Algebra 2 Algebra 2 Honors MATHEMATICS. Florida Standards

Transcription

1 Algebra 2 Algebra 2 Honors MATHEMATICS Florida Standards

2 Florida Standards Standards for Mathematical Practice 1. Make sense of problems and persevere in solving them. (MAFS.K12.MP.1) Solving a mathematical problem involves making sense of what is known and applying a thoughtful and logical process which sometimes requires perseverance, flexibility, and a bit of ingenuity. 2. Reason abstractly and quantitatively. (MAFS.K12.MP.2) The concrete and the abstract can complement each other in the development of mathematical understanding: representing a concrete situation with symbols can make the solution process more efficient, while reverting to a concrete context can help make sense of abstract symbols. 3. Construct viable arguments and critique the reasoning of others. (MAFS.K12.MP.3) A well-crafted argument/critique requires a thoughtful and logical progression of mathematically sound statements and supporting evidence. 4. Model with mathematics. (MAFS.K12.MP.4) Many everyday problems can be solved by modeling the situation with mathematics. 5. Use appropriate tools strategically. (MAFS.K12.MP.5) Strategic choice and use of tools can increase reliability and precision of results, enhance arguments, and deepen mathematical understanding. 6. Attend to precision. (MAFS.K12.MP.6) Attending to precise detail increases reliability of mathematical results and minimizes miscommunication of mathematical explanations. 7. Look for and make use of structure. (MAFS.K12.MP.7) Recognizing a structure or pattern can be the key to solving a problem or making sense of a mathematical idea. 8. Look for and express regularity in repeated reasoning. (MAFS.K12.MP.8) Recognizing repetition or regularity in the course of solving a problem (or series of similar problems) can lead to results more quickly and efficiently.

3 Algebra 2 & Algebra 2 Honors: Florida Standards Building on their work with linear, quadratic, and exponential functions, students extend their repertoire of functions to include polynomial, rational, and radical functions. Students work closely with the expressions that define the functions, and continue to expand and hone their abilities to model situations and to solve equations, including solving quadratic equations over the set of complex numbers and solving exponential equations using the properties of logarithms. The Mathematical Practice Standards apply throughout each course and, together with the content standards, prescribe that students experience mathematics as a coherent, useful, and logical subject that makes use of their ability to make sense of problem situations. The critical areas for this course, organized into five units, are as follows: Polynomial, Rational, and Radical Relationships: This unit develops the structural similarities between the system of polynomials and the system of integers. Students draw on analogies between polynomial arithmetic and base-ten computation, focusing on properties of operations, particularly the distributive property. Students connect multiplication of polynomials with multiplication of multi-digit integers, and division of polynomials with long division of integers. Students identify zeros of polynomials, including complex zeros of quadratic polynomials, and make connections between zeros of polynomials and solutions of polynomial equations. The unit culminates with the fundamental theorem of algebra. A central theme of this unit is that the arithmetic of rational expressions is governed by the same rules as the arithmetic of rational numbers. Trigonometric Functions: Building on their previous work with functions, and on their work with trigonometric ratios and circles in Geometry, students now use the coordinate plane to extend trigonometry to model periodic phenomena. Modeling with Functions: In this unit students synthesize and generalize what they have learned about a variety of function families. They extend their work with exponential functions to include solving exponential equations with logarithms. They explore the effects of transformations on graphs of diverse functions, including functions arising in an application, in order to abstract the general principle that transformations on a graph always have the same effect regardless of the type of the underlying function. They identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit. The description of modeling as the process of choosing and using mathematics and statistics to analyze empirical situations, to understand them better, and to make decisions is at the heart of this unit. The narrative discussion and diagram of the modeling cycle should be considered when knowledge of functions, statistics, and geometry is applied in a modeling context. Inferences and Conclusions from Data: In this unit, students see how the visual displays and summary statistics they learned in earlier grades relate to different types of data and to probability distributions. They identify different ways of collecting data including sample surveys, experiments, and simulations and the role that randomness and careful design play in the conclusions that can be drawn. Applications of Probability: Building on probability concepts that began in the middle grades, students use the languages of set theory to expand their ability to compute and interpret theoretical and experimental probabilities for compound events, attending to mutually exclusive events, independent events, and conditional probability. Students should make use of geometric probability models wherever possible. They use probability to make informed decisions.

