Unit 1 – Congruence, Similarity & Transformations
• Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using
• Verify experimentally the properties of rotations, reflections, and translations:
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.
• 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.
• 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.
Unit 2 - Rational and Irrational Numbers
• Understand real numbers that are not rational are called irrational.
• Understand that every rational number has a decimal expansion that either repeats or terminates.
• Convert a repeating decimal to a fraction.
• Evaluate square roots of small perfect squares.
• Evaluate cube roots of small perfect cubes.
• Solve equations involving square and cube roots.
• Know the square root of a non-perfect square is irrational & estimate the value of irrational square
• Locate irrational numbers on a number line.
Unit 3 - 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 &
mathematical problems in two and three dimensions.
• Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Unit 4 - Slope
• Graph proportional relationships on a coordinate plane.
• Understand that rate of change is slope.
• Compare proportional relationships represented in different ways.
• Derive the formula y = mx + b (understand m is slope and b is y-intercept) by visual representations of
• Understand that any two points on a line can be used to find the slope.
• Distinguish between linear and nonlinear functions.
Unit 5 - Linear Functions
• Definition of a function (for each input there is exactly one output).
• Use a function rule to generate ordered pairs.
• Graph a function using ordered pairs.
• Determine rate of change of one function represented in multiple ways (table, graph, description,
• Describe and compare the rate of change of two functions.
• Compare the properties of two functions in different representations (graphs, tables, and verbal
• Construct a function to model a linear relationship between two quantities.
• Given a graph or a table, interpret what rate of change and initial value mean in the context of the linear
• Given a real-world situation, describe what the rate of change and initial value means.
Unit 6 - Solving Equations with One Variable
• Solve linear equations in one variable with one solution, infinitely many solutions, or no solution.
• Solve linear equations in one variable including use of the distributive property.
• Solve linear equations in one variable including rational number coefficients (whole numbers, decimals,
fractions, and their opposites).
• Solve linear equations in one variable combining like terms (on one or both sides of the equal sign).
Unit 7 - Systems of Equations
• Understand a solution to a system is the intersection of their graphs.
• Demonstrate that the point of intersection satisfies both equations.
• After graphing a system of equations, estimation may be used to determine the solution.
• Solve simple cases shown graphically or algebraically (one solution, no solution, infinite solutions) by
inspection (i.e., by sight).
• Solve systems graphically.
• Solve systems algebraically (data tables, substitution, and elimination through addition/subtraction).
• Given a system of equations, create a real world situation. (contextualize)
• Given a real-world situation, create and solve a system of equations. (decontextualize)
Unit 8 - Volume
• Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world
and mathematical problems.
Unit 9 – Angles
• 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. For example, 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.
Unit 10 - Rules of Exponents and Scientific Notation
• 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
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.
Unit 11 - Patterns 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. 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.
• 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.
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?
Unit 1: Integers
• Describe situations in which opposites combine to make zero
• Represent integer addition on a number line
• Evaluate integer sums
• Understand subtraction of a number as adding the additive inverse
• Show that the distance between two integers is the absolute value of their difference in real world contexts
• Evaluate integer products
• Understand quotients of integers with non-zero divisors
• Apply properties of operations to integers
• Solve real-world problems involving
Unit 2: Adding & Subtracting Rational Numbers
• Convert between different forms of rational numbers (decimals and fractions) • Understand that rational numbers in decimal form either terminate or repeat
• Show that a rational number and its opposite have a sum of zero (additive inverses)
• Evaluate rational sums
• Understand subtraction of rational numbers as adding the additive inverse
• Show that the distance between two rational numbers is the absolute value of their difference in real-world contexts
• Interpret sums and differences of rational numbers in real-world contexts
• Solve real-world problems involving addition and subtraction of rational numbers
Unit 3: Multiplying & Dividing Rational Numbers
• Evaluate rational products and quotients • Understand why the denominator cannot equal zero
• Apply properties (including distributive property) when performing operations with rational numbers
• Interpret products and quotients of rational numbers in real-world contexts
• Solve real-world problems using the four operations with rational numbers
Unit 4: Expressions
• Identify coefficients, variables, constants, and like terms.
