# Courses of Study : Mathematics

Number and Quantity
Together, irrational numbers and rational numbers complete the real number system, representing all points on the number line, while there exist numbers beyond the real numbers called complex numbers.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
1. Extend understanding of irrational and rational numbers by rewriting expressions involving radicals, including addition, subtraction, multiplication, and division, in order to recognize geometric patterns.
Unpacked Content Evidence Of Student Attainment:
Students:
• Rewrite given expressions involving radicals.
• Use the operations of addition, subtraction, multiplication, and division, with radicals within expressions.
Teacher Vocabulary:
• Rational numbers
• Irrational numbers
• Geometric Patterns
Knowledge:
Students know:
• Order of operations, Algebraic properties, Number sense.
• Computation with whole numbers and integers.
• Rational and irrational numbers.
• Measuring length and finding perimeter and area of rectangles and squares.
• Volume and capacity.
• Pythagorean theorem.
Skills:
Students are able to:
• Use the operations of addition, subtraction, division, and multiplication, with radicals.
• Demonstrate an understanding of radicals as they apply to problems involving squares, perfect squares, and square roots (e.g., the Pythagorean Theorem, circle geometry, volume, and area).
Understanding:
Students understand that:
• rewriting radical expressions of rational and irrational numbers can help in recognizing geometric patterns.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.1.1: Define rational and irrational numbers and radicals.
GEO.1.2: Identify the product of a nonzero rational number and an irrational number as irrational.
GEO.1.3: Identify the sum of a rational number and an irrational number is irrational.
GEO.1.4: Discuss why the product of two rational numbers is rational.
GEO.1.5: Describe the properties of addition and multiplication rational and irrational numbers and radicals.
GEO.1.6: Apply properties of fractions to add, subtract, multiply, and divide rational numbers.

Prior Knowledge Skills:
• Define rational number.
• Define rational numbers, horizontal, and vertical.
• Recall how to extend a horizontal number line.
• Recall how to extend a vertical number line.
• Demonstrate addition and subtraction of whole numbers using a horizontal or vertical number line.
• Give examples of rational numbers.
• Define absolute value and additive inverse.
• Explain that the sum of a number and it's opposite is zero.
• Locate positive, negative, and zero numbers on a number line.
• Recall properties of addition and subtraction.
• Model addition and subtraction using manipulatives.
• Show addition and subtraction of 2 or more rational numbers using a number line within real world context.
• Define absolute value and additive inverse.
• Show subtraction as the additive inverse.
• Give examples of the opposite of a given number.
• Show addition and subtraction using a number line.
• Discuss various strategies for solving real-world and mathematical problems.
• Identify properties of operations for addition and subtraction.
• Recall the steps for solving addition and subtraction of rational numbers.
• Identify the difference between two rational numbers on a number line.
• Recall the steps for solving multiplication and division of fraction problems.
• Recall the steps for solving multiplication and division of whole number problems.
• Define distributive property, rational numbers, product.
• Solve problems using the distributive property.
• Recall basic multiplication facts using manipulatives.
• Identify the properties of operations for multiplication.
• Define quotient, divisor, and integer.
• Recall the rules for multiplying integers.
• Solve real-world problems.
• Recall the steps of division.
• Discuss various strategies for solving real-world and mathematical problems.
• Identify properties of operations for multiplication.
• Define terminating decimals.
• Give examples of equivalent fractions and decimals.
• Recall the steps for dividing decimals.
• Recall the steps of division.
• Discuss various strategies for solving real-world and mathematical problems.
• Recall steps for solving fractional problems.
• Identify properties of operations for addition and multiplication.
• Recall the rules for multiplication and division of rational numbers.
• Recall the rules for addition and subtraction of rational numbers.Analyze the given word problem to set up a mathematical problem.
• Recognize the mathematical operations of rational numbers in any form, including converting between forms. (Ex. 0.25=1/4 =25%)
• Recognize the rules of operations of positive and negative numbers.
• Recognize properties of numbers (Distributive, Associative, Commutative).
• Recall problem solving methods.
• Define expanding decimals, terminating decimals, rational number, and irrational number.
• Identify and give examples of rational numbers.
• Demonstrate how to convert fractions to decimals.
• Recall steps for division of fractions.
• Recognize rational and irrational numbers.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.9.1 Solve real-world problems involving addition and/or subtraction of rational numbers (whole numbers of decimals) using models when needed.

Quantitative reasoning includes and mathematical modeling requires attention to units of measurement.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
2. Use units as a way to understand problems and to guide the solution of multi-step problems.

a. Choose and interpret units consistently in formulas.

b. Choose and interpret the scale and the origin in graphs and data displays.

c. Define appropriate quantities for the purpose of descriptive modeling.

d. Choose a level of accuracy appropriate to limitations of measurements when reporting quantities.
Unpacked Content Evidence Of Student Attainment:
Students:
• Interpret and make sense of problems through analyzing units for agreement using dimensional analysis (e.g., knowing that there are 5,280 feet in a mile we might change 20 miles/hour to inches per second by 20 miles/hour x 5280 feet/mile x 12 inches/foot x 1 hour/60 minutes x 1 minute/60 seconds to yield just over 352 inches per second).
• Model contextual problem situations with appropriately chosen units or derived units, analyze the data using those units, and interpret the solution (e.g., problems involving per capita income, person hours, heat degree days, or currency conversions).
• Interpret and evaluate, with and without appropriate technology, graphical and tabular data displays for consistency with the data and precisely determine and interpret a scale and origin that is useful in examining the problem of interest.
• Choose appropriate quantities for descriptively modeling important features of the phenomenon being investigated (e.g., find a good measure of overall highway safety: propose and debate such measures as fatalities per year, fatalities per driver per year, or fatalities per vehicle mile driven).
• Given contextual situations involving measurements, report direct measurements and measurements gained by combining direct measurements to levels of accuracy allowed by the units on the quantities and will not report combined or converted results with accuracy beyond that of the original measurements (e.g., if one side of a rectangle is measured to the nearest meter, and the other side to the nearest centimeter, the perimeter can only be accurate to the nearest meter).
Teacher Vocabulary:
• Units
• Scales
• Descriptive modeling
• Justify
• Interpret
• Identify
• Quantities
• Dimensional analysis
• Formulas
• Scale
• Consistency
• Precise
• Accuracy
• Margin of error
• Perimeter
• Volume
• Area
• Direct measurement
Knowledge:
Students know:
• Techniques for dimensional analysis,
• Uses of technology in producing graphs of data.
• Criteria for selecting different displays for data (e.g., knowing how to select the window on a graphing calculator to be able to see the important parts of the graph.
• Descriptive models .
• Attributes of measurements including precision and accuracy and techniques for determining each.
Skills:
Students are able to:
• Choose the appropriate known conversions to perform dimensional analysis to convert units.
• Correctly use graphing window and other technology features to precisely determine features of interest in a graph.
• Determine when a descriptive model accurately portrays the phenomenon it was chosen to model.
• Justify their selection of model and choice of quantities in the context of the situation modeled and critique the arguments of others concerning the same situation.
• Determine and distinguish the accuracy and precision of measurements.
Understanding:
Students understand that:
• The relationships of units to each other and how using a chain of conversions allows one to reach a desired unit or rate.
• Different models reveal different features of the phenomenon that is being modeled.
• Calculations involving measurements can't produce results more accurate than the least accuracy in the original measurements.
• The margin of error in a measurement, (often expressed as a tolerance limit), varies according to the measurement, tool used, and problem context.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.2.1: Interpret units consistently in formulas.
GEO.2.2: Choose units consistently in formulas.
GEO.2.3: Use units as a way to guide the solution of multi-step problems.
GEO.2.4: Use units as a way to understand problems.
GEO.2.5: Convert between units of measurement within the same system.
GEO.2.6: Choose the scale and the origin in graphs.
GEO.2.7: Interpret the scale and the origin in data displays.
GEO.2.8: Define units of measurement.
GEO.2.9: Identify appropriate units of measure to best describe a real-world application.
GEO.2.10: Recognize the limitations for each type of measurement tool.
GEO.2.11: Determine the level of precision needed for real-world measurements.
GEO.2.12: Relate how rounding effects the accuracy of the measurement.

Prior Knowledge Skills:
• Convert like measurement units within a given system. (Example: 120 min = 2 hrs).
• Know relative sizes of measurement units within one system of units, including km, m, cm; kg, g; lb, oz; l, ml; and hr, min, sec.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.9.2 Given a real-world scenario, identify the appropriate unit to obtain the most accurate measurement. (Ex: When baking a cake, should you measure 1 cup of sugar with a teaspoon or a measuring cup?)

Algebra and Functions
Focus 1: Algebra
The structure of an equation or inequality (including, but not limited to, one-variable linear and quadratic equations, inequalities, and systems of linear equations in two variables) can be purposefully analyzed (with and without technology) to determine an efficient strategy to find a solution, if one exists, and then to justify the solution.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
3. Find the coordinates of the vertices of a polygon determined by a set of lines, given their equations, by setting their function rules equal and solving, or by using their graphs.
Unpacked Content Evidence Of Student Attainment:
Students:
• Given the equations to a set of lines.
• Use simultaneous linear equations. to produce the coordinates of the vertices of a polygon.
Teacher Vocabulary:
• vertices
• Function rules
• linear equations
• System of equations
Knowledge:
Students know:
• Substitution, Elimination, and Graphing methods to solve simultaneous linear equations.
Skills:
Students are able to:
• Find the coordinates of the vertices of a polygon given a set of lines and their equations by setting their function rules equal and solving or by using their graph.
Understanding:
Students understand that:
• Given the equations to a set of lines you can find the coordinates of the vertices of a polygon by setting their function rules equal and solving or by using their graph.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.3.1: Define systems of equations, constraints, viable solution, and nonviable solution.
GEO.3.2: Determine if a solution to a system of equations or inequalities is viable or nonviable.
GEO.3.3: Create a system of equations or inequalities to represent the given constraints (linear).
GEO.3.4: Create an equation or inequality to represent the given constraints (linear).
GEO.3.5: Determine if there is one solution, infinite solutions, or no solutions to a system of equations or inequalities.

Prior Knowledge Skills:
• Recall how to solve linear equations.
• Demonstrate how to graph solutions to linear equations.
• Recall how to graph ordered pairs on a Cartesian plane.
• Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
• Define simultaneous.
• Recall how to solve linear equations.
• Recall properties of operations for addition and multiplication.
• Discover that the intersection of two lines on a coordinate plane is the solution to both equations.
• Define point of intersection.
• Recall how to solve linear equations.
• Demonstrate how to graph on the Cartesian plane.
• Identify ordered pairs.
• Recall how to solve linear equations in two variables by using substitution.
• Create a word problem from given information.
• Recall how to solve linear equations.
• Explain how to write an equation to solve real-world mathematical problems.
Expressions, equations, and inequalities can be used to analyze and make predictions, both within mathematics and as mathematics is applied in different contexts - in particular, contexts that arise in relation to linear, quadratic, and exponential situations.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
4. Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.

