M.8: Grade 8

Grade 8

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Grade 8 bridges arithmetic and algebra: irrational numbers and integer exponents establish the number system, while linear equations, their graphs, and systems of linear equations form the algebraic core. Functions are introduced formally, with emphasis on linear and nonlinear models. Geometric transformations, congruence, similarity, the Pythagorean theorem, and volume of three-dimensional figures complete the year.

M.8.1Know 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.M.8.10Analyze and solve pairs of simultaneous linear equations.M.8.11Understand 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.M.8.12Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). (e.g., 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.)M.8.13Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. (e.g., 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.)M.8.14Construct 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.M.8.15Describe 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.M.8.16Verify experimentally the properties of rotations, reflections and translations:M.8.17Understand 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.M.8.18Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates.M.8.19Understand 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.M.8.2Use 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 such as π^2. (e.g., By truncating the decimal expansion of √2, 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.)M.8.20Use 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. (e.g., Arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.)M.8.21Explain a proof of the Pythagorean Theorem and its converse.M.8.22Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.M.8.23Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.M.8.24Know the formulas for the volumes of cones, cylinders and spheres and use them to solve real-world and mathematical problems.M.8.25Construct 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.M.8.26Know 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.M.8.27Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. (e.g., 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.)M.8.28Understand 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. (e.g., 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?)M.8.3Know and apply the properties of integer exponents to generate equivalent numerical expressions. (e.g., 3^2 × 3^–5 = 3^–3 = 1/3^3 = 1/27.)M.8.4Use 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.M.8.5Use 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. (e.g., 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.)M.8.6Perform 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.M.8.7Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. (e.g., Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.)M.8.8Use 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.M.8.9Solve linear equations in one variable.M.8.EEExpressions and EquationsM.8.FFunctionsM.8.GGeometryM.8.NSThe Number SystemM.8.SPStatistics and Probability
Example Problems
Rewrite the following decimal as a simplified fraction:
Tony is 5 years older than Beth. He is also 8 years older than half of Beth's age.

Let be Tony's age and be Beth's age.

Write a system of equations to represent this situation.
Kade is training for a race. He plans to run 5 kilometers each session.

He wrote this equation to find the total distance (x) he runs after how many sessions (m):


What is the dependent variable?
Does the following rule describe the relationship between the x- and y-coordinates in the table?


xy
62
93
124
Graph:
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