High School Math OLS Standards

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Represent and solve equations and inequalities graphically

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Solve systems of equations.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Solve equations and inequalities in one variable.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Understand solving equations as a process of reasoning and explain the reasoning.

Reasoning with Equations and Inequalities Standards

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Create equations that describe numbers or relationships.

Creating Equations Standards

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Rewrite rational expressions.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Understand the relationship between zeros and factors of polynomials.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Perform arithmetic operations on polynomials.

Arithmetic with Polynomials and Rational Expression Standards

Learning Progression

Complexity c

Complexity b

Complexity a

Write expressions in equivalent forms to solve problems.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Interpret the structure of expressions.

Seeing Structure in Expressions

Understand that polynomials form a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials. a. Focus on polynomial expressions that simplify to forms that are linear or quadratic. (A1, M2) b. Extend to polynomial expressions beyond those expressions that simplify to forms that are linear or quadratic. (A2, M3)

Add and subtract linear and/or quadratic polynomials. Models may be used.

Add and subtract linear polynomials. Models may be used.

Add linear polynomials. Models may be used.

Understand and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x − a is p(a). In particular, p(a) = 0 if and only if (x – a) is a factor of p(x).

Multiply two binomials.

Multiply a variable by a binomial.

Identify a polynomial (binomials only).

Identify zeros of polynomials, when factoring is reasonable, and use the zeros to construct a rough graph of the function defined by the polynomial.

Find the zeros of a polynomial when the polynomial is factored (e.g., x2 – 9 = 0 and x2 = 3x = 2 = 0).

Identify a polynomial (trinomial) (e.g., x2 = 3x = 2 = 0).

Identify a polynomial (binomial) (e.g., x2 – 9 = 0).

Rewrite simple rational expressions in different forms; write a(x)/ b(x) in the form q(x) + r(x)/ b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Identify a rational expression (e.g., 6x/ 3 = 2x).

Rewrite expressions in different forms (e.g., x2 + 1 = (x * x) + 1).v

Given a visual model, identify an expression (e.g. 2 * 2 * 2 = 23 ).

Create equations and inequalities in one variable and use them to solve problems. Include equations and inequalities arising from linear, quadratic, simple rational, and exponential functions. a. Focus on applying linear and simple exponential expressions. (A1, M1) b. Focus on applying simple quadratic expressions. (A1, M2) c. Extend to include more complicated function situations with the option to solve with technology. (A2, M3)

Represent and solve a realworld situation with a two-step linear equation or inequality (e.g., Abby has $15 to spend on a snack and two matching T-shirts. If she spends $3 on the snack, what is the maximum price of each T-shirt? Key: x ≤ 5).

Represent and solve a real-world problem with a onestep linear equation or inequality (e.g., Abby has $5, and she wants to buy a T-shirt for $8. How much more money does she need? Key: 5 + x = 8)

Represent a realworld problem with a linear equation, using concrete objects, models and pictures (see example below).

Demonstrate or act out the situation.

Recognize the unknown.

Show what addition, subtraction, multiplication and division represents using manipulatives.

Know that a symbol  or letter can represent a missing value.

Count physical objects up to 30.

Relate a picture or objects to a number sentence.

Engagement Statements (demonstration of engaged in the topic)

Interact with concrete objects.

Represent numbers.

Interact with representations of the unknown.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. a. Focus on applying linear and simple exponential expressions. (A1, M1) b. Focus on applying simple quadratic expressions. (A1, M2) c. Extend to include more complicated function situations with the option to graph with technology. (A2, M3)

Create an equation with two variables to represent a linear relationship between quantities in a given context (e.g., y = 2x + 4).

Using a two-variable equation describing a realworld situation, given the value of one variable, find and interpret the value of the other variable (e.g., Sally starts with $4 and gets an allowance of $2 each week. After x weeks, she has y = 2x + 4 dollars. When x = 3, find y and interpret the result).

Identify the meaning of each number and/ or variable in a given two-variable equation that describe a realworld situation (e.g., Sally starts with $4 and gets an allowance of $2 each week. After x weeks she has y = 2x + 4 dollars. What does 4 represent? What does x represent?).

Recognize variables.

Recognize that one variable affects the other variable.

Recognize that a variable can represent any number.

Demonstrate that a coefficient represents a constant decrease or increase.

Demonstrate or act out the situation.

Recognize the unknown.

Show what addition, subtraction, multiplication and division represents using manipulatives.

Know that a symbol � or letter can represent a missing value.

Engagement Statements (demonstration of engaged in the topic)

Interact with concrete objects.

Represent quantities.

Interact with representations of the unknown.

Represent constraints by equations or inequalities, and by systems of equations and/ or inequalities, and interpret solutions as viable or nonviable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. (A1, M1) a. While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations. (A2, M3)

Represent a constraint with an equation or inequality in two variables (e.g., x + y ≤ 8, describing the number of boys and girls in an 8-passenger van).

Create a one-variable constraint using an inequality (e.g., x ≤ 6).

Demonstrate a constraint using words or models (e.g., how many students can fit at this table?).

[Between c and b:

Identify the symbols <, >, and =.]

Recognize the unknown.v

Demonstrate quantities using manipulatives.

Understand the order of numbers.

Identify limitations of a real-world situation.

Compare quantities using manipulatives or pictures to identify which is more and which is less.

Engagement Statements (demonstration of engaged in the topic)

Demonstrate or act out a situation.

Interact with real-world situations with restraints, e.g., seats at a table, passengers in a bus, eggs in a carton.

Interact with situations involving greater than, equal to, or less than.

Interact with a model of a real-world situation.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. a. Focus on formulas in which the variable of interest is linear or square. For example, rearrange Ohm’s law V = IR to highlight resistance R, or rearrange the formula for the area of a circle A = (π)r2 to highlight radius r. (A1) b. Focus on formulas in which the variable of interest is linear. For example, rearrange Ohm’s law V = IR to highlight resistance R. (M1) c. Focus on formulas in which the variable of interest is linear or square. For example, rearrange the formula for the area of a circle A = (π)r2 to highlight radius r. (M2) d. While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations. (A2, M3)

Rearrange a onestep formula to highlight a quantity (e.g., use the formula a=lw to highlight the length of the rectangle by rearranging it to l=a/w).

Rearrange a one-step equation to solve for a variable (e.g., solve for x: y=2+x, y/2=x).

Match a formula to a given situation (e.g., recognize that a=lw is the formula for the area of a rectangle)

Recognize that a formula represents a situation.

Recognize what the variables represent in the formula.

Model a situation with manipulatives.

Relate a picture or objects to a number sentence.

Read and interpret a traditional one-step number sentence (2 × 3 = � ).

Identify a number sentence.

Recognize the symbols for addition (+), subtraction, (–), multiplication (×), division

(÷), and equals (=).

Recognize a numerical expression with and without variables.

Demonstrate or act out a situation.

Engagement Statements (demonstration of engaged in the topic)

Interact with real-world situations.

Interact with a model of a real-world situation.

Interact with representations of the unknown.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Order a given sequence of steps to solve an equation (e.g., 2x + 5 = 13). Solve two-step equations with integer coefficients and solutions, explaining the steps.

Determine a step needed to solve a two-step equation.

Determine the step needed to solve a one-step equation (e.g., to solve x + 5 = 13, subtract 5 from both sides).

[Between C and B:

Identify the first step in solving a two-step problem involving addition and subtraction.]

Demonstrate the meaning of the solution.

Understand that the variable represents the unknown.

Understand that the operations addition and subtraction are inverse operations.

Understand the the operations multiplication and divisions are inverse operations.

Understand that the equal sign is a balance.

Understand that an operation applied to one side needs to be applied to the other.

Use manipulatives to understand the situation.

Demonstrate or act out a situation.

Engagement Statements (demonstration of engaged in the topic)

Interact with manipulatives that represent a situation.

Interact with real-world situations.

Interact with a model of a real-world situation.

Interact with representations of the unknown.

Interact with a balance scale.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Given a graph and an equation, fill out three points on a corresponding table of values.

Given a table of values, graph the line on the coordinate plane.

Identify a point on a line on a coordinate plane.

Locate an ordered pair (x, y) as a point on the coordinate plane.

Plot a point on the coordinate plane.

Identify the x- and y- axis.

Identify that the x- and y- axes are number lines.

Identify a point.

Demonstrate that on a number line and on the coordinate plane the spaces need to be counted not the grid lines.

Identify a line.

Locate the origin at the point (0, 0) on the coordinate plane.

Engagement Statements (demonstration of engaged in the topic)

Interact with the coordinate grid.

Explain why the x-coordinates of the points where the graphs of the equation y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately (e.g., using technology to graph the functions, making tables of values, or finding successive approximations).

Locate the coordinate at which two lines intersect. Using the x coordinate of the intersection point, substitute it back into the original equation to show that it is a solution of the equation.

Locate the coordinate point on the graph at which two lines intersect.

Identify whether two lines intersect.

Identify a line.

Identify crossing lines in the real-world.

Show that non-parallel lines can be extended so they eventually cross.

Engagement Statements (demonstration of engaged in the topic)

Interact with the coordinate grid.

Interact with manipulatives representing lines.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Given a graph of an inequality including the shaded region, identify three points that make the inequality true.

Identify on a graph of a line ≤, ≥ is represented by a solid line; and < and > are represented by a dotted line.

Identify the graph of a linear inequality has a shaded region.

Identify ordered pairs that satisfy an inequality

Plot ordered pairs that satisfy an inequality.

Understand that a linear inequality has more than one solution.

Understand that the solutions to a linear equation in two variables are all the points on a straight line.

Understand that the solutions to a linear inequality in two variables are points on a plane.

Each point on a coordinate plane is associated to a (x,y) ordered pair.

Using trial error explore ordered pairs that satisfy an inequality

Using trial error explore ordered pairs that satisfy an equation.

Identify a shaded region on a coordinate plane.

Identify a shaded region in a picture or of an object.

Engagement Statements (demonstration of engaged in the topic)

Interact with a graph

Interact with the coordinate grid.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve linear equations with more than one step.

