Sunday, December 4, 2022, 7:01 AM
Site: Dearborn Educational Curriculum
Course: Dearborn Educational Curriculum (DEC)
Glossary: Mi Math Standards
1

#### 1.G.A

Grade 1 » Geometry » Reason with shapes and their attributes.

#### 1.G.A.1

Distinguish between defining attributes (e.g., triangles are closed and three-sided) versus non-defining attributes (e.g., color, orientation, overall size); build and draw shapes to possess defining attributes.

01

#### 1.G.A.2

Compose two-dimensional shapes (rectangles, squares, trapezoids, triangles, half-circles, and quarter-circles) or three-dimensional shapes (cubes, right rectangular prisms, right circular cones, and right circular cylinders) to create a composite shape, and compose new shapes from the composite shape.1

01

#### 1.G.A.3

Partition circles and rectangles into two and four equal shares, describe the shares using the words halves, fourths, and quarters, and use the phrases half of, fourth of, and quarter of. Describe the whole as two of, or four of the shares. Understand for these examples that decomposing into more equal shares creates smaller shares.

01

#### 1.MD.A

Grade 1 » Measurement & Data » Measure lengths indirectly and by iterating length units.

#### 1.MD.A.1

Order three objects by length; compare the lengths of two objects indirectly by using a third object.

01

#### 1.MD.A.2

Express the length of an object as a whole number of length units, by laying multiple copies of a shorter object (the length unit) end to end; understand that the length measurement of an object is the number of same-size length units that span it with no gaps or overlaps. Limit to contexts where the object being measured is spanned by a whole number of length units with no gaps or overlaps.

01

#### 1.MD.B

Grade 1 » Measurement & Data » Tell and write time.

#### 1.MD.B.3

Tell and write time in hours and half-hours using analog and digital clocks.

01

#### 1.MD.C

Grade 1 » Measurement & Data » Represent and interpret data.

#### 1.MD.C.4

Organize, represent, and interpret data with up to three categories; ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

01

#### 1.NBT.A

Grade 1 » Number & Operations in Base Ten » Extend the counting sequence.

#### 1.NBT.A.1

Count to 120, starting at any number less than 120. In this range, read and write numerals and represent a number of objects with a written numeral.

01

#### 1.NBT.B

Grade 1 » Number & Operations in Base Ten » Understand place value.

#### 1.NBT.B.2

Understand that the two digits of a two-digit number represent amounts of tens and ones. Understand the following as special cases:

01

#### 1.NBT.B.2a

10 can be thought of as a bundle of ten ones — called a "ten."

01

#### 1.NBT.B.2b

The numbers from 11 to 19 are composed of a ten and one, two, three, four, five, six, seven, eight, or nine ones.

01

#### 1.NBT.B.2c

The numbers 10, 20, 30, 40, 50, 60, 70, 80, 90 refer to one, two, three, four, five, six, seven, eight, or nine tens (and 0 ones).

01

#### 1.NBT.B.3

Compare two two-digit numbers based on meanings of the tens and ones digits, recording the results of comparisons with the symbols >, =, and <.

01

#### 1.NBT.C

Grade 1 » Number & Operations in Base Ten » Use place value understanding and properties of operations to add and subtract.

#### 1.NBT.C.4

Add within 100, including adding a two-digit number and a one-digit number, and adding a two-digit number and a multiple of 10, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used. Understand that in adding two-digit numbers, one adds tens and tens, ones and ones; and sometimes it is necessary to compose a ten.

01

#### 1.NBT.C.5

Given a two-digit number, mentally find 10 more or 10 less than the number, without having to count; explain the reasoning used.

01

#### 1.NBT.C.6

Subtract multiples of 10 in the range 10-90 from multiples of 10 in the range 10-90 (positive or zero differences), using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

01

#### 1.OA.A

Grade 1 » Operations & Algebraic Thinking » Represent and solve problems involving addition and subtraction.

#### 1.OA.A.1

Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.1

01

#### 1.OA.A.2

Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

01

#### 1.OA.B

Grade 1 » Operations & Algebraic Thinking » Understand and apply properties of operations and the relationship between addition and subtraction.

#### 1.OA.B.3

Apply properties of operations as strategies to add and subtract.2Examples: If 8 + 3 = 11 is known, then 3 + 8 = 11 is also known. (Commutative property of addition.) To add 2 + 6 + 4, the second two numbers can be added to make a ten, so 2 + 6 + 4 = 2 + 10 = 12. (Associative property of addition.)

