut Curricula: Functions are a mathematical way to describe relationships between two quantities that vary. Concepts What Students Need to Know Bloom s Taxonomy Levels Skills What Students Need To Be Able To Do Rate of change Constant additive change Slope x-intercept y-intercept slope-intercept form point-slope form standard form velocity Find (slope given two points) Interpret (slope as rate of change) Determine (whether function is linear) Find (equation of a line) Rearrange (equation of a line into a different form) Model (linear function) Graph (linear function) Find (slopes of parallel and perpendicular lines) Interpret (parameters of linear function) 27

28 Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills Ratios can be used to show a relationship between changing quantities, such as vertical and Level 1: Recall/Reproduction Level 1: To find rates of change from tables To find slope (C1.1) Applying existing knowledge to generate new ideas, products, or processes horizontal change. Recall/Reproduction (C1.4) Identifying trends and forecast If the ratio of two variables is Level 1: To write and graph an equation of a possibilities constant, then the variables have a Recall/Reproduction direct variation (C2.3) Collecting, analyzing, and special relationship, called a direct Level 2: Skills/Concepts To write and graph linear equations evaluation data and/or sources to variation. using slope-intercept form identify solutions and/or make To write and graph linear equations informed decisions using point-slope form (C2.7) Synthesizing and making connections between information and arguments Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills A line on a graph can be represented by a linear equation. Forms of linear equations include the slope-intercept, point-slope, and standard forms. The relationship between two line can be determined by comparing their slopes and y-intercepts. Velocity. Level 1: Recall/Reproduction To graph linear equations using intercepts Level 2: Skills/Concepts To write linear equations in standard form Level 1: Recall/Reproduction To determine whether lines are parallel, perpendicular or neither. (C2.8) Interpreting information and drawing conclusions based on the best analysis (C3/4.1) Interacting, collaborating, and publishing with peers, experts, or others (C5.2) Participating actively, as well as being reliable and punctual Level 2: Skills/Concepts To write equations of parallel lines and perpendicular lines Assessment Evidence 28

29 Assured Assessments: (Performance Based/Curriculum Based) Other Evidence: Unit 4 Test o Advanced o Academic o Algebra 1-A Entrance slips Exit slips Observations Learning Plan Days CCSS DOK Instructional Practices Ensured Assignments # of Days per Section Differentiation/ Resources F.LE.1b Rate of Change and 2 D: Slope CCSS 5.1 CCSS: Pg. 298 #9-25 odd 32 PH 6.1 PH: Pg. 312 #1-6, 27-29, 40, 54 Teacher Notes Worksheets/etc. : CCSS: Pg. 298 #26-49, D: 33 PH: Pg. 313 #7-26, 30-39, 41-53, N.Q.2 A.CED.2 Direct Variation CCSS 5.2 : CCSS: Pg. 304 #9-29 odd, #30-43, D: 34 PH 5.2 PH: Pg. 280 #1-45 odd Worksheets/etc. : 35 A.SSE.1a F.IF.4 F.BF.1a Investigating y = mx+b CCSS CB 5.3 CCSS: Pg. 307 #1-7 1 D: 29

30 A.SSE.2 A.CED.2 F.BF.3 F.IF.7a PH Activity 6.2 PH: Pg. 316 #1-8 Worksheets/etc. : 36 A.SSE.1a F.IF.4 F.BF.1a A.SSE.2 A.CED.2 F.BF.3 F.IF.7a Slope-Intercept Form CCSS 5.3 PH 6.2 CCSS: Pg. 312 #7-21 odd, #36-44 PH: Pg. 320 #1-27 odd, #41-55 odd, # D: Worksheets/etc. : 37 CCSS: Pg. 312 #23-35 odd, #245-58,63-77 PH: Pg. 320 #29-39, 56, odd D: : 38 A.SSE.1a F.IF.4 F.BF.1a A.SSE.2 A.CED.2 F.BF.3 F.IF.7a Point-Slope Form and Writing Linear Equations CCSS 5.4 PH 6.5 CCSS: Pg. 318 #9-15 odd, #24-26 PH: Pg. 339 #1-29 odd, #37-53 odd, # D: Worksheets/etc. : 30

