No records

1.OA.1 
1 
Operations & Algebraic Thinking 
Represent and solve problems involving addition and subtraction. Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 


1.OA.2 
1 
Operations & Algebraic Thinking 
Represent and solve problems involving addition and subtraction. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 


1.OA.3 
1 
Operations & Algebraic Thinking 
Understand and apply properties of operations and the relationship between addition and subtraction. Apply properties of operations as strategies to add and subtract. Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.) (Students need not use formal terms for these properties.). 


1.OA.4 
1 
Operations & Algebraic Thinking 
Understand and apply properties of operations and the relationship between addition and subtraction. Understand subtraction as an unknownaddend problem. For example, subtract 10  8 by finding the number that makes 10 when added to 8. 


1.OA.5 
1 
Operations & Algebraic Thinking 
Add and subtract within 20. Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 


1.OA.6 
1 
Operations & Algebraic Thinking 
Add and subtract within 20. Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13  4 = 13  3  1 = 10  1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12  8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13). 


1.OA.7 
1 
Operations & Algebraic Thinking 
Work with addition and subtraction equations. Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8  1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2. 


1.OA.8 
1 
Operations & Algebraic Thinking 
Work with addition and subtraction equations. Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _  3, 6 + 6 = _. 


1.NBT.1 
1 
Number & Operations in Base Ten 
Extend the counting sequence. Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral. 


1.NBT.2 
1 
Number & Operations in Base Ten 
Understand place value. Understand that the two digits of a twodigit number represent amounts of tens and ones. Understand the following as special cases:. 


1.NBT.3 
1 
Number & Operations in Base Ten 
Understand place value. Compare two twodigit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <. 


1.NBT.4 
1 
Number & Operations in Base Ten 
Use place value understanding and properties of operations to add and subtract. Add within 100, including adding a twodigit number and a onedigit number, and adding a twodigit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding twodigit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten. 


1.NBT.5 
1 
Number & Operations in Base Ten 
Use place value understanding and properties of operations to add and subtract. Given a twodigit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used. 


1.NBT.6 
1 
Number & Operations in Base Ten 
Use place value understanding and properties of operations to add and subtract. Subtract multiples of 10 in the range 1090 from multiples of 10 in the range 1090 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. 


1.MD.1 
1 
Measurement & Data 
Measure lengths indirectly and by iterating length units. Order three objects by length; compare the lengths of two objects indirectly by using a third object. 


1.MD.2 
1 
Measurement & Data 
Measure lengths indirectly and by iterating length units. Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of samesize length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps. 


1.MD.3 
1 
Measurement & Data 
Tell and write time. Tell and write time in hours and halfhours using analog and digital clocks. 


1.MD.4 
1 
Measurement & Data 
Represent and interpret data. Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another. 


1.G.1 
1 
Geometry 
Reason with shapes and their attributes. Distinguish between defining attributes (e.g., triangles are closed and threesided) versus nondefining attributes (e.g., color, orientation, overall size); for a wide variety of shapes; build and draw shapes to possess defining attributes. 


1.G.2 
1 
Geometry 
Reason with shapes and their attributes. Compose twodimensional shapes (rectangles, squares, trapezoids, triangles, halfcircles, and quartercircles) or threedimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape. (Students do not need to learn formal names such as right rectangular prism.). 


1.G.3 
1 
Geometry 
Reason with shapes and their attributes. Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares. 


K12.MP.1 
1 
Mathematical Practice 
Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, Does this make sense? They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. 


K12.MP.2 
1 
Mathematical Practice 
Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. 


K12.MP.3 
1 
Mathematical Practice 
Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. 


K12.MP.4 
1 
Mathematical Practice 
Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, twoway tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. 


K12.MP.5 
1 
Mathematical Practice 
Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. 


K12.MP.6 
1 
Mathematical Practice 
Attend to precision. Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. 


K12.MP.7 
1 
Mathematical Practice 
Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5  3(x  y)^2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. 


