Organization: SAVVAS Product Name: enVision Mathematics 2021 Accelerated Grade 7 Product Version: 1 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-ec9c7dee-e072-323e-812a-bb5d0f3697d7 Timestamp: Tuesday, July 27, 2021 11:29 AM 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 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 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 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 Solve unit rate problems including those involving unit pricing and constant speed. - M06.A-R.1.1.4 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 Use informal arguments to establish that the sum of the interior angles of a triangle is 180 degrees. - 7A.38a 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 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 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 Evaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000). - 7A.15a Explain that the square root of a non-perfect square is irrational. - 7A.15b Compute the least common multiple (LCM) of two numbers both less than or equal to 12. - 6.NS.4b 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 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 Write, interpret, and explain statements of order for rational numbers in real-world contexts. - NC.6.NS.7.b 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 Know and apply the properties of integer exponents to generate equivalent numerical expressions. - NY-8.EE.1 Perform multiplication and division with numbers expressed in scientific notation, including problems where both standard decimal form and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. Interpret scientific notation that has been generated by technology. - NY-8.EE.4 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. - NY-8.EE.5 Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = p, where p is a positive rational number. Know square roots of perfect squares up to 225 and cube roots of perfect cubes up to 125. Know that the square root of a non-perfect square is irrational. - NY-8.EE.2 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. - NY-8.EE.3 Apply the properties of operations to generate equivalent expressions. - M06.B-E.1.1.5 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. - 7A.16a Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities. - 7A.16b 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 Interpret scientific notation that has been generated by technology. - 7A.16c Write and evaluate numerical expressions involving whole-number exponents. - M06.B-E.1.1.1 Understand, explain, and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3² × 3-5 = 3-3 = 1/3³ = 1/27. - 8.EE.1 Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational. - 8.EE.2 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 × 108; and the population of the world as 7 × 109; and determine that the world population is more than 20 times larger. - 8.EE.3 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are useUse scientific notation and choose units of appropriate size for measurements of very large or very small quantities, e.g., use millimeters per year for seafloor spreading. Interpret scientific notation that has been generated by technology. - 8.EE.4 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 8.EE.5 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - 8.EE.6 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 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - NY-8.EE.6 Solve linear equations in one variable. - NY-8.EE.7 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 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 The student will recognize and represent patterns with whole number exponents and perfect squares. - 6.4 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 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 7A.12 Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers. - 7A.11 Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions. - 7A.14 Generate expressions in equivalent forms based on context and explain how the quantities are related. - 7A.13 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 Define the real number system as composed of rational and irrational numbers. - 7A.10 Apply properties of operations as strategies to multiply and divide rational numbers. - 7.NS.A.2c Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats. - 7.NS.A.2d Create equations in two variables to represent relationships between quantities in context; graph equations on coordinate axes with labels and scales and use them to make predictions. Limit to contexts arising from linear functions. - 7A.19 Estimate and compare very large or very small numbers in scientific notation. - 7A.16 Use square root and cube root symbols to represent solutions to equations. - 7A.15 Use variables to represent quantities in a real-world or mathematical problem and construct algebraic expressions, equations, and inequalities to solve problems by reasoning about the quantities. - 7A.18 Solve multi-step real-world and mathematical problems involving rational numbers (integers, signed fractions and decimals), converting between forms as needed. Assess the reasonableness of answers using mental computation and estimation strategies. - 7A.17 Find all possible outcomes of a compound event. - A7.DP.9.5 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 Find the probability of a compound event. - A7.DP.9.6 Use multiplicative