Organization: SAVVAS Product Name: envisionmath2.0 Custom Grades 6-8 Grade 7 Virginia Product Version: 1 Source: IMS Online Validator Profile: 1.2.0 Identifier: realize-470bb4e4-fd77-3488-9728-377f778e6431 Timestamp: Thursday, December 17, 2020 11:35 AM EST 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: Divide rational numbers. - 7.NC.1.9 Divide integers. - 7.NC.1.8 Multiply rational numbers. - 7.NC.1.7 Multiply integers. - 7.NC.1.6 Add and subtract rational numbers. - 7.NC.1.5 Mathematical Modeling: Probability - 7.DP.7MM Subtract integers. - 7.NC.1.4 Represent real-world or mathematical situations using equations and inequalities involving variables and rational numbers. - 7.A.3.3 Add integers. - 7.NC.1.3 Represent, write, solve, and graph problems leading to linear inequalities with one variable in the form x + p > q and x + p < q, where p, and q are nonnegative rational numbers. - 7.A.3.2 Recognize rational numbers and write them in decimal form. - 7.NC.1.2 Write and solve problems leading to linear equations with one variable in the form px + q = r and p(x + q) = r where p, q, and r are rational numbers. - 7.A.3.1 Relate integers, their opposites, and their absolute values. - 7.NC.1.1 Solve equations resulting from proportional relationships in various contexts. - 7.2.4.2 Represent relationships in various contexts with equations involving variables and positive and negative rational numbers. Use the properties of equality to solve for the value of a variable. Interpret the solution in the original context. - 7.2.4.1 Test for equivalent ratios to decide whether quantities are in a proportional relationship. - 7.P.2.3 Reporting the number of observations. - M.7.20a Use the constant of proportionality in an equation to represent a proportional relationship. - 7.P.2.4 Use ratio concepts and reasoning to solve multi-step problems. - 7.P.2.1 Find unit rates with ratios of fractions and use them to solve problems. - 7.P.2.2 Use a graph to determine whether two quantities are proportional. - 7.P.2.5 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. Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. - M.7.20c Determine whether a relationship is proportional and use representations to solve problems. - 7.P.2.6 Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. - M.7.20b The student will determine unknown side lengths or angle measures of quadrilaterals. - 7.6b The student will compare and contrast quadrilaterals based on their properties. - 7.6a Use proportions and ratios to solve problems involving scale drawings and conversions of measurement units. - 7.3.2.3 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 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. - MAFS.7.EE.2.3 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. - MAFS.7.EE.2.4 Describe the likelihood that an event will occur. - 7.DP.7.1 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. - M.7.10 Simulate a compound event to approximate its probability. - 7.DP.7.7 Find the probability of a compound event. - 7.DP.7.6 Determine the experimental probability of an event. - 7.DP.7.3 Determine the theoretical probability of an event. - 7.DP.7.2 Find all possible outcomes of a compound event. - 7.DP.7.5 Construct viable arguments and critique the reasoning of others. Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. - MAFS.K12.MP.3.1 Use probability models to find probabilities of events. - 7.DP.7.4 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. (e.g., 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.) - M.7.18 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. - M.7.17 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. - M.7.16 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. - M.7.15 The student will solve practical problems involving operations with rational numbers. - 7.2 The student will solve single-step and multistep practical problems, using proportional reasoning. - 7.3 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. - M.7.14 Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - M.7.13 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. - M.7.12 The student will solve problems, including practical problems, involving the relationship between corresponding sides and corresponding angles of similar quadrilaterals and triangles. - 7.5 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. - M.7.11 The student will apply translations and reflections of right triangles or rectangles in the coordinate plane. - 7.7 The student, given data in a practical situation, will represent data in a histogram. - 7.9a 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. - M.7.19 Use proportional reasoning to assess the reasonableness of solutions. - 7.A.2.4 Use proportional reasoning to solve real-world and mathematical problems involving ratios. - 7.A.2.3 Solve multi-step problems involving proportional relationships involving distance-time, percent increase or decrease, discounts, tips, unit pricing, similar figures, and other real-world and mathematical situations. - 7.A.2.2 Represent proportional relationships with tables, verbal descriptions, symbols, and graphs; translate from one representation to another. Determine and compare the unit rate (constant of proportionality, slope, or rate of change) given any of these representations. - 7.A.2.1 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. (e.g., 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.) - M.7.21 Use proportions to solve percent problems. - 7.P.3.2 Represent and solve percent problems using equations. - 7.P.3.3 Understand, find, and analyze percents of numbers. - 7.P.3.1 Apply percent reasoning to solve simple interest problems. - 7.P.3.6 Solve problems involving percent change and percent error. - 7.P.3.4 Solve problems involving percent markup and markdown. - 7.P.3.5 The student will investigate and describe the difference between the experimental probability and