Unit A - Cracking Codes Patterns and Repetition in our World

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1 Unit A - Cracking Codes Patterns and Repetition in our World Overview Student will be engaged in learning mathematical skills within the context of interesting problems that connect to real world issues. Students will be expected to use the 21st Century Skill: Analyzing to independently use their learning in new situations. They will learn to interpret patterns in the real world and use mathematics to evaluate complex situations, apply properties of patterns to inform decisions by analyzing information, and design and create representations of data. Students coming from Pre-Algebra should be proficient in combining like terms, order of operations, the rules of integer operations, graphing on a coordinate plane, finding rate of change and the meaning of the intercepts. The students begin with hands-on activities building concrete models which helps them generalize to tables, graphs and equations. They will recognize and extend addition, multiplication and other patterns including fractions and decimals. They are asked to write explicit and recursive rules for the patterns. Later students will explore rules for arithmetic and geometric sequences that will help them find values and terms that extend the usefulness of patterns. They will explore the rules that will reach a higher understanding of the patterns to explain real world phenomenon. Students will develop strategies to select the most efficient method to solve problems involving arithmetic or geometric sequences. Challenges for students can be: 1) distinguishing between arithmetic and geometric sequences, 2) accurately representing data on a graph, 3) independently transferring skills to new situations. In the next unit students will extend pattern rules to function relationships. The students will use technology to generate and display patterns. 21 st Century Capacities: Analyzing ESTABLISHED GOALS/ STANDARDS MP2 Reason abstractly and quantitatively MP4 Model with Mathematics MP5 Use appropriate tools strategically MP7 Look for and make use of structure Stage 1 - Desired Results Transfer: Students will be able to independently use their learning in new situations to Interpret patterns in the real world and use mathematics to evaluate complex situations. (Analyzing) 2. Apply properties of patterns to inform decisions by analyzing information. (Analyzing) Madison Public Schools June

2 MP8 Look for and express regularity in repeated reasoning F-IF 3. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. 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 both recursively and with an explicit formula, use them to model situations, and translate between the two forms. UNDERSTANDINGS: Students will understand that: 1. A pattern can be described using a variety of mathematical means. 2. Representations of patterns can be used and extended to draw conclusions and/or find solutions. 3. Patterns can be used to explain real world phenomena. Students will know 1. Format for explicit rules a + d(x) where the first input is 0 a + d(x - c) where the first input is c ar (x) where the first input is 0 ar (x-1) where the first input is 1 2. The format for recursive rules as sentences: Starting with (first output), each term increases/decreases by or Where the first input is multiplied by. 3. How to use tables, graphs, and/or lists to display patterns 4. What a fractal is 5. That sequences are functions 6. Vocabulary: explicit, recursive, geometric, arithmetic, fractal Meaning: ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. How can we mathematically model the world? B. Which tools can I use to solve this problem and which are best? C. What is the most efficient way to solve this problem? D. How do you express and describe a pattern and use it to make predictions or solve problems? Acquisition: Students will be skilled at 1. Writing the recursive rule for a set of data 2. Writing the explicit rule for a set of data 3. Determine if a pattern is arithmetic or geometric 4. Finding the explicit and/ or recursive rule for a word problem or visual pattern 5. Identifying patterns from real world context 6. Representing patterns using tables, graphs and equations 7. Creating an appropriate scale to represent data within a pattern on x and y axes 8. Labeling x and y axes 9. Creating and extending fractals Madison Public Schools June

