### 6.6.RP Ratios and Proportional Relationships

6 Understand ratio concepts and use ratio reasoning to solve problems.

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

Interpret ratios of two quantities Write a ratio to describe objects in a picture (Sixth grade – R.1)

Use ratios to solve word problems (Sixth grade – R.3)

#### 6.6.RP.2 Understand the concept of a unit rate a/b associated with a ratio a:b with b is not equal to 0, and use rate language in the context of a ratio relationship.

Interpret and calculate nit rates and equivalent rates (Sixth grade – R.8)

Calculate unit rates. Unit rates: word problems (Sixth grade – R.9)

#### 6.6.RP.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.

#### 6.6.RP.3.a 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.

Determine ratio tables (Sixth grade – R.2)

Equivalent ratios (Sixth grade – R.4)

Equivalent ratios: word problems (Sixth grade – R.5)

Compare ratios using tables: word problems (Sixth grade – R.6)

Coordinate graphs review (Sixth grade – W.1)

#### 6.6.RP.3.b Solve unit rate problems including those involving unit pricing and constant speed.

Unit rates and equivalent rates (Sixth grade – R.8)

Unit rates: word problems (Sixth grade – R.9)

Unit prices with fractions and decimals (Sixth grade – U.3)

Unit prices with customary unit conversions (Sixth grade – U.4)

#### 6.6.RP.3.c 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.

Percents of numbers and money amounts (Sixth grade – R.14)

Percents of numbers: word problems (Sixth grade – R.15)

Which is the better coupon? (Sixth grade – U.1)

Unit prices: which is the better buy? (Sixth grade – U.2)

Sale prices (Sixth grade – U.5)

Sale prices: find the original price (Sixth grade – U.6)

Percents – calculate tax, tip, mark-up, and more (Sixth grade – U.7)

#### 6.6.RP.3.d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Convert and compare customary units (Sixth grade – S.3)

Convert, compare, add, and subtract mixed customary units (Sixth grade – S.4)

Multiply and divide mixed customary units (Sixth grade – S.5)

Customary unit conversions involving fractions and mixed numbers (Sixth grade – S.6)

Convert and compare metric units (Sixth grade – S.7)

Convert between customary and metric systems (Sixth grade – S.8)

Unit prices with customary unit conversions (Sixth grade – U.4)

### 6.6.NS The Number System

#### 6 Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

#### 6.6.NS.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.

Divide whole numbers by unit fractions using models (Sixth grade – L.1)

Reciprocals (Sixth grade – L.2)

Divide whole numbers and unit fractions (Sixth grade – L.3)

Divide fractions (Sixth grade – L.5)

Estimate quotients when dividing mixed numbers (Sixth grade – L.6)

Divide fractions and mixed numbers (Sixth grade – L.7)

Divide fractions and mixed numbers: word problems (Sixth grade – L.8)

Add, subtract, multiply, or divide two fractions (Sixth grade – O.7)

Add, subtract, multiply, or divide two fractions: word problems (Sixth grade – O.8)

#### 6 Compute fluently with multi-digit numbers and find common factors and multiples.

#### 6.6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.

Divisibility rules (Sixth grade – C.1)

Division patterns with zeroes (Sixth grade – C.2)

Divide numbers ending in zeroes: word problems (Sixth grade – C.3)

Estimate quotients (Sixth grade – C.4)

Divide whole numbers – 2-digit divisors (Sixth grade – C.5)

Divide whole numbers – 3-digit divisors (Sixth grade – C.6)

Add, subtract, multiply, or divide two whole numbers (Sixth grade – O.1)

Add, subtract, multiply, or divide two whole numbers: word problems (Sixth grade – O.2)

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

Add and subtract decimal numbers (Sixth grade – G.1)

Add and subtract decimals: word problems (Sixth grade – G.2)

Estimate sums and differences of decimals (Sixth grade – G.3)

Maps with decimal distances (Sixth grade – G.4)

Multiply decimals (Sixth grade – H.1)

Estimate products of decimal numbers (Sixth grade – H.2)

Inequalities with decimal multiplication (Sixth grade – H.3)

Divide decimals by whole numbers (Sixth grade – H.4)

Divide decimals by whole numbers: word problems (Sixth grade – H.5)

Multiply and divide decimals by powers of ten (Sixth grade – H.6)

Division with decimal quotients (Sixth grade – H.7)

Inequalities with decimal division (Sixth grade – H.8)

Add, subtract, multiply, or divide two decimals (Sixth grade – O.4)

Add, subtract, multiply, or divide two decimals: word problems (Sixth grade – O.5)

Perform multiple operations with decimals (Sixth grade – O.6)

#### 6.6.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.

Identify factors (Sixth grade – E.4)

Greatest common factor (Sixth grade – E.7)

Least common multiple (Sixth grade – E.8)

GCF and LCM: word problems (Sixth grade – E.9)

#### 6 Apply and extend previous understandings of numbers to the system of rational numbers.

#### 6.6.NS.5 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, 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.