01

#### 1.OA.B.4

Understand subtraction as an unknown-addend problem. For example, subtract 10 - 8 by finding the number that makes 10 when added to 8.

01

#### 1.OA.C

Grade 1 » Operations & Algebraic Thinking » Add and subtract within 20.

#### 1.OA.C.5

Relate counting to addition and subtraction (e.g., by counting on 2 to add 2).

01

#### 1.OA.C.6

Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 - 4 = 13 - 3 - 1 = 10 - 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 - 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).

01

#### 1.OA.D

Grade 1 » Operations & Algebraic Thinking » Work with addition and subtraction equations.

#### 1.OA.D.7

Understand the meaning of the equal sign, and determine if equations involving addition and subtraction are true or false. For example, which of the following equations are true and which are false? 6 = 6, 7 = 8 - 1, 5 + 2 = 2 + 5, 4 + 1 = 5 + 2.

01

#### 1.OA.D.8

Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11, 5 = _ - 3, 6 + 6 = _.

01

2

#### 2.G.A

Grade 2 » Geometry » Reason with shapes and their attributes.

#### 2.G.A.1

Recognize and draw shapes having specified attributes, such as a given number of angles or a given number of equal faces.1 Identify triangles, quadrilaterals, pentagons, hexagons, and cubes.

02

#### 2.G.A.2

Partition a rectangle into rows and columns of same-size squares and count to find the total number of them.

02

#### 2.G.A.3

Partition circles and rectangles into two, three, or four equal shares, describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths. Recognize that equal shares of identical wholes need not have the same shape.

02

#### 2.MD.A

Grade 2 » Measurement & Data » Measure and estimate lengths in standard units.

#### 2.MD.A.1

Measure the length of an object by selecting and using appropriate tools such as rulers, yardsticks, meter sticks, and measuring tapes.

02

#### 2.MD.A.2

Measure the length of an object twice, using length units of different lengths for the two measurements; describe how the two measurements relate to the size of the unit chosen.

02

#### 2.MD.A.3

Estimate lengths using units of inches, feet, centimeters, and meters.

02

#### 2.MD.A.4

Measure to determine how much longer one object is than another, expressing the length difference in terms of a standard length unit.

02

#### 2.MD.B

Grade 2 » Measurement & Data » Relate addition and subtraction to length.

#### 2.MD.B.5

Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

02

#### 2.MD.B.6

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.

02

#### 2.MD.C

Grade 2 » Measurement & Data » Work with time and money.

#### 2.MD.C.7

Tell and write time from analog and digital clocks to the nearest five minutes, using a.m. and p.m.

02

07

#### 7.EE.B.4

Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities.

07

#### 7.EE.B.4a

Solve word problems leading to equations of the form px + q = r and p(x + q) = r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. For example, the perimeter of a rectangle is 54 cm. Its length is 6 cm. What is its width?

07

#### 7.EE.B.4b

Solve word problems leading to inequalities of the form px + q > r or px + q < r, where p, q, and r are specific rational numbers. Graph the solution set of the inequality and interpret it in the context of the problem. For example: As a salesperson, you are paid $50 per week plus$3 per sale. This week you want your pay to be at least \$100. Write an inequality for the number of sales you need to make, and describe the solutions.

07

#### 7.G.A

Grade 7 » Geometry » Draw construct, and describe geometrical figures and describe the relationships between them.

#### 7.G.A.1

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

07

#### 7.G.A.2

Draw (freehand, with ruler and protractor, and with technology) geometric shapes with given conditions. Focus on constructing triangles from three measures of angles or sides, noticing when the conditions determine a unique triangle, more than one triangle, or no triangle.

07

#### 7.G.A.3

Describe the two-dimensional figures that result from slicing three-dimensional figures, as in plane sections of right rectangular prisms and right rectangular pyramids.

07

#### 7.G.B

Grade 7 » Geometry » Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

#### 7.G.B.4

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

07

#### 7.G.B.5

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

07

#### 7.G.B.6

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

07

#### 7.NS.A

Grade 7 » The Number System » Apply and extend previous understandings of operations with fractions.

#### 7.NS.A.1

Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.

07

#### 7.NS.A.1a

Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged.

07

#### 7.NS.A.1b

Understand p + q as the number located a distance |q| from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing real-world contexts.

07

#### 7.NS.A.1c

Understand subtraction of rational numbers as adding the additive inverse, p - q = p + (-q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts.

