Organization: SAVVAS Product Name: enVision Mathematics 2021 Common Core Grade 6 Product Version: 1 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-76ee1467-4513-3f34-bfa5-6c49b886ce33 Timestamp: Tuesday, July 27, 2021 12:37 PM EDT Status: VALID! Conformant: true ----- VALID! ----- Resource Validation Results The document is valid. ----- VALID! ----- Schema Location Results Schema locations are valid. ----- VALID! ----- Schema Validation Results The document is valid. ----- VALID! ----- Schematron Validation Results The document is valid. Curriculum Standards: Interpret Results: Draw logical conclusions from the data based on the original question. (GAISE Model, step 4) - 6.SP.1d Design and use a plan to collect appropriate data to answer a statistical question. (GAISE Model, step 2) - 6.SP.1b Find and position pairs of rational numbers on a coordinate plane. - NC.6.NS.6.b.3 Analyze Data: Select appropriate graphical methods and numerical measures to analyze data by displaying variability within a group, comparing individual to individual, and comparing individual to group. (GAISE Model, step 3) - 6.SP.1c Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. - NC.6.NS.6.b.2 Understand signs of numbers in ordered pairs as indicating locations in quadrants. - NC.6.NS.6.b.1 Formulate Questions: Recognize and formulate a statistical question as one that anticipates variability and can be answered with quantitative datFor example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because of the variability in students’ ages. (GAISE Model, step 1) - 6.SP.1a Create a line plot to represent a given or generated data set, and analyze the data to answer questions and solve problems, recognizing the outliers and generating the median. - 5.DS.A.2 Create a line graph to represent a data set, and analyze the data to answer questions and solve problems. - 5.DS.A.1 Solve unit rate problems including those involving unit pricing and constant speed. - M06.A-R.1.1.4 Construct tables of equivalent ratios relating quantities with whole-number measurements, find missing values in the tables, and/or plot the pairs of values on the coordinate plane. Use tables to compare ratios. - M06.A-R.1.1.3 Find the unit rate a/b associated with a ratio a:b (with b not equal to 0) and use rate language in the context of a ratio relationships. - M06.A-R.1.1.2 Use ratio language and notation (such as 3 to 4, 3:4, ¾) to describe a ratio relationship between two quantities. - M06.A-R.1.1.1 Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percentage. - M06.A-R.1.1.5 Solve equations involving positive rational numbers using number sense, properties of arithmetic and the idea of maintaining equality on both sides of the equation. Interpret a solution in the original context and assess the reasonableness of results. - 6.2.3.2 Represent real-world or mathematical situations using equations and inequalities involving variables and positive rational numbers. - 6.2.3.1 Summarize numerical data sets in relation to their context, such as by: - MAFS.6.SP.2.5 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. - MAFS.6.SP.2.4 Solve problems involving addition and subtraction of fractions and mixed numbers with unlike denominators, and justify the solution. - 5.NF.B.6 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. - 5.OA.3 Calculate the area of quadrilaterals. Quadrilaterals include squares, rectangles, rhombuses, parallelograms, trapezoids and kites. When formulas are used, be able to explain why they are valid. - 6.3.1.2 Calculate the surface area and volume of prisms and use appropriate units, such as cm² and cm³. Justify the formulas used. Justification may involve decomposition, nets or other models. - 6.3.1.1 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. - 5.OA.2 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. - 5.OA.1 English language learners communicate information, ideas and concepts necessary for academic success in the content area of Mathematics. - ELD.K12.ELL.MA.1 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. - MAFS.K12.MP.3.1 Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. - MAFS.5.MD.2.2 Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^3 and A = 6s^2 to find the volume and surface area of a cube with sides of length s = ½. - 6.EE.2c Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. - 6.EE.2b Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. - 6.EE.2a Compute the least common multiple (LCM) of two numbers both less than or equal to 12. - 6.NS.4b Solve multi-step problems involving division with unit fractions. - 5.NC.9.7 Compute the greatest common factor (GCF) of two numbers both less than or equal to 100. - 6.NS.4a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. - NC.6.NS.7.a Express sums of two whole numbers, each less than or equal to 100, using the distributive property to factor out a common factor of the original addends. - 6.NS.4c Write, interpret, and explain statements of order for rational numbers in real-world contexts. - NC.6.NS.7.b Apply the properties of operations to generate equivalent expressions. - M06.B-E.1.1.5 Identify parts of an expression using mathematical terms (e.g., sum, term, product, factor, quotient, coefficient, quantity). - M06.B-E.1.1.3 Evaluate expressions at specific values of their variables, including expressions that arise from formulas used in real-world problems. - M06.B-E.1.1.4 Write and evaluate numerical expressions involving whole-number exponents. - M06.B-E.1.1.1 Write algebraic expressions from verbal descriptions. - M06.B-E.1.1.2 Plot and interpret points in the first quadrant of the Cartesian coordinate plane. - 5.GM.C.7 Use and evaluate variables in expressions, equations, and inequalities that arise from various contexts, including determining when or if, for a given value of the variable, an equation or inequality involving a variable is true or false. - 6.A.1.3 Represent relationships between two varying quantities involving no more than two operations with rules, graphs, and tables; translate between any two of these representations. - 6.A.1.2 Plot integer- and rational-valued (limited to halves and fourths) ordered-pairs as coordinates in all four quadrants and recognize the reflective relationships among coordinates that differ only by their signs. - 6.A.1.1 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 3/4-cup 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? - MAFS.6.NS.1.1 The student will represent relationships between quantities using ratios, and will use appropriate notations, such as a/b, a to b, and a:b. - 6.1 The student will recognize and represent patterns with whole number exponents and perfect squares. - 6.4 Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. - 6.NS.6b Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3, and that 0 is its own opposite. - 6.NS.6a Interpret numerical expressions without evaluating them. - 5.OA.13.3 Plot rational numbers on number lines and ordered pairs on coordinate planes. - 6.NS.6d Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. - 6.NS.6c Use the order of operations to evaluate expressions. - 5.OA.13.1 Create and evaluate expressions involving variables and whole number exponents. - 6.EEI.A.2 Identify and generate equivalent algebraic expressions using mathematical properties. - 6.EEI.A.3 Describe the difference between an expression and an equation. - 6.EEI.A.1 Apply the relationship between ratios, equivalent fractions and percents to solve problems in various contexts, including those involving mixture and concentrations. - 6.N.3.3 Use multiplicative reasoning and representations to solve ratio and unit rate problems. - 6.N.3.4 Find products when bases from –6 to 6 are squared and cubed, using a calculator. - MAFS.8.EE.1.AP.2b Represent a problem situation with a mathematical model. - 5.NC.7.12 Read, write, and compare decimals to thousandths. Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and <, symbols to record the results of comparisons. - NY-5.NBT.3b Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem and/or represent solutions of such inequalities on number lines. - M06.B-E.2.1.4 Represent three-dimensional figures using nets made of rectangles and triangles. - M06.C-G.1.1.5 Given coordinates for the vertices of a polygon in the plane, use the coordinates to find side lengths and area of the polygon (limited to triangles and special quadrilaterals). Formulas will be provided. - M06.C-G.1.1.4 Determine the volume of right rectangular prism with fractional edge lengths. Formulas will be provided. - M06.C-G.1.1.3 Determine the area of irregular or compound polygons. Example: Find the area of a room in the shape of an irregular polygon by composing and/or decomposing. - M06.C-G.1.1.2 Determine the area of triangles and special quadrilaterals (i.e., square rectangle, parallelogram, rhombus, and trapezoid). Formulas will be provided. - M06.C-G.1.1.1 Use substitution to determine whether a given number in a specified set makes an equation or inequality true. - M06.B-E.2.1.1 Write algebraic expressions to represent real-world or mathematical problems. - M06.B-E.2.1.2 Identify and use ratios to compare quantities. Recognize that multiplicative comparison and additive comparison are different. - 6.N.3.1 Determine the unit rate for ratios. - 6.N.3.2 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q, and x are all non-negative rational numbers. - M06.B-E.2.1.3 Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units. - 6.3.3.1 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. - MAFS.K12.MP.5.1 Summarize quantitative data sets in relation to their context. - NY-6.SP.5 Display quantitative data in plots on a number line, including dot plots, and histograms. - NY-6.SP.4 Recognize that a measure of center for a quantitative 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. - NY-6.SP.3 Understand that a set of quantitative data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. - NY-6.SP.2 Determine the surface area of triangular and rectangular prisms (including cubes). Formulas will be provided. - M06.C-G.1.1.6 Recognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths; - 5.NSF.4a 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. - MAFS.5.OA.2.3 Solve problems in various real-world and mathematical contexts that require the conversion of weights, capacities, geometric measurements, and time within the same measurement systems using appropriate units. - 6.GM.3.2 Understand additive inverses when adding and subtracting integers. - NC.6.NS.9.a Apply the order of operations to evaluate numerical expressions, e.g., 6 + 8 ÷ 2; (6 + 8) ÷ 2. Note: Exponents and nested grouping symbols are not included. - NY-5.OA.1 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. e.g., Express the calculation “add 8 and 7, then multiply by 2” as (8 + 7) × 2. Recognize that 3 × (18,932 + 921) is three times as large as 18,932 + 921 without having to calculate the indicated sum or product. - NY-5.OA.2 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. e.g., 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. - NY-5.OA.3 Use dot plots, histograms, and box plots to represent data. - NC.6.SP.4.a Compare the attributes of different representations of the same data. - NC.6.SP.4.b Analyze and describe the properties of prisms and pyramids. - 5.GM.A.3 Classify figures in a hierarchy based on properties. - 5.GM.A.2 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. - 6.G.A.4 Represent real-world or mathematical situations using expressions, equations and inequalities involving variables and rational numbers. - 6.A.3.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. - 6.G.A.1 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. - 6.G.A.3 Use number sense and properties of operations and equality to solve real-world and mathematical problems involving equations in the form x + p = ! and px = q, where x, p, and q are nonnegative rational numbers. Graph the solution on a number line, interpret the solution in the original context, and assess the reasonableness of the solution. - 6.A.3.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l x w x h and V = b x h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. - 6.G.A.2 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 real-world contexts, explaining the meaning of 0 in each situation. - MAFS.6.NS.3.5 Solve real-world 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. - MAFS.6.NS.3.8 Solve multistep problems with fractions and decimals. - 6.NC.1.7 Divide with mixed numbers. - 6.NC.1.6 Display numerical data in plots on a number line. - NC.6.SP.4 Summarize numerical data sets in relation to their context. - NC.6.SP.5 Divide whole numbers and decimals. - 6.NC.1.2 Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. - 5.MD.5c Divide a fraction by another fraction. - 6.NC.1.5 Use models and equations to represent fraction division. - 6.NC.1.4 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. - NC.6.SP.1 Add, subtract, and multiply decimals. - 6.NC.1.1 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. - NC.6.SP.2 Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. - 5.MD.5b Understand that both a measure of center and a description of variability should be considered when describing a numerical data set. - NC.6.SP.3 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. - 5.MD.5a Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order: Order of Operations (e.g., use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2). - M.6.13c Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. (e.g., Describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.) - M.6.13b Write expressions that record operations with numbers and with letters standing for numbers. (e.g., Express the calculation, “Subtract y from 5” as 5 – y.) - M.6.13a Unfold three-dimensional figures into two-dimensional rectangles and triangles (nets) to find the surface area and to solve real-world and mathematical problems. - 6.GM.4 Use visual models (e.g., model by packing) to discover that the formulas for the volume of a right rectangular prism (𝑉=𝑉𝑙𝑉𝑙𝑤ℎ,𝑉𝑙𝑤𝑉=𝑉𝑙𝑤𝑉𝐵ℎ) are the same for whole or fractional edge lengths. Apply these formulas to solve real-world and mathematical problems. - 6.GM.2 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. - 6.GM.1 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 x² + 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. - MAFS.K12.MP.7.1 Generate two numeric patterns given two rules. - 5.RA.A.1a Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. - MAFS.6.EE.3.9 Mathematical Modeling: Solve Area, Surface Area, and Volume Problems - 6.G.7MM Identify the relationship between two numeric patterns. - 5.RA.A.1d Write an equation to express the relationship between the dependent and independent variables. - M06.B-E.3.1.1 Analyze the relationship between the dependent and independent variables using graphs and tables and/or relate these to an equation. - M06.B-E.3.1.2 Develop and use formulas for the area of squares and parallelograms using a variety of methods including but not limited to the standard algorithm. - 6.GM.1.1 Develop and use formulas to determine the area of triangles. - 6.GM.1.2 Find the area of right triangles, other triangles, special quadrilaterals, and polygons that can be decomposed into triangles and other shapes to solve real-world and mathematical problems. - 6.GM.1.3 Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. - 5.NF.4b Distinguish comparisons of absolute value from statements