---
title: "Math 6"
url: "https://books.hrgrvs.net/2/standards/74/math-6"
---

# Standards for Mathematical Content Grade 6

## Ratios and Proportional Relationships

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

-   6.RP.1. Write and describe the relationship in real life context
    between two quantities using ratio language. *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.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) and apply it to solve real world
    problems (e.g., unit pricing, constant speed).


-   *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."*


-   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).

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, and understand equivalencies.

b)  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?*

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.

d)  Use ratio reasoning to convert measurement units between given
    measurement systems (e.g., convert kilometers to miles); manipulate
    and transform units appropriately when multiplying or dividing
    quantities.

## The Number System

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

-   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). *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.2. Fluently multiply and divide multi-digit whole numbers using
    the standard algorithm. Express the remainder as a whole number,
    decimal, or simplified fraction; explain or justify your choice
    based on the context of the problem.

-   6.NS.3. Fluently add, subtract, multiply, and divide multi-digit
    decimals using the standard algorithm for each operation. Express
    the remainder as a terminating decimal, or a repeating decimal, or
    rounded to a designated place value.

-   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).*

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

-   6.NS.5 Understand that positive and negative numbers 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, explain the
    meaning of 0 in each situation.

-   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.

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.

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.

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.

-   6.NS.7. Understand ordering and absolute value of rational numbers.

a)  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.*

b)  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.*

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

d)  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.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.

## Expressions and Equations

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

-   6.EE.1. Write and evaluate numerical expressions involving
    whole-number exponents *For example multiply by powers of 10 and
    products of numbers using exponents.*

> *(7^3^ = 7•7•7)*

-   6.EE.2. Write, read, and evaluate expressions in which letters stand
    for numbers.

a)  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.*

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. *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.*

c)  Evaluate expressions and formulas. Include formulas used in
    real-world problems. Perform arithmetic operations, including those
    involving whole number exponents, in the conventional order with or
    without parentheses. (Order of Operations)

-   6.EE.3. Apply the properties of operations to generate equivalent
    expressions. Model (e.g., manipulatives, graph paper) and apply the
    distributive, commutative, identity, and inverse properties with
    integers and variables by writing equivalent expressions. *For
    example, apply the distributive property to the expression 3 (2 + x)
    to produce the equivalent expression 6 + 3x.*

-   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). *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.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.

*For example: does 5 make 3x \> 7 true?*

-   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.

-   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.

-   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.

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

-   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. *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.*

## Geometry

### 

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

-   6.G.1. Find the area of right triangles, other triangles, special
    quadrilaterals, and polygons by composing or decomposing into other
    polygons (e.g., rectangles and triangles). Apply these techniques in
    the context of solving real-world and mathematical problems.

-   6.G.2. Apply the standard formulas to find volumes of prisms. Use
    the attributes and properties (including shapes of bases) of prisms
    to identify, compare or describe three-dimensional figures including
    prisms and cylinders.

-   6.G.3. Draw polygons in the coordinate plane given coordinates for
    the vertices; determine the length of a side joining the coordinates
    of vertices with the same first or the same second coordinate. Apply
    these techniques in the context of solving real-world and
    mathematical problems.

-   6.G.4. Represent three-dimensional figures (e.g., prisms) 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.

-   6.G.5. Identify, compare or describe attributes and properties of
    circles (radius, and diameter). (L)

## Statistics and Probability

### Develop understanding of statistical variability.

-   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. *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.2 Understand that a set of data has a distribution which can be
    described by its center (mean, median, or mode), spread (range), and
    overall shape and can be used to answer a statistical question.

-   6.SP.3 Recognize that a measure of center (mean, median, or mode)
    for a numerical data set summarizes all of its values with a single
    number, while a measure of variation (range) describes how its
    values vary with a single number.

### Summarize and describe distributions.

-   6.SP.4. Display numerical data in plots on a number line, including
    dot or line plots, histograms and box (box and whisker) plots.