Example: Rearrange the formula for the area of a trapezoid to highlight one of the bases.
Unpacked Content Evidence Of Student Attainment:
Students:
• Rearrange formulas which arise in contextual situations to isolate variables that are of interest for particular problems.
For example, if the electric company charges for power by the formula COST = 0.03 KWH + 15, a consumer may wish to determine how many kilowatt hours they may use to keep the cost under particular amounts, by considering KWH< (COST - 15)/0.03 which would yield to keep the monthly cost under $75, they need to use less than 2000 KWH. Teacher Vocabulary: • Literal equations • Variable • Constant Knowledge: Students know: • Properties of equality and inequality Skills: Students are able to: • Accurately rearrange equations or inequalities to produce equivalent forms for use in resolving situations of interest. Understanding: Students understand that: • The structure of mathematics allows for the procedures used in working with equations to also be valid when rearranging formulas, The isolated variable in a formula is not always the unknown and rearranging the formula allows for sense-making in problem solving. Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.4.1: Accurately rearrange equations and inequalities to produce equivalent forms for use in resolving situations of interest. Prior Knowledge Skills: • Define equivalent, simplify, term, distributive property, associative property of addition and multiplication, and the commutative property of addition and multiplication. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.9.4 Solve one-step equations or inequalities. Focus 2: Connecting Algebra to Functions Graphs can be used to obtain exact or approximate solutions of equations, inequalities, and systems of equations and inequalities?including systems of linear equations in two variables and systems of linear and quadratic equations (given or obtained by using technology).  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 5. Verify that the graph of a linear equation in two variables is the set of all its solutions plotted in the coordinate plane, which forms a line. Unpacked Content Evidence Of Student Attainment: Students: Given an equation in two variables, • Verify that any ordered pair that makes the equation true is a point on the graph. • Show that there are an infinite number of ordered pairs that satisfy the equation. Teacher Vocabulary: • Graphically Finite solutions • Infinite solutions Knowledge: Students know: • Appropriate methods to find ordered pairs that satisfy an equation. • Techniques to graph the collection of ordered pairs to form a line Skills: Students are able to: • Accurately find ordered pairs that satisfy the equation. • Accurately graph the ordered pairs and form a line Understanding: Students understand that: • An equation in two variables has an infinite number of solutions (ordered pairs that make the equation true), and those solutions can be represented by the graph of a Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.5.1: Define ordered pair and coordinate plane. GEO.5.2: Create linear equations with two variables. GEO.5.3: Graph linear equations on coordinate axes with labels and scales. GEO.5.4: Identify an ordered pair and plot it on the coordinate plane. Prior Knowledge Skills: • Define linear functions, nonlinear functions, slope, and y-intercept. • Recall how to solve problems using the distributive property. • Recognize linear equations. • Identify ordered pairs. • Recognize ordered pairs. • Define similar triangles, intercept, slope, vertical, horizontal, and origin. • Recognize similar triangles. • Generate the slope of a line using given ordered pairs. • Analyze the graph to determine the rate of change. • Demonstrate how to plot points on a coordinate plane using ordered pairs from table. • Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept. • Graph a function given the slope-intercept form of an equation. • Recognize that two sets of points with the same slope may have different y-intercepts. • Graph a linear equation given the slope-intercept form of an equation. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.9.5 Interpret the meaning of a point on the graph of a line. (Ex.: On a graph of milkshake purchases, trace the graph to a point and tell the number of milkshakes purchased and the total cost.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 1 Classroom Resources: 1 6. Derive the equation of a circle of given center and radius using the Pythagorean Theorem. a. Given the endpoints of the diameter of a circle, use the midpoint formula to find its center and then use the Pythagorean Theorem to find its equation. b. Derive the distance formula from the Pythagorean Theorem. Unpacked Content Evidence Of Student Attainment: Students: Given the center (h,k) and radius (r) of a circle, • Explain and justify that every point on the circle is a combination of a horizontal and vertical shift from the center with a length equal to the radius. • Create a right triangle from the center of a circle to a general point on the circle, and show that the legs of the right triangle are the absolute values of x-h and y-k, and the hypotenuse is r, then apply Pythagorean theorem to show that r2 = (x - h)2 + (y - k)2. Given the endpoint of the diameter of the circle, • Find the center of the circle using the midpoint formula, and write the equation of the circle in standard form using the Pythagorean Theorem. • Analyze distance in the coordinate plane and use distance to relate points and lines. • Calculate the distance between two points using the Pythagorean Theorem. • Generalize methods for determining the distance between two coordinate points. • Derive the distance formula using a right triangle and the Pythagorean Theorem. Teacher Vocabulary: • Pythagorean theorem • Radius • Translation Knowledge: Students know: • Key features of a circle. • The Pythagorean Theorem, Midpoint Formula, Distance Formula. Skills: Students are able to: • Create a right triangle in a circle using the horizontal and vertical shifts from the center as the legs and the radius of the circle as the hypotenuse. • Write the equation of the circle in standard form when given the endpoints of the diameter of a circle, using the midpoint formula to find the circle's center, and then use the Pythagorean Theorem to find the equation of the circle. • Find the distance between two points when using the Pythagorean Theorem and use that process to create the Distance Formula. Understanding: Students understand that: • Circles represent a fixed distance in all directions in a plane from a given point, and a right triangle may be created to show the relationship of the horizontal and vertical shift to the distance, • Circles written in standard form are useful for recognizing the center and radius of a circle. • The distance formula and Pythagorean Theorem can both be used to find length measurements of segments (or sides of a geometric figure) Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.6.1: Define radius, diameter, midpoint and Pythagorean Theorem. GEO.6.2: Apply the Pythagorean Theorem to find the distance from the center to a point on the circle. GEO.6.3: Derive the equation of a circle given the center and the radius. GEO.6.4: Use the midpoint formula to find the center of a circle based on the endpoints of the diameter. Prior Knowledge Skills: • Identify parts of a circle. • Recall how to find circumference of a circle. • Recall the meaning of a radius and diameter. • Identify all types of angles. • Recognize the attributes of a circle. • Identify and label parts of a circle. • Define diameter, radius, circumference, area of a circle, and formula. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.9.6 Using real-world models (Ex. Pizza or Pie) on a coordinate grid, determine the length of the radius. Data Analysis, Statistics, and Probability Focus 1: Quantitative Literacy Mathematical and statistical reasoning about data can be used to evaluate conclusions and assess risks.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 7. Use mathematical and statistical reasoning with quantitative data, both univariate data (set of values) and bivariate data (set of pairs of values) that suggest a linear association, in order to draw conclusions and assess risk. Example: Estimate the typical age at which a lung cancer patient is diagnosed, and estimate how the typical age differs depending on the number of cigarettes smoked per day. Unpacked Content Evidence Of Student Attainment: Students: • Given both univariate and bivariate data that suggest a linear association. • Use mathematical and statistical reasoning to draw conclusions and asses risk. Teacher Vocabulary: • Mathematical reasoning • Statistical reasoning • Univariate data • bivariate data • quantitative data • linear association • Scatter plots • linear model • Slope • bar graphs, Pie graphs, Histograms • Mean, median, mode • Standard deviation Knowledge: Students know: • Patterns found on scatter plots of bivariate data. • Strategies for determining slope and intercepts of a linear model. • Strategies for informally fitting straight lines to bivariate data with a linear relationship. • Methods for finding the distance between two points on a coordinate plane and between a point and a line. Skills: Students are able to: • Construct a scatter plot to represent a set of bivariate data. • Use mathematical vocabulary to describe and interpret patterns in bivariate data. • Use logical reasoning and appropriate strategies to draw a straight line to fit data that suggest a linear association. • Use mathematical vocabulary, logical reasoning, and closeness of data points to a line to judge the fit of the line to the data. • Find a central value using mean, median and mode. • Find how spread out the univariate data is using range, quartiles and standard deviation. • Make plots like Bar Graphs, Pie Charts and Histograms. Understanding: Students understand that: • Using different representations and descriptors of a data set can be useful in seeing important features of the situation being investigated, • When visual examination of a scatter plot suggests a linear association in the data, fitting a straight line to the data can aid in interpretation and prediction. • Modeling bivariate data with scatter plots and fitting a straight line to the data can aid in interpretation of the data and predictions about unobserved data. • A set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape. • Using different representations and descriptors of a data set can be useful in seeing important features of the situation being investigated. • Statistical measures of center and variability that describe data sets can be used to compare data sets and answer questions. Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.7.1: Define bivariate scatter plot, quantitative data, outlier, cluster, linear, nonlinear, and positive and negative association. GEO.7.2: Describe patterns found in a scatter plot. GEO.7.3: Demonstrate how to label and plot information on a scatter plot (dot plot). GEO.7.4: Distinguish the difference between positive and negative correlation. GEO.7.5: Recall how to describe the spread of the scatter plot (dot plot). Prior Knowledge Skills: • Define bivariate scatter plot, outlier, cluster, linear, nonlinear, and positive and negative association. • Describe patterns found in a scatter plot. • Demonstrate how to label and plot information on a scatter plot (dot plot). • Distinguish the difference between positive and negative correlation. • Recall how to describe the spread of the scatter plot (dot plot). Focus 2: Visualizing and Summarizing Data Data arise from a context and come in two types: quantitative (continuous or discrete) and categorical. Technology can be used to "clean" and organize data, including very large data sets, into a useful and manageable structure - a first step in any analysis of data.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 8. Use technology to organize data, including very large data sets, into a useful and manageable structure. Unpacked Content Evidence Of Student Attainment: Students: • Given quantitative (continuous or discrete) and categorical data. • Use technology to organize data, including a very large set of data into a useful and manageable structure. Teacher Vocabulary: • Continuous data • Discrete data • quantitative • Categorical • line of best fit • Curve of best fit • Scatter plot Knowledge: Students know: • How to use technology to create graphical models of data in scatterplots or frequency distributions. • How to use technology to graph scatter plots given a set of data and estimate the equation of best fit. • How to distinguish between independent and dependent variables. Skills: Students are able to: • recognize patterns, trends, clusters, and gaps in the organized data. Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.8.1: Solve equations for y. GEO.8.2: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations. GEO.8.3: Analyze data from tables. GEO.8.4: Summarize categorical data for two categories in two-way frequency tables. GEO.8.5: Recognize possible associations and trends in the data. GEO.8.6: Create a scatter plot and line of best fit using data from a spreadsheet. GEO.8.7: Organize numerical data in a spreadsheet. GEO.8.8: Create graphical representations from classroom-generated data to model consumer costs. GEO.8.9: Create graphical representations from classroom-generated data to predict future outcomes. GEO.8.10: Create graphical representations from equations to model consumer costs. GEO.8.11: Create graphical representations from equations to predict future outcomes. GEO.8.12: Create graphical representations from tables to model consumer costs. GEO.8.13: Create graphical representations from tables to predict future outcomes. Prior Knowledge Skills: • Demonstrate how to plot points on a Cartesian plane using ordered pairs. • Recall how to complete input/output tables. • Recognize numeric patterns. • Given a function, create a rule. • Define linear equation, coefficient, distributive property.and variable. • Recall how to solve equations for a missing variable. • Recall properties of operation for addition and multiplication. • Solve mulit-step equations. • Identify properties of operations. Distributions of quantitative data (continuous or discrete) in one variable should be described in the context of the data with respect to what is typical (the shape, with appropriate measures of center and variability, including standard deviation) and what is not (outliers), and these characteristics can be used to compare two or more subgroups with respect to a variable.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 9. Represent the distribution of univariate quantitative data with plots on the real number line, choosing a format (dot plot, histogram, or box plot) most appropriate to the data set, and represent the distribution of bivariate quantitative data with a scatter plot. Extend from simple cases by hand to more complex cases involving large data sets using technology. Unpacked Content Evidence Of Student Attainment: Students: Given numerical data in any form (e.g., all real numbers), • Organize and display univariate quantitative data using plots on a real number line, using dot plots, histograms, or box plots that is most appropriate to the given data set. • Organize and display bivariate quantitative data using a scatter plot, and extend from simple cases by hand to more complex cases involving a large data set using technology. Teacher Vocabulary: • Dot plots • Histograms • Box plots • Scatter plots • Univariate data • Bivariate data Knowledge: Students know: • Techniques for constructing dot plots, histograms, scatter plots and box plots from a set of data. Skills: Students are able to: • Choose from among data display (dot plots, histograms, box plots, scatter plots) to convey significant features of data. • Accurately construct dot plots, histograms, and box plots. • Accurately construct scatter plots using technology to organize and analyze the data. Understanding: Students understand that: • Sets of data can be organized and displayed in a variety of ways each of which provides unique perspectives of the data set. • Data displays help in conceptualizing ideas and in solving problems. Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.9.1: Organize and display univariate quantitavite data using plots on a real number line, using dot plots, histograms, or box plots that is most appropriate to the given data set. GEO.9.2: Organize and display bivariate quatitative data using a scatter plot, and extend from simple cases by hand to more complex cases involving a large data set using technology. Prior Knowledge Skills: • Define dot plots, line plot, stem and leaf plots, upper quartile, lower quartile, median, histograms, and box plots. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.9.9 10 After collecting data or with given data, construct a simple graph (line, pie, bar, picture, etc.) or table and interpret the data in terms of range and mode.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 10. Use statistics appropriate to the shape of the data distribution to compare and contrast two or more data sets, utilizing the mean and median for center and the interquartile range and standard deviation for variability. a. Explain how standard deviation develops from mean absolute deviation. b. Calculate the standard deviation for a data set, using technology where appropriate. Unpacked Content Evidence Of Student Attainment: Students: Given two or more different data sets, • Compare the center (median, mean) and the spread (interquartile range, standard deviation) of the data sets to describe differences and similarities of the data sets. • Explain how standard deviation develops from mean absolute deviation. • Calculate the standard deviation for a data set, and use technology where it is appropriate. Teacher Vocabulary: • Center • Median • Mean • Spread • Interquartile range • Standard deviation • Absolute mean deviation Knowledge: Students know: • Techniques to calculate the center and spread of data sets. • Techniques to calculate the mean absolute deviation and standard deviation. • Methods to compare data sets based on measures of center (median, mean) and spread (interquartile range and standard deviation) of the data sets. Skills: Students are able to: • Accurately find the center (median and mean) and spread (interquartile range and standard deviation) of data sets. • -Present viable arguments and critique arguments of others from the comparison of the center and spread of multiple data sets. • Explain their reasoning on how standard deviation develops from the mean absolute deviation. Understanding: Students understand that: • Multiple data sets can be compared by making observations about the center and spread of the data. • The center and spread of multiple data sets are used to justify comparisons of the data. • Both the mean and the median are used to calculate the mean absolute and standard deviations Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.10.1: Accurately find the center (median and mean) and spread (interquartile range and standard deviation) of data sets. GEO.10.2: Present viable arguments and critique arguments of others from the comparison of the center and spread of multiple data sets. GEO.10.3: Reason how standard deviation develops from the mean absolute deviation. Prior Knowledge Skills: • Define measure of variability, distribution, and measure of center. • Compare the measure of center and measure of variability of two distributions. • Relate the measure of variation with the concept of range. • Relate the measure of the center with the concept of mean. • Recall how to calculate measure of center and measure of variability. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.9.9 10 After collecting data or with given data, construct a simple graph (line, pie, bar, picture, etc.) or table and interpret the data in terms of range and mode.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 11. Interpret differences in shape, center, and spread in the context of data sets, accounting for possible effects of extreme data points (outliers) on mean and standard deviation. Unpacked Content Evidence Of Student Attainment: Students: • Given multiple data sets, recognize and explain the differences in shape, center, and spread, including effects of outliers on mean and standard deviation. Teacher Vocabulary: • Outliers • Center • Shape • Spread • Mean • Standard deviation Knowledge: Students know: • Techniques to calculate the center and spread of data sets. • Methods to compare attributes (e.g. shape, median, mean, interquartile range, and standard deviation) of the data sets. • Methods to identify outliers. Skills: Students are able to: • Accurately identify differences in shape, center, and spread when comparing two or more data sets. • Accurately identify outliers for the mean and standard deviation. • Explain, with justification, why there are differences in the shape, center, and spread of data sets. Understanding: Students understand that: • Differences in the shape, center, and spread of data sets can result from various causes, including outliers and clustering. Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.11.1: Identify differences in shape, center, and spread when comparing two or more data sets, GEO.11.2: Identify outliers for the mean and standard deviation. GEO.11.3: Justify why there are differences in the shape, center, and spread of data sets. Prior Knowledge Skills: • Define measure of variability, distribution, and measure of center. • Compare the measure of center and measure of variability of two distributions. • Relate the measure of variation with the concept of range. • Relate the measure of the center with the concept of mean. • Recall how to calculate measure of center and measure of variability. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.9.11 Interpret general trends on a graph. (Limited to increase and decrease.) Scatter plots, including plots over time, can reveal patterns, trends, clusters, and gaps that are useful in analyzing the association between two contextual variables.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 12. Represent data of two quantitative variables on a scatter plot, and describe how the variables are related. a. Find a linear function for a scatter plot that suggests a linear association and informally assess its fit by plotting and analyzing residuals, including the squares of the residuals, in order to improve its fit. b. Use technology to find the least-squares line of best fit for two quantitative variables. Unpacked Content Evidence Of Student Attainment: Students: Given a data set of two quantitative variables, • Create a scatter plot (with and without technology). • Create a linear function which best fits the data. • Compare the graphs of the scatter plot and function to see the fit to the original data. • Fit a linear function to the data if the scatter plot indicates a linear association. • Use technology to find the least-squares line of best fit for two quantitative variable. Teacher Vocabulary: • Quantitative variables • Scatter plot • Residuals Knowledge: Students know: • Techniques for creating a scatter plot, • Techniques for fitting linear functions to data. • Methods for using residuals to judge the closeness of the fit of the linear function to the original data. Skills: Students are able to: • Accurately create a scatter plot of data. • Make reasonable assessments on the fit of the function to the data by examining residuals. • Accurately fit a function to data when there is evidence of a linear association. • Use technology to find the least-squares line of best fit for two quantitative variable. Understanding: Students understand that: • Functions are used to create equations representative of ordered pairs of data. • Residuals may be examined to analyze how well a function fits the data. • When a linear association is suggested, a linear function can be fit to the scatter plot to aid in modeling the relationship. Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.12.1: Create a scatter plot of data. GEO.12.2: Calculate the fit of the function to the data by examining residuals. GEO.12.3: Describe a function to its data when there is evidence of a linear association. GEO.12.4: Use technology to find the least-squares line of best fit for two quantitative varible. Prior Knowledge Skills: • Define bivariate scatter plot, outlier, cluster, linear, nonlinear, and positive and negative association. • Describe patterns found in a scatter plot. • Demonstrate how to label and plot information on a scatter plot (dot plot). • Distinguish the difference between positive and negative correlation. Analyzing the association between two quantitative variables should involve statistical procedures, such as examining (with technology) the sum of squared deviations in fitting a linear model, analyzing residuals for patterns, generating a least-squares regression line and finding a correlation coefficient, and differentiating between correlation and causation.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 13. Compute (using technology) and interpret the correlation coefficient of a linear relationship. Unpacked Content Evidence Of Student Attainment: Students: Given a data set of two quantitative variables, • Use technology to compute and interpret the correlation coefficient of a linear relationship. Teacher Vocabulary: • Interpret • Correlation coefficient • linear relationship Knowledge: Students know: • Techniques for creating a scatter plot using technology. • Techniques for fitting linear functions to data. • Accurately fit a function to data when there is evidence of a linear association. Skills: Students are able to: • use technology to graph different data sets • Use the correlation coefficient to assess the strength and direction of the relationship between two data sets. Understanding: Students understand that: • using technology to graph some data and look at the regression line that technology can generate for a scatter plot. Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.13.1: Define mean, standard deviation, population, sample, and correlation coefficient. GEO.13.2: Calculate the correlation coefficient. Prior Knowledge Skills: • Define measure of variability, distribution, and measure of center. • Analyze the skew of the distributions and recognize the difference in measure of center and variability. • Compare the measure of center and measure of variability of two distributions. • Relate the measure of variation with the concept of range. • Relate the measure of the center with the concept of mean. • Recall how to calculate measure of center and measure of variability. • Discuss how to read and interpret a graph. • Define measure of variability, measure of center, inference and mean absolute deviation. • Recall how to calculate measure of center and measure of variability. • Recall that center is related to measure of center and measure of variability is related to variation. • Compare and contrast the measure of center and measure of variability of two numerical data sets. • Calculate the mean absolute deviation of a data set in context.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 14. Distinguish between correlation and causation. Unpacked Content Evidence Of Student Attainment: Students: • Given situations where two variables are correlated, explain why the correlation does not mean that changes in one variable cause the changes in the other variable. Teacher Vocabulary: • Correlation • Causation Knowledge: Students know: • How to read and analyze scatter plots. • To use scatter plots to look for trends, and to find positive and negative correlations. • The key differences between correlation and causation. Skills: Students are able to: • distinguish between correlation and causation Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.14.1: Define correlation and causation. Prior Knowledge Skills: • Define bivariate scatter plot, outlier, cluster, linear, nonlinear, and positive and negative association. • Describe patterns found in a scatter plot. • Demonstrate how to label and plot information on a scatter plot (dot plot). • Distinguish the difference between positive and negative correlation. • Recall how to describe the spread of the scatter plot (dot plot). Data analysis techniques can be used to develop models of contextual situations and to generate and evaluate possible solutions to real problems involving those contexts.  Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0 15. Evaluate possible solutions to real-life problems by developing linear models of contextual situations and using them to predict unknown values. a. Use the linear model to solve problems in the context of the given data. b. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the given data. Unpacked Content Evidence Of Student Attainment: Students: Given real life problems from a data set of two quantitative variables, • Create a scatter plot (with and without technology). • Create a function which best fits the data of linear models of contextual situations and using them to predict unknown values. • Compare the graphs of the scatter plot and function to see the fit to the original data. • Fit a linear function to the data if the scatter plot indicates a linear association. Given a contextual situation that yields a data set of ordered pairs that suggests a linear relationship, • Fit a linear function to the data. • Determine the slope and intercept of that function. • Interpret the slope and intercept of the linear function in the context of the data. Teacher Vocabulary: • Quantitative variables • Scatter plot • Residuals • Slope • Rate of change • Intercepts • Constant • Ordered pairs • Horizontal lines • Vertical lines Knowledge: Students know: • Techniques for creating a scatter plot. • Techniques for fitting a linear function to a scatter plot. • Methods to find the slope and intercept of a linear function. • Techniques for fitting various functions (linear, quadratic, exponential) to data. • Methods for using residuals to judge the closeness of the fit of the function to the original data. Skills: Students are able to: • Accurately create a scatter plot of data. • Correctly choose a function to fit the scatter plot. • Make reasonable assessments on the fit of the function to the data by examining residuals. • Accurately fit a linear function to data when there is evidence of a linear association. • Accurately fit linear functions to scatter plots. • Correctly find the slope and intercept of linear functions. • Justify and explain the relevant connections slope and intercept of the linear function to the data. Understanding: Students understand that: • Functions are used to create equations representative of ordered pairs of data. • Residuals may be examined to analyze how well a function fits the data. • When a linear association is suggested, a linear function can be fit to the scatter plot to aid in modeling the relationship. • Linear functions are used to model data that have a relationship that closely resembles a linear relationship. • The slope and intercept of a linear function may be interpreted as the rate of change and the zero point (starting point). Diverse Learning Needs: Essential Skills: Learning Objectives: GEO.15.1: Define slope as a rate of change. GEO. 15.2: Understand that the y-intercept is the initial amount in the context of the data. GEO.15.3: Understand that rate of change in the context of the data is the label of the y-axis divided by the label of the x-axis. GEO.15.4: Demonstrate use of a graphing calculator, including using a table, making a graph, and finding successive approximations. GEO.15.5: Given a contextual situation, interpret and defend the solution in the context of the original problem. Prior Knowledge Skills: • Demonstrate how to plot points on a coordinate plane using ordered pairs from table. • Analyze the graph to determine the rate of change. • Generate the slope of a line using given ordered pairs. • Draw and label a coordinate plane. Alabama Alternate Achievement Standards AAS Standard: M.G.AAS.9.15 When given a real-world scenario, choose the independent or dependent variable. Ex.: If I buy 5 books that cost$8.00 each, the total cost is \$40. Which variable is independent?