4 Fluency Recommendations A/G- Algebra I students become fluent in solving characteristic problems involving the analytic geometry of lines, such as writing down the equation of a line given a point and a slope. Such fluency can support them in solving less routine mathematical problems involving linearity, as well as in modeling linear phenomena (including modeling using systems of linear inequalities in two variables). A-APR.1- Fluency in adding, subtracting, and multiplying polynomials supports students throughout their work in Algebra, as well as in their symbolic work with functions. Manipulation can be more mindful when it is fluent. A-SSE.1b- Fluency in transforming expressions and chunking (seeing parts of an expression as a single object) is essential in factoring, completing the square, and other mindful algebraic calculations. The following Mathematics and English Language Arts Florida Standards should be taught throughout the course: MAFS.912. 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 o their parts as a single entity. MAFS.912.N-Q.1.2: Define appropriate quantities for the purpose of descriptive modeling. MAFS.912.F-IF.3.9: Compare properties of two functions each represented in a different way(algebraically, graphically, numerically in tables, or by verbal descriptions) LAFS.910.RST.1.3: Follow precisely a complex multistep procedure when carrying out experiments, taking measurements or performing tasks, attending to special cases or exceptions defined in the text. LAFS.910.RST.2.4: Determine the meaning of symbols, key terms, and other domain-specific words and phrases as they are used in context and topics. LAFS.910.RST.3.7: Translate quantitative or technical information expressed in words in a text into visual form and translate information expressed visually or mathematically into words. LAFS.910.SL.1.1: Initiate and participate effectively in a range of collaborative discussions with diverse partners. LAFS.910.SL.1.2: Integrate multiple sources of information presented in diverse media or formats evaluating the credibility and accuracy of each source. LAFS.910.SL.1.3: Evaluate a speaker s point of view, reasoning, and use of evidence and rhetoric, identifying any fallacious reasoning or exaggerated or distorted evidence. LAFS.910.SL.2.4: Present information, findings and supporting evidence clearly, concisely, and logically such that listeners can follow the line of reasoning. LAFS.910.WHST.1.1: Write arguments focused on discipline-specific content. LAFS.910.WHST.2.4: Produce clear and coherent writing in which the development, organization, and style are appropriate to task, purpose, and audience. LAFS.910.WHST.3.9: Draw evidence from informational texts to support analysis, reflection, and research.

5 Algebra 2 & Algebra 2 Honors: Florida State Standards At A Glance First Quarter Second Quarter Third Quarter Fourth Quarter SMT Unit 0 Linear & Quadratic Functions (Prerequisite) MAFS.912.A.-REI.1.1 MAFS.912.A-CED.1.4 MAFS.912.F-IF.2.6 MAFS.912.A-REI.2.4 MAFS.912.A-SSE.2.3a &b DIA 1 MAFS.912.N-CN.1.1 MAFS.912.N-CN.1.2 MAFS.912.N-CN.3.7 *MAFS.912.N-CN.3.8 MAFS.912.F-IF.3.8a MAFS.912.A-REI.4.11 MAFS.912.A-REI.3.7 MAFS.912.F-BF.2.3 MAFS.912.F-IF.2.5 MAFS.912.F-IF.3.7 MAFS.912.A-CED.1.1 MAFS.912.A-CED.1.2 MAFS.912.A-CED.1.3 MAFS.912.G-GPE.1.2 DIA 2 * Standards highlighted are for HONORS only* Unit 1- Polynomial Functions MAFS.912.A-APR.1.1 MAFS.912.A-APR.2.3 MAFS.912.A-SSE.1.1 MAFS.912.A-SSE.1.2 MAFS.912.A-APR.3.4 MAFS.912.A-APR.2.2 *MAFS.912.N-CN.3.9 *MAFS.912.A-APR.3.5 MAFS.912.F-BF.2.3 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.5 MAFS.912.A-CED.1.2 MAFS.912.A-CED.1.3 DIA 3 Unit 2-Statistitics MAFS.912.S-IC.1.1 MAFS.912.S-IC.1.2 MAFS.912.S-IC.2.3 MAFS.912.S-IC.2.4 MAFS.912.S-IC.2.5 MAFS.912.S-ID.1.4 MAFS.912.S-ID.2.6 MAFS.912.S-IC.2.6 Unit 3- Probability MAFS.912.S-CP.1.1 MAFS.912.S-CP.1.2 MAFS.912.S-CP.1.3 MAFS.912.S-CP.1.4 MAFS.912.S-CP.1.5 MAFS.912.S-CP.2.6 MAFS.912.S-CP.2. 7 *MAFS.912.S-CP.2.8 *MAFS.912.S-CP.2.9 *MAFS.912.S-MD.2.6 *MAFS.912.S-MD.2. 7 Unit 4 - Radical Functions MAFS.912.N-RN.1.1 MAFS.912.N-RN.1.2 MAFS.912.A-REI.1.2 MAFS.912.F-BF.2.4a MAFS.912.F-BF.2.3 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.5 MAFS.912.F-IF.3.7b MAFS.912.A-CED.1.2 *MAFS.912.F-BF.2.4b *MAFS.912.F-BF.2.4c *MAFS.912.F-BF.2.4d DIA 4 Unit 5- Exponential and Logarithmic Functions MAFS.912.A-SSE.2.3 MAFS.912.A-REI.1.1 MAFS.912.F-LE.1.4 MAFS.912.F-BF.1.2 MAFS.912.F-IF.1.3 MAFS.912.F-LE.1.2 MAFS.912.F-BF.1.1a MAFS.912.A-SSE.2.4 MAFS.912.F-BF.2.3 MAFS.912.F-IF.2.4 MAFS.912.F-LE.2.5 MAFS.912.F-IF.3.7e MAFS.912.A-CED.1.2 MAFS.912.F-IF.3.8b DIA 5 Unit 6 - Rational Functions MAFS.912.A-APR.4.6 MAFS.912.A.-REI.1.2 MAFS.912.F-BF.2.3 MAFS.912.F-IF.2.4 MAFS.912.F-IF.2.5 MAFS.912.A-CED.1.2 *MAFS.912.F-IF.3.7d DIA 6 Unit 7 Trigonometric Functions MAFS.912.F-TF.1.1 MAFS.912.F-TF.1.2 MAFS.912.F-TF.3.8 MAFS.912.F-TF.2.5 DIA 7 EOC Exam SMT

