Evidence Outcomes:

1. Apply 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.” (CCSS: 6.RP.A.1)

2. Apply the concept of a unit rate a/b associated with a ratio a: b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is ¾ 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 non-complex fractions.) (CCSS: 6.RP.A.2)

3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations. (CCSS: 6.RP.A.3) 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. (CCSS: 6.RP.A.3.a) 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? (CCSS: 6.RP.A.3.b) 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. (CCSS: 6.RP.A.3.c) Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities. (CCSS: 6.RP.A.3.d)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Analyze scenarios to communicate the relationship between quantities

• Reason with quantities to compare

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Make meaning of a scenario to provide insight about a relationship between quantities

• Reason with numbers to construct arguments about the relationship between multiple quantities

In order to meet these essential understandings, students must know...

• The concept of a ratio

• The meaning of a unit rate

• The concept of a percentage

• How to use equivalent ratios, tape diagrams, double number line diagrams, and equations

• How to critically read a real-world problem

In order to meet these essential understandings, students must be able to...

• Describe the relationship between two quantities

• Convert rates to unit rates using simple fractions

• Make tables of equivalent ratios relating quantities with whole-number measurements, find missing values in tables, and plot the pairs of values on the coordinate plane

• Compare ratios using tables

• Solve problems involving unit rates

• Solve percentage problems involving finding the whole, given a part, and the percentage

• Convert measurement units

Evidence Outcomes:

1. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for 2/3 ÷ 3/4 and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that 2/3 ÷ 3/4 = 8/9 because 3/4 of 8/9 is 2/3. (In general, a/b ÷ c/d = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb. of chocolate equally? How many 3/4- cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area ½ square mi? (CCSS: 6.NS.A.1)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Fluidly move between contextualizing and decontextualizing during the problem solving process

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Interpret and compute quotients of fractions

• Contextualize fractional quotient problems to make sense of the standard algorithm

• Decontextualize real-world scenarios to preform necessary calculations for fraction division problems

In order to meet these essential understandings, students must know...

• The standard algorithm for dividing fractions

• Applications and uses of fractions

• How to critically read a real-world problem

In order to meet these essential understandings, students must be able to...

• Use/create visual fraction models to show the quotient

• Create a story context for a given fraction division problem

• Set up and solve a fraction division problem

Evidence Outcomes:

1. Fluently divide multi-digit numbers using the standard algorithm. (CCSS: 6.NS.B.2)

2. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. (CCSS: 6.NS.B.3)

3. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1 – 100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4(9 + 2). (CCSS: 6.NS.B.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Examine and apply a variety of strategies to accurately and efficiently solve decimal problems

• Discern patterns to articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Demonstrate flexibility and explain the methods used for computing with multi-digit decimal problems

• Produce accurate answers efficiently of multi-digit decimal problems

• Justify answers to multi-digit decimal problems by communicating clearly and accurately

• Apply an understanding of place-value to find common factors and multiples

In order to meet these essential understandings, students must know...

• Conceptual understanding of dividing multi-digit numbers

• Strategies for dividing multi-digit numbers (box method, partial quotients, long division, mental math)

• The algorithm for dividing multi-digit numbers (long division)

• Conceptual understanding of all operations for decimals

• Strategies for computing with decimals

  • Adding/Subtracting - expanded form, place value blocks, standard

  • Multiplying - grid, lattice, change into whole numbers, change into fractions, area model, standard

  • Dividing - place value blocks, long division

• The algorithms for operating with decimals

• The various symbols used in math to represent multiplication and division

• Meaning of the greatest common factor

• Meaning of the least common multiple

• The Distributive Property

In order to meet these essential understandings, students must be able to...