K12.MP.8 
1 
Mathematical Practice 
Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y  2)/(x 1) = 3. Noticing the regularity in the way terms cancel when expanding (x  1)(x + 1), (x  1)(x^2 + x + 1), and (x  1)(x^3 + x^2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. 


No records

912.N.RN.1 
10 
High School: Number and Quantity 
Extend the properties of exponents to rational exponents. Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing 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) x 3] to hold, so [5^(1/3)]^3 must equal 5. 


912.N.RN.2 
10 
High School: Number and Quantity 
Extend the properties of exponents to rational exponents. Rewrite expressions involving radicals and rational exponents using the properties of exponents. 


912.N.RN.3 
10 
High School: Number and Quantity 
Use properties of rational and irrational numbers. Explain why the sum or product of 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. 


912.N.Q.1 
10 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Use units as a way to understand problems and to guide the solution of multistep problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays. 


912.N.Q.2 
10 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Define appropriate quantities for the purpose of descriptive modeling. 


912.N.Q.3 
10 
High School: Number and Quantity 
Reason quantitatively and use units to solve problems. Choose a level of accuracy appropriate to limitations on measurement when reporting quantities. 


912.N.CN.1 
10 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. 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. 


912.N.CN.2 
10 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Use the relation i^2 = 1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers. 


912.N.CN.3 
10 
High School: Number and Quantity 
Perform arithmetic operations with complex numbers. Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers. 


912.N.CN.4 
10 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number. 


912.N.CN.5 
10 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (1 + ?3i)^3 = 8 because (1 + ?3i) has modulus 2 and argument 120Â°. 


912.N.CN.6 
10 
High School: Number and Quantity 
Represent complex numbers and their operations on the complex plane. Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. 


912.N.CN.7 
10 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Solve quadratic equations with real coefficients that have complex solutions. 


912.N.CN.8 
10 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Extend polynomial identities to the complex numbers. For example, rewrite x^2 + 4 as (x + 2i)(x  2i). 


912.N.CN.9 
10 
High School: Number and Quantity 
Use complex numbers in polynomial identities and equations. Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials. 


912.N.VM.1 
10 
High School: Number and Quantity 
Represent and model with vector quantities. Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes (e.g., v(bold), v, v, v(not bold)). 


912.N.VM.2 
10 
High School: Number and Quantity 
Represent and model with vector quantities. Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point. 


912.N.VM.3 
10 
High School: Number and Quantity 
Represent and model with vector quantities. Solve problems involving velocity and other quantities that can be represented by vectors. 


912.N.VM.4 
10 
High School: Number and Quantity 
Perform operations on vectors. Add and subtract vectors. 


912.N.VM.4a 
10 
High School: Number and Quantity 
Add vectors endtoend, componentwise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes. 


912.N.VM.4b 
10 
High School: Number and Quantity 
Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum. 


912.N.VM.4c 
10 
High School: Number and Quantity 
Understand vector subtraction v  w as v + (w), where (w) is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction componentwise. 


912.N.VM.5 
10 
High School: Number and Quantity 
Perform operations on vectors. Multiply a vector by a scalar. 


912.N.VM.5a 
10 
High School: Number and Quantity 
Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication componentwise, e.g., as c(v(sub x), v(sub y)) = (cv(sub x), cv(sub y)). 


912.N.VM.5b 
10 
High School: Number and Quantity 
Compute the magnitude of a scalar multiple cv using cv = cv. Compute the direction of cv knowing that when cv =? 0, the direction of cv is either along v (for c > 0) or against v (for c < 0). 


912.N.VM.6 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network. 


912.N.VM.7 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled. 


912.N.VM.8 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Add, subtract, and multiply matrices of appropriate dimensions. 


912.N.VM.9 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties. 


912.N.VM.10 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Understand that the zero and identity matrices play a role in matrix addition and multiplication similar to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse. 


912.N.VM.11 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors. 


912.N.VM.12 
10 
High School: Number and Quantity 
Perform operations on matrices and use matrices in applications. Work with 2 X 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area. 


912.A.SSE.1 
10 
Algebra 
Interpret the structure of expressions. Interpret expressions that represent a quantity in terms of its context. 