Solve 1-step linear equations.

Solve for the missing number within a given number sentence involving addition or subtraction of numbers less than 10.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve a two- or three-step linear equation in one variable. Models may be used.

Solve a one-step linear equation in one variable. Models may be used.

Given a linear equation in one-variable and a list of possible solutions, identify the solution of the equation.

Understand the idea of what a solution to an equation means.

Identify the solution of the equation using manipulatives.

Using trial error explore the idea of the equal sign being a balance.

Identify a number sentence.

Recognize the symbols for addition (+), subtraction, (–), multiplication (×), division (÷), and equals (=).

Read and interpret a traditional one-step number sentence (2 × 3 = �).

Relate a picture or objects to a number sentence.

Know that a symbol � or letter can represent a missing value.

Count to 30.

Count physical objects up to 30.

Recognize a numerical expression with and without variables.

Identify a numerical expression without exponents.

Engagement Statements (demonstration of engaged in the topic)

Interact with a balance scale.

Interact with physical objects (blocks) or drawings that represent an expression or an equation.

Interact with physical objects (blocks) or drawings representing addition, subtraction, or multiplication word problems.

Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x − p)² = q that has the same solutions. b. Solve quadratic equations as appropriate to the initial form of the equation by inspection, e.g., for x² = 49; taking square roots; completing the square; applying the quadratic formula; or utilizing the Zero-Product Property after factoring.

Identify or create perfect squares (e.g., square root of 25 = 5).

Identify equivalent expressions that are cubes (e.g., m3 = m x m x m).

Identify equivalent expressions that are squared (e.g., m2 = m x m).

Solve systems of linear equations algebraically and graphically. a. Limit to pairs of linear equations in two variables. (A1, M1) b. Extend to include solving systems of linear equations in three variables, but only algebraically. (A2, M3)

Identify the coordinate at which two lines intersect.

Locate the point on the graph at which two lines intersect.

Identify whether two lines intersect.

Identify a line.

Identify crossing lines in the real-world.

Show that the length of intersecting lines can be extended so they will eventually cross.

Engagement Statements (demonstration of engaged in the topic)

Interact with manipulatives representing lines.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = −3x and the circle x2 + y2 = 3.

Locate the coordinate of the point(s) at which a line intersects a quadratic function (e.g., at which two coordinates does the line intersect the parabola?).

Locate the point(s) on the graph at which a line intersects a quadratic function (e.g., identify on the graph where the line intersects the parabola).

Identify whether a line intersects a quadratic function (e.g., does the line intersect the parabola at one or two points? Does the line intersect the parabola?).

Interpret expressions that represent a quantity in terms of its context. a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity.

Represent a realworld situation with an expression, both numerals and variables. Recognize parts of the expression in the real-world situation.

Represent a real-world situation with a numeric expression. Recognize parts of the expression in the real-world situation.

Represent a realworld situation with a model using concrete objects.

Demonstrate or act out the situation

Show what addition, subtraction, multiplication and division represents using manipulatives.

Recognize the symbols for addition (+), subtraction, (–), multiplication (×), division (÷), and equals (=).

Engagement Statements (demonstration of engaged in the topic)

Interact with concrete objects.

Interact with models.

Use the structure of an expression to identify ways to rewrite it. For example, to factor 3x(x − 5) + 2(x − 5), students should recognize that the “x − 5” is common to both expressions being added, so it simplifies to (3x + 2)(x − 5); or see x4 − y4 as (x2 )2 − (y2 )2 , thus recognizing it as a difference of squares that can be factored as (x2 − y2 )(x2 + y2 ).

Simplify expressions involving variables (e.g., (2(x + 4) = 2x + 8)).

Identify the equivalent numeric expression (e.g., 7 + 5 = 5 + 7).

Identify equivalent expressions with whole numbers less than 10 using concrete objects (e.g., objects, dots, etc.).

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression. a. Factor a quadratic expression to reveal the zeros of the function it defines. (A1, M2) b. Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines. (A1, M2) c. Use the properties of exponents to transform expressions for exponential functions. For example, 8t can be written as 23t.

Apply properties of integer exponents to generate equivalent variable expressions (e.g., b2 x b4 = b6 ).

Apply properties of integer exponents to generate equivalent numerical expressions (e.g., 52 x 54 = 56 ).

Interpret numerical expressions with exponents (e.g., 54 means 5 x 5 x 5 x 5).

Demonstrate or act out the situation.

Show what addition, subtraction, multiplication and division represents using manipulatives.

Identify the base and the exponent of an exponential expression.

Recognize the symbols for addition (+), subtraction, (–), multiplication (×), division (÷), and equals (=).

Count physical objects up to 30.

Engagement Statements (demonstration of engaged in the topic)

Interact with concrete objects

Interact with models and pictures.

Interact with representations of the unknown.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Prove and apply trigonometric identities.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Model periodic phenomena with trigonometric functions.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Extend the domain of trigonometric functions using the unit circle.

Trigonometric Functions Standards

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Interpret expressions for functions in terms of the situation they model.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Construct and compare linear, quadratic, and exponential models, and solve problems.

Linear Quadratic and Exponential Models Standards

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Build new functions from existing functions.

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Build a function that models a relationship between two quantities.

Building Functions Standards

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Analyze functions using different representations.

Learning Progression

Complexity c

Complexity b

Complexity a

Analyze functions using different representations.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Interpret functions that arise in applications in terms of the context.

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Understand the concept of a function, and use function notation.

Interpreting Functions Standards

Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from context. i. Focus on linear and exponential functions. (A1, M1) ii. Focus on situations that exhibit quadratic or exponential relationships. (A1, M2) b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. (A2, M3) c. Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Create a linear function that represents a linear relationship between quantities in a given context.

Given a linear function that describes a realworld situation and given the value of one variable, find and interpret the value of the other variable.

Identify the meaning of each number and/ or variable in a linear function that describes a realworld situation.

Count physical objects.

Relate the input and the output to a function machine.

Answer questions about a table.

Identify what the input is in a table.

Interpret variables and numbers in the linear function that describes the real - world situation

Know that a variable can represent an unknown value.

Identify a number sentence.

Recognize that the signs and numbers are not variables.

Identify numbers.

Relate a picture or objects to a number sentence.

Identify a number sentence.

Engagement Statements (demonstration of engaged in the topic)

Interact with no more than 3 answer choices be able to select 1 from different positions.

Interact with a visual model.

Interact with real-world situations described by linear functions

Interact with a model of a real-world situation.

Interact with representations of the unknown.

Write arithmetic and geometric sequences both recursively and with an explicit formula. Use them to model situations and translate between the two forms.

Identify the rule for a pattern.

Identify the next term in a pattern.

Determine if a given set represents a pattern.

Experience patterns and non-patterns.

Discover patterns in the real world.

Build patterns using manipulatives.

Draw the patterns.

Translate the patterns to numbers.

Recognize the informal “rule” for the pattern.

Engagement Statements (demonstration of engaged in the topic)

Engage with patterns using manipulatives

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them. (A2, M3) a. Focus on transformations of graphs of quadratic functions, except for f(kx. (A1, M2)

Identify a line reflected over the y-axis on a coordinate grid.

Identify a line reflected over the x-axis on a coordinate grid.

Identify a line on a coordinate grid.

Find inverse functions. a. Informally determine the input of a function when the output is known. (A1, M1)

Solve for x when y is given (e.g. y = x+3; what is the value of x when y is 5?).

Identify the input and output of a function.

Identify a function.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Determine if a relation is a function.

Complete an input-output table when given the function rule and values.

Identify the input or output of a function given in table form.

[Between b and c:

Interact with real-world situations that can be represented by functions.

Interact with relations that are not functions (e.g., Talk about pets, making Dog, Cat,and Horse as the input, ask the question “Which student has which pet?” and establish the correspondence, to decide if the relation is or is not a function.)

Establish the ordered pairs between the corresponding elements of the input and output.]

Show real world relations in table form.

Know the meaning of the words input and output.

Relate input and output to a function machine.

Understand that a table is made up of columns and rows.

Know that input is in the left column and the output is in right column.

Columns are up and down.

Rows are left and right.

Engagement Statements (demonstration of engaged in the topic)

Interact with constructing tables.

Interact with real-world situations.

Interact with a function machine.

Interact with technology.

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Given a linear equation using function notation, complete a table of values.

Represent an equation in y= form with f(x).

Understand that f(x)=y.

Experience real world concepts that are represented by a symbol. (e.g., street signs, symbols on technology tools, etc.)

Recognize that a function can be written as abstract mathematical symbolic language

Levels F.IF.1a-c need to be mastered.

Engagement Statements (demonstration of engaged in the topic)

Interact with constructing tables.

Interact with real-world situations.

Interact with a function machine.

Interact with technology.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. For example, the Fibonacci sequence is defined recursively by f(0) = f(1) = 1, f(n + 1) = f(n) + f(n − 1) for n ≥ 1.

Given a sequence, determine the functional rule.

Predict the next three terms in an arithmetic or geometric sequence (e.g., 3,6,9 …).

Identify the common ratio or common difference in a sequence.

Describe the “rule” for a pattern.

Identify the “rule” for a pattern.

Discover patterns in the real world

Build or draw patterns using technology or manipulatives.

Translate patterns to numbers

Replicate a given pattern.

Engagement Statements (demonstration of engaged in the topic)

Interact with patterns

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity. (A2, M3) a. Focus on linear and exponential functions. (M1) b. Focus on linear, quadratic, and exponential functions. (A1, M2)

Given a function made up of several linear functions, determine where the function is increasing, decreasing, or flat

Given a graph of a linear equation, identify the y and/or x intercept.

Determine whether the linear function is increasing, decreasing, or flat.

Identify the x- and y- axis.

Read the graph from left to right.

Recognize that the x- and y- axes are number lines.

Recognize a line.

Recognize patterns of the line going up or down, or staying at the same level, reading from left to right.

Experience the creation of graphs using science probes

Demonstrate stories of a given graph.

Create a graph to a story using technology.

Know that flat means not going up or down.