07

#### 7.NS.A.1d

Apply properties of operations as strategies to add and subtract rational numbers.

07

#### 7.NS.A.2

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

07

#### 7.NS.A.2a

Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (-1)(-1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts.

07

#### 7.NS.A.2b

Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then -(p/q) = (-p)/q = p/(-q). Interpret quotients of rational numbers by describing real-world contexts.

07

#### 7.NS.A.2c

Apply properties of operations as strategies to multiply and divide rational numbers.

07

#### 7.NS.A.2d

Convert a rational number to a decimal using long division; know that the decimal form of a rational number terminates in 0s or eventually repeats.

07

#### 7.NS.A.3

Solve real-world and mathematical problems involving the four operations with rational numbers.1

07

#### 7.RP.A

Grade 7 » Ratios & Proportional Relationships » Analyze proportional relationships and use them to solve real-world and mathematical problems.

#### 7.RP.A.1

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

07

#### 7.RP.A.2

Recognize and represent proportional relationships between quantities.

07

#### 7.RP.A.2a

Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin.

07

#### 7.RP.A.2b

Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships.

07

#### 7.RP.A.2c

Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn.

07

#### 7.RP.A.2d

Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate.

07

#### 7.RP.A.3

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.

07

#### 7.SP.A

Grade 7 » Statistics & Probability » Use random sampling to draw inferences about a population.

#### 7.SP.A.1

Understand that statistics can be used to gain information about a population by examining a sample of the population; generalizations about a population from a sample are valid only if the sample is representative of that population. Understand that random sampling tends to produce representative samples and support valid inferences.

07

#### 7.SP.A.2

Use data from a random sample to draw inferences about a population with an unknown characteristic of interest. Generate multiple samples (or simulated samples) of the same size to gauge the variation in estimates or predictions. For example, estimate the mean word length in a book by randomly sampling words from the book; predict the winner of a school election based on randomly sampled survey data. Gauge how far off the estimate or prediction might be.

07

#### 7.SP.B

Grade 7 » Statistics & Probability » Draw informal comparative inferences about two populations.

#### 7.SP.B.3

Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

07

#### 7.SP.B.4

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.

07

#### 7.SP.C

Grade 7 » Statistics & Probability » Investigate chance processes and develop, use, and evaluate probability models.

#### 7.SP.C.5

Understand that the probability of a chance event is a number between 0 and 1 that expresses the likelihood of the event occurring. Larger numbers indicate greater likelihood. A probability near 0 indicates an unlikely event, a probability around 1/2 indicates an event that is neither unlikely nor likely, and a probability near 1 indicates a likely event.

07

#### 7.SP.C.6

Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times.

07

#### 7.SP.C.7

Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy.

07

#### 7.SP.C.7a

Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected.

07

#### 7.SP.C.7b

Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

07

#### 7.SP.C.8

Find probabilities of compound events using organized lists, tables, tree diagrams, and simulation.

07

#### 7.SP.C.8a

Understand that, just as with simple events, the probability of a compound event is the fraction of outcomes in the sample space for which the compound event occurs.

07

#### 7.SP.C.8b

Represent sample spaces for compound events using methods such as organized lists, tables and tree diagrams. For an event described in everyday language (e.g., "rolling double sixes"), identify the outcomes in the sample space which compose the event.

07

#### 7.SP.C.8c

Design and use a simulation to generate frequencies for compound events. For example, use random digits as a simulation tool to approximate the answer to the question: If 40% of donors have type A blood, what is the probability that it will take at least 4 donors to find one with type A blood?

07

8

#### 8.EE.A

Grade 8 » Expressions & Equations » Expressions and Equations Work with radicals and integer exponents.

#### 8.EE.A.1

Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 32 × 3-5 = 3-3 = 1/33 = 1/27.

08

#### 8.EE.A.2

Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that √2 is irrational.

08

#### 8.EE.A.3

Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and to express how many times as much one is than the other. For example, estimate the population of the United States as 3 times 108 and the population of the world as 7 times 109, and determine that the world population is more than 20 times larger.

08

#### 8.EE.A.4

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

08

#### 8.EE.B

Grade 8 » Expressions & Equations » Understand the connections between proportional relationships, lines, and linear equations.

#### 8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

08

#### 8.EE.B.6

Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation y = mx + b for a line intercepting the vertical axis at b.

08

#### 8.EE.C

Grade 8 » Expressions & Equations » Analyze and solve linear equations and pairs of simultaneous linear equations.