reasoning and representations to solve ratio and unit rate problems. - 6.N.3.4 Determine the experimental probability of an event. - A7.DP.9.3 Use probability models to find probabilities of events. - A7.DP.9.4 Apply the Pythagorean Theorem to determine unknown side lengths of right triangles, including real-world applications. - 8A.50 Describe the likelihood that an event will occur. - A7.DP.9.1 Determine the theoretical probability of an event. - A7.DP.9.2 Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. - 7.NS.A.2a Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real- world contexts. - 7.NS.A.2b Explain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates. - 7A.10a Convert a decimal expansion that repeats into a rational number. - 7A.10b Simulate a compound event to approximate its probability. - A7.DP.9.7 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem 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 Identify and use ratios to compare quantities. Recognize that multiplicative comparison and additive comparison are different. - 6.N.3.1 Write algebraic expressions to represent real-world or mathematical problems. - M06.B-E.2.1.2 Determine the unit rate for ratios. - 6.N.3.2 Describe and analyze distributions. - 7.SP.3 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 Solve problems in various contexts involving conversion of weights, capacities, geometric measurements and times within measurement systems using appropriate units. - 6.3.3.1 Broaden statistical reasoning by using the GAISE model. - 7.SP.2 Summarize quantitative data sets in relation to their context. - NY-6.SP.5 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihooA probability near 0 indicates an unlikely event; a probability around 1/2 indicates an event that is neither unlikely nor likely; and a probability near 1 indicates a likely event. - 7.SP.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 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. - 7.SP.7 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 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. - 7.SP.6 Determine the surface area of triangular and rectangular prisms (including cubes). Formulas will be provided. - M06.C-G.1.1.6 Apply the Pythagorean Theorem to find the distance between two points in a coordinate plane. - 8A.49 Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. - 7.1.1.5 Informally justify the Pythagorean Theorem and its converse. - 8A.48 Understand that division of two integers will always result in a rational number. Use this information to interpret the decimal result of a division problem when using a calculator. - 7.1.1.2 Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. Recognize that π is not rational, but that it can be approximated by rational numbers such as 22/7 and 3.14. - 7.1.1.1 Understand that statistics can be used to gain information about a population by examining a sample of the population. - 7.SP.1 Compare positive and negative rational numbers expressed in various forms using the symbols <, >, =, ≤, ≥. - 7.1.1.4 Locate positive and negative rational numbers on the number line, understand the concept of opposites, and plot pairs of positive and negative rational numbers on a coordinate grid. - 7.1.1.3 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 Construct geometric shapes (freehand, using a ruler and a protractor, and using technology), given a written description or measurement constraints with an emphasis on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - 7A.34 Solve problems involving scale drawings of geometric figures, including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale. - 7A.33 Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle. - 7A.36 Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections. - 7A.35 Define and develop a probability model, including models that may or may not be uniform, where uniform models assign equal probability to all outcomes and non-uniform models involve events that are not equally likely. - 7A.30 Find probabilities of simple and compound events through experimentation or simulation and by analyzing the sample space, representing the probabilities as percents, decimals, or fractions. - 7A.32 Approximate the probability of an event using data generated by a simulation (experimental probability) and compare it to the theoretical probability. - 7A.31 Analyze and apply properties of parallel lines cut by a transversal to determine missing angle measures. - 7A.38 Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure. - 7A.37 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms. - 7A.39 Explain in context the meaning of a point (x,y) on the graph of a proportional relationship, with special attention to the points (0,0) and (1, r) where r is the unit rate. - 7A.2c Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book. - 7.SP.B.4 Identify the constant of proportionality (unit rate) and express the proportional relationship using multiple representations including tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - 7A.2b Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. - 7.G.A.1 Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - 7.G.A.2 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable. - 7.SP.B.3 Use equivalent ratios displayed in a table or in a graph of the relationship in the coordinate plane to determine whether a relationship between two quantities is proportional. - 7A.2a Use dot plots, histograms, and