6 Florida Standards Conceptual Categories and Domains for Pre Calculus Honors Conceptual Category: Number and Quantity Domain: N-RN: The Real Number System Domain: N-CN: Complex Numbers Conceptual category: Algebra Domain: SSE: Seeing Structure in Expressions Domain: APR: Arithmetic with Polynomials and Rational Expressions Domain: CED: Creating Equations Conceptual category: Geometry Domain: G-GPE: Expressing Geometric Properties with Equations Conceptual category: Statistics and Probability Domain: S-IC: Making Inferences and Justifying Conclusions Domain: S-CP: Conditional Probability and the Rules of Probability Domain: S-MD: Using Probability to Make Decisions Domain: REI: Reasoning with Equations and Inequalities Conceptual category: Functions Domain: F-IF: Interpreting Functions Domain: F-BF: Building Functions Domain: F-LE: Linear, Quadratic, and Exponential Models Domain: F-TF: Trigonometric Functions Throughout the map, in the resource column, the Flipbook was used as a reference. The mapping committee recommended printing the flipbook as the pages referenced are the actual paper page numbers, not the file page numbers. The document also provided directions to print and create your flipbook:

7 Course: Algebra 2 & Algebra 2 Honors Unit 0: Linear & Quadratics Functions: Prerequisite In what ways can the problem be solved, and why should one method be chosen over the other? How can algebra describe the relationship between sets of numbers? How can the relationship between quantities best be represented? Standard The students will: I can: Remarks Resources MAFS.912.A-REI.1.1 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. MAFS.912.A-CED.1.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. SMP #4 MAFS.912.F-IF.2.6 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. SMP # 4,#5 apply order of operations and inverse operations to solve equations. construct an argument to justify my solution process. Provide examples for how the same equation might be solved in a variety of ways as long as equivalent quantities are added or subtracted to both sides of the equation, the order of steps taken will not matter. solve formulas for a specified variable. Give students formulas, such as area and volume (or from science or business), and have students solve the equations for each of the different variables in the formula. define interval, rate of change, and average rate of change. explain the connection between average rate of change and the slope formula. calculate the average rate of change of a function, represented either by function notation, a graph, or a table, over a specific input interval. compare the rates of change of two or more functions when they are represented with function notation, with a graph, or with a table. interpret the meaning of the average rate of change (using units) as it relates to a real-world problem. Book pgs Book pgs Book pgs 98-99

8 Can the student model and solve quadratic equations using a variety of algebraic methods? Why structure expressions in different ways? Standard The students will: I can: Remarks Resources MAFS.912.A-REI.2.4 Solve quadratic equations in one variable. SMP #4 MAFS.912.N-CN.1.1 Know there is a complex number i such that i 2 = -1, and every complex number has the form a + bi with a and b real. SMP #6 MAFS.912.N-CN.1.2 Use the relation i 2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. SMP #7, #8 MAFS.912.N-CN.3.7 Solve quadratic equations with real coefficients that have complex solutions. SMP #1, #7 MAFS.912.N-CN.3.8 *Honors Only* Extend polynomial identities to the complex numbers. SMP #7, #8 solve quadratic equations by inspection (e.g., for x² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. 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, and derive the quadratic equation from this form. know there is a complex number i such that i² = 1, and every complex number has the form a + bi with a and b real. use the relation i² = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. solve quadratic equations with real coefficients that have complex solutions. express the solution of a quadratic equation as a complex number, a + bi. Write the factors of polynomials using complex numbers. use complex numbers to rewrite a sum of squares, a 2 + b 2 as the product of a complex number and its conjugate. show that factored quadratics have real coefficients when written in standard form. Some students may think that rewriting equations into various forms (taking square roots, completing the square, using quadratic formula and factoring) are isolated techniques within a unit of quadratic equations. Teachers should help students see the value of these skills in the context of solving higher degree equations and examining different families of functions. Includes recognition of the cycles of i (i 2 =-1, i 3 =i 2 i = -i, etc.). In the cases of quadratic equations, when the use of quadratic formula is not critical, students sometime ignore the negative solutions. For example, for the equation x 2 = 9, students may mention 3 and forget about ( 3), or mention 3i and forget about (- 3i) for the equation x 2 = - 9. If this misconception persists, advise students to solve this type of quadratic equation either by factoring or by the quadratic formula. For example, rewrite x² + 4 as (x + 2i)(x 2i) Book pgs Book pgs 27-30

