Lesson 1
Introduction To Ratio TablesLesson 2
Summary PreparationLesson 5
Understanding A Gallery ProblemLesson 6
Gallery Problems Exercise
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 Subject:
 Mathematics
 Material Type:
 Full Course
 Level:
 Middle School
 Grade:
 7
 Provider:
 Pearson
 Tags:

 2D
 3D
 7th Grade Mathematics
 Absolute Value
 Addition
 Algebraic Expressions
 Angles
 Architecture
 Area
 Assessment
 BranchED SC
 Building
 Buildings
 Causality
 Circles
 Circumference
 Concepts
 Cones
 Cooking
 Cube Structure
 Cubes
 Cylinders
 Decimals
 Design
 Diameter
 Dimensions
 Distance
 Distributive Property
 Division
 Equations
 Equilateral
 Estimation
 Experimental Probability
 Expressions
 Expresssions
 Folding Paper
 Formulas
 Fractions
 Goals
 Graphing
 Group Projects
 Height
 Independent and Dependent Events
 Inequalities
 Integers
 Iowa K12 ECurriculum
 Law of Large Numbers
 Likelihood
 Mean Absolute Deviation
 Measurements and Data
 Missing Angles
 Models
 Money
 Multiplication
 Multiplication Property of Equality
 Negative Numbers
 Number Line
 Obtuse
 Operations
 Paper Folding
 Parallelograms
 Percentages
 Perimeter
 Pi
 Planar Slices
 Polygons
 Presentations
 Prisms
 Probability
 ProblemSolving
 Problems.
 Projects.
 Proportional Relationship
 Proportional Relationships
 Proportionality
 Proportions
 Pyramids
 Quadrilaterals
 Rain
 Rates
 Ratio
 Rational Numbers
 Ratios
 Rectangles
 Rhombus
 Right Prisms
 Sampling
 Scale
 Scale Drawing
 Slope
 Strouhal Numbers
 Subtraction
 Surface Area
 Taxation
 Theoretical Probability
 Time
 Travel
 Triangles
 Variables
 Volume
 Weight
 License:
 Creative Commons Attribution NonCommercial
 Language:
 English
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Learning Domain: Ratio and Proportional Relationships
Standard: Compute unit rates associated with ratios of fractions to solve realworld and mathematical problems.
Learning Domain: Ratios and Proportional Relationships
Standard: Compute unit rates, including those involving complex fractions, with like or different units.
Learning Domain: Ratios and Proportional Relationships
Standard: Recognize and represent proportional relationships between quantities.
Learning Domain: Ratios and Proportional Relationships
Standard: Decide whether two quantities in a table or graph are in a proportional relationship.
Learning Domain: Ratios and Proportional Relationships
Standard: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Learning Domain: Ratios and Proportional Relationships
Standard: Represent proportional relationships with equations.
Learning Domain: Ratios and Proportional Relationships
Standard: 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.
Learning Domain: Ratios and Proportional Relationships
Standard: Solve multistep real world and mathematical problems involving ratios and percentages.
Learning Domain: The Number System
Standard: 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. Explore the real number system and its appropriate usage in realworld situations.
Learning Domain: The Number System
Standard: 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.
Learning Domain: Expressions and Equations
Standard: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Learning Domain: Expressions and Equations
Standard: 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%"ť is the same as "multiply by 1.05."ť
Learning Domain: Expressions and Equations
Standard: Solve multistep reallife 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 as strategies 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.
Learning Domain: Expressions and Equations
Standard: Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Learning Domain: Expressions and Equations
Standard: 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?
Learning Domain: Expressions and Equations
Standard: 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.
Learning Domain: Geometry
Standard: 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.
Learning Domain: Geometry
Standard: 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.
Learning Domain: Geometry
Standard: Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Learning Domain: Geometry
Standard: 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.
Learning Domain: Geometry
Standard: Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.
Learning Domain: Geometry
Standard: Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Learning Domain: The Number System
Standard: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Learning Domain: The Number System
Standard: 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.
Learning Domain: The Number System
Standard: 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 realworld contexts.
Learning Domain: The Number System
Standard: 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 realworld contexts.
Learning Domain: The Number System
Standard: Apply properties of operations as strategies to add and subtract rational numbers.
Learning Domain: The Number System
Standard: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Learning Domain: The Number System
Standard: 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 realworld contexts.
Learning Domain: The Number System
Standard: Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero 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 realworld contexts.
Learning Domain: The Number System
Standard: Apply properties of operations as strategies to multiply and divide rational numbers.
Learning Domain: The Number System
Standard: 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.
Learning Domain: The Number System
Standard: Solve realworld and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
Learning Domain: Ratios and Proportional Relationships
Standard: 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.
Learning Domain: Ratios and Proportional Relationships
Standard: Recognize and represent proportional relationships between quantities.