Understanding integers (Sixth grade – M.1)

Working with temperatures above and below zero (Sixth grade – S.9)

#### 6.6.NS.6 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.

#### 6.6.NS.6.a 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.

Absolute value and opposite integers (Sixth grade – M.2)

Integers on number lines (Sixth grade – M.3)

#### 6.6.NS.6.b 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.

Coordinate graphs review (Sixth grade – W.1)

Graph points on a coordinate plane (Sixth grade – W.2)

Reflections: graph the image (Sixth grade – BB.18)

#### 6.6.NS.6.c 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.

Decimal number lines (Sixth grade – F.9)

Integers on number lines (Sixth grade – M.3)

Graph integers on horizontal and vertical number lines (Sixth grade – M.4)

Rational numbers: find the sign (Sixth grade – P.6)

Coordinate graphs review (Sixth grade – W.1)

Graph points on a coordinate plane (Sixth grade – W.2)

Coordinate graphs as maps (Sixth grade – W.3)

Translations: graph the image (Sixth grade – BB.17)

#### 6.6.NS.7 Understand ordering and absolute value of rational numbers.

#### 6.6.NS.7.a Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram.

Write inequalities from number lines (Sixth grade – Z.2)

#### 6.6.NS.7.b Write, interpret, and explain statements of order for rational numbers in real-world contexts.

Compare rational numbers (Sixth grade – P.1)

Put rational numbers in order (Sixth grade – P.2)

#### 6.6.NS.7.c 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 real-world situation.

Absolute value and opposite integers (Sixth grade – M.2)

Absolute value of rational numbers (Sixth grade – P.3)

#### 6.6.NS.7.d Distinguish comparisons of absolute value from statements about order.

Put rational numbers in order (Sixth grade – P.2)

Absolute value of rational numbers (Sixth grade – P.3)

#### 6.6.NS.8 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.

Coordinate graphs review (Sixth grade – W.1)

Graph points on a coordinate plane (Sixth grade – W.2)

Coordinate graphs as maps (Sixth grade – W.3)

Distance between two points (Sixth grade – W.4)

Relative coordinates (Sixth grade – W.5)

### 6.6.EE Expressions and Equations

#### 6 Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Write multiplication expressions using exponents (Sixth grade – D.1)

Evaluate exponents (Sixth grade – D.2)

Find the missing exponent or base (Sixth grade – D.3)

Exponents with decimal bases (Sixth grade – D.4)

Exponents with fractional bases (Sixth grade – D.5)

#### 6.6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.

#### 6.6.EE.2.a Write expressions that record operations with numbers and with letters standing for numbers.

Write variable expressions (Sixth grade – X.1)

Write variable expressions: word problems (Sixth grade – X.2)

Write a two-variable equation (Sixth grade – AA.6)

#### 6.6.EE.2.b 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.

Identify terms and coefficients (Sixth grade – X.6)

#### 6.6.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, in the conventional order when there are no parentheses to specify a particular order (Order of Operations).

Perform multiple operations with whole numbers (Sixth grade – O.3)

Convert between Celsius and Fahrenheit (Sixth grade – S.10)

Evaluate variable expressions with whole numbers (Sixth grade – X.3)

Evaluate multi-variable expressions (Sixth grade – X.4)

Evaluate variable expressions with decimals, fractions, and mixed numbers (Sixth grade – X.5)

Complete a table for a two-variable relationship (Sixth grade – AA.5)

#### 6.6.EE.3 Apply the properties of operations to generate equivalent expressions.

Properties of addition (Sixth grade – X.7)

Properties of multiplication (Sixth grade – X.8)

Distributive property (Sixth grade – X.9)

Write equivalent expressions using properties (Sixth grade – X.11)

#### 6.6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them).

Add and subtract like terms (Sixth grade – X.12)

Identify equivalent expressions (Sixth grade – X.13)

#### 6 Reason about and solve one-variable equations and inequalities.

#### 6.6.EE.5 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.

Does x satisfy an equation? (Sixth grade – Y.1)

Find the solution from a set (Sixth grade – Y.2)

Solve one-step equations with whole numbers (Sixth grade – Y.6)

Solutions to inequalities (Sixth grade – Z.1)

Solve one-step inequalities (Sixth grade – Z.4)