9 Can the student model and solve quadratic equations using a variety of algebraic methods? Why structure expressions in different ways? Standard The students will: MAFS.912.F-IF.3.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a) 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. SMP #7, #8 MAFS.912.A-SSE.2.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. SMP #7 I can: Remarks Resources explain that there are three forms of quadratic functions: Use various representations of standard form, vertex form, and factored form. the same function to apply completing the square to rewrite a quadratic emphasize different Book pgs characteristics of that function function in vertex form. For example, the y-intercept of factor a quadratic expression to find the zeroes of the the function y = x 2-4x 12 is Graphs function it represents. easy to recognize as (0, -12). (2007) identify and factor perfect-square trinomials. However, rewriting the function define an exponential function. as y = (x 6)(x + 2) reveals Quadratics rewrite exponential functions using the properties of zeros at (6, 0) and at ( -2, 0). (2009) exponents. Building factor a quadratic expression to find the zeroes of the Connections function it represents. (NCTM) identify and factor perfect-square trinomials. define an exponential function. rewrite exponential functions using the properties of exponents. Furthermore, completing the square allows the equation to be written as y = (x 2) 2 16, which shows that the vertex (and minimum point) of the parabola is at (2, -16). Students often confuse the k value with the y-intercept. Be sure to explain that the y- intercept is not (0, k). Show them how to substitute 0 for x and solve for the y-intercept. Book pgs Book pgs MAFS.912.A-REI.4.11 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. SMP #5 explain that a point of intersection on the graph of a system of equations represents a solution to both equations. use a graphing calculator to determine the approximate solutions to a system of equations. Students may believe that the graph of a function is simply a line or curve connecting the dots, without recognizing that the graph represents all solutions to the equation. Book pg. 83 (refer to page 82 as indicated) MAFS.912.A-REI.3.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. SMP #8 determine the approximate solution of a system of equations in which on equation is linear and one equation is quadratic by graphing and estimating the point(s) of intersection Provide students opportunities to practice linear vs. non-linear systems; consistent vs. inconsistent systems; pure computational vs. real-world contextual problems (e.g., chemistry and physics CCSS Flip Book pgs 79-80

10 applications encountered in science classes). MAFS.912.G-GPE.1.2 Derive the equation of a parabola given a focus and directrix. SMP #2, #3, #7 define a parabola. determine the distance from a point on the parabola to the directrix. determine the distance from a point on the parabola to the focus using the distance formula (Pythagorean Theorem). equate the two distance expressions for a parabola to write its equation. identify the focus and directrix of a parabola when given its equation. Define a parabola as a set of points satisfying the condition that their distance from a fixed point (focus) equals their distance from a fixed line (directrix). Start with a horizontal directrix and a focus on the y-axis, and use the distance formula to obtain an equation of the resulting parabola in terms of y and x2. Next use a vertical directrix and a focus on the x-axis to obtain an equation of a parabola in terms of x and y2. CCSS Flip Book pg 181 MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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 explain why f(x) + k translates the original graph of f(x) up k units and why f(x)-k translates the original graph of f(x) down k units. explain why f(x + k) translates the original graph of f(x) left k units and why f(x-k) translates the original graph of f(x) right k units. explain why kf(x) vertically stretches or shrinks the graph of f(x) by a factor of k and predict whether a given value of k will cause a stretch or shrink. explain why f(kx) horizontally stretches or shrinks the graph of f(x) by a factor of 1/k and predict whether a given value of k will cause a stretch or a shrink. describe the transformation that changed a graph of f(x) into a different graph when given pictures of the pre-image and image. determine the value of k given the graph of a transformed function. graph the listed transformations when given a graph of f(x) and a value of k ( f(x) k, f(x k), kf(x), and f(kx). use a graphing calculator to generate examples of functions with different k values. Expose students to absolute value and piecewise defined functions as you teach the transformations. In this standard, k is referred to as a constant. Traditionally, a is the factor that causes a vertical stretch, h is the horizontal translation, and k is the vertical translation CCSS Flip Book pgs

