Maryland College and Career-Ready Standards 8th Grade Math Activities

MD.MP. Mathematical Practices

MP.1. Make sense of problems and persevere in solving them.

MP.2. Reason abstractly and quantitatively.

MD.MA.8.NS. The Number System (NS)

Know that there are numbers that are not rational, and approximate them by rational numbers.

8.NS.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.

8.NS.1.1. Knowledge of differences between rational and irrational.

8.NS.1.2. Knowledge of definition and description of rational and irrational.

8.NS.1.3. Ability to identify and provide examples of rational versus irrational numbers, of the real number system.

MD.MA.8.EE. Expressions and Equations (EE)

Work with radicals and integer exponents.

8.EE.1. Know and apply the properties of integer exponents to generate equivalent numerical expressions. For example, 3^2 x 3^-5 = 3^-3 = 1/3^3 = 1/27.

8.EE.1.1. Ability to recognize and apply the following properties of integer exponents:

8.EE.1.1.2. Negative Exponents

8.EE.1.1.4. Power of Powers

8.EE.1.2. Ability to apply a combination of properties to show equivalency.

8.EE.2. Use square root and cube root symbols to represent solutions to equations of the form x^2 = p and x^3 = 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.

8.EE.2.1. Ability to recognize and apply the following:

8.EE.2.1.1. Perfect Squares

8.EE.2.1.3. Square Roots (Symbol Notation)

8.EE.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 3x10^8 and the population of the world as 7x10^9, and determine that the world population is more than 20 times larger.

8.EE.3.1. Ability to compare large and small numbers using properties of integer exponents (see 8.EE.1).

8.EE.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.

8.EE.4.1. Ability to compare units of measure.

8.EE.4.2. Ability to read scientific notation on a calculator.

Understand the connections between proportional relationships, lines, and linear equations.

8.EE.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.

8.EE.5.1. Ability to relate and compare graphic, symbolic, numerical representations of proportional relationships.

8.EE.5.2. Ability to calculate constant rate of change/slope of a line graphically.

8.EE.5.3. Ability to understand that all proportional relationships start at the origin.

8.EE.5.4. Ability to recognize and apply direct variation.

8.EE.6. Use similar triangles to explain why the slope m is the same between any two distinct points on a nonvertical 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.

8.EE.6.1. Ability to understand that similar right triangles (provide diagram of graphical notation) can be used to establish that slope is constant for a non-vertical line (see 8.G.1).

Analyze and solve linear equations and pairs of simultaneous linear equations.

8.EE.7. Solve linear equations in one variable.

8.EE.7b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

8.EE.7b.1. See the skills and knowledge that are stated in the Standard.

MD.MA.8.F. Functions (F)

Define, evaluate, and compare functions.

8.F.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 (function notation is not required in Grade 8).

8.F.1.1. Ability to recognize functional relationships and apply the following:

8.F.1.1.2. Vertical Line Test

8.F.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.

8.F.2.2. Ability to calculate slope/rate of change of a line graphically from a table or verbal description.

Use functions to model relationships between quantities.

8.F.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.

8.F.4.1. Ability to calculate and interpret constant rate of change/slope from a scenario, table, graph, or two points.

8.F.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.

8.F.5.1. Ability to distinguish rate of change within an interval of a function.

8.F.5.2. Ability to interpret directionality and steepness of the graph of a function.

8.F.5.3. Ability to sketch a graph given algebraic context or a scenario (slope and initial value).

MD.MA.8.G. Geometry (G)

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

8.G.1. Verify experimentally the properties of rotations, reflections, and translations.

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

8.G.1a.1. Ability to conduct experiments which show that rotations, reflections, and translations of lines and line segments are rigid.

8.G.1a.2. Ability to use transformation notation (A➞A’➞A”).

8.G.1b. Angles are taken to angles of the same measure.

8.G.1b.1. Ability to conduct experiments which show that rotations, reflections, and translations of angles are rigid.

8.G.1b.2. Ability to use transformation notation (∠A➞∠A’➞∠A”).

8.G.1c. Parallel lines are taken to parallel lines.

8.G.1c.1. Ability to conduct experiments which show that rotations, reflections, and translations of parallel lines are rigid.

8.G.1c.2. Ability to use transformation notation (A➞A’➞A”).

8.G.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.

8.G.2.1. Ability to use a sequence of transformations and map one figure to a second figure to show congruency.

8.G.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.

