From my Twitter feed, I came across a blog
post from Tara Linney
about integrating technology into the ELA Common Core Standards. I thought that
it might be useful for mathematics also. Here it is.

Some integration will be implicit and inferred and
other ideas will be very explicit. The standards are taken directly from http://CoreStandards.org. The use of
algebraic thinking and models within the standards, while not explicitly
stating the use of technology, commonly utilizes technology to assist students
in developing or presenting their explanations. As we progress into high school,
the areas for technology integration increase dramatically due to the
complexity of the mathematics involved.

__From the Standards for Mathematical Practice__

Make sense of problems and persevere in solving
them.

"Mathematically proficient students can
explain correspondences between equations, verbal descriptions, tables, and
graphs or draw diagrams of important features and relationships, graph data,
and search for regularity or trends. Younger students might rely on using
concrete objects or pictures to help conceptualize and solve a problem."

2. Reason abstractly and quantitatively

"the ability to

*decontextualize*—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to

*contextualize*, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved."

4. Model with mathematics

"They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas."

5. Use appropriate tools strategically

"Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software...When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts."

__Operations and Algebraic Thinking__

__Kindergarten__

CCSS.Math.Content.K.OA.A.2
Solve addition and subtraction word problems, and add and subtract within 10,
e.g., by using objects or drawings to represent the problem.

CCSS.Math.Content.K.OA.A.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

CCSS.Math.Content.K.OA.A.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

CCSS.Math.Content.2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

CCSS.Math.Content.2.OA.C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

CCSS.Math.Content.3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

CCSS.Math.Content.4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

CCSS.Math.Content.4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

CCSS.Math.Content.K.OA.A.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).

CCSS.Math.Content.K.OA.A.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

__First Grade__CCSS.Math.Content.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}CCSS.Math.Content.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.

__Second Grade__CCSS.Math.Content.2.OA.A.1 Use addition and subtraction within 100 to solve one- and two-step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

^{1}CCSS.Math.Content.2.OA.C.3 Determine whether a group of objects (up to 20) has an odd or even number of members, e.g., by pairing objects or counting them by 2s; write an equation to express an even number as a sum of two equal addends.

CCSS.Math.Content.2.OA.C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

__Third Grade__CCSS.Math.Content.3.OA.A.3 Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

^{1}__Fourth Grade__CCSS.Math.Content.4.OA.A.2 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison.

^{1}CCSS.Math.Content.4.OA.C.5 Generate a number or shape pattern that follows a given rule. Identify apparent features of the pattern that were not explicit in the rule itself.

*For example, given the rule “Add 3” and the starting number 1, generate terms in the resulting sequence and observe that the terms appear to alternate between odd and even numbers. Explain informally why the numbers will continue to alternate in this way*.__Fifth Grade__
CCSS.Math.Content.5.OA.B.3
Generate two numerical patterns using two given rules. Identify apparent
relationships between corresponding terms. Form ordered pairs consisting of
corresponding terms from the two patterns, and graph the ordered pairs on a
coordinate plane.

*For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so*.__Number & Operations in Base Ten____First Grade__

CCSS.Math.Content.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.

CCSS.Math.Content.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.

__Second Grade__

CCSS.Math.Content.2.NBT.B.7 Add and subtract within 1000, 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. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.

__Third Grade__

CCSS.Math.Content.3.NBT.A.2 Fluently add and subtract within 1000 using strategies and algorithms based on place value, properties of operations, and/or the relationship between addition and subtraction.

__Fourth Grade__

CCSS.Math.Content.4.NBT.B.4 Fluently add and subtract multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.4.NBT.B.5 Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

CCSS.Math.Content.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

__Fifth Grade__CCSS.Math.Content.5.NBT.B.5 Fluently multiply multi-digit whole numbers using the standard algorithm.

CCSS.Math.Content.5.NBT.B.6 Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

CCSS.Math.Content.5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths, 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.

__Number & Operations -- Fractions Third Grade__CCSS.Math.Content.3.NF.A.3b Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3). Explain why the fractions are equivalent, e.g., by using a visual fraction model. CCSS.Math.Content.3.NF.A.3d Compare two fractions with the same numerator or the same denominator by reasoning about their size. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.