about order. (e.g., recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.) - M.6.10d Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. (e.g., interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.) - M.6.10a Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. - MAFS.5.MD.3.5.b Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. - MAFS.5.MD.3.5.a Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. (e.g., for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars). - M.6.10c Write, interpret, and explain statements of order for rational numbers in real-world contexts (e.g., write –3o C > –7o C to express the fact that –3o C is warmer than –7o C). - M.6.10b Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. - MAFS.5.MD.3.5.c Summarize numerical data sets in relation to their context. - 6.SP.5 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.1 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.2 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. - 6.SP.3 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. - 6.SP.4 Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. Solve real-world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., using visual fraction models and equations to represent the problem. e.g., How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? - NY-5.NF.7c Apply and extend previous understandings of the volume of a right rectangular prism to find the volume of right rectangular prisms with fractional edge lengths. Apply this understanding to the context of solving real-world and mathematical problems. - NC.6.G.2 Represent right prisms and right pyramids using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. - NC.6.G.4 Understand a ratio as a comparison of two quantities and represent these comparisons. - 6.RP.A.1 Understand the concept of a unit rate associated with a ratio, and describe the meaning of unit rate. - 6.RP.A.2 Solve problems involving ratios and rates. - 6.RP.A.3 Mathematical Modeling: Understand and Use Percent - 6.P.6MM Identify parts of an expression using mathematical terms and view one or more of those parts as a single entity. - NC.6.EE.2.b Write expressions that record operations with numbers and with letters standing for numbers. - NC.6.EE.2.a Evaluate expressions at specific values of their variables using expressions that arise from formulas used in real-world problems. - NC.6.EE.2.c Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. - MAFS.5.G.2.3 Classify and organize two-dimensional figures into Venn diagrams based on the attributes of the figures. - MAFS.5.G.2.4 Find the volume of rectangular prisms using a formula. - 5.MD.11.2 Use models, prior knowledge of volume and previously learned strategies to solve word problems involving volume. - 5.MD.11.4 Interpret and compute quotients of fractions (including mixed numbers), and solve word problems involving division of fractions by fractions. - M06.A-N.1.1.1 Write and solve a multiplication or division equation. - 6.AF.4.4 Write and solve an addition or subtraction equation. - 6.AF.4.3 Understand and write an inequality that describes a real-world situation. - 6.AF.4.6 Write and solve equations that involve rational numbers. - 6.AF.4.5 Identify dependent and independent variables. - 6.AF.4.8 Write and represent solutions of inequalities. - 6.AF.4.7 Use patterns to write and solve equations with variables. - 6.AF.4.9 Identify parts of an expression using mathematical terminology. - 6.EEI.A.2a Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. - M.6.16 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number or depending on the purpose at hand, any number in a specified set. - M.6.17 Evaluate non-negative rational number expressions. - 6.EEI.A.2c Apply the properties of operations to generate equivalent expressions (e.g., apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y). - M.6.14 Evaluate expressions at specific values of the variables. - 6.EEI.A.2b Identify when two expressions are equivalent; i.e., when the two expressions name the same number regardless of which value is substituted into them. (e.g., The expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.) - M.6.15 Understand the meaning of the variable in the context of the situation. - 6.EEI.A.2e Write and evaluate numerical expressions involving whole-number exponents. - M.6.12 Write and evaluate algebraic expressions. - 6.EEI.A.2d Reason abstractly and quantitatively. - 5.MP.2 Use the properties of equality to write equivalent equations. - 6.AF.4.2 Determine if a value for a variable makes an equation true. - 6.AF.4.1 Solve real-world 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. - M.6.11 Model with mathematics. - 5.MP.4 Analyze the relationship between dependent and independent variables in tables, graphs, and equations. - 6.AF.4.10 Use unit rates to solve problems. - 6.P.5.7 Use ratio reasoning to convert customary measurements. - 6.P.5.8 Use unit rates to convert metric measurements. - 6.P.5.9 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers. - M.6.18 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. - M.6.19 Compare ratios to solve problems. - 6.P.5.3 Solve ratio problems by using tables and graphs to show equivalent ratios. - 6.P.5.4 Solve problems involving rates. - 6.P.5.5 Compare unit rates to solve problems. - 6.P.5.6 Giving quantitative measures of center, describing variability, and any overall pattern, and noting any striking deviations. - NC.6.SP.5.b.1 Mathematical Modeling: Integers and Rational Numbers - 6.NC.2MM Justifying the appropriate choice of measures of center using the shape of the data distribution. - NC.6.SP.5.b.2 Use a ratio to describe the relationship between two quantities. - 6.P.5.1 Use multiplication and division to find equivalent ratios. - 6.P.5.2 Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. (e.g., In a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.) - M.6.20 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number. - M.6.27 Display numerical data in plots on a number line, including dot plots, histograms and box plots. - M.6.28 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. (e.g., “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.) - M.6.25 Through informal observation, understand that a set of data collected to answer a statistical question has a distribution which can be described by its center (mean/ median), spread (range), and overall shape. - M.6.26 Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. - M.6.23 Use tables to identify relationships between patterns. - 5.OA.15.2 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. - M.6.24 Find the area of right triangles, other triangles, special quadrilaterals and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. - M.6.21 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = B h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. - M.6.22 Create and use rules, tables, spreadsheets and graphs to describe patterns of change and solve problems. - 5.2.1.1 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. - NY-6.EE.5 Identify when two expressions are equivalent. - NY-6.EE.4 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q; x – p = q; px = q; and x/p = q for cases in which p, q, and x are all nonnegative rational numbers. - NY-6.EE.7 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem. Understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. - NY-6.EE.6 Use variables to represent two quantities in a real-world problem that change in relationship to one another.Given a verbal context and an equation, identify the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. - NY-6.EE.9 Write an inequality of the form x > c, x ≥ c, x ≤ c, or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of these