Geometry and Measurement
Focus 1: Measurement
Areas and volumes of figures can be computed by determining how the figure might be obtained from simpler figures by dissection and recombination.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 3 Classroom Resources: 3
16. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.
Unpacked Content Evidence Of Student Attainment:
Students:

Given a circle,
• Use repeated reasoning from multiple examples of the ratio of circle circumference to the diameter, to informally conjecture that the circumference of any circle is a little more than three times the diameter.
• Divide the circle into an equal number of sectors, and rearrange the sectors to form a shape that is approaching a parallelogram.
• Make conjectures about the area and perimeter of the new shape as the number of sectors becomes larger, and relate those conjectures to the original circle.

• Given a cylinder, explain how a cylinder could be divided into an infinite number of circles, and the area of those circles multiplied by the height is the volume of the cylinder, and use Cavalieri's Principle to demonstrate that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
• Given a pyramid or cone, explain that the shapes could be divided into cross-sections, and the area of the cross-sections is decreasing as the cross-sections become further away from the base, and the area of an infinite number of cross-sections is the volume of a pyramid or cone.
Teacher Vocabulary:
• Dissection arguments
• Cavalieri's Principle
• Cylinder
• Pyramid
• Cone
• Ratio
• Circumference
• Parallelogram
• Limits
• Conjecture
• Cross-section
Knowledge:
Students know:
• Techniques to find the area and perimeter of parallelograms.
• Techniques to find the area of circles or polygons.
Skills:
Students are able to:
• Accurately decompose circles, cylinders, pyramids, and cones into other geometric shapes.
• Explain and justify how the formulas for circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone may be created from the use of other geometric shapes.
Understanding:
Students understand that:
• Geometric shapes may be decomposed into other shapes which may be useful in creating formulas.
• Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.16.1: Define two-dimensional objects and three-dimensional objects.
GEO.16.2: Identify the two-dimensional figures that result from slicing three-dimensional figures as in plane section of right rectangular prisms and right rectangular pyramids.
GEO.16.3: Identify three-dimensional objects generated by rotations of two-dimensional objects (as in rotating a circle to create a sphere).
GEO.16.4: Distinguish between two-dimensional and three-dimensional objects.

Prior Knowledge Skills:
• Define three-dimensional figures and nets.
• Identify three-dimensional figures.
• Select and create a three-dimensional figure using manipulatives.
• Define two-dimensional figure, three-dimensional figure, and plane section.
• List attributes of three-dimensional figures.
• List attributes of two-dimensional figures.
• Describe the relationship between two- and three-dimensional figures.
• Recognize symmetry.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.16 Given a cross section of a three-dimensional object, identify the shapes of two-dimensional cross sections (limited to sphere, rectangular prism, or triangular prism).

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 8 Classroom Resources: 8
17. Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.

a. Give an informal argument for the formulas for the surface area and volume of a sphere, cylinder, pyramid, and cone using dissection arguments, Cavalieri's Principle, and informal limit arguments.

b. Apply geometric concepts to find missing dimensions to solve surface area or volume problems.
Unpacked Content Evidence Of Student Attainment:
Students:

(17a) Given a sphere,
• Explain how surface area is the total area for the surface of a sphere, and that if we could "unroll" the sphere and show it as a rectangle, the rectangle would have a width that is equivalent to the diameter of the sphere. Its length would be the same as the circumference of the sphere.
• Explain how we could find the volume of spheres by using pyramids., understanding the radius of the sphere would be the height of the pyramid.

• Given a cylinder, explain how a cylinder could be divided into an infinite number of circles, and the area of those circles multiplied by the height is the volume of the cylinder, and use Cavalieri's Principle to demonstrate that if two solids have the same height and the same cross-sectional area at every level, then they have the same volume.
• Given a pyramid or cone, explain that the shapes could be divided into cross-sections, and the area of the cross-sections is decreasing as the cross-sections become further away from the base, and the area of an infinite number of cross-sections is the volume of a pyramid or cone.
• (17b) Given a formula, explain how to solve for the missing linear dimension using opposite operations.
Teacher Vocabulary:
• Dissection arguments
Principle
• Cylinder
• Pyramid
• Cone
• Ratio
• Circumference
• Parallelogram
• Limits
• Conjecture
• Cross-section
• Surface Area
• Knowledge:
Students know:
• Techniques to find the area and perimeter of parallelograms, Techniques to find the area of circles or polygons
Skills:
Students are able to:
• Accurately decompose circles, spheres, cylinders, pyramids, and cones into other geometric shapes.
• Explain and justify how the formulas for surface area, and volume of a sphere, cylinder, pyramid, and cone may be created from the use of other geometric shapes.
Understanding:
Students understand that:
• Geometric shapes may be decomposed into other shapes which may be useful in creating formulas.
• Geometric shapes may be divided into an infinite number of smaller geometric shapes, and the combination of those shapes maintain the area and volume of the original shape.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.17.1: Define Cavalieri's principle, circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone, oblique, radius, diameter, and height, base.
GEO.17.2: Compare surface areas of similar figures and volumes of similar figures to determine a relationship using dissection arguments, Cavalieri's principle, and informal limit arguments.
GEO.17.3: Compare the characteristics and volume of oblique and right solids.
GEO.17.4: Describe the properties of a given object (cylinder, pyramid, prism, and cone).
GEO.17.5: Identify the necessary characteristics of a given solid to solve for its volume and surface area(cylinder, pyramid, prism, and cone).
GEO.17.6: Calculate the surface area of three-dimensional figures (cylinder, pyramid, prism, and cone).
GEO.17.7: Calculate the volume of a cylinder, pyramid, prism, and cone.
GEO.17.8: Calculate the area of a circle.
GEO.17.9: Calculate the circumference of a circle.
GEO.17.10: Calculate the area of the base shape.
GEO.17.11: Identify the relationship of geometric representations to real-life objects.
GEO.17.12: Identify the base shape.

Prior Knowledge Skills:
• Define three-dimensional figures, surface area, and nets.
• Identify three-dimensional figures.
• Evaluate how to apply using surface area of a three-dimensional figure to solving real-world and mathematical problems.
• Draw nets to find the surface area of a given three-dimensional figure.
• Recall how to calculate the area of a rectangle and triangle.
• Select and create a three-dimensional figure using manipulatives.
• Define diameter, radius, circumference, area of a circle, and formula.
• Identify and label parts of a circle.
• Recognize the attributes of a circle.
• Apply the formula of area and circumference to real world mathematical situations.
• Define formula, volume, cone, cylinders, spheres, and height.
• Discuss the measure of volume and give examples.
• Solve problems with exponents.
• Recall how to find circumference of a circle.
• Identify parts of a circle.
• Calculate the volume of three-dimensional figures.
• Solve real-world problems using the volume formulas for three-dimensional figures.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.17 Compare and contrast the volume of real-world geometric figures.

Constructing approximations of measurements with different tools, including technology, can support an understanding of measurement.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 5 Learning Activities: 3 Classroom Resources: 2
18. Given the coordinates of the vertices of a polygon, compute its perimeter and area using a variety of methods, including the distance formula and dynamic geometry software, and evaluate the accuracy of the results.
Unpacked Content Evidence Of Student Attainment:
Students:
Given the coordinates of the vertices of a polygon,
• Use the distance formula to find the measure of the sides to aide in computing the perimeter and area using a variety of methods.

Methods may include:
• Using the distance formula to find the length of each side.
• Using slopes of perpendicular lines to determine when a side of a triangle or quadrilateral may be used as the height.
• Using a system of equations and the distance formula to find the height of a non right triangle.
• Using geometry software such as geogebra.org to find the perimeter and area of polygons to aide in the accuracy of results.
Teacher Vocabulary:
• Coordinates
• vertices
• perimeter
• Area
• Distance formula
• Evaluate
• Accuracy
Knowledge:
Students know:
• The distance formula and its applications.
• Techniques for coordinate graphing.
• Techniques for using geometric software for coordinate graphing and to find the perimeter and area.
Skills:
Students are able to:
• Create geometric figures on a coordinate system from a contextual situation.
• Accurately find the perimeter of polygons and the area of polygons such as triangles and rectangles from the coordinates of the shapes.
• Explain and justify solutions in the original context of the situation.
Understanding:
Students understand that:
• Contextual situations may be modeled in a Cartesian coordinate system.
• Coordinate modeling is frequently useful to visualize a situation and to aid in solving contextual problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.18.1: Define area, perimeter, regular polygons, inscribed polygons, circumscribed polygons, and vertices.
GEO.18.2: Analyze the given information to develop a logical process to calculate area or perimeter.
GEO.18.3: Create equations for area and perimeter based on given information.
GEO.18.4: Illustrate graphically an inscribed or circumscribed polygon.
GEO.18.5: Solve equations given the area and perimeter.
GEO.18.6: Plot given coordinates on the Cartesian plane.