Learning Domain: Ratios and Proportional Relationships
Standard: 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.
Learning Domain: Ratios and Proportional Relationships
Standard: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Learning Domain: Ratios and Proportional Relationships
Standard: 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.
Learning Domain: Ratios and Proportional Relationships
Standard: 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.
Learning Domain: Ratios and Proportional Relationships
Standard: 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.
Learning Domain: Statistics and Probability
Standard: 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.
Learning Domain: Statistics and Probability
Standard: Use random sampling to draw inferences about a population. 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.
Learning Domain: Statistics and Probability
Standard: 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.
Learning Domain: Statistics and Probability
Standard: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventhgrade science book are generally longer than the words in a chapter of a fourthgrade science book.
Learning Domain: Statistics and Probability
Standard: 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.
Learning Domain: Statistics and Probability
Standard: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Learning Domain: Statistics and Probability
Standard: 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.
Learning Domain: Statistics and Probability
Standard: 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.
Learning Domain: Statistics and Probability
Standard: Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land openend down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Learning Domain: Statistics and Probability
Standard: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Learning Domain: Statistics and Probability
Standard: 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.
Learning Domain: Statistics and Probability
Standard: 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"ť), identify the outcomes in the sample space which compose the event.
Learning Domain: Statistics and Probability
Standard: 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?
Learning Domain: The Number System
Standard: 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.
Learning Domain: The Number System
Standard: 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., ěŰ^2). For example, by truncating the decimal expansion of ‰ö_2 (square root of 2), show that ‰ö_2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.
Learning Domain: Mathematical Practices
Standard: 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?"ť They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.
Learning Domain: Mathematical Practices
Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize"Óto 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"Óand 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.
Learning Domain: Mathematical Practices
Standard: 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"Óif there is a flaw in an argument"Óexplain 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.
Learning Domain: Mathematical Practices
Standard: 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, twoway 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.
Learning Domain: Mathematical Practices
Standard: 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.
Learning Domain: Mathematical Practices
Standard: 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.
Learning Domain: Mathematical Practices
Standard: 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 x 8 equals the well remembered 7 x 5 + 7 x 3, in preparation for learning about the distributive property. In the expression x^2 + 9x + 14, older students can see the 14 as 2 x 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.
Learning Domain: Mathematical Practices
Standard: 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)(x^2 + x + 1), and (x  1)(x^3 + x^2 + 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.
Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them.
Standard: 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.
Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them.
Standard: 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.
Cluster: Draw, construct, and describe geometrical figures and describe the relationships between them.
Standard: Describe the twodimensional figures that result from slicing threedimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.
Cluster: Solve reallife and mathematical problems involving angle measure, area, surface area, and volume
Standard: 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.
Cluster: Solve reallife and mathematical problems involving angle measure, area, surface area, and volume
Standard: Use facts about supplementary, complementary, vertical, and adjacent angles in a multistep problem to write and solve simple equations for an unknown angle in a figure.
Cluster: Solve reallife and mathematical problems involving angle measure, area, surface area, and volume
Standard: Solve realworld and mathematical problems involving area, volume and surface area of two and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
Cluster: Analyze proportional relationships and use them to solve realworld and mathematical problems
Standard: 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.
Cluster: Analyze proportional relationships and use them to solve realworld and mathematical problems
Standard: Recognize and represent proportional relationships between quantities.
Cluster: Analyze proportional relationships and use them to solve realworld and mathematical problems
Standard: 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.
Cluster: Analyze proportional relationships and use them to solve realworld and mathematical problems
Standard: Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
Cluster: Analyze proportional relationships and use them to solve realworld and mathematical problems
Standard: 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.
Cluster: Analyze proportional relationships and use them to solve realworld and mathematical problems
Standard: 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.
Cluster: Analyze proportional relationships and use them to solve realworld and mathematical problems
Standard: 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.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: 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.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: 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 realworld contexts.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: 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 realworld contexts.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: Apply properties of operations as strategies to add and subtract rational numbers.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: 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 realworld contexts.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with nonzero 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 realworld contexts.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: Apply properties of operations as strategies to multiply and divide rational numbers.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: 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.
Cluster: Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers
Standard: Solve realworld and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
Cluster: Use properties of operations to generate equivalent expressions
Standard: Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
Cluster: Use properties of operations to generate equivalent expressions
Standard: 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.”
Cluster: Solve reallife and mathematical problems using numerical and algebraic expressions and equations
Standard: Solve multistep reallife 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 as strategies 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.
Cluster: Solve reallife and mathematical problems using numerical and algebraic expressions and equations
Standard: Use variables to represent quantities in a realworld or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.
Cluster: Solve reallife and mathematical problems using numerical and algebraic expressions and equations
Standard: 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?
Cluster: Solve reallife and mathematical problems using numerical and algebraic expressions and equations
Standard: 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.