forms have infinitely many solutions; represent solutions of such inequalities on a number line. - NY-6.EE.8 Locate rational numbers on a horizontal or vertical number line. - 6.NS.C.6a Add and subtract fractions (including mixed numbers) with unlike denominators. (May include multiple methods and representations.) - M05.A-F.1.1.1 Divide fractions using visual fraction models. - MAFS.6.NS.1.AP.1a Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane. - 6.NS.C.6c Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. - 6.NS.C.6b Describe the qualitative aspects of the data (e.g., how it was measured, units of measurement). - 6.DS.5b Give measures of center (median, mean). - 6.DS.5c Find measures of variability (interquartile range, mean absolute deviation) using a number line. - 6.DS.5d Describe the overall pattern (shape) of the distribution. - 6.DS.5e State the sample size. - 6.DS.5a Summarize numerical data sets in relation to the context. - 6.DSP.B.5 Solve problems involving operations (+, -, x, and divided by) with whole numbers, decimals (through thousandths), straight computation, or word problems. - M06.A-N.2.1.1 Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and <, symbols to record the results of comparisons. - 5.NBT.3b Justify the choices for measure of center and measure of variability based on the shape of the distribution. - 6.DS.5f Display and interpret data. - 6.DSP.B.4 Finding the whole, given a part and the percent. - NC.6.RP.4.c 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. - 5.OA.B.3 Using equivalent ratios, such as benchmark percents (50%, 25%, 10%, 5%, 1%), to determine a part of any given quantity. - NC.6.RP.4.b Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write -3 degrees C > -7 degrees C to express the fact that -3 degrees C is warmer than -7 degrees C. - 6.NS.C.7b Understanding and finding a percent of a quantity as a ratio per 100. - NC.6.RP.4.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret -3 > -7 as a statement that -3 is located to the right of -7 on a number line oriented from left to right. - 6.NS.C.7a Write and evaluate numerical expressions involving whole-number exponents. - NY-6.EE.1 Apply the properties of operations to generate equivalent expressions. - NY-6.EE.3 Write, read, and evaluate expressions in which letters stand for numbers. - NY-6.EE.2 Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than -30 dollars represents a debt greater than 30 dollars. - 6.NS.C.7d Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. - 6.NS.C.7c Follow precisely a multistep procedure when carrying out experiments, taking measurements, or performing technical tasks. - LAFS.68.RST.1.3 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.B.3 Add decimals to hundredths using familiar strategies, such as partial sums. - 5.NC.2.4 Identify parts of the coordinate plane (x-axis, y-axis, and the origin) and the ordered pair (x-coordinate and y-coordinate). Limit the coordinate plane to quadrant I. - M05.C-G.1.1.1 Represent real-world and mathematical problems by plotting points in quadrant I of the coordinate plane and interpret coordinate values of points in the context of the situation. - M05.C-G.1.1.2 Model sums and differences of decimals. - 5.NC.2.3 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) divided by (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) divided by (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) divided by(c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup 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? - 6.NS.A.1 Classify two-dimensional figures in a hierarchy based on properties. - 5.G.B.4 Solve real world problems involving multiplication of fractions and mixed numbers. e.g., using visual fraction models or equations to represent the problem. - NY-5.NF.6 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 to estimate mentally and assess the reasonableness of answers. - MAFS.5.NF.1.2 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems. - NY-6.G.4 Draw polygons in the coordinate plane given coordinates for the vertices. Use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. - NY-6.G.3 Mathematical Modeling: Represent and Solve Equations and Inequalities - 6.NC.4MM Find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. - NY-6.G.2 Find area of triangles, trapezoids, and other polygons by composing into rectangles or decomposing into triangles and quadrilaterals. Apply these techniques in the context of solving real-world and mathematical problems. - NY-6.G.1 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 to estimate mentally and assess the reasonableness of answers, e.g., recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. - NY-5.NF.2 Write an equation to express one quantity, the dependent variable, in terms of the other quantity, the independent variable. - 6.EEI.C.9a Analyze the relationship between the dependent and independent variables using graphs, tables and equations and relate these representations to each other. - 6.EEI.C.9b Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. - 5.MD.C.4 Make sense of problems and persevere in solving them. - MP.1 Locate points on a coordinate grid. - 5.G.14.1 Reason abstractly and quantitatively. - MP.2 Look for and make use of structure. - MP.7 Look for and express regularity in repeated reasoning. - MP.8 Construct viable arguments and critique the reasoning of others. - MP.3 Model with mathematics. - MP.4 Use reasoning to solve problems by making sense of quantities and relationships in the situation. - 5.G.14.4 Use appropriate tools strategically. - MP.5 Use coordinates to find the side lengths of polygons drawn in quadrant I of a coordinate plane. - MAFS.6.G.1.AP.3b Attend to precision. - MP.6 Draw polygons on a coordinate plane given the coordinates of the vertices. - MAFS.6.G.1.AP.3a Locate and plot integers and other rational numbers on a horizontal or vertical number line; locate and plot pairs of integers and other rational numbers on a coordinate plane. - M06.A-N.3.1.3 Represent quantities in real-world contexts using positive and negative numbers, explaining the meaning of 0 in each situation (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge). - M06.A-N.3.1.1 Determine the opposite of a number and recognize that the opposite of the opposite of a number is the number itself (e.g., -(-3) = 3; 0 is its own opposite). - M06.A-N.3.1.2 Convert measurement units within and between two systems of measurement. - 6.RP.A.3d Solve unit rate problems. - 6.RP.A.3b Solve percent problems. - 6.RP.A.3c Create tables of equivalent ratios, find missing values in the tables and plot the pairs of values on the Cartesian coordinate plane. - 6.RP.A.3a Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. - MAFS.6.EE.1.4 Match a two-dimensional net to its corresponding three-dimensional figure. - MAFS.6.G.1.AP.4a Engage effectively in a range of collaborative discussions (one-on-one, in groups, and teacher-led) with diverse partners on grade 6 topics, texts, and issues, building on others’ ideas and expressing their own clearly. (a) Come to discussions prepared, having read or studied required material; explicitly draw on that preparation by referring to evidence on the topic, text, or issue to probe and reflect on ideas under discussion. (b) Follow rules for collegial discussions, set specific goals and deadlines, and define individual roles as needed. (c) Pose and respond to specific questions with elaboration and detail by making comments that contribute to the topic, text, or issue under discussion. (d) Review the key ideas expressed and demonstrate understanding of multiple perspectives through reflection and paraphrasing. - LAFS.6.SL.1.1 Relating the choice of measures of center to the shape of the data