Know that increasing is up and decreasing is down.

Recognize that in real-world increasing is adding quantities. For example, buying more objects increases the price. Connecting these situations with the vocabulary of “positive slope”.

Recognize that in real-world decreasing is removing quantities. For example, buying more objects decreases the amount of money you have. Connecting these situations with the vocabulary of “negative slope”.

Recognize that in real-world there are situations where there is no change in the dependent variable. For example, an adult height does not change over time.

Connecting these situations with the vocabulary of “zero slope”.

Engagement Statements (demonstration of engaged in the topic)

Interact with real-world examples of lines.

Relate the domain of a function to its graph, and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. a. Focus on linear and exponential functions. (M1) b. Focus on linear, quadratic, and exponential functions. (A1, M2) c. Emphasize the selection of a type of function for a model based on behavior of data and context. (A2, M3)

Given the graph represented by a linear function, determine the domain.

Given a context of a linear equation, describe the domain.

Given a table, state the input values.

Given a real world situation, e.g., When I buy 1 chocolate bar - I pay 2 dollars, when I buy 2 chocolate bars - I pay 4 dollars, match the values in the situation with the values in the table.

Count physical objects.

Relate the input and the output to a function machine.

Answer questions about a table.

Identify what the input is in a table.

Engagement Statements (demonstration of engaged in the topic)

Interact with real-world situations that can be represented by functions.

Interact with constructing tables.

Interact with relations that are functions.

Explore a chart or a table.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. (A2, M3)

Identify the slope of a line when the equation is written in slope intercept form.

Identify the slope of a line when given in graph form.

Determine whether a slope is present on a given visual graph.

Graph functions expressed symbolically and indicate key features of the graph, by hand in simple cases and using technology for more complicated cases. Include applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate. a. Graph linear functions and indicate intercepts. (A1, M1) b. Graph quadratic functions and indicate intercepts, maxima, and minima. (A1, M2) c. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. (A2, M3) d. Graph polynomial functions, identifying zeros, when factoring is reasonable, and indicating end behavior. (A2, M3) e. Graph simple exponential functions, indicating intercepts and end behavior. (A1, M1) f. Graph exponential functions, indicating intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. (A2, M3)

Graph a linear function using a graph with a scale of 1.

Determine whether an ordered pair is a viable solution to a given linear function.

Determine the y intercept point for a linear graph.

Determine whether the line is increasing (going up), decreasing (going down), or flat.

Identify two point on a linear graph.

Classify graphs of functions as linear or nonlinear.

Identify the x- and y- axis.

Recognize that the x- and y- axes are number lines.

Recognize a line vs. a curve.

Draw lines and curves.

Use technology to explore graphing.

Recognize where the lines intersect the axes

Engagement Statements (demonstration of engaged in the topic)

Interact with graphs.

Interact with technology.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. (A2, M3) i. Focus on completing the square to quadratic functions with the leading coefficient of 1. (A1, M2) b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of changeG in functions such as y = (1.02)t , and y = (0.97)t and classify them as representing exponential growth or decay. (A2, M3) i. Focus on exponential functions evaluated at integer inputs. (A1, M2)

Identify equivalent expressions (e.g. 2x + 2x = 4x, x2 * x2 =x4 ).

Identify equivalent expressions (limit to three terms) (e.g. x + x + x = 3x, x*x*x=x3 ).

Identify equivalent expressions (limit to two terms) (e.g. x + x = 2x).

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. (A2, M3) a. Focus on linear and exponential functions. (M1) b. Focus on linear, quadratic, and exponential functions. (A1, M2)

Compare a function given in table form to another function given in graphical form. For example, which one is increasing?

Match a function given in table form to its graph.

Match a function given as a verbal description to its graph. For example, which is an increasing function?

Demonstrate stories to match an increasing/decreasing graph.

Match an increasing/decreasing graph to a story.

Connect to F.IF.4 and F.IF.3

Engagement Statements (demonstration of engaged in the topic)

Interact with graphs.

Interact with technology.

Experience and act out real world situation that can be graphed (e.g., distance/time graphs).

Distinguish between situations that can be modeled with linear functions and with exponential functions. a. Show that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals. b. Recognize situations in which one quantity changes at a constant rate per unit interval relative to another. c. Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Identify a situation that represents a linear and/or exponential function.

Identify the graph of a linear function and an exponential function.

Identify a graph of a linear function.

Sort graphs based on their shape.

Draw lines and curves

Recognize a line vs. a curve

Observe graphs of different functions with technology

Engagement Statements (demonstration of engaged in the topic

Interact with technology

Interact with no more than 3 answer choices be able to select 1 from different positions.

Interact with a visual model.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two inputoutput pairs (including reading these from a table).

After creating a sequence, make a graph that represents that sequence.

Create a geometric sequence of at least 5 numbers with models.

Create an arithmetic sequence with a model.

Translate patterns to numbers

Build patterns using manipulatives

Draw the patterns

Recognize the informal “rule” for the pattern

Experience patterns and non-patterns.

Discover patterns in the real world.

Engagement Statements (demonstration of engaged in the topic

Engage with patterns using manipulatives

Interact with models of patterns.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically. (A1, M2)

Observe a situation that shows increasing and decreasing linear and exponential events. (e.g., doubling a penny every day gives you more money than receiving $100 a day for a month)

Identify an exponential function.

Identify if a linear function is increasing or decreasing.

For exponential models, express as a logarithm the solution to abct=d where a, c, and d are numbers, and the base b is 2, 10, or e; evaluate the logarithm using technology.

Identify equivalent expressions with exponents.

Identify equivalent expressions with exponents (limit to power 3 expressions) (e.g. Which is the same as m x m x m?).

Identify equivalent expressions with exponents (limit to power 2 expressions) (e.g. Which is the same as m x m?).

Interpret the parameters in a linear or exponential function in terms of a context.

Given a context, interpret the parameters of an exponential function (e.g., during x weeks, the existing number of fish in the pond has been doubled). This situation is modeled by the exponential function y = 100(2^x), where 100 is the initial number of fish in the pond, 2 is a growth factor, (2^x) is the number by which the initial number of fish, 100, is multiplied for every increase in x, and y is the total number of fish in the pond.

Given a context, interpret the parameters of a linear function (e.g., Marsha has $10 already saved and saves an additional $5 a week for x number of weeks). This situation is modeled by a linear function f(x) = 5x +10, where 10 is the initial amount that has been saved, 5 is the weekly saving, 5x is the amount of money saved during x weeks, and f(x) is a total amount of money saved including the initial amount.

Identify the constant in a linear or exponential equation. OR When given the graph of a function, identify the y intercept.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Identify the measure of a central angle on a circle when the measure of the arc is given.

Identify the measure of an angle.

Identify an angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counterclockwise around the unit circle.

Identify the measure of a central angle on a circle when the measure of the arc is given.

Identify the measure of an angle.

Identify an angle.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Identify the measure of a central angle on a circle when the measure of the arc is given.

Identify the measure of an angle.

Identify an angle.

Find the hypotenuse when the length of the sides is given.

Identify the parts of a right triangle.

Identify a right triangle.

Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

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Complexity c

Complexity b

Complexity a

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Complexity c

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Apply geometric concepts in modeling situations.

Modeling with Geometry

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Complexity c

Complexity b

Complexity a

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Complexity c

Complexity b

Complexity a

Understand the relationships between length, area, and volume.

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Complexity c

Complexity b

Complexity a

Visualize relationships between two-dimensional and three-dimensional objects.

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Complexity c

Complexity b

Complexity a

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Learning Progression

Complexity c

Complexity b

Complexity a

Explain volume formulas, and use them to solve problems.

Geometric Measurement and Dimension

Learning Progression

Complexity c

Complexity b

Complexity a

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Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

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Learning Progression

Complexity c

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Use coordinates to prove simple geometric theorems algebraically and to verify specific geometric statements.

Not on BP

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Complexity c

Complexity b

Complexity a

Translate between the geometric description and the equation for a conic section.

Expressing Geometric Properties with Equations

Learning Progression

Complexity c

Complexity b

Complexity a

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Learning Progression

Complexity c

Complexity b

Complexity a

Find arc lengths and areas of sectors of circles.

BP??

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Complexity c

Complexity b

Complexity a

Learning Progression

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Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Understand and apply theorems about circles.

Circles

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Complexity c

Complexity b

Complexity a

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Complexity c

Complexity b

Complexity a

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Define trigonometric ratios, and solve problems involving right triangles.

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Learning Progression

Complexity c

Complexity b

Complexity a

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Learning Progression

Complexity c

Complexity b

Complexity a

Prove and apply theorems both formally and informally involving similarity using a variety of methods.

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Complexity c

Complexity b

Complexity a

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Learning Progression

Complexity c

Complexity b

Complexity a

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Use complex numbers in polynomial identities and equations.

Similarity Right Triangles and Trigonometry

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

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Complexity a

Make geometric constructions.

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Prove geometric theorems both formally and informally using a variety of methods.

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Understand congruence in terms of rigid motions.

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Complexity b

Complexity a

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Learning Progression

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Complexity a

Experiment with transformations in the plane.

Congruence

Prove that all circles are similar using transformational arguments.

Compare two circles and determine how to change one to make it the same as the other (e.g., circle A needs to be enlarged to match circle B).

Label the parts of a circle.

Locate circles in the environment.

Not on BP

Identify and describe relationships among angles, radii, chords, tangents, and arcs, and use them to solve problems. Include the relationship between central, inscribed, and circumscribed angles and their intercepted arcs; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Use the radius of a circle to determine the length of the diameter and vice versa.

Identify parts of a circle (radius, diameter, circumference, chord, and arc).

Locate circles in the environment.

Between level c and b:

Recognize the difference between the circle (the points equal distance from the midpoint) and the area of the circle.

Recognize circles in a collection of different shapes.

Recognize that a circle is not a sphere.

Know that a circle has no straight lines.

Know that a circle has no vertices/corners).

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of shapes.

Construct the inscribed and circumscribed circles of a triangle, prove and apply the property tht opposite angles are supplementary for a quadrilateral inscribed in a circle.

