Grade 8 Math Michigan standards Standards

Functions

Statistics and Probability

Expressions and Equations

The Number System

Geometry

Standards for Mathematical Practice

Functions

Solving Expressions and Equations

Students solve increasingly complex mathematical problems, making productive use of algebra and functions.

Statistics and Probability

Students demonstrate increasingly complex understanding of measurement, data and analytic procedures.

Geometry

Students demonstrate increasingly complex spatial reasoning and understanding of geometric principles.

Expressions and Equations

The Number System

Students demonstrate increasingly complex understanding of number sense.

Work with radicals and integer exponents.

Know and apply the properties of integer exponents to generate equivalent numerical expressions.

Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = 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.

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.

Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology.

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

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

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.

Solve linear equations in one variable.

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).

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

Analyze and solve pairs of simultaneous linear equations.

Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection.

Solve real-world and mathematical problems leading to two linear equations in two variables.

Define, evaluate, and compare functions.

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

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.

Use functions to model relationships between quantities.

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.

Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally.

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

Verify experimentally the properties of rotations, reflections, and translations:

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

Angles are taken to angles of the same measure.

Parallel lines are taken to parallel lines.

Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibits the congruence between them.

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

Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two-dimensional figures, describe a sequence that exhibits the similarity between them.

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Understand and apply the Pythagorean Theorem.

Explain a proof of the Pythagorean Theorem and its converse.

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

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.

Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

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

Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).

Investigate patterns of association in bivariate data.

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.

Know that 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.

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

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.

Identify a geometric sequence of whole numbers with a whole number common ratio.

Identify the meaning of an exponent (limited to exponents of 2 and 3).

Compose and decompose whole numbers up to 999.

Graph a simple ratio by connecting the origin to a point representing the ratio in the form of y/x. For example, when given a ratio in standard form (2:1), convert to 2/1, and plot the point (1,2).

Solve simple algebraic equations with one variable using addition and subtraction.

The student can compose and decompose whole numbers.

The student can solve algebraic equations with one variable using addition and subtraction.

The student can differentiate between a single unit or a few single blocks and a group of 10.

The student can identify which of two choices is needed to answer a question or solve a problem.

The student can compose and decompose two-digit whole numbers with base-10 blocks.

The student can select appropriate numbers in order to solve addition and subtraction problems and solve as appropriate.

Given a function table containing at least 2 complete ordered pairs, identify a missing number that completes another ordered pair (limited to linear functions).

Determine the values or rule of a function using a graph or a table.

Describe how a graph represents a relationship between two quantities.

The student can identify the missing number that completes an ordered pair in a function table.

The student can describe a relationship between two quantities shown on a graph using increasing, decreasing, or maintaining.

The student can extend or describe patterns involving objects or symbols.

The student can identify the topic of information presented in a picture graph or bar graph (e.g., use a model of a graph to identify the topic of favorite pizza toppings).

The student can recognize a growing or shrinking pattern in a data table that contains at least 3 data points.

The student can identify a relationship between two quantities shown in a graph as more than or less than.

Recognize translations, rotations, and reflections of shapes.

Identify shapes that are congruent.

Identify similar shapes with and without rotation.

Compare any angle to a right angle, and describe the angle as greater than, less than, or congruent to a right angle.

Use the formulas for perimeter, area, and volume to solve real-world and mathematical problems (limited to perimeter and area of rectangles and volume of rectangular prisms).

The student can identify shapes that are congruent.

The student can identify similar shapes and/or letters with and without rotation.

The student can recognize an angle as being greater than, less than, or equal to a right angle when given a model of a right angle.

When given the formulas for perimeter, area, and volume, the student can use the formulas to solve real-world and mathematical problems, limited to rectangles and rectangular prisms.

The student can match similar two-dimensional shapes, limited to circle, square, rectangle, and triangle when presented without rotation.

The student can match similar two-dimensional shapes, limited to circle, square, star, and triangle with the same shape highlighted within a functional context (e.g., a square and square window frame).

The student can identify a corner when compared with another attribute or shape without a corner.

The student can use informal units to determine the perimeter of a rectangle of 8 units or less without using the word perimeter (e.g., how many steps, blocks, etc.).

The student can identify similar shapes without rotation.

The student can identify similar shapes and/or letters with and without rotation (limited to quarter- and half-turns).

The student can recognize a right angle.

The student can identify the area or perimeter using models and dimensions of rectangles, limited to single-digit numbers (e.g., 2 + 3 + 2 + 3) with a shape that also contains unit squares.

Subtract fractions with like denominators (halves, thirds, fourths, and tenths) with minuends less than or equal to one.

Express a fraction with a denominator of 100 as a decimal.

Compare quantities represented as decimals in real-world examples to hundredths.

The student can add and subtract fractions with common denominators (limited to halves, thirds, fourths and tenths) with sums and differences less than or equal to one.

The student can express a fraction with a denominator of 100 as a decimal. Functional skill is expressing money as a fraction/decimal of a dollar (e.g. 23/100 of a dollar = $0.23, etc.)

The student can recognize that subtracting one-half from one whole equals one-half.

The student can differentiate coins and bills from each other and from other similar objects.

The student can add and subtract fractions with common denominators with sums or differences less than or equal to one and limited to halves, thirds, and fourths (fractions shown as models).

The student can identify a combination of coins and bills up to $5 using decimal notation (e.g., $2.50 is equal to two one-dollar bills and two quarters.)

Construct a graph or table from given categorical data, and compare data categorized in the graph or table.

The student can match given data with a graph or table that shows this data and compare data categorized in a graph or table.

The student can sort given data into two groups.

The student can identify a graph or table that matches given information, limited to 3 data entries.