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

a)  Reporting the number of observations (occurrences).

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

c)  Giving quantitative measures of center (median and/or mean) and
    variability (interquartile range), as well as describing any overall
    pattern and any outliers with reference to the context in which the
    data were gathered.

d)  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.

-   6.SP.6 Analyze whether a game is mathematically fair or unfair by
    explaining the probability of all possible outcomes.

-   6.SP.7. Solve or identify solutions to problems involving possible
    combinations (e.g., if ice cream sundaes come in 3 flavors with 2
    possible toppings, how many different sundaes can be made using only
    one flavor of ice cream with one topping?)

# Standards for Mathematical Practice

Instruction around the Standards of Mathematical Practices is delivered
across all grades K-12. These eight standards define experiences that
build understanding of mathematics and ways of thinking through which
students develop, apply, and assess their knowledge.

## Make sense of problems and persevere in solving them.

-   explain correspondences between a new problem and previous problems

-   represent algebraic expressions numerically, graphically,
    concretely/with manipulatives, verbally/written

-   explain connections between the multiple representations

-   determine the question that needs to be answered

-   analyze a problem and make a plan for solving it

-   choose a reasonable strategy

-   identify the knowns and unknowns in a problem

-   use previous knowledge and skills to simplify and solve problems

-   break a problem into manageable parts or simpler problems

-   solve a problem in more than one way

## Reason abstractly and quantitatively.

-   represent a situation symbolically and carry out its operations

-   create a coherent representation of the problem

-   translate an algebraic problem to a real world context

-   explain the relationship between the symbolic abstraction and the
    context of the problem

-   compute using different properties

-   consider the quantitative values, including units, for the numbers
    in a problem

## Construct viable arguments and critique the reasoning of others.

-   construct arguments using both concrete and abstract explanations

-   justify conclusions, communicate conclusions, and respond to the
    arguments

-   listen to arguments, critique their viability, and ask questions to
    clarify the argument

-   compare effectiveness of two arguments by identifying and explaining
    both logical and/or flawed reasoning

-   recognize general mathematical truths and use statements to justify
    the conjectures

-   identify special cases or counter-examples that don't follow the
    mathematical rules

-   infer meaning from data and make arguments using its context

## Model with Mathematics.

-   apply mathematics to solve problems arising in everyday life and
    society

-   identify important quantities in a practical situation and map their
    relationships using such tools as diagrams, two-way tables, graphs,
    and formulas

-   interpret their mathematical results in the context of the situation
    and reflect on whether the results make sense

-   make assumptions and approximations to simplify a situation,
    realizing the final solution will need to be revised

-   analyze quantitative relationships to draw conclusions

-   reflect on whether their results make sense

-   improve the model if it has not served its purpose

## Use appropriate tools strategically.

-   select and use tools appropriate to the task: pencil and paper,
    protractor, visual and physical fraction models, algebra tiles,
    geometric models, calculator, spreadsheet, and interactive geometry
    software.

-   use estimation and other mathematical knowledge to confirm the
    accuracy of their problem solving

-   identify relevant external and digital mathematical resources and
    use them to pose or solve problems

-   represent and compare possibilities visually with technology when
    solving a problem

-   explore and deepen their understanding of concepts through the use
    of technological tools

## Attend to precision.

-   use clear definitions in explanations

-   understand and use specific symbols accurately and consistently:
    equality, inequality, ratios, parenthesis for multiplication and
    division, absolute value, square root

-   specify units of measure, and label axes to clarify the
    correspondence with quantities in a problem

-   calculate accurately and efficiently, express numerical answers with
    a degree of precision appropriate for the problem context

## Look for and make use of structure.

-   discern a pattern or structure

-   understand complex structures as single objects or as being composed
    of several objects

-   check if the answer is reasonable

## Look for and express regularity in repeated reasoning.

-   identify if calculations or processes are repeated

-   use alternative and traditional methods to solve problems

-   evaluate the reasonableness of their intermediate results, while
    attending to the details