Notes on Reading the Washington State Mathematics Standards Transition Documents

Transcription

1 Notes on Reading the Washington State Mathematics Standards Transition Documents This document serves as a guide to translate between the 2008 Washington State K-8 Learning Standards for Mathematics and the Common Core State Standards (CCSS) for Mathematics. It begins with the Standards for Mathematical Practice which are the backbone of the CCSS for Mathematics. These practices highlight the change in focus, through instructional practices, of developing these habits of mind in our students. One or more of these Standards for Mathematical Practice should be intentionally incorporated in the development of any concept or procedure taught. The Standards for Mathematical Practice are followed by the key critical areas of focus for a grade-level. These critical areas are the overarching concepts and procedures that must be learned by students to be successful at the next grade level and beyond. As units are planned, one should always reflect back on these critical areas to ensure that the concept or fluency developed in the unit is tied directly to one of these. The CCSS were developed around these critical areas in order for instruction to be deep and focused on a few key topics each year. By narrowing the focus and deepening the understanding, increases in student achievement will be realized. The body of this document includes a two-column table which indicates the alignment of the 2008 Washington State K-8 Learning Standards for Mathematics to the CCSS for Mathematics at a grade level. It is meant to be read from left to right across the columns. The right column contains all of the CCSS for Mathematics for that grade. The left column indicates the grade-level Washington standard that most aligns to it. Bolded words are used to describe the degree of alignment between these sets of standards. If the words bolded are Continue to, this indicates that the CCSS standard and the Washington standard are closely aligned. The teacher should read the wording carefully on the CCSS standard because often there is a more in-depth development of the aligned Washington standard and often there are more than one standard that address a particular Washington standard. If the word extend is bolded that indicates that the Washington and CCSS standards are similar but the CCSS takes the concept further than the Washington standard. Lastly, if the words Move students to are bolded, then the CCSS standards take the Washington standard to a deeper or further understanding of this particular cluster concept. If the left-hand side is blank, the CCSS is new material that does not match the Washington standards at this grade level. Sometimes there can be a page or more of these unaligned standards. One is reminded that while this is new material for this grade level, other standards currently taught at this grade level in the 2008 Washington standards will have moved to other grades. The movement of these unaligned standards is laid out on the last pages of this document. Comments on CCSS Integrated Mathematics 2 standards are often found in italics at the end of a cluster. Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 1

2 Washington State Geometry Mathematics Standards Transition Document This document serves as one guide to translate between the 2008 Washington State Standards for Mathematics and the Common Core State Standards for Mathematics. The Standards for Mathematical Practice describes varieties of expertise that mathematics educators at all levels should seek to develop in their students. These standards should be integrated throughout the teaching and learning of the content standards of the Common Core State Standards. 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. Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 2

3 With full implementation of the Common Core State Standards for mathematics in Integrated Mathematics 2, instructional time should focus on six critical areas: 1) extending the laws of exponents to rational exponents and exploring distinctions between rational and irrational numbers; 2) comparing key characteristics of quadratic functions to those of linear and exponential function, modeling phenomena by these functions, identifying solutions of quadratic equations, and learning to anticipate the features of graphs of quadratic functions; 3) focusing on the structure of expressions, rewriting expressions to clarify and reveal aspects of the relationship they represent, creating and solving equations, inequalities, and systems of equations involving exponential and quadratic expressions; 4) using the languages of set theory to compute and interpret theoretical and experimental probability for compound events, attending to mutually exclusive events, independent events, and conditional probability; 5) building a formal understanding of similarity; identifying criteria for similarity of triangles, using similarity to solve problems, and applying similarity in right triangles to understand right triangle trigonometry, with particular attention to special right triangles and the Pythagorean Theorem; 6) proving basic theorems about circles, using the distance formula to write the equation of a circle when given the radius and the coordinates of its center, writing the equation of a parabola, drawing the graph of an equation of a circle in the coordinate plane, applying techniques for solving quadratic equations to determine intersections between lines and circles or parabolas and between two circles, and developing informal arguments justifying common formulas for circumference, area, and volume of geometric objects. (+) denotes mathematics students should learn in order to take advanced courses, included to increase coherence, not for high stakes assessments denotes a Modeling standard throughout this document, both a conceptual category and a mathematical practice Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 3