11 analyze similarities and differences between functions with different k value MAFS.912.F-IF.2.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. explain how the domain of a function is represented in its graph. identify the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domain CCSS Flip Book page 96 MAFS.912.F-IF.3.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. SMP #6, #7 graph quadratic functions showing intercepts, maxima, and minima. use technology to graph a quadratic function and to find precise values for the intercepts, maximum, or minimum. graph absolute value functions and piecewise-defined functions including step functions. explain that there are three forms of quadratic functions: standard form, vertex form, and factored form. explain that the graph of all three forms of quadratic functions is a parabola. find the x-intercepts of a quadratic to find the axis of symmetry. identify the line of symmetry. sketch a graph of a parabola written in vertex form. apply completing the square to rewrite a quadratic function in vertex form. Use various representations of the same function to emphasize different characteristics of that function. For example, the y-intercept of the function y = x 2 4x 12 is easy to recognize as (0, -12). However, rewriting the function as y = (x 6)(x + 2) reveals zeros at (6, 0) and at ( -2, 0). CCSS Flip Book pgs Graphs (2007) Quadratics (2009) Building Connections (NCTM)

12 MAFS.912.A-CED.1.1 Create equations and inequalities in one variable and use them to solve problems. SMP #4 identify the variables and quantities represented in a real-world problem. determine the best model for the real-world problem. write the equation or inequality that best models the problem. solve the equation or inequality. interpret the solution in the context of the problem. Begin with simple equations and inequalities and build up to more complex equations. CCSS Flip Book pgs MAFS.912.A-CED.1.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales SMP #4 identify the variables and quantities represented in a real-world problem. determine the best model for the real-world problem. write the equation that best models the problem. determine appropriate scale and label the axes. graph equations on coordinate axes with appropriate labels and scales. CCSS Flip Book pgs MAFS.912.A-CED.1.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. SMP #4 identify the variables and quantities represented in a real-world problem. determine the best models for the real-world problem. write the system of equations and/or inequalities that best models the problem. graph the system on coordinate axes with appropriate labels and scales. interpret solutions in the context of the situation modeled and decide if they are reasonable While this standard is initially introduced in the context of linear functions, it applies to all units and should be addressed continually. The standard specifically addresses solving a system by graphing. CCSS Flip Book pgs 60-61

13 Course: Algebra 2 & Algebra 2 Honors Unit 1: Polynomials Functions How can properties of the real number system be useful when working with polynomials expressions? Standard Remarks The students will: I can: MAFS.912.A-APR.1.1 apply the definition of an integer to explain Consider mentioning that like Understand that polynomials form a closure. integers, polynomials are closed system analogous to the integers, apply the definition of a polynomial to explain under addition, subtraction, and namely, they are closed under the closure. multiplication, but not division. operations of addition, subtraction, add and subtract polynomials. and multiplication; add, subtract and multiply polynomials. When adding, subtracting, multiply polynomials. multiplying and dividing focus on polynomial expressions that simplify SMP #2, #7 to forms that are linear or quadratic in a positive integer power of x. Resources Book pg MAFS.912.A-APR.2.3 Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial. SMP #1, #8 identify the zeros of factored polynomials. identify the multiplicity of the zeros of a factored polynomial. explain how the multiplicity of the zeros provides a clue as to how the graph will behave as it approaches and leave the x- intercept. sketch a rough graph using the zeros and other easily identifiable points (y-intercept). Use graphing technology to expedite the exploration of a multitude of functions so students can more easily observe the relationships. Book pg. 50