Identify a circumscribed circle about a triangle.

Identify a circle inscribed in a triangle.

Identify a circle.

Find arc lengths and areas of sectors of circles. a. Apply similarity to relate the length of an arc intercepted by a central angle to the radius. Use the relationship to solve problems. b. Derive the formula for the area of a sector, and use it to solve problems.

Identify the central angle of a circle. OR Apply the formula to the area of a sector (e.g., area of a slice of pie).

Identify the sector of a circle.

Identify the arc of a circle.

Derive formulas that relate degrees and radians, and convert between the two. (A2, M3)

Identify the central angle of a circle.

Identify the sector of a circle.

Identify the arc of a circle.

Not on BP

Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and arc length.

Identify points, lines, line segments, angles (right, acute, obtuse, and order by size), and perpendicular and parallel lines.

Identify points, lines, line segments and angles (right, acute, obtuse, and order by size).

Identify points, lines, and line segments, and order angles by size.

Understand that an angle is created by two (straight) rays that meet at a point

Draw geometric figures (e.g. points, lines, line segments and angles)

Recognize different angles in real - world situations

Use manipulatives to create angles and recognize that the size of the angle is related to the amount of rotation of the initial ray.

Engagement Statements (demonstration of engaged in the topic)

Interact with geometric figures (e.g. points, lines, rays, line segments and angles).

Interact with line segments and angles in the real world.

Prove and apply theorems about triangles. Theorems include but are not restricted to the following: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Determine the sum of the measures of the interior angles of a triangle. Identify congruent angles in isosceles and equilateral triangles.

Identify right, equilateral, and isosceles triangles.

Identify a triangle.

Understand that a triangle is a closed shape.

Understanding a triangle has three points (vertices) and three sides.

Sort a variety of shapes to recognize triangles.

Count the number of sides of a shape.

Count the number of angles of a shape.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of shapes to recognize triangles.

Prove and apply theorems about parallelograms. Theorems include but are not restricted to the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

Identify the congruent sides and angles of a parallelogram.

Identify congruent sides on a parallelogram.

Identify a rectangle and a parallelogram.

Sort quadrilaterals based on the presence or absence of right angles.

Discover right angles/square corners.

Understand that a quadrilateral is a closed shape.

Sort a variety of figures to recognize quadrilaterals.

Sort quadrilaterals and describe their rule.

Count the number of sides of a shape.

Count the number of angles of a shape.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of shapes to recognize quadrilaterals.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct a circle given a center and a radius. Copy a segment.

Construct a line segment given its endpoints.

Identify geometric tools (e.g., straightedge, protractor, and ruler) and their uses.

Match tools with drawings and their names.

Observe the usage of geometric tools, e.g., straight edge, ruler, protractor, and compass.

Explore/use geometric tools.

Engagement Statements (demonstration of engaged in the topic)

Interact with geometric tools (e.g., straightedge, protractor, compass, and ruler)

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Construct an equilateral triangle.

Construct a circle given a center and point on the circle.

Given three congruent line segments (or sticks), make an equilateral triangle.

Understand “congruent triangles” as being of equal sidelength and angle measures.

Interact with geometric tools (e.g., straightedge, protractor, and ruler).

Recognize and identify triangles with equal side lengths.

Sort triangles by equal or different side lengths.

Sort triangles by equal or different angle measures.

Use manipulatives to create triangles.

Recognize triangles in the real-world.

Engagement Statements (demonstration of engaged in the topic)

Interact with triangles in the real-world.

Classify twodimensional figures in a hierarchy based on properties.

Classify two-dimensional shapes based on their properties.

Sort different types of quadrilaterals.

Sort different types of triangles.

Recognize and identify triangles with equal side lengths.

Sort triangles by the presence or absence of a right angle, an angle larger than a right angle, or three angles smaller than a right angle.

Sort triangles by the number of equal side lengths.

Sort triangles by the number of equal angle measures.

Use manipulatives to create triangles.

Recognize triangles in the real-world.

Engagement Statements (demonstration of engaged in the topic)

Interact with triangles in the real-world.

Interact with and sort a variety of shapes to recognize triangles.

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not, e.g., translation versus horizontal stretch.

Demonstrate that a rotation (turn), a reflection (flip), or a translation (slide) maps a figure onto another.

Identify whether a rotation (turn), a reflection (flip), or a translation (slide) can map a figure onto another.

Match shapes in different orientations. (i.e., shapes = 2D)

[Between levels b and c:

Recognize the orientation of objects using terms such as above, below, in front of, behind, and next to.]

Match drawings of shapes having the same orientation.

Match shapes using manipulatives.

Recognize objects in the environment using names of shapes.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of shapes (e.g. using pattern blocks)

Identify the symmetries of a figure, which are the rotations and reflections that carry it onto itself. a. Identify figures that have line symmetry; draw and use lines of symmetry to analyze properties of shapes. b. Identify figures that have rotational symmetry; determine the angle of rotation, and use rotational symmetry to analyze properties of shapes.

Show that two figures have symmetry on a coordinate plane.

Identify figures that have line symmetry or rotational symmetry, using concrete objects or on a coordinate plane.

Given visual models, determine which figures have line symmetry. (i.e., figure =3D)

Sort photos of real - world shapes with and without line symmetry.

Identify shapes without line symmetry.

Explore the concept of line symmetry in the real - world.

Engagement Statements (demonstration of engaged in the topic)

Interact with shapes.

Interact with visual models of shapes with line symmetry.

Observe a demonstration of shape folding, using shapes with and without line symmetry.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Identify that a translation requires a direction and distance; a rotation requires a center and angle; and a reflection requires a line.

Identify whether a transformed figure is a “translation,” “reflection,” or “rotation.”

Identify whether a transformed figure is a “slide,” “flip,” or “turn.”

Match drawings of figures having the same and different orientation.

Experience different rigid transformations in combination with the vocabulary.

Match the figures’ orientation with the type of the transformation.

Demonstrate with hand movement or technology what the terms, slide, flip, and turn mean.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of 2D - shapes (e.g. using pattern blocks) including angles and line segments.

Observe a demonstration of transformations using technology and manipulatives.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using items such as graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure.

Given visuals or real-world items, demonstrate a rotation (turn), a reflection (flip), or a translation (slide).

Match shapes in different orientations.

[Between levels b and c:

Recognize the orientation of shapes using terms such as slide, flip, and turn.]

Match drawings of shapes having the same orientation.

Match shapes using manipulatives.

Recognize shapes in the environment using the names of shapes.

Experience different rigid transformations in combination with the vocabulary.

Match the shapes’ orientation with the type of the transformation

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of shapes (e.g. using pattern blocks)

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent

Identify a basic rigid motion (a rotation (turn), a reflection (flip), or a translation (slide) that maps one figure onto another. (Restrict to situations in which a single basic rigid motion suffices.)

Show two figures are congruent by demonstrating that a rotation (turn), a reflection (flip), or a translation (slide) maps one onto the other.

Match shapes to show congruence by placing one figure on top of the other.

Recognize congruent shapes.

Experience different rigid transformations

Match the shapes’ orientation with the type of the transformation, in combination with the vocabulary, flip, slide, and turn.

Recognize when two similar shapes are the same size or not.

Recognize shapes that are the same.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of 2D - shapes, including angles and line segments

Observe demonstrations of shapes mapping onto each other or not, using technology or manipulatives.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Identify whether a rotation (turn), a reflection (flip), or a translation (slide) is required to show that a triangle is congruent to another triangle on a coordinate plane. Limit to one transformation.

Identify whether a rotation (turn), a reflection (flip), or a translation (slide) is required to show that a triangle is congruent to another triangle. Limit to one transformation.

Match triangles in different orientations.

[Between levels c and b:

Recognize congruent triangles.]

Experience different rigid transformations.

Match the triangles’ orientation with the type of the transformation, in combination with the vocabulary, flip, slide, and turn.

Match drawings of triangles having the same orientations.

Recognize when two similar triangles are the same size or not.

Recognize triangles that are the same shape.

Identify triangles.

G.CO.10 needs to be taught before G.CO.7

Observe demonstrations of triangles mapping onto each other or not, using technology or manipulatives.

Interact with a variety of triangles, including angles and line segments

Engagement Statements (demonstration of engaged in the topic)

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Determine whether two triangles are congruent using ASA, SAS, or SSS.

Match corresponding parts (sides and angles) of congruent triangles.

Match one corresponding part (side or angle) of two congruent triangles.

Given two congruent triangles with different orientations, identify a corresponding angle or side.

Recognize corresponding parts - sides and angles

Recognize angles and sides of a triangle.

Understand that an angle in a triangle is created by two sides that meet at a point (vertex)

G.CO.10 needs to be taught first.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of congruent triangles.

Observe demonstrations of triangles mapping onto each other or not, using technology or manipulatives.

Prove and apply theorems about lines and angles. Theorems include but are not restricted to the following: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

Identify a pair of vertical, complementary, supplementary, corresponding, alternative interior, or alternate exterior angles.

Find a missing angle measure for situations involving vertical, complementary, supplementary, corresponding, alternative interior, and alternative exterior angles.

Given a pair of vertical angles and a missing angle measurement, find the missing angle measure.

Bisect a line segment using a ruler, compass, technology, or other means and label the midpoint.

Create a pair of perpendicular lines using a ruler, compass, technology or other means. Include the right-angle marking.

Identify vertical angles.

Identify a set of perpendicular lines.

Identify the midpoint of a line segment. Identify a right angle.

Recognize or draw intersecting lines.

Sort intersecting lines by angles that meet perpendicular or not.

Among intersecting lines recognize the special case of perpendicular lines.

Discover vertical angles.

Discover right angles/square corners.

Understand that angles are created by two lines that meet at a point.

Recognize angles formed by intersecting lines.

Understand that the midpoint is equal distance from each end point.

Manipulate a drawn line segment, drawn on e.g., patty paper, to recognize the midpoint.

Identify a line segment.