box plots to represent data. - NC.6.SP.4.a Describe the two-dimensional figures that result from slicing three- dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - 7.G.A.3 Informally derive the formula for area of a circle. - 7A.36a Solve area and circumference problems in real-world and mathematical situations involving circles. - 7A.36b Understand that rewriting an expression in different forms in real-world and mathematical problems can reveal and explain how the quantities are related. - NY-7.EE.2 Construct a function to model the linear relationship between two variables. - 7A.23 Analyze and describe the properties of prisms and pyramids. - 5.GM.A.3 Add, subtract, factor, and expand linear expressions with rational coefficients by applying the properties of operations. - NY-7.EE.1 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - NY-7.EE.4 Solve multi-step real-world and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate. Assess the reasonableness of answers using mental computation and estimation strategies. - NY-7.EE.3 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 Solve multi-step linear equations in one variable, including rational number coefficients, and equations that require using the distributive property and combining like terms. - 7A.21 Represent constraints by equations and/or inequalities, and interpret solutions as viable or nonviable options in a modeling context. Limit to contexts arising from linear. - 7A.20 Mathematical Modeling: Use Sampling to Draw Inferences about Populations - A7.DP.8MM 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 Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. - 7A.27 Examine a sample of a population to generalize information about the population. - 7A.26 Use a number from 0 to 1 to represent the probability of a chance event occurring, explaining that larger numbers indicate greater likelihood of the event occurring, while a number near zero indicates an unlikely event. - 7A.29 Make informal comparative inferences about two populations using measures of center and variability and/or mean absolute deviation in context. - 7A.28 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 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 a fraction by another fraction. - 6.NC.1.5 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 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 Add, subtract, and multiply decimals. - 6.NC.1.1 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 Mathematical Modeling: Solve Problems Using Equations and Inequalities - A7.AF.6MM Collect and use data to predict probabilities of events. - 7A.30a Compare probabilities from a model to observed frequencies, explaining possible sources of discrepancy. - 7A.30b 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 Find the y-intercept of a graph and explain what it means. - A7.AF.7.8 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 Derive the equation y = mx + b. - A7.AF.7.9 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 Solve equations with variables on both sides of the equal sign. - A7.AF.7.2 Solve multistep equations and pairs of equations using more than one approach. - A7.AF.7.3 Solve equations that have like terms on one side. - A7.AF.7.1 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 Understand the slope of a line. - A7.AF.7.6 Write equations to describe linear relationships. - A7.AF.7.7 Determine the number of solutions an equation has. - A7.AF.7.4 Compare proportional relationships represented in different ways. - A7.AF.7.5 In a problem context, understand that rewriting an expression in an equivalent form can reveal and explain properties of the quantities represented by the expression and can reveal how those quantities are relateFor example, a discount of 15% (represented by p − 0.15p) is equivalent to (1 − 0.15)p, which is equivalent to 0.85p or finding 85% of the original price. - 7.EE.2 Write an equation to express the relationship between the dependent and independent variables. - M06.B-E.3.1.1 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 7.EE.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 Observe the relative frequency of an event over the long run, using simulation or technology, and use those results to predict approximate relative frequency. - 7A.31a 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 Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - 7.EE.4 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 Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 93/4 inches long in the center of a door that is 271/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. - 7.EE.3 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 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 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 Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.” - 7.EE.A.2 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - 7.EE.A.1 Use measures of center and measures of variability for quantitative data from random samples or populations to draw informal comparative inferences about the populations. - NY-7.SP.4 Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams, and determine the probability of an event by finding the fraction of outcomes in the sample space for which the compound event occurred. - 7A.32a Design and use a simulation to generate frequencies for compound events. - 7A.32b Construct and interpret box-plots, find the interquartile range, and