distribution and the context in which the data were gathered. - M.6.29d Make a line plot to display a data set of measurements in fractions of a unit (1/2 , 1/4 , 1/8). Use operations on fractions for this grade to solve problems involving information presented in line plots. For example, given different measurements of liquid in identical beakers, find the amount of liquid each beaker would contain if the total amount in all the beakers were redistributed equally. - 5.MD.2 Write, read, and evaluate expressions in which letters stand for numbers. - MAFS.6.EE.1.2 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. - MAFS.6.EE.1.3 Measure volumes by counting unit cubes, using cubic cm, cubic in, cubic ft, and improvised units. - 5.MD.4 Reporting the number of observations. - M.6.29a Write and evaluate numerical expressions involving whole-number exponents. - MAFS.6.EE.1.1 Interpret information presented in diverse media and formats (e.g., visually, quantitatively, orally) and explain how it contributes to a topic, text, or issue under study. - LAFS.6.SL.1.2 Giving quantitative measures of center (median and/or mean), 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. - M.6.29c Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. - 5.MD.1 Delineate a speaker’s argument and specific claims, distinguishing claims that are supported by reasons and evidence from claims that are not. - LAFS.6.SL.1.3 Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. - M.6.29b Find the surface area of the three dimensional figure by adding the areas of the shapes forming the two-dimensional nets. - MAFS.6.G.1.AP.4b Integrate quantitative or technical information expressed in words in a text with a version of that information expressed visually (e.g., in a flowchart, diagram, model, graph, or table). - LAFS.68.RST.3.7 Multiply decimals using partial products and models. - 5.NC.4.6 Solve real-world 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. - 6.NS.C.8 Find the area of quadrilaterals using models. - MAFS.6.G.1.AP.1c Understand that the absolute value of a rational number is its distance from 0 on the number line. - 6.NS.C.7 Use positive and negative numbers to represent quantities. - 6.NS.C.5 Use place-value understanding and an algorithm for multiplying whole numbers to multiply a decimal and a whole number. - 5.NC.4.4 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. Apply the formulas V = l × w × h and V = B × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. - NY-5.MD.5b Decompose complex shapes (polygon, trapezoid, and pentagon) into simple shapes (rectangles, squares, triangles) to measure area. - MAFS.6.G.1.AP.1b Compose rectangles to find areas of right triangles using graph paper. - MAFS.6.G.1.AP.1a Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. - NY-5.MD.5a Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. - NY-5.MD.5c Classify two-dimensional figures in a hierarchy based on properties. - M05.C-G.2.1.1 Interpret and compute quotients of fractions. - NC.6.NS.1.a Solve real-world and mathematical problems involving division of fractions. - NC.6.NS.1.b Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category. e.g., All rectangles have four right angles and squares are rectangles, so all squares have four right angles. Note: The inclusive definition of a trapezoid will be utilized, which defines a trapezoid as “A quadrilateral with at least one pair of parallel sides.” - NY-5.G.3 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. - NY-5.G.2 Classify two-dimensional figures in a hierarchy based on properties. - NY-5.G.4 Find the fractional length and volume of a rectangular prism with edges using models. - MAFS.6.G.1.AP.2a Convert among different-sized standard measurement units within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. - 5.MD.A.1 Apply and extend previous understandings of multiplication to multiply a fraction by a whole number or a fraction. Find the area of a rectangle with fractional side lengths by tiling it with rectangles of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. e.g., A shaded portion shows the rectangle with the appropriate unit fraction side lengths. The area of a 2/3 × 3/4 rectangle is 6/12 because the whole is partitioned into 12 parts with 6 of them shaded. - NY-5.NF.4b 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., x-axis and x-coordinate, y-axis and y-coordinate). - NY-5.G.1 Use nets to find the surface area of threedimensional figures whose sides are made up of rectangles and triangles. - 6.GM.A.4b Represent three-dimensional figures using nets made up of rectangles and triangles. - 6.GM.A.4a Evaluate real-world and algebraic expressions for specific values using the Order of Operations. Grouping symbols should be limited to parentheses, braces, and brackets. Exponents should be limited to whole-numbers. - 6.EEI.2c Translate between algebraic expressions and verbal phrases that include variables. - 6.EEI.2a Investigate and identify parts of algebraic expressions using mathematical terminology, including term, coefficient, constant, and factor. - 6.EEI.2b Convert between equivalent representations of positive rational numbers. - 6.1.1.7 Determine greatest common factors and least common multiples. Use common factors and common multiples to calculate with fractions and find equivalent fractions. - 6.1.1.6 Factor whole numbers; express a whole number as a product of prime factors with exponents. - 6.1.1.5 Determine equivalences among fractions, decimals and percents; select among these representations to solve problems. - 6.1.1.4 Understand that percent represents parts out of 100 and ratios to 100. - 6.1.1.3 Compare positive rational numbers represented in various forms. Use the symbols <, = and >. - 6.1.1.2 Compare two numbers from billions to thousandths using the symbols >, = or <, and justify the solution. - 5.NBT.A.2 Make and analyze frequency tables and histograms. - 6.DP.8.4 Write arguments focused on discipline-specific content. (a) Introduce claim(s) about a topic or issue, acknowledge and distinguish the claim(s) from alternate or opposing claims, and organize the reasons and evidence logically. (b) Support claim(s) with logical reasoning and relevant, accurate data and evidence that demonstrate an understanding of the topic or text, using credible sources. (c) Use words, phrases, and clauses to create cohesion and clarify the relationships among claim(s), counterclaims, reasons, and evidence. (d) Establish and maintain a formal style. (e) Provide a concluding statement or section that follows from and supports the argument presented. - LAFS.68.WHST.1.1 Use measures of variability to describe a data set. - 6.DP.8.5 Select and use appropriate statistical measures. - 6.DP.8.6 Summarize numerical data sets. - 6.DP.8.7 Identify and write statistical questions. - 6.DP.8.1 Identify the mean, median, mode, and range of a data set. - 6.DP.8.2 Make and interpret box plots. - 6.DP.8.3 Use models to visualize the relationship between division and multiplication to divide decimals by 2-digit whole numbers. - 5.NC.6.4 Use models to divide a decimal by a decimal. - 5.NC.6.5 Apply V = l · w · h and V = Bh to find the volume of right rectangular prisms. - 6.GM.A.2b Understand that the volume of a right rectangular prism can be found by filling the prism with multiple layers of the base. - 6.GM.A.2a Convert measurements within a single system of measurement: customary (i.e., in., ft., yd., oz., lb., sec., min., hr.) or metric (i.e., mm, cm, m, km, g, kg, mL, L) from a larger to a smaller unit and a smaller to a larger unit. - 5.MDA.1 Create a line plot consisting of unit fractions and use operations on fractions to solve problems related to the line plot. - 5.MDA.2 Divide multi-digit whole numbers and decimals