8.G.4.2. Ability to show that similar figures maintain shape but alter size through dilation (scale factor).

8.G.4.3. Ability to demonstrate that congruency is a special case of similarity (scale factor of 1).

Understand and apply the Pythagorean Theorem.

8.G.7. Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions (SC 8).

8.G.7.1. See the skills and knowledge that are stated in the Standard.

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

8.G.9. Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

8.G.9.1. See the skills and knowledge that are stated in the Standard.

MD.MA.8.SP. Statistics and Probability (SP)

Investigate patterns of association in bivariate data.

8.SP.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.

8.SP.1.1. Ability to integrate technology and relate the scenarios to authentic student-centered situations.

8.SP.1.2. Ability to keep paired data organized in relation to one another within two sets of quantities.

8.SP.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.

8.SP.2.1. See the skills and knowledge that are stated in the Standard.

8.SP.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?

8.SP.4.1. Ability to integrate technology and to relate the scenarios to authentic student-centered situations.

MD.MA.AI. Algebra I

Unit 1: Relationships between Quantities and Reasoning with Equations

HSN-Q.A. Reason quantitatively and use units to solve problems.

N.Q.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.

N.Q.1.2. Ability to convert units of measure using dimensional analysis.

HSA-SSE.A. Interpret the structure of expressions.

A.SSE.1a. Interpret expressions that represent a quantity in terms of its context – Interpret parts of an expression, such as terms, factors, and coefficients.

A.SSE.1a.1. Ability to make connections between symbolic representations and proper mathematics vocabulary.

A.SSE.1b. Interpret expressions that represent a quantity in terms of its context – 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.

A.SSE.1b.1. Ability to interpret and apply rules for order of operations.

HSA-CED.A. Create equations that describe numbers or relationships.

A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.2.3. Ability to determine unknown parameters needed to create an equation that accurately models a given situation.

A.CED.3. Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

A.CED.3.1. Ability to distinguish between a mathematical solution and a contextual solution.

A.CED.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.

A.CED.4.1. Ability to recognize/create equivalent forms of literal equations.

HSA-REI.B. Solve equations and inequalities in one variable.

A.REI.3. Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters.

A.REI.3.1. Ability to analyze the structure of an equation to determine the sequence of steps that need to be applied to arrive at a solution.

A.REI.3.2. Ability to accurately perform the steps needed to solve a linear equation/inequality.

Unit 2: Linear and Exponential Relationships

HSA-REI.D. Represent and solve equations and inequalities graphically.

A.REI.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.

A.REI.12.1. Ability to explain why a particular shaded region represents the solution of a given linear inequality or system of linear inequalities.

A.REI.12.2. Ability to convey the mathematics behind the dotted versus solid boundary lines used when graphing the solutions to linear inequalities.

HSF-IF.A. Understand the concept of a function and use function notation.

F.IF.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).

F.IF.1.1. Ability to determine if a relation is a function.

F.IF.1.3. Ability to use of function notation.

F.IF.1.4. Knowledge of and ability to apply the vertical line test.

F.IF.2. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context.

F.IF.2.1. Ability to make connections between context and algebraic representations which use function notation.

HSF-IF.B. Interpret functions that arise in applications in terms of a context.

F.IF.4. For a function that models a relationship between two quantities, interpret key features of the graph and the table in terms of the quantities, and sketch the graph 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.

F.IF.4.1. Ability to translate from algebraic representations to graphic or numeric representations and identify key features using the various representations.

F.IF.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.

F.IF.6.1. Knowledge that the rate of change of a function can be positive, negative or zero.

F.IF.6.3. Ability to identify the rate of change from multiple representations.

HSF-IF.C. Analyze functions using different representations.

F.IF.7a. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases – Graph linear and quadratic functions and show intercepts, maxima, and minima.

F.IF.7a.1. See the skills and knowledge that are stated in the Standard.

F.IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.IF.9.2. Ability to recognize common attributes of a function from various representations.

HSF-LE.A. Construct and compare linear, quadratic, and exponential models and solve problems.

F.LE.1a. Distinguish between situations that can be modeled with linear functions and with exponential functions – Prove that linear functions grow by equal differences over equal intervals; and that exponential functions grow by equal factors over equal intervals.

F.LE.1a.1. See the skills and knowledge that are stated in the Standard.

HSF-LE.B. Interpret expressions for functions in terms of the situation they model.

F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.

