## 6.EE: Expressions & Equations

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

6.EE.A.1

Write and evaluate numerical expressions involving whole-number exponents.

Prime Factorization I
Prime Factorization II
Order of Operations (with Exponents)
More Order of Operations (with Exponents)
Order of Operations (Four Basic Operations and Exponents)
Order of Operations (with Parentheses)
Order of Operations (Four Basic Operations, Exponents, and Parentheses)
Order of Operations: Evaluating Numerical Expressions
More Word Problems with Whole Numbers
Knowledge is the Key
Order of Operations (Using Division for Grouping)
Exponents: Squares and Cubes
Exponents: Powers with Positive Bases
Writing and Evaluating Numerical Expressions
6.EE.A.2

Write, read, and evaluate expressions in which letters stand for numbers.

Parts of an Expression
6.EE.A.2a

Write expressions that record operations with numbers and with letters standing for numbers.

Writing More Algebraic Expressions
Writing Algebraic Expressions
Introduction to Writing Algebraic Expressions
Evaluate Algebraic Expressions
*For example, express the calculation “Subtract y from 5” as 5 – y*.
6.EE.A.2b

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*.
6.EE.A.2c

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).

Patterns and Sequences (Using Formulas)
Sequences: Using a Formula to Find Terms
Evaluating Decimal Expressions and Functions
The Secret Rule
Input/Output Tables
Evaluate Algebraic Expressions
*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
6.EE.A.3

Apply the properties of operations to generate equivalent expressions.

Parts of an Expression
Simplifying Algebraic Expressions: Like Terms
Equivalent Expressions and the Distributive Property
*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*.
6.EE.A.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).

*For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.*.### Reason about and solve one-variable equations and inequalities.

6.EE.B.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.

6.EE.B.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.

Writing Algebraic Expressions
Writing Algebraic Expressions: Representing Quantities in Real World Situations
6.EE.B.7

Solve real-world and mathematical problems by writing and solving equations of the form

Solving One-Step Equations
Using One-Step and Two-Step Equations to Solve Problems
Solving Problems with Fractions
Solving One-Step and Two-Step Equations
Solving One-Step and Two-Step Equations with Rationals
One Step Equations: Introduction
Introduction to Writing Algebraic Expressions
Writing Algebraic Equations to Solve Problems
Solving Real World Problems by Writing and Solving Equations
*x*+*p*=*q*and*px*=*q*for cases in which*p*,*q*and*x*are all nonnegative rational numbers.
6.EE.B.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.### Represent and analyze quantitative relationships between dependent and independent variables.

6.EE.C.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. 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.

Representing Situations as Tables of Values (Using One Operation)
Representing Situations as Tables of Values
Representing Situations as Tables of Values (Using More Than One Operation)
Completing a Table of Values II
Completing a Table of Values I
Representing Situations with Two variables
## 6.G: Geometry

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

6.G.A.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.

Exploring Areas of Triangles
Finding the Areas of Triangles on a Grid
Finding the Areas of Triangles
Areas of Parallelograms and Rhombi
Areas of Rhombi and Trapezoids
Area of Polygons
Diagonals and Lines of Symmetry in Regular Polygons
Perimeter and Area of Combined Polygons II
Perimeter and Area of Polygons Challenging Problems
Mystery in the Sand of Egypt
Measure Under Pressure
6.G.A.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

Volume of Rectangular Prisms with Fractional Edges
*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.
6.G.A.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.

Plotting Points and Figures on a Coordinate Grid
6.G.A.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.

Surface Area of Prisms
## 6.NS: The Number System

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

6.NS.A.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.

Dividing Fractions (Using Patterns with Multiplication)
What is a Reciprocal?
Dividing Fractions
Dividing Mixed Numbers
*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 1/2 square mi?*.### Compute fluently with multi-digit numbers and find common factors and multiples.

6.NS.B.2

Fluently divide multi-digit numbers using the standard algorithm.

