Lesson 1
Evaluate ExpressionsLesson 3
Properties of OperationsLesson 5
Mathematical CharacterLesson 6
Gallery Problems
Please log in to save materials. Log in
 Subject:
 Mathematics
 Material Type:
 Full Course
 Level:
 Middle School
 Grade:
 6
 Provider:
 Pearson
 Tags:

 5 Number Summary
 6th Grade Mathematics
 Absolute Value
 Absolute Values
 Addition Property of Equality
 Algebraic Expressions
 Box Plot
 Brackets
 BranchED SC
 BranchEd
 Center
 Coefficient
 Comparing Numbers
 Comparison
 Composite Figures
 Concepts
 Constant Ratio
 Cooking
 Coordinate Plane
 Cube
 Cubes
 Data
 Decimals
 Definitions
 Distance
 Distributive Property
 Dividing Fractions
 Division
 Double Number Line
 Double Number Lines
 Edges
 Equations
 Equivalence
 Equivalent Expressions
 Equivalent Ratios
 Exponents
 Expressions
 Expresssions
 Faces
 Five Number Summary
 Formulas
 Fractions
 Fuel Economy
 Gallery
 Glide Ratios
 Graphing
 Graphing Points
 Graphs
 Greatest Common Factor
 Grids
 Histogram
 Idaho Math
 Inequalities
 Inequality
 Ingredients
 Integers
 Iowa K12 ECurriculum
 Iowa Video
 Line Plot
 Line Plots
 Long Division
 Math Tricks
 Mean
 Mean Absolute Deviation
 Measurements and Data
 Median
 Mixtures
 Mode
 Models
 Multiplication Property of Equality
 Multiplicative Structure
 Multiplying and Dividing Fractions
 Negative Numbers
 Number Line
 Numbers
 Numerical Expressions
 Order of Operations
 Parallelograms
 Parentheses
 Percent Problems
 Percent Statements
 Population Density
 Price
 Prices
 Prisms
 ProblemSolving
 Projects.
 Properties of Operations
 Pyramid
 Quantity
 Range
 Rate
 Rates
 Ratio Tables
 Rectangles
 Rectangular Prisms
 Size
 Speed
 Spread
 Square
 Statements
 Statistical Measures
 Tables
 Tape Diagrams
 Terminology
 Time
 Trapezoids
 Triangles
 Unit Conversion
 Value
 Values
 Variability
 Variables
 Vertices
 Volume
 WholePart Statements
 Word Problems
 educ 134
 googleclassroom
 License:
 Creative Commons Attribution NonCommercial
 Language:
 English
Education Standards
 1
 2
 3
 4
 5
 ...
 6
 7
 8
 9
 10
 11
 12
 13
 14
 15
 16
 17
 18
 19
 20
 21
 22
 23
 24
 25
 26
 27
 28
 29
 30
 31
 32
 33
 34
 35
 36
 37
 38
 39
 40
 41
 42
 43
 44
 45
 46
 47
 48
 49
 50
 51
 52
 53
 54
 55
 56
 57
 58
 59
 60
 61
 62
 63
 64
 65
 66
 67
 68
 69
 70
 71
 72
 73
 74
 75
 76
 77
 78
 79
 80
 81
 82
 83
 84
 85
 86
 87
 88
 89
 90
 91
 92
 93
 94
 95
 96
 97
 98
 99
 100
 101
 102
 103
 104
 105
 106
 107
 108
 109
 110
 111
 112
 113
 114
 115
 116
 117
 118
 119
 120
 121
 122
 123
Learning Domain: Operations and Algebraic Thinking
Standard: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Learning Domain: Operations and Algebraic Thinking
Standard: Write simple expressions requiring parentheses that record calculations with numbers, and interpret numerical expressions without evaluating them.
Learning Domain: Ratios and Proportional Relationships
Standard: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.
Learning Domain: Ratios and Proportional Relationships
Standard: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship.
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio and rate reasoning to solve realworld and mathematical problems.