determine if a data point is an outlier. - NY-7.SP.1 Informally assess the degree of visual overlap of two quantitative data distributions. - NY-7.SP.3 Represent events described in everyday language in terms of outcomes in the sample space which composed the event. - 7A.32c Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts. - 7.NS.A.1b Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - 7.NS.A.1c Apply properties of operations as strategies to add and subtract rational numbers. - 7.NS.A.1d 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 Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. - 7.NS.A.1a 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 Evaluate expressions at specific values of their variables using expressions that arise from formulas used in real-world problems. - NC.6.EE.2.c Find probabilities of compound events using organized lists, sample space tables, tree diagrams, and simulation. - NY-7.SP.8 Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? - 7.EE.4a Solve word problems leading to inequalities of the form px +q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example, as a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions. - 7.EE.4b 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 Understand and write an inequality that describes a real-world situation. - 6.AF.4.6 Identify dependent and independent variables. - 6.AF.4.8 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. - 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 Write and evaluate numerical expressions involving whole-number exponents. - M.6.12 Understand the meaning of the variable in the context of the situation. - 6.EEI.A.2e Write and evaluate algebraic expressions. - 6.EEI.A.2d 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 Determine if a value for a variable makes an equation true. - 6.AF.4.1 Use a sequence of translations, reflections, and rotations to show that figures are congruent. - A7.G.11.5 Dilate two-dimensional figures. - A7.G.11.6 Rotate a two-dimensional figure. - A7.G.11.3 Describe and perform a sequence of transformations. - A7.G.11.4 Translate two-dimensional figures. - A7.G.11.1 Reflect two-dimensional figures. - A7.G.11.2 Use unit rates to convert metric measurements. - 6.P.5.9 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 Solve problems involving rates. - 6.P.5.5 Compare unit rates to solve problems. - 6.P.5.6 Find the interior and exterior angle measures of a triangle. - A7.G.11.9 Giving quantitative measures of center, describing variability, and any overall pattern, and noting any striking deviations. - NC.6.SP.5.b.1 Use a sequence of transformations, including dilations, to show that figures are similar. - A7.G.11.7 Use a ratio to describe the relationship between two quantities. - 6.P.5.1 Identify and find the measures of angles formed by parallel lines and a transversal. - A7.G.11.8 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. - 8.EE.B.5 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 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b. - 8.EE.B.6 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number. - M.6.27 Mathematical Modeling: Solve Problems Involving Geometry - A7.G.10MM 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 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 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 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 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 Graph proportional relationships. - 7A.5 Summarize numerical data sets in relation to the context. - 6.DSP.B.5 Determine whether a relationship between two variables is proportional or non-proportional. - 7A.4 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 proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems. - 7A.7 Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept. - 7A.6 Solve real-world and mathematical problems involving the four operations of rational numbers, including complex fractions. Apply properties of operations as strategies where applicable. - 7A.9 Justify the choices for measure of center and measure of variability based on the shape of the distribution. - 6.DS.5f Apply and extend knowledge of operations of whole numbers, fractions, and decimals to add, subtract, multiply, and divide rational numbers including integers, signed fractions, and decimals. - 7A.8 Display and interpret data. - 6.DSP.B.4 Finding the whole, given a part and the percent. - NC.6.RP.4.c 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 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 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 Understanding and finding a percent of a quantity as a ratio per 100. - NC.6.RP.4.a Calculate unit rates of length, area, and other quantities measured in like or different units that include ratios or fractions. - 7A.1 Write and evaluate numerical expressions involving whole-number exponents. - NY-6.EE.1 Represent a relationship between two quantities and determine whether the two quantities are related proportionally. - 7A.2 Write, read, and evaluate expressions in which letters stand for numbers. - NY-6.EE.2 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 Compute and interpret quotients of positive fractions. - 6.NS.A.1 Determine whether a relationship is proportional and use representations to solve problems. - A7.P.3.6 Use a graph to determine whether two quantities are proportional. - A7.P.3.5 Use the constant of proportionality