#### 8.EE.C.7

Solve linear equations in one variable.

08

#### 8.EE.C.7a

Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).

08

#### 8.EE.C.7b

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

08

#### 8.EE.C.8

Analyze and solve pairs of simultaneous linear equations.

08

#### 8.EE.C.8a

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

08

#### 8.EE.C.8b

Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x + 2y = 6 have no solution because 3x + 2y cannot simultaneously be 5 and 6.

08

#### 8.EE.C.8c

Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

08

#### 8.F.A

Grade 8 » Functions » Define, evaluate, and compare functions.

#### 8.F.A.1

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output.1

08

#### 8.F.A.2

Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

08

#### 8.F.A.3

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.

08

#### 8.F.B

Grade 8 » Functions » Use functions to model relationships between quantities.

#### 8.F.B.4

Construct a function to model a linear relationship between two quantities. Determine the rate of change  and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.

08

#### 8.F.B.5

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

08

#### 8.G.A

Grade 8 » Geometry » Understand congruence and similarity using physical models, transparencies, or geometry software.

#### 8.G.A.1

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

08

#### 8.G.A.1a

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

08

#### 8.G.A.1b

Angles are taken to angles of the same measure.

08

#### 8.G.A.1c

Parallel lines are taken to parallel lines.

08

#### 8.G.A.2

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

08

#### 8.G.A.3

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

08

#### 8.G.A.4

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

08

#### 8.G.A.5

Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles. For example, arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

08

#### 8.G.B

Grade 8 » Geometry » Understand and apply the Pythagorean Theorem.

#### 8.G.B.6

Explain a proof of the Pythagorean Theorem and its converse.

08

#### 8.G.B.7

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

08

#### 8.G.B.8

Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

08

#### 8.G.C

Grade 8 » Geometry » Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

#### 8.G.C.9

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

08

#### 8.NS.A

Grade 8 » The Number System » Know that there are numbers that are not rational, and approximate them by rational numbers.

#### 8.NS.A.1

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

08

#### 8.NS.A.2

Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

08

#### 8.SP.A

Grade 8 » Statistics & Probability » Investigate patterns of association in bivariate data.

#### 8.SP.A.1

Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

08

#### 8.SP.A.2

Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

08

#### 8.SP.A.3

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

08

#### 8.SP.A.4

Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

08

H

#### HSA-APR.A.1

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

09, 10, 11, 12

#### HSA-APR.B.2

Know and apply the Remainder Theorem: For a polynomial p(x) and a number a, the remainder on division by x - a is p(a), so p(a) = 0 if and only if (x - a) is a factor of p(x).

09, 10, 11, 12

#### HSA-APR.B.3

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to construct a rough graph of the function defined by the polynomial.

09, 10, 11, 12

#### HSA-APR.C.4

Prove polynomial identities and use them to describe numerical relationships. For example, the polynomial identity (x2 + y2)2 = (x2 - y2)2 + (2xy)2 can be used to generate Pythagorean triples.

09, 10, 11, 12

#### HSA-APR.C.5

(+) 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, with coefficients determined for example by Pascal's Triangle.1

09, 10, 11, 12

#### HSA-APR.D.6

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.

09, 10, 11, 12

#### HSA-APR.D.7

(+) 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.

09, 10, 11, 12

#### HSA-CED.A.1

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

09, 10, 11, 12

#### HSA-CED.A.2

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

09, 10, 11, 12

#### HSA-CED.A.3

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.

09, 10, 11, 12

#### HSA-CED.A.4

Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law V = IR to highlight resistance R.

09, 10, 11, 12

#### HSA-REI.A.1

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.

09, 10, 11, 12

#### HSA-REI.A.2

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

09, 10, 11, 12

#### HSA-REI.B.3

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

09, 10, 11, 12

#### HSA-REI.B.4

Solve quadratic equations in one variable.

09, 10, 11, 12

#### HSA-REI.B.4a

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. Derive the quadratic formula from this form.

09, 10, 11, 12

#### HSA-REI.B.4b

Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b.

09, 10, 11, 12

#### HSA-REI.C.5

Prove 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.

09, 10, 11, 12

#### HSA-REI.C.6

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

09, 10, 11, 12

#### HSA-REI.C.7

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.

09, 10, 11, 12

#### HSA-REI.C.8

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

09, 10, 11, 12

#### HSA-REI.C.9

(+) 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).

09, 10, 11, 12

#### HSA-REI.D.10

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).