Identify a line.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of intersecting lines.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

Compare the volume of two objects with the same base but different heights and vice versa (e.g., Which cup can hold more water: the shorter or the taller cup; given the choice of different sized cubes, identify which would hold more).

Distinguish between objects that do and do not have volume.

Sort three-dimensional objects (cones, cylinders, spheres).

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Compare the volume of two objects with the same base but different heights and vice versa (e.g., Which cup can hold more water: the shorter or the taller cup; given the choice of different sized cubes, identify which would hold more).

Distinguish between objects that do and do not have volume.

Sort three-dimensional objects (cones, cylinders, spheres, pyramids).

Identify the shapes of two-dimensional crosssections of three-dimensional objects, and identify three-dimensional objects generated by rotations of twodimensional objects.

Identify cross-sections of three- dimensional shapes.

Identify faces of threedimensional shapes.

Identify two- and threedimensional shapes.

Understand how and when changes to the measures of a figure (lengths or angles) result in similar and non-similar figures.

Compare the volume of threedimensional shapes.

Compare the area of shapes.

Compare similar shapes.

When figures are similar, understand and apply the fact that when a figure is scaled by a factor of k, the effect on lengths, areas, and volumes is that they are multiplied by k, k2, and k3, respectively.

Find the volume of two similar threedimensional shapes.

Find the area of two similar shapes.

Identify similar shapes.

Derive the equation of a circle of given center and radius using the Pythagorean theorem; complete the square to find the center and radius of a circle given by an equation.

Identify the diameter of a circle.

Identify the radius of a circle.

Identify a circle.

Use coordinates to prove simple geometric theorems algebraically and to verify geometric relationships algebraically, including properties of special triangles, quadrilaterals, and circles. For example, determine if a figure defined by four given points in the coordinate plane is a rectangle; determine if a specific point lies on a given circle. (G, M2)

Find the perimeter of quadrilaterals drawn on a coordinate grid.

Identify shapes on a coordinate grid.

Identify special triangles, quadrilaterals, and circles.

Justify the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems, e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point.

Describe the “rise and run” relationships between two perpendicular lines.

Identify the slopes of parallel and perpendicular lines in a coordinate grid.

Identify parallel and perpendicular lines in a coordinate grid.

Use manipulatives to create parallel and perpendicular lines.

Identify parallel lines in the real-world.

Sort lines in to parallel and perpendicular.

Sort intersecting lines by angles that meet perpendicular or not.

Among intersecting lines recognize the special case of perpendicular lines.

Discover right angles/square corners.

Understand that angles are created by two lines that meet at a point.

Recognize angles formed by intersecting lines.

Identify a line.

Engagement Statements (demonstration of engaged in the topic)

Interact with manipulatives, eg., straws and skewers.

Interact with a coordinate grid.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Find the midpoint of a vertical or horizontal line on a coordinate grid.

Find the length of a vertical or horizontal line on a coordinate grid.

Identify points, lines, and line segments.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Find the area and perimeter of shapes given on a coordinate grid. (Restrict to shapes with sides that are vertical or horizontal line segments.)

Find the perimeter of shapes given on a coordinate grid. (Restrict to shapes with sides that are vertical or horizontal line segments.)

Identify shapes on a coordinate grid.

Use manipulatives to create shapes.

Identify shapes in the real-world.

Sort shapes.

Engagement Statements (demonstration of engaged in the topic)

Interact with manipulatives.

Interact with a coordinate grid.

Interact with physical objects that represent 2-D shapes in the real-world, e.g., the lid on a sandwich box, a window, etc.

Use geometric shapes, their measures, and their properties to describe objects, e.g., modeling a tree trunk or a human torso as a cylinder.

Connect the shape of real-world objects to twodimensional and three-dimensional shapes (e.g., the trunk of a tree is cylindrical in shape; a car is cube in shape; the center of a sunflower is circular in shape; a bookshelf is rectangular prism in shape).

Connect the shape of realworld objects to twodimensional shapes (e.g., a window is rectangular in shape, a wheel is circular in shape, and a table can be of many different shapes).

Connect two-dimensional shapes with realworld objects.

Apply concepts of density based on area and volume in modeling situations, e.g., persons per square mile, BTUs per cubic foot.

Calculate and compare the densities of two datasets in the same modeling situation (e.g, Is the population density of Ohio or New York greater?).

Calculate the density of a given situation.

Given representations of density, identify the one with the greatest or least density (e.g., If 3 squares of the same size have different numbers of dots in them, which one has the greatest number of dots?).

Apply geometric methods to solve design problems, e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios.

Compare the volume of realworld objects.

Compare the area of real-world objects.

Sort shapes that model a real-world object (e.g., a baseball is a sphere, a can of soup is a cylinder).

Verify experimentally the properties of dilations given by a center and a scale factor. a. A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged.

Determine the dimensions of a figure after dilation.

Determine if a figure is bigger or smaller after dilation.

Compare 2 figures to determine if a dilation has occurred.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles, and the proportionality of all corresponding pairs of sides.

Determine if figures are similar; describe or select why two figures are or are not similar.

Determine if two rectangles or triangles are similar.

Identify similar triangles.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Identify similar triangles in different orientations.

Identify similar triangles.

Identify a triangle.

Prove and apply theorems about triangles. Theorems include but are not restricted to the following: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean theorem proved using triangle similarity.

Identify different types of triangles.

Identify parts of a right triangle.

Identify a right angle in the environment.

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures that can be decomposed into triangles.

Identify if triangles are similar, or not, in a given geometric figure;

Identify if triangles are similar or not in a decomposed polygon; e.g., is triangle ABD similar to triangle DCA?

Identify similar triangles.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Identify parts of a right triangle.

Identify right triangles.

Identify a triangle.

Not on BP

Explain and use the relationship between the sine and cosine of complementary angles.

Identify parts of a right triangle.

Identify right triangles.

Identify an angle of a triangle.

Solve problems involving right triangles. a. Use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems if one of the two acute angles and a side length is given. (G, M2) b. Use trigonometric ratios and the Pythagorean theorem to solve right triangles in applied problems. (A2, M3)

Construct a right triangle on a coordinate plane and label the parts.

Identify the parts of a right triangle (right angle, legs, and hypotenuse).

Given an assortment of triangles, identify right triangles.

Conditional Probability And The Rules Of Probability

Making Inferences And Justifying Conclusions

Interpreting Categorical And Quantitative Data

High School — Statistics and Probablity

Modeling With Geometry

Geometric Measurement And Dimension

Expressing Geometric Properties With Equations

Circles

Similarity, Right Triangles, And Trigonometry

Congruence

High School — Geometry

Trigonometric Functions

Linear, Quadratic, And Exponential Models

Building Functions

Interpreting Functions

High School — Functions

Reasoning With Equations And Inequalities

Creating Equations

Arithmetic With Polynomials And Rational Expressions

Seeing Structure In Expressions

High School — Algebra

Vector And Matrix Quantities

The Complex Number System

Quantities

The Real Number System

High School — Number and Quantity

Standards for Mathematical Practice

Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers.

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Derive the quadratic formula using the method of completing the square.

Represent a system of linear equations as a single matrix equation in a vector variable.

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Compose functions.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Verify by composition that one function is the inverse of another.

Find the inverse of a function algebraically, given that the function has an inverse.

Produce an invertible function from a non-invertible function by restricting the domain.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Graph rational functions, identifying zeros and asymptotes when factoring is reasonable, and indicating end behavior.

Graph logarithmic functions, indicating intercepts and end behavior.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π − x, π + x, and 2π − x in terms of their values for x, where x is any real number.

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.

Construct a tangent line from a point outside a given circle to the circle.

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Derive the equation of a parabola given a focus and directrix.

Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Explain proofs of the Laws of Sines and Cosines and use the Laws to solve problems.

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles, e.g., surveying problems, resultant forces.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Find the conjugate of a complex number; use conjugates to find magnitudes and quotients of complex numbers.

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Calculate the distance between numbers in the complex plane as the magnitude of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Extend polynomial identities to the complex numbers.

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes, e.g., v, |v|, ||v||, <img src="https://purl.org/ASN/resources/images/D2784929/N.VM.1.gif" alt="vec{x}" />.

Understand that the zero and identity matrices play a role in matrix addition and multiplication analogous to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Solve problems involving velocity and other quantities that can be represented by vectors.

Add and subtract vectors.

Multiply a vector by a scalar

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c (v_x, v_y) = (cv _x, cv_y).

Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Add, subtract, and multiply matrices of appropriate dimensions.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)·P(B|A) = P(B)·P(A|B), and interpret the answer in terms of the model.

Use permutations and combinations to compute probabilities of compound events and solve problems.

Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.

Calculate the expected value of a random variable; interpret it as the mean of the probability distribution.

Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value.

Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value.

Weigh the possible outcomes of a decision by assigning probabilities to payoff values and finding expected values.

Use probabilities to make fair decisions, e.g., drawing by lots, using a random number generator.

Analyze decisions and strategies using probability concepts, e.g., product testing, medical testing, pulling a hockey goalie at the end of a game.

Understand that polynomials form a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Focus on polynomial expressions that simplify to forms that are linear or quadratic.

Extend to polynomial expressions beyond those expressions that simplify to forms that are linear or quadratic.

Understand and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x − a is p(a). In particular, p(a) = 0 if and only if (x – a) is a factor of p(x).

Identify zeros of polynomials, when factoring is reasonable, and use the zeros to construct a rough graph of the function defined by the polynomial.

Prove polynomial identities and use them to describe numerical relationships.

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Create equations and inequalities in one variable and use them to solve problems. Include equations and inequalities arising from linear, quadratic, simple rational, and exponential functions.

Focus on applying linear and simple exponential expressions.

Focus on applying simple quadratic expressions.

Extend to include more complicated function situations with the option to solve with technology.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Focus on applying linear and simple exponential expressions.

Focus on applying simple quadratic expressions.

Extend to include more complicated function situations with the option to graph with technology.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Focus on formulas in which the variable of interest is linear or square.

Focus on formulas in which the variable of interest is linear.

Focus on formulas in which the variable of interest is linear or square.