to the hundredths place using up to two-digit divisors and four-digit dividends, and justify the solution. - 5.NBT.A.8 x + p = q in which p, q and x are all nonnegative rational numbers; and, - NC.6.EE.7.a Use place value understanding to round decimals to any place. - 5.NBT.A.4 Round numbers from billions to thousandths place. - 5.NBT.A.5 Add and subtract multi-digit whole numbers and decimals to the thousandths place, and justify the solution. - 5.NBT.A.6 p ∙ x = q for cases in which p, q and x are all nonnegative rational numbers. - NC.6.EE.7.b Multiply multi-digit whole numbers and decimals to the hundredths place, and justify the solution. - 5.NBT.A.7 Understand signs of numbers in ordered pairs as indicating locations in quadrants of the Cartesian coordinate plane. - 6.GM.A.3a 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. - MAFS.6.SP.2.5d 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. - MAFS.6.SP.2.5c Construct polygons in the Cartesian coordinate plane. - 6.GM.A.3d Find distances between points with the same first coordinate or the same second coordinate. - 6.GM.A.3c Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. - MAFS.6.SP.2.5b Reporting the number of observations. - MAFS.6.SP.2.5a Recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. - 6.GM.A.3b Locate positive rational numbers on a number line and plot pairs of positive rational numbers on a coordinate grid. - 6.1.1.1 Multiply a fraction (including mixed numbers) by a fraction. - M05.A-F.2.1.2 Divide unit fractions by whole numbers and whole numbers by unit fractions. - M05.A-F.2.1.4 Add and subtract integers; use efficient and generalizable procedures including but not limited to standard algorithms. - 6.N.2.3 Solve real-world and mathematical problems requiring addition, subtraction, multiplication and division of multi-digit whole numbers. Use various strategies, including the inverse relationships between operations, the use of technology, and the context of the problem to assess the reasonableness of results. - 5.1.1.4 Estimate solutions to arithmetic problems in order to assess the reasonableness of results. - 5.1.1.3 Write expressions that record operations with numbers and with letters standing for numbers. - NY-6.EE.2a Evaluate expressions given specific values for their variables. Include expressions that arise from formulas in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order (Order of Operations). - NY-6.EE.2c Identify parts of an expression using mathematical terms (term, coefficient, sum, difference, product, factor, and quotient); view one or more parts of an expression as a single entity. - NY-6.EE.2b Write, read, and evaluate algebraic expressions. - NC.6.EE.2 Write and evaluate numerical expressions, with and without grouping symbols, involving whole-number exponents. - NC.6.EE.1 Identify when two expressions are equivalent and justify with mathematical reasoning. - NC.6.EE.4 Apply the properties of operations to generate equivalent expressions without exponents. - NC.6.EE.3 Analyze the choice of measures of center and variability based on the shape of the data distribution and/or the context of the data. - 6.DSP.B.5d Use variables to represent numbers and write expressions when solving a real-world or mathematical problem. - NC.6.EE.6 Make sense of problems and persevere in solving them. - 6.MP.1 Recognize and draw a net for a three-dimensional figure. - 5.3.1.2 Use substitution to determine whether a given number in a specified set makes an equation true. - NC.6.EE.5 Reason abstractly and quantitatively. - 6.MP.2 Describe and classify three-dimensional figures including cubes, prisms and pyramids by the number of edges, faces or vertices as well as the types of faces. - 5.3.1.1 Consider the context in which a problem is situated to select the most useful form of the quotient for the solution and use the context to interpret the quotient appropriately. - 5.1.1.2 Divide multi-digit numbers, using efficient and generalizable procedures, based on knowledge of place value, including standard algorithms. Recognize that quotients can be represented in a variety of ways, including a whole number with a remainder, a fraction or mixed number, or a decimal. - 5.1.1.1 Estimate solutions to problems with whole numbers, fractions and decimals and use the estimates to assess the reasonableness of results in the context of the problem. - 6.1.3.5 Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers. - 6.1.3.4 Calculate the percent of a number and determine what percent one number is of another number to solve problems in various contexts. - 6.1.3.3 Use the meanings of fractions, multiplication, division and the inverse relationship between multiplication and division to make sense of procedures for multiplying and dividing fractions. - 6.1.3.2 Multiply and divide decimals and fractions, using efficient and generalizable procedures, including standard algorithms. - 6.1.3.1 Evaluate numerical expressions involving grouping symbols (i.e., parentheses, brackets, braces). - 5.ATO.1 Translate verbal phrases into numerical expressions and interpret numerical expressions as verbal phrases. - 5.ATO.2 Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations. - 6.2.1.2 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. - MAFS.5.NBT.2.6 Understand that a variable can be used to represent a quantity that can change, often in relationship to another changing quantity. Use variables in various contexts. - 6.2.1.1 Add, subtract, multiply, and divide decimals to hundredths, 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. - MAFS.5.NBT.2.7 Fluently multiply multi-digit whole numbers using the standard algorithm. - MAFS.5.NBT.2.5 Convert measurements of capacity, length and weight within a given measurement system. - 5.GM.D.8 Explain and justify which measure of central tendency (mean, median, or mode) would provide the most descriptive information for a given set of data. - 6.D.1.2 Use appropriate tools strategically. - 6.MP.5 Calculate the mean, median, and mode for a set of real-world data. - 6.D.1.1 Attend to precision. - 6.MP.6 Construct viable arguments and critique the reasoning of others. - 6.MP.3 Model with mathematics. - 6.MP.4 Report the number of observations. - 6.DSP.B.5a 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. - MAFS.K12.MP.1.1 Write, evaluate and interpret numeric expressions using the order of operations. - 5.RA.B.3 Give 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 of the data. - 6.DSP.B.5c Look for and make use of structure. - 6.MP.7 Translate written expressions into algebraic expressions. - 5.RA.B.4 Create and analyze box and whisker plots observing how each segment contains one quarter of the data. - 6.D.1.3 Look for and express regularity in repeated reasoning. - 6.MP.8 Describe the nature of the attribute under investigation, including how it was measured and its units of measurement. - 6.DSP.B.5b Analyze the relationship between quantities in different representations (context, equations, tables, and graphs). - NC.6.EE.9.b Describe quantities having opposite directions or values. - NC.6.NS.5.a Represent quantities in real-world contexts, explaining the meaning of 0 in each situation. - NC.6.NS.5.b Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. - MAFS.5.NF.2.4.b Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. - 6.EE.C.9 Summarize numerical data sets in relation to their context. - 6.SP.B.5 Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation. - 6.NS.3 Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). - 6.NS.4 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 3/4-cup 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? - 6.NS.1 Display numerical data in plots on a number line, including dot plots, histograms, and