theoretical probability of an event. - 7.8b The student will determine the theoretical and experimental probabilities of an event. - 7.8a 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. (e.g., 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.) - M.7.24 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 ½ indicates an event that is neither unlikely nor likely and a probability near 1 indicates a likely event. - M.7.23 Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. (e.g., 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.) - M.7.22 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. - MAFS.7.G.2.5 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. - MAFS.7.G.2.6 Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients. - MAFS.7.EE.1.1 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.” - MAFS.7.EE.1.2 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. - MAFS.7.G.2.4 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 Reason abstractly and quantitatively. Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. - MAFS.K12.MP.2.1 The student, given data in a practical situation, will compare histograms with the same data represented in stem-and-leaf plots, line plots, and circle graphs. - 7.9c The student, given data in a practical situation, will make observations and inferences about data represented in a histogram. - 7.9b 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). - M.7.2a Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation. Focus special attention on the points (0, 0) and (1, r) where r is the unit rate. - M.7.2d Represent proportional relationships by equations. (e.g., 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.) - M.7.2c Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams and verbal descriptions of proportional relationships. - M.7.2b Solve inequalities using multiplication or division. - 7.AF.5.5 Determine what the cross section looks like when a 3D figure is sliced. - 7.G.8.7 Solve problems involving the area of a circle. - 7.G.8.6 Write and solve two-step inequalities. - 7.AF.5.6 Use the Distributive Property to solve equations. - 7.AF.5.3 Solve problems involving radius, diameter, and circumference of circles. - 7.G.8.5 Solve problems involving angle relationships. - 7.G.8.4 Solve inequalities using addition or subtraction. - 7.AF.5.4 Use the area of the base of a three-dimensional figure to find its volume. - 7.G.8.9 Solve inequalities that require multiple steps. - 7.AF.5.7 Find the area and surface area of 2-dimensional composite shapes and 3-dimensional prisms. - 7.G.8.8 Represent a problem with a two-step equation. - 7.AF.5.1 Solve a problem with a two-step equation. - 7.AF.5.2 Solve problems with rational numbers. - 7.NC.1.10 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. - MAFS.7.NS.1.2d Apply properties of operations as strategies to multiply and divide rational numbers. - MAFS.7.NS.1.2c 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. - MAFS.7.G.1.2 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. - MAFS.7.NS.1.2b Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids. - MAFS.7.G.1.3 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. - MAFS.7.NS.1.2a 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. - MAFS.7.G.1.1 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. - MAFS.7.SP.3.6 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. - MAFS.7.SP.3.7 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. - MAFS.7.SP.3.5 Draw triangles when given information about their side lengths and angle measures. - 7.G.8.3 Draw figures with given conditions. - 7.G.8.2 Use the key in a scale drawing to find missing measures. - 7.G.8.1 Use appropriate tools strategically. Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. - MAFS.K12.MP.5.1 Mathematical Modeling: Generate Equivalent Expressions - 7.AF.4MM Recognize and generate equivalent representations of positive and negative rational numbers, including equivalent fractions. - 7.1.1.5 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 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 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. - M.7.4c 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. (i.e., To add “p + q” on the number line, start at “0” and move to “p” then move |q| 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. - M.7.4b Apply properties of operations as strategies to add and subtract rational numbers. - MAFS.7.NS.1.1d Describe situations in which opposite quantities combine to make 0. (e.g., A hydrogen atom has 0 charge because its two constituents are oppositely charged.) - M.7.4a 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. - MAFS.7.NS.1.1c 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. - MAFS.7.NS.1.1b 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. - MAFS.7.NS.1.1a Mathematical Modeling: Solve Problems Involving Geometry - 7.G.8MM Apply understanding of order of operations and grouping symbols when using calculators and other technologies. - 7.A.4.2 Use properties of operations (limited to associative, commutative, and distributive) to generate equivalent numerical and algebraic expressions containing rational numbers, grouping symbols and whole number exponents. - 7.A.4.1 Mathematical Modeling: Use Sampling to Draw Inferences about Populations - 7.DP.6MM Apply properties of operations as strategies to add and subtract rational numbers. - M.7.4d Add expressions that represent real-world problems. - 7.AF.4.6 Subtract expressions using properties of operations. - 7.AF.4.7 Expand expressions using the Distributive Property. - 7.AF.4.4 Use common factors and the Distributive Property to factor expressions. - 7.AF.4.5 Use an equivalent expression to find new information. - 7.AF.4.8 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. - MAFS.7.RP.