14 Course: Algebra 2 & Algebra Honors Unit 1: Polynomials Functions How can properties of the real number system be useful when working with polynomials expressions? Standard Remarks The students will: I can: MAFS.912.A-SSE.1.1 define expression, term, factor and coefficient Students should recognize that in the Interpret expressions that represent group the parts of an expression differently in expression 2x + 1, 2 is the a quantity in terms of its context. order to better interpret their meaning coefficient, 2 and x are factors, and a) Interpret parts of an expression, 1 is a constant, as well as 2x and 1 being terms of the binomial such as terms, factors and expression. coefficients b) Interpret complicated expressions by viewing one or more of their parts as a single entity. SMP #7 MAFS.912.A-SSE.1.2 Use the structure of an expression to identify ways to rewrite the polynomial function. (ex. Difference of Squares, Perfect Squares, etc.) SMP #7 identify clues in the structure of expressions to rewrite the function: like terms, common factors, difference of squares, perfect squares, etc. explain why equivalent expressions are equivalent. apply models for factoring and multiplying polynomials to rewrite expressions. Have students create their own expressions that meet specific criteria (e.g., number of terms factorable, difference of two squares, etc.) and verbalize how they can be written and rewritten in different forms. Additionally, pair/group students to share their expressions and rewrite one another s expressions. Resources Book pg (refer to pg. 34 and 35 as indicated) MAFS.912.A-APR.3.4 Prove polynomial identities and use them to describe numerical relationships. SMP #8 MAFS.912.A-APR.2.2 Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x a is p(a), so p(a) = 0 if and only if (x a) is a factor if p(x). SMP #8 verify polynomial identities: difference of squares, difference of cubes, sum of cubes. factor polynomials completely. divide polynomials using long division and synthetic division. apply the Remainder Theorem, when appropriate, to check the answer. apply the Remainder Theorem to determine if a divisor is a factor of a polynomial. Illustrate how polynomial identities are used to determine numerical relationships such as: 25 2 =(20 + 5) 2 = Students can benefit from exploring the rational root theorem, which can be used to find all of the possible rational roots (i.e., zeros) of a polynomial with integer coefficients. When the goal is to identify all roots of a polynomial, including irrational or complex roots, it is useful to graph the polynomial function to determine the most likely candidates for the roots of the polynomial that are the x-intercepts of the graph. Book pg Book pg

15 Course: Algebra 2 & Algebra 2 Honors Unit 1: Polynomials Functions How can properties of the real number system be useful when working with polynomials expressions? Standard Remarks The students will: I can: MAFS.912.N-CN.3.9 *Honors Only* apply the Fundamental Theorem of Algebra to While this is only an Honors Know the Fundamental Theorem of demonstrate that the number of linear factors standard, Regular students would Algebra; show that is it true for all a polynomial has is equal to the degree of that also benefit from this knowledge. polynomials. polynomial. SMP #7 MAFS.912.A-APR.3.5 *Honors Only* Know and apply the Binomial Theorem for the expansion of (x + y) n in powers of x and y for a positive integer n, where x and y are any numbers, with coefficients determined for example by Pascal s Triangle. SMP #7 generate Pascal s Triangle to find the coefficients of a binomial expansion. Return to these learning targets after teaching combinations in Unit 4 Statistics and Probability: apply the combination formula. write the binomial expansion of (a + b) n by applying the Binomial Theorem (a + b) n = n C 0 a n b 0 + n C 1 a n-1 b 1 + n C 2 a n-2 b nc n a 0 b n Resources MAFS.912.F-BF.2.3 Identify the effect on the graph of replacing f(x) by f(x) + k, kf(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 explain why f(x) + k translates the original graph of f(x) up k units and why f(x)-k translates the original graph of f(x) down k units. explain why f(x + k) translates the original graph of f(x) left k units and why f(x-k) translates the original graph of f(x) right k units. explain why kf(x) vertically stretches or shrinks the graph of f(x) by a factor of k and predict whether a given value of k will cause a stretch or shrink. explain why f(kx) horizontally stretches or shrinks the graph of f(x) by a factor of 1/k and predict whether a given value of k will cause a stretch or a shrink. describe the transformation that changed a graph of f(x) into a different graph when given pictures of the pre-image and image. determine the value of k given the graph of a transformed function. graph the listed transformations when given a graph of f(x) and a value of k ( f(x) k, f(x k), kf(x), Expose students to absolute value and piecewise defined functions as you teach the transformations. In this standard, k is referred to as a constant. Traditionally, a is the factor that causes a vertical stretch, h is the horizontal translation, and k is the vertical translation CCSS Flip Book pgs

16 and f(kx). use a graphing calculator to generate examples of functions with different k values. analyze similarities and differences between functions with different k value MAFS.912.F-IF. 2.4 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. SMP #1, #7, #8 MAFS.912.F-IF.2.5 Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. MAFS.912.A-CED.1.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales SMP #4 identify the x-intercepts, increasing/decreasing intervals. define and identify the relative maximum/minimum. use the problem situation to explain the end behavior of the function. explain how the domain of a function is represented in its graph. identify the appropriate domain of a function that represents a problem situation, defend my choice, and explain why other numbers might be excluded from the domain. identify the variables and quantities represented in a real-world problem. determine the best model for the real-world problem. write the equation that best models the problem. determine appropriate scale and label the axes. graph equations on coordinate axes with appropriate labels and scales. Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domains. Students may believe that the slope of a linear function is merely a number used to sketch the graph of the line. Apply the real world meaning for slopes as a rate of change. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. Students may believe that it is reasonable to input any x-value into a function, so they will need to examine multiple situations in which there are various limitations to the domain CCSS Flip Book pg CCSS Flip Book page 96 CCSS Flip Book pgs 58-59