09, 10, 11, 12

#### HSA-REI.D.11

Explain why the x-coordinates of the points where the graphs of the equations 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, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.*

09, 10, 11, 12

#### HSA-REI.D.12

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.

09, 10, 11, 12

#### HSA-SSE.A.1

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

09, 10, 11, 12

#### HSA-SSE.A.1a

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

09, 10, 11, 12

#### HSA-SSE.A.1b

Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

09, 10, 11, 12

#### HSA-SSE.A.2

Use the structure of an expression to identify ways to rewrite it. For example, 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).

09, 10, 11, 12

#### HSA-SSE.B.3

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

09, 10, 11, 12

#### HSA-SSE.B.3a

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

09, 10, 11, 12

#### HSA-SSE.B.3b

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

09, 10, 11, 12

#### HSA-SSE.B.3c

Use the properties of exponents to transform expressions for exponential functions. For example the expression 1.15t can be rewritten as (1.151/12)12t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

09, 10, 11, 12

#### HSA-SSE.B.4

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. For example, calculate mortgage payments.*

09, 10, 11, 12

#### HSA.APR.A

High School: Algebra » Arithmetic with Polynomials & Rational Expressions » Perform arithmetic operations on polynomials.

#### HSA.APR.B

High School: Algebra » Arithmetic with Polynomials & Rational Expressions » Understand the relationship between zeros and factors of polynomials.

#### HSA.APR.C

High School: Algebra » Arithmetic with Polynomials & Rational Expressions » Use polynomial identities to solve problems.

#### HSA.APR.D

High School: Algebra » Arithmetic with Polynomials & Rational Expressions » Rewrite rational expressions.

#### HSA.CED.A

High School: Algebra » Creating Equations✭ » Create equations that describe numbers or relationships.

#### HSA.REI.A

High School: Algebra » Reasoning with Equations & Inequalities » Understand solving equations as a process of reasoning and explain the reasoning.

#### HSA.REI.B

High School: Algebra » Reasoning with Equations & Inequalities » Solve equations and inequalities in one variable.

#### HSA.REI.C

High School: Algebra » Reasoning with Equations & Inequalities » Solve systems of equations.

#### HSA.REI.D

High School: Algebra » Reasoning with Equations & Inequalities » Represent and solve equations and inequalities graphically.

#### HSA.SSE.A

High School: Algebra » Seeing Structure in Expressions » Interpret the structure of expressions.

#### HSA.SSE.B

High School: Algebra » Seeing Structure in Expressions » Write expressions in equivalent forms to solve problems.

#### HSF-BF.A.1

Write a function that describes a relationship between two quantities.*

09, 10, 11, 12

#### HSF-BF.A.1a

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

09, 10, 11, 12

#### HSF-BF.A.1b

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.

09, 10, 11, 12

#### HSF-BF.A.1c

(+) 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.

09, 10, 11, 12

#### HSF-BF.A.2

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

09, 10, 11, 12

#### HSF-BF.B.3

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(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.

09, 10, 11, 12

#### HSF-BF.B.4

Find inverse functions.

09, 10, 11, 12

#### HSF-BF.B.4a

Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x-1) for x ≠ 1.

09, 10, 11, 12

#### HSF-BF.B.4b

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

09, 10, 11, 12

#### HSF-BF.B.4c

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

09, 10, 11, 12

#### HSF-BF.B.4d

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

09, 10, 11, 12

#### HSF-BF.B.5

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

09, 10, 11, 12

#### HSF-IF.A.1

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).

09, 10, 11, 12

#### HSF-IF.A.2

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

09, 10, 11, 12

#### HSF-IF.A.3

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.

09, 10, 11, 12

#### HSF-IF.B.4

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: intercepts; intervals where the function is increasing, decreasing, positive, or negative; relative maximums and minimums; symmetries; end behavior; and periodicity.*

09, 10, 11, 12

#### HSF-IF.B.5

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.*

09, 10, 11, 12

#### HSF-IF.B.6

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.*

09, 10, 11, 12

#### HSF-IF.C.7

Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

09, 10, 11, 12

#### HSF-IF.C.7a

Graph linear and quadratic functions and show intercepts, maxima, and minima.

09, 10, 11, 12

#### HSF-IF.C.7b

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

09, 10, 11, 12

#### HSF-IF.C.7c

Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

09, 10, 11, 12

#### HSF-IF.C.7d

(+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior.