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. - MAFS.7.RP.1.1 Recognize and represent proportional relationships between quantities. - MAFS.7.RP.1.2 Write equivalent expressions for given expressions. - 7.AF.4.2 Use properties of operations to simplify expressions. - 7.AF.4.3 Write and evaluate algebraic expressions. - 7.AF.4.1 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. - M.7.5d Apply properties of operations as strategies to multiply and divide rational numbers. - M.7.5c 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. - M.7.5b 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. - M.7.5a Make inferences about a population from a sample data set. - 7.DP.6.2 Determine if a sample is representative of a population. - 7.DP.6.1 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. - MAFS.7.SP.2.3 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. - MAFS.7.SP.2.4 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 The student will graph a line representing a proportional relationship between two quantities given the slope and an ordered pair, or given the equation in y = mx form where m represents the slope as rate of change. - 7.10b The student will determine the y-intercept, b, in an additive relationship between two quantities and write an equation in the form y = x + b to represent the relationship. - 7.10c The student will graph a line representing an additive relationship between two quantities given the y-intercept and an ordered pair, or given the equation in the form y = x + b, where b represents the y-intercept. - 7.10d The student will make connections between and among representations of a proportional or additive relationship between two quantities using verbal descriptions, tables, equations, and graphs. - 7.10e Compare populations using the mean, median, mode, range, interquartile range, and mean absolute deviation. - 7.DP.6.4 Draw comparative inferences about two populations using median and interquartile range (IQR). - 7.DP.6.3 The student will determine the slope, m, as rate of change in a proportional relationship between two quantities and write an equation in the form y = mx to represent the relationship. - 7.10a 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 Estimate solutions to multiplication and division of integers in order to assess the reasonableness of results. - 7.N.2.1 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? - MAFS.7.EE.2.4a Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. - MAFS.7.RP.1.2b 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. - MAFS.7.RP.1.2a Illustrate multiplication and division of integers using a variety of representations. - 7.N.2.2 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. - MAFS.7.RP.1.2d 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. - MAFS.7.RP.1.2c 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. - MAFS.7.EE.2.4b The student will solve one- and two-step linear inequalities in one variable, including practical problems, involving addition, subtraction, multiplication, and division, and graph the solution on a number line. - 7.13 The student will solve two-step linear equations in one variable, including practical problems that require the solution of a two-step linear equation in one variable. - 7.12 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. - MAFS.7.SP.1.1 The student will evaluate algebraic expressions for given replacement values of the variables. - 7.11 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. - MAFS.7.SP.1.2 Look for and make use of structure. Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x² + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y. - MAFS.K12.MP.7.1 Explain the relationship between the absolute value of a rational number and the distance of that number from zero on a number line. Use the symbol for absolute value. - 7.N.2.6 Solve real-world and mathematical problems involving addition, subtraction, multiplication and division of rational numbers; use efficient and generalizable procedures including but not limited to standard algorithms. - 7.N.2.3 Solve real-world and mathematical problems involving the four operations with rational numbers. - MAFS.7.NS.1.3 Recognize and represent proportional relationships between quantities. - M.7.2 Use proportional relationships to solve multistep ratio and percent problems (e.g., simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, and/or percent error). - M.7.3 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, 𝑥 and 𝑥𝑦, is proportional if it can be expressed in the form 𝑥𝑦𝑦/𝑥𝑦𝑦𝑥 = k or 𝑥𝑦𝑦𝑥𝑦 = k𝑥𝑦𝑦𝑥𝑦𝑥. Distinguish proportional relationships from other relationships, including inversely proportional relationships (xy = k or y = k/x). - 7.2.1.1 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. - MAFS.7.SP.3.7a 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? - MAFS.7.SP.3.7b Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. (e.g., If a person walks ½ mile in each ¼ hour, compute the unit rate as the complex fraction ½ /¼ miles per hour, equivalently 2 miles per hour.) - M.7.1 Solve real-world and mathematical problems involving the four operations with rational numbers. Instructional Note: Computations with rational numbers extend the rules for manipulating fractions to complex fractions. - M.7.6 Apply properties of operations as strategies to add, subtract, factor and expand linear expressions with rational coefficients. - M.7.7 Use reasoning with proportions to display and interpret data in circle graphs (pie charts) and histograms. Choose the appropriate data display and know how to create the display using a spreadsheet or other graphing technology. - 7.D.1.2 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. - MAFS.7.SP.3.8a 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. - MAFS.7.SP.3.8b 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? - MAFS.7.SP.3.8c The student will compare and order rational numbers. - 7.1c 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 The student will compare