Geometry 
4.G: Draw And Identify Lines And Angles, And Classify Shapes By Properties Of Their Lines And Angles. 
4.G.2  Classify twodimensional figures based on the presence or absence of parallel or perpendicular lines, or the presence or absence of angles of a specified size. Recognize right triangles as a category, and identify right triangles. 
5.G: Classify TwoDimensional Figures Into Categories Based On Their Properties. 
5.G.3  Understand that attributes belonging to a category of two dimensional figures also belong to all subcategories of that category. For example, all rectangles have four right angles and squares are rectangles, so all squares have four right angles. 

5.G.4  Classify twodimensional figures in a hierarchy based on properties. 
Graph Points On The Coordinate Plane To Solve RealWorld And Mathematical Problems. 
5.G.1  Use a pair of perpendicular number lines, called axes, to define a coordinate system, with the intersection of the lines (the origin) arranged to coincide with the 0 on each line and a given point in the plane located by using an ordered pair of numbers, called its coordinates. Understand that the first number indicates how far to travel from the origin in the direction of one axis, and the second number indicates how far to travel in the direction of the second axis, with the convention that the names of the two axes and the coordinates correspond (e.g., xaxis and xcoordinate, yaxis and ycoordinate). 
4.NBT: Generalize Place Value Understanding For MultiDigit Whole Numbers. 
4.NBT.2  Read and write multidigit whole numbers using baseten numerals, number names, and expanded form. Compare two multidigit numbers based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. 

4.NBT.3  Use place value understanding to round multidigit whole numbers to any place. 
Use Place Value Understanding And Properties Of Operations To Perform MultiDigit Arithmetic. 
4.NBT.4  Fluently add and subtract multidigit whole numbers using the standard algorithm. 

4.NBT.5  Multiply a whole number of up to four digits by a onedigit whole number, and multiply two twodigit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. 
4.NF: Understand Decimal Notation For Fractions, And Compare Decimal Fractions. 
4.NF.7  Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model. 
Number And Operations—Fractions 
5.NF: Apply And Extend Previous Understandings Of Multiplication And Division To Multiply And Divide Fractions. 
5.NF.6  Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem. 
Use Equivalent Fractions As A Strategy To Add And Subtract Fractions. 
5.NF.1  Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, 2/3 + 5/4 = 8/12 + 15/12 = 23/12. (In general, a/b + c/d = (ad + bc)/bd.) 

5.NF.2  Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g., by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions 
Operations And Algebraic Thinking 
4.OA: 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: 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. 
Write And Interpret Numerical Expressions. 
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. 
Ratios And Proportional Relationships 
6.RP: Understand Ratio Concepts And Use Ratio Reasoning To Solve Problems. 
6.RP.3  Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. 
7.RP: Analyze Proportional Relationships And Use Them To Solve RealWorld And Mathematical Problems. 
7.RP.3  Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. 
Statistics And Probability 
6.SP: Summarize And Describe Distributions. 
6.SP.4  Display numerical data in plots on a number line, including dot plots, histograms, and box plots. 
7.SP: Draw Informal Comparative Inferences About Two Populations. 
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.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. 
The Number System 
6.NS: Apply And Extend Previous Understandings Of Multiplication And Division To Divide Fractions By Fractions. 
6.NS.1  Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi? 
Apply And Extend Previous Understandings Of Numbers To The System Of Rational Numbers. 
6.NS.5  Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation. 

6.NS.8  Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. 
Compute Fluently With MultiDigit Numbers And Find Common Factors And Multiples. 
6.NS.3  Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation. 
7.NS: Apply And Extend Previous Understandings Of Operations With Fractions To Add, Subtract, Multiply, And Divide Rational Numbers. 
7.NS.1  Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. 

7.NS.2  Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. 