Core Content Connectors by Common Core State Standards: Algebra
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2+2+2+2+2+5 = (20+5) = 20 +2• 20•5+5 | 2+2+2+2+2+5 = (20+5) = 20 +2• 20•5+5 | ||
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− | |colspan=2|5. Know and apply the Binomial Theorem for the expansion of <math>(x+y)^n\ in\ powers\ of\ x\ and\ y\ for\ a\ positive\ integer\ n,\ where\ x\ and\ y\ are\ any\ numbers,\ with\ | + | |colspan=2|5. Know and apply the Binomial Theorem for the expansion of <math>(x+y)^n\ in\ powers\ of\ x\ and\ y\ for\ a\ positive\ integer\ n,\ where\ x\ and\ y\ are\ any\ numbers,\ with\ coe\!f\!ficients\ determined\ for\ example\ by\ Pascal's\ Triangle</math> |
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||'''''CCCs linked to A-APR.5''''' | ||'''''CCCs linked to A-APR.5''''' |
Revision as of 15:19, 20 May 2014
Mathematics High School—Algebra Overview
- Seeing Structure in Expressions
- Interpret the structure of expressions
- Write expressions in equivalent forms to solve problems
- Arithmetic with Polynomials and Rational Expressions
- Perform arithmetic operations on polynomials
- Understand the relationship between zeros and factors of polynomials
- Use polynomial identities to solve problems
- Rewrite rational expressions
- Creating Equations
- Create equations that describe numbers or relationships
- Reasoning with Equations and Inequalities
- Understand solving equations as a process of reasoning and explain the reasoning
- Solve equations and inequalities in one variable
- Solve systems of equations
- Represent and solve equations and inequalities graphically
Seeing Structure in Expressions | A-SSE |
Interpret the structure of expressions | |
1. Interpret expressions that represent a quantity in terms of its context. | |
a. Interpret parts of an expression, such as terms, factors, and coefficients. | |
b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret ![]() ![]() ![]() | |
CCCs linked to A-SSE.1 | H.PRF.2a1 Translate an algebraic expression into a word problem. |
2. Use the structure of an expression to identify ways to rewrite it. For example, see ![]() ![]() | |
CCCs linked to A-SSE.2 | H.NO.2c1 Simplify expressions that include exponents. |
H.NO.2c2 Rewrite expressions that include rational exponents. | |
Write expressions in equivalent forms to solve problems | |
3. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. | |
a. Factor a quadratic expression to reveal the zeros of the function it defines. | |
b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. | |
c. Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. | |
CCCs linked to A-SSE.3 | H.NO.1a1 Represent quantities and expressions that use exponents. |
H.PRF.2a2 Factor a quadratic expression. | |
H.PRF.2a3 Given a quadratic expression, explain the meaning of the zeros graphically. That is for an expression ![]() | |
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. For example, calculate mortgage payments. | |
CCCs linked to A-SSE.4 | H.PRF.2a4 Use the formula to solve real world problems such as calculating the height of a tree after n years given the initial height of the tree and the rate the tree grows each year. |
Arithmetic with Polynomials and Rational Expressions | A-APR |
Perform arithmetic operations on polynomials | |
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. | |
CCCs linked to A-APR.1 | H.NO.2a2 Understand the definition of a polynomial. |
H.NO.2a3 Understand the concepts of combining like terms and closure. | |
H.NO.2a4 Add, subtract, and multiply polynomials and understand how closure applies under these operations. | |
Understand the relationship between zeros and factors of polynomials | |
2. Know and apply the Remainder Theorem: ![]() | |
CCCs linked to A-APR.2 | H.NO.2.a5 Understand and apply the Remainder Theorem. |
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. | |
CCCs linked to A-APR.3 | H.NO.2a6 Find the zeros of a polynomial when the polynomial is factored. |
Use polynomial identities to solve problems | |
4. Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity ![]() | |
CCCs linked to A-APR.4 | H.NO.3a6 Prove polynomial identities by showing steps and providing reasons. |
H.NO.3a7 Illustrate how polynomial identities are used to determine numerical relationships such as
2+2+2+2+2+5 = (20+5) = 20 +2• 20•5+5 | |
5. Know and apply the Binomial Theorem for the expansion of ![]() | |
CCCs linked to A-APR.5 | None |
Rewrite rational expressions | |
6. Rewrite simple rational expressions in different forms; write ![]() ![]() ![]() ![]() ![]() | |