• Use models to uncover how division works

• Demonstrate flexibility in choosing methods to perform multi-digit number division

• Explain the various methods for dividing multi-digit numbers

• Produce accurate answers efficiently to multi-digit division problems

• Use models to uncover how operations with decimals work

• Demonstrate flexibility in choosing methods to perform the four operations with decimals

• Explain the various methods for operating with decimals

• Produce accurate answers efficiently to operations with decimals problems

• Discern patterns from examples and use knowledge of the structure of the base-ten number system to find the greatest common factor of two numbers

• Discern patterns from examples and use knowledge of the structure of the base-ten number system to find the least common multiple of two numbers

• Use the distributive property to express a sum of two whole numbers (1-100) by applying properties of multiplication and division

Evidence Outcomes:

1. Explain why 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, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real- world contexts, explaining the meaning of 0 in each situation. (CCSS: 6.NS.C.5)

2. Describe 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. (CCSS: 6.NS.C.6) Use opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; identify that the opposite of the opposite of a number is the number itself, e.g., -(-3) = 3 , and that 0 is its own opposite. (CCSS: 6.NS.C.6.a) Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; explain that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. (CCSS: 6.NS.C.6.b) 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. (CCSS: 6.NS.C.6.c)

3. Order and find absolute value of rational numbers. (CCSS: 6.NS.C.7) 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. (CCSS: 6.NS.C.7.a) Write, 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. (CCSS: 6.NS.C.7.b) Define the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of -30 dollars, write |-30| = 30 to describe the size of the debt in dollars. (CCSS: 6.NS.C.7.c) 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. (CCSS: 6.NS.C.7.d)

4. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate. (CCSS: 6.NS.C.8)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Fluidly move between contextualizing and decontextualizing during the problem solving process

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Contextualize positive numbers, negative numbers, and the absolute value

• Explain and justify the order of rational numbers using their location of the number line

• Apply appropriate tools to plot points on a number line and plot ordered pairs on a coordinate plane

In order to meet these essential understandings, students must know...

• The meaning of negative numbers

• Opposites

• How to write negative numbers

• Definition of the absolute value

• How to critically read a real-world problem

In order to meet these essential understandings, students must be able to...

• Explain why positive and negative numbers are used and related

• Decontextualize real-world problems by using positive and negative numbers and explaining the meaning of 0

• Extend the number line to be able to plot negative numbers

• Plot positive and negative points on a number line

• Explain what taking the opposite of an opposite generates

• Demonstrate an understanding of the meaning of a sign in front of a number in an ordered pair

• Explain what it means when an ordered pair only differs by signs

• Plot integers and other rational numbers on a horizontal or vertical number line

• Plot pairs of integers and other rational numbers (ordered pairs) on a coordinate grid

• Order and find the absolute value of rational numbers

• Interpret statements of inequality as it relates to their position on the number line

• Contextualize statements of order for rational numbers

• Solve and interpret solutions to real-world absolute value problems

• Distinguish comparisons of absolute value from statements about order

Evidence Outcomes:

1. Write and evaluate numerical expressions involving whole-number exponents. (CCSS: 6.EE.A.1)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Recognize that expressions can be written in multiple forms and describe cause-and-effect relationships and patterns

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Communicate a justification of why expressions are equivalent using arguments about properties of operations and whole numbers

• Visualize complex expressions as being composed of several more simple expressions

• Analyze the structure of expressions and determine equivalence among expressions

In order to meet these essential understandings, students must know...

• The meaning of an expression

• The function of an exponent

• What a variable is and what it represents

• Vocabulary of Expressions

  • Sum

  • Term

  • Product

  • Factor

  • Quotient

  • Coefficient

• Different symbols used to represent multiplication and division

• Order of Operations

• Properties of Operations

  • Commutative Property

  • Associative Property

  • Distributive Property

In order to meet these essential understandings, students must be able to...