#### 6.6.EE.6 Use 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.

Convert between Celsius and Fahrenheit (Sixth grade – S.10)

Write variable expressions: word problems (Sixth grade – X.2)

Write an equation from words (Sixth grade – Y.3)

Solve word problems involving two-variable equations (Sixth grade – AA.4)

#### 6.6.EE.7 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 nonnegative rational numbers.

Model and solve equations using algebra tiles (Sixth grade – Y.4)

Write and solve equations that represent diagrams (Sixth grade – Y.5)

Solve one-step equations with whole numbers (Sixth grade – Y.6)

Solve one-step equations with decimals, fractions, and mixed numbers (Sixth grade – Y.7)

Solve one-step equations: word problems (Sixth grade – Y.8)

#### 6.6.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.

Write inequalities from number lines (Sixth grade – Z.2)

Graph inequalities on number lines (Sixth grade – Z.3)

#### 6 Represent and analyze quantitative relationships between dependent and independent variables.

#### 6.6.EE.9 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.

Solve one-step equations: word problems (Sixth grade – Y.8)

Identify independent and dependent variables (Sixth grade – AA.2)

Find a value using two-variable equations (Sixth grade – AA.3)

Solve word problems involving two-variable equations (Sixth grade – AA.4)

Complete a table for a two-variable relationship (Sixth grade – AA.5)

Write a two-variable equation (Sixth grade – AA.6)

Identify the graph of an equation (Sixth grade – AA.7)

Graph a two-variable equation (Sixth grade – AA.8)

Interpret a graph: word problems (Sixth grade – AA.9)

Write an equation from a graph using a table (Sixth grade – AA.10)

### 6.6.G Geometry

#### 6 Solve real-world and mathematical problems involving area, surface area, and volume.

#### 6.6.G.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.

Area of compound figures (Sixth grade – BB.24)

Compare area and perimeter of two figures (Sixth grade – BB.28)

#### 6.6.G.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 = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Volume of cubes and rectangular prisms (Sixth grade – BB.36)

#### 6.6.G.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.

Coordinate graphs review (Sixth grade – W.1)

Distance between two points (Sixth grade – W.4)

#### 6.6.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.

Nets of 3-dimensional figures (Sixth grade – BB.35)

Surface area of cubes and rectangular prisms (Sixth grade – BB.37)

Volume and surface area of triangular prisms (Sixth grade – BB.38)

### 6.6.SP Statistics and Probability

#### 6 Develop understanding of statistical variability.

#### 6.6.SP.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers.

Identify representative, random, and biased samples (Sixth grade – DD.4)

#### 6.6.SP.2 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.

Stem-and-leaf plots (Sixth grade – CC.3)

Create line plots (Sixth grade – CC.5)

Interpret box-and-whisker plots (Sixth grade – CC.19)

#### 6.6.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.

Calculate mean, median, mode, and range (Sixth grade – DD.1)

Interpret charts to find mean, median, mode, and range (Sixth grade – DD.2)

Mean, median, mode, and range: find the missing number (Sixth grade – DD.3)

#### 6 Summarize and describe distributions.

#### 6.6.SP.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.

Interpret pictographs (Sixth grade – CC.1)

Create pictographs (Sixth grade – CC.2)

Stem-and-leaf plots (Sixth grade – CC.3)

Interpret line plots (Sixth grade – CC.4)

Create line plots (Sixth grade – CC.5)

Create frequency tables (Sixth grade – CC.7)

Interpret bar graphs (Sixth grade – CC.8)

Create bar graphs (Sixth grade – CC.9)

Interpret double bar graphs (Sixth grade – CC.10)

Create double bar graphs (Sixth grade – CC.11)

Create histograms (Sixth grade – CC.13)

Circle graphs with fractions (Sixth grade – CC.14)

Interpret line graphs (Sixth grade – CC.15)

Create line graphs (Sixth grade – CC.16)

Interpret double line graphs (Sixth grade – CC.17)

Create double line graphs (Sixth grade – CC.18)

Interpret box-and-whisker plots (Sixth grade – CC.19)

Choose the best type of graph (Sixth grade – CC.20)

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

#### 6.6.SP.5.a Reporting the number of observations.

Create frequency tables (Sixth grade – CC.7)

Create histograms (Sixth grade – CC.13)

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

Identify representative, random, and biased samples (Sixth grade – DD.4)

#### 6.6.SP.5.c 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.

Calculate mean, median, mode, and range (Sixth grade – DD.1)

Interpret charts to find mean, median, mode, and range (Sixth grade – DD.2)

Mean, median, mode, and range: find the missing number (Sixth grade – DD.3)