in an equation to represent a proportional relationship. - A7.P.3.4 Test for equivalent ratios to decide whether quantities are in a proportional relationship. - A7.P.3.3 Find unit rates with ratios of fractions and use them to solve problems. - A7.P.3.2 Use ratio concepts and reasoning to solve multi-step problems. - A7.P.3.1 Find the volumes of cones. - A7.G.13.3 Find the volume of a sphere and use it to solve problems. - A7.G.13.4 Find the surface areas of cylinders, cones, and spheres. - A7.G.13.1 Use what I know about finding volumes of rectangular prisms to find the volume of a cylinder. - A7.G.13.2 Use scientific notation to write very large or very small quantities. - A7.NC.2.9 Estimate large and small quantities using a power of 10. - A7.NC.2.8 Write a number with a negative or zero exponent a different way. - A7.NC.2.7 Use the properties of exponents to write equivalent expressions. - A7.NC.2.6 Solve equations involving squares or cubes. - A7.NC.2.5 Find square roots and cube roots of rational numbers. - A7.NC.2.4 Recognize when linear equations in one variable have one solution, infinitely many solutions, or no solutions. Give examples and show which of these possibilities is the case by successively transforming the given equation into simpler forms. - NY-8.EE.7a Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and combining like terms. - NY-8.EE.7b Compare and order rational and irrational numbers. - A7.NC.2.3 Identify a number that is irrational. - A7.NC.2.2 Design simple experiments and collect data. Determine mean, median and range for quantitative data and from data represented in a display. Use these quantities to draw conclusions about the data, compare different data sets, and make predictions. - 7.4.1.1 Write repeating decimals as fractions. - A7.NC.2.1 Describe the impact that inserting or deleting a data point has on the mean and the median of a data set. Know how to create data displays using a spreadsheet to examine this impact. - 7.4.1.2 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 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 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 Model with mathematics. - MP.4 Use appropriate tools strategically. - MP.5 Attend to precision. - MP.6 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 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 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 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 Write and evaluate numerical expressions involving whole-number exponents. - MAFS.6.EE.1.1 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 Mathematical Modeling: Analyze and Solve Percent Problems - A7.P.4MM 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 Understand that the absolute value of a rational number is its distance from 0 on the number line. - 6.NS.C.7 Mathematical Modeling: Generate Equivalent Expressions - A7.AF.5MM 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 Know and apply the properties of positive and negative integer exponents to generate equivalent numerical expressions. - 8.1.1.4 Express approximations of very large and very small numbers using scientific notation; understand how calculators display numbers in scientific notation. Multiply and divide numbers expressed in scientific notation, express the answer in scientific notation, using the correct number of significant digits when physical measurements are involved. - 8.1.1.5 Classify real numbers as rational or irrational. Know that when a square root of a positive integer is not an integer, then it is irrational. Know that the sum of a rational number and an irrational number is irrational, and the product of a non-zero rational number and an irrational number is irrational. - 8.1.1.1 Compare real numbers; locate real numbers on a number line. Identify the square root of a positive integer as an integer, or if it is not an integer, locate it as a real number between two consecutive positive integers. - 8.1.1.2 Determine rational approximations for solutions to problems involving real numbers. - 8.1.1.3 Use random numbers generated by a calculator or a spreadsheet or taken from a table to simulate situations involving randomness, make a histogram to display the results, and compare the results to known probabilities. - 7.4.3.1 Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions. - 7.4.3.2 Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on probabilities. - 7.4.3.3 Use the Distributive Property to solve equations. - A7.AF.6.3 Solve inequalities using addition or subtraction. - A7.AF.6.4 Represent a problem with a two-step equation. - A7.AF.6.1 Solve a problem with a two-step equation. - A7.AF.6.2 Solve inequalities that require multiple steps. - A7.AF.6.7 Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - 8.G.B.7 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - 8.G.B.8 Solve inequalities using multiplication or division. - A7.AF.6.5 Explain a proof of the Pythagorean Theorem and its converse. - 8.G.B.6 Write and solve two-step inequalities. - A7.AF.6.6 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 Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. - 7.SP.C.7 Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event. - 7.SP.C.5 Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. - 7.SP.C.6 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 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 Verify experimentally parallel lines are mapped to parallel