Overview of Math Standards


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1 ManhattanOgden USD 383 Math Year at a Glance Algebra 1 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse audiences, Quality Producers who create intellectual, artistic and practical products which reflect high standards Complex Thinkers who identify, access, integrate, and use available resources Collaborative Workers who use effective leadership and group skills to develop positive relationships within diverse settings. Community Contributors who use time, energies and talents to improve the welfare of others SelfDirected Learners who create a positive vision for their future, set priorities and assume responsibility for their actions. Click here for more. Overview of Math Teams of teachers and administrators comprised the pk12+ Vertical Alignment Team to draft the maps below. The full set of Kansas College and Career () for Math, adopted in 2010, can be found here. To reach these standards, teachers use Holt curriculum, resources, assessments and supplemented instructional interventions. of Mathematical Practice 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. Click here for more. Additionally, educators strive to provide math instruction centered on: 1: Focus  Teachers significantly narrow and deepen the scope of how time and energy is spent in the math classroom. They do so in order to focus deeply on only the concepts that are prioritized in the standards. 2: Coherence  Principals and teachers carefully connect the learning within and across grades so that students can build new understanding onto foundations. 3: Fluency  Students are expected to have speed and accuracy with simple calculations; teachers structure class time and/or homework time for students to memorize, through repetition, core functions. 4: Deep Understanding  Students deeply understand and can operate easily within a math concept before moving on. They learn more than the trick to get the answer right. They learn the math. 5: Application  Students are expected to use math concepts and choose the appropriate strategy for application even when they are not prompted. 6: Dual Intensity  Students are practicing and understanding. There is more than a balance between these two things in the classroom both are occurring with intensity. Click here for more. 1
2 When appropriate: ManhattanOgden USD 383 Math Year at a Glance Algebra 1 Use multiple representations to enforce concepts (graph, table, equation). Model real life situations that compliment the mathematical techniques. Notes: Vocabulary terms are listed only in the unit they are first introduced. 1. Solving Linear Equations 1.1 Solving Simple Equations 1.2 Solving MultiStep Equations 1.3 Solving Equations with Variables on Both Sides 1.4 Solving Absolute Value Equations 1.5 Rewriting Equations and Formulas 2. Solving Linear Inequalities 2.1 Writing & Graphing Inequalities 2.2 Solving Inequalities Using + and 2.3 Solving Inequalities Using x and 1 ACED.A.1 AREI.A.1 AREI.B.3 NQ.A.1 ACED.A.4 ACED.A.1 AREI.B.3 Conjecture Rule Theorem Equation (linear) Solution Inverse operations Equivalent equations Identity Absolute value equation Extraneous solution Literal equation Formula Inequality Solution set Equivalent inequalities Compound inequality How can you use equations to solve reallife problems? How can you solve an equation with variables on both sides? How can you solve an absolute value equation? How can you use a formula for one measurement to write a formula for a different measurement? How can you use an inequality to describe a reallife statement? How can you solve simple and multistep inequalities using the four basic operations? Solve simple and multistep equations in one variable using the properties of equality. Solve equations containing an absolute value. Manipulate equations to solve for different variables. Write, graph, and solve single and multistep linear inequalities using properties of inequalities Might be necessary to incorporate some review work including number sense and order of operations. Use inequality notation, ex. x < 3 but show how to read set builder, and if appropriate, show interval
3 2.4 Solving MultiStep Inequalities 2.5 Solving Compound Inequalities 2.6 Solving Absolute Value Inequalities Absolute value inequality Absolute deviation How can you use inequalities to describe intervals on the real number line? How can you solve an absolute value inequality? Write, graph and solve compound inequalities using properties of inequalities notation for advanced classes Write, graph, and solve absolute value inequalities 3. Graphing Linear Functions 3.1 Functions 3.2 Linear Functions 3.3 Function Notation 3.4 Graphing Linear Equations in Standard Form 3.5 Graphing Linear Equations in Slope Intercept Form 3.6 Transformations of Linear Functions 3.7 Graphing Absolute Value Functions 2 FIF.A.1 ACED.A.2 AREI.C.10 FIF.B.5 FIF.C.7a FLE.A.1b FIF.A.2 FIF.C.7a FIF.C.9 FIF.B.4 FLE.B.5 FBF.B.3 AREI.D.10 FIF.C.7b Relation Function Domain Range Independent variable Dependent variable Linear equation Linear function Nonlinear function Function notation x and y intercepts Slope (rise and run) Standard form Slopeintercept form Constant function Family of functions Parent function Transformation Translation Reflection What is a function? How can you determine whether a function is linear or nonlinear? How can you use function notation to represent a function? How can you describe the graph of a linear function in both slopeintercept and standard form? How is an absolute value function graphed? How do translations and reflections affect the parent Determine whether a relation is a function, and if so name its domain and range. Represent situations using function notation. Differentiate between linear and nonlinear functions. Graph and interpret linear functions when When teaching transformations, include only translations and reflections. Consider going over linear and absolute value graphs, and then do transformations on both at the same time (horizontal shifts are easier to see on absolute value functions)