__Fourth Grade__CCSS.Math.Content.4.NF.A.1 Explain why a fraction

*a*/

*b*is equivalent to a fraction (

*n*×

*a*)/(

*n*×

*b*) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.

CCSS.Math.Content.4.NF.A.2 Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model.CCSS.Math.Content.4.NF.B.3b Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model.

*Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8*.CCSS.Math.Content.4.NF.B.3d Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem.

CCSS.Math.Content.4.NF.B.4a
Understand a fraction

*a*/*b*as a multiple of 1/*b*.*For example, use a visual fraction model to represent 5/4 as the product 5 × (1/4), recording the conclusion by the equation 5/4 = 5 × (1/4)*.
CCSS.Math.Content.4.NF.B.4b
Understand a multiple of a/b as a multiple of 1/b, and use this understanding
to multiply a fraction by a whole number.

*For example, use a visual fraction model to express 3 × (2/5) as 6 × (1/5), recognizing this product as 6/5. (In general, n × (a/b) = (n × a)/b.)*
CCSS.Math.Content.4.NF.B.4c
Solve word problems involving multiplication of a fraction by a whole number,
e.g., by using visual fraction models and equations to represent the problem.

*For example, if each person at a party will eat 3/8 of a pound of roast beef, and there will be 5 people at the party, how many pounds of roast beef will be needed? Between what two whole numbers does your answer lie?*
CCSS.Math.Content.4.NF.C.7
Compare two decimals to hundredths by reasoning about their size. Recognize
that comparisons are valid only when the two decimals refer to the same whole.
Record the results of comparisons with the symbols >, =, or <,
and justify the conclusions, e.g., by using a visual model.

__Fifth Grade__

CCSS.Math.Content.5.NF.A.2
Solve word problems involving addition and subtraction of fractions referring
to the same whole, including cases of unlike denominators, e.g., by using
visual fraction models or equations to represent the problem. Use benchmark
fractions and number sense of fractions to estimate mentally and assess the
reasonableness of answers.

*For example, recognize an incorrect result 2/5 + 1/2 = 3/7, by observing that 3/7 < 1/2*.
CCSS.Math.Content.5.NF.B.4a
Interpret the product (

*a*/*b*) ×*q*as a parts of a partition of*q*into*b*equal parts; equivalently, as the result of a sequence of operations*a*×*q*÷*b*.*For example, use a visual fraction model to show (2/3) × 4 = 8/3, and create a story context for this equation. Do the same with (2/3) × (4/5) = 8/15. (In general, (a/b) × (c/d) = ac/bd.)*
CCSS.Math.Content.5.NF.B.6
Solve real world problems involving multiplication of fractions and mixed
numbers, e.g., by using visual fraction models or equations to represent the
problem.

*CCSS.Math.Content.5.NF.B.7a Interpret division of a unit fraction by a non-zero whole number, and compute such quotients. For example, create a story context for (1/3) ÷ 4, and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that (1/3) ÷ 4 = 1/12 because (1/12) × 4 = 1/3.*

*CCSS.Math.Content.5.NF.B.7b Interpret division of a whole number by a unit fraction, and compute such quotients. For example, create a story context for 4 ÷ (1/5), and use a visual fraction model to show the quotient. Use the relationship between multiplication and division to explain that 4 ÷ (1/5) = 20 because 20 × (1/5) = 4.**CCSS.Math.Content.5.NF.B.7c Solve real world problems involving division of unit fractions by non-zero whole numbers and division of whole numbers by unit fractions, e.g., by using visual fraction models and equations to represent the problem. For example, how much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 1/3-cup servings are in 2 cups of raisins?*__Measurement & Data__

__First Grade__

CCSS.Math.Content.1.MD.B.3
Tell and write time in hours and half-hours using analog and digital clocks.

__Second Grade__

CCSS.Math.Content.2.MD.C.7
Tell and write time from analog and digital clocks to the nearest five minutes,
using a.m. and p.m.

__Third Grade__

CCSS.Math.Content.3.MD.C.7c
Use tiling to show in a concrete case that the area of a rectangle with
whole-number side lengths

*a*and*b*+*c*is the sum of*a*×*b*and*a*×*c*. Use area models to represent the distributive property in mathematical reasoning.__High School__

Calculate expected values and use them to solve
problems

CCSS.Math.Content.HSS-MD.A.1
(+) 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.

