Core Content Connectors by Common Core State Standards: Algebra

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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 $P(1+r)n$ as the product of $P$ and a factor not depending on $P$.
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 $x^4 - y^4\ as\ (x^2)^2 - (y^2)^2$, thus recognizing it as a difference of squares that can be factored as $(x^2-y^2)(x^2+y^2)$.
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 $1.15^t$ can be rewritten as $(1.151^\frac{1}{12})^12t \approx 1.012^12t$ 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 $(x-a)(x-c),\ a\ and\ c\ correspond\ to\ the\ x\ intercepts\ (if\ a\ and\ c\ are\ real).$
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: $For\ a\ polynomial\ p(x)\ and\ a\ number\ a,\ the\ remainder\ on\ division\ by\ x-a\ is\ p(a),\ so\ p(a)=0\ if\ and\ only\ if\ (x-a)\ is\ a\ factor\ of\ p(x)$.
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 $(x^2+y^2)^2 =(x^2-y^2)^2 + (2^xy)^22$ can be used to generate Pythagorean triples.
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. For example the polynomial identity $(a+b)^2=a^2 + 2ab + b^2$ can be used to rewrite $(25)^2 = (20+5)^2 = 20^2 + 2(20*5) + 5^2$
5. Know and apply the Binomial Theorem for the expansion of $(x+y)^n\ in\ powers\ of\ x\ and\ y\ for\ a\ positive\ integer\ n,\ where\ x\ and\ y\ are\ any\ numbers,\ with\ coe\!f\!ficients\ determined\ for\ example\ by\ Pascal's\ Triangle$
CCCs linked to A-APR.5	None
Rewrite rational expressions
6. Rewrite simple rational expressions in different forms; write $\frac{a(x)}{b(x)}$ in the form $\frac{q(x)+r(x)}{b(x)}$, where $a(x),\ b(x),\ q(x),\ and\ r(x)$ are polynomials with the degree of $r(x)$ less than the degree of $b(x)$, using inspection, long division, or, for the more complicated examples, a computer algebra system.
CCCs linked to A-APR.6	H.PRF.2a5 Rewrite rational expressions, $\frac{a(x)}{b(x)}$, in the form $\frac{q(x)+r(x)}{b(x)}$ by using factoring, long division, or synthetic division.
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 $V =IR$ to highlight resistance $R$.
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 $x$ into an equation of the form $(x-p)^2=q$ that has the same solutions. Derive the quadratic formula from this form.
b. Solve 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 $a\pm bi\ for\ real\ numbers\ a\ and\ b$.
CCCs linked to A-REI.4	H.PRF.2b3 Transform a quadratic equation written in standard form to an equation in vertex form $(x-p)=q^2$ by completing the square.
	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 $y=-3x$ and the circle $x^2+y^2=3$
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.