Prior Knowledge Skills:
• Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
• Demonstrate an understanding of an extended coordinate plane.
• Draw and label a 4 quadrant coordinate plane.
• Draw and extend vertical and horizontal number lines.
• Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
• Recall how to graph points in all four quadrants of the coordinate plane.
• Define area.
• Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
• Apply area formulas to solve real-world mathematical problems.
• Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
• Explain how to find the area for rectangles.
• Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
• Recognize and demonstrate that two right triangles make a rectangle.
• Define vertices.
• Apply absolute value to find the length of a side joining points with the same first coordinate or the same second coordinate.
• Plot points on a coordinate plane., then connect points for the vertices to sketch a polygon.
• Identify ordered pairs.
• Recognize polygons.
• Define perimeter and area.
• Identify the length between vertices on a coordinate plane.
• Calculate the perimeter and area using the distance between the vertices.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.18 Find the perimeter or area of a square, rectangle, or equilateral triangle to solve real-world problems when given the length of at least one side.

When an object is the image of a known object under a similarity transformation, a length, area, or volume on the image can be computed by using proportional relationships.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
19. Derive and apply the relationships between the lengths, perimeters, areas, and volumes of similar figures in relation to their scale factor.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a pair of similar polygons, -Find the ratio of the perimeters, the ratio of the areas, and the ratio of the volumes of similar figures from the scale factor. -Use the ratios to find the missing perimeters, areas and volume .
Teacher Vocabulary:
• Derive
• Apply
• Scale Factor
• Similar figures
• Ratio of length
• Ratio of perimeter
• Ratio of area
• Ratio of volume
Knowledge:
Students know:
• Scale factors of similar figures.
• The ratio of lengths, perimeter, areas, and volumes of similar figures.
• Similar figures.
Skills:
Students are able to:
• Find the scale factor of any given set of similar figures.
• Find the ratios of perimeter, area, and volume
Understanding:
Students understand that:
• Just as their corresponding sides are in the same proportion, perimeters and areas of similar polygons have a special relationship. Perimeters: The ratio of the perimeters is the same as the scale factor. If the scale factor of the sides of two similar polygons is m/n, then the ratio of the areas is (m/n)2
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.19.1: Define scale factor, similarity and proportions.
GEO.19.2: Compare two figures in terms of similarity.
GEO.19.3: Create proportional equations from given information.
GEO.19.4: Solve proportional equations.
GEO.19.5: Prove that equivalent ratios are proportions.

Prior Knowledge Skills:
• Define unit rate, proportion, and rate.
• Create a ratio or proportion from a given word problem.
• Calculate unit rate by using ratios or proportions.
• Interpret a fraction as division of the numerator by the denominator.
Example: (a/b=a divides;b).
• Write a ratio as a fraction.
• Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
• Create a ratio or proportion from a given word problem, diagram, table, or equation.
• Calculate unit rate or rate by using ratios or proportions.
• Restate real world problems or mathematical problems.
• Construct a graph from a set of ordered pairs given in the table of equivalent ratios.
• Compute the unit rate, unit price, and constant speed.
• Create a proportion or ratio from a given word problem.
• Identify the two units being compared.
• Calculate a proportion for missing information.
• Identify a proportion from given information.
• Solve a proportion using part over whole equals percent over 100.
• Form a ratio.
• Define proportions and proportional relationships.
• Demonstrate how to write ratios as a fraction.
• Solve proportional problems.
• Define proportional and nonproportional.
• Recall that for two relationships to be proportional they must have the same unit rate and pass through the origin on a coordinate plane.
• Apply the rule of proportional relationship to real world context.
• Recall how to solve proportions using cross products.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.18 Find the perimeter or area of a square, rectangle, or equilateral triangle to solve real-world problems when given the length of at least one side.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 1 Classroom Resources: 1
20. Derive and apply the formula for the length of an arc and the formula for the area of a sector.
Unpacked Content Evidence Of Student Attainment:
Students:
• Given an arc intercepted by an angle,
• Use dilations to create arcs intercepted by the same central angle with radii of various sizes (including using dynamic geometry software), and use the ratios of the arc lengths and radii to make conjectures regarding possible relationship between the arc length and the radius.
• Justify the conjecture for the formula for any arc length (i.e., since 2πr is the circumference of the whole circle, a piece of the circle is reduced by the ratio of the arc angle to a full angle (360)).
• Find the ratio of the arc length to the radius of each intercepted arc and use the ratio to name the angle calling this the radian measure of the angle by extending the definition of one radian as the angle which intercepts an arc of the same length as the radius.
• Develop the formula for the area of a sector by interpreting a circle as a complete revolution and a sector as a fractional part of a revolution.
Teacher Vocabulary:
• Similarity
• Constant of proportionality
• Sector
• Arc
• Derive
• Arc length
• Area of sector
• Central angle
• Dilation
Knowledge:
Students know:
• Techniques to use dilations (including using dynamic geometry software) to create circles with arcs intercepted by same central angles.
• Techniques to find arc length.
• Formulas for area and circumference of a circle.
Skills:
Students are able to:
• Reason from progressive examples using dynamic geometry software to form conjectures about relationships among arc length, central angles, and the radius.
• Use logical reasoning to justify (or deny) these conjectures and critique the reasoning presented by others.
• Interpret a sector as a portion of a circle, and use the ratio of the portion to the whole circle to create a formula for the area of a sector.
Understanding:
Students understand that:
• Radians measure the ratio of the arc length to the radius for an intercepted arc.
• The ratio of the area of a sector to the area of a circle is proportional to the ratio of the central angle to a complete revolution.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.20.1: Define arc length, radian measure, and sector.
GEO.20.2: Prove the length of the arc intercepted by an angle is proportional to the radius by similarity.
GEO.20.3: Prove the formula for the area of the sector.
GEO.20.4: Illustrate an arc of a circle by constructing the radii of a circle.

Prior Knowledge Skills:
• Identify parts of a circle.
• Recall the meaning of a radius and diameter.
• Identify all types of angles.
• Recognize the attributes of a circle.
• Identify and label parts of a circle.
• Define diameter, radius, circumference, area of a circle, and formula.
Focus 2: Transformations
Applying geometric transformations to figures provides opportunities for describing the attributes of the figures preserved by the transformation and for describing symmetries by examining when a figure can be mapped onto itself.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 2 Classroom Resources: 2
21. Represent transformations and compositions of transformations in the plane (coordinate and otherwise) using tools such as tracing paper and geometry software.

a. Describe transformations and compositions of transformations as functions that take points in the plane as inputs and give other points as outputs, using informal and formal notation.

b. Compare transformations which preserve distance and angle measure to those that do not.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a variety of transformations (translations, rotations, reflections, and dilations),
• Represent the transformations and compositions of transformations in the plane using a variety of methods (e.g., technology, transparencies, semi-transparent mirrors (MIRAs), patty paper, compass).
• Describe transformations and compositions of transformations functions that take points in the plane as inputs and give other points as outputs, explain why this satisfies the definition of a function, and adapt function notation to that of a mapping [e.g., f(x,y) → f(x+a, y+b)].
• Compare transformations that preserve distance and angle to those that do not.
Teacher Vocabulary:
• Transformation
• Reflection
• Translation
• Rotation
• Dilation
• Isometry
• Composition
• Horizontal stretch
• Vertical stretch
• Horizontal shrink
• Vertical shrink
• Clockwise
• Counterclockwise
• Symmetry
• Preimage
• Image
Knowledge:
Students know:
• Characteristics of transformations (translations, rotations, reflections, and dilations).
• Methods for representing transformations.
• Characteristics of functions.
• Conventions of functions with mapping notation.
Skills:
Students are able to:
• Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology.
• Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane, including when the transformation preserves distance and angle.
• Use the language and notation of functions as mappings to describe transformations.
Understanding:
Students understand that:
• Mapping one point to another through a series of transformations can be recorded as a function.
• Some transformations (translations, rotations, and reflections) preserve distance and angle measure, and the image is then congruent to the pre-image, while dilations preserve angle but not distance, and the pre-image is similar to the image.
• Distortions, such as only a horizontal stretch, preserve neither.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.21.1: Define distance, angle, input, output, plane, translation, reflection, rotation, and dilation.
GEO.21.2: Compare transformation that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).
GEO.21.3: Describe transformations as functions that take points in a plane as inputs and give other points as outputs.
GEO.21.4: Represent transformation in the plane.
GEO.21.5: Generate an input output table.
GEO.21.6: Compare the distance and angles of the figures from the pre-image to the image.
GEO.21.7: Measure distance and angle(s) of an image.

Prior Knowledge Skills:
• Define rotation, reflection, and translation.
• Recognize translations (slides), rotations (turns), and reflections (flips).
• Distinguish between lines and line segments.
• Demonstrate how to measure length.
• Demonstrate how to use a protractor to measure angles.
• Identify parallel lines.
• Define square root, cube root, inverse, perfect square, perfect cube, and irrational number.
• Define square root, expressions, and approximations.
• Identify perfect squares and square roots.
• Demonstrate how to locate points on a vertical or horizontal number line.
• Define ordered pairs.
• Show how to plot points on a Cartesian plane.
• Locate the origin on the coordinate plane.
• Identify the length between vertices on a coordinate plane.
• Recall how to read a graph or table.
• Draw and label a coordinate plane.
• Plot independent (input) and dependent (output) values on a coordinate plane.
• Plot pairs of integers and/or rational numbers on a coordinate plane.
• Arrange integers and/or rational numbers on a horizontal or vertical number line.
• Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
• Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
• Calculate the distances between points having the same first or second coordinate using absolute value.
• Define number line.
• Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
• Calculate missing input and/or output values in a table.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 2 Classroom Resources: 2
22. Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.

a. Given a geometric figure and a rotation, reflection, or translation, draw the image of the transformed figure using graph paper, tracing paper, or geometry software.

b. Specify a sequence of rotations, reflections, or translations that will carry a given figure onto another.

c. Draw figures with different types of symmetries and describe their attributes.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a geometric figure,
• Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.
• Produce the image of the figure under a rotation, reflection, or translation using graph paper, tracing paper, or geometry software.
• Describe and justify the sequence of transformations that will carry a given figure onto another.
• Draw figures such as rectangles, parallelograms, trapezoids, or regular polygons.
• Identify which figures that have rotations or reflections that carry the figure onto itself.
• Perform and communicate rotations and reflections that map the object to itself.
• Distinguish these transformations from those which do not carry the object back to itself.
• Describe the relationship of these findings to symmetry.
Teacher Vocabulary:
• Transformation
• Reflection
• Translation
• Rotation
• Dilation
• Isometry
• Composition
• horizontal stretch
• vertical stretch
• horizontal shrink
• vertical shrink
• Clockwise
• Counterclockwise
• Symmetry
• Trapezoid
• Square
• Rectangle
• Regular polygon
• parallelogram
• Mapping
• preimage
• Image
Knowledge:
Students know:
• Characteristics of transformations (translations, rotations, reflections, and dilations).
• Techniques for producing images under transformations using graph paper, tracing paper, or geometry software.
• Characteristics of rectangles, parallelograms, trapezoids, and regular polygons.
Skills:
Students are able to:
• Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology.
• Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane.
Understanding:
Students understand that:
• Mapping one point to another through a series of transformations can be recorded as a function.
• Since translations, rotations and reflections preserve distance and angle measure, the image is then congruent.
• The same transformation may be produced using a variety of tools, but the geometric sequence of steps that describe the transformation is consistent.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.22.1: Define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.22.2: Describe the effects of rotations, reflection, and translations on two dimensional figures using coordinates.
GEO.22.3: Illustrate figures transformed by a rotation, reflection or translation.
GEO.22.4: Describe the process of transforming a given figure.
GEO.22.5: Graph a figure on a coordinate plane.