Learning Domain: Ratios and Proportional Relationships
Standard: Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Learning Domain: Ratios and Proportional Relationships
Standard: Solve unit rate problems including those involving unit pricing and constant speed.
Learning Domain: Ratios and Proportional Relationships
Standard: Understand that a percentage is a rate per 100 and use this to solve problems involving wholes, parts, and percentages.
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio reasoning to convert measurement units; convert units appropriately when multiplying or dividing quantities.
Learning Domain: Operations and Algebraic Thinking
Standard: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Learning Domain: Operations and Algebraic Thinking
Standard: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation "add 8 and 7, then multiply by 2"ť as 2 x (8 + 7). Recognize that 3 x (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Learning Domain: Expressions and Equations
Standard: Write and evaluate numerical expressions involving wholenumber exponents.
Learning Domain: Expressions and Equations
Standard: Write, read, and evaluate expressions in which letters stand for numbers.
Learning Domain: Expressions and Equations
Standard: Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation "Subtract y from 5"ť as 5  y.
Learning Domain: Expressions and Equations
Standard: Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
Learning Domain: Expressions and Equations
Standard: Evaluate expressions at specific values for their variables. Include expressions that arise from formulas in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^3 and A = 6 s^2 to find the volume and surface area of a cube with sides of length s = 1/2.
Learning Domain: Expressions and Equations
Standard: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
Learning Domain: Expressions and Equations
Standard: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Learning Domain: Expressions and Equations
Standard: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Learning Domain: Expressions and Equations
Standard: Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Learning Domain: Expressions and Equations
Standard: Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Learning Domain: Expressions and Equations
Standard: Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Learning Domain: Expressions and Equations
Standard: Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Learning Domain: Geometry
Standard: Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.
Learning Domain: Geometry
Standard: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
Learning Domain: Geometry
Standard: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.
Learning Domain: Geometry
Standard: Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.
Learning Domain: The Number System
Standard: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) Ö (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) Ö (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) Ö (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Learning Domain: The Number System
Standard: Fluently divide multidigit numbers using the standard algorithm.
Learning Domain: The Number System
Standard: Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.
Learning Domain: The Number System
Standard: Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Learning Domain: The Number System
Standard: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.
Learning Domain: The Number System
Standard: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Learning Domain: The Number System
Standard: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., (3) = 3, and that 0 is its own opposite.
Learning Domain: The Number System
Standard: Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Learning Domain: The Number System
Standard: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
Learning Domain: The Number System
Standard: Understand ordering and absolute value of rational numbers.
Learning Domain: The Number System
Standard: Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret 3 > 7 as a statement that 3 is located to the right of 7 on a number line oriented from left to right.
Learning Domain: The Number System
Standard: Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write 3ĺˇC > 7ĺˇC to express the fact that 3ĺˇC is warmer than 7ĺˇC.
Learning Domain: The Number System
Standard: Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of 30 dollars, write 30 = 30 to describe the size of the debt in dollars.
Learning Domain: The Number System
Standard: Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than 30 dollars represents a debt greater than 30 dollars.
Learning Domain: The Number System
Standard: Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Learning Domain: Ratios and Proportional Relationships
Standard: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak."ť "For every vote candidate A received, candidate C received nearly three votes."ť
Learning Domain: Ratios and Proportional Relationships
Standard: Understand the concept of a unit rate a/b associated with a ratio a:b with b ‰äĘ 0 (b not equal to zero), and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." (Expectations for unit rates in this grade are limited to noncomplex fractions.)
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Learning Domain: Ratios and Proportional Relationships
Standard: Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Learning Domain: Ratios and Proportional Relationships
Standard: Solve unit rate problems including those involving unit pricing and constant speed. For example, If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Learning Domain: Ratios and Proportional Relationships
Standard: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent.
Learning Domain: Ratios and Proportional Relationships
Standard: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Learning Domain: Statistics and Probability
Standard: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "How old am I?"ť is not a statistical question, but "How old are the students in my school?"ť is a statistical question because one anticipates variability in students' ages.
Learning Domain: Statistics and Probability
Standard: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
Learning Domain: Statistics and Probability
Standard: 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.