F.LE.5.1. Ability to interpret the slope and y-intercept of a linear model in terms of context.

Unit 3: Descriptive Statistics

HSS-ID.A. Summarize, represent, and interpret data on a single count or measurement variable.

S.ID.1. Represent data with plots on the real number line (dot plots, histograms, and box plots).

S.ID.1.1. Ability to determine the best data representation to use for a given situation.

S.ID.1.2. Knowledge of key features of each plot.

S.ID.1.4. Ability to analyze data given in different formats.

S.ID.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.

S.ID.2.1. Ability to interpret measures of center and spread (variability) as they relate to several data sets.

S.ID.3. Interpret differences in shape, center, and spread in the context of the data sets, accounting for possible effects of extreme data points (outliers).

S.ID.3.1. Ability to recognize gaps, clusters, and trends in the data set.

HSS-ID.B. Summarize, represent, and interpret data on two categorical and quantitative variables.

S.ID.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.

S.ID.5.2. Ability to read and use a two-way frequency table.

S.ID.5.4. Ability to read a segmented bar graph.

S.ID.6a. Represent data on two quantitative variables on a scatter-plot, and describe how the variables are related – Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

S.ID.6a.1. Ability to recognize types of relationships that lend themselves to linear and exponential models.

S.ID.6a.2. Ability to create and use regression models to represent a contextual situation.

S.ID.6b. Represent data on two quantitative variables on a scatter-plot, and describe how the variables are related – Informally assess the fit of a function by plotting and analyzing residuals.

S.ID.6b.1. Ability to create a graphic display of residuals.

S.ID.6b.2. Ability to recognize patterns in residual plots.

S.ID.6b.3. Ability to calculate error margins (residuals) with a calculator.

S.ID.6b.4. Ability to analyze the meaning of patterns in residual plots.

S.ID.6c. Represent data on two quantitative variables on a scatter-plot, and describe how the variables are related – Fit a linear function for a scatter plot that suggests a linear association.

S.ID.6c.1. Ability to recognize a linear relationship displayed in a scatter plot.

HSS-ID.C. Interpret linear models.

S.ID.7. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the data.

S.ID.7.1. See the skills and knowledge that are stated in the Standard.

Unit 4: Expressions and Equations

HSN-RN.B. Use properties of rational and irrational Numbers.

N.RN.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.

N.RN.3.1. Ability to perform operations on both rational and irrational numbers.

HSA-SSE.B. Write expressions in equivalent forms to solve problems.

A.SSE.3a. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression – Factor a quadratic expression to reveal the zeros of the function it defines.

A.SSE.3a.1. Ability to connect the factors, zeros and x-intercepts of a graph.

A.SSE.3a.2. Ability to connect the factors, zeros and x-intercepts of a graph.

A.SSE.3c. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression – Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^t can be rewritten as (1.15^1/12)12^t ≈ 1.01212t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

A.SSE.3c.1. Ability to connect experience with properties of exponents from Unit 2 of this course to more complex expressions.

HSA-APR.A. Perform arithmetic operations on polynomials.

A.APR.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.

A.APR.1.1. Ability to show that when polynomials are added, subtracted or multiplied that the result is another polynomial.

HSA-CED.A. Create equations that describe numbers or relationships.

A.CED.2. Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

A.CED.2.3. Ability to determine unknown parameters needed to create an equation that accurately models a given situation.

A.CED.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.

A.CED.4.2. Ability to recognize and create different forms of literal equations.

HSA-REI.B. Solve equations and inequalities in one variable.

A.REI.4a. Solve quadratic equations in one variable – Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x–p)^2 = q that has the same solutions. Derive the quadratic formula from this form.

A.REI.4a.1. Ability to solve literal equations for a variable of interest.

Unit 5: Quadratic Functions and Modeling

HSF-IF.B. Interpret functions that arise in applications in terms of a context.

F.IF.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.

F.IF.4.2. Ability to connect experiences with linear and exponential functions from Unit 2 of this course to quadratic, square root, cube root, absolute value, step and piecewise defined models.

F.IF.4.3. Ability to connect appropriate function to context.

F.IF.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.

F.IF.5.3. Ability to connect experiences with linear and exponential functions from Unit 2 of this course to quadratic, square root, cube root, absolute value, step and piecewise defined models.

F.IF.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.

F.IF.6.2. Knowledge that the rate of change of a function can be positive, negative, zero or can have no change.

F.IF.6.3. Ability to identify the rate of change from multiple representations.