Prior Knowledge Skills:
• Recognize dilations.
• Recognize translations.
• Recognize rotations.
• Recognize reflections.
• Define rotation, reflection, and translation.
• Recognize translations (slides), rotations (turns), and reflections (flips).
• Distinguish between lines and line segments.
• Identify parallel lines.
• Demonstrate how to locate points on a vertical or horizontal number line.
• Define ordered pairs.
• Show how to plot points on a Cartesian plane.
• Locate the origin on the coordinate plane.
• Identify the length between vertices on a coordinate plane.
• Recall how to read a graph or table.
• Draw and label a coordinate plane.
• Plot independent (input) and dependent (output) values on a coordinate plane.
• Plot pairs of integers and/or rational numbers on a coordinate plane.
• Arrange integers and/or rational numbers on a horizontal or vertical number line.
• Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
• Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
• Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
• Calculate the distances between points having the same first or second coordinate using absolute value.
• Define number line.
• Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
23. Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
Unpacked Content Evidence Of Student Attainment:
Students:
• Use geometric terminology (angles, circles, perpendicular lines, parallel lines, and line segments) to describe the series of steps necessary to produce a rotation, reflection, or translation.
• Use these descriptions to communicate precise definitions of rotation, reflection, and translation.
Teacher Vocabulary:
• Transformation
• Reflection
• Translation
• Rotation
• Dilation
• Isometry
• Composition
• Clockwise
• Counterclockwise
• Preimage
• Image
Knowledge:
Students know:
• Characteristics of transformations (translations, rotations, reflections, and dilations).
• -Properties of a mathematical definition, i.e., the smallest amount of information and properties that are enough to determine the concept. (Note: may not include all information related to concept).
Skills:
Students are able to:
• Accurately perform rotations, reflections, and translations on objects with and without technology.
• Communicate the results of performing transformations on objects.
• Use known and developed definitions and logical connections to develop new definitions.
Understanding:
Students understand that:
• Geometric definitions are developed from a few undefined notions by a logical sequence of connections that lead to a precise definition.
• A precise definition should allow for the inclusion of all examples of the concept and require the exclusion of all non-examples.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.23.1: Define rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.23.2: Describe the effects of rotations, reflection, and translations on two dimensional figures using coordinates.
GEO.23.3: Describe the effects of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
GEO.23.4: Describe the process of transforming a given figure.
GEO.23.5: Illustrate figures transformed by a rotation, reflection or translation.
GEO.23.6: Recognize the type of transformation from a pre-image to an image.

Prior Knowledge Skills:
• Recognize dilations.
• Recognize translations.
• Recognize rotations.
• Recognize reflections.
• Analyze an image and its dilation to determine if the two figures are similar.
• Define dilation.
• Recall how to find scale factor.
• Give examples of scale drawings.
• Recognize translations.
• Recognize reflections.
• Recognize rotations.
• Identify parallel lines.
• Compare translations to reflections.
• Compare reflections to rotations.
• Compare rotations to translations.
• Define diameter, radius, circumference, area of a circle, and formula.
• Identify and label parts of a circle.
• Recognize the attributes of a circle.
• Define rotation, reflection, and translation.
• Recognize translations (slides), rotations (turns), and reflections (flips).
• Distinguish between lines and line segments.
• Identify parallel lines.
• Define square root, cube root, inverse, perfect square, perfect cube, and irrational number.
• Define square root, expressions, and approximations.
• Demonstrate how to locate points on a vertical or horizontal number line.
• Define ordered pairs.
• Show how to plot points on a Cartesian plane.
• Locate the origin on the coordinate plane.
• Identify the length between vertices on a coordinate plane.
• Recall how to read a graph or table.
• Draw and label a coordinate plane.
• Plot independent (input) and dependent (output) values on a coordinate plane.
• Plot pairs of integers and/or rational numbers on a coordinate plane.
• Arrange integers and/or rational numbers on a horizontal or vertical number line.
• Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
• Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
• Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
• Calculate the distances between points having the same first or second coordinate using absolute value.
• Define number line.
• Demonstrate the location of positive and negative numbers on a vertical and horizontal number line.
• Calculate missing input and/or output values in a table.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.21 Identify and/or model characteristics of a geometric figure that has undergone a transformation (reflection, rotation, translation) by drawing, explaining, or using manipulatives.

Showing that two figures are congruent involves showing that there is a rigid motion (translation, rotation, reflection, or glide reflection) or, equivalently, a sequence of rigid motions that maps one figure to the other.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 1 Classroom Resources: 1
24. Define congruence of two figures in terms of rigid motions (a sequence of translations, rotations, and reflections); show that two figures are congruent by finding a sequence of rigid motions that maps one figure to the other.
Example: △ABC is congruent to △XYZ since a reflection followed by a translation maps △ABC onto △XYZ. Unpacked Content Evidence Of Student Attainment:
Students:
• Given two geometric figures, determine if a sequence of rotations, reflections, and translations will carry the first to the second, and if so justify their congruence by the definition of congruence in terms of rigid motions.
Teacher Vocabulary:
• Rigid motions
• Congruence
Knowledge:
Students know:
• Characteristics of translations, rotations, and reflections including the definition of congruence.
• Techniques for producing images under transformations using graph paper, tracing paper, compass, or geometry software.
• Geometric terminology (e.g., angles, circles, perpendicular lines, parallel lines, and line segments) which describes the series of steps necessary to produce a rotation, reflection, or translation.
Skills:
Students are able to:
• Use geometric descriptions of rigid motions to accurately perform these transformations on objects.
• Communicate the results of performing transformations on objects.
Understanding:
Students understand that:
• Any distance preserving transformation is a combination of rotations, reflections, and translations.
• If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.24.1: Define congruence.
GEO.24.2: Applying the definition of congruence determine if two figures are congruent.
GEO.24.3: Illustrate a sequence of rigid motions on a coordinate plane that maps one figure to another.
GEO.24.4: Illustrate a vertical and horizontal shift on a coordinate plane.
Example: Rectangle PQRS has vertices P(-3,5), Q(-4,2), R (3,0), 5(4,3). Translate PQRS vertically 3 units.
GEO.24.5: Recognize composition of transformations.
GEO.24.6: Graph a figure on a coordinate plane.

Prior Knowledge Skills:
• Recognize dilations.
• Recognize translations.
• Recognize rotations.
• Recognize reflections.
• Analyze an image and its dilation to determine if the two figures are similar.
• Define dilation.
• Recall how to find scale factor.
• Give examples of scale drawings.
• Recognize translations.
• Recognize reflections.
• Recognize rotations.
• Identify parallel lines.
• Define congruent and sequence.
• Compare translations to reflections.
• Compare reflections to rotations.
• Compare rotations to translations.
• Identify congruent figures.
• Define diameter, radius, circumference, area of a circle, and formula.
• Identify and label parts of a circle.
• Recognize the attributes of a circle.
• Define rotation, reflection, and translation.
• Recognize translations (slides), rotations (turns), and reflections (flips).
• Distinguish between lines and line segments.
• Identify parallel lines.
• Define square root, cube root, inverse, perfect square, perfect cube, and irrational number.
• Define square root, expressions, and approximations.
• Demonstrate how to locate points on a vertical or horizontal number line.
• Define ordered pairs.
• Show how to plot points on a Cartesian plane.
• Locate the origin on the coordinate plane.
• Identify the length between vertices on a coordinate plane.
• Recall how to read a graph or table.
• Draw and label a coordinate plane.
• Plot independent (input) and dependent (output) values on a coordinate plane. M. 6.13.2: Plot pairs of integers and/or rational numbers on a coordinate plane. M. 6.13.3: Arrange integers and/or rational numbers on a horizontal or vertical number line. M. 6.13.4: Locate the position of integers and/or rational numbers on a horizontal or vertical number line. M. 6.11.1: Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection. M. 6.11a.3: Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes. M. 6.11d.1: Calculate the distances between points having the same first or second coordinate using absolute value. M. 6.10a.1: Define number line. M. 6.10a.2: Demonstrate the location of positive and negative numbers on a vertical and horizontal number line. M. 6.3.6: Calculate missing input and/or output values in a table.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 3 Lesson Plans: 1 Classroom Resources: 2
25. Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another.

a. Verify that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

b. Verify that two triangles are congruent if (but not only if) the following groups of corresponding parts are congruent: angle-side-angle (ASA), side-angle-side (SAS), side-side-side (SSS), and angle-angle-side (AAS).

Example: Given two triangles with two pairs of congruent corresponding sides and a pair of congruent included angles, show that there must be a sequence of rigid motions will map one onto the other.
Unpacked Content Evidence Of Student Attainment:
Students:
• Given a triangle and its image under a sequence of rigid motions (translations, reflections, and translations), verify that corresponding sides and corresponding angles are congruent.
• Given two triangles that have the same side lengths and angle measures, find a sequence of rigid motions that will map one onto the other.
• Use rigid motions and the basic properties of rigid motions (that they preserve distance and angle), which are assumed without proof to establish that the usual triangle congruence criteria make sense and can then be used to prove other theorems.
Teacher Vocabulary:
• Corresponding sides and angles
• Rigid motions
• If and only if
• Triangle congruence
• Angle-Side-Angle (ASA)
• Side-Angle-Side (SAS)
• Side-Side->Side (SSS)
Knowledge:
Students know:
• Characteristics of translations, rotations, and reflections including the definition of congruence.
• Techniques for producing images under transformations.
• Geometric terminology which describes the series of steps necessary to produce a rotation, reflection, or translation.
• Basic properties of rigid motions (that they preserve distance and angle).
• Methods for presenting logical reasoning using assumed understandings to justify subsequent results.
Skills:
Students are able to:
• Use geometric descriptions of rigid motions to accurately perform these transformations on objects.
• Communicate the results of performing transformations on objects.
• Use logical reasoning to connect geometric ideas to justify other results.
• Perform rigid motions of geometric figures.
• Determine whether two plane figures are congruent by showing whether they coincide when superimposed by means of a sequence of rigid motions (translation, reflection, or rotation).
• Identify two triangles as congruent if the lengths of corresponding sides are equal (SSS criterion), if the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if two pairs of corresponding angles are congruent and the lengths of the corresponding sides between them are equal (ASA criterion).
• Apply the SSS, SAS, and ASA criteria to verify whether or not two triangles are congruent.
Understanding:
Students understand that:
• If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent.
• It is beneficial to have minimal sets of requirements to justify geometric results (e.g., use ASA, SAS, or SSS instead of all sides and all angles for congruence).
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.25.1: Define congruent, corresponding, triangles, angles, and the concept of if and only if.
GEO.25.2: Compare angles and sides of two triangles to determine congruency.
GEO.25.3: Determine the lengths of sides and the measures of angles in triangles.
GEO.25.4: Identify corresponding parts of triangles.