3 Unit B - Relationships (Equations, Inequalities and Functions) Overview Students begin the unit with a quick review of solving equations and inequalities. Focus should be on solving more difficult equations (ie. equations with a variable on both sides) and inequalities (those with negative numbers and with the variable on the right hand side of the inequality). Students are asked to justify their work using math and words. Asking for step by step explanations in words will help them in justify steps in proof writing in Geometry. The concept of solving equations is expanded to solving literal equations, solving compound equations and solving absolute value equations and inequalities. Students learn function notation during the second part of this unit. The goal is for them to be able to move fluently between representations of a function. 21 st Century Capacities: Analyzing, Product Creation Stage 1 - Desired Results ESTABLISHED GOALS/ STANDARDS MP 3 Construct viable arguments and critiques the reasoning of others. MP 4 Model with Mathematics MP 8 Look for and express regularity in repeated reasoning. 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. A-REI 3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 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. b. Interpret complicated expressions by viewing one or more of their parts as a single entity... A-SSE 3.Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. A-CED 1. Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear... functions Transfer: Students will be able to independently use their learning in new situations to Analyze the real world and use mathematics to model complex situations. (Analyzing) 2. Demonstrate fluency with math facts, computation and concepts 3. Justify reasoning using clear and appropriate mathematical language. (Product Creation) Meaning: UNDERSTANDINGS: Students will understand that: 1. The properties of equality are used to manipulate equations while maintaining equality. 2. Functions represent unique relationships between inputs and outputs that can be represented algebraically, graphically or as a table. ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. How do I know if my answer is correct? B. Do my answers make sense? C. Could another person understand the steps to my solution? Madison Public Schools June

4 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. 8.F.1 and F.IF.1 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1 8.F.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). 8F 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 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 10. Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). F-IF 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). F-IF 2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. F-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... F-IF 5. Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. 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). 3. Functions are used to represent real world situations including those with one answer, a range of answers or no answer. 4. Creating a clear argument for a solution benefits you and your audience. Acquisition: Students will know 1. The properties of equality and inequality 2. How to state a value of an equation that has no, one, multiple, infinite, or bounded solutions. 3. How to identify the independent and dependent variable 4. Vocabulary: algebraic expression, associative property, coefficient, constant, commutative property, distributive property, evaluate, inequality symbol, integers, inverse operations, literal equations, absolute value, real numbers, variable, dependent variable, domain, range,function, input, output, linear function, mapping diagram, non-linear function, relation, table, vertical line test D. How do I interpret this function? Students will be skilled at 1. Solving equations including those with a variable on both sides, with fractions, with distributing, combining like terms, with x 2 or x 3 (review) 2. Solving literal equations (review) 3. Solving inequalities on the number line (review) 4. Modeling real world rate/work problems (especially those that involve adding two scenarios to get a whole or setting two scenarios equal to each other) 5. Solving absolute value equations 6. Solving compound inequalities 7. Determining whether a relationship is a function. 8. Applying function notation to solve for inputs and outputs 9. Modeling with function notation, or a table or a graph Madison Public Schools June

5 Unit C - What s In A Line? - Elements of Linear Equations Overview 10. Interpreting graphs In this unit students will learn how to model with, interpret and graph linear functions. The ability to fluidly move between different representations of linear relationships is a skill that students will continue to use and to build upon in Algebra and later math courses. Students will use technology to experiment with changing the parameters of a linear equation and noting how those changes affect the graph of the relationship. 21st Century Capacities: Analyzing, Synthesizing, Product Creation ESTABLISHED GOALS/ STANDARDS MP4 Model with Mathematics MP7 Look for and make use of structure MP8 Look for and express regularity in repeated reasoning A.SSE.1 Interpret expressions that represent a quantity in terms of its context. A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. 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.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. Stage 1 - Desired Results Transfer: Students will be able to independently use their learning in new situations to Model relationships among quantities; (Analyzing) 2. Represent and interpret patterns in numbers, data and objects; (Product Creation) 3. Draw conclusions about graphs, shapes, equations, or objects. (Synthesizing) UNDERSTANDINGS: Students will understand that: 1. There are many ways to represent a function. 2. Changing the parameters of a function changes key features of the relationship 3. Linear functions are characterized by a constant rate of change Meaning: ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. How can mathematics model observed relationships? B. Which representation best communicates what I want the audience to understand? C. What are the different ways a linear function can be represented? D. What is the significance of the slope and the intercepts of a function? Madison Public Schools June