• Add and subtract like terms including rational constants and rational coefficients
• Use properties of operations (including the distributive property) to expand linear expressions with rational numbers
• Use properties of operations (including the distributive property) to factor linear expressions with rational numbers
• Rewrite an expression in different forms in a problem context to recognize how the quantities are related
Unit 5: Equations & Inequalities
• Fluently solve equations in the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers
• Solve word problems by writing and solving equations
• Compare algebraic and arithmetic solutions by identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width? (To solve algebraically: 2w + 2(6) = 54; To solve arithmetically: [54 – 2(6)] ÷ 2 is the width.)
• Solve inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. This includes ≤ and ≥
• Graph the solution set of the inequality
• Solve word problems by writing and solving inequalities. Interpret the solution set in the context of the problem. For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions
Unit 6: Ratios & Proportions
• Use various strategies to determine whether two quantities are proportional (e.g., by constructing a rate table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin)
• Compute unit rates with ratios of fractions including ratios of lengths, areas, and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.
• Identify the unit rate (constant of proportionality) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.
• Write equations for proportional relationships using the unit rate. For example: d = 65t
• Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the origin and (1, r) where r is the unit rate.
Unit 7: Application of Percents
•Solve multi-step ratio and percent problems involving:
• simple interest • tax • markups and markdowns
• gratuities and commissions • fees • percent increase and percent decrease
• percent error
• Solve multi-step real-world problems with positive and negative rational numbers in any form (whole numbers, fractions, decimals)
• Apply properties to numbers in any form
• convert between forms of numbers as appropriate
• assess reasonableness of answers using mental computation and estimation
Unit 8: Geometric Drawing & Scaling
• Solve problems involving scale drawings of geometric figures
• Reproduce a scale drawing at a different scale
• Compute actual lengths and areas from a scale drawing
• Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions
• Focus 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
Unit 9: Angle Relationships
• Recognize angle relationships, specifically supplementary, complementary, vertical, and adjacent angles.
• Apply knowledge of angle relationships in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
• Solve simple geometric problems using equations.
Unit 10: 2D & 3D Geometry
• Find the area of two-dimensional polygons (including composite figures).
• Solve real-world and mathematical problems involving area of polygons.
• Know and use the formulas to find the area and circumference of a circle.
• Solve real-world and mathematical problems involving area and circumference of circles.
• Informally derive the relationship between the circumference and area of a circle.
• Describe the two-dimensional figures that result from slicing right rectangular prisms
• Describe the two-dimensional figures that result from slicing right rectangular pyramids
• Solve real-world and mathematical problems involving volume of right prisms
• Solve real-world and mathematical problems involving surface area of figures composed of triangles, quadrilaterals, polygons, cubes, and right prisms
Unit 11: Sampling & Statistics
l• Examine a sample population
• Generalize about a population
• Determine validity of samples
• Understand random sampling, representative samples and making inferences
• Use random samples to draw inferences
• Generate multiple samples (or simulated) to gauge variation
• Assess degree of overlap of two data distributions with similar variability
• Use measures of center to draw inferences
• Use measures of variability to draw inferences
Unit 12: Theoretical & Experimental Probability
• Understand that the probability of a chance event is a number between 0 and 1.
• 0 indicates an impossible occurrence (0% probability) and 1 indicates a certainty (100% probability).
• A probability near 0 indicates an unlikely event, a probability around ½ indicates an event that is
neither unlikely nor likely, and a probability near 1 indicates a likely event.
• Approximate the probability of a chance event by collecting data.
• Develop a probability model and use it to find probabilities of events.
• Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation the observed frequencies?
• Understand that the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.
• Represent sample spaces for compound events using methods such as organized lists, tables, and tree diagrams. Identify the outcomes in the sample space that compose the event.
• Design and use a simulation to generate frequencies for compound events.