lines. - NY-8.G.1c Verify experimentally angles are mapped to angles of the same measure. - NY-8.G.1b 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 Understand that the graph of a proportional relationship is a line through the origin whose slope is the unit rate (constant of proportionality). Know how to use graphing technology to examine what happens to a line when the unit rate is changed. - 7.2.1.2 Understand that a relationship between two variables, 𝑉𝑙𝑤𝑉𝐵𝑥 and 𝑉𝑙𝑤𝑉𝐵𝑥𝑦, is proportional if it can be expressed in the form 𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦/𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦𝑥 = k or 𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦𝑥𝑦 = k𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦𝑥𝑦𝑥. Distinguish proportional relationships from other relationships, including inversely proportional relationships (xy = k or y = k/x). - 7.2.1.1 Recognize and represent proportional relationships between quantities. - 7.RP.2 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2) /(1/4) miles per hour, equivalently 2 miles per hour. - 7.RP.1 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. - 7.RP.3 Verify experimentally lines are mapped to lines, and line segments to line segments of the same length. - NY-8.G.1a Understand signs of numbers in ordered pairs as indicating locations in quadrants of the Cartesian coordinate plane. - 6.GM.A.3a 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 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 Understand that a function is linear if it can be expressed in the form f(x) = mx + b or if its graph is a straight line. - 8.2.1.3 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 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 Use variables to represent numbers and write expressions when solving a real-world or mathematical problem. - NC.6.EE.6 Use substitution to determine whether a given number in a specified set makes an equation true. - NC.6.EE.5 Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - 7A.18a 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 word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality, and interpret it in the context of the problem. - 7A.18b Solve real-world and mathematical problems requiring arithmetic with decimals, fractions and mixed numbers. - 6.1.3.4 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 Represent the relationship between two varying quantities with function rules, graphs and tables; translate between any two of these representations. - 6.2.1.2 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 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 Model with mathematics. - 6.MP.4 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 Create and analyze box and whisker plots observing how each segment contains one quarter of the data. - 6.D.1.3 Analyze the relationship between quantities in different representations (context, equations, tables, and graphs). - NC.6.EE.9.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 Use properties of algebra to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. Properties of algebra include associative, commutative and distributive laws. - 7.2.3.1 Summarize numerical data sets in relation to their context. - 6.SP.B.5 Apply understanding of order of operations and grouping symbols when using calculators and other technologies. - 7.2.3.3 Evaluate algebraic expressions containing rational numbers and whole number exponents at specified values of their variables. - 7.2.3.2 Fluently add, subtract, multiply, and divide multi-digit decimals using the 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 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 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 Calculate the volume and surface area of cylinders and justify the formulas used. - 7.3.1.2 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 Demonstrate an understanding of the proportional relationship between the diameter and circumference of a circle and that the unit rate (constant of proportionality) is π. Calculate the circumference and area of circles and sectors of circles to solve problems in various contexts. - 7.3.1.1 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 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 Evaluate algebraic expressions, including expressions containing radicals and absolute values, at specified values of their variables. - 8.2.3.1 Justify steps in generating equivalent expressions by identifying the properties used, including the properties of algebra. Properties include the associative, commutative and distributive laws, and the order of operations, including grouping symbols. - 8.2.3.2 Know that real numbers are either rational or irrational. Understand informally that every number has a decimal expansion which is repeating, terminating, or is non-repeating and non-terminating. - 8.NS.1 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions, e.g., π². For example, by truncating the decimal expansion of √2, , show that √2, is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. - 8.NS.2 Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood? - 7.SP.C.8c Use ratio and rate reasoning to solve real-world and mathematical problems. - NY-6.RP.3 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, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - 7.SP.C.8a Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. - NY-6.RP.1 Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., “rolling double sixes”), identify the outcomes in the sample space which compose the event. - 7.SP.C.8b Use data from a random sample to draw inferences about a population with an unknown characteristic of interest, generating multiple samples to gauge variation and making predictions or conclusions about the population. - 7A.26d Differentiate between a sample and a population. - 7A.26a Compare sampling techniques to determine whether a sample is random and thus representative of a population, explaining that random sampling tends to produce representative samples and support valid inferences. - 7A.26b Determine whether conclusions and generalizations can be made about a population based on a sample. - 7A.26c Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. - 7.RP.A.2c Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - 7.RP.A.2b Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. - 7.RP.A.2a 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 Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies? - 7.SP.C.7b Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. - 7.RP.A.2d Represent proportional relationships within and between similar figures. - 7.G.1b Compute actual lengths and areas from a scale drawing and reproduce a scale drawing at a different scale. - 7.G.1a Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. - 7.SP.C.7a Focus on constructing quadrilaterals with given conditions noticing types and properties of resulting quadrilaterals and whether it is possible to construct different quadrilaterals using the same conditions. - 7.G.2b Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - 7.G.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 Write and evaluate numerical expressions involving whole-number exponents. - 6.EE.A.1 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line, and estimate the value of expressions. - NY-8.NS.2 Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion eventually repeats. Know that other numbers that are not rational are called irrational. - NY-8.NS.1 Interpret absolute value as magnitude for a positive or negative quantity in a real-world context. - NC.6.NS.5.c.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. - MAFS.6.SP.1.2 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 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 Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour. - 7.RP.A.1 Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error. - 7.RP.A.3 Recognize and represent proportional relationships between quantities. - 7.RP.A.2 Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs. - NY-7.SP.8a Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. - 7.EE.B.4 Represent sample spaces for compound events using methods such as organized lists, sample space tables, and tree diagrams.For an event described in everyday language, identify the outcomes in the sample space which compose the event. - NY-7.SP.8b Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. - 7.EE.B.3 Design and use a simulation to generate frequencies for compound events. - NY-7.SP.8c Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences. - 7.SP.A.1 Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be. - 7.SP.A.2 Mathematical Modeling: Probability - A7.DP.9MM Parallel lines are taken to parallel lines. - 8.G.A.1c 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 Angles are taken to angles of the same measure. - 8.G.A.1b Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. - MAFS.6.NS.2.3 Lines are taken to lines, and line segments to line segments of the same length. - 8.G.A.1a Interpret the rate of change (slope) and initial value of the linear function from a description of a relationship or from two points in a table or graph. - 7A.23a Solve problems involving division of fractions by fractions. - 6.NS.A.1a Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. - NY-7.NS.2 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 Solve real-world and mathematical problems involving the four operations with rational numbers.Note: Computations with rational numbers extend the rules for manipulating fractions to complex fractions limited to (a/b)/(c/d) where a, b, c, and d are integers and b, c, and d ≠ 0. - NY-7.NS.3 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 Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers. Represent addition and subtraction on a horizontal or vertical number line. - NY-7.NS.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 𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦𝑥𝑦𝑥𝑥>𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦𝑥𝑦𝑥𝑥𝑐 or 𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦𝑥𝑦𝑥𝑥𝑐𝑥<𝑉𝑙𝑤𝑉𝐵𝑥𝑦𝑦𝑥𝑦𝑥𝑥𝑐𝑥𝑐 and graph the solution set on a number line. - 6.EEI.8a 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 Recognize that inequalities have infinitely many solutions. - 6.EEI.8b 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 Compare populations using the mean, median, mode, range, interquartile range, and mean absolute deviation. - A7.DP.8.4 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 Make inferences about a population from a sample data set. - A7.DP.8.2 Draw comparative inferences about two populations using median and interquartile range (IQR). - A7.DP.8.3 Determine if a sample is representative of a population. - A7.DP.8.1 Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. - NY-7.EE.4a Solve word problems leading to inequalities of