6 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. 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). 8.F.3 Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; 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; relative maximums and minimums; symmetries; end behavior; and periodicity F.IF.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. F.BF.1 Write a function that describes a relationship between two quantities. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation Students will know 1. Know what makes a function linear (from a graph, table, word problem) 2. The meaning of slope, the intercepts and any point in context 3. A horizontal line has zero slope and a vertical line has no slope 4. The effects of changing the parameters of an equation in the form y=mx+b 5. Vocabulary: parallel, perpendicular, direct variation Acquisition: Students will be skilled at 6. Reading a graph (distance vs time) 7. Graphing equations using a table 8. Determining if a graph is increasing or decreasing 9. Finding the slope of line (from two points, from a graph) 10. Finding unit rates (in context, from a graph) 11. Using the magnitude of slope to compare two functions 12. Graphing and writing equations of vertical and horizontal lines 13. Graphing lines in slope intercept form 14. Getting equations into y=mx+b form 15. Determining if two lines are parallel or perpendicular 16. Given a point, writing an equation perpendicular or parallel to another 17. Determining break even points of two scenarios 18. Identifying direct variation situations 19. Determining the slope of a direct variation situation 20. Using direct variation in context 21. Using proportion to find missing parts of direct variations 22. Getting an equation in standard form 23. Finding the intercepts of two functions 24. Fluidly moving between the forms of equation of line and use that form to interpret the graph slope intercept (context of m and b) standard (context of intercepts) Madison Public Schools June

7 of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. F.LE.1b Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. F.LE.1c Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another. F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table). F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. point slope 25. Identifying when to use each form to solve a problem 26. Write the equation of a line in slopeintercept form, point slope form, or standard form given: the slope and y intercept the slope and one ordered pair on the line two ordered pairs an ordered pair and an equation or a parallel or perpendicular line 27. Write a linear equation to describe a set of points 28. Application of all of the above Madison Public Schools June

8 Unit D - Describing Data - Identifying Trends and Making Decisions Overview Describing Data extends linear thinking to statistical modeling. First, students develop measures of central tendency by studying dispersion through the 5-number summary and the corresponding box and whisker graph. Students then make frequency tables and histograms that shape discussions about skewness. Next, students compare two quantities in scatterplots and add context to Unit C concepts of slope and line of best fit. Students model linear relationships both manually with trend-lines and digitally with graphing calculators or software. Students use models to make predictions both inside and outside of the known range and understand limitations of those predictions. Students describe strength of fit using correlation coefficients, which strengthen understandings of slope from Unit C. Students are challenged to explain the difference between correlation and causation. Students explain the impact of an outlier on linear models. Students expand their notions of linear models to piecewise functions. This is a prelude to other nonlinear modeling, including exponential and quadratic models which will resurface later in the course. 21 st Century Capacities: Analyzing, Synthesizing ESTABLISHED GOALS/ STANDARDS MP 1 Make sense of problems and persevere in solving them MP3 Construct viable arguments and critique the reasoning of others MP5 Use appropriate tools strategically MP6 Attend to precision S.ID.1 Represent data with plots on the real number line (histograms, dot plots, box plots). S.ID.2-3 Use statistics appropriate to the shape of the data distribution to compare Stage 1 - Desired Results Transfer: Students will be able to independently use their learning in new situations to Represent, summarize, and interpret patterns in data (Analyzing) 2. Use appropriate tools/methods to make mathematical concepts more concrete and accessible 3. Make sense of a problem, initiate a plan, execute it, and evaluate the reasonableness of the solution (Synthesizing) Meaning: UNDERSTANDINGS: Students will understand that: 1. Mathematicians select and use appropriate statistical methods and tools to analyze data, show trends, and describe or make predictions; ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. As consumers of information, how do we analyze the validity of statistics? B. How does technology help to create meaning out of the data? Madison Public Schools June