CCSS.Math.Content.HSS-MD.A.2
(+) Calculate the expected value of a random variable; interpret it as the mean
of the probability distribution.

CCSS.Math.Content.HSS-MD.A.3
(+) Develop a probability distribution for a random variable defined for a
sample space in which theoretical probabilities can be calculated; find the
expected value.

*For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes.*
CCSS.Math.Content.HSS-MD.A.4
(+) Develop a probability distribution for a random variable defined for a
sample space in which probabilities are assigned empirically; find the expected
value.

*For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households?*
Use probability to evaluate outcomes of decisions

CCSS.Math.Content.HSS-MD.B.5
(+) Weigh the possible outcomes of a decision by assigning probabilities to
payoff values and finding expected values.

CCSS.Math.Content.HSS-MD.B.5a
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.*
CCSS.Math.Content.HSS-MD.B.5b
Evaluate and compare strategies on the basis of expected values.

*For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident.*
CCSS.Math.Content.HSS-MD.B.6
(+) Use probabilities to make fair decisions (e.g., drawing by lots, using a
random number generator).

CCSS.Math.Content.HSS-MD.B.7
(+) Analyze decisions and strategies using probability concepts (e.g., product
testing, medical testing, pulling a hockey goalie at the end of a game).

__Geometry__

__Kindergarten__

CCSS.Math.Content.K.G.B.5
Model shapes in the world by building shapes from components (e.g., sticks and
clay balls) and drawing shapes.

CCSS.Math.Content.K.G.B.6
Compose simple shapes to form larger shapes.

*For example, “Can you join these two triangles with full sides touching to make a rectangle*?”__Fifth Grade__

CCSS.Math.Content.5.G.A.2
Represent real world and mathematical problems by graphing points in the first
quadrant of the coordinate plane, and interpret coordinate values of points in
the context of the situation.

__Seventh Grade__

Draw construct, and
describe geometrical figures and describe the relationships between them.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

Solve real-life and
mathematical problems involving angle measure, area, surface area, and volume.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

__Eighth Grade__

Understand congruence
and similarity using physical models, transparencies, or geometry software.

CCSS.Math.Content.8.G.A.1
Verify experimentally the properties of rotations, reflections, and
translations:

CCSS.Math.Content.8.G.A.1a
Lines are taken to lines, and line segments to line segments of the same
length.

CCSS.Math.Content.8.G.A.1b
Angles are taken to angles of the same measure.

CCSS.Math.Content.8.G.A.1c
Parallel lines are taken to parallel lines.

CCSS.Math.Content.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.

CCSS.Math.Content.8.G.A.3
Describe the effect of dilations, translations, rotations, and reflections on
two-dimensional figures using coordinates.

CCSS.Math.Content.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.

CCSS.Math.Content.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*.
Understand and apply
the Pythagorean Theorem.

CCSS.Math.Content.8.G.B.6
Explain a proof of the Pythagorean Theorem and its converse.

CCSS.Math.Content.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.

CCSS.Math.Content.8.G.B.8
Apply the Pythagorean Theorem to find the distance between two points in a
coordinate system.

Solve real-world and
mathematical problems involving volume of cylinders, cones, and spheres.

CCSS.Math.Content.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.

__Ratios & Proportional Relationships__

__Seventh Grade__

CCSS.Math.Content.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.

CCSS.Math.Content.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.__The Number System__

__Sixth Grade__

CCSS.Math.Content.6.NS.A.1
Interpret and compute quotients of fractions, and solve word problems involving
division of fractions by fractions, e.g., by using visual fraction models and
equations to represent the problem.

*For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?*.
CCSS.Math.Content.6.NS.C.8
Solve real-world and mathematical problems by graphing points in all four
quadrants of the coordinate plane. Include use of coordinates and absolute
value to find distances between points with the same first coordinate or the
same second coordinate.

__Eigth Grade__

CCSS.Math.Content.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.

CCSS.Math.Content.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., π

^{2}).*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*.__Expressions & Equations__

__Sixth Grade__

CCSS.Math.Content.6.EE.C.9
Use variables to represent two quantities in a real-world problem that change
in relationship to one another; write an equation to express one quantity,
thought of as the dependent variable, in terms of the other quantity, thought
of as the independent variable. Analyze the relationship between the dependent
and independent variables using graphs and tables, and relate these to the
equation. For example, in a problem involving motion at constant speed, list
and graph ordered pairs of distances and times, and write the equation d = 65t
to represent the relationship between distance and time.