* Division of Whole Numbers (Sept. 2013)
6.NS.B.3

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

Multiplication of Decimals
More Multiplication of Decimals
Addition and Subtraction of Decimals (including More Than Two Addends)
Review of Multiplication with Decimals
Division of Decimals
Measure Under Pressure
More Review of Multiplication with Decimals
Operations with Decimals
6.NS.B.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.

Understanding GCF and LCM
Greatest Common Factor and Least Common Multiple
Greatest Common Factor and Least Common Multiple: Applications and Alternative Methods
Multiples and Consecutive Numbers
LCM and GCF Problems
Completing a Table of Values I
More Word Problems with Whole Numbers
Greatest Common Factor
Least Common Multiple
Greatest Common Factor
*For example, express 36 + 8 as 4 (9 + 2).*.### Apply and extend previous understandings of numbers to the system of rational numbers.

6.NS.C.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.

Using Integers to Represent Situations
More Integer Addition and Subtraction Concepts to Solve Problems I
Using Integer Addition and Subtraction Concepts to Solve Problems II
6.NS.C.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.NS.C.6a

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.

6.NS.C.6b

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.

6.NS.C.6c

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.

Locating and Identifying Integers I
Locating and Identifying Integers II
Locating and Identifying Integers III
Locating and Identifying Integers V
Plotting Points and Figures on a Coordinate Grid
Locating and Identifying Integers VI
Locating and Identifying Integers IV
6.NS.C.7

Understand ordering and absolute value of rational numbers.

Comparing and Ordering Integers
Integers and Divisibility
Multiples and Integers
Understanding Integers: One Greater
Introduction to Absolute Value
6.NS.C.7a

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

Using Integer Addition and Subtraction Concepts to Solve Problems II
*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*.
6.NS.C.7b

Write, interpret, and explain statements of order for rational numbers in real-world contexts.

*For example, write –3*.^{o}C > –7^{o}C to express the fact that –3^{o}C is warmer than –7^{o}C
6.NS.C.7c

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
*For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars*.
6.NS.C.7d

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*.
6.NS.C.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.

Problem Solving on a Coordinate Plane
Plotting Points and Figures on a Coordinate Grid
Mystery in the Sand of Egypt
Polygons in the Coordinate Plane
## 6.RP: Ratios & Proportional Relationships

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

6.RP.A.1

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

Ratios
Writing Ratios and Solving Proportions
Using Proportions
What is a Ratio
Rates and Ratios
*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.”*
6.RP.A.2

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.

Rates and Ratios
Solving Unit Rate Problems
*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.”*^{1}
6.RP.A.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.

Representing Situations as Tables of Values (Using One Operation)
Circle Graphs
Using Proportions
The Lost Talisman
Measure Under Pressure
6.RP.A.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.

Equivalent Ratios in Tables and Graphs
Representing Situations as Tables of Values
Completing a Table of Values II
Input/Output Tables
6.RP.A.3b

Solve unit rate problems including those involving unit pricing and constant speed.

Using Proportions to Solve Problems
More Word Problems with Whole Numbers
Solving Unit Rate Problems
*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?*
6.RP.A.3c

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.

Using Proportions to Solve Problems
Fractions, Decimals, and Percents II
Percent Concepts
Finding Percent of a Quantity
6.RP.A.3d

Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.

Using Ratio Reasoning to Convert Measurement Units
## 6.SP: Statistics & Probability

### Develop understanding of statistical variability.

6.SP.A.1

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

Statistical Questioning and Data Distribution
*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*.
6.SP.A.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.

Statistical Questioning and Data Distribution
6.SP.A.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.

### Summarize and describe distributions.

6.SP.B.4

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

Line Plots
Histograms
6.SP.B.5

Summarize numerical data sets in relation to their context, such as by:

Statistical Questioning and Data Distribution
6.SP.B.5a

Reporting the number of observations.

6.SP.B.5b

Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

6.SP.B.5c

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.

Median and Mode
Range, Minimum, Maximum and Mean
More Range, Minimum, Maximum and Mean
Reaching the Possible
Range, Interquartile Range and Box Plots
Mean and Mean Absolute Deviation
6.SP.B.5d

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.