Learning Domain: Statistics and Probability
Standard: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Learning Domain: Statistics and Probability
Standard: Summarize and describe distributions. Summarize numerical data sets in relation to their context, such as by:
Learning Domain: Statistics and Probability
Standard: Reporting the number of observations.
Learning Domain: Statistics and Probability
Standard: Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
Learning Domain: Statistics and Probability
Standard: 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 was gathered.
Learning Domain: Statistics and Probability
Standard: Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data was gathered.
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: Write and interpret numerical expressions
Standard: Use parentheses, brackets, or braces in numerical expressions, and evaluate expressions with these symbols.
Cluster: Write and interpret numerical expressions
Standard: Write simple expressions that record calculations with numbers, and interpret numerical expressions without evaluating them. For example, express the calculation “add 8 and 7, then multiply by 2” as 2 × (8 + 7). Recognize that 3 × (18932 + 921) is three times as large as 18932 + 921, without having to calculate the indicated sum or product.
Cluster: Solve realworld and mathematical problems involving area, surface area, and volume
Standard: Find area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving realworld and mathematical problems.
Cluster: Solve realworld and mathematical problems involving area, surface area, and volume
Standard: Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving realworld and mathematical problems.
Cluster: Solve realworld and mathematical problems involving area, surface area, and volume
Standard: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving realworld and mathematical problems.
Cluster: Solve realworld and mathematical problems involving area, surface area, and volume
Standard: Represent threedimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving realworld and mathematical problems.
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0 (b not equal to zero), and use rate language in the context of a ratio relationship. For example, "This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar." "We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger." (Expectations for unit rates in this grade are limited to noncomplex fractions.)
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Use ratio and rate reasoning to solve realworld and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Make tables of equivalent ratios relating quantities with wholenumber measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Solve unit rate problems including those involving unit pricing and constant speed. For example, If it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole given a part and the percent.
Cluster: Understand ratio concepts and use ratio reasoning to solve problems
Standard: Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Cluster: Apply and extend previous understandings of multiplication and division to divide fractions by fractions
Standard: Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?
Cluster: Compute fluently with multidigit numbers and find common factors and multiples
Standard: Fluently divide multidigit numbers using the standard algorithm.
Cluster: Compute fluently with multidigit numbers and find common factors and multiples
Standard: Fluently add, subtract, multiply, and divide multidigit decimals using the standard algorithm for each operation.
Cluster: Compute fluently with multidigit numbers and find common factors and multiples
Standard: Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, debits/credits, positive/negative electric charge); use positive and negative numbers to represent quantities in realworld contexts, explaining the meaning of 0 in each situation.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Understand ordering and absolute value of rational numbers.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Write, interpret, and explain statements of order for rational numbers in realworld contexts. For example, write –3°C > –7°C to express the fact that –3°C is warmer than –7°C.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a realworld situation. For example, for an account balance of –30 dollars, write –30 = 30 to describe the size of the debt in dollars.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.
Cluster: Apply and extend previous understandings of numbers to the system of rational numbers
Standard: Solve realworld and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.
Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard: Write and evaluate numerical expressions involving wholenumber exponents.
Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard: Write, read, and evaluate expressions in which letters stand for numbers.
Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard: Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5 – y.
Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard: Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard: Evaluate expressions at specific values for their variables. Include expressions that arise from formulas in realworld problems. Perform arithmetic operations, including those involving wholenumber exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^3 and A = 6 s^2 to find the volume and surface area of a cube with sides of length s = 1/2.
Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
Cluster: Apply and extend previous understandings of arithmetic to algebraic expressions
Standard: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Cluster: Reason about and solve onevariable equations and inequalities
Standard: Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
Cluster: Reason about and solve onevariable equations and inequalities
Standard: Use variables to represent numbers and write expressions when solving a realworld or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
Cluster: Reason about and solve onevariable equations and inequalities
Standard: Solve realworld and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.
Cluster: Reason about and solve onevariable equations and inequalities
Standard: Write an inequality of the form x > c or x < c to represent a constraint or condition in a realworld or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.
Cluster: Represent and analyze quantitative relationships between dependent and independent variables
Standard: Use variables to represent two quantities in a realworld problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.
Cluster: Develop understanding of statistical variability
Standard: Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, “How old am I?” is not a statistical question, but “How old are the students in my school?” is a statistical question because one anticipates variability in students’ ages.
Cluster: Develop understanding of statistical variability
Standard: Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.
Cluster: Develop understanding of statistical variability
Standard: 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.
Cluster: Summarize and describe distributions
Standard: Display numerical data in plots on a number line, including dot plots, histograms, and box plots.
Cluster: Summarize and describe distributions
Standard: Summarize and describe distributions. Summarize numerical data sets in relation to their context, such as by:
Cluster: Summarize and describe distributions
Standard: Reporting the number of observations.
Cluster: Summarize and describe distributions
Standard: Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.
Cluster: Summarize and describe distributions
Standard: 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 was gathered.
Cluster: Summarize and describe distributions
Standard: Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data was gathered.
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 6
Lesson 1
Above and Below Sea LevelLesson 2
Opposite of a NumberLesson 4
Possible or Impossible?Lesson 5
InequalitiesLesson 6
Assess and ReviseLesson 7
Gallery ProblemsLesson 8
Coordinate PlaneLesson 10
ReflectionsLesson 11
Peer Review and ReviseLesson 12
Gallery Problems
Lesson 1
Cooking with FractionsLesson 2
DivisionLesson 3
Whole Number DivisionLesson 4
Divide a Fraction by a FractionLesson 7
Gallery ProblemsLesson 10
Where Does the Decimal Point Go?Lesson 11
Multiplying and DividingLesson 12
Self CheckLesson 13
Gallery Problems
Lesson 1
Equivalent RatiosLesson 2
Comparing Numbers with RatiosLesson 3
Expressing RatiosLesson 4
Tape DiagramsLesson 5
Double Number LinesLesson 6
Double Number Line for ModelingLesson 7
Expressing Ratios NumericallyLesson 8
Student ProjectLesson 10
Relate Ratio Tables to GraphsLesson 11
Glide RatioLesson 12
Apply Your Knowledge about RatiosLesson 13
Finding PercentsLesson 14
Percent Statements about DataLesson 15
Percents Greater than 100%Lesson 16
Student Self CheckLesson 17
Gallery ProblemsLesson 18
Student PresentationsLesson 19
Peer Review
Lesson 1
Math TricksLesson 2
Evaluating ExpressionsLesson 3
Expressions in Words & SymbolsLesson 5
Substituting Numbers for LettersLesson 6
Mathematical VocabularyLesson 9
Common MultiplesLesson 11
Peer ReviewLesson 12
Gallery Problems Exercise
Lesson 2
Symbolic RepresentationLesson 4
Reasoning to Identify SolutionsLesson 7
Problem Solving ExerciseLesson 11
Self Check ReviewLesson 12
Gallery Problems Exercise
Lesson 5
Defining Rate DiscussionLesson 7
Reviewing Conversion FactorsLesson 9
Using Rates To Solve ProblemsLesson 10
Gallery Problems ExerciseLesson 13
Reviewing Quantitative RelationshipsLesson 14
Rules For Computing A ValueLesson 15
Gallery Problems Exercise
Lesson 7
Gallery Problems ExerciseLesson 8
Classroom Project Presentation
Lesson 1
Reviewing Statistical QuestionsLesson 2
Collecting & Organizing DataLesson 3
Outlining A Project ProposalLesson 4
Construction of A Line PlotLesson 7
Measures & Data SetsLesson 8
Matching Stats With Line PlotsLesson 10
Manipulating Data PointsLesson 13
Self Check ExerciseLesson 14
Characteristics Of DataLesson 15
Reviewing Data SetsLesson 16
Classroom Project Presentation
Lesson 1
Comparing Surface Area & VolumeLesson 4
Basic & Composite FiguresLesson 7
Identifying Nets For CubesLesson 10
Gallery Problems Exercise (Groups)