17 MAFS.912.A-CED.1.3 Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. SMP #4 identify the variables and quantities represented in a real-world problem. determine the best models for the real-world problem. write the system of equations and/or inequalities that best models the problem. graph the system on coordinate axes with appropriate labels and scales. interpret solutions in the context of the situation modeled and decide if they are reasonable While this standard is initially introduced in the context of linear functions, it applies to all units and should be addressed continually. The standard specifically addresses solving a system by graphing. CCSS Flip Book pgs 60-61

18 Course: Algebra 2 & Algebra 2 Honors Unit 2: Statistics How can a population be described when it is large, it would be nearly impossible to collect all the data? In what ways does one event impact the probability of another event ofsurring? Standard The students will: MAFS.912.S-IC.1.1 Understand statistics as a process for making inferences about population parameters based on a random sample from that population. SMP #2 MAFS.912.S-IC.1.2 Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. SMP # 1, 3, 4 MAFS.912.S-IC.2.3 Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each. SMP #2 I can: Remarks Resources define populations, population parameter, random sample, and inference. explain why randomization is used to draw a sample that represents a population well. recognize that statistics involves drawing conclusions about a population based on the results obtained from a random sample of the population. choose a probability model for a problem situation. conduct a simulation of the model and determine which results are typical of the model and which results are considered outliers (possible, but unexpected). decide if the data collected is consistent with the selected model or if another model is required. pose a question that suggest a model and a means of collecting data and answer my question. define sample survey, experiment, observation study, and randomization. describe the purpose of a sample survey, and experiment, and an observational study. describe the differences among sample surveys, experiments, and observational studies. explain the role of randomization in sample surveys, experiments, and observational studies. apply random sampling techniques to draw a sample from a population. Students may believe that population parameters and sample statistics are one in the same, e.g., that there is no difference between the population mean which is a constant and the sample mean which is a variable. FSSS Flip Book pgs FSSS Flip Book pgs FSSS Flip Book pgs

19 Standard The students will: MAFS.912.S-IC.2.4 Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling. SMP # 3, 4, 6 Course: Algebra 2 & Algebra 2 Honors Unit 2: Statistics How can a population be described when it is large, it would be nearly impossible to collect all the data? In what ways does one event impact the probability of another event ofsurring? I can: Remarks Resources define population mean, sample mean, population Students may use computer proportion and sample proportion. generated simulation models calculate the sample mean and proportion. based upon the results of sample use sample means and sample proportions to surveys to estimate population estimate population values. statistics and margins of error. defend the statement, The population mean or proportion is close to the sample mean or Standard 4 addresses estimation proportion when the sample is randomly selected of the population proportion and large enough to represent the population well. parameter and the population infer that the population mean or proportion is mean parameter. equal to the sample mean or proportion and Data need not come from just a conduct simulation to determined which sample survey to cover this topic. A results are typical of this model and which results margin-of-error formula cannot be are considered outliers (possible, but unexpected). developed through simulation, but choose an appropriate margin of error for the students can discover that as the sample mean or proportion and create a sample size is increased, the confidence interval based on the results of the empirical distribution of the simulation conducted. sample proportion and the sample determine how often the true population mean or mean tend toward a certain shape proportion is within the margin of error of each (the Normal distribution), and the sample mean or proportion. standard error of the statistics decreases (i.e. the variation) in the pose a question regarding the mean or proportion models becomes smaller. The of a population, use statistical techniques to actual formulas will need to be estimate the parameter, and design an appropriate stated. product to summarize the process and report the estimate. Explain what the results mean in terms of variability in a population and use results to calculate the error for these estimates. FSSS Flip Book pgs

20 Course: Algebra 2 & Algebra 2 Honors Unit 2: Statistics How can a population be described when it is large, it would be nearly impossible to collect all the data? In what ways does one event impact the probability of another event ofsurring? Standard The students will: MAFS.912.S-IC.2.5 Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant SMP #3, 4, 6 MAFS.912.S-ID.1.4 Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets and tables to estimate areas under the normal curve. I can: Remarks Resources calculate the sample mean and standard deviation of Standard 5 addresses testing whether the two treatment groups and the difference of the some characteristic of two paired or means. independent groups is the same or different by the use of re-sampling conduct a simulation for each treatment group using techniques. Conclusions are based on the sample results as the parameters for the the concept of p-value. Re-sampling distributions. procedures can begin by hand but calculate the difference of means for each simulation typically will require technology to and represent those differences in a histogram. gather enough observations for which a apply the results of the simulation to create a p-value calculation will be meaningful. confidence interval for the difference of means. use the confidence interval to determine if the Use a variety of devices as appropriate to carry out simulations: number cubes, parameters are significantly different based on the cards, random digit tables, graphing original difference of means. calculators, computer programs. pose a question regarding the means or proportions of two populations, use statistical techniques to estimate the difference, and design an appropriate product to summarize the process and report the estimate. use the mean and standard deviation of a set of data to fit the data to a normal curve. use the Rule to estimate the percent of a normal population that falls within 1, 2, or 3 standard deviations of the mean. recognize that normal distributions are only appropriate for unimodal and symmetric shapes. estimate the area under a normal curve using a calculator, table, or spreadsheet. Book pg 232

