Grade 6 Math Colorado standards Standards

Geometry

Statistics & Probability

Expressions & Equations

The Number System

Ratios & Proportional Relationships

Standards for Mathematical Practice

Apply and extend previous understandings of arithmetic to algebraic expressions.

Expressions & Equations: Apply and extend previous understandings of arithmetic to algebraic expressions.

Write and evaluate numerical expressions involving whole-number exponents.

Write and evaluate numerical expressions involving single-digit integer factors, like -4 x 3 can mean 3 groups of -4, which totals -12.

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

Read and evaluate expressions in which letters stand for numbers.

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

Identify parts of an expression that indicate operations (add, subtract, multiply, divide).

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.

Evaluate expressions at specific values of their variables.

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 arithmetic operations in the conventional order.

For example, if x = 4, x + 1 should be interpreted as 4 + 1, which has a value of 5.

Apply the properties of operations to generate equivalent expressions.

Apply the properties of operations to identify given equivalent expressions using single-digit integers.

For example, apply the distributive property to the expression 3(2 + 4) to identify the equivalent expression 6 + 12.

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

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.

Expressions & Equations: Reason about and solve one-variable equations and inequalities.

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.

Answer the question: Which values from a specified set make the equation true?

Use substitution to determine whether a given number in a specified set makes an equation true.

For example, for the equation 5x = 1-, does x = 1 make the equation true? Does x = 2? x = 3? x = 4?

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.

Identify or match an expression using variables to a real-world problem.

For example, a number of apples plus a number of bananas could match the expression a + b.

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.

Solve real-world and mathematical problems by solving equations of the form x ± p = q and px = q for cases in which p, q and x are all whole numbers.

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.

Evaluate an inequality of the form x > c or x < c that represents a real-world or mathematical problem.

For example, if you have more than 8 dollars, the inequality x > 8 would be true for any amount of money (x) greater than 8 dollars.

Represent and analyze quantitative relationships between dependent and independent variables.

Expressions & Equations: Represent and analyze quantitative relationships between dependent and independent variables.

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.

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

Analyze the relationship between variables using a table, such as a 2- column table with on column labeled with each variable.

For example, a table of people and shoes in the classroom might have a column p, for people, and a column s, for shoes, and although the number of people and shoes can vary, there is always twice as many people as shoes.

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

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

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.

Find the area of right triangles, other triangles, parallelograms, and trapezoids by composing into rectangles or decomposing into triangles and other shapes.

Apply these techniques in the context of solving real-world and mathematical problems.

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.

Find the volume of a right rectangular prism with whole number edge lengths by packing it with unit cubes and show that the volume is the same as would be found by multiplying the edge lengths of the prism.

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.

Create polygons in the coordinate plane given coordinates for the vertices.

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.

Represent three-dimensional figures using nets made up of rectangles, and use the nets to find the surface area of these figures.

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

The Number System: Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

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.

Illustrate quotients of fractions, and solve word problems involving division of a larger benchmark fraction by a smaller benchmark fraction resulting in whole-number quotients, e.g., by using visual fraction models to represent division in problems like 3 divided by 1 .

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

The Number System: Compute fluently with multi-digit numbers and find common factors and multiples.

Fluently divide multi-digit numbers using the standard algorithm.

Divide multi-digit numbers using illustrations or demonstrations of fair share and equal share strategies.

For example, students can divide 36 cookies into 12 boxes by counting out one cookie per box until there are 3 in each box.

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

Add, subtract, multiply, and divide numbers less than ten with one or two decimal places using appropriate strategies for each operation and/or a calculator.

For example, do simple calculations with amounts of money.

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.

Find common factors of two whole numbers less than or equal to 20 and common multiples of two whole numbers less than or equal to 10.

For example, if there are 8 hot dogs in a package and 6 buns in a package, how many packages of each do you buy to not have hot dogs or buns left over?

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

The Number System: Apply and extend previous understandings of numbers to the system of rational numbers.

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.

Identify positive and negative numbers that 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).

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.

Identify integers on a number line.

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.

Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative integer coordinates.

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.

Use opposite signs of numbers indicating locations on opposite sides of 0 on the number line.

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.

Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane.

Find and position integers on a horizontal or vertical number line diagram.

Find and position pairs of integers on a coordinate plane.

Order and find absolute value of rational numbers.

Order and find absolute value of integers between -10 and 10.

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

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

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

For example, identify that -2 is greater than -5 because -2 is located to the right of -5 on a number line.

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.

Relate negative values to a real-world situation, such as -2 could represent owing someone two dollars.

Distinguish comparisons of absolute value from statements about order.

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.

Solve real-world and mathematical problems represented by graphed integer-coordinate points in all four quadrants of the coordinate plane.

Find distances between points with the same first coordinate or the same second coordinate.

For example, if Shelby lives at 1st St. and 6th Ave., and Lisa les at 4th St. and 6th Ave., how many blocks away from each other do they live?

Understand ratio concepts and use ratio reasoning to solve problems.

Ratios & Proportional Relationships: Understand ratio concepts and use ratio reasoning to solve problems.

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

Apply the concept of a ratio and use ratio language or actions to describe a ratio relationship between two quantities up to at least 5 each.

For example, count out two shoes for each one person, or indicate there are four wheels for each car in the parking lot.

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.

Apply the concept of a unit rate and use rate language and/or written representations in the context of a ratio relationship.

For example, student uses ‘per’ and ‘for each’ language, such as “There are 24 hours for each day” or “There are two gloves per student.” (Expectations for unit rates in this grade are limited to non-complex fractions)

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.

Use ratio and rate reasoning to solve real-world problems, e.g., by reasoning about tables of equivalent rations.

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 should be simple, like 1:2 is equivalent to 2:4, or 2:3 is equivalent to 4:6.

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

For example, students should be able to make a table relating the two socks they wear each day and use that to find the number of socks they need for a 3-day trip.

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.

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

Develop understanding of statistical variability.

Statistics & Probability: Develop understanding of statistical variability.

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

Match statistical questions to appropriate sources of data. For example, match the question, “How tall are 6th graders?” to a sample of 6th grade students.

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.

Match statistical questions to a distribution of data that can be described by its center, spread, and overall shape.

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.

Match given measures of center (mean or median) and variation (range) to a display of a data distribution.

Summarize and describe distributions.

Statistics & Probability: Summarize and describe distributions.

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

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

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

Summarize numerical data sets by counting the number of observations, identifying the largest and smallest observations, and informally identifying observations near the center.

Reporting the number of observations.

Interpret these values in a real-world context.

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

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.

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.

Algebra and Functions

Data, Statistics, and Probability

Geometry

Make sense of problems and persevere in solving them.

Reason abstractly and quantitatively.

Construct viable arguments and critique the reasoning of others.

Model with mathematics.

Use appropriate tools strategically.

Attend to precision.

Look for and make use of structure.

Look for and express regularity in repeated reasoning.

Number and Quantity