HSF-IF.C. Analyze functions using different representations.

F.IF.9. Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).

F.IF.9.2. Ability to recognize common attributes of a function from multiple representations.

MD.MA.AII. Algebra II

Unit 1: Polynomial, Rational, and Radical Relationships

HSA-SSE.B. Write expressions in equivalent forms to solve problems.

A.SSE.3c. Choose and produce an equivalent form of an expression to reveal and explain properties of the quantity represented by the expression – Use the properties of exponents to transform expressions for exponential functions. For example, the expression 1.15^t can be rewritten as (1.15^1/12)^12t to reveal the approximate equivalent monthly interest rate if the annual rate is 15%.

A.SSE.3c.2. Ability to connect experience with properties of exponents from Unit 4 of Algebra I to more complex expressions

HSF-IF.C. Analyze functions using different representations.

F.IF.7c. Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases – Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.

F.IF.7c.2. Ability to identify key features of a function: max, min, intercepts, zeros, and end behaviors.

HSG-GPE.A. Translate between the geometric description and the equation for a conic section.

G.GPE.2. Derive the equation of a parabola given a focus and directrix.

G.GPE.2.2. Ability to connect the algebraic and geometric definitions of a parabola.

Unit 2: Trigonometric Functions

HSS-ID.B. Summarize, represent, and interpret data on two categorical and quantitative variables.

S.ID.6a. Represent data on two quantitative variables on a scatter-plot, and describe how the variables are related – Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear and exponential models.

S.ID.6a.1. Ability to recognize types of relationships that lend themselves to linear and exponential models.

S.ID.6a.2. Ability to create and use regression models to represent a contextual situation.

Unit 3: Modeling with Functions

HSF-IF.B. Interpret functions that arise in applications in terms of a context.

F.IF.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.

F.IF.4.3. Ability to connect appropriate function to context.

F.IF.4.4. Knowledge of the key features of linear, exponential, polynomial, root, absolute value, piece-wise, simple rational, logarithmic and trigonometric functions.

F.IF.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.

F.IF.6.4. Ability to apply this skill to linear, quadratic, polynomial, root and simple rational functions.

HSF-IF.C. Analyze functions using different representations.

F.IF.8. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function.

F.IF.8.1. Ability to connect experience with writing linear, quadratic and exponential functions in various forms from Algebra I to writing all functions in various forms.

F.IF.8b. Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function – 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.

F.IF.8b.1. Ability to connect experience with properties of exponents from Algebra I Unit 2: Linear and Exponential Relationships to more complex expressions.

HSF-BF.A. Build a function that models a relationship between two quantities.

F.BF.1a. Write a function that describes a relationship between two quantities – Determine an explicit expression, a recursive process, or steps for calculation from a context.

F.BF.1a.1. Ability to connect experience with linear and exponential functions from Algebra I Unit 2 to quadratic functions.

HSF-LE.B. Interpret expressions for functions in terms of the situation they model.

F.LE.5. Interpret the parameters in a linear or exponential function in terms of a context.

F.LE.5.1. Ability to interpret the slope and y-intercept of a linear model in terms of context.

F.LE.5.2. Ability to identify the initial amount present in an exponential model (f(0) = b^0+k = 1+k).

F.LE.5.3. Ability to interpret the rate of increase/decrease in an exponential model.

Unit 4: Inferences and Conclusions from Data

HSS-IC.A. Understand and evaluate random processes underlying statistical experiments.

S.IC.1. Understand statistics as a process for making inferences about population parameters based on a random sample from that population.

S.IC.1.1. Knowledge of various sampling methods (e.g., simple random, convenience, stratified…).

S.IC.1.2. Ability to select an appropriate sampling technique for a given situation.

S.IC.1.3. Ability to explain in context the difference between values describing a population and a sample.

S.IC.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?

S.IC.2.1. Ability to calculate and analyze theoretical and experimental probabilities accurately.

S.IC.2.2. Knowledge of various types of sampling procedures and ability to select and carry out the appropriate process for a given situation.

S.IC.2.4. Ability to explain and use the Law of Large Numbers.

HSS-IC.B. Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

S.IC.3. Recognize the purposes of and differences among sample surveys, experiments, and observational studies; explain how randomization relates to each.

S.IC.3.1. Ability to conduct sample surveys, experiments and observational studies.

S.IC.5. Use data from a randomized experiment to compare two treatments; use simulations to decide if differences between parameters are significant.