CCCs linked to A-APR.6 | H.PRF.2a5 Rewrite rational expressions, ![]() ![]() |
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. | |
CCCs linked to A-APR.7 | None |
Creating Equations | A-CED |
Create equations that describe numbers or relationships | |
1. 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. | |
CCCs linked to A-CED.1 | H.PRF.2b1 Translate a real-world problem into a one variable linear equation. |
2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. | |
CCCs linked to A-CED.2 | H.PRF.2b2 Solve equations with one or two variables using equations or graphs. |
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. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. | |
CCCs linked to A-CED.3 | H.PRF.2a6 Write and use a system of equations and/or inequalities to solve a real world problem. |
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law ![]() ![]() | |
CCCs linked to A-CED.4 | H.PRF.1b2 Solve multi-variable formulas or literal equations, for a specific variable. |
Reasoning with Equations and Inequalities | A-REI |
Understand solving equations as a process of reasoning and explain
the reasoning | |
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. | |
CCCs linked to A-REI.1 | H.PRF.2b2 Solve equations with one or two variables using equations or graphs |
2. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. | |
CCCs linked to A-REI.2 | H.NO.2a1 Solve simple equations using rational numbers with one or more variables. |
Solve equations and inequalities in one variable | |
3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. | |
CCCs linked to A-REI.3 | H.PRF.2b2 Solve equations with one or two variables using equations or graphs. |
H.ME.1b2 Solve a linear equation to find a missing attribute given the area, surface area, or volume and the other attribute. | |
4. Solve quadratic equations in one variable. | |
a. Use the method of completing the square to transform and quadratic equation in ![]() ![]() | |
b. Solve quadratic equations by inspection (e.g., for ![]() ![]() | |
CCCs linked to A-REI.4 | H.PRF.2b3 Transform a quadratic equation written in standard form to an equation in vertex form ![]() |
H.PRF.2b4 Derive the quadratic formula by completing the square on the standard form of a quadratic equation. | |
H.PRF.2b5 Solve quadratic equations in one variable by simple inspection, taking the square root, factoring, and completing the square. | |
Solve systems of equations | |
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. | |
CCCs linked to A-REI.5 | H.PRF.2b6 Solve systems of equations using the elimination method (sometimes called linear combinations). |
H.PRF.2b7 Solve a system of equations by substitution (solving for one variable in the first equation and substitution it into the second equation). | |
6. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. | |
CCCs linked to A-REI.6 | H.PRF.2b8 Solve systems of equations using graphs. |
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 ![]() ![]() | |
CCCs linked to A-REI.7 | H.PRF.2b9 Solve a system containing a linear equation and a quadratic equation in two variables graphically and symbolically. |
8. Represent a system of linear equations as a single matrix equation in a vector variable. | |
CCCs linked to A-REI.8 | None |
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 × 3 or greater). | |
CCCs linked to A-REI.9 | None |
Represent and solve equations and inequalities graphically | |
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). | |
CCCs linked to A-REI.10 | H.PRF.2b10 Understand that all solutions to an equation in two variables are contained on the graph of that equation. |
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. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions. | |
CCCs linked to A-REI.11 | H.PRF. 2d1 Explain why the intersection of y = f(x) and y = g(x) is the solution of f(x) = g(x) for any combination of linear or exponential. Find the solution(s) by: Using technology to graph the equations and determine their point of intersection, Using tables of values, or Using successive approximations that become closer and closer to the actual value. |
12. Graph the solutions to a linear inequality in two variables as a halfplane (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. | |
CCCs linked to A-REI.12 | H.PRF.2b11 Graph the solutions to a linear inequality in two variables as a half-plane, excluding the boundary for non-inclusive inequalities. |
H.PRF.2b12 Graph the solution set to a system of linear inequalities in two variables as the intersection of their corresponding half-planes. |