While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equation y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, making tables of values, or finding successive approximations.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations in one variable.

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x − p)² = q that has the same solutions.

Solve quadratic equations as appropriate to the initial form of the equation by inspection, e.g., for x² = 49; taking square roots; completing the square; applying the quadratic formula; or utilizing the Zero-Product Property after factoring.

Verify that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve systems of linear equations algebraically and graphically.

Limit to pairs of linear equations in two variables.

Extend to include solving systems of linear equations in three variables, but only algebraically.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Interpret expressions that represent a quantity in terms of its context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions. For example, 8t can be written as 2³t.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from context.

Focus on linear and exponential functions.

Focus on situations that exhibit quadratic or exponential relationships.

Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Focus on transformations of graphs of quadratic functions, except for f(kx);

Find inverse functions.

Informally determine the input of a function when the output is known.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Focus on linear and exponential functions.

Focus on linear, quadratic, and exponential functions.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Focus on linear and exponential functions.

Focus on linear, quadratic, and exponential functions.

Emphasize the selection of a type of function for a model based on behavior of data and context.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph functions expressed symbolically and indicate key features of the graph, by hand in simple cases and using technology for more complicated cases. Include applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate.

Graph linear functions and indicate intercepts.

Graph quadratic functions and indicate intercepts, maxima, and minima.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros, when factoring is reasonable, and indicating end behavior.

Graph simple exponential functions, indicating intercepts and end behavior.

Graph exponential functions, indicating intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Focus on completing the square to quadratic functions with the leading coefficient of 1.

Use the properties of exponents to interpret expressions for exponential functions.

Focus on exponential functions evaluated at integer inputs.

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

Focus on linear and exponential functions.

Focus on linear, quadratic, and exponential functions.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Show that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Interpret the parameters in a linear or exponential function in terms of a context.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counter-clockwise around the unit circle.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Prove that all circles are similar using transformational arguments.

Identify and describe relationships among angles, radii, chords, tangents, and arcs and use them to solve problems. Include the relationship between central, inscribed, and circumscribed angles and their intercepted arcs; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct the inscribed and circumscribed circles of a triangle; prove and apply the property that opposite angles are supplementary for a quadrilateral inscribed in a circle.

Find arc lengths and areas of sectors of circles.

Apply similarity to relate the length of an arc intercepted by a central angle to the radius. Use the relationship to solve problems.

Derive the formula for the area of a sector, and use it to solve problems.

Derive formulas that relate degrees and radians, and convert between the two.

Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and arc length.

Prove and apply theorems about triangles. Theorems include but are not restricted to the following: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Prove and apply theorems about parallelograms.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Classify two-dimensional figures in a hierarchy based on properties.

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not, e.g., translation versus horizontal stretch.

Identify the symmetries of a figure, which are the rotations and reflections that carry it onto itself.

Identify figures that have line symmetry; draw and use lines of symmetry to analyze properties of shapes.

Identify figures that have rotational symmetry; determine the angle of rotation, and use rotational symmetry to analyze properties of shapes.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using items such as graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove and apply theorems about lines and angles. Theorems include but are not restricted to the following: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Understand how and when changes to the measures of a figure (lengths or angles) result in similar and non-similar figures.

When figures are similar, understand and apply the fact that when a figure is scaled by a factor of k, the effect on lengths, areas, and volumes is that they are multiplied by k, k², and k³, respectively.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Use coordinates to prove simple geometric theorems algebraically and to verify geometric relationships algebraically, including properties of special triangles, quadrilaterals, and circles.

Justify the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems, e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Use geometric shapes, their measures, and their properties to describe objects, e.g., modeling a tree trunk or a human torso as a cylinder.

Apply concepts of density based on area and volume in modeling situations, e.g., persons per square mile, BTUs per cubic foot.

Apply geometric methods to solve design problems, e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios.

Verify experimentally the properties of dilations given by a center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove and apply theorems about triangles.

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures that can be decomposed into triangles.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Solve problems involving right triangles.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems if one of the two acute angles and a side length is given.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

Rewrite rational expressions.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations and inequalities graphically.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Understand the concept of a function, and use function notation.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Construct and compare linear, quadratic, and exponential models, and solve problems.

Interpret expressions for functions in terms of the situation they model.

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

Prove and apply trigonometric identities.

Understand and apply theorems about circles.

Find arc lengths and areas of sectors of circles.

Experiment with transformations in the plane.

Understand congruence in terms of rigid motions.

Prove geometric theorems both formally and informally using a variety of methods.

Make geometric constructions.

Classify and analyze geometric figures.

Explain volume formulas, and use them to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects.

Understand the relationships between lengths, area, and volumes.

Translate between the geometric description and the equation for a conic section.

Use coordinates to prove simple geometric theorems algebraically and to verify specific geometric statements.

Apply geometric concepts in modeling situations.

Understand similarity in terms of similarity transformations.

Prove and apply theorems both formally and informally involving similarity using a variety of methods.

Define trigonometric ratios, and solve problems involving right triangles.

Apply trigonometry to general triangles.

Perform arithmetic operations with complex numbers.

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Reason quantitatively and use units to solve problems.

Extend the properties of exponents to rational exponents.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Perform operations on vectors.

Perform operations on matrices, and use matrices in applications.

Understand independence and conditional probability, and use them to interpret data.

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Calculate expected values, and use them to solve problems.

Use probability to evaluate outcomes of decisions.

Understand and evaluate random processes underlying statistical experiments.

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Summarize, represent, and interpret data on a single count or measurement variable.

Summarize, represent, and interpret data on two categorical and quantitative variables.

Interpret linear models.

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.

Know there is a complex number i such that i² = −1, and every complex number has the form a + bi with a and b real.

Use the relation i² = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Solve quadratic equations with real coefficients that have complex solutions.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Define appropriate quantities for the purpose of descriptive modeling.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Understand vector subtraction v − w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events ("or," "and," "not").

Understand that two events A and B are independent if and only if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations.

Find the conditional probability of A given B as the fraction of B's outcomes that also belong to A, and interpret the answer in terms of the model.

Apply the Addition Rule, P(A or B) = P(A) + P(B) − P(A and B), and interpret the answer in terms of the model.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Decide if a specified model is consistent with results from a given data-generating process, e.g., using simulation.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between sample statistics are statistically significant.

Evaluate reports based on data.

Represent data with plots on the real number line (dot plots, histograms, and box plots) in the context of real-world applications using the GAISE model.

In the context of real-world applications by using the GAISE model, use statistics appropriate to the shape of the data distribution to compare center (median and mean) and spread (mean absolute deviation, interquartile range, and standard deviation) of two or more different data sets.

In the context of real-world applications by using the GAISE model, interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.

Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Represent data on two quantitative variables on a scatter plot, and describe how the variables are related.

Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions, or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models.

Informally assess the fit of a function by discussing residuals.

Fit a linear function for a scatterplot that suggests a linear association.

Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

Compute (using technology) and interpret the correlation coefficient of a linear fit.

Distinguish between correlation and causation.

Find the expected payoff for a game of chance. For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant.

Evaluate and compare strategies on the basis of expected values.

Reasoning With Equations And Inequalities

Creating Equations

Arithmetic With Polynomials And Rational Expressions

Seeing Structure In Expressions

Standards for Mathematical Practice

Know and apply the Binomial Theorem for the expansion of (x + y)n in powers of x and y for a positive integer n, where x and y are any numbers.

Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.

Derive the quadratic formula using the method of completing the square.

Represent a system of linear equations as a single matrix equation in a vector variable.

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems.

Understand that polynomials form a system analogous to the integers, namely, that they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

Focus on polynomial expressions that simplify to forms that are linear or quadratic.

Extend to polynomial expressions beyond those expressions that simplify to forms that are linear or quadratic.

Understand and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x − a is p(a). In particular, p(a) = 0 if and only if (x – a) is a factor of p(x).

Identify zeros of polynomials, when factoring is reasonable, and use the zeros to construct a rough graph of the function defined by the polynomial.

Prove polynomial identities and use them to describe numerical relationships.

Rewrite simple rational expressions in different forms; write a(x)/b(x) in the form q(x) + r(x)/b(x), where a(x), b(x), q(x), and r(x) are polynomials with the degree of r(x) less than the degree of b(x), using inspection, long division, or, for the more complicated examples, a computer algebra system.

Create equations and inequalities in one variable and use them to solve problems. Include equations and inequalities arising from linear, quadratic, simple rational, and exponential functions.

Focus on applying linear and simple exponential expressions.

Focus on applying simple quadratic expressions.

Extend to include more complicated function situations with the option to solve with technology.

Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

Focus on applying linear and simple exponential expressions.

Focus on applying simple quadratic expressions.

Extend to include more complicated function situations with the option to graph with technology.

Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations.

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Focus on formulas in which the variable of interest is linear or square.

Focus on formulas in which the variable of interest is linear.

Focus on formulas in which the variable of interest is linear or square.

While functions will often be linear, exponential, or quadratic, the types of problems should draw from more complicated situations.

Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.

Understand that the graph of an equation in two variables is the set of all its solutions plotted in the coordinate plane, often forming a curve (which could be a line).

Explain why the x-coordinates of the points where the graphs of the equation y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, making tables of values, or finding successive approximations.

Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes.

Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise.

Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

Solve quadratic equations in one variable.

Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x − p)² = q that has the same solutions.

Solve quadratic equations as appropriate to the initial form of the equation by inspection, e.g., for x² = 49; taking square roots; completing the square; applying the quadratic formula; or utilizing the Zero-Product Property after factoring.

Verify that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Solve systems of linear equations algebraically and graphically.

Limit to pairs of linear equations in two variables.

Extend to include solving systems of linear equations in three variables, but only algebraically.

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically.

Interpret expressions that represent a quantity in terms of its context.

Interpret parts of an expression, such as terms, factors, and coefficients.

Interpret complicated expressions by viewing one or more of their parts as a single entity.

Use the structure of an expression to identify ways to rewrite it.

Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression.

Factor a quadratic expression to reveal the zeros of the function it defines.