Prior Knowledge Skills:
• Define congruent and sequence.
• Identify congruent figures.
• Recognize attributes of geometric shapes.
• Identify the length between vertices on a coordinate plane.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)

Showing that two figures are similar involves finding a similarity transformation (dilation or composite of a dilation with a rigid motion) or, equivalently, a sequence of similarity transformations that maps one figure onto the other.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 2 Classroom Resources: 2
26. Verify experimentally the properties of dilations given by a center and a scale factor.

a. Verify that 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.

b. Verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a center of dilation, a scale factor, and a polygonal image,
• Create a new image by extending a line segment from the center of dilation through each vertex of the original figure by the scale factor to find each new vertex.
• Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the same line.
• Find the ratio of the length of the line segment from the center of dilation to each vertex in the new image and the corresponding segment in the original image and compare that ratio to the scale factor.
• Conjecture a generalization of these results for all dilations.
Teacher Vocabulary:
• Dilations
• Center
• Scale factor
Knowledge:
Students know:
• Methods for finding the length of line segments (both in a coordinate plane and through measurement).
• Dilations may be performed on polygons by drawing lines through the center of dilation and each vertex of the polygon then marking off a line segment changed from the original by the scale factor.
Skills:
Students are able to:
• Accurately create a new image from a center of dilation, a scale factor, and an image.
• Accurately find the length of line segments and ratios of line segments.
• Communicate with logical reasoning a conjecture of generalization from experimental results.
Understanding:
Students understand that:
• A dilation uses a center and line segments through vertex points to create an image which is similar to the original image but in a ratio specified by the scale factor.
• The ratio of the line segment formed from the center of dilation to a vertex in the new image and the corresponding vertex in the original image is equal to the scale factor.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.26.1: Define dilation and scale factor.
GEO.26.2: Apply a scale factor.
GEO.26.2: Illustrate when given an original figure with a line (e.g., m) through it, not through the center, a parallel line to m will be created when the dilation is performed.
Example: Given a line x=, dilate the graph and line by 2. What happened to the line?
GEO.26.3: Illustrate when given an original figure with a line (e.g., m) through its center the line will remain unchanged when the dilation is performed.
GEO.26.4: Illustrate dilation.
Example: Find the distance of line AB, given A (0,0) and B (2,3), after dilating AB by a scale factor of 1/2.
GEO.26.5: Determine the change in length of a line segment after dilation.
GEO.26.6: Discuss the properties of parallel lines.
GEO.26.7: Dilate a line segment.
GEO.26.8: Recognize whether a dilation is an enlargement or a reduction.

Prior Knowledge Skills:
• Recall how to name points on a Cartesian plane using ordered pairs.
• Recognize ordered pairs (x, y).
• Define similar.
• Recognize dilations.
• Recognize translations.
• Recognize rotations.
• Recognize reflections.
• Identify similar figures.
• Analyze an image and its dilation to determine if the two figures are similar.
• Define dilation.
• Recall how to find scale factor.
• Give examples of scale drawings.
• Identify parts of the Cartesian plane.
• Recognize ordered pairs.
• Define function, ordered pairs, input, output.
• Demonstrate how to plot points on a Cartesian plane using ordered pairs.���������������������

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 1 Classroom Resources: 1
27. Given two figures, determine whether they are similar by identifying a similarity transformation (sequence of rigid motions and dilations) that maps one figure to the other.
Unpacked Content Evidence Of Student Attainment:
Students:
• Given two figures, determine if they are similar by demonstrating whether one figure can be obtained from the other through a dilation and a combination of translations, reflections, and rotations.
Teacher Vocabulary:
• Similarity transformation
• Similarity
• Proportionality
• Corresponding pairs of angles
• Corresponding pairs of sides
• Rigid Motion
Knowledge:
Students know:
• Properties of rigid motions and dilations.
• Definition of similarity in terms of similarity transformations.
• Techniques for producing images under a dilation and rigid motions.
Skills:
Students are able to:
• Apply rigid motion and dilation to a figure.
• Explain and justify whether or not one figure can be obtained from another through a combination of rigid motion and dilation.
Understanding:
Students understand that:
• A figure that may be obtained from another through a dilation and a combination of translations, reflections, and rotations is similar to the original.
• When a figure is similar to another the measures of all corresponding angles are equal, and all of the corresponding sides are in the same proportion.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.27.1: Establish a sequence of similarity transformations between two similar polygons.
GEO.27.2: Determine if two triangles are similar based on their corresponding parts.
GEO.27.3: Develop a similarity statement for two similar polygons.
GEO.27.4: Identify corresponding angles and sides based on similarity statements.

Prior Knowledge Skills:
• Recognize dilations.
• Recognize translations.
• Recognize rotations.
• Recognize reflections.
• Define rotation, reflection, and translation.
• Recognize translations (slides), rotations (turns), and reflections (flips).
• Distinguish between lines and line segments.
• Identify parallel lines.
• Demonstrate how to locate points on a vertical or horizontal number line.
• Define ordered pairs.
• Show how to plot points on a Cartesian plane.
• Locate the origin on the coordinate plane.
• Identify the length between vertices on a coordinate plane.
• Recall how to read a graph or table.
• Draw and label a coordinate plane.
• Plot independent (input) and dependent (output) values on a coordinate plane.
• Plot pairs of integers and/or rational numbers on a coordinate plane.
• Arrange integers and/or rational numbers on a horizontal or vertical number line.
• Locate the position of integers and/or rational numbers on a horizontal or vertical number line.
• Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
• Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
• Calculate the distances between points having the same first or second coordinate using absolute.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
28. Verify criteria for showing triangles are similar using a similarity transformation (sequence of rigid motions and dilations) that maps one triangle to another.

a. Verify that two triangles are similar if and only if corresponding pairs of sides are proportional and corresponding pairs of angles are congruent.

b. Verify that two triangles are similar if (but not only if) two pairs of corresponding angles are congruent (AA), the corresponding sides are proportional (SSS), or two pairs of corresponding sides are proportional and the pair of included angles is congruent (SAS).

Example: Given two triangles with two pairs of congruent corresponding sides and a pair of congruent included angles, show there must be a set of rigid motions that maps one onto the other.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a triangle,
• Produce a similar triangle through a dilation and a combination of translations, rotations, and reflections.
• Demonstrate that a dilation and a combination of translations, reflections, and rotations maintain the measure of each angle in the triangles and all corresponding pairs of sides of the triangles are proportional.

Given two triangles,
• Explain why if the measures of two angles from one triangle are equal to the measures of two angles from another triangle, then measures of the third angles must be equal to each other.
• Use this established fact and the properties of a similarity transformation to demonstrate that the corresponding sides are proportional (SSS), two pairs of corresponding sides are proportional and the pair of included angles is congruent (SAS), or Angle-Angle (AA) criterion for similar triangles are sufficient.
Teacher Vocabulary:
• Similarity transformation
• Similarity
• Proportionality
• Corresponding pairs of angles
• Corresponding pairs of sides
• Similarity criteria for triangles
• Rigid Motion
Knowledge:
Students know:
• The sum of the measures of the angles of a triangle is 180 degrees.
• Properties of rigid motions and dilations.
• Definition of similarity in terms of similarity transformations.
• Techniques for producing images under a dilation and rigid motions.
Skills:
Students are able to:
• Apply rigid motion and dilation to a figure.
• Explain and justify whether or not one figure can be obtained from another through a combination of rigid motion and dilation.
Understanding:
Students understand that:
• A figure that may be obtained from another through a dilation and a combination of translations, reflections, and rotations is similar to the original.
• When a figure is similar to another the measures of all corresponding angles are equal, and all of the corresponding sides are in the same proportion.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.28.1: Define corresponding, similarity and proportions.
GEO.28.2: Evaluate the properties of the triangles to prove congruency.
GEO.28.3: Create proportional equations from given information.
GEO.28.4: Evaluate the angle-side-angle (ASA), side-angle-side (SAS), and side-side-side (SSS), Theorems to prove similarity.
GEO.28.5: Evaluate the AA postulate to prove similarity.
GEO.28.6: Compare two figures in terms of similarity.
GEO.28.7: Demonstrate that equivalent ratios are proportions.
GEO.28.8: Solve proportional equations.

Prior Knowledge Skills:
• Apply properties to find missing angle measures.
• Identify a transversal.
• Identify exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.
• Identify attributes of triangles.
• Define exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, corresponding angles, and transversal.
• Discover the Angle Sum Theorem (sum of the interior angles of a triangle equal 180 degrees).
• Identify parallel lines.
• Demonstrate how to use a protractor to measure angles.
• Demonstrate how to measure length.
• Distinguish between lines and line segments.
• Recognize translations (slides), rotations (turns), and reflections (flips).
• Define rotation, reflection, and translation.
• Recognize attributes of geometric shapes.
• Identify the length between vertices on a coordinate plane.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.24 When given two congruent triangles that have been transformed (limit to a translation), determine the congruent parts. (Ex: Determine which leg on Triangle A is congruent to which leg on Triangle B.)

Focus 3: Geometric Arguments, Reasoning, and Proof
Using technology to construct and explore figures with constraints provides an opportunity to explore the independence and dependence of assumptions and conjectures.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 0
29. Find patterns and relationships in figures including lines, triangles, quadrilaterals, and circles, using technology and other tools.

a. Construct figures, using technology and other tools, in order to make and test conjectures about their properties.

b. Identify different sets of properties necessary to define and construct figures.

Unpacked Content Evidence Of Student Attainment:
Students:
• Discover patterns and relationships in figures using technology and other tools.
• Construct figures, using technology and other tools, to make and test conjectures about their properties.
• Identify different properties necessary to define and construct figures
Teacher Vocabulary:
• Conjectures
• Construct
• Congruent
• Compass
• Straightedge
Knowledge:
Students know:
• Use technology and other tools to discover patterns and relationships in figures.
• Use patterns. relationships and properties to construct figures.
Understanding:
Students understand that:
• Many of the constructions build on the relationships among the objects and are justified by the properties used during the construction. Technology can be used as a means to explain the properties and definitions by developing procedures to carry out the construction.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.29.1: Construct a copy of a segment, copy of an angle, the bisection of a segment, the bisection of an angle, perpendicular line, perpendicular bisector of a line segment, and parallel lines.
GEO.29.2: Describe a specific construction process.
GEO.29.3: Demonstrate the proper use of a geometric construction tools.