31 39 CCSS: Pg. 318 #17-23 odd, #27-29, PH: Pg. 340 #31-35, 54, 55, 60, D: 40 F.IF.9 F.IF.4 F.BF.1a A.SSe.2 A.CED.2 F.IF.7a Standard Form CCSS 5.5 PH 6.4 : CCSS: Pg. 326 #9-29 odd, #39-44 PH: Pg. 333 #1-25 odd, #39-45 odd, # D: 41 Worksheets/etc. : CCSS: Pg. 326 #31-37 odd, #45-58, PH: Pg. 333 #27-37, 47, 48, D: G.GPE.5 Parallel and Perpendicular Lines CCSS 5.6 : CCSS: Pg. 334 #7-25 odd 2 D: 42 PH 6.6 PH: Pg. 346 #1-17 odd, #40,41, 44, 45, Worksheets/etc. : 31

32 43 CCSS: Pg. 334 #27-35,38-47 PH: Pg. 346 #19-39 odd, #42, 43, 46-50, : Common Misunderstandings D: Interdisciplinary Connection Unit Vocabulary Technology Resources 32

33 Constant of Variation for a Direct Variation Direct Variation Linear Equation Linear Parent Function Opposite Reciprocals Parallel Lines Parent Function Perpendicular Lines Point-Slope Form Rate of Change Slope Slope-Intercept Form Standard Form of a Linear Equation X-Intercept Y Intercept Math behind the scenes video Activity labs Lessons Chapter reviews Editable worksheets (remediate/enriched; ELL support) Portable study center Math videos Kuta Infinite Software Pre-Algebra Algebra I Geometry Algebra II 33

34 Subject(s) Grade/Course Unit of Study Unit Type(s) Pacing Algebra I 9 th grade #5: Scatter Plots and Trend Lines Skills-based 3 weeks plus 1 week re-teaching Overarching Standards: Analyze functions using different representations Summarize, represent, and interpret data on a single count or measurement variable Summarize, represent, and interpret data on two categorical and quantitative variables Interpret linear models Priority and Supporting Standards: (CCSS/Content Standards, ELL) F-IF 7b. Graph... piecewise-defined functions, including step functions and absolute value functions. S-ID 1 Represent data with plots on the real number line (dot plots, histograms, and box plots) S-ID 2. Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. S-ID 3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers). Math Practices: (Mathematical Practices #1 and #3 describe a classroom environment that encourages thinking mathematically and are critical for quality teaching and learning.) 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. Explanations and Examples: Piecewise linear functions may model sets of data. For example the world women s swimming records for the 100 meter free style exhibit two distinct linear trends; one for the years , the other for Students may compare and contrast the advantage of each of these representations. Students may use spreadsheets, graphing calculators and statistical software for calculations, summaries, and comparisons of data sets. Example: Given a set of test scores: 99, 96, 94, 93, 90, 88, 86, 77, 70, 68, find the mean, median and standard deviation. Explain how the values vary about the mean and median. What information does this give the teacher? Example: After the NBA season LeBron James switched teams from the Cleveland Cavaliers to the Miami Heat, and he remained the top scorer (in points per game) in his first year in Miami. Compare team statistics for Cleveland ( ) and Miami ( ) for all players who averaged at least 10 minutes per game. Using the 1.5 X IQR rule, determine for which team and year James s performance may be considered an outlier. 34