box plots. - 6.SP.B.4 Fluently divide multi-digit numbers using the standard algorithm. - 6.NS.2 Solve real-world 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. - 6.NS.8 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 real-world contexts, explaining the meaning of 0 in each situation. - 6.NS.5 Create and interpret circle graphs. - 6.DSP.B.4b Use dot plots, histograms and box plots to display and interpret numerical data. - 6.DSP.B.4a Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” - MAFS.6.RP.1.1 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” - MAFS.6.RP.1.2 Investigate and translate among multiple representations of rational numbers (fractions, decimal numbers, percentages). Fractions should be limited to those with denominators of 2, 3, 4, 5, 8, 10, and 100. - 6.NS.9 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. - MAFS.6.RP.1.3 Using variables to represent two quantities in a real-world or mathematical context that change in relationship to one another. - NC.6.EE.9.a Write and/or solve equations of the form x+p = q and px = q in which p and q are rational numbers. - 7.EEI.B.4a Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. - MAFS.6.EE.1.2b Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s³ and A = 6s² to find the volume and surface area of a cube with sides of length s = 1/2. - MAFS.6.EE.1.2c Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y. - MAFS.6.EE.1.2a Model addition and subtraction of fractions and decimals using a variety of representations. - 5.1.3.2 Add and subtract decimals and fractions, using efficient and generalizable procedures, including standard algorithms. - 5.1.3.1 Solve real-world and mathematical problems requiring addition and subtraction of decimals, fractions and mixed numbers, including those involving measurement, geometry and data. - 5.1.3.4 Estimate sums and differences of decimals and fractions to assess the reasonableness of results. - 5.1.3.3 Find the area of a rectangle using fractions and diagrams. - 5.NC.8.6 Use models, equations and previously learned strategies to multiply mixed numbers. - 5.NC.8.7 Use a rule or table to represent ordered pairs of positive integers and graph these ordered pairs on a coordinate system. - 5.2.1.2 Convert between customary and metric units. - 6.P.5.10 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., x-axis and x-coordinate, y-axis and y-coordinate). - 5.G.1 Use ratio and rate reasoning to solve real-world and mathematical problems. - NY-6.RP.3 Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. - 5.G.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio relationship.Note: Expectations for unit rates in this grade are limited to non-complex fractions. - NY-6.RP.2 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.3 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. - NY-6.RP.1 Classify two-dimensional figures in a hierarchy based on properties. - 5.G.4 Generate equivalent expressions and evaluate expressions involving positive rational numbers by applying the commutative, associative, and distributive properties and order of operations to solve real-world and mathematical problems. - 6.A.2.1 Apply the formulas V = l × w × h and V = B × h for volume of right rectangular prisms with whole-number edge lengths. - 5.GM.B.5 Write or select an algebraic expression that represents a real-world situation. - MAFS.6.EE.1.AP.2a Find and position rational numbers on a horizontal or vertical number line. - NC.6.NS.6.a.2 Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. - 6.EE.A.3 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. - 6.EE.A.4 Recognize opposite signs of numbers as indicating locations on opposite sides of 0 and that the opposite of the opposite of a number is the number itself. - NC.6.NS.6.a.1 Write and evaluate numerical expressions involving whole-number exponents. - 6.EE.A.1 Write, read, and evaluate expressions in which letters stand for numbers. - 6.EE.A.2 Interpret absolute value as magnitude for a positive or negative quantity in a real-world context. - NC.6.NS.5.c.1 Distinguish comparisons of absolute value from statements about order. - NC.6.NS.5.c.2 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., x-axis and x-coordinate, y-axis and y-coordinate). - MAFS.5.G.1.1 Solve numerical expressions involving whole-number bases and exponents (e.g., 5 + 24 x 6 = 101) - MAFS.6.EE.1.AP.1a Represent real world and mathematical problems by graphing points in the first quadrant of the coordinate plane, and interpret coordinate values of points in the context of the situation. - MAFS.5.G.1.2 Identify what an exponent represents (e.g., 8³= 8 x 8 x 8). - MAFS.6.EE.1.AP.1b 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. - MAFS.6.SP.1.2 Convert metric units of length. - 5.MD.12.4 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. - MAFS.6.SP.1.1 Convert customary units of weight. - 5.MD.12.3 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. - MAFS.6.SP.1.3 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 to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2. - 5.NF.A.2 Convert metric units of capacity. - 5.MD.12.5 Fluently multiply multi-digit whole numbers using strategies to include a standard algorithm. - 5.NSBT.5 Round decimals to any given place value within thousandths. - 5.NSBT.4 Read and write decimals in standard and expanded form. Compare two decimal numbers to the thousandths using the symbols >, =, or <. - 5.NSBT.3 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They 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. - MAFS.K12.MP.2.1 Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. - MAFS.5.MD.3.5 Add, subtract, multiply, and divide decimal numbers to hundredths using concrete area models and drawings. - 5.NSBT.7 Divide up to a four-digit dividend by a two-digit divisor, using strategies based on place value, the properties of operations, and the relationship between multiplication and division. - 5.NSBT.6 Evaluate whether sides of an equation are equal using models - MAFS.6.EE.1.AP.4a Divide multi-digit whole numbers by a single-digit number. - MAFS.6.NS.2.AP.2a Divide multi-digit whole numbers by a two-digit number with the quotient having no remainders. - MAFS.6.NS.2.AP.2b Use properties to produce equivalent expressions. - MAFS.6.EE.1.AP.3a Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons. - MAFS.5.NBT.1.3.b Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2). - MAFS.6.NS.2.4 Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. - MAFS.6.NS.2.3 Fluently divide multi-digit numbers using the standard algorithm. - MAFS.6.NS.2.2 Solve one-step, addition, subtraction, multiplication, or division problems involving decimals whose place value ranges from the thousand to the thousandths places. - MAFS.6.NS.2.AP.3a Solve problems involving division of fractions by fractions. - 6.NS.A.1a Understand that if any solutions exist, the solution set for an equation or inequality consists of values that make the equation or inequality true. - 6.EEI.B.5 Write and solve equations using variables to represent quantities, and understand the meaning of the variable in the context of the situation. - 6.EEI.B.6 Solve one-step linear equations in one variable involving non-negative rational numbers. - 6.EEI.B.7 Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. - MAFS.5.OA.1.2 Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols. - MAFS.5.OA.1.1 Use substitution to determine whether a given number in a specified set makes a one-variable equation or inequality true. - 6.EEI.B.4 Write an inequality of the form 𝑉𝑙𝑤𝑉𝐵𝑥>𝑉𝑙𝑤𝑉𝐵𝑥𝑐 or 𝑉𝑙𝑤𝑉𝐵𝑥𝑐𝑥<𝑉𝑙𝑤𝑉𝐵𝑥𝑐𝑥𝑐 and graph the solution set on a number line. - 6.EEI.8a Recognize that inequalities have infinitely many solutions. - 6.EEI.8b Understand that