4 Absolute value function Vertex Vertex form graph of a linear or absolute value function? presented in both standard and slopeintercept form. Graph and interpret an absolute value function in vertex form. Translate and reflect linear and absolute value functions to form many other graphs all related to the parent graph. 4. Writing Linear Functions 4.1 Writing Equations in SlopeIntercept Form 4.2 Writing Equations in PointSlope Form 4.3 Writing Equations of Parallel & Perpendicular Lines 4.4 Scatter Plots & Lines of Best Fit 4.5 Analyzing Lines of Fit ACED.A.2 FBF.A.1a FLE.A.1b FLE.A.2 FLE.B.5 SID.B.6a SID.B.6b SID.B.6c SID.C.7 SID.C.8 S.ID.C.9 Linear model Pointslope form Parallel lines Perpendicular lines Scatter plot Correlation Line of best fit Piecewise function (optional) When given information about a linear function, how can you write its equation? When are two lines parallel or perpendicular? How can you use a scatter plot and a line of best fit to make conclusions and predictions about data? Write an equation for a linear function in both pointslope and slopeintercept form. Determine when two lines are parallel or perpendicular. Stress pointslope form and the relationships between the various forms of a line. Students should be able to draw a line of best fit, and estimate its 3
5 4.7 Piecewise Functions (teacher discretion) FIf.A.3 FBF.A.1a FBFA.2 AREI.D.10 FIF.C.7b Draw a line of best fit, find a suitable equation, and use it to interpret and predict. equation using pointslope techniques, understand the difference between positive, negative, and no correlation and make predictions based on the line of best fit. 5. Solving Systems of Linear Equations , 5.5 Solving Systems of Linear Equations by Graphing, Substitution, and Elimination ACED.A.3 AREI.C.6 AREI.C.5 AREI.D.11 AREI.D.12 System of linear equations Substitution Elimination Halfplane How can a system of linear equations be solved? How many solutions can a system of linear equations have? Solve a linear system in two variables using the techniques of graphing, substitution, an elimination. 4.7 could be used to reinforce earlier concepts such as domain, range, but completely at teacher s discretion (not tested) More emphasis should be placed on systems of two lines, but absolute value functions 4
6 5.4 Solving Special Systems of Linear Equations 5.6 Graphing Linear Inequalities in Two Variables 5.7 Systems of Linear Inequalities **SEMESTER BREAK** How can a linear inequality or system of linear inequalities be solved? Realize how many solutions are possible just by looking at the equations of the system (ex. two lines could intersect once, infinitely many times, or not at all. should be used as well. May choose to combine or change order of sections to best suit class needs/teaching style. Interpret a special solution (infinite or no solutions). Solve a single linear inequality or a system of linear inequalities by graphing. Stress multiple ways to solve systems; encourage efficiency, but ultimately the correct solution (by whichever method) is most important. 6. Exponential Functions and Sequences 6.1 Properties of Exponents 6.2 Radicals & Rational Exponents NRN.A.2 NRN.A.1 ACED.A.2 FIf.B.4 FIF.C.7e FIF.C.9 FBF.A.1a n th root Radical Index of radical Exponential function Exponential growth, decay Exponential equation What are the basic rules of exponents? How are roots and powers used to make a rational exponent? Apply the basic rules of exponents to expressions containing numbers and variables. 6.1 will require thorough review and time. Students should understand that a rational 5
7 6.3 Exponential Functions 6.4 Exponential Growth & Decay 6.5 Solving Exponential Equations FBF.B.3 FLE.A.1a FLE.A.2 ASSE.B.3c FIF.C.8b FLE.A.1c ACED.A.1 AREI.A.1 AREI.D.11 FIF.A.3 FBF.A.2 What are the characteristics of an exponential function, and how is it like and different from a linear function? What is the difference between exponential growth and exponential decay? How is an exponential equation solved? Convert between roots/powers and rational exponents. Graph exponential growth and decay functions and transform them using translations and reflections. Solve an exponential equations with like bases. exponent is another way of writing a root and a power; can limit to square and cube roots. Students should recognize and understand a very basic exponential function, both growth and decay, with previously learned transformations applied. Students should be able to solve an exponential equation with LIKE bases; teachers can experiment with unlike bases if applicable to their students. 6
8 7. Polynomial Equations and Factoring Algebra AAPR. A.1: Understand that polynomials form a system similar to integers in being closed. AAPR.B.3: Identify zeros of polynomials in factored form. AREI.B.4b: Solve quadratic equations in one variable by inspectionexample x 2 =49, quadratic formula, factoring. ASSE.A.2: Use the structure of an expression to identify ways to rewrite it. AREI.B.3a: Solve linear equations and inequalities in one variable including equations with coefficients represented by letters. Polynomial (also monomial, binomial, trinomial) Degree Factored form Standard form Zeroproduct property Leading coefficient FOIL method (special case of distribution) Root (also repeated root) Closed How are polynomials added, subtracted, multiplied and divided? How is a polynomial equation solved? How is factoring used to break a trinomial into a product of two binomials? How do you recognize and factor special products? Perform the four basic operations on polynomial expressions (including long and synthetic division). Solve a polynomial equation through factoring and application of the zeroproduct property and recognize the the solutions are the x intercepts o the polynomial equation. Factor a polynomial expression Consider breaking the unit into two parts and testing separately (operations and then factoring). Review factoring out a GCF from a polynomial. Spend plenty of time factoring as it is an important component for many algebra 2 skills. 7
