---
title: "Math 8"
url: "https://books.hrgrvs.net/2/standards/76/math-8"
---



# Standards for Mathematical Content Grade 8

## The Number System

### Know that there are numbers that are not rational, and approximate them by rational numbers.

-   8.NS.1. Classify real numbers as either rational (the ratio of two
    integers, a terminating decimal number, or a repeating decimal
    number) or irrational.

-   8.NS.2. Order real numbers, using approximations of irrational
    numbers, locating them on a number line. *For example, show that*
    √*2 is between 1 and 2, then between 1.4 and 1.5, and explain how to
    continue on to get better approximations.*

-   8.NS.3. Identify or write the prime factorization of a number using
    exponents.

## Expressions and Equations

### 

### Work with radicals and integer exponents.

-   8.EE.1. Apply the properties (product, quotient, power, zero,
    negative exponents, and rational exponents) of integer exponents to
    generate equivalent numerical expressions. *For example, 3^2^ ×
    3^--5^ = 3^--3^ = 1/3^3^ = 1/27.*

-   8.EE.2. Use square root and cube root symbols to represent solutions
    to equations of the form *x*^2^ = *p* and *x*^3^ = *p*, where *p* is
    a positive rational number. Evaluate square roots of small perfect
    squares and cube roots of small perfect cubes. Know that √2 is
    irrational.

-   8.EE.3. Use numbers expressed in the form of a single digit times an
    integer power of 10 to estimate very large or very small quantities,
    and to express how many times as much one is than the other. *For
    example, estimate the population of the United States as 3 × 10^8^
    and the population of the world as 7 × 10^9^, and determine that the
    world population is more than 20 times larger.*

-   8.EE.4. Perform operations with numbers expressed in scientific
    notation, including problems where both standard notation and
    scientific notation are used. Use scientific notation and choose
    units of appropriate size for measurements of very large or very
    small quantities. Interpret scientific notation that has been
    generated by technology.

### Understand the connections between proportional relationships, lines, and linear equations.

-   8.EE.5. Graph linear equations such as *y=mx+b*, interpreting *m* as
    the slope or rate of change of the graph and *b* as the
    *y*-intercept or starting value. Compare two different proportional
    relationships represented in different ways. *For example, compare a
    distance-time graph to a distance-time equation to determine which
    of two moving objects has greater speed.*

-   8.EE.6. Use similar triangles to explain why the slope *m* is the
    same between any two distinct points on a non-vertical line in the
    coordinate plane; derive the equation *y* = *mx* for a line through
    the origin and the equation *y* = *mx* + *b* for a line intercepting
    the vertical axis at *b*.

### Analyze and solve linear equations and pairs of simultaneous linear equations.

-   8.EE.7. Solve linear equations in one variable.

a)  Give examples of linear equations in one variable with one solution,
    infinitely many solutions, or no solutions. Show which of these
    possibilities is the case by successively transforming the given
    equation into simpler forms, until an equivalent equation of the
    form *x* = *a*, *a* = *a*, or *a* = *b* results (where *a* and *b*
    are different numbers).

b)  Solve linear equations with rational coefficients, including
    equations whose solutions require expanding expressions using the
    distributive property and combining like terms.

-   8.EE.8. Analyze and solve systems of linear equations.

a)  Show that the solution to a system of two linear equations in two
    variables is the intersection of the graphs of those equations
    because points of intersection satisfy both equations
    simultaneously.

b)  Solve systems of two linear equations in two variables and estimate
    solutions by graphing the equations. Simple cases may be done by
    inspection. *For example, 3x + 2y = 5 and 3x + 2y = 6 have no
    solution because 3x + 2y cannot simultaneously be 5 and 6.*

c)  Solve real-world and mathematical problems leading to two linear
    equations in two variables. *For example, given coordinates for two
    pairs of points, determine whether the line through the first pair
    of points intersects the line through the second pair.*

## Functions

### 

### Define, evaluate, and compare functions.

-   8.F.1. Understand that a function is a rule that assigns to each
    input (the domain) exactly one output (the range). The graph of a
    function is the set of ordered pairs consisting of an input and the
    corresponding output. *For example, use the vertical line test to
    determine functions and non-functions*.