__Seventh Grade__

CCSS.Math.Content.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*.__Eigth Grade__

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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*.__Functions__

__Eigth Grade__

Define, evaluate, and
compare functions.

CCSS.Math.Content.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}
CCSS.Math.Content.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*.
CCSS.Math.Content.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 = s*.^{2}giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line
Use functions to model
relationships between quantities.

CCSS.Math.Content.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.
CCSS.Math.Content.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.

__Statistics & Probability__

__Seventh Grade__

CCSS.Math.Content.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.

CCSS.Math.Content.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*.
CCSS.Math.Content.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?*__Eigth Grade__

CCSS.Math.Content.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.

CCSS.Math.Content.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*.__High School Number & Quantity__

CCSS.Math.Content.HSN-Q.A.2
Define appropriate quantities for the purpose of descriptive modeling.

Perform arithmetic operations with complex numbers.

CCSS.Math.Content.HSN-CN.A.1
Know there is a complex number

*i*such that*i*^{2}= –1, and every complex number has the form*a + bi*with*a*and*b*real.
CCSS.Math.Content.HSN-CN.A.2
Use the relation

*i*^{2}= –1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers.
CCSS.Math.Content.HSN-CN.A.3
(+) Find the conjugate of a complex number; use conjugates to find moduli and
quotients of complex numbers.

Represent complex numbers and their operations on
the complex plane.

CCSS.Math.Content.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.

CCSS.Math.Content.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°.
CCSS.Math.Content.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.

Use complex numbers in polynomial identities and
equations.

CCSS.Math.Content.HSN-CN.C.7
Solve quadratic equations with real coefficients that have complex solutions.

CCSS.Math.Content.HSN-CN.C.8
(+) Extend polynomial identities to the complex numbers.

*For example, rewrite x*.^{2}+ 4 as (x + 2i)(x – 2i)
CCSS.Math.Content.HSN-CN.C.9
(+) Know the Fundamental Theorem of Algebra; show that it is true for quadratic
polynomials.

Represent and model with vector quantities.

CCSS.Math.Content.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*).
CCSS.Math.Content.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.

CCSS.Math.Content.HSN-VM.A.3
(+) Solve problems involving velocity and other quantities that can be
represented by vectors.

Perform operations on vectors.

CCSS.Math.Content.HSN-VM.B.4
(+) Add and subtract vectors.

CCSS.Math.Content.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.

CCSS.Math.Content.HSN-VM.B.4b
Given two vectors in magnitude and direction form, determine the magnitude and
direction of their sum.

CCSS.Math.Content.HSN-VM.B.4c
Understand vector subtraction

**–***v***as***w***+ (–***v***), 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.***w*
CCSS.Math.Content.HSN-VM.B.5
(+) Multiply a vector by a scalar.

CCSS.Math.Content.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*(*v*,_{x}*v*) = (_{y}*cv*,_{x}*cv*)._{y}
CCSS.Math.Content.HSN-VM.B.5b
Compute the magnitude of a scalar multiple

*c*using ||**v***c*|| = |**v***c*|**. Compute the direction of***v**c*knowing that when |**v***c*|**≠ 0, the direction of***v**c*is either along**v****(for***v**c*> 0) or against**(for***v**c*< 0).
Perform operations on matrices and use matrices in
applications.

CCSS.Math.Content.HSN-VM.C.6
(+) Use matrices to represent and manipulate data, e.g., to represent payoffs
or incidence relationships in a network.

CCSS.Math.Content.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.

CCSS.Math.Content.HSN-VM.C.8
(+) Add, subtract, and multiply matrices of appropriate dimensions.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

__High School: Algebra__

CCSS.Math.Content.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.*
CCSS.Math.Content.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.

CCSS.Math.Content.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*.
CCSS.Math.Content.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.

CCSS.Math.Content.HSA-REI.C.8
(+) Represent a system of linear equations as a single matrix equation in a
vector variable.

CCSS.Math.Content.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).

CCSS.Math.Content.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).

CCSS.Math.Content.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
CCSS.Math.Content.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.