9 completely using appropriate strategies. Graph a quadratic function and transform the parent function to form many other functions. 8. Graphing Quadratic Functions 8.1 Graphing Quadratic Functions (with Vertical Stretch/Compression) 8.2 Graphing Quadratic Functions (with Vertical Translations) 8.3 Graphing in Standard Form 8.4 Graphing in Vertex Form 8.5 Using Intercept Form 8.6 Comparing Linear, Exponential, and Quadratic Functions Algebra ACED.A.2: Create equations in two or more variables to represent relationships, graph equations on coordinate axes with labels and scales. FIF.C.7a: Graph linear and quadratic functions and show intercepts, maxima, and minima. FBF.B.3: Identify the effect on the graph by replacing f(x) by f(x) + k. kf(x), f(kx) and f(x+k) for specific values of k, experiment using technology. FIF.C.9: Compare properties of two Quadratic function Parabola Vertex Axis of symmetry Vertical stretch, shrink Maximum and minimum value Zero Vertex form Intercept form What are some characteristics of the graph of a quadratic function? How do transformations affect the graph of ff(xx) = xx 2? How do you graph a quadratic function when given in standard, vertex, or intercept form? How can you compare the growth rates of linear, exponential, and quadratic functions? Graph a quadratic function when presented in different forms. Make connections among the different forms of a quadratic function. Compare and contrast features of the linear, exponential, and quadratic functions. Teach VERTICAL stretches and shrinks in this unit (not previously taught). Observe that all quadratics have symmetry, but no need to even/odd terminology. 8
10 functions represented in a different way (algebraically, graphically, numerically in tables, or by verbal description) FIF.B.4: Interpret and sketch graphs. FBF.A.1: Write a function that describes a relationship between two quantities. ASSE.B.3: Factor a quadratic to find zeros. AAPR.B.3: Identify zeros of polynomials when factored and use zeros to construct a rough graph. ACED.A.2: Create equations to represent relationships, graph 9
11 equations on coordinate axes. FIF.C.8: Write a function in equivalent and explain properties of the function. FIF.B.6: Calculate and interpret the average rate of change. FLE.A.3: Observe using graphs and tables increasing functions. 9. Solving Quadratic Equations 9.1 Properties of Radicals 9.2 Solving Quadratic Equations by Graphing 9.3 Solving Quadratic Equations by Square Roots NRN.A.2 NRN.B.3 AREI.D.11 FIF.C.7a ACED.A.1 ACED.A.4 AREI.B.4b ASSE.B.3b AREI.B.4a FIF.C.8a AREI.C.7 Radical expression Rationalize Quadratic equation Quadratic formula Discriminant How are the four basic operations performed on square and cube roots? How are quadratic equations solved? How can you determine the number of real solutions to a quadratic equation? How many solutions are possible when you have Simplify radical expressions. Solve quadratic equations using the methods of graphing, square roots, and quadratic formula. Students are expected to rationalize denominators, but exclude use of the conjugate. Limit radicals to square and cube roots (include both 10
12 9.5 Solving Quadratic Equations by the Quadratic Formula 9.6 Systems on Non Linear Equations system on nonlinear equations? Determine the number of real solutions to a quadratic equation through graphing or evaluating the discriminant. Determine how many solutions exist when given the graph of a nonlinear system. variables and numbers). Stress multiple ways to solve quadratic equations; encourage efficiency, but ultimately the correct solution (by whichever method) is most important. Students are not expected to solve a system of nonlinear equations, only note the number of possible solutions and understand why; you might discuss how to solve, but the expectation is not to carry out 11
13 substitution or elimination for specific solution sets. 10. Radical Functions and Equations 10.1 Graphing Square Root Functions 10.3 Solving Square Root Equations ACED.A.2 FIF.B.4 F.IF.B.6 FIF.C.7b F.IF.C.9 ACED.A.1 Square root function What are some characteristics of the graph of a square root function? How can you solve an equation involving square roots? Graph and describe a square root function, including ones that have been transformed. If time allows, introduce the square root function (graphs and equations) Solve an equation containing a square root. 11. Data Analysis and Displays 11.1 Measures of Center and Variation 11.2 BoxandWhisker Plots SID.A.3 SID.A.1 Mean Median Mode Range Interquartile range Outlier Quartile Boxandwhisker plot How can you describe the center and variation of a data set? How can a boxandwhisker plot be used to describe a data set? I can choose an appropriate measure of center. I can use a boxandwhisker plot to interpret a data set. If time allows and teacher feels there is a need, students should be able to use the three measures of center, range as a measure of variation, and displaying onevariable data in boxand 12
14 whisker (or other) plots. 13
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