• Write and evaluate numerical expressions involving exponents

• Write expressions where a letter represents an unknown number

• Communicate the parts of an expression using appropriate vocabulary

• Explain how the different terms within an expression can be viewed as single entities (i.e. 2(8+7) can be described as the product of two factors where (8+7) is both a single entity and a sum of two terms)

• Evaluate expressions when given a specific value for the variable, including those arising from real-world scenarios (applying order of operations)

• Apply properties of operations to generate equivalent expressions

• Justify when two expressions are equivalent using repeated reasoning

Evidence Outcomes:

1. Write, read, and evaluate expressions in which letters stand for numbers. (CCSS: 6.EE.A.2) 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. (CCSS: 6.EE.A.2.a) 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. (CCSS: 6.EE.A.2.b) Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s^3 and A = 6s^2 to find the volume and surface area of a cube with sides of length s = ½.

2. Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3(2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6(4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y. (CCSS: 6.EE.A.3)

3. Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for. (CCSS: 6.EE.A.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Investigate unknown values to form hypothesis, make observations, and draw conclusions

• Make sense of quantities in the real-world and determine how they relate to one another in a problem

• Fluidly move between contextualizing and decontextualizing during the problem solving process

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Decontextualize a problem by representing the relationship as an expression

• Ensure problem solving is making sense by contextualizing the problem back into the real-world meaning

• Represent the situation, consider the units involved, attend to the meaning of quantities, and use properties of equality to solve one-variable equations and inequalities

In order to meet these essential understandings, students must know...

• Properties of Equality

  • Additive Inverse Property

  • Multiplicative Inverse Property

• What a variable is and what it represents

• What an inequality is

• Procedures for solving single-step equations and inequalities

• Rules for solving single-step equations and inequalities

• How to critically read a real-world problem

In order to meet these essential understandings, students must be able to...

• Describing 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?

• Determine whether a given number in a specified set makes an equation or inequality using substitution

• Decontextualize real-world problems using variables to represent numbers and write expressions

• Solve real-world and mathematical problems that result in one-step equations

• Write an inequality to represent constraints or conditions occurring in real-world and mathematical problems

• Explain and justify why certain inequalities have infinitely many solutions

• Represent solutions to inequalities on a number line

Evidence Outcomes:

1. Describe 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. (CCSS: 6.EE.B.5)

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

3. Solve real-world 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 non-negative rational numbers. (CCSS: 6.EE.B.7)

4. Write an inequality of the form x > c, x ≥ c, x < c, or x ≤ c to represent a constraint or condition in a real-world or mathematical problem. Show that inequalities of the form x > c, x ≥ c, x < c, or x ≤ c have infinitely many solutions; represent solutions of such inequalities on number line diagrams. (CCSS: 6.EE.B.8)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Articulate how mathematical concepts relate to one another in the context of a problem or abstract relationships.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Analyze the relationship between the independent and dependent variable in a given scenario

• Reason about the operations that relate constant and variable quantities in equations with dependent and independent variables

• Model with mathematics to represent, analyze, make predictions, or provide insights into mathematical representations to draw conclusions about real-world problems arising in everyday life and society

In order to meet these essential understandings, students must know...

• Independent variable

• Dependent variable

In order to meet these essential understandings, students must be able to...

• Use variables to represent two quantities in a real-world problem that change in relationship to one another

• Write an equation to express one quantity (dependent variable) in terms of the other (independent variable)

• Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation

Evidence Outcomes:

1. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time. (CCSS: 6.EE.C.9)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Develop understanding of statistical variability.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Make sense of practical problems by turning them into statistical investigations

• Decontextualize data collected from statistical investigations by describing the data’s center, spread, and shape

• Model variability in data collected to answer statistical questions and draw conclusions based on center, spread, and shape

In order to meet these essential understandings, students must know...

• Statistical question

• Distribution

• Measures of central tendencies (mean, median, mode)

• Variation (range)

In order to meet these essential understandings, students must be able to...

• Identify a statistical question as one that anticipate variability in the data

• Demonstrate that a set of collected data that answers a statistical question has a distribution that can be described by its measures of central tendencies

• Determine the difference between a measure of central tendency and the variation of a data set

Evidence Outcomes:

1. Identify 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. (CCSS: 6.SP.A.1)

2. Demonstrate that a set of data collected to answer a statistical question has a distribution that can be described by its center, spread, and overall shape. (CCSS: 6.SP.A.2)

3. Explain 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. (CCSS: 6.SP.A.3)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Communicate effectively based on purpose, task, and audience using appropriate vocabulary.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Move from context to abstraction and back to context while finding and using measures of central and variability and describing what they mean in the context of the data

• Analyze data sets and draw conclusions based on the data display and measures of center and/or variability

• Communicate how different data displays communicate different meanings

In order to meet these essential understandings, students must know...