912.A.SSE.1a 
10 
Algebra 
Interpret parts of an expression, such as terms, factors, and coefficients. 


912.A.SSE.1b 
10 
Algebra 
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. 


912.A.SSE.2 
10 
Algebra 
Interpret the structure of expressions. 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). 


912.A.SSE.3 
10 
Algebra 
Write expressions in equivalent forms to solve problems. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. 


912.A.SSE.3a 
10 
Algebra 
Factor a quadratic expression to reveal the zeros of the function it defines. 


912.A.SSE.3b 
10 
Algebra 
Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. 


912.A.SSE.3c 
10 
Algebra 
Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15^t can be rewritten as [1.15^(1/12)]^(12t) ? 1.012^(12t) to reveal the approximate equivalent monthly interest rate if the annual rate is 15%. 


912.A.SSE.4 
10 
Algebra 
Write expressions in equivalent forms to solve problems. 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. 


912.A.APR.1 
10 
Algebra 
Perform arithmetic operations on polynomials. 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. 


912.A.APR.2 
10 
Algebra 
Understand the relationship between zeros and factors of polynomial. 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). 


912.A.APR.3 
10 
Algebra 
Understand the relationship between zeros and factors of polynomials. 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. 


912.A.APR.4 
10 
Algebra 
Use polynomial identities to solve problems. 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 + (2xy)^2 can be used to generate Pythagorean triples. 


912.A.APR.5 
10 
Algebra 
Use polynomial identities to solve problems. Know and apply that the Binomial Theorem gives 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. (The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument.). 


912.A.APR.6 
10 
Algebra 
Rewrite rational expressions. Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form 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. 


912.A.APR.7 
10 
Algebra 
Rewrite rational expressions. 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. 


912.A.CED.1 
10 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.2 
10 
Algebra 
Create equations that describe numbers or relationship. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. 


912.A.CED.3 
10 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.CED.4 
10 
Algebra 
Create equations that describe numbers or relationship. 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. 


912.A.REI.1 
10 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. 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. 


912.A.REI.2 
10 
Algebra 
Understand solving equations as a process of reasoning and explain the reasoning. Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 


912.A.REI.3 
10 
Algebra 
Solve equations and inequalities in one variable. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 


912.A.REI.4 
10 
Algebra 
Solve equations and inequalities in one variable. Solve quadratic equations in one variable. 


912.A.REI.4a 
10 
Algebra 
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. 


912.A.REI.4b 
10 
Algebra 
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 Â± bi for real numbers a and b. 


912.A.REI.5 
10 
Algebra 
Solve systems of equations. 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. 


912.A.REI.6 
10 
Algebra 
Solve systems of equations. Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables. 


912.A.REI.7 
10 
Algebra 
Solve systems of equations. 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. 


912.A.REI.8 
10 
Algebra 
Solve systems of equations. Represent a system of linear equations as a single matrix equation in a vector variable. 


912.A.REI.9 
10 
Algebra 
Solve systems of equations. 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). 


912.A.REI.10 
10 
Algebra 
Represent and solve equations and inequalities graphically. 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). 


912.A.REI.11 
10 
Algebra 
Represent and solve equations and inequalities graphically. Explain why the xcoordinates 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. 


912.A.REI.12 
10 
Algebra 
Represent and solve equations and inequalities graphically. 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 halfplanes. 


912.F.IF.1 
10 
Functions 
Understand the concept of a function and use function notation. Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x). 


912.F.IF.2 
10 
Functions 
Understand the concept of a function and use function notation. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. 


912.F.IF.3 
10 
Functions 
Understand the concept of a function and use function notation. Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n+1) = f(n) + f(n1) for n ? 1 (n is greater than or equal to 1). 


912.F.IF.4 
10 
Functions 
Interpret functions that arise in applications in terms of the context. 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 behavior; and periodicity. 


912.F.IF.5 
10 
Functions 
Interpret functions that arise in applications in terms of the context. 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 personhours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. 