35 Priority and Supporting Standards: (CCSS/Content Standards, ELL) S-ID 6. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the context of the data... Explanations and Examples: Students may use spreadsheets, graphing calculators, and statistical software to represent data, describe how the variables are related, fit functions to data, perform regressions, and calculate residuals. Example: Make a scatter plot of data showing the rise in sea level over the past century. Fit a trend line and use it to predict the sea level in the year c. Fit a linear function for a scatter plot that suggests a linear association S-ID 7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data. Students may use spreadsheets or graphing calculators to create representations of data sets and create linear models. Example: Lisa lights a candle and records its height in inches every hour. The results recorded as (time, height) are (0, 20), (1, 18.3), (2, 16.6), (3, 14.9), (4, 13.2), (5, 11.5), (7, 8.1), (9, 4.7), and (10, 3). Express the candle s height (h) as a function of time (t) and state the meaning of the slope and the intercept in terms of the burning candle. S-ID 8. Compute (using technology) and interpret the correlation coefficient of a linear fit. S-ID 9. Distinguish between correlation and causation. Example: Collect height, shoe-size, and wrist circumference data for each student. Determine the best way to display the data. Answer the following questions: Is there a correlation between any two of the three indicators? Is there a correlation between all three indicators? What patterns and trends are apparent in the data? What inferences can be made from the data? Some data leads observers to believe that there is a cause and effect relationship when a strong relationship is observed. Students should be careful not to assume that correlation implies causation. The determination that one thing causes another requires a controlled randomized experiment. Example: Diane did a study for a health class about the effects of a student s end-of-year math test scores on height. Based on a graph of her data, she found that there was a direct relationship between students math scores and height. She concluded that doing well on your end-of-course math tests makes you tall. Is this conclusion justified? Explain any flaws in Diane s reasoning. 35

36 Essential Question(s): Enduring Understanding(s): How do we make predictions and informed decisions based on current numerical information? What are the advantages and disadvantages of analyzing data by hand versus by using technology? What is the potential impact of making a decision from data that contains one or more outliers? Although scatter plots and trend lines may reveal a pattern, the relationship of the variables may indicate a correlation, but not causation. Exact from State of Connecticut Curricula: Concepts What Students Need to Know Bloom s Taxonomy Levels Skills What Students Need To Be Able To Do Measures of central tendency (mean, median, mode) Measures of spread (range, interquartile range, standard deviation) Outlier Histogram Box plot Scatter plot Trend line Line of best fit Correlation Correlation coefficient Causation Interpolation Extrapolation Piecewise linear function Calculate (mean, median, mode, interquartile range) Fit (trend line to scatter plot) Use (technology to find standard deviation, line of best fit, correlation coefficient) Interpret (correlation) Predict (using interpolation, extrapolation) Identify (outliers) Evaluate (piecewise function) Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills Sometimes it is helpful to order numerical data into intervals. Frequency tables and histograms display numerical data organized into intervals. Three measures of central tendency of a set of data are mean, median and mode. Compare and contrast data Box and whisker plot displays the maximum, minimum, and quartiles of a data set. Level 2: Skills/Concepts To make and interpret frequency tables and histograms Level 1: Recall/Reproduction To find mean, median, mode and range (C1.4) Identifying trends and forecast possibilities (C2.3) Collecting, analyzing, and evaluation data and/or sources to identify solutions and/or make informed decisions (C2.8) Interpreting information and drawing conclusions based on the best analysis (C3/4.6) Articulating thoughts and ideas effectively using oral, written and nonverbal communication skills in a variety of forms and contexts 36

37 Concepts (Students will ) DOK Skills (What students need to be able to do) 21 st Century Skills/ Life and Career Skills Graphing ordered pairs is a way to determine whether to sets of numerical data are related. If two sets of data are related, it may be possible to use a line to estimate or predict values. Additional concepts: Outlier Correlation Coefficient Causation Interpolation Extrapolation Piecewise Linear Functions Level 1: Recall/Reproduction Level 3: Thinking/Reasoning Level 1: Recall/Reproduction To find the standard deviation of a data set To make and interpret box and whisker plots To find quartiles and percentiles Level 2: Skills/Concepts To write an equation of a trend line and line of best fit Level 4: Extended Thinking To use a trend line and a line of best fit to make future predictions Assured Assessments: (Performance Based/Curriculum Based) Unit 5 Test o Advanced o Academic o Algebra 1-B Assessment Evidence Other Evidence: Entrance slips Exit slips Observations PBA Stroop Test or Monopoly Score using the NHS School- Wide Critical Thinking & Problem Solving Rubric o Advanced o Academic PBA You Be The Statistician Score using the NHS School- Wide Critical Thinking & Problem Solving Rubric o Algebra 1-B Learning Plan Days CCSS DOK Instructional Practices Ensured Assignments # of Days per Section Differentiation/ Resources N.Q.1 Frequency and 1 D: 44 S.ID.1 Histograms CCSS CCSS: Pg. 735 #7-19 odd, Teacher Notes 37