• Different number line data displays

  1. Dot plots

  2. Histograms

  3. Box plots

• Quantitative measures of central (median and/or mean)

• Quantitative variability (interquartile range and/or mean absolute deviation)

In order to meet these essential understandings, students must be able to...

• Display numerical data in plots on a number line

• Write informative texts about data sets in relation to their context

  • Reporting number of observations

  • Describing the nature of the attribute under investigation (how it was measured and its unit of measure)

  • Giving the measures of central tendency and variability

  • Describing overall patterns and deviations of the data (referencing context)

  • Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

Evidence Outcomes:

1. Display numerical data in plots on a number line, including dot plots, histograms, and box plots. (CCSS: 6.SP.B.4)

2. Summarize numerical data sets in relation to their context, such as by: (CCSS: 6.SP.B.5) Reporting the number of observations. (CCSS: 6.SP.B.5.a) Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. (CCSS: 6.SP.B.5.b) Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered. (CCSS: 6.SP.B.5.c) Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered. (CCSS: 6.SP.B.5.d)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Communicate effectively based on purpose, task, and audience using appropriate vocabulary.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Move from context to abstraction and back to context while finding and using measures of central and variability and describing what they mean in the context of the data

• Analyze data sets and draw conclusions based on the data display and measures of center and/or variability

• Communicate how different data displays communicate different meanings

In order to meet these essential understandings, students must know...

• Different number line data displays

  • Dot plots

  • Histograms

  • Box plots

• Quantitative measures of central (median and/or mean)

• Quantitative variability (interquartile range and/or mean absolute deviation)

In order to meet these essential understandings, students must be able to...

• Display numerical data in plots on a number line

• Write informative texts about data sets in relation to their context

  • Reporting number of observations

  • Describing the nature of the attribute under investigation (how it was measured and its unit of measure)

  • Giving the measures of central tendency and variability

  • Describing overall patterns and deviations of the data (referencing context)

  • Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered

Evidence Outcomes:

1. Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems. (CCSS: 6.G.A.1)

2. Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = 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. (CCSS: 6.G.A.2)

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

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. (CCSS: 6.G.A.4)

Transfer Goals: Based on the Evidence Outcomes, what will students transfer to new contexts/situations?

• Based on an understanding of any problem, initiate a plan, execute it, and evaluate the reasonableness of the solution.

Essential Understandings: In order to meet these transfer goals, the essential ideas and core processes students must understand are...

• Make sense of a problem involving area, surface area, and volume by determining the meaning of the problem, monitoring their progress, and adjusting work when necessary

• Apply geometric thinking, concepts, and formulas to solve real-world problems amount area, surface area, and volume

• Utilize tools to support modeling real-world geometry problems.

In order to meet these essential understandings, students must know...

• Area Formulas

  • Triangles

  • Quadrilaterals

  • Polygons composed of rectangles and triangles

• Volume of right rectangular prisms formula

• The use of a nets

In order to meet these essential understandings, students must be able to...

• Find the area of triangles and special quadrilaterals (parallelogram, rectangle, square, rhombus, trapezoid)

• Decompose composite polygons comprised of rectangles and triangles to find the area

• Find the volume of a right rectangular prisms with fractional edges using unit cubes

• Derive the volume formula for a rectangular prism

• Apply the formula find the volume of rectangular prisms in real-world contexts

• Solve problems that require drawing polygons in the coordinate plane when given the points of the vertices

• Use coordinates to find the lengths of sides of a polygon drawn in the coordinate plane to solve problems

• FInd the surface area of prisms comprised of rectangles and triangles using nets