21 Course: Algebra 2 & Algebra 2 Honors Unit 2: Statistics How can a population be described when it is large, it would be nearly impossible to collect all the data? In what ways does one event impact the probability of another event ofsurring? Standard The students will: MAFS.912.S-ID.2.6 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 date to solve problems in the context of the data. b. informally assess the fit of a function by plotting and analyzing residuals. c. fit a linear function for a scatter plot that suggests a linear association. MAFS.912.S-IC.2.6 Evaluate reports based on data. SMP #2, 3 I can: Remarks Resources determine when linear, quadratic, and exponential Use given functions or choose a models should be used to represent a data set. function suggested by the context. determine whether linear and exponential models are Emphasize linear, quadratic, and increasing or decreasing. exponential models. sketch the function of best fit on the scatter plot. use technology to find the function of best fit for a scatter plot. use the function of best fit to make predictions. compute the residuals for the set of data and the function of best fit. construct a scatter plot of the residuals. analyze the residual plot to determine whether the function is an appropriate fit. sketch a line of best fit on a scatter plot that appears linear. write the equation of the line of best fit using technology or by using two points on the best fit line. identify the variables as quantitative or categorical. describe how the data was collected. indicate any potential biases or flaws. identify inferences the author of the report made from sample data. write or present a summary of a data-based report addressing the sampling techniques used, inferences made, and any flaws or biases. Book pg 233

22 Standard The students will: MAFS.912.S- CP.1.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ( or, and, not ). SMP #1 MAFS.912.S-CP.1.2 Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent. SMP #1, 3 MAFS.912.S-CP.1.3 Understand the conditional probability of A given B as P (A and B)/P (B), and interpret independence of A and B as saying that the conditional probability A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. SMP #1, 6 Course: Algebra 2 & Algebra 2 Honors Unit 3: Probability How can a population be described when it is large, it would be nearly impossible to collect all the data? In what ways does one event impact the probability of another event occurring? Remarks I can: define a sample space and events within the sample space. establish events as subsets of a sample space based on the union, intersection, and/or complement of other events. use correct set notation, with appropriate symbols, to identify sets and subsets. define union, intersection, and complement. draw Venn diagrams that show relationships between sets within a sample space. define and identify independent events. explain and provide an example to illustrate that for two independent events, the probability of the events occurring together is the product of the probability of each event. calculate probabilities for events, including joint probabilities, using various methods (e.g., Venn diagrams, frequency table). Predict if two events are independent, explain reasoning and check my statement by calculating P (A and B) and P(A) x P(B). define dependent events and conditional probability. explain that conditional probability is the probability of an event occurring given the occurrence of some other event and give examples that illustrate conditional probabilities. explain that for two events A and B, the probability of event A occurring given the occurrence of event B is: P (A\B) = P(A and B) and give examples to P(B) show how to use the formula. explain that A and B are independent events if the occurance of A does not impact the probability of B occurring and vice versa. determine if two events are independent and justify the conclusion. Students may believe that multiplying across branches of a tree diagram has nothing to do with conditional probability and that independence of events and mutually exclusive events are the same thing. Resources Book pgs Book pg 236 Book pg 237

23 Course: Algebra 2 & Algebra 2 Honors Unit 3: Probability How can a population be described when it is large, it would be nearly impossible to collect all the data? In what ways does one event impact the probability of another event ofsurring? Standard Remarks Resources The students will: MAFS.912.S- CP.1.4 Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. SMP # 1, 2, 3, 4, 6, 7 MAFS.912.S-CP.1.5 Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. SMP #1, 4 I can: determine when a two-way frequency table is an appropriate display for a set of data. collect data from a random sample. construct a two-way frequency table for the data using the appropriate categories for each variable. decide if events are independent of each other by comparing P(B/A) and P(B) or P(A/B) and P(A). calculate the conditional probability of A given B using the formula P(A/B) = P(A and B). P(B) pose a question for which a two-way frequency is appropriate, use statistical techniques to sample the population, and design an appropriate product to summarize the process and report the results. illustrate the concept of conditional probability using everyday examples of dependent events. illustrate the concept of independence using everyday examples of independent events. Students may use spreadsheets, graphing calculators, and simulations to create frequency tables and conduct analyses to determine if events are independent or determine approximate conditional probabilities. For example, collect: data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results FSSS Flip Book pgs FSSS Flip Book pg 240