9 center (median, mean) and spread (iqr, standard deviation) and interpret differences between multiple sets including outliers. CC.8.SP.1/S.ID.6 Construct and interpret bivariate data to observe patterns and describe how variables are related. CC.8.SP.2/S.ID6abc/S.ID.8 Informally (trendline) and formally (regression) fit and judge the fit of a straight line to a data set. CC.8.SP.3/S.ID.7 Use linear models to make statements in the context of the problem, ie. slope, y-intercept, and making predictions. S.ID.9 Differentiate between correlation and causation 2. Mathematicians analyzed data to evaluate inferences, make predictions and/or communicate an decision; 3. Correlation does not imply causation. Students will know 1. How to find and interpret measures of center as well as measures of spread 2. How to create and interpret a dot plot, histogram, and box-and-whisker plot 3. How to fit a trend line to data, write an equation for the trend line, and use the equation to interpolate or extrapolate. 4. The contextual meaning of the parameters of the trend line equation. 5. How to find the equation for the line of best fit using technology 6. The difference between one variable being correlated to the other and one variable causing the other to occur; 7. How to use technology to calculate the regression equation and the correlation coefficient; 8. That outliers can affect the accuracy of a prediction made with a regression line; 9. That some relationships do not take a linear form. 10. Vocabulary: Correlation coefficient, Distribution, Extrapolation, Histogram, Interpolation, Linear regression, Linear relationship/model, Non-linear relationship / model, Piecewise function, Regression equation, Skewed distribution, Trend line. C. How can I best communicate to an audience what the statistics say? Acquisition: Students will be skilled at 1. Constructing a frequency table and histogram; 2. Finding the five-number summary, range, and IQR, and constructing a box-and-whisker plot to compare sets of data; 3. Drawing a trend line through points, determining the equation of the trend line, interpreting the slope of the trend line in the context of the problem, and using the equation of the trend line to make a prediction; 4. Matching a graph with a possible value of r; 5. Using technology to calculate the regression equation and the correlation coefficient; 6. Making predictions based on the regression equation; 7. Stating the impact of an outlier. Madison Public Schools June

10 Unit E - Linear Systems: Points In Common Overview In this unit students will use previously learned skills in graphing equations and extend those to graph systems of equations and graph inequalities and graph systems of inequalities. Students will model using systems of equations or inequalities. Students will also solve systems of equations using substitution or elimination. Students will be encouraged to analyze a system before solving it to determine the most efficient method to use to solve the system. 21 st Century Capacities: Synthesizing, Product Creation ESTABLISHED GOALS/ STANDARDS MP4 Model with Mathematics MP5 Use appropriate tools strategically MP6 Attend to precision MP7 Look for and make use of structure 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.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. 8.EE.8 Analyze and solve pairs of simultaneous linear equations. a. Understand that solutions to a system of two linear equations in two variables Stage 1 - Desired Results Transfer: Students will be able to independently use their learning in new situations to Demonstrate fluency with math facts, computation and concepts. (synthesizing) 2. Use appropriate strategies to make reaching solutions more efficient, accessible and accurate. 3. Justify reasoning using clear and appropriate mathematical language. (product creation) UNDERSTANDINGS: Students will understand that: 1. Effective problem solvers work to make sense of the problem before trying to solve it. 2. Linear relationships can have more or less than one solution. Meaning: ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. How do we make decisions based on multiple constraints? B. What are the advantages and disadvantages of each method of solving a system of linear equations? C. What does the solution tell me? Madison Public Schools June

11 correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously. b. Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. A.REI.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. Students will know 1. How to apply systems of equations to solve problems 2. How to apply systems of inequalities to solve problems 3. Vocabulary: system, consistent (independent, dependent), inconsistent, Acquisition: Students will be skilled at 1. Solving systems of equations by graphing 2. Solving systems of equations by substitution, 3. Solving systems of equations by elimination 4. Modeling using a system of equations 5. Graphing inequalities with two variables 6. Graphing systems of inequalities A.REI.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. A.REI.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. Madison Public Schools June