09, 10, 11, 12

#### HSF-IF.C.7e

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

09, 10, 11, 12

#### HSF-IF.C.8

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

09, 10, 11, 12

#### HSF-IF.C.8a

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.

09, 10, 11, 12

#### HSF-IF.C.8b

Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)ᵗ, y = (0.97)ᵗ, y = (1.01)12ᵗ, y = (1.2)ᵗ/10, and classify them as representing exponential growth or decay.

09, 10, 11, 12

#### HSF-IF.C.9

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.

09, 10, 11, 12

#### HSF-LE.A.1

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

09, 10, 11, 12

#### HSF-LE.A.1a

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

09, 10, 11, 12

#### HSF-LE.A.1b

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

09, 10, 11, 12

#### HSF-LE.A.1c

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

09, 10, 11, 12

#### HSF-LE.A.2

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).

09, 10, 11, 12

#### HSF-LE.A.3

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

09, 10, 11, 12

#### HSF-LE.A.4

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.

09, 10, 11, 12

#### HSF-LE.B.5

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

09, 10, 11, 12

#### HSF-TF.A.1

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

09, 10, 11, 12

#### HSF-TF.A.2

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.

09, 10, 11, 12

#### HSF-TF.A.3

(+) 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 x, π + x, and 2π - x in terms of their values for x, where x is any real number.

09, 10, 11, 12

#### HSF-TF.A.4

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

09, 10, 11, 12

#### HSF-TF.B.5

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

09, 10, 11, 12

#### HSF-TF.B.6

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

09, 10, 11, 12

#### HSF-TF.B.7

(+) 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.*

09, 10, 11, 12

#### HSF-TF.C.8

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.

09, 10, 11, 12

#### HSF-TF.C.9

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

09, 10, 11, 12

#### HSF.BF.A

High School: Functions » Building Functions » Build a function that models a relationship between two quantities.

#### HSF.BF.B

High School: Functions » Building Functions » Build new functions from existing functions.

#### HSF.IF.A

High School: Functions » Interpreting Functions » Understand the concept of a function and use function notation.

#### HSF.IF.B

High School: Functions » Interpreting Functions » Interpret functions that arise in applications in terms of the context.

#### HSF.IF.C

High School: Functions » Interpreting Functions » Analyze functions using different representations.

#### HSF.LE.B

High School: Functions » Linear, Quadratic, & Exponential Models* » Interpret expressions for functions in terms of the situation they model.

#### HSF.TF.A

High School: Functions » Trigonometric Functions » Extend the domain of trigonometric functions using the unit circle.

#### HSF.TF.B

High School: Functions » Trigonometric Functions » Model periodic phenomena with trigonometric functions.

#### HSF.TF.C

High School: Functions » Trigonometric Functions » Prove and apply trigonometric identities.

#### HSG-C.A.1

Prove that all circles are similar.

09, 10, 11, 12

#### HSG-C.A.2

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

09, 10, 11, 12

#### HSG-C.A.3

Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

09, 10, 11, 12

#### HSG-C.A.4

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

09, 10, 11, 12

#### HSG-C.B.5

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

09, 10, 11, 12

#### HSG-CO.A.1

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

09, 10, 11, 12

#### HSG-CO.A.2

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).

09, 10, 11, 12

#### HSG-CO.A.3

Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

09, 10, 11, 12

#### HSG-CO.A.4

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

09, 10, 11, 12

#### HSG-CO.A.5

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

09, 10, 11, 12

#### HSG-CO.B.6

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.

09, 10, 11, 12

#### HSG-CO.B.7

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.

09, 10, 11, 12

#### HSG-CO.B.8

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

09, 10, 11, 12

#### HSG-CO.C.10

Prove theorems about triangles. Theorems include: 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.

09, 10, 11, 12

#### HSG-CO.C.11

Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

09, 10, 11, 12

#### HSG-CO.C.9

Prove theorems about lines and angles. Theorems include: 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.

09, 10, 11, 12

#### HSG-CO.D.12

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.

09, 10, 11, 12

#### HSG-CO.D.13

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

09, 10, 11, 12

#### HSG-GMD.A.1

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

09, 10, 11, 12

#### HSG-GMD.A.2

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

09, 10, 11, 12

#### HSG-GMD.A.3

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

09, 10, 11, 12

#### HSG-GMD.B.4

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

09, 10, 11, 12

#### HSG-GPE.A.1

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.