-   8.F.2. Compare properties of two functions, each represented in a
    different way (algebraically, graphically, numerically in tables, or
    by verbal descriptions). *For example, given a linear function
    represented by a table of values and a linear function represented
    by an algebraic expression, determine which function has the greater
    rate of change.*

-   8.F.3. Interpret the equation *y* = *mx* + *b* as defining a linear
    function, whose graph is a straight line; give examples of functions
    that are not linear. *For example, the function A = s^2^ giving the
    area of a square as a function of its side length is not linear
    because its graph contains the points (1,1), (2,4) and (3,9), which
    are not on a straight line.*

### Use functions to model relationships between quantities.

-   8.F.4. Construct a function to model a linear relationship between
    two quantities.

    -   Determine the rate of change and initial value of the function
        from a description of a relationship or from two (*x*, *y*)
        values, including reading these from a table or from a graph.

    -   Interpret the rate of change and initial value of a linear
        function in terms of the situation it models, and in terms of
        its graph or a table of values.

-   8.F.5. Given a verbal description between two quantities, sketch a
    graph. Conversely, given a graph, describe a possible real-world
    example. *For example, graph the position of an accelerating car or
    tossing a ball in the air.*

## Geometry

### 

### Understand congruence and similarity using physical models, transparencies, or geometry software.

-   8.G.1. Through experimentation, verify the properties of rotations,
    reflections, and translations (transformations) to figures on a
    coordinate plane).

a)  Lines are taken to lines, and line segments to line segments of the
    same length.

b)  Angles are taken to angles of the same measure.

c)  Parallel lines are taken to parallel lines.

-   8.G.2. Demonstrate understanding of congruence by applying a
    sequence of translations, reflections, and rotations on
    two-dimensional figures. Given two congruent figures, describe a
    sequence that exhibits the congruence between them.

-   8.G.3 .Describe the effect of dilations, translations, rotations,
    and reflections on two-dimensional figures using coordinates.

-   8.G.4. Demonstrate understanding of similarity, by applying a
    sequence of translations, reflections, rotations, and dilations on
    two-dimensional figures. Describe a sequence that exhibits the
    similarity between them.

-   8.G.5. Justify using informal arguments to establish facts about

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-   the angle sum of triangles (sum of the interior angles of a triangle
    is 180º)

-   measures of exterior angles of triangles,

-   angles created when parallel lines are cut be a transversal (e.g.,
    alternate interior angles) and

-   angle-angle criterion for similarity of triangles.

### Understand and apply the Pythagorean Theorem.

-   8.G.6. Explain the Pythagorean Theorem and its converse.

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-   8.G.7. Apply the Pythagorean Theorem to determine unknown side
    lengths in right triangles in real-world and mathematical problems
    in two and three dimensions.

-   8.G.8. Apply the Pythagorean Theorem to find the distance between
    two points in a coordinate system.

### Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

-   8.G.9. Identify and apply the formulas for the volumes of cones,
    cylinders, and spheres and use them to solve real-world and
    mathematical problems.

## Statistics and Probability

### 

### Investigate patterns of association in bivariate data.

-   8.SP.1. Construct and interpret scatter plots for bivariate
    measurement data to investigate patterns of association between two
    quantities. Describe patterns such as clustering, outliers, positive
    or negative association, linear association, and nonlinear
    association.

-   8.SP.2. Explain why straight lines are widely used to model
    relationships between two quantitative variables. For scatter plots
    that suggest a linear association, informally fit a straight line,
    and informally assess the model fit by judging the closeness of the
    data points to the line.

-   8.SP.3. Use the equation of a linear model to solve problems in the
    context of bivariate measurement data, interpreting the slope and
    y-intercept. *For example, in a linear model for a biology
    experiment, interpret a slope of 1.5 cm/hr as meaning that an
    additional hour of sunlight each day is associated with an
    additional 1.5 cm in mature plant height.*

-   8.SP.4. Construct and interpret a two-way table summarizing data on
    two categorical variables collected from the same subjects and use
    relative frequencies to describe possible association between the
    two variables. *For example, collect data from students in your
    class on whether or not they have a curfew on school nights and
    whether or not they have assigned chores at home. Is there evidence
    that those who have a curfew also tend to have chores?*

# 

# 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.

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-   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