12 Unit F - Beyond Straight Lines - Quadratic and Absolute Value Functions Overview In this unit students work with quadratic expressions, quadratic equations, radicals and rational expressions to see how changing the form of an expression or equation can give the item a clearer meaning and can make it easier to work with. By the end of the unit students should be able to fluently solve quadratic equations. They should be able to fluently identify transformations made to the parent function so they are able to visualize the graph to make estimations and to check to see if their solution makes sense. The unit ends with students using their new factoring skills to simplify rational expressions and equations into manageable problems. 21 st Century Capacities: Analyzing, Product Creation ESTABLISHED GOALS/ STANDARDS MP4 Model with Mathematics MP5 Use appropriate tools strategically MP7 Look for and make use of structure A.SSE.1a Interpret parts of an expression, such as terms, factors, and coefficients. A.SSE.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. A.CED.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. 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). 8.F.2 Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. CC.8.F.5 Describe qualitatively the functional relationship Stage 1 - Desired Results Transfer: Students will be able to independently use their learning in new situations to Explain real world phenomena mathematically for events that are parabolic in nature; 2. Draw conclusions about graphs and equations;(analyzing) 3. Manipulate equations/expressions or objects to create order and establish relationships. (Analyzing)(Product Creation) Meaning: UNDERSTANDINGS: Students will understand that: 1. Quadratics functions can be used to model real world relationships. 2. Changing the parameters of a function relates to transformations on the coordinate plane. 3. Key points in quadratic functions have meaning in real-world context. ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. Why do I need nonlinear functions? B. How do changes to the parent quadratic/absolute function change the graph? C. What can the characteristics of a quadratic function tell you about real world events? Madison Public Schools June

13 between two quantities by reading a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. 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; relative maximums and minimums; symmetries; end behavior; and periodicity F.IF.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Analyze functions using different representations. 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 F.IF.7a Graph linear and quadratic functions and show intercepts, maxima, and minima. F.IF.7b Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. F.IF.8a Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. F.IF.9 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. Build a function that models a relationship between two quantities. 4. Expressions and equations can be written in different but equivalent forms to build meaning or ease of use. Students will know 1. Some binomials (x-a)(x+a) and (x + c) 2 can be quickly multiplied using patterns 2. That the discriminant of an expression can give insight about factoring that expression 3. That the discriminant of an equation can give insight about the solutions to that equation 4. The meaning of the vertex of an ax 2 + bx + c = d in context 5. The meaning of the x and y intercepts of ax 2 + bx + c = d in context 6. What changing the parameters of ax 2 + bx + c = y does to the graph of the parent function. 7. Vocabulary: zeros, real roots, perfect square trinomial, binomial, polynomial, vertex, discriminant, line of symmetry, leading coefficient, restrictions, rational expression D. What is another way to represent this? Acquisition: Students will be skilled at 1. Adding and subtracting polynomials 2. Multiplying monomials and polynomials 3. Multiplying binomials 4. Factoring (distributive property) 5. Factoring x 2 + bx + c expressions 6. Factoring ax 2 + bx + c expressions, where a >1 7. Factoring special products 8. Solving ax 2 + bx + c = d by factoring 9. Solving ax 2 + bx + c = d with the quadratic formula 10. Solving ax 2 + bx + c = d by graphing 11. Finding the vertex of ax 2 + bx + c = d 12. Finding the x and y intercepts of ax 2 + bx + c = d 13. Completing the square of a quadratic equation to find the max or min of the function (vertex) 14. Using a quadratic equation to model real world (ex. projectile motion 15. Applying transformations to graph quadratics functions in the form f(x) = a(x-h) 2 + k 16. Applying transformations to graph absolute value functions in the form f(x) = a x-h + k 17. Using factoring skills to simplify, Madison Public Schools June

