Standards - Mathematics

Find probabilities of simple and compound events through experimentation or simulation and by analyzing the sample space, representing the probabilities as percents, decimals, or fractions.

Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams, and determine the probability of an event by finding the fraction of outcomes in the sample space for which the compound event occurred.

Design and use a simulation to generate frequencies for compound events.

Represent events described in everyday language in terms of outcomes in the sample space which composed the event.

Solve problems involving scale drawings of geometric figures, including computation of actual lengths and areas from a scale drawing and reproduction of a scale drawing at a different scale.

Construct geometric shapes (freehand, using a ruler and a protractor, and using technology), given a written description or measurement constraints with an emphasis on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

Describe the two-dimensional figures created by slicing three-dimensional figures into plane sections.

Explain the relationships among circumference, diameter, area, and radius of a circle to demonstrate understanding of formulas for the area and circumference of a circle.

Informally derive the formula for area of a circle.

Solve area and circumference problems in real-world and mathematical situations involving circles.

Use facts about supplementary, complementary, vertical, and adjacent angles in multi-step problems to write and solve simple equations for an unknown angle in a figure.

Solve real-world and mathematical problems involving area, volume, and surface area of two- and three- dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right rectangular prisms.

Define the real number system as composed of rational and irrational numbers.

Explain that every number has a decimal expansion; for rational numbers, the decimal expansion repeats or terminates.

Convert a decimal expansion that repeats into a rational number.

Locate rational approximations of irrational numbers on a number line, compare their sizes, and estimate the values of the irrational numbers.

Develop and apply properties of integer exponents to generate equivalent numerical and algebraic expressions.

Use square root and cube root symbols to represent solutions to equations.

Evaluate square roots of perfect squares (less than or equal to 225) and cube roots of perfect cubes (less than or equal to 1000).

Explain that the square root of a non-perfect square is irrational.

Estimate and compare very large or very small numbers in scientific notation.

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.

Interpret scientific notation that has been generated by technology.

Determine whether a relationship between two variables is proportional or non-proportional.

Graph proportional relationships.

Interpret the unit rate of a proportional relationship, describing the constant of proportionality as the slope of the graph which goes through the origin and has the equation y = mx where m is the slope.

Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept.

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in a coordinate plane.

Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.

Graph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the initial value.

Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts.

Compare proportional and non-proportional linear relationships represented in different ways (algebraically, graphically, numerically in tables, or by verbal descriptions) to solve real-world problems.

Solve multi-step linear equations in one variable, including rational number coefficients, and equations that require using the distributive property and combining like terms.

Determine whether linear equations in one variable have one solution, no solution, or infinitely many solutions of the form x = a, a = a, or a = b (where a and b are different numbers).