Complete the square in a quadratic expression to reveal the maximum or minimum value of the function it defines.

Use the properties of exponents to transform expressions for exponential functions. For example, 8t can be written as 2³t.

Perform arithmetic operations on polynomials.

Understand the relationship between zeros and factors of polynomials.

Use polynomial identities to solve problems.

Rewrite rational expressions.

Create equations that describe numbers or relationships.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Solve systems of equations.

Represent and solve equations and inequalities graphically.

Interpret the structure of expressions.

Write expressions in equivalent forms to solve problems.

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.

Trigonometric Functions

Linear, Quadratic, And Exponential Models

Building Functions

Interpreting Functions

Standards for Mathematical Practice

Compose functions.

Read values of an inverse function from a graph or a table, given that the function has an inverse.

Verify by composition that one function is the inverse of another.

Find the inverse of a function algebraically, given that the function has an inverse.

Produce an invertible function from a non-invertible function by restricting the domain.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.

Graph rational functions, identifying zeros and asymptotes when factoring is reasonable, and indicating end behavior.

Graph logarithmic functions, indicating intercepts and end behavior.

Use inverse functions to solve trigonometric equations that arise in modeling contexts; evaluate the solutions using technology, and interpret them in terms of the context.

Use special triangles to determine geometrically the values of sine, cosine, tangent for π/3, π/4, and π/6, and use the unit circle to express the values of sine, cosine, and tangent for π − x, π + x, and 2π − x in terms of their values for x, where x is any real number.

Use the unit circle to explain symmetry (odd and even) and periodicity of trigonometric functions.

Understand that restricting a trigonometric function to a domain on which it is always increasing or always decreasing allows its inverse to be constructed.

Prove the addition and subtraction formulas for sine, cosine, and tangent, and use them to solve problems.

Write a function that describes a relationship between two quantities.

Determine an explicit expression, a recursive process, or steps for calculation from context.

Focus on linear and exponential functions.

Focus on situations that exhibit quadratic or exponential relationships.

Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.

Identify the effect on the graph of replacing f(x) by f(x) + k, kf(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Focus on transformations of graphs of quadratic functions, except for f(kx);

Find inverse functions.

Informally determine the input of a function when the output is known.

Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).

Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers.

For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include the following: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.

Focus on linear and exponential functions.

Focus on linear, quadratic, and exponential functions.

Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes.

Focus on linear and exponential functions.

Focus on linear, quadratic, and exponential functions.

Emphasize the selection of a type of function for a model based on behavior of data and context.

Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph.

Graph functions expressed symbolically and indicate key features of the graph, by hand in simple cases and using technology for more complicated cases. Include applications and how key features relate to characteristics of a situation, making selection of a particular type of function model appropriate.

Graph linear functions and indicate intercepts.

Graph quadratic functions and indicate intercepts, maxima, and minima.

Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions.

Graph polynomial functions, identifying zeros, when factoring is reasonable, and indicating end behavior.

Graph simple exponential functions, indicating intercepts and end behavior.

Graph exponential functions, indicating intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context.

Focus on completing the square to quadratic functions with the leading coefficient of 1.

Use the properties of exponents to interpret expressions for exponential functions.

Focus on exponential functions evaluated at integer inputs.

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

Focus on linear and exponential functions.

Focus on linear, quadratic, and exponential functions.

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Show that linear functions grow by equal differences over equal intervals and that exponential functions grow by equal factors over equal intervals.

Recognize situations in which one quantity changes at a constant rate per unit interval relative to another.

Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly or quadratically.

For exponential models, express as a logarithm the solution to abct = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.

Interpret the parameters in a linear or exponential function in terms of a context.

Understand radian measure of an angle as the length of the arc on the unit circle subtended by the angle.

Explain how the unit circle in the coordinate plane enables the extension of trigonometric functions to all real numbers, interpreted as radian measures of angles traversed counter-clockwise around the unit circle.

Choose trigonometric functions to model periodic phenomena with specified amplitude, frequency, and midline.

Prove the Pythagorean identity sin²(θ) + cos²(θ) = 1, and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.

Build a function that models a relationship between two quantities.

Build new functions from existing functions.

Understand the concept of a function, and use function notation.

Interpret functions that arise in applications in terms of the context.

Analyze functions using different representations.

Construct and compare linear, quadratic, and exponential models, and solve problems.

Interpret expressions for functions in terms of the situation they model.

Extend the domain of trigonometric functions using the unit circle.

Model periodic phenomena with trigonometric functions.

Prove and apply trigonometric identities.

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.

Modeling With Geometry

Geometric Measurement And Dimension

Expressing Geometric Properties With Equations

Circles

Similarity, Right Triangles, And Trigonometry

Congruence

Standards for Mathematical Practice

Construct a tangent line from a point outside a given circle to the circle.

Give an informal argument using Cavalieri's principle for the formulas for the volume of a sphere and other solid figures.

Derive the equation of a parabola given a focus and directrix.

Derive the equations of ellipses and hyperbolas given the foci, using the fact that the sum or difference of distances from the foci is constant.

Explain proofs of the Laws of Sines and Cosines and use the Laws to solve problems.

Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles, e.g., surveying problems, resultant forces.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

Derive the formula A = ½ ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

Prove that all circles are similar using transformational arguments.

Identify and describe relationships among angles, radii, chords, tangents, and arcs and use them to solve problems. Include the relationship between central, inscribed, and circumscribed angles and their intercepted arcs; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

Construct the inscribed and circumscribed circles of a triangle; prove and apply the property that opposite angles are supplementary for a quadrilateral inscribed in a circle.

Find arc lengths and areas of sectors of circles.

Apply similarity to relate the length of an arc intercepted by a central angle to the radius. Use the relationship to solve problems.

Derive the formula for the area of a sector, and use it to solve problems.

Derive formulas that relate degrees and radians, and convert between the two.

Know precise definitions of ray, angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and arc length.

Prove and apply theorems about triangles. Theorems include but are not restricted to the following: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

Prove and apply theorems about parallelograms.

Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

Classify two-dimensional figures in a hierarchy based on properties.

Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not, e.g., translation versus horizontal stretch.

Identify the symmetries of a figure, which are the rotations and reflections that carry it onto itself.

Identify figures that have line symmetry; draw and use lines of symmetry to analyze properties of shapes.

Identify figures that have rotational symmetry; determine the angle of rotation, and use rotational symmetry to analyze properties of shapes.

Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using items such as graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent.

Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

Prove and apply theorems about lines and angles. Theorems include but are not restricted to the following: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.

Give an informal argument for the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri's principle, and informal limit arguments.

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

Understand how and when changes to the measures of a figure (lengths or angles) result in similar and non-similar figures.

When figures are similar, understand and apply the fact that when a figure is scaled by a factor of k, the effect on lengths, areas, and volumes is that they are multiplied by k, k², and k³, respectively.

Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

Use coordinates to prove simple geometric theorems algebraically and to verify geometric relationships algebraically, including properties of special triangles, quadrilaterals, and circles.

Justify the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems, e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point.

Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.

Use geometric shapes, their measures, and their properties to describe objects, e.g., modeling a tree trunk or a human torso as a cylinder.

Apply concepts of density based on area and volume in modeling situations, e.g., persons per square mile, BTUs per cubic foot.

Apply geometric methods to solve design problems, e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios.

Verify experimentally the properties of dilations given by a center and a scale factor:

A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged.

The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides.

Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

Prove and apply theorems about triangles.

Use congruence and similarity criteria for triangles to solve problems and to justify relationships in geometric figures that can be decomposed into triangles.

Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Explain and use the relationship between the sine and cosine of complementary angles.

Solve problems involving right triangles.

Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems if one of the two acute angles and a side length is given.

Understand and apply theorems about circles.

Find arc lengths and areas of sectors of circles.

Experiment with transformations in the plane.

Understand congruence in terms of rigid motions.

Prove geometric theorems both formally and informally using a variety of methods.

Make geometric constructions.

Classify and analyze geometric figures.

Explain volume formulas, and use them to solve problems.

Visualize relationships between two-dimensional and three-dimensional objects.

Understand the relationships between lengths, area, and volumes.

Translate between the geometric description and the equation for a conic section.

Use coordinates to prove simple geometric theorems algebraically and to verify specific geometric statements.

Apply geometric concepts in modeling situations.

Understand similarity in terms of similarity transformations.

Prove and apply theorems both formally and informally involving similarity using a variety of methods.

Define trigonometric ratios, and solve problems involving right triangles.

Apply trigonometry to general triangles.

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.

Vector And Matrix Quantities

The Complex Number System

Quantities

The Real Number System

Standards for Mathematical Practice

Find the conjugate of a complex number; use conjugates to find magnitudes and quotients of complex numbers.

Represent complex numbers on the complex plane in rectangular and polar form (including real and imaginary numbers), and explain why the rectangular and polar forms of a given complex number represent the same number.

Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation.

Calculate the distance between numbers in the complex plane as the magnitude of the difference, and the midpoint of a segment as the average of the numbers at its endpoints.

Extend polynomial identities to the complex numbers.

Know the Fundamental Theorem of Algebra; show that it is true for quadratic polynomials.

Recognize vector quantities as having both magnitude and direction. Represent vector quantities by directed line segments, and use appropriate symbols for vectors and their magnitudes, e.g., v, |v|, ||v||, <img src="https://purl.org/ASN/resources/images/D2784929/N.VM.1.gif" alt="vec{x}" />.

Understand that the zero and identity matrices play a role in matrix addition and multiplication analogous to the role of 0 and 1 in the real numbers. The determinant of a square matrix is nonzero if and only if the matrix has a multiplicative inverse.

Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.

Work with 2 × 2 matrices as transformations of the plane, and interpret the absolute value of the determinant in terms of area.

Find the components of a vector by subtracting the coordinates of an initial point from the coordinates of a terminal point.

Solve problems involving velocity and other quantities that can be represented by vectors.

Add and subtract vectors.

Multiply a vector by a scalar

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c (v_x, v_y) = (cv _x, cv_y).