14 F.BF.1 Write a function that describes a relationship between two quantities Build new functions from existing functions. F.BF.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. A.SSE.2 Use the structure of an expression to identify ways to rewrite it. A.SSE.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression A.SSE.3a Factor a quadratic expression to reveal the zeros of the function it defines. A.SSE.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. A.APR.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials A.REI.4 Solve quadratic equations in one variable. A.REI.4a Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x p)2 = q that has the same solutions. Derive the quadratic formula from this form. (done in this unit as an extension) A.REI.4b Solve quadratic equations by inspection (e.g., for x 2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. multiply and divide rational expressions 18. Using factoring to simplify rational equations before solving 19. Finding the restrictions on the variable of a rational expression Madison Public Schools June

15 Unit G - Growth and Decay - Understanding Exponential Functions Overview This unit builds on concepts of a function and patterns of change as students work with interesting and significant relationships that are exponential in nature. Students study rules of exponents and develop meaning for negative and rational exponents. Then they will apply those rules to exponential functions. Students will transform functions as they did with linear, quadratic, and absolute value models. When comparing an exponential model with a linear model, the question is not if the exponential model will generate very large or very small inputs, but rather when. Students will gain an appreciation for the power of mathematics in identifying and addressing solutions and making predictions and decisions about significant real world problems. 21 st Century Skills: Product Creations, Synthesizing ESTABLISHED GOALS/ STANDARDS MP4 Model with Mathematics MP5 Use appropriate tools strategically MP7 Look for and make use of structure A.SSE.1b. Interpret complicated expressions by viewing one or more of their parts as a single entity. N.RN.1 Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents. N.RN.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple Stage 1 - Desired Results Transfer: Students will be able to independently use their learning in new situations to Model relationships among quantities. (synthesizing) 2. Manipulate expressions to create order and establish relationships. (product creation) 3. Draw conclusions about graphs, shapes, equations, or objects. (synthesizing) 4. Justify reasoning using clear and appropriate mathematical language. (product creation) UNDERSTANDINGS: Students will understand that: 1. When comparing an exponential model with a linear model, the exponential model will eventually generate very large or very small inputs 2. Mathematicians create or use models to examine, describe, solve and/or make Meaning: ESSENTIAL QUESTIONS: Students will explore & address these recurring questions: A. What are the similarities and differences between linear, quadratic and exponential functions? B. How do changes to the parent quadratic/absolute function change the Madison Public Schools June

16 cases and using technology for more complicated cases. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. F.BF.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. F.LE.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; and that exponential functions grow by equal factors over equal intervals F.LE.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (include reading these from a table). F.LE.3 Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. F.LE.5 Interpret the parameters in a linear or exponential function in terms of a context. A-SSE3c. Use the properties of exponents to transform expressions for exponential functions F.IF.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. b. Use the properties of exponents to interpret expressions for exponential functions. predictions. 3. Mathematicians argue the relationships between problem scenarios and mathematical representation. Students will know 1. y = ar x and/or y = ab x 2. The effects of parameters on an exponential function 3. a m a n = a m+n 4. a m /a n = a m-n 5. (a m ) n = a mn 6. a = a 1/2 7. Applications such as compound interest, doubling time, half life 8. Applications to geometry including area and volume 9. Vocabulary: exponential, decay, graph? C. What can the characteristics of an exponential function tell you about real world events? D. What is another way to represent this? Acquisition: Students will be skilled at 1. Distinguishing between linear, exponential and quadratic growth in tables, graphs or equations; 2. Write a recursive or explicit rule for an exponential function; 3. Using exponent rules to simplify expressions 4. Rational exponents; 5. Using a recursive feature of a graphing calculator to model exponential growth; 6. Distinguishing between exponential growth and decay in real world situations; 7. Describing the effects of the parameters in exponential functions; 8. Fitting an exponential function to a set of data; 9. Determining the growth or decay factor and writing an explicit equation for an exponential function; 10. Determining the percent rate of change and the growth or decay factor. Madison Public Schools June