09, 10, 11, 12

#### HSG-GPE.A.2

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

09, 10, 11, 12

#### HSG-GPE.A.3

(+) 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.

09, 10, 11, 12

#### HSG-GPE.B.4

Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

09, 10, 11, 12

#### HSG-GPE.B.5

Prove 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).

09, 10, 11, 12

#### HSG-GPE.B.6

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

09, 10, 11, 12

#### HSG-GPE.B.7

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

09, 10, 11, 12

#### HSG-MG.A.1

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

09, 10, 11, 12

#### HSG-MG.A.2

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

09, 10, 11, 12

#### HSG-MG.A.3

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).*

09, 10, 11, 12

#### HSG-SRT.A.1

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

09, 10, 11, 12

#### HSG-SRT.A.1a

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.

09, 10, 11, 12

#### HSG-SRT.A.1b

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

09, 10, 11, 12

#### HSG-SRT.A.2

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.

09, 10, 11, 12

#### HSG-SRT.A.3

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

09, 10, 11, 12

#### HSG-SRT.B.4

Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

09, 10, 11, 12

#### HSG-SRT.B.5

Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

09, 10, 11, 12

#### HSG-SRT.C.6

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.

09, 10, 11, 12

#### HSG-SRT.C.7

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

09, 10, 11, 12

#### HSG-SRT.C.8

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

09, 10, 11, 12

#### HSG-SRT.D.10

(+) Prove the Laws of Sines and Cosines and use them to solve problems.

09, 10, 11, 12

#### HSG-SRT.D.11

(+) 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).

09, 10, 11, 12

#### HSG-SRT.D.9

(+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

09, 10, 11, 12

#### HSG.C.A

High School: Geometry » Circles » Understand and apply theorems about circles

#### HSG.C.B

High School: Geometry » Circles » Find arc lengths and areas of sectors of circles

#### HSG.CO.A

High School: Geometry » Congruence » Experiment with transformations in the plane

#### HSG.CO.B

High School: Geometry » Congruence » Understand congruence in terms of rigid motions

#### HSG.CO.C

High School: Geometry » Congruence » Prove geometric theorems

#### HSG.CO.D

High School: Geometry » Congruence » Make geometric constructions

#### HSG.GMD

High School: Geometry » Geometric Measurement & Dimension

#### HSG.GMD.A

High School: Geometry » Geometric Measurement & Dimension » Explain volume formulas and use them to solve problems

#### HSG.GMD.B

High School: Geometry » Geometric Measurement & Dimension » Visualize relationships between two-dimensional and three-dimensional objects

#### HSG.GPE.A

High School: Geometry » Expressing Geometric Properties with Equations » Translate between the geometric description and the equation for a conic section

#### HSG.GPE.B

High School: Geometry » Expressing Geometric Properties with Equations » Use coordinates to prove simple geometric theorems algebraically

#### HSG.MG.A

High School: Geometry » Modeling with Geometry » Apply geometric concepts in modeling situations

#### HSG.SRT.A

High School: Geometry » Similarity, Right Triangles, & Trigonometry » Understand similarity in terms of similarity transformations

#### HSG.SRT.B

High School: Geometry » Similarity, Right Triangles, & Trigonometry » Prove theorems involving similarity

#### HSG.SRT.C

High School: Geometry » Similarity, Right Triangles, & Trigonometry » Define trigonometric ratios and solve problems involving right triangles

#### HSG.SRT.D

High School: Geometry » Similarity, Right Triangles, & Trigonometry » Apply trigonometry to general triangles

#### HSN-CN.A.1

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.

09, 10, 11, 12

#### HSN-CN.A.2

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

09, 10, 11, 12

#### HSN-CN.A.3

(+) Find the conjugate of a complex number; use conjugates to find moduli and quotients of complex numbers.

09, 10, 11, 12

#### HSN-CN.B.4

(+) 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.

09, 10, 11, 12

#### HSN-CN.B.5

(+) Represent addition, subtraction, multiplication, and conjugation of complex numbers geometrically on the complex plane; use properties of this representation for computation. For example, (-1 + √3 i)3 = 8 because (-1 + √3 i) has modulus 2 and argument 120°.

09, 10, 11, 12

#### HSN-CN.B.6

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

09, 10, 11, 12

#### HSN-CN.C.7

Solve quadratic equations with real coefficients that have complex solutions.

09, 10, 11, 12

#### HSN-CN.C.8

(+) Extend polynomial identities to the complex numbers. For example, rewrite x2 + 4 as (x + 2i)(x - 2i).