Compute the magnitude of a scalar multiple cv using ||cv|| = |c|v. Compute the direction of cv knowing that when |c|v ≠ 0, the direction of cv is either along v (for c > 0) or against v (for c < 0).

Use matrices to represent and manipulate data, e.g., to represent payoffs or incidence relationships in a network.

Multiply matrices by scalars to produce new matrices, e.g., as when all of the payoffs in a game are doubled.

Add, subtract, and multiply matrices of appropriate dimensions.

Understand that, unlike multiplication of numbers, matrix multiplication for square matrices is not a commutative operation, but still satisfies the associative and distributive properties.

Perform arithmetic operations with complex numbers.

Represent complex numbers and their operations on the complex plane.

Use complex numbers in polynomial identities and equations.

Reason quantitatively and use units to solve problems.

Extend the properties of exponents to rational exponents.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Perform operations on vectors.

Perform operations on matrices, and use matrices in applications.

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.

Know there is a complex number i such that i² = −1, and every complex number has the form a + bi with a and b real.

Use the relation i² = −1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.

Solve quadratic equations with real coefficients that have complex solutions.

Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.

Define appropriate quantities for the purpose of descriptive modeling.

Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.

Explain how the definition of the meaning of rational exponents follows from extending the properties of integer exponents to those values, allowing for a notation for radicals in terms of rational exponents.

Rewrite expressions involving radicals and rational exponents using the properties of exponents.

Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational.

Add vectors end-to-end, component-wise, and by the parallelogram rule. Understand that the magnitude of a sum of two vectors is typically not the sum of the magnitudes.

Given two vectors in magnitude and direction form, determine the magnitude and direction of their sum.

Understand vector subtraction v − w as v + (-w), where -w is the additive inverse of w, with the same magnitude as w and pointing in the opposite direction. Represent vector subtraction graphically by connecting the tips in the appropriate order, and perform vector subtraction component-wise.

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Use the rules of probability to compute probabilities of compound events in a uniform probability model.

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Understand independence and conditional probability, and use them to interpret data.

Conditional Probability and The Rules of Probability

Not on BP. This standard is taught in Algebra 2.

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP. This standard is taught in Algebra 2.

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP. This standard is taught in Algebra 2.

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP. This standard is taught in Algebra 2.

Learning Progression

Complexity c

Complexity b

Complexity a

Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

Not on BP. This standard is taught in Algebra 2.

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP. This standard is taught in Algebra 2. See http://education.ohio.gov/Topics/Learning-in-Ohio/ Mathematics/Ohio-s-Learning-Standards-in-Mathematics/ Transitioning-to-the-2017-Learning-Standards-in-Ma

Learning Progression

Complexity c

Complexity b

Complexity a

Understand and evaluate random processes underlying statistical experiments.

Making Inferences and Justifying Conclusions

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Not on BP

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Summarize, represent, and interpret data on two categorical and quantitative variables.

Not on BP

Learning Progression

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Learning Progression

Complexity c

Complexity b

Complexity a

Summarize, represent, and interpret data on a single count or measurement variable.

Interpreting Categorical and Quantitative Data

Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

Calculate the probability of given event.

List all possible outcomes of an event.

Choose the possible outcomes of an event (e.g., 4 possible colors spun on a 4-section spinner).

Play games involving probability with or without technology.

In a given context identify the number of likely outcomes to be able to choose a fitting simulation, e.g., Will it be a boy or a girl? - I can use a penny (heads/tails) to simulate the situation. There are four colors of a candy in a bag, what is the probability of picking a red candy?

Know what an outcome is.

Know the language of “likely” and “not likely”.

Engagement Statements (demonstration of engaged in the topic)

Observe and engage with demonstrations of equal probability situations.

Engage with a group of students and interact with e.g., spinners and number cubes, to experience outcomes.

Interact with 2, 4 or 6 outcomes with equal probability from real-world situations.

Understand that two events A and B are independent if and only if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

Create a Venn Diagram given categorical data.

Calculate probabilities based on a Venn Diagram.

Arrange given data into a Venn Diagram (e.g., sports teams).

Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

Calculate conditional probabilities of events from Venn Diagrams using the addition rule (e.g., the probability of students who like horror or comedy movies who also like pizza).

Calculate conditional probabilities of events. (e.g., chance of drawing an ace or face card).

Arrange given data into a Venn Diagram.

Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in 10th grade. Do the same for other subjects and compare the results.

Create a two-way frequency table when given data and calculate the probability of an event.

Given a two-way frequency table, calculate the probability of an event.

Complete a two-way frequency table given data.

Should be worked on in conjunction with S.ID.5.

Enter the data numerically in the two - way table.

Create a table with labels for columns and rows.

Sort cards with data into 4 cells of the table

Gather the data on e.g. index cards.

Recognize a question that involves two categories (e.g. gender/music style, gender/pet ownership.)

Engagement Statements (demonstration of engaged in the topic)

Interact with a two-way frequency table.

Engage with a group of students to gather real-world data and observe the data being organized into a two-way table.

Interact with situations involving greater than, equal to, or less than.

Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

Given a real-world scenario, student will name the conditional probabilities and their effects.

Given a real-world scenario, student will name the conditional probabilities.

Given a real-world scenario, student will determine if the situation or event is conditional or independent.

Should be worked on in conjunction with S.CP.1.

Experience and engage in a context where the two events have conditional probability as well as in a context where the two events are independent, e.g., What is the chance of receiving a reward if you follow the rules? (dependent) and What is the chance of not getting homework if you compliment the teacher’s attire? (independent)

In a familiar context identify the number of possible outcomes.

Engagement Statements (demonstration of engaged in the topic)

Observe and engage with demonstrations of equal probability situations.

Engage with a group of students and interact with e.g., spinners and number cubes, to experience outcomes.

Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

Given a Venn Diagram or a table with “a given b” statement, identify “a” and “b”.

Given a Venn Diagram or a table, distinguish dependent and independent events.

Given a Venn Diagram or a table and data, correctly input missing data on the table.

Apply the Addition Rule, P(A or B) = P(A) + P(B) − P(A and B), and interpret the answer in terms of the model.

Write the Addition Rule equation given a complete two-way table.

Name/ identify 2 missing condition variables and input into equation.

Given a two-way table and the Addition Rule with missing condition, student will identify one missing variable.

Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

Determine if the given data could come from a specific datagenerating device (spinner, coin, number cube).

Determine the likelihood (likely, impossible, unlikely, and certain) of outcomes from a data-generating device.

Determine the likelihood (certain or impossible) of an outcome from a data-generating device.

Decide if a specified model is consistent with results from a given datagenerating process, e.g., using simulation. For example, a model says a spinning coin falls heads up with probability 0.5. Would a result of 5 tails in a row cause you to question the model?

Understand a probability of 0 as impossible, a probability of 1 as certain, a probability near 0 as unlikely, near 1 as likely, and near 1/2 as equally likely.

Understand a probability near 0 as unlikely and near 1 as likely.

Understand a probability near 0 as unlikely and near 1 as likely using a number line.

Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

Identify sample surveys, experiments, and observational studies.

Identify a sample survey and an experiment.

Identify a sample survey.

Use data from a sample survey to estimate a population mean or proportion; develop a margin of error through the use of simulation models for random sampling.

Estimate the mean of data given in a sample survey.

Determine the mean of data given in a sample survey.

Match the mean of data given in a sample survey.

Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between sample statistics are statistically significant.

Compare data of a randomized experiments to determine outcome differences.

Determine if a given treatment changed the outcome of an experiment.

Match the given treatment that changed the outcome (e.g., bleach changed the stain, water did not).

Evaluate reports based on data.

Evaluate if data supports the claim/results. Evaluate given data to determine results.

Determine if data supports the results/claim.

Match the data to the given results.

Represent data with plots on the real number line (dot plots, histograms, and box plots) in the context of real-world applications using the GAISE model.

Collect data in real-world context to create a dot plot, histogram, or box plot to represent collected data.

Create a dot plot, histogram, or a box plot to represent given or collected data.

Match given data to a given dot plot, histogram, or box plot.

[Between level c and b:

Identify a missing whole number value on a number line marked with whole number up to 10.]

Sort representations of different data displays.

Recognize different data representations, table, dot plot, histogram and box plot.

Gather data, e.g., the height of the people in the classroom (at least 11 heights) and organize the same data in table, histogram, and box plot.

Recognize that on a number line the spaces need to be counted not the grid lines, assuming a scale of 1.

Understand that 2 is the distance from 0 to 2 and 3 is the distance from 0 to 3 using standard units for all lengths from 1 to 10.

Identify 0 on a number line.

Recognize a point on a number line.

Know what a number line is.

Know the order of the numbers from 0 to 10.

Engagement Statements (demonstration of engaged in the topic)

Interact with a variety of data representations, i.e. dot plot, histogram or box plot.

Engage with graphing technology.

Engage with a group of students to gather real-world data and observe the data being organized into different displays.

In the context of real-world applications by using the GAISE model, use statistics appropriate to the shape of the data distribution to compare center (median and mean) and spread (mean absolute deviation, interquartile range, and standard deviation) of two or more different data sets.

Compare mean, median, and mode of 2 or more given graphs or collected data sets.

Compute mean, median, or mode of a given graph or collected data set involving numbers less than 100.

Identify the median and mode of a graph or a given data set involving numbers less than 50.

Order a set of an odd quantity of numbers, e.g., 11 data points, from least to greatest.

Gather real-world data points.

Interact with arithmetic operations.

Recognize the symbols for addition (+), subtraction, (–), multiplication (×), division (÷), and equals (=).

Count up to 50.

Engagement Statements (demonstration of engaged in the topic)

Interact with technology

Engage with a group of students to gather real-world data and observe the data being organized into different displays.

In the context of realworld applications by using the GAISE model, interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

Interpret a dot plot.

Interpret a histogram.

Complete an incomplete dot plot, box plot, or histogram (e.g., adding missing labels and missing data points).

S.ID.1b: Create a dot plot, histogram, or a box plot to represent given or collected data.

Match the vocabulary to the correct display.

Understand the labels on a data display.

Recognize a point on a number line.