Cluster: Use random sampling to draw inferences about a population
Standard: 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.
Cluster: Use random sampling to draw inferences about a population
Standard: Use random sampling to draw inferences about a population. 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.
Cluster: Draw informal comparative inferences about two populations
Standard: 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.
Cluster: Draw informal comparative inferences about two populations
Standard: Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventhgrade science book are generally longer than the words in a chapter of a fourthgrade science book.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: 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.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its longrun relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: 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.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: 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.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land openend down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: 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.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: 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.
Cluster: Investigate chance processes and develop, use, and evaluate probability models
Standard: 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?
Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers
Standard: 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.
Cluster: Know that there are numbers that are not rational, and approximate them by rational numbers
Standard: 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., π^2). For example, by truncating the decimal expansion of √2 (square root 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: Reason abstractly and quantitatively. Mathematically proficient students make sense of the quantities and their relationships in problem situations. Students 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: 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, twoway 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: 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 x^2 + 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.
Common Core State Standards Math
Cluster: Mathematical practices
Standard: 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)(x^2 + x + 1), and (x – 1)(x^3 + x^2 + 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.
Math, Grade 7
Four fullyear digital course, built from the ground up and fullyaligned to the Common Core State Standards, for 7th grade Mathematics. Created using researchbased approaches to teaching and learning, the Open Access Common Core Course for Mathematics is designed with studentcentered learning in mind, including activities for students to develop valuable 21st century skills and academic mindset.
Lesson 2
Model Integer AdditionLesson 3
Model Integer SubtractionLesson 7
Self Check ExerciseLesson 8
Gallery Problems ExerciseLesson 9
Model Integers MultiplicationLesson 12
Simplifying Numerical ExpressionsLesson 13
Understanding Rational NumbersLesson 14
Self Check ExerciseLesson 15
Gallery Problems Exercise
Lesson 5
Graphing A Table Of ValuesLesson 7
Expressing Ratios As A Unit RateLesson 8
Identifying Verbal DescriptionsLesson 11
Solution StrategiesLesson 12
Gallery Problems ExerciseLesson 14
Creating Equations, Tables & GraphsLesson 15
Percent Increase ProblemsLesson 16
Percent Decrease ProblemsLesson 17
Identifying Errors In ReasoningLesson 18
Understanding Percent ChangeLesson 19
Solution Strategies (Feedback)Lesson 20
Gallery Problems Exercise
Lesson 1
Shape CharacteristicsLesson 2
Four Types Of AnglesLesson 4
Diagonals Of A RhombusLesson 6
Classifying TrianglesLesson 7
Exploring PolygonsLesson 8
PreAssessmentLesson 9
Review QuizLesson 10
Gallery Problems Exercise
Lesson 1
House PlansLesson 2
PolygonsLesson 3
Building with PolygonsLesson 4
Measuring CirclesLesson 5
Area of a CircleLesson 6
ScaleLesson 7
Changing ScaleLesson 8
Applying Scale to ProjectLesson 9
Self Check ExerciseLesson 10
Gallery Problems ExerciseLesson 11
Calculating VolumeLesson 12
Applying The Volume FormulaLesson 13
Exploring CrossSectionsLesson 14
Surface Area Of PrismsLesson 16
Unit ReviewLesson 17
Self Check ExerciseLesson 18
Gallery Problems ExerciseLesson 19
Unit Concepts Project PresentationLesson 20
Project Presentations (Feedback)
Lesson 1
Algebraic ReasoningLesson 3
Geometric ExpressionsLesson 7
Matching Equations To ProblemsLesson 9
Peer ReviewLesson 10
Gallery Problems ExerciseLesson 11
Solving & Graphing InequalitiesLesson 15
Self Check ExerciseLesson 16
Gallery Problems Exercise
Lesson 4
Experimental ProbabilityLesson 5
The Law Of Large NumbersLesson 6
Compound Events & Sample SpacesLesson 7
Fundamental Counting PrincipleLesson 10
Project ProposalLesson 11
Self Check ExerciseLesson 12
Sampling In Relation To ProbabilityLesson 13
Collecting & Analyzing DataLesson 14
Effects of A Nonrandom SampleLesson 15
Sampling ExperimentsLesson 16
Comparing Sets of DataLesson 18
Self Check ExerciseLesson 19
Gallery Problems ExerciseLesson 20
Group Project PresentationLesson 21
Project Presentations Feedback
Lesson 2
Project ProposalLesson 5
Interpreting Graphs & DiagramsLesson 6
Linear MeasurementsLesson 7
Refining Problem Solving SkillsLesson 8
Project Rubric & CriteriaLesson 9
Gallery Problems ExerciseLesson 10
Project PresentationLesson 11
Project Presentation (Continued)Lesson 12
Project Presentation (Feedback)