09, 10, 11, 12

#### HSN-CN.C.9

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

09, 10, 11, 12

#### HSN-Q.A.1

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.

09, 10, 11, 12

#### HSN-Q.A.2

Define appropriate quantities for the purpose of descriptive modeling.

09, 10, 11, 12

#### HSN-Q.A.3

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

09, 10, 11, 12

#### HSN-RN.A.1

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. For example, we define 51/3 to be the cube root of 5 because we want (51/3)3 = 5(1/3)3 to hold, so (51/3)3 must equal 5.

09, 10, 11, 12

#### HSN-RN.A.2

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

09, 10, 11, 12

#### HSN-RN.B.3

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.

09, 10, 11, 12

#### HSN-VM.A.1

(+) 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||, v).

09, 10, 11, 12

#### HSN-VM.A.2

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

09, 10, 11, 12

#### HSN-VM.A.3

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

09, 10, 11, 12

09, 10, 11, 12

#### HSN-VM.B.4a

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.

09, 10, 11, 12

#### HSN-VM.B.4b

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

09, 10, 11, 12

#### HSN-VM.B.4c

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.

09, 10, 11, 12

#### HSN-VM.B.5

(+) Multiply a vector by a scalar.

09, 10, 11, 12

#### HSN-VM.B.5a

Represent scalar multiplication graphically by scaling vectors and possibly reversing their direction; perform scalar multiplication component-wise, e.g., as c(vx, vy) = (cvx, cvy).

09, 10, 11, 12

#### HSN-VM.B.5b

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).

09, 10, 11, 12

#### HSN-VM.C.10

(+) Understand that the zero and identity matrices play a role in matrix addition and multiplication similar 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.

09, 10, 11, 12

#### HSN-VM.C.11

(+) 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.

09, 10, 11, 12

#### HSN-VM.C.12

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

09, 10, 11, 12

#### HSN-VM.C.6

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

09, 10, 11, 12

#### HSN-VM.C.7

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

09, 10, 11, 12

#### HSN-VM.C.8

(+) Add, subtract, and multiply matrices of appropriate dimensions.

09, 10, 11, 12

#### HSN-VM.C.9

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

09, 10, 11, 12

#### HSN.CN.A

High School: Number and Quantity » The Complex Number System » Perform arithmetic operations with complex numbers.

#### HSN.CN.B

High School: Number and Quantity » The Complex Number System » Represent complex numbers and their operations on the complex plane.

#### HSN.CN.C

High School: Number and Quantity » The Complex Number System » Use complex numbers in polynomial identities and equations.

#### HSN.Q.A

High School: Number and Quantity » Quantities* » Reason quantitatively and use units to solve problems.

#### HSN.RN

High School: Number and Quantity » The Real Number System

#### HSN.RN.A

High School: Number and Quantity » The Real Number System » Extend the properties of exponents to rational exponents.

#### HSN.RN.B

High School: Number and Quantity » The Real Number System » Use properties of rational and irrational numbers.

#### HSN.VM.A

High School: Number and Quantity » Vector & Matrix Quantities » Represent and model with vector quantities.

#### HSN.VM.B

High School: Number and Quantity » Vector & Matrix Quantities » Perform operations on vectors.

#### HSN.VM.C

High School: Number and Quantity » Vector & Matrix Quantities » Perform operations on matrices and use matrices in applications.

#### HSS-CP.A.1

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").

09, 10, 11, 12

#### HSS-CP.A.2

Understand that two events A and B are independent 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.

09, 10, 11, 12

#### HSS-CP.A.3

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.

09, 10, 11, 12

#### HSS-CP.A.4

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 tenth grade. Do the same for other subjects and compare the results.

09, 10, 11, 12

#### HSS-CP.A.5

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.

09, 10, 11, 12

#### HSS-CP.B.6

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.

09, 10, 11, 12

#### HSS-CP.B.7

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.

09, 10, 11, 12

#### HSS-CP.B.8

(+) 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.

09, 10, 11, 12

#### HSS-CP.B.9

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

09, 10, 11, 12

#### HSS-IC.A.1

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

09, 10, 11, 12

#### HSS-IC.A.2

Decide if a specified model is consistent with results from a given data-generating 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?

09, 10, 11, 12

#### HSS-IC.B.3

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

09, 10, 11, 12

#### HSS-IC.B.4

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.

09, 10, 11, 12

#### HSS-IC.B.5

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