4 Unit 1 Extending the Number System Aligned current WA standards Integrated Mathematics 2 Common Core State Standards Students currently: Students need to: Extend the properties of exponents to rational exponents. There are no WA Integrated Mathematics 2 mathematics standards that match these CCSS. Move students to explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allow for a notation for radicals in terms of rational exponents. For example, we define 5 1/3 to be the cube root of 5 because we want (5 1/3 ) 3 = 5 (1/3)3 to hold, so (5 1/3 ) 3 must equal 5. (N.RN.1) Rewrite expressions involving radicals and rational exponents using the properties of exponents. (N.RN.2) Use properties of rational and irrational numbers. Move students to explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. (N.RN.3) Connect N.RN.3 to physical situation, e.g., finding the perimeter of a square of area 2. Perform arithmetic operations with complex numbers. Move students to 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. (N.CN.1) Use the relation i 2 = 1 and the commutative, associative, and Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 4

5 distributive properties to add, subtract, and multiply complex numbers. (N.CN.2) Limit to multiplications that involve i² as the highest power of i. Perform arithmetic operations on polynomials. M2.5.A Use algebraic properties to factor and combine like terms in polynomials. Continue to 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. APR.1) Aligned current WA standards Students currently: Focus on polynomial expressions that simplify to forms that are linear or quadratic in a positive integer power of x. Unit 2 Quadratic Functions and Modeling Integrated Mathematics 2 Common Core State Standards Students need to: Interpret functions that arise in applications in terms of a context. Continue to relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. (F.IF.5*) M2.2.A Represent a quadratic function with a symbolic expression, as a graph, in a table, and with a description, and make connections among the representations. 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 Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 5

6 behavior; and periodicity. (F.IF.4*) 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.6*) Focus on quadratic functions; compare with linear and exponential functions studied in Integrated Mathematics 1. Analyze functions using different representations. M2.1.D Solve problems that can be represented by exponential functions and equations. M2.2.C Translate between the standard form of a quadratic function, the vertex form, and the factored form; graph and interpret the meaning of each form. Continue to 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.7*) Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. (F.IF.8) 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.IF.9) M2.2.A Represent a quadratic function with a symbolic expression, a a graph, in a table, and with a description, and make connections among the representations. M2.2.B Sketch the graph of a quadratic function, describe the effects that changes in the parameters have on the graph, and interpret the x-intercepts as solutions to a quadratic equation. Continue to graph linear and quadratic functions and show intercepts, maxima, and minima. (F.IF.7a*) Graph square root, cube root, and extend to piecewise-defined functions, including step functions and absolute value functions. (F.IF.7b*) Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 6

7 M2.2.D Solve quadratic equations that can be factored as (ax + b)(cx + d) where a, b, c, and d are integers. M2.2.F Solve quadratic equations that have real roots by completing the square and by using the quadratic formula. 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.8a) Move students to use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02) t, y = (.097) t, y = (1.01) 12t, y = (1.2) t/10, and classify them as representing exponential growth or decay. (F.IF.8b) For F.IF.7b, compare and contrast absolute value, step and piecewise defined functions with linear, quadratic, and exponential functions. Highlight issues of domain, range, and usefulness when examining piecewise defined functions. Note that this unit, and in particular in F.IF.8b, extends the work begun in Integrated Mathematics 1 on exponential functions with integer exponents. For F.IF.9, focus on expanding the types of functions considered to include linear, exponential, and quadratic. Extend work with quadratics to include the relationship between coefficients and roots, and that once roots are known, a quadratic equation can be factored. Build a function that models a relationship between two quantities. M2.1.A Select and justify functions and equations to model and solve problems.. Continue to write a function that describes a relationship between two quantities. (F.BF.1*) Determine an explicit expression, a recursive process, or steps for calculation from a context. (F.BF.1a*) Extend to combine standard function types using arithmetic Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 7

8 . operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (F.BF.1b*) Focus on situations that exhibit a quadratic or exponential relationship. Build new functions from existing functions. M2.2.B Sketch the graph of a quadratic function, describe the effects that changes in the parameters have on the graph, and interpret the x-intercepts as solutions to a quadratic equation. Continue to 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. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (F.BF.3) Move students to find inverse functions. (F.BF.4) Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2x 3 or f(x) = (x + 1)/(x - 1) for x 1. (F.BF.4a) For F.BF.3, focus on quadratic functions, and consider including absolute value functions. For F.BF.4a, focus on linear functions but consider simple situations where the domain of the function must be restricted in order for the inverse to exist, such as f(x) = x2, x>0. Construct and compare linear, quadratic, and exponential models and solve problems. M2.2.H Determine if a bivariate data set can be better modeled with an exponential or a quadratic function and use the model to make predictions. Extend to observing using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function. Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 8

9 (F.LE.3) Compare linear and exponential growth to quadratic growth. Aligned current WA standards Students currently: Unit 3 Expressions and Equations Integrated Mathematics 2 Common Core State Standards Students need to: Interpret the structure of expressions. M2.5.A Use algebraic properties to factor and combine like terms in polynomials. Continue to use the structure of an expression to identify ways to rewrite it. For example, see x 4 y 4 as (x 2 ) 2 (y 2 ) 2, thus recognizing it as a difference of squares that can be factored as (x 2 - y 2 )(x 2 + y 2 ). Focus on quadratic and exponential expressions. (A.SSE.2) Move students to interpret expressions that represent a quantity in terms of its context. (A.SSE.1*) Interpret parts of an expression, such as terms, factors, and coefficients. (A.SSE.1a*) Interpret complicated expression by viewing one or more of their parts as a single entity. For example, interpret P(1 + r) n as the product of P and a factor not depending on P. (A.SSE.1b*) Focus on quadratic and exponential expressions. For A.SSE.1b, exponents are extended from the integer exponents found in Integrated Mathematics 1 to rational exponents focusing on those that represent square or cube roots. Write expressions in equivalent forms to solve problems. Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 9

10 M2.2.D Solve quadratic equations that can be factored as (ax + b)(cx + d) where a, b, c, and d are integers M2.5.A Use algebraic properties to factor and combine like terms in polynomials. M2.2.F Solve quadratic equations that have real roots by completing the square and by using the quadratic formula. Continue to factor a quadratic expression to reveal the zeros of the function it defines. (A.SSE.3a*) Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expressions. (A.SSE.3*) Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (A.SSE.3b*) Extend to using the properties of exponents to transform expression for exponential functions. For example the expression 1.15 t can be rewritten as (1.15 1/12 ) 12t = t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. (A.SSE.3c*) It is important to balance conceptual understanding and procedural fluency in work with equivalent expressions. For example, development of a skill in factoring and completing the square goes hand-in-hand with understanding what different forms of a quadratic expression reveal. Create equations that describe numbers or relationships. M2.1.A Select and justify functions and equations to model and solve problems. Continue to create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. (A.CED.2) Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions. (A.CED.1) Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 10

11 Extend to rearranging 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.CED.4) Extend to formulas involving squared variables. Extend work on linear and exponential equations in Unit 1 to quadratic equations. Solve equations and inequalities in one variable. M2.2.D Solve quadratic equations that can be factored as (ax + b)(cx + d) where a, b, c, and d are integers. M2.2.F Solve quadratic equations that have real roots by completing the square and by using the quadratic formula. M2.1.C Solve problems that can be represented by quadratic functions, equations, and inequalities. Continue to solve quadratic equations in one variable. (A.REI.4) 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. (A.REI.4a) Extend to solving 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 ± b for real numbers a and b. (A.REI.4b) Students should learn of the existence of the complex number system, but will not solve quadratics with complex solutions until Algebra II. Use complex numbers in polynomial identities and equations. M2.2.G Solve quadratic equations and inequalities including equations with complex roots. Continue to solve quadratic equations with real coefficients that have complex solutions. (N.CN.7) Extend polynomial identities to the complex numbers. For example, Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 11

12 rewrite x as (x + 2i)(x 2i). (N.CN.8) (+) M2.2.E Determine the number and nature of the roots of a quadratic equation. M2.1.B Solve problems that can be represented by systems of equations and inequalities. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. (N.CN.9) (+) Solve systems of equations. Continue to solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = -3x and the circle x 2 + y 2 = 3. (A.REI.7) Include systems consisting of one linear and one quadratic equation. Include systems that lead to work with fractions. For example, finding the intersections between x 2 + y 2 = 1 and y = (x+1)/2 leads to the point (3/5, 4/5) on the unit circle, corresponding to the Pythagorean triple = 5 2. Aligned current WA standards Students currently: Unit 4 Applications of Probability Integrated Mathematics 2 Common Core State Standards Students need to: Understand independence and conditional probability and use them to interpret data. M2.4.B Given a finite sample space consisting of equally likely outcomes and containing events A and B, determine whether A and B are independent or dependent, and find the conditional probability of A given B. Continue to 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 ). (S.CP.1) 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. (S.CP.2) 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 Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 12

13 probability of 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. (S.CP.3) Continue to 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. 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. (S.CP.4) Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. (S.CP.5) Build on work with two-way tables from Integrated Mathematics 1, Unit 4 (S.ID.5) to develop understanding of conditional probability and independence. Use the rules of probability to compute probabilities of compound events in a uniform probability model. M2.4.A Apply the fundamental counting principle and the ideas of order and replacement to calculate probabilities in situations arising from two-stage experiments (compound events). M2.4.C Compute permutations and combinations, and use the result to calculate probabilities. Continue to find the conditional probability of A given B as the fraction of B s outcomes that also belong to A, and interpret the answer in terms of the model. (S.CP.6) Apply the Addition Rule, P(A or B) = P(A) + P(B) P(A and B), and interpret the answer in terms of the model. (S.CP.7) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B A) = P(B)P(A B), and interpret the answer in terms of the model. (S.CP.8) (+) Use permutations and combinations to compute probabilities of Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 13

14 M2.1.E Solve problems involving combinations and permutations compound events and solve problems. (S.CP.9) (+) Use probability to evaluate outcomes of decisions. Extend to using probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). (S.MD.6) (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). (S.MD.7) (+) Aligned current WA standards Students currently: This unit sets the stage for work in Integrated Mathematics 3, where the ideas of statistical inference are introduced. Evaluating the risks associated with conclusions drawn from sample data (i.e., incomplete information) requires an understanding of probability concepts. Unit 5 Right Triangle Trigonometry, and Proof Integrated Mathematics 2 Common Core State Standards Students need to: Understand similarity in terms of similarity transformations. Move students to verify experimentally the properties of dilations given by a center and a scale factor. (G.SRT.1) a) A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b) The dilation of a line segment is longer or shorter in the ratio given by the scale factor. M2.3.F Determine and prove triangle similarity.. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 14

15 similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. (G.SRT.2) Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. (G.SRT.3) Prove geometric theorems. M2.3.A Use deductive reasoning to prove that a valid geometric statement is true. M2.3.E Know, explain, and apply basic postulates and theorems about triangles and the special lines, line segments, and rays associated with a triangle. M2.3.F Determine and prove triangle congruence, triangle similarity, and other properties of triangles. Continue to prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment s endpoints. (G.CO.9) Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 ; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point. (G.CO.10) M2.3.J Know, prove and apply basic theorems about parallelograms M2.3.K Know, prove and apply theorems about properties of quadrilaterals and other polygons Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals. (G.CO.11) Encourage multiple ways of writing proofs, such as in narrative paragraphs, using flow diagrams, in two-column format, and using diagrams without words. Students should be encouraged to focus on the validity of the underlying reasoning while exploring a variety of formats for expressing that reasoning. Implementation of G.CO.10 may be extended to include concurrence of Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 15

16 perpendicular bisectors and angle bisector as preparation for G.C.3 in Unit 6. M2.3.G Know, prove, and apply the Pythagorean Theorem and its converse. M2.3.F Determine and prove triangle congruence and other properties of triangles. Prove theorems involving similarity. Continue to prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity. (G.SRT.4) Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. (G.SRT.5) Use coordinates to prove simple geometric theorems algebraically. M2.3.L Determine the coordinates of a point that is described geometrically. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. (G.GPE.6) Define trigonometric ratios and solve problems involving right triangles. M2.3.H Solve problems involving the basic trigonometric ratios of sine, cosine, and tangent. M2.3.I Use the properties of special right triangles (30ᵒ- 60ᵒ - 90ᵒ and 45ᵒ- 45ᵒ- 90ᵒ) to solve problems. Continue to understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. (G.SRT.6) Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. (G.SRT.8) Extend to explaining and using the relationship between the sine and cosine of complementary angles. (G.SRT.7) Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 16

17 Aligned current WA standards Students currently: Prove and apply trigonometric identities. Move students to prove the Pythagorean identity sin 2 (θ) + cos 2 (θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. (F.TF.8) In this course, limit Ө to angles between 0 and 90 degrees. Connect with the Pythagorean theorem and the distance formula. A course with a greater focus on trigonometry could include the (+) standard F.TP.9: Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems. This could continue to be limited to acute angles in Integrated mathematics 2. Extension of trigonometric functions to other angles through the unit circle is included in Integrated Mathematics 3. Unit 6 Circles With and Without Coordinates Integrated Mathematics 2 Common Core State Standards Students need to: Understand and apply theorems about circles. There are no 2008 WA Integrated Mathematics 2 standards that match these CCSS geometry standards. Move students to identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. (G.C.2) Prove that all circles are similar. (G.C.1) Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. (G.C.3) Construct a tangent line from a point outside a given circle to the circle. (G.C.4) (+) Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 17

18 Find arc lengths and areas of sectors of circles. Move students to derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. (G.C.5) Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Translate between the geometric description and the equation for a conic section. Move students to derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. (G.GPE.1) Derive the equation of a parabola given a focus and directrix. (G.GPE.2) Connect the equations of circles and parabolas to prior work with quadratic equations. The directrix should be parallel to a coordinate axis. Use coordinates to prove simple geometric theorems algebraically. M2.3.L Determine the coordinates of a point that is described geometrically. M2.3.M Verify and apply properties of triangles and quadrilaterals Continue to use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3) lies on the circle centered at the origin and containing the point (0, 2). (G.GPE.4) Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 18

19 In the coordinate plane. Include simple proofs involving circles. Explain volume formulas and use them to solve problems. Continue to use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* (G.GMD.3) There are no 2008 WA Integrated Mathematics 2 standards that match these CCSS volume standards Extend to giving an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri s principle, and informal limit arguments. (G.GMD.1) Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is k 2 times the area of the first. Similarly, volumes of solid figures scale by k 3 under a similarity transformation with scale factor k. Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 19

20 With full implementation of the CCSS, Integrated Mathematics 2 teachers will no longer be responsible for teaching students the standard listed below. The course levels where this standard will be emphasized are in parentheses. M2.5.D Find the terms and partial sums of arithmetic and geometric series and the infinite sum for geometric series. (Integrated Mathematics 1, Integrated Mathematics 3) While not Integrated Mathematics 2 CCSS, attention to these 2008 WA Integrated Mathematics 2 learning standard will be a natural part of developing understanding of concepts. M2.3.B Identify errors or gaps in a mathematical argument and develop counterexamples to refute invalid statements about geometric relationships. M2.3.C Write the converse, inverse, and contrapositive of a valid proposition and determine their validity. M2.3.D Distinguish between definitions and undefined geometric terms and explain the role of definitions, undefined terms, postulates (axioms), and theorems. M2.5.B Use different degrees of precision in measurement, explain the reason for using a certain degree of precision and apply estimation strategies to obtain reasonable measurements with appropriate precision for a given purpose. M2.5.C Solve problems involving measurement conversions within and between systems, including those involving derived units, and analyze solutions in terms of reasonableness of solutions and appropriate units. Collaboratively developed by OSPI and the ESD Regional Mathematics Coordinators Updated 12/05/11 Page 20