a mean is a measure of center that represents a balance point or fair share of a data set and can be influenced by the presence of extreme values within the data set. - NC.6.SP.3.a.1 Understand the median as a measure of center that is the numerical middle of an ordered data set. - NC.6.SP.3.a.2 Illustrate multiplication and division of fractions and decimals to show connections to fractions, whole number multiplication, and inverse relationships. - 6.N.4.2 Multiply and divide fractions and decimals using efficient and generalizable procedures. - 6.N.4.3 Solve and interpret real-world and mathematical problems including those involving money, measurement, geometry, and data requiring arithmetic with decimals, fractions and mixed numbers. - 6.N.4.4 Analyze the relationship between independent and dependent variables using graphs and tables. - 6.EEI.9b Translate among graphs, tables, and equations. - 6.EEI.9c Estimate solutions to problems with whole numbers, decimals, fractions, and mixed numbers and use the estimates to assess the reasonableness of results in the context of the problem. - 6.N.4.1 Write an equation that models a relationship between independent and dependent variables. - 6.EEI.9a Use distances between two points that are either vertical or horizontal to each other (not requiring the distance formula) to solve real-world and mathematical problems about congruent two-dimensional figures. - 6.GM.4.3 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, two-way 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. - MAFS.K12.MP.4.1 Convert among different-sized standard measurement units (i.e., km, m, cm; kg, g; lb, oz.; l, ml; hr, min, sec) within a given measurement system (e.g., convert 5 cm to 0.05 m), and use these conversions in solving multi-step, real world problems. - MAFS.5.MD.1.1 Find the area of special quadrilaterals and polygons by decomposing into triangles or rectangles. - NC.6.G.1.b Find the area of triangles by composing into rectangles and decomposing into right triangles. - NC.6.G.1.a Relate volume to the operations of multiplication and addition and solve real world and mathematical problems involving volume. Find the volume of a right rectangular prism with whole-number side lengths by packing it with unit cubes, and show that the volume is the same as would be found by multiplying the edge lengths, equivalently by multiplying the height by the area of the base. Represent threefold whole-number products as volumes, e.g., to represent the associative property of multiplication. - 5.MD.C.5a Recognize volume as additive. Find volumes of solid figures composed of two non-overlapping right rectangular prisms by adding the volumes of the non-overlapping parts, applying this technique to solve real world problems. - 5.MD.C.5c Apply the formulas V = l × w × h and V = b × h for rectangular prisms to find volumes of right rectangular prisms with whole-number edge lengths in the context of solving real world and mathematical problems. - 5.MD.C.5b Classify quadrilaterals by their properties. - 5.G.16.2 Evaluate an algebraic expression with whole numbers, decimals, and fractions. - 6.AF.3.5 Use variables to write algebraic expressions. - 6.AF.3.4 Combine like terms in algebraic expressions. - 6.AF.3.7 Recognize the relationship between multiplying fractions and finding the areas of rectangles with fractional side lengths. - 5.NF.B.7a Identify and write equivalent algebraic expressions. - 6.AF.3.6 Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins? - 5.NF.B.7c Determine the measure of center of a data set and understand that it is a single number that summarizes all the values of that data set. - NC.6.SP.3.a Understand that describing the variability of a data set is needed to distinguish between data sets in the same scale, by comparing graphical representations of different data sets in the same scale that have similar measures of center, but different spreads. - NC.6.SP.3.b Write and evaluate numbers with exponents. - 6.AF.3.1 Use the order of operations to evaluate numerical expressions. - 6.AF.3.3 Write the prime factorization and find the greatest common factor and the least common multiple of two numbers. - 6.AF.3.2 Find the area of polygons by composing or decomposing the shapes into rectangles or triangles. - 6.GM.A.1 Mathematical Modeling: Use Positive Rational Numbers - 6.NC.1MM Plot the data. - MAFS.6.SP.2.AP.5b Define the mean, mode, and range of the data. - MAFS.6.SP.2.AP.5c Collect real-world data by surveying. - MAFS.6.SP.2.AP.5a Define the second number in an ordered pair as the vertical distance from the origin. - 5.GM.C.6d Recognize that a statistical question is one that anticipates variability in the data related to the question and accounts for it in the answers. - NY-6.SP.1a Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation 'Subtract y from 5' as 5 - y. - 6.EE.A.2a Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2 (8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms. - 6.EE.A.2b Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^3 and A = 6s^2 to find the volume and surface area of a cube with sides of length s = 1/2. - 6.EE.A.2c Represent the axes as scaled perpendicular number lines that both intersect at 0, the origin. - 5.GM.C.6a Identify any point on the Cartesian coordinate plane by its ordered pair coordinates. - 5.GM.C.6b Define the first number in an ordered pair as the horizontal distance from the origin. - 5.GM.C.6c Find areas of trapezoids and kites. - 6.G.7.3 Find the areas of triangles. - 6.G.7.2 Represent solid figures using nets. - 6.G.7.5 Find the areas of polygons. - 6.G.7.4 Use the distributive property to express the sum of two whole numbers. - MAFS.6.NS.2.AP.4c Use what I know about areas of rectangles to find the areas of parallelograms and rhombuses. - 6.G.7.1 Find the greatest common factor of two numbers that are less than or equal to 50. - MAFS.6.NS.2.AP.4a Find the least common multiple of two whole numbers that are less than or equal to 10. - MAFS.6.NS.2.AP.4b Display data on a line plot, such as dot plots, histograms or box plots. - MAFS.6.SP.2.AP.4a Add, subtract, multiply, and divide decimals to hundredths, using concrete models (to include, but not limited to: base ten blocks, decimal tiles, etc.) 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. - 5.NBT.7 Understand that ratios can be expressed as equivalent unit ratios by finding and interpreting both unit ratios in context. - NC.6.RP.2 Draw a net of a pyramid and use it to find the pyramid's surface area. - 6.G.7.7 Draw a net of a prism and use it to find the prism's surface area. - 6.G.7.6 Find the volume of a rectangular prism with fractional edge lengths. - 6.G.7.8 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. - MAFS.K12.MP.6.1 Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all non-negative rational numbers. - MAFS.6.EE.2.7 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. - MAFS.6.EE.2.8 Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true. - MAFS.6.EE.2.5 Find the area of a rectangle with fractional side lengths by tiling it with unit squares of the appropriate unit fraction side lengths, and show that the area is the same as would be found by multiplying the side lengths. Multiply fractional side lengths to find areas of rectangles, and represent fraction products as rectangular areas. - 5.NF.B.4b Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set. - MAFS.6.EE.2.6 Explain patterns in the number of zeros of the product when multiplying a number by powers of 10, and explain patterns in the placement of the decimal point when a decimal is multiplied or divided by a power of 10. Use whole-number exponents to denote powers of 10. - 5.NBT.2 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrat