middle school math solution: course 1 - carnegie learning

Ohio Correlation: Course 1 | 104/13/18

Middle School Math Solution: Course 1Ohio Correlation

6.MP MATHEMATICAL PRACTICESThe standards for Mathematical Practice describe the skills that mathematics educators should seek to develop in their students. The descriptions of the mathematical practices in this document provide examples of how student performance will change and grow as they engage with and master new and more advanced mathematical ideas across the grade levels. (Standards 6.MP.1–8).Standard Correlation

6.MP.1 Make Sense of problems and persevere in solving them.

In Grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for effi cient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most effi cient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a diff erent way?”. Students can explain the relationships between equations, verbal descriptions, tables, and graphs. Mathematically profi cient students check their answers to problems using diff erent methods.

This practice, as well as all practices are evident in every lesson. Icons indicate which practice is emphasized in the lesson.

6.MP.2 Reason abstractly and quantitatively.In Grade 6, students represent a wide variety of real-world contexts through the use of real numbers and variables in mathematical expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations or other meaningful moves. To reinforce students’ reasoning and understanding, teachers might ask, “How do you know?” or “What is the relationship of the quantities?”.

Activities that use this practice have an icon located throughout the book. Please reference Teachers Manual FM20-FM21.

Example: On page M1-54, Activity 5.2 has icon in header.

6.MP.3 Construct viable arguments and critique the reasoning of others.

In Grade 6, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (e.g., box plots, dot plots, histograms, etc.). They further refi ne their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like “How did you get that?”, “Why is that true?” “Does that always work?” They explain their thinking to others and respond to others’ thinking.





Standard Correlation

6.MP.4 Model with mathematics.In Grade 6, students model problem situations symbolically, graphically, in tables, contextually, and with drawings of quantities as needed. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students begin to explore covariance and represent two quantities simultaneously. Students use number lines to compare numbers and represent inequalities. They use measures of center and variability and data displays (i.e., box plots and histograms) to draw inferences about and make comparisons between data sets. Students need many opportunities to connect and explain the connections between the diff erent representations. They should be able to use all of these representations as appropriate and apply them to a problem context. Students should be encouraged to answer questions such as “What are some ways to represent the quantities?” or “What formula might apply in this situation?”



6.MP.5 Use appropriate tools strategically.Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in Grade 6 may decide to represent fi gures on the coordinate plane to calculate area. Number lines are used to create dot plots, histograms, and box plots to visually compare the center and variability of the data. Visual fraction models can be used to represent situations involving division of fractions. Additionally, students might use physical objects or applets to construct nets and calculate the surface area of three-dimensional fi gures. Students should be encouraged to answer questions such as “What approach did you try fi rst?” or “Why was it helpful to use?”



6.MP.6 Attend to precision.In Grade 6, students continue to refi ne their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to rates, ratios, geometric fi gures, data displays, and components of expressions, equations, or inequalities. When using ratio reasoning in solving problems, students are careful about specifying units of measure and labeling axes to clarify the correspondence with quantities in a problem. Students also learn to express numerical answers with an appropriate degree of precision when working with rational numbers in a situational problem. Teachers might ask, “What mathematical language, defi nitions, or properties can you use to explain ___?”






6.MP.7 Look for and make use of structure.Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent expressions (e.g., 6 + 2n = 2 (3 + n) by distributive property) and solve equations (e.g., 3c = 15 so c = 5 by division property of equality). Students compose and decompose two- and three-dimensional fi gures to solve real-world problems involving area and volume. Teachers might ask, “What do you notice when ___?” or “What parts of the problem might you eliminate, simplify, or ___?”



6.MP.8 Look for an express regularity in repeated reasoning.

In Grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns. Given multiple opportunities to solve and model problems, they may notice that aa bb ÷ cc dd = aadd bbcc and construct other examples and models that confi rm their generalization. Students connect place value and their prior work with operations to understand algorithms to fl uently divide multi-digit numbers and perform all operations with multi-digit decimals. Students informally begin to make connections between covariance, rates, and representations showing the relationships between quantities. Students should be encouraged to answer questions such as, “How would we prove that ___?” or “How is this situation similar and/or diff erent from other situations?”





6.RP RATIOS AND PROPORTIONAL RELATIONSHIPSUnderstand ratio concepts and use ratio reasoning to solve problems (Standards 6.RP.1–3).Standard Correlation

6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.

The following are examples of ratio language: “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every two wings there was one beak.” “For every vote candidate A received, candidate C received nearly three votes.”

TEXTBOOK: Module 2Topic/Lesson: T1L1: Introduction to Ratio and Ratio Reasoning (M2-7 thru M2-24)T1L2: Comparing Ratios to Solve Problems (M2-25 thru M2-36)T1L3: Determining Equivalent Ratios (M2-37 thru M2-56) T1L4: Using Tables to Represent Equivalent Ratios (M2-57 thru M2-68)T1L5: Graphs and Ratios (M2-69 thru M2-84)T1L6: One Is Not Enough (M2-85 thru M2-98)

SKILLS PRACTICE: Module 2, Relating QuantitiesTopic 1: Ratios (pp. 35–42)Topic 2: Percents (pp. 43–48) Topic 3: Unit Rates and Conversions (pp. 49–55)

MATHia: Module, Relating QuantitiesUnit: Ratio ReasoningWorkspace: Understanding Ratio Relationships; Equivalent Ratios

6.RP.2Understand 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.

The following are examples of rate language: “This recipe has a ratio of four cups of flour to two cups of sugar, so the rate is two cups of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (In sixth grade, unit rates are limited to non-complex fractions.)

TEXTBOOK: Module 2 Topic/Lesson: T3L2: Introduction to Unit Rates (M2-185 thru M2-198)


MATHia: Module, Relating QuantitiesUnit: Ratio ReasoningWorkspace: Understanding Ratio Relationships; Equivalent Ratios; Multiple Representations of Ratios




6.RP.3Use ratio and rate reasoning to solve real-world (with a context) and mathematical (void of context) problems, using strategies such as reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations involving unit rate problems.

6.RP.3a Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.

TEXTBOOK: Module 2Topic/Lesson: T1L2: Comparing Ratios to Solve Problems (M2-25 thru M2-36) T1L3: Determining Equivalent Ratios (M2-37 thru M2-56) T1L4: Using Tables to Represent Equivalent Ratios (M2-57 thru M2-68)T1L5: Graphs and Ratios (M2-69 thru M2-84)T1L6: One Is Not Enough (M2-85 thru M2-98)

SKILLS PRACTICE: Module 2, Relating QuantitiesTopic 1: Ratios (pp. 35–42)Topic 2: Percents (pp. 43–48)Topic 3: Unit Rates and Conversions (pp. 49–55)

MATHia: Module, Relating QuantitiesUnit: Problem Solving Using Ratio and Rate Reasoning; Introduction to Percent; Rate Reasoning; Ratio Reasoning to Convert Units; Problem Solving with Equivalent Ratios and Rates using Tables; Problem Solving with Equivalent Ratios and Rates using Double Number Lines; Problem Solving with Equivalent Ratios and Rates using Graphs; Percent Models; Fraction, Decimal, Percent Conversions; Determining a Part Given a Percent and a Whole; Determining a Whole Given a Percent and a Part; Fractional Rates; Comparing Rates; Converting Within Systems; Converting Between Systems

6.RP.3bSolve unit rate problems including those involving unit pricing and constant speed.

For example, if it took four hours to mow eight lawns, how many lawns could be mowed in 32 hours? What is the hourly rate at which lawns were being mowed?

TEXTBOOK: Module 2Topic/Lesson: T3L2: Introduction to Unit Rates (M2-185 thru M2-198)T3L3: Multiple Representation of Unit Rates (M2-199 thru M2-208)


MATHia: Module, Relating QuantitiesUnit: Problem Solving Using Ratio and Rate Reasoning; Introduction to Percent; Rate Reasoning; Ratio Reasoning to Convert UnitsWorkspace: Problem Solving with Equivalent Ratios and Rates using Tables; Problem Solving with Equivalent Ratios and Rates using Double Number Lines; Problem Solving with Equivalent Ratios and Rates using Graphs; Percent Models; Fraction, Decimal, Percent Conversions; Determining a Part Given a Percent and a Whole; Determining a Whole Given a Percent and a Part; Fractional Rates; Comparing Rates; Converting Within Systems; Converting Between Systems




6.RP.3c Find a percent of a quantity as a rate per 100. Solve problems involving finding the whole, given a part and the percent.

(For example, 30% of a quantity means 30/100 times the quantity.)

TEXTBOOK: Module 2Topic/Lesson: T2L1: Percent, Fraction, and Decimal Equivalence (M2-109 thru M2-122)T2L2: Using Estimation and Benchmark Percents (M2-123 thru M2-136)T2L3: Determining the Part and the Whole in Percent Problems (M2-137 thru M2-156)

SKILLS PRACTICE: Module 2, Relating QuantitiesTopic 1: Ratios (pp. 35–42), Topic 2: Percents (pp. 43–48), Topic 3: Unit Rates and Conversions (pp. 49–55)

MATHia: Module, Relating QuantitiesUnit: Problem Solving Using Ratio and Rate Reasoning, Introduction to Percent, Rate Reasoning, Ratio Reasoning to Convert UnitsWorkspace: Problem Solving with Equivalent Ratios and Rates using Tables; Problem Solving with Equivalent Ratios and Rates using Double Number Lines; Problem Solving with Equivalent Ratios and Rates using Graphs; Percent Models; Fraction, Decimal, Percent Conversions; Determining a Part Given a Percent and a Whole; Determining a Whole Given a Percent and a Part; Fractional Rates; Comparing Rates; Converting Within Systems; Converting Between Systems

6.RP.3dUse ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

TEXTBOOK: Module 2Topic/Lesson: T3L1: Using Ratio Reasoning to Convert Units (M2-165 thru M2-184)

SKILLS PRACTICE: Module 2, Relating QuantitiesTopic 1: Ratios (pp. 35–42), Topic 2: Percents (pp. 43–48), Topic 3: Unit Rates and Conversions (pp. 49–55)

MATHia: Module, Relating QuantitiesUnit: Problem Solving Using Ratio and Rate Reasoning; Introduction to Percent; Rate Reasoning; Ratio Reasoning to Convert UnitsWorkspace: Problem Solving with Equivalent Ratios and Rates using Tables; Problem Solving with Equivalent Ratios and Rates using Double Number Lines; Problem Solving with Equivalent Ratios and Rates using Graphs; Percent Models; Fraction, Decimal, Percent Conversions; Determining a Part Given a Percent and a Whole; Determining a Whole Given a Percent and a Part; Fractional Rates; Comparing Rates; Converting Within Systems; Converting Between Systems



6.NS THE NUMBER SYSTEMApply and extend previous understandings of multiplication and division of whole numbers to divide fractions by fractions (Standard 6.NS.1). Compute (add, subtract, multiply and divide) fluently with multi-digit numbers and decimals and find common factors and multiples (Standards 6.NS.2–4). Apply and extend previous understandings of numbers to the system of rational numbers (Standards 6.NS.5–8).Standard Correlation

6.NS.1Interpret and compute quotients of fractions.

6.NS.1.a Compute quotients of fractions by fractions.

For example, by applying strategies such as visual fraction models, equations, and the relationship between multiplication and division, to represent problems.

TEXTBOOK: Module 1Topic/Lesson: T2L1: Identifying and Ordering Rational Numbers (M1-71 thru M1-82)T2L2: Multiplying and Dividing with Fractions (M1-83 thru M1-92)T2L3: Fractions by Fraction Division (M1-93 thru M1-106)

SKILLS PRACTICE: Module 1, Composing and DecomposingTopic 2: Positive Rational Numbers (pp. 14–20)

MATHia: Module, Composing and DecomposingUnit: Fraction DivisionWorkspace: Representing Fraction Division; Interpreting Remainders Using Models; Developing the Fraction Division Algorithm

6.NS.1.bSolve real-world problems involving division of fractions by fractions.

For example, how much chocolate will each person get if three people share 1/2 pound of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mile and area 1/2 square mile?

TEXTBOOK: Module 1Topic/Lesson: T2L1: Identifying and Ordering Rational Numbers (M1-71 thru M1-82)T2L2: Multiplying and Dividing with Fractions (M1-83 thru M1-92)T2L3: Fraction by Fraction Division (M1-93 thru M1-106)






6.NS.1.c Explain the meaning of quotients in fraction division problems.

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.)

TEXTBOOK: Module 1Topic/Lesson: T2L1: Identifying and Ordering Rational Numbers (M1-71 thru M1-82)T2L2: Multiplying and Dividing with Fractions (M1-83 thru M1-92)T2L3: Fraction by Fraction Division (M1-93 thru M1-106)



6.NS.2Fluently divide multi-digit numbers using a standard algorithm.

TEXTBOOK: Module 1Topic/Lesson: T3L1: Deepening Understanding of Volume (M1-115 thru M1-130)T3L4: Dividing with Volume and Surface Area (M1-165 thru M1-176)

SKILLS PRACTICE: Module 1, Composing and DecomposingTopic 3: Decimals and Volume (pp. 21–34)

MATHia: Module, Composing and DecomposingUnit: Fraction DivisionWorkspace: Multiplying and Dividing Rational Numbers

6.NS.3Fluently add, subtract, multiply, and divide multi-digit decimals using a standard algorithm for each operation.

TEXTBOOK: Module 1Topic/Lesson: T3L1: Deepening Understanding of Volume (M1-115 thru M1-130)T3L2: Volume Composition and Decomposition (M1-131 thru M1-142)T3L3: Surface Area of Rectangular Prisms and Pyramids (M1-143 thru M1-164)T3L4: Dividing with Volume and Surface Area (M1-165 thru M1-175)


MATHia: Module, Composing and DecomposingUnit: Decimal OperationsWorkspace: Converting Fractions to Decimals; Adding and Subtracting Decimals; Decimal Sums and Differences; Exploring Decimal Facts; Multiplying and Dividing Decimals; Decimal Products and Quotients




6.NS.4 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).

TEXTBOOK: Module 1Topic/Lesson: T1L4: Common Factors and Common Multiples (M1-39 thru M1-50) T1L5: Least Common Multiple and Greatest Common Factor (M1-51 thru M1-60)

SKILLS PRACTICE: Module 1, Composing and DecomposingTopic 1: Factors and Area (pp. 9–13)

6.NS.5Understand that positive and negative numbers are used together to describe quantities having opposite directions or values.

(For example, temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of zero in each situation.

TEXTBOOK: Module 4Topic/Lesson: T1L1: Introduction to Negative Numbers (M4-7 thru M4-22)

SKILLS PRACTICE: Module 4, Moving Beyond Positive QuantitiesTopic 1: Signed Numbers (pp. 96–100)

MATHia: Module, Moving Beyond Positive QuantitiesUnit: IntegersWorkspace: Introduction to Negative Numbers; Representing Integers on Number Lines; Using Absolute Value

6.NS.6Understand 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.

6.NS.6.aRecognize opposite signs of numbers as indicating locations on opposite sides of zero on the number line; recognize that the opposite of the opposite of a number is the number itself.

For example, –(–3) = 3, and zero is its own opposite.

TEXTBOOK: Module 4Topic/Lesson: T1L1: Introduction to Negative Numbers (M4-7 thru M4-22)T1L3: Rational Number Systems (M4-35 thru M4-46)


MATHia: Module, Moving Beyond Positive QuantitiesUnit: The Coordinate PlaneWorkspace: Exploring Symmetry on the Coordinate Plane; Identifying and Interpreting Ordered Pairs; Plotting Points




6.NS.6.b Understand that the signs of numbers in ordered pairs indicate their location 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.

TEXTBOOK: Module 4Topic/Lesson: T2L1: Extending the Coordinate Plane (M4-57 thru M4-72)

SKILLS PRACTICE: Module 4, Moving Beyond Positive QuantitiesTopic 2: The Four Quadrants (pp. 101–109)


6.NS.6.cFind 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.

TEXTBOOK: Module 4Topic/Lesson: T1L1: Introduction to Negative Numbers (M4-7 thru M4-22)T2L1: Extending the Coordinate Plane (M4-57 thru M4-72)

SKILLS PRACTICE: Module 4, Moving Beyond Positive QuantitiesTopic 1: Signed Numbers (pp. 96–100)Topic 2: The Four Quadrants (pp. 101–109)


6.NS.7Understand ordering and absolute value or rational numbers.

6.NS.7.aInterpret 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.

TEXTBOOK: Module 4Topic/Lesson: T1L1: Introduction to Negative Numbers (M4-7 thru M4-22)T1L2: Absolute Value (M4-23 thru M4-34)






6.NS.7.bWrite, interpret, and explain statements of order for rational numbers in real-world contexts.

For example, write –3° C > –7° C to express the fact that –3° C is warmer than –7° C.

TEXTBOOK: Module 4Topic/Lesson: T1L1: Introduction to Negative Numbers (M4-7 thru M4-22)T1L2: Absolute Value (M4-23 thru M4-34)



6.NS.7.cUnderstand the absolute value of a rational number as its distance from zero on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world context.

For example, for an account balance of –30 dollars, write|–30|= 30 to describe the size of the debt in dollars.

TEXTBOOK: Module 4Topic/Lesson: T1L2: Absolute Value (M4-23 thru M4-34)



6.NS.7.dDistinguish 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.

TEXTBOOK: Module 4Topic/Lesson: T1L2: Absolute Value (M4-23 thru M4-34)






6.NS.8Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same x coordinate or the same y-coordinate.

TEXTBOOK: Module 4 Topic/Lesson: T2L1: Extending the Coordinate Plane (M4-57 thru M4-72)T2L2: Graphing Geometric Figures (M4-73 thru M4-86)T2L3: Problem Solving on the Coordinate Plane (M4-87 thru M4-111)

SKILLS PRACTICE: Module 4, Moving Beyond Positive QuantitiesTopic 2: The Four Quadrants (pp. 101–109)

MATHia: Module, Moving Beyond Positive QuantitiesUnit: Multiple RepresentationsWorkspace: Solving One-Step Equations Using Multiple Representations in Four Quadrants



6.EE EXPRESSIONS AND EQUATIONSApply and extend previous understandings of arithmetic to algebraic expressions involving exponents and variables (Standards 6.EE.1–4). They reason about and solve one-variable equations and inequalities (Standards 6.EE.5–8). Represent and analyze quantitative relationships between dependent and independent variables in a real-world context (Standard 6.EE.9)Standard Correlation

6.EE.1 Write and evaluate numerical expressions involving whole-number exponents.

TEXTBOOK: Module 3 Topic/Lesson: T1L1: Evaluating Numeric Expressions (M3-7 thru M3-22)

SKILLS PRACTICE: Module 3, Determining Unknown QuantitiesTopic 1: Expressions (pp. 56–75)

MATHia: Module, Reasoning with Expressions and EquationsUnit: Reasoning with Expressions and EquationsWorkspace: Using Picture Algebra with Addition, Subtraction, and Multiplication

6.EE.2Write, read, and evaluate expressions in which letters represent numbers.

6.EE.2.aWrite expressions that record operations with numbers and with letters representing numbers.

For example, express the calculation “Subtract y from 5” as 5 – y and express “Jane had $105.00 in her bank account. One year later, she had x dollars more. Write an expression that shows her new balance” as $105.00 + x.

TEXTBOOK: Module 3 Topic/Lesson: T1L2: Introduction to Algebraic Expression (M3-23 thru M3-34)T1L3: Equivalent Expressions (M3-35 thru M3-66)T1L5: Using Algebraic Expressions to Analyze and Solve Problems (M3-67 thru M3-74)


MATHia: Module, Reasoning with Expressions and Equations; Determining Unknown QuantitiesUnit: Reasoning with Expressions and Equations; Number Properties; Algebraic ExpressionsWorkspace: Using Picture Algebra with Multiplication, Total Given

6.EE.2.bIdentify parts of an expression using mathematical terms (for example, sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity and a sum of two terms.

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.

TEXTBOOK: Module 1; Module 3 Topic/Lesson: M1T1L1: Writing Equivalent Expressions Using the Distributive Property (M1-7 thru M1-14)M3T1L2: Introduction to Algebraic Expressions (M3-23 thru M3-34)

SKILLS PRACTICE: Module 1, Composing and Decomposing; Module 3, Determining Unknown Quantities:Module 1, Topic 1: Factors and Area (pp. 1–13)Module 3, Topic 1: Expressions (pp. 56–75)

MATHia: Module, Reasoning with Expressions and Equations; Determining Unknown QuantitiesUnit: Reasoning with Expressions and Equations; Number Properties; Algebraic ExpressionsWorkspace: Using Picture Algebra with Addition and Subtraction, Total Given




6.EE.2.c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, using the algebraic order of operations when there are no parentheses to specify a particular order.

For example, use the formulas V = s3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1/2.

TEXTBOOK: Module 3Topic/Lesson: T1L2: Introduction to Algebraic Expressions (M3-23 thru M3-34)T1L5: Using Algebraic Expressions to Analyze and Solve Problems (M3-67 thru M3-74)


MATHia: Module, Reasoning with Expressions and Equations; Determining Unknown QuantitiesUnit: Reasoning with Expressions and Equations; Number Properties; Algebraic ExpressionsWorkspace: Patterns and One-Step Expressions

6.EE.3Apply 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.

TEXTBOOK: Module 1; Module 3Topic/Lesson: M1T1L1: Writing Equivalent Expressions Using the Distributive Property (M1-7 thru M1-14) M3T1L3: Equivalent Expressions (M3-35 thru M3-66)M3T1L5: Using Algebraic Expressions to Analyze and Solve Problems (M3-67 thru M3-74)

SKILLS PRACTICE: Module 1, Composing and Decomposing; Module 3, Determining Unknown Quantities:Module 1, Topic 1: Factors and Area (pp. 1–13)Module 3, Topic 1: Expressions (pp. 56–75)

MATHia: Module, Reasoning with Expressions and EquationsUnit: Reasoning with Expressions and EquationsWorkspace: Using Picture Algebra with Addition, Subtraction, and Multiplication; Using Picture Algebra with Multiplication, Total Given; Using Picture Algebra with Addition and Subtraction, Total Given; Patterns and One-Step Expressions

6.EE.4Identify when two expressions are equivalent.

For example, the expressions y + y + y and 3y are equivalent because they name the same number, regardless of which number y represents.

TEXTBOOK: Module 3 Topic/Lesson: T1L4: Verifying Equivalent Expressions (M3-53 thru M2-66)


MATHia: Module, Determining Unknown QuantitiesUnit: Equivalent Algebraic ExpressionsWorkspace: Modeling Equivalent Algebraic Expressions; Exploring the Distributive Property with Algebraic Expressions; Simplifying Algebraic Expressions (No Type In); Simplifying Algebraic Expressions (Type In)




6.EE.5 Understand solving an equation or inequality as a process of answering the 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.

TEXTBOOK: Module 3 Topic/Lesson: T2L1: Reasoning with Equal Expressions (M3-87 thru M3-106)

SKILLS PRACTICE: Module 3, Determining Unknown QuantitiesTopic 2: Equations (pp. 76–85)

MATHia: Module, Determining Unknown QuantitiesUnit: Equivalent Algebraic ExpressionsWorkspace: Modeling Equivalent Algebraic Expressions; Exploring the Distributive Property with Algebraic Expressions; Simplifying Algebraic Expressions (No Type In); Simplifying Algebraic Expressions (Type In)

6.EE.6Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

TEXTBOOK, Module 3Topic/Lesson: T1L5: Using Algebraic Expressions to Analyze and Solve Problems (M3-67 thru M3-74) T2L2: Solving One-Step Addition Equations (M3-107 thru M3-118)T2L3: Solving One-Step Multiplication Equations (M3-119 thru M3-134)

SKILLS PRACTICE: Module 3, Determining Unknown QuantitiesTopic 1: Expressions (pp. 56–75)Topic 2: Equations (pp. 76–85)

MATHia: Module, Determining Unknown QuantitiesUnit: Problem Solving with One-Step EquationsWorkspace: Patterns and One-Step Equations; Problem Solving Using Multiple Representations in the First Quadrant; Problem Solving with Decimals

6.EE.7Solve real-world and mathematical problems by writing and solving equations of the form x + a = b and ax = b for cases in which a, b and x are all non-negative rational numbers.

TEXTBOOK: Module 3Topic/Lesson: T2L2: Solving One-Step Addition Equations (M3-107 thru M3-118)T2L3: Solving One-Step Multiplication Equations (M3-119 thru M3-134)


MATHia: Module, Determining Unknown QuantitiesUnit: Problem Solving with One-Step Equations, Solving One-Step EquationsWorkspace: Patterns and One-Step Equations; Problem Solving Using Multiple Representations in the First Quadrant; Problem Solving with Decimals; Solving One-Step Equations with a Balance; Representing One-Step Equations; Using Substitution to Identify Solutions to Equations; Solving with Multiplication and Division (No Type In); Solving One-Step Equations (Type In)




6.EE.8 Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

TEXTBOOK: Module 3 Topic/Lesson: T2L1: Reasoning with Equal Expressions (M3-87 thru M3-106)


MATHia: Module, Determining Unknown Quantities; Moving Beyond Positive QuantitiesUnit: Solving One-Step Inequalities, IntegersWorkspace: Graphing Inequalities with Positive Rational Numbers; Graphing Inequalities with Rational Numbers

6.EE.9Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation.

For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

TEXTBOOK: Module 3; Module 4Topic/Lesson: M3T2L4: Solving Equations to Solve Problems (M3-135 thru M3-144)M3T3L1: Independent and Dependent Variables (M3-155 thru M3-182)M3T3L2: Using Graphs to Solve One-Step Equations (M3-183 thru M3-192)M3T3L3: Multiple Representations of Equations (M3-193 thru M3-206)M3T3L4: Relating Distance, Rate, and Time (M3-207 thru M3-220)M4T2L3: Problem Solving on the Coordinate Plane (M4-87 thru M4-111)

SKILLS PRACTICE: Module 3, Determining Unknown Quantities; Module 4, Moving Beyond Positive QuantitiesModule 3, Topic 2: Equations (pp. 76–85), Module 3, Topic 3: Graphing Quantitative Relationship (pp. 86–95), Module 4, Topic 2 The Four Quadrants (pp. 101–109)

MATHia: Module, Determining Unknown QuantitiesUnit: Solving One-Step InequalitiesWorkspace: Solving One-Step Equations with a Balance; Representing One-Step Equations; Using Substitution to Identify Solutions to Equations; Solving with Multiplication and Division (No Type In); Solving One-Step Equations (Type In)



6.G GEOMETRYSolve real-world and mathematical problems involving area, surface area, and volume (Standards 6.G.1–4).Standard Correlation

6.G.1 Through composition into rectangles or decomposition into triangles, find the area of right triangles, other triangles, special quadrilaterals, and polygons; apply these techniques in the context of solving real-world and mathematical problems.

TEXTBOOK: Module 1Topic/Lesson: T1L2: Area of Triangles and Quadrilaterals (M1-15 thru M1-28)T1L3: Composite Figures (M1-29 thru M1-38)

SKILLS PRACTICE: Module 1, Composing and DecomposingTopic 1: Factors and Area (pp. 1–13)

MATHia: Module, Composing and DecomposingUnit: AreaWorkspace: Developing Area Formulas; Calculating Area of Various Figures; Solving Area Problems; Calculating Area of Composite Figures

6.G.2Find the volume of a right rectangular prism with appropriate unit fraction edge lengths by packing it with cubes of the appropriate unit fraction edge lengths.

For example, 3½ x 2 x 6), and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = lwh and V = Bh to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems. (Note: Model the packing using drawings and diagrams).

TEXTBOOK: Module 1Topic/Lesson: T3L1: Deepening Understanding of Volume (M1-115 thru M1-130)T3L2: Volume Composition and Decomposition (M-131 thru M1-142)


MATHia: Module, Composing and DecomposingUnit: Volume and Surface AreaWorkspace: Calculating Volume of Right Prisms; Using Volume of Right Prisms

6.G.3Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same x coordinate or the same y coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

TEXTBOOK: Module 4Topic/Lesson: T2L2: Graphing Geometric Figures (M4-73 thru M4-86)

SKILLS PRACTICE:Background information addressed in: Module 4, Moving Beyond Positive QuantitiesTopic 2, The Four Quadrants (pp. 101–109)





6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real world and mathematical problems.

TEXTBOOK: Module 1 Topic/Lesson: T3L3: Surface Area of Rectangular Prisms and Pyramids (M1-143 thru M1-164)


MATHia: Module, Composing and DecomposingUnit: Volume and Surface AreaWorkspace: Calculating Surface Area of Right Prisms



6.SP STATISTICS AND PROBABILITYDevelop understanding of statistical problem solving. (Standards 6.SP.1–3). Summarize and describe distributions (Standards 6.SP.4–5).Standard Correlation

6.SP.1 Develop statistical reasoning by using the GAISE model:

6.SP.1.aFormulate Questions: Recognize and formulate a statistical question as one that anticipates variability and can be answered with quantitative data.

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 of the variability in students’ ages (GAISE Model, step 1).

TEXTBOOK: Module 5 Topic/Lesson: T1L1: Understanding the Statistical Process (M5-7 thru M5-24)

6.SP.1.bCollect Data: Design and use a plan to collect appropriate data to answer a statistical question (GAISE Model, step 2).


6.SP.1.cAnalyze Data: Select appropriate graphical methods and numerical measures to analyze data by displaying variability within a group, comparing individual to individual, and comparing individual to group (GAISE Model, step 3).


6.SP.1.dInterpret Results: Draw logical conclusions from the data based on the original question (GAISE Model, step 4).


6.SP.2Understand that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread/range and overall shape.

TEXTBOOK: Module 5 Topic/Lesson: T1L2: Analyzing Numerical Data Displays (M5-25 thru M5-46)T1L3: Using Histograms to Display Data (M5-47 thru M5-60)T2L1: Analyzing Data Using Measures of Center (M5-71 thru M5-86)T2L2: Displaying the Five-Number Summary (M5-87 thru M5-104)

SKILLS PRACTICE: Module 5, Describing Variability of QuantitiesTopic 1: The Statistical Process (pp. 110–121)Topic 2: Numerical Summaries of Data (pp. 122–138)

MATHia: Module, Describing Variability of QuantitiesUnit: Measures of Central TendencyWorkspace: Calculating Mean, Median, Mode, and Range; Determining Appropriate Measures; Measuring the Effects of Changing Data Sets




6.SP.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

TEXTBOOK: Module 5 Topic/Lesson: T2L1: Analyzing Data Using Measures of Center (M5-71 thru M5-86) T2L2: Displaying the Five-Number Summary (M5-87 thru M5-104)

SKILLS PRACTICE: Module 5, Describing Variability of QuantitiesTopic 2: Numerical Summaries of Data (pp. 122–138)


6.SP.4 Display numerical data in plots on a number line, including dot plots (line plots), histograms, and box plots (GAISE Model, step 3).

TEXTBOOK: Module 5 Topic/Lesson: T1L2: Analyzing Numerical Data Displays (M5-25 thru M5-46)T1L3: Using Histograms to Display Data (M5-47 thru M5-60)T2L2: Displaying the Five-Number Summary (M5-87 thru M5-104)


MATHia: Module, Describing Variability of QuantitiesUnit: Displays of Numerical Data, Box PlotsWorkspace: Creating and Interpreting Stem Plots; Creating and Interpreting Dot Plots; Creating and Interpreting Histograms; Constructing Box Plots; Interpreting Box Plots

6.SP.5 Summarize numerical data sets in relation to their context, such as by:

6.SP.5.a Report the number of observations.

TEXTBOOK: Module 5 Topic/Lesson: T1L2: Analyzing Numerical Data Displays (M5-25 thru M5-46)T1L3: Using Histograms to Display Data (M5-47 thru M5-60)






6.SP.5.b Describe the nature of the attribute under investigation, including how it was measured and its units of measurement.

TEXTBOOK: Module 5Topic/Lesson: T1L2: Analyzing Numerical Data Displays (M5-25 thru M5-46)T1L3: Using Histograms to Display Data (M5-47 thru M5-60)



6.SP.5.c Find the quantitative measures of center (median and/or mean) for a numerical data set and recognize that this value summarizes the data set with a single number. Interpret mean as an equal or fair share. Find measures of variability (range and interquartile range) as well as informally describe the shape and the presence of clusters, gaps, peaks, and outliers in a distribution.

TEXTBOOK: Module 5 Topic/Lesson: T1L2: Analyzing Numerical Data Displays (M5-25 thru M5-46)T1L3: Using Histograms to Display Data (M5-47 thru M5-60)T2L1: Analyzing Data Using Measures of Center (M5-71 thru M5-86)T2L2: Displaying the Five-Number Summary (M5-87 thru M5-104)T2L4: Choosing Appropriate Measures (Mean Absolute Deviation is above grade level in this Lesson) (M5-117 thru M5-130)



6.SP.5.d Choose the measures of center and variability, based on the shape of the data distribution and the context in which the data were gathered.

TEXTBOOK: Module 5Topic/Lesson: T2L4: Choosing Appropriate Measures (Mean Absolute Deviation is above grade level in this Lesson) (M5-117 thru M5-130)

SKILLS PRACTICE: Module 5, Describing Variability of QuantitiesTopic 2: Numerical Summaries of Data (pp. 122–138)


middle school math solution: course 1 - carnegie learning

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