__High School: Functions__

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

CCSS.Math.Content.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*.
CCSS.Math.Content.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.*
CCSS.Math.Content.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.

**Analyze functions using different representations.**

CCSS.Math.Content.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.

CCSS.Math.Content.HSF-IF.C.7a
Graph linear and quadratic functions and show intercepts, maxima, and minima.

CCSS.Math.Content.HSF-IF.C.7b
Graph square root, cube root, and piecewise-defined functions, including step
functions and absolute value functions.

CCSS.Math.Content.HSF-IF.C.7c
Graph polynomial functions, identifying zeros when suitable factorizations are
available, and showing end behavior.

CCSS.Math.Content.HSF-IF.C.7d
(+) Graph rational functions, identifying zeros and asymptotes when suitable
factorizations are available, and showing end behavior.

CCSS.Math.Content.HSF-IF.C.7e
Graph exponential and logarithmic functions, showing intercepts and end
behavior, and trigonometric functions, showing period, midline, and amplitude.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.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*.**Build a function that models a relationship between two quantities.**

CCSS.Math.Content.HSF-BF.A.1
Write a function that describes a relationship between two quantities.

CCSS.Math.Content.HSF-BF.A.1a
Determine an explicit expression, a recursive process, or steps for calculation
from a context.

CCSS.Math.Content.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*.
CCSS.Math.Content.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*.
CCSS.Math.Content.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.

**Build new functions from existing functions.**

CCSS.Math.Content.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.
CCSS.Math.Content.HSF-BF.B.4
Find inverse functions.

CCSS.Math.Content.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*.
CCSS.Math.Content.HSF-BF.B.4b
(+) Verify by composition that one function is the inverse of another.

CCSS.Math.Content.HSF-BF.B.4c
(+) Read values of an inverse function from a graph or a table, given that the
function has an inverse.

CCSS.Math.Content.HSF-BF.B.4d
(+) Produce an invertible function from a non-invertible function by
restricting the domain.

CCSS.Math.Content.HSF-BF.B.5
(+) Understand the inverse relationship between exponents and logarithms and
use this relationship to solve problems involving logarithms and exponents.

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

CCSS.Math.Content.HSF-LE.A.1
Distinguish between situations that can be modeled with linear functions and
with exponential functions.

CCSS.Math.Content.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.

CCSS.Math.Content.HSF-LE.A.1b
Recognize situations in which one quantity changes at a constant rate per unit
interval relative to another.

CCSS.Math.Content.HSF-LE.A.1c
Recognize situations in which a quantity grows or decays by a constant percent
rate per unit interval relative to another.

CCSS.Math.Content.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).

CCSS.Math.Content.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.

CCSS.Math.Content.HSF-LE.A.4
For exponential models, express as a logarithm the solution to

*ab*=^{ct}*d*where*a*,*c*, and*d*are numbers and the base*b*is 2, 10, or*e*; evaluate the logarithm using technology.**Interpret expressions for functions in terms of the situation they model.**

CCSS.Math.Content.HSF-LE.B.5
Interpret the parameters in a linear or exponential function in terms of a
context.

**Extend the domain of trigonometric functions using the unit circle.**

CCSS.Math.Content.HSF-TF.A.1
Understand radian measure of an angle as the length of the arc on the unit
circle subtended by the angle.

CCSS.Math.Content.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.

CCSS.Math.Content.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.
CCSS.Math.Content.HSF-TF.A.4
(+) Use the unit circle to explain symmetry (odd and even) and periodicity of
trigonometric functions.

**Model periodic phenomena with trigonometric functions.**

CCSS.Math.Content.HSF-TF.B.5
Choose trigonometric functions to model periodic phenomena with specified
amplitude, frequency, and midline.

CCSS.Math.Content.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.

CCSS.Math.Content.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.

**Prove and apply trigonometric identities.**

CCSS.Math.Content.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.

CCSS.Math.Content.HSF-TF.C.9
(+) Prove the addition and subtraction formulas for sine, cosine, and tangent
and use them to solve problems.

__High School: Modeling__

One
of the insights provided by mathematical modeling is that essentially the same
mathematical or statistical structure can sometimes model seemingly different
situations. Models can also shed light on the mathematical structures
themselves, for example, as when a model of bacterial growth makes more vivid
the explosive growth of the exponential function.

