Operations And Algebraic Thinking 
K.OA: Understand Addition As Putting Together And Adding To, And Under Stand Subtraction As Taking Apart And Taking From. 
K.OA.1  Represent addition and subtraction with objects, fingers, mental images, drawings2, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations. 

K.OA.2  Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem. 

K.OA.3  Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1). 

K.OA.4  For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation. 

K.OA.5  Fluently add and subtract within 5. 
1.OA: Represent And Solve Problems Involving Addition And Subtraction. 
1.OA.1  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.2 

1.OA.2  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. 
Understand And Apply Properties Of Operations And The Relationship Between Addition And Subtraction. 
1.OA.3  Apply properties of operations as strategies to add and subtract.3 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.) 

1.OA.4  Understand subtraction as an unknownaddend problem. For example, subtract 10 – 8 by finding the number that makes 10 when added to 8. 
Add And Subtract Within 20. 
1.OA.5  Relate counting to addition and subtraction (e.g., by counting on 2 to add 2). 

1.OA.6  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). 
Work With Addition And Subtraction Equations. 
1.OA.7  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  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 = �. 
2.OA: Represent And Solve Problems Involving Addition And Subtraction. 
2.OA.1  Use addition and subtraction within 100 to solve one and twostep word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 
Add And Subtract Within 20. 
2.OA.2  Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two onedigit numbers. 
Work With Equal Groups Of Objects To Gain Foundations For Multiplication. 
2.OA.3  Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends. 

2.OA.4  Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. 
3.OA: Represent And Solve Problems Involving Multiplication And Division. 
3.OA.1  Interpret products of whole numbers, e.g., interpret 5 × 7 as the total number of objects in 5 groups of 7 objects each. For example, describe a context in which a total number of objects can be expressed as 5 × 7. 

3.OA.2  Interpret wholenumber quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. 

3.OA.3  Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.1 

3.OA.4  Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = � ÷ 3, 6 × 6 = ?. 
Understand Properties Of Multiplication And The Relationship Between Multiplication And Division. 
3.OA.5  Apply properties of operations as strategies to multiply and divide.2 Examples: If 6 × 4 = 24 is known, then 4 × 6 = 24 is also known. (Commutative property of multiplication.) 3 × 5 × 2 can be found by 3 × 5 = 15, then 15 × 2 = 30, or by 5 × 2 = 10, then 3 × 10 = 30. (Associative property of multiplication.) Knowing that 8 × 5 = 40 and 8 × 2 = 16, one can find 8 × 7 as 8 × (5 + 2) = (8 × 5) + (8 × 2) = 40 + 16 = 56. (Distributive property.) 

3.OA.6  Understand division as an unknownfactor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. 
Multiply And Divide Within 100. 
3.OA.7  Fluently multiply and divide within 100, using strategies such as the relationship between multiplication and division (e.g., knowing that 8 × 5 = 40, one knows 40 ÷ 5 = 8) or properties of operations. By the end of Grade 3, know from memory all products of two onedigit numbers. 
Solve Problems Involving The Four Operations, And Identify And Explain Patterns In Arithmetic. 
3.OA.8  Solve twostep word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3 

3.OA.9  Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends. 
4.OA: Use The Four Operations With Whole Numbers To Solve Problems. 
4.OA.1  Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 × 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations. 

4.OA.2  Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.1 

4.OA.3  Solve multistep word problems posed with whole numbers and having wholenumber answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. 
Gain Familiarity With Factors And Multiples. 
4.OA.4  Find all factor pairs for a whole number in the range 1–100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given whole number in the range 1–100 is a multiple of a given onedigit number. Determine whether a given whole number in the range 1–100 is prime or composite. 
Generate And Analyze Patterns. 
4.OA.5  Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself. For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way. 
5.OA: Write And Interpret Numerical Expressions. 
5.OA.1  Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. 

5.OA.2  Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product. 
Analyze Patterns And Relationships. 
5.OA.3  Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so. 
Reasoning With Equations And Inequalities 
HSA.REI: Solve Equations And Inequalities In One Variable 
HSA.REI.3  Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. 

HSA.REI.4  Solve quadratic equations in one variable. 

HSA.REI.4.a  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. 

HSA.REI.4.b  Solve quadratic equations by inspection (e.g., for x2 = 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 ± bi for real numbers a and b. 
Understand Solving Equations As A Process Of Reasoning And Explain The Reasoning 
HSA.REI.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. 

HSA.REI.2  Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. 
Solve Systems Of Equations 
HSA.REI.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. 

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

HSA.REI.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 x2 + y2 = 3. 

HSA.REI.8  (+) Represent a system of linear equations as a single matrix equation in a vector variable. 

HSA.REI.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). 
Represent And Solve Equations And Inequalities Graphically 
HSA.REI.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). 

HSA.REI.11  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.★ 

HSA.REI.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 halfplanes. 
Statistics And Probability 
6.SP: Develop Understanding Of Statistical Variability. 
6.SP.1  Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages. 

6.SP.2  Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. 

6.SP.3  Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number. 
Summarize And Describe Distributions. 
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 

6.SP.5  Summarize numerical data sets in relation to their context, such as by: 

6.SP.5.a  Reporting the number of observations. 

6.SP.5.b  Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 

6.SP.5.c  Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. 

6.SP.5.d  Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. 
7.SP: Use Random Sampling To Draw Inferences About A Population. 
7.SP.1  Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. 

7.SP.2  Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. 
Draw Informal Comparative Inferences About Two Populations. 
7.SP.3  Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. 

7.SP.4  Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventhgrade science book are generally longer than the words in a chapter of a fourthgrade science book. 
Investigate Chance Processes And Develop, Use, And Evaluate Probability Models. 
7.SP.5  Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. 

7.SP.6  Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 

7.SP.7  Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. 

7.SP.7.a  Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. 

7.SP.7.b  Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land openend down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? 

7.SP.8  Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation. 

7.SP.8.a  Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. 

7.SP.8.b  Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. 

7.SP.8.c  Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? 
8.SP: Investigate Patterns Of Association In Bivariate Data. 
8.SP.1  Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association. 

8.SP.2  Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line. 

8.SP.3  Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height. 

8.SP.4  Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a twoway table. Construct and interpret a twoway table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores? 
Using Probability To Make Decisions 
HSS.MD: Calculate Expected Values And Use Them To Solve Problems 
HSS.MD.1  (+) Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. 

HSS.MD.2  (+) Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. 

HSS.MD.3  (+) Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on 
All Five Questions Of A MultipleChoice Test Where Each Question Has Four Choices, And Find The Expected Grade Under Various Grading Schemes. 
HSS.MD.4  (+) Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? 
Use Probability To Evaluate Outcomes Of Decisions: Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. 
HSS.MD.5  (+) Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values. 

HSS.MD.5.a  Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast food restaurant. 

HSS.MD.5.b  Evaluate and compare strategies on the basis of expected values. For example, compare a highdeductible versus a lowdeductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. 

HSS.MD.6  (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). 

HSS.MD.7  (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). 