and order numbers greater than zero written in scientific notation. - 7.1b 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. (e.g., a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”) - M.7.8 The student will identify and describe absolute value of rational numbers. - 7.1e 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. (e.g., 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.) - M.7.9 The student will determine square roots of perfect squares. - 7.1d Use proportional reasoning to solve problems involving ratios in various contexts. - 7.1.2.5 Solve problems in various contexts involving calculations with positive and negative rational numbers and positive integer exponents, including computing simple and compound interest. - 7.1.2.4 Mathematical Modeling: Solve Problems Using Equations and Inequalities - 7.AF.5MM Demonstrate an understanding of the relationship between the absolute value of a rational number and distance on a number line. Use the symbol for absolute value. - 7.1.2.6 Add, subtract, multiply and divide positive and negative rational numbers that are integers, fractions and terminating decimals; use efficient and generalizable procedures, including standard algorithms; raise positive rational numbers to whole-number exponents. - 7.1.2.1 Understand that calculators and other computing technologies often truncate or round numbers. - 7.1.2.3 The student will investigate and describe the concept of negative exponents for powers of ten. - 7.1a Use real-world contexts and the inverse relationship between addition and subtraction to explain why the procedures of arithmetic with negative rational numbers make sense. - 7.1.2.2 Solve multi-step problems involving proportional relationships in numerous contexts. - 7.2.2.2 Represent proportional relationships with tables, verbal descriptions, symbols, equations and graphs; translate from one representation to another. Determine the unit rate (constant of proportionality or slope) given any of these representations. - 7.2.2.1 Mathematical Modeling: Analyze and Solve Percent Problems - 7.P.3MM Represent real-world or mathematical situations using equations and inequalities involving variables and positive and negative rational numbers. - 7.2.2.4 Recognize that the graph of a proportional relationship is a line through the origin and the coordinate (1, r), where both r and the slope are the unit rate (constant of proportionality, k). - 7.A.1.2 Use knowledge of proportions to assess the reasonableness of solutions. - 7.2.2.3 Describe that the relationship between two variables, x and y, is proportional if it can be expressed in the form y/x = k or y = kx; distinguish proportional relationships from other relationships, including inversely proportional relationships ( xy = k or y = k/x ). - 7.A.1.1 Calculate probability as a fraction of sample space or as a fraction of area. Express probabilities as percents, decimals and fractions. - 7.D.2.2 Determine the theoretical probability of an event using the ratio between the size of the event and the size of the sample space; represent probabilities as percents, fractions and decimals between 0 and 1. - 7.D.2.1 Use proportional reasoning to draw conclusions about and predict relative frequencies of outcomes based on probabilities. - 7.D.2.3 Make sense of problems and persevere in solving them. Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. - MAFS.K12.MP.1.1 Describe the properties of similarity, compare geometric figures for similarity, and determine scale factors resulting from dilations. - 7.GM.4.1 Apply proportions, ratios, and scale factors to solve problems involving scale drawings and determine side lengths and areas of similar triangles and rectangles. - 7.GM.4.2 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 Recognize and generate equivalent representations of rational numbers, including equivalent fractions. - 7.N.1.3 Know that every rational number can be written as the ratio of two integers or as a terminating or repeating decimal. - 7.N.1.1 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 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. (e.g., 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.) - M.7.10b 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. (e.g., The perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? An arithmetic solution similar to “54 – 6 – 6 divided by 2” may be compared with the reasoning involved in solving the equation 2w – 12 = 54. An arithmetic solution similar to “54/2 – 6” may be compared with the reasoning involved in solving the equation 2(w – 6) = 54.) - M.7.10a 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 The student will solve problems, including practical problems, involving the volume and surface area of rectangular prisms and cylinders. - 7.4b The student will describe and determine the volume and surface area of rectangular prisms and cylinders. - 7.4a Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x – 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x² + x + 1), and (x – 1)(x³ + x² + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. - MAFS.K12.MP.8.1 Calculate the circumference and area of circles to solve problems in various contexts, in terms of ! and using approximations for π. - 7.GM.3.2 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 and can be approximated by rational numbers such as 22/7 and 3.14. - 7.GM.3.1 List of all Files Validated: imsmanifest.xml I_000b3fef-985e-3c21-8044-3a7470f29a2e_1_R/BasicLTI.xml I_000f1050-9f7a-3d6f-87bf-3830284426f9_R/BasicLTI.xml I_0026ec1b-f503-30ae-bbd9-861cd5ca3d47_R/BasicLTI.xml I_0030f685-2236-3b29-9bc3-4a6939df001e_R/BasicLTI.xml I_003ac775-0bc9-31a2-85c1-7262a03d27f4_1_R/BasicLTI.xml I_00414ad3-b9c2-3262-929c-250bfd8a3bce_R/BasicLTI.xml I_004a20c2-ec3a-3c64-bdfb-89b24dd17eae_1_R/BasicLTI.xml I_0074686d-c68d-3234-9f0e-80cc44d37bf7_R/BasicLTI.xml 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