the form px + q > r, px + q ≥ r, px + q ≤ r, or px + q < r, where p, q, and r are rational numbers. Graph the solution set of the inequality on the number line and interpret it in the context of the problem. - NY-7.EE.4b 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 Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number. - 8.NS.A.1 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 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., pi^2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations. - 8.NS.A.2 Represent a proportional relationship using an equation. - NY-7.RP.2c Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. - NY-7.RP.2d 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 Decide whether two quantities are in a proportional relationship. - NY-7.RP.2a Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - NY-7.RP.2b 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 Solve real-world and mathematical problems involving area, volume, and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. - 7.G.6 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - 7.G.3 Draw (freehand, with ruler and protractor, and with technology) geometric figures with given conditions. - 7.G.2 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - 7.G.5 Work with circles. - 7.G.4 Solve problems involving similar figures with right triangles, other triangles, and special quadrilaterals. - 7.G.1 Evaluate an algebraic expression with whole numbers, decimals, and fractions. - 6.AF.3.5 Identify and write equivalent algebraic expressions. - 6.AF.3.6 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 Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. - 7.G.B.6 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - NY-7.G.5 Apply the formulas for the area and circumference of a circle to solve problems. - NY-7.G.4 Given a pair of two-dimensional figures, determine if a series of rigid motions maps one figure onto the other, recognizing that if such a sequence exists the figures are congruent; describe the transformation sequence that verifies a congruence relationship. - 7A.42a Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids.Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids.Find the volume of right triangular prisms, and solve volume problems involving three-dimensional objects composed of right rectangular prisms. - NY-7.G.6 Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. - NY-7.G.1 Write and evaluate numbers with exponents. - 6.AF.3.1 Describe the two-dimensional shapes that result from slicing three-dimensional solids parallel or perpendicular to the base. - NY-7.G.3 Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle. - 7.G.B.4 Draw triangles when given measures of angles and/or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle. - NY-7.G.2 Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure. - 7.G.B.5 Write the prime factorization and find the greatest common factor and the least common multiple of two numbers. - 6.AF.3.2 Solve real-world and mathematical problems involving the four operations with rational numbers. Computations with rational numbers extend the rules for manipulating fractions to complex fractions. - 7.NS.3 Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers). - 8.EE.7a Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. - 8.G.C.9 Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms. - 8.EE.7b Find the area of polygons by composing or decomposing the shapes into rectangles or triangles. - 6.GM.A.1 Use proportional relationships to solve multistep ratio and percent problems. - NY-7.RP.3 Compute unit rates associated with ratios of fractions. - NY-7.RP.1 Recognize and represent proportional relationships between quantities. - NY-7.RP.2 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 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 solid figures using nets. - 6.G.7.5 Solve problems with rational numbers. - A7.NC.1.10 Use what I know about areas of rectangles to find the areas of parallelograms and rhombuses. - 6.G.7.1 Understand that ratios can be expressed as equivalent unit ratios by finding and interpreting both unit ratios in context. - NC.6.RP.2 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 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 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 Determine whether linear equations in one variable have one solution, no solution, or infinitely many solutions of the form x = a, a = a, or a = b (where a and b are different numbers). - 7A.21a Represent and solve real-world and mathematical problems with equations and interpret each solution in the context of the problem. - 7A.21b Given the formulas for the volume of cones, cylinders, and spheres, solve mathematical and real-world problems. - NY-8.G.9 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system. - NY-8.G.8 Using coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. - NC.6.G.3.b Drawing polygons in the coordinate plane given coordinates for the vertices. - NC.6.G.3.a Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions. - NY-8.G.7 Apply properties of operations as strategies to add and subtract rational numbers. - NY-7.NS.1d 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.5 Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. - NY-7.NS.1c Write expressions using variables to represent quantities in real-world and mathematical situations. Understand the meaning of the variable in the context of t