1.2 http://corestandards.org/Math/Content/HSA/APR/A/1/ 7 Standard CCSS.Math.Content.HSA-APR.A.1 Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. 09 10 11 12 B62C1C106873438AA0126760075A65A3 86029C6936A24246AC435CA696634800 http://corestandards.org/Math/Content/HSA/APR/B/2/ 7 Standard CCSS.Math.Content.HSA-APR.B.2 Know and apply the Remainder Theorem: For a polynomial <i>p</i>(<i>x</i>) and a number <i>a</i>, the remainder on division by <i>x - a</i> is <i>p</i>(<i>a</i>), so <i>p</i>(<i>a</i>) = 0 if and only if (<i>x - a</i>) is a factor of <i>p</i>(<i>x</i>). 09 10 11 12 B62C1C106873438AA0126760075A65A3 F807C53F9EDE43FDB34DF614CA072F87 http://corestandards.org/Math/Content/HSA/APR/B/3/ 7 Standard CCSS.Math.Content.HSA-APR.B.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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 F807C53F9EDE43FDB34DF614CA072F87 http://corestandards.org/Math/Content/HSA/APR/C/4/ 7 Standard CCSS.Math.Content.HSA-APR.C.4 Prove polynomial identities and use them to describe numerical relationships. <i>For example, the polynomial identity (x<sup>2</sup> + y<sup>2</sup>)<sup>2</sup> = (x<sup>2</sup> - y<sup>2</sup>)<sup>2</sup> + (2xy)<sup>2</sup> can be used to generate Pythagorean triples.</i></li> 09 10 11 12 B62C1C106873438AA0126760075A65A3 A2539A99653A47DBAFA1740D2F0BE124 http://corestandards.org/Math/Content/HSA/APR/C/5/ 7 Standard CCSS.Math.Content.HSA-APR.C.5 (+) Know and apply the Binomial Theorem for the expansion of (<i>x</i> + <i>y</i>)<sup><i>n</i></sup> in powers of <i>x</i> and <i>y</i> for a positive integer <i>n</i>, where <i>x</i> and <i>y</i> are any numbers, with coefficients determined for example by Pascal's Triangle.<sup>1</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 A2539A99653A47DBAFA1740D2F0BE124 http://corestandards.org/Math/Content/HSA/APR/D/6/ 7 Standard CCSS.Math.Content.HSA-APR.D.6 Rewrite simple rational expressions in different forms; write <sup><i>a</i>(<i>x</i>)</sup>/<sub><i>b</i>(<i>x</i>)</sub> in the form <i>q</i>(<i>x</i>) + <sup><i>r</i>(<i>x</i>)</sup>/<sub><i>b</i>(<i>x</i>)</sub>, where <i>a</i>(<i>x</i>), <i>b</i>(<i>x</i>), <i>q</i>(<i>x</i>), and <i>r</i>(<i>x</i>) are polynomials with the degree of <i>r</i>(<i>x</i>) less than the degree of <i>b</i>(<i>x</i>), using inspection, long division, or, for the more complicated examples, a computer algebra system. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0EC032FAF5F34E04B00A864C2E588A09 http://corestandards.org/Math/Content/HSA/APR/D/7/ 7 Standard CCSS.Math.Content.HSA-APR.D.7 (+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 0EC032FAF5F34E04B00A864C2E588A09 http://corestandards.org/Math/Content/HSA/CED/A/1/ 7 Standard CCSS.Math.Content.HSA-CED.A.1 Create equations and inequalities in one variable and use them to solve problems. <i>Include equations arising from linear and quadratic functions, and simple rational and exponential functions</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/CED/A/2/ 7 Standard CCSS.Math.Content.HSA-CED.A.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/CED/A/3/ 7 Standard CCSS.Math.Content.HSA-CED.A.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. <i>For example, represent inequalities describing nutritional and cost constraints on combinations of different foods</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/CED/A/4/ 7 Standard CCSS.Math.Content.HSA-CED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. <i>For example, rearrange Ohm's law V = IR to highlight resistance R</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 2B387BDF43CB43B58983E460B46BE605 http://corestandards.org/Math/Content/HSA/REI/A/1/ 7 Standard CCSS.Math.Content.HSA-REI.A.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. 09 10 11 12 B62C1C106873438AA0126760075A65A3 76A52762F6C540829E34B961AC57ED32 http://corestandards.org/Math/Content/HSA/REI/A/2/ 7 Standard CCSS.Math.Content.HSA-REI.A.2 Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 09 10 11 12 B62C1C106873438AA0126760075A65A3 76A52762F6C540829E34B961AC57ED32 http://corestandards.org/Math/Content/HSA/REI/B/3/ 7 Standard CCSS.Math.Content.HSA-REI.B.3 Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 09 10 11 12 B62C1C106873438AA0126760075A65A3 7F7DAA3BD975420E9A21319845B45A2A http://corestandards.org/Math/Content/HSA/REI/B/4/ 7 Standard CCSS.Math.Content.HSA-REI.B.4 Solve quadratic equations in one variable. 09 10 11 12 B62C1C106873438AA0126760075A65A3 7F7DAA3BD975420E9A21319845B45A2A 5BD6F36D021A429E951EE42C678F1CF5 617091A725F44F42A823052465447328 http://corestandards.org/Math/Content/HSA/REI/B/4/a/ 8 Component CCSS.Math.Content.HSA-REI.B.4a Use the method of completing the square to transform any quadratic equation in <i>x</i> into an equation of the form (<i>x</i> - <i>p</i>)<sup>2</sup> = <i>q</i> that has the same solutions. Derive the quadratic formula from this form. 09 10 11 12 B62C1C106873438AA0126760075A65A3 87B0241DE22D4A18943F3BC6CBF5DA82 http://corestandards.org/Math/Content/HSA/REI/B/4/b/ 8 Component CCSS.Math.Content.HSA-REI.B.4b Solve quadratic equations by inspection (e.g., for <i>x</i><sup>2</sup> = 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 <i>a</i> &plusmn; <i>bi</i> for real numbers <i>a</i> and <i>b</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 87B0241DE22D4A18943F3BC6CBF5DA82 http://corestandards.org/Math/Content/HSA/REI/C/5/ 7 Standard CCSS.Math.Content.HSA-REI.C.5 Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/6/ 7 Standard CCSS.Math.Content.HSA-REI.C.6 Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/7/ 7 Standard CCSS.Math.Content.HSA-REI.C.7 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 <i>y</i> = -3<i>x</i> and the circle <i>x</i><sup>2</sup> + <i>y</i><sup>2</sup> = 3. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/8/ 7 Standard CCSS.Math.Content.HSA-REI.C.8 (+) Represent a system of linear equations as a single matrix equation in a vector variable. 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/C/9/ 7 Standard CCSS.Math.Content.HSA-REI.C.9 (+) Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 &times; 3 or greater). 09 10 11 12 B62C1C106873438AA0126760075A65A3 DA79BD4E58434CAABB4C784D994D19E8 http://corestandards.org/Math/Content/HSA/REI/D/10/ 7 Standard CCSS.Math.Content.HSA-REI.D.10 Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line). 09 10 11 12 B62C1C106873438AA0126760075A65A3 A65B3884A9784D21B409AC13B11DFD9B http://corestandards.org/Math/Content/HSA/REI/D/11/ 7 Standard CCSS.Math.Content.HSA-REI.D.11 Explain why the <i>x</i>-coordinates of the points where the graphs of the equations <i>y</i> = <i>f</i>(<i>x</i>) and <i>y</i> = <i>g</i>(<i>x</i>) intersect are the solutions of the equation <i>f</i>(<i>x</i>) = <i>g</i>(<i>x</i>); find the solutions approximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where <i>f</i>(<i>x</i>) and/or <i>g</i>(<i>x</i>) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 A65B3884A9784D21B409AC13B11DFD9B http://corestandards.org/Math/Content/HSA/REI/D/12/ 7 Standard CCSS.Math.Content.HSA-REI.D.12 Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. 09 10 11 12 B62C1C106873438AA0126760075A65A3 A65B3884A9784D21B409AC13B11DFD9B http://corestandards.org/Math/Content/HSA/SSE/A/1/ 7 Standard CCSS.Math.Content.HSA-SSE.A.1 Interpret expressions that represent a quantity in terms of its context.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 16247FEF153B4404A115600AF3C81E03 529B17211FDB499292FE9DCD0CAAA928 5155CED2475A42D88AD48B7A676C7943 http://corestandards.org/Math/Content/HSA/SSE/A/1/a/ 8 Component CCSS.Math.Content.HSA-SSE.A.1a Interpret parts of an expression, such as terms, factors, and coefficients. 09 10 11 12 B62C1C106873438AA0126760075A65A3 137C90D24DE949558F17713B84161B4D http://corestandards.org/Math/Content/HSA/SSE/A/1/b/ 8 Component CCSS.Math.Content.HSA-SSE.A.1b Interpret complicated expressions by viewing one or more of their parts as a single entity. <i>For example, interpret P(1+r)<sup>n</sup> as the product of P and a factor not depending on P</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 137C90D24DE949558F17713B84161B4D http://corestandards.org/Math/Content/HSA/SSE/A/2/ 7 Standard CCSS.Math.Content.HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. <i>For example, see x<sup>4</sup> - y<sup>4</sup> as (x<sup>2</sup>)<sup>2</sup> - (y<sup>2</sup>)<sup>2</sup>, thus recognizing it as a difference of squares that can be factored as (x<sup>2</sup> - y<sup>2</sup>)(x<sup>2</sup> + y<sup>2</sup>)</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 16247FEF153B4404A115600AF3C81E03 http://corestandards.org/Math/Content/HSA/SSE/B/3/ 7 Standard CCSS.Math.Content.HSA-SSE.B.3 Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.<sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 BEA7E3EF2A4C4C8F85F12D9743C25155 194A6E971ED74A3695C6CE468E337562 C057EE31BA98447FB241F2F732B896BD D07C6A24D9F045EFACB20582472FC7EB http://corestandards.org/Math/Content/HSA/SSE/B/3/a/ 8 Component CCSS.Math.Content.HSA-SSE.B.3a Factor a quadratic expression to reveal the zeros of the function it defines. 09 10 11 12 B62C1C106873438AA0126760075A65A3 C705AD3CCF9641C9A3101BDB10737B11 http://corestandards.org/Math/Content/HSA/SSE/B/3/b/ 8 Component CCSS.Math.Content.HSA-SSE.B.3b Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 09 10 11 12 B62C1C106873438AA0126760075A65A3 C705AD3CCF9641C9A3101BDB10737B11 http://corestandards.org/Math/Content/HSA/SSE/B/3/c/ 8 Component CCSS.Math.Content.HSA-SSE.B.3c Use the properties of exponents to transform expressions for exponential functions. <i>For example the expression 1.15<sup>t</sup> can be rewritten as (1.15<sup>1/12</sup>)<sup>12t</sup> &asymp; 1.012<sup>12t</sup> to reveal the approximate equivalent monthly interest rate if the annual rate is 15%</i>. 09 10 11 12 B62C1C106873438AA0126760075A65A3 C705AD3CCF9641C9A3101BDB10737B11 http://corestandards.org/Math/Content/HSA/SSE/B/4/ 7 Standard CCSS.Math.Content.HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. <i>For example, calculate mortgage payments.</i><sup>*</sup> 09 10 11 12 B62C1C106873438AA0126760075A65A3 BEA7E3EF2A4C4C8F85F12D9743C25155