S.IC.5.1. Ability to set up and conduct a randomized experiment or investigation, collect data and interpret the results.

Unit 5: Applications of Probability

HSS-CP.A. Understand independence and conditional probability and use them to interpret data.

S.CP.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”).

S.CP.1.2. Understanding of and ability to use set notation, key vocabulary and graphic organizers linked to this standard.

S.CP.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.

S.CP.2.3. Ability to determine if two events are dependent or independent.

S.CP.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.

S.CP.5.1. Ability to make connections between statistical concepts and real world situations.

HSS-CP.B. Use the rules of probability to compute probabilities of compound events in a uniform probability model.

S.CP.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.

S.CP.8.1. Ability to analyze a situation to determine the probability of a described event.

S.CP.8.2. Ability to make connections between numeric results and context.

S.CP.9. (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

S.CP.9.1. Ability to use formulas containing factorial notation.

S.CP.9.2. Ability to analyze a situation to determine the probability of a described event.

HSS-MD.A. Use probability to evaluate outcomes of decisions.

S.MD.6. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator).

S.MD.6.3. Knowledge of the Law of Large Numbers.

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

S.MD.7.2. Knowledge of and ability to use a variety of data collection techniques.

S.MD.7.4. Knowledge of the Law of Large Numbers.

MD.MA.G. Geometry

Unit 1: Congruence, Proof, and Constructions

HSG-CO.A. Experiment with transformations in the plane.

G.CO.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.

G.CO.1.1. Ability to use mathematical vocabulary accurately.

G.CO.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.

G.CO.2.1. Ability to see parallels between function transformations (F.BF.3) and geometric transformations.

G.CO.2.2. Knowledge that rigid transformations preserve the size and shape of a figure.

G.CO.3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

G.CO.3.1. Ability to use appropriate vocabulary to describe rotations and reflections.

G.CO.3.2. Ability to use the characteristics of a figure to determine and then describe what happens to the figure as it is rotated (such as axis of symmetry, congruent angles or sides…).

G.CO.4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

G.CO.4.1. Ability to construct a definition for each term based upon a synthesis of experiences.

HSG-CO.B. Understand congruence in terms of rigid motions.

G.CO.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.

G.CO.6.1. Ability to recognize the effects of rigid motion on orientation and location of a figure.

G.CO.6.2. Ability to use rigid motions to map one figure onto another.

G.CO.6.3. Ability to use the definition of congruence as a test to see if two figures are congruent.

G.CO.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.

G.CO.7.1. Knowledge of vocabulary corresponding parts and the connection to the given triangles.

G.CO.7.2. Ability to identify the corresponding parts of two triangles.

Unit 2: Similarity, Proof, and Trigonometry

HSG-SRT.A. Understand similarity in terms of similarity transformations.

G.SRT.1a. Verify experimentally the properties of dilations given by a center and a scale factor – A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged.

G.SRT.1a.1. Ability to connect experiences with dilations and orientation to experiences with lines.

G.SRT.1b. Verify experimentally the properties of dilations given by a center and a scale factor – The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

G.SRT.1b.1. Ability to develop a hypothesis based on observations.

G.SRT.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.

G.SRT.2.1. Ability to make connections between the definition of similarity and the attributes of two given figures.

G.SRT.2.2. Ability to set up and use appropriate ratios and proportions.

HSG-SRT.C. Define trigonometric ratios and solve problems involving right triangles.

G.SRT.8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

G.SRT.8.1. See the skills and knowledge that are stated in the Standard.

HSG-MG.A. Apply geometric concepts in modeling situations.

G.MG.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).

G.MG.1.1. See the skills and knowledge that are stated in the Standard.

G.MG.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).

G.MG.3.1. See the skills and knowledge that are stated in the Standard.

Unit 3: Extending to Three Dimensions

HSG-GMD.A. Explain volume formulas and use them to solve problems.

G.GMD.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.

G.GMD.1.1. See the skills and knowledge that are stated in the Standard.

G.GMD.3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.

G.GMD.3.1. See the skills and knowledge that are stated in the Standard.

Unit 4: Connecting Algebra and Geometry Through Coordinates

HSG-GPE.B. Use coordinates to prove simple geometric theorems algebraically.

G.GPE.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.

G.GPE.4.1. Ability to use distance, slope and midpoint formulas…

G.GPE.6. Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

G.GPE.6.1. Ability to use the slope formula.