The
basic modeling cycle is summarized in the diagram. It involves (1) identifying
variables in the situation and selecting those that represent essential
features, (2) formulating a model by creating and selecting geometric,
graphical, tabular, algebraic, or statistical representations that describe
relationships between the variables, (3) analyzing and performing operations on
these relationships to draw conclusions, (4) interpreting the results of the
mathematics in terms of the original situation, (5) validating the conclusions
by comparing them with the situation, and then either improving the model or,
if it is acceptable, (6) reporting on the conclusions and the reasoning behind
them.

__High School: Geometry__

CCSS.Math.Content.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.

CCSS.Math.Content.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).

CCSS.Math.Content.HSG-CO.A.3
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.

CCSS.Math.Content.HSG-CO.A.4
Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.

CCSS.Math.Content.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.

CCSS.Math.Content.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*.
CCSS.Math.Content.HSG-CO.D.13
Construct an equilateral triangle, a square, and a regular hexagon inscribed in
a circle.

CCSS.Math.Content.HSG-SRT.A.1
Verify experimentally the properties of dilations given by a center and a scale
factor:

CCSS.Math.Content.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.

CCSS.Math.Content.HSG-SRT.A.1b
The dilation of a line segment is longer or shorter in the ratio given by the
scale factor.

CCSS.Math.Content.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.

CCSS.Math.Content.HSG-SRT.A.3
Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.

CCSS.Math.Content.HSG-SRT.C.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.

*CCSS.Math.Content.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.

CCSS.Math.Content.HSG-C.A.4
(+) Construct a tangent line from a point outside a given circle to the circle.

CCSS.Math.Content.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..

CCSS.Math.Content.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.

CCSS.Math.Content.HSG-GPE.A.2
Derive the equation of a parabola given a focus and directrix.

CCSS.Math.Content.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.

CCSS.Math.Content.HSG-GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and
rectangles, e.g., using the distance formula.

CCSS.Math.Content.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).

CCSS.Math.Content.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).

CCSS.Math.Content.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).

__High School: Statistics & Probability__

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

CCSS.Math.Content.HSS-ID.A.1
Represent data with plots on the real number line (dot plots, histograms, and
box plots).

CCSS.Math.Content.HSS-ID.A.2
Use statistics appropriate to the shape of the data distribution to compare
center (median, mean) and spread (interquartile range, standard deviation) of
two or more different data sets.

CCSS.Math.Content.HSS-ID.A.3
Interpret differences in shape, center, and spread in the context of the data
sets, accounting for possible effects of extreme data points (outliers).

CCSS.Math.Content.HSS-ID.A.4
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, represent, and interpret data on two categorical and quantitative variables**

CCSS.Math.Content.HSS-ID.B.5
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.

CCSS.Math.Content.HSS-ID.B.6
Represent data on two quantitative variables on a scatter plot, and describe
how the variables are related.

CCSS.Math.Content.HSS-ID.B.6a
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.

CCSS.Math.Content.HSS-ID.B.6b
Informally assess the fit of a function by plotting and analyzing residuals.

CCSS.Math.Content.HSS-ID.B.6c
Fit a linear function for a scatter plot that suggests a linear association.

**Interpret linear models**

CCSS.Math.Content.HSS-ID.C.7
Interpret the slope (rate of change) and the intercept (constant term) of a
linear model in the context of the data.

CCSS.Math.Content.HSS-ID.C.8
Compute (using technology) and interpret the correlation coefficient of a
linear fit.

CCSS.Math.Content.HSS-ID.C.9
Distinguish between correlation and causation.

CCSS.Math.Content.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*?
CCSS.Math.Content.HSS-IC.B.3
Recognize the purposes of and differences among sample surveys, experiments,
and observational studies; explain how randomization relates to each.

CCSS.Math.Content.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.

CCSS.Math.Content.HSS-IC.B.5
Use data from a randomized experiment to compare two treatments; use
simulations to decide if differences between parameters are significant.

CCSS.Math.Content.HSS-IC.B.6
Evaluate reports based on data.

CCSS.Math.Content.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.*

CCSS.Math.Content.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.
CCSS.Math.Content.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.

CCSS.Math.Content.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.

CCSS.Math.Content.HSS-CP.B.9
(+) Use permutations and combinations to compute probabilities of compound
events and solve problems.

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