Prior Knowledge Skills:
• Demonstrate how to use a protractor to draw an angle.
• Construct segments of a given length using a ruler.
• Recognize attributes of geometric shapes.
Proof is the means by which we demonstrate whether a statement is true or false mathematically, and proofs can be communicated in a variety of ways (e.g., two-column, paragraph).
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 2 Classroom Resources: 2
30. Develop and use precise definitions of figures such as angle, circle, perpendicular lines, parallel lines, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
Unpacked Content Evidence Of Student Attainment:
Students:
Given undefined notions of point, line, distance along a line, and distance around a circular arc,
• Develop precise definitions of angle, circle, perpendicular line, parallel line, and line segment.
• Identify examples and non-examples of angles, circles, perpendicular lines, parallel lines, and line segments.
Teacher Vocabulary:
• Point
• Line
• Segment
• Angle
• Perpendicular line
• Parallel line
• Distance
• Arc length
• Ray
• Vertex
• Endpoint
• Plane
• Collinear
• Coplanar
• Skew
Knowledge:
Students know:
• Undefined notions of point, line, distance along a line, and distance around a circular arc.
• Properties of a mathematical definition, i.e., the smallest amount of information and properties that are enough to determine the concept. (Note: may not include all information related to concept).
Skills:
Students are able to:
• Use known and developed definitions and logical connections to develop new definitions.
Understanding:
Students understand that:
• Geometric definitions are developed from a few undefined notions by a logical sequence of connections that lead to a precise definition, A precise definition should allow for the inclusion of all examples of the concept, and require the exclusion of all non-examples.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.30.1: Define angle, circle, perpendicular line, parallel line, line segment, and distance.
GEO.30.2: Describe angle, circle, perpendicular line, parallel line, line segment, and distance.
GEO.30.3: Illustrate a point, line, distance along a line, and distance around a circular arc.
GEO.30.4: Identify angle, circle, perpendicular line, parallel line, line segment, and distance.

Prior Knowledge Skills:
• Define exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, corresponding angles, and transversal.
• Identify exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.
• Identify a transversal.
• Apply properties to find missing angle measures.
• Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
• Identify all types of angles.
• Identify right angles and straight angles.
• Demonstrate how to use a protractor to draw an angle.
• Define vertices.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.30 Demonstrate perpendicular lines, parallel lines, line segments, angles, and circles by drawing, modeling, identifying or creating.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 3 Classroom Resources: 3
31. Justify whether conjectures are true or false in order to prove theorems and then apply those theorems in solving problems, communicating proofs in a variety of ways, including flow chart, two-column, and paragraph formats.

a. Investigate, prove, and apply theorems about lines and angles, including but not limited to: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; the points on the perpendicular bisector of a line segment are those equidistant from the segment's endpoints.

b. Investigate, prove, and apply theorems about triangles, including but not limited to: the sum of the measures of the interior angles of a triangle is 180?; the base angles of isosceles triangles are congruent; the segment joining the midpoints of two sides of a triangle is parallel to the third side and half the length; a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem using triangle similarity.

c. Investigate, prove, and apply theorems about parallelograms and other quadrilaterals, including but not limited to both necessary and sufficient conditions for parallelograms and other quadrilaterals, as well as relationships among kinds of quadrilaterals.

Example: Prove that rectangles are parallelograms with congruent diagonals.

Unpacked Content Evidence Of Student Attainment:
Students:
• Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology.
• Explain the requirements of a mathematical proof.
1. Present a complete mathematical proof of geometry theorems including the following: vertical angles are congruent. when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent. points on a perpendicular bisector of a line segment are exactly those equidistant from the segment's endpoints.
Critique proposed proofs made by others.
2. Present a complete mathematical proof of geometry theorems about triangles, including the following: measures of interior angles of a triangle sum to 180o. base angles of isosceles triangles are congruent. the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length. the medians of a triangle meet at a point.
Critique proposed proofs made by others.
3. Present a complete mathematical proof of geometry theorems about parallelograms, including the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.
Critique proposed proofs made by others.
Teacher Vocabulary:
• Same side interior angle
• Consecutive interior angle
• Vertical angles
• Linear pair
• Complementary angles
• Supplementary angles
• Perpendicular bisector
• Equidistant
• Theorem Proof
• Prove
• Transversal
• Alternate interior angles
• Corresponding angles
• Interior angles of a triangle
• Isosceles triangles
• Equilateral triangles
• Base angles
• Median
• Exterior angles
• Remote interior angles
• Centroid
• Parallelograms
• Diagonals
• Bisect
Knowledge:
Students know:
• Requirements for a mathematical proof.
• Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
• Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
• Generate a conjecture about geometric relationships that calls for proof.
Understanding:
Students understand that:
• Proof is necessary to establish that a conjecture about a relationship in mathematics is always true, and may also provide insight into the mathematics being addressed.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.31.1: Define vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelogram, bisector, and converse.
GEO.31.2: Develop a process that demonstrates the logical order of properties to form a proof.
GEO.31.3: Arrange statements to form a logical order.
GEO.31.4: Identify measures of vertical angles, alternate interior angles, corresponding angles, measures of interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, and median.
GEO.31.5: Illustrate vertical angle, transversal, parallel lines, alternate interior angles, corresponding angles, perpendicular bisector, line segment, equidistant, endpoints, interior angles of a triangle, base angles of isosceles triangles, isosceles triangles, midpoint, median, intersection, opposite sides, opposite angles, diagonals, parallelograms, bisectors, and their properties.
GEO.31.6: Find the measure of the third interior angle of a triangle when given the measure of the other two interior angles.

Prior Knowledge Skills:
• Define a right angle, Pythagorean Theorem, converse, and proof.
• Define exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, corresponding angles, and transversal.
• Identify attributes of triangles.
• Identify exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.
• Identify a transversal.
• Apply properties to find missing angle measures.
• Discover the Angle Sum Theorem (sum of the interior angles of a triangle equal 180 degrees).
• Identify parallel lines.
• Define supplementary angles, complementary angles, vertical angles, adjacent angles, parallel lines, perpendicular lines, and intersecting lines.
• Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
• Recognize and demonstrate that two right triangles make a rectangle.
• Recognize polygons.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Proofs of theorems can sometimes be made with transformations, coordinates, or algebra; all approaches can be useful, and in some cases one may provide a more accessible or understandable argument than another.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 4 Classroom Resources: 4
32. Use coordinates to prove simple geometric theorems algebraically.
Unpacked Content Evidence Of Student Attainment:
Students:
• Given coordinates and geometric theorems and statements defined on a coordinate system, use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others.
Teacher Vocabulary:
• Simple geometric theorems
• Simple geometric figures
Knowledge:
Students know:
• Relationships (e.g. distance, slope of line) between sets of points.
• Properties of geometric shapes.
• Coordinate graphing rules and techniques.
• Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
• Accurately determine what information is needed to prove or disprove a statement or theorem.
• Accurately find the needed information and explain and justify conclusions.
• Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
Understanding:
Students understand that:
• Modeling geometric figures or relationships on a coordinate graph assists in determining truth of a statement or theorem.
• Geometric theorems may be proven or disproven by examining the properties of the geometric shapes in the theorem through the use of appropriate algebraic techniques.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.32.1: Apply formulas, and properties of polygons, angles, and lines to draw conclusions from the given information.
GEO.32.2: Identify properties of perpendicular and parallel lines, properties of polygons.
GEO.32.3: Illustrate polygons created by given coordinates on a coordinate plane.
GEO.32.4: Identify distance formula, circle formula, Pythagorean Theorem, midpoint.

Prior Knowledge Skills:
• Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
• Demonstrate an understanding of an extended coordinate plane.
• Draw and label a 4 quadrant coordinate plane.
• Draw and extend vertical and horizontal number lines.
• Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
• Recall how to graph points in all four quadrants of the coordinate plane.
• Define ordered pairs.
• Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
• Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
• Identify which signs indicate the location of a point in a coordinate plane.
• Recall how to plot ordered pairs on a coordinate plane.
• Identify the length between vertices on a coordinate plane.
• Calculate the perimeter and area using the distance between the vertices.
• Define a right angle, Pythagorean Theorem, converse, and proof.
• Recognize examples of right triangles.
• Demonstrate how to find square roots.
• Solve problems with exponents.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 3 Lesson Plans: 2 Classroom Resources: 1
33. Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.

Example: Find the equation of a line parallel or perpendicular to a given line that passes through a given point.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a line,
• Create lines parallel to the given line and compare the slopes of parallel lines by examining the rise/run ratio of each line.
• Create lines perpendicular to the given line by rotating the line 90 degrees and compare the slopes by examining the rise/run ratio of each line.
• Use understandings of similar triangles and logical reasoning to prove that parallel lines have equal slopes and the slopes of perpendicular lines are negative reciprocals.Given a geometric problem involving parallel or perpendicular lines.
• Apply the appropriate slope criteria to solve the problem and justify the solution including finding equations of lines parallel or perpendicular to a given line.
Teacher Vocabulary:
• Parallel lines
• Perpendicular lines
• Slope
• Slope triangle
Knowledge:
Students know:
• Techniques to find the slope of a line.
• Key features needed to solve geometric problems.
• Techniques for presenting a proof of geometric theorems.
Skills:
Students are able to:
• Explain and justify conclusions reached regarding the slopes of parallel and perpendicular lines.
• Apply slope criteria for parallel and perpendicular lines to accurately find the solutions of geometric problems and justify the solutions.
• Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.
Understanding:
Students understand that:
• Relationships exist between the slope of a line and any line parallel or perpendicular to that line.
• Slope criteria for parallel and perpendicular lines may be useful in solving geometric problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.33.1: Define slope, point slope formula, slope-intercept formula, standard form of a line, parallel lines, and perpendicular lines.
GEO.33.2: Demonstrate and explain algebraically how perpendicular lines have only one common point.
GEO.33.3: Demonstrate and explain algebraically how parallel lines have no common points.
GEO.33.4: Write and solve equations of parallel and perpendicular lines.
GEO.33.5: Illustrate graphically how perpendicular lines have only one common point.
GEO.33.6: Illustrate graphically how parallel lines have no common points.
GEO.33.7: Write an equation of a line in slope intercept form.
GEO.33.8: Find the slope of a given line.

Prior Knowledge Skills:
• Define slope, intercept, linear, equation, and bivariate.
• Recall how to determine the rate of change (slope) from a graph.
• Identify the parts of the slope-intercept form of an equation.
• Recognize how to read a graph.
• Recall how to write an equation in slope-intercept form.
• Apply the identification of the slope and the y-intercept to a real-world situation.
• Create a graph to model a real-word situation.
• Define proportional relationships, unit rate, and slope.
• Demonstrate how to graph on a Cartesian plane.
• Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
• Define linear functions, nonlinear functions, slope, and y-intercept.
• Recognize linear equations.
• Identify ordered pairs.
• Recognize ordered pairs.
• Generate the slope of a line using given ordered pairs.
• Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
• Graph a function given the slope-intercept form of an equation.
• Recognize that two sets of points with the same slope may have different y-intercepts.
• Graph a linear equation given the slope-intercept form of an equation.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Focus 4: Solving Applied Problems and Modeling in Geometry
Recognizing congruence, similarity, symmetry, measurement opportunities, and other geometric ideas, including right triangle trigonometry, in real-world contexts provides a means of building understanding of these concepts and is a powerful tool for solving problems related to the physical world in which we live.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 2 Classroom Resources: 2
34. Use congruence and similarity criteria for triangles to solve problems in real-world contexts.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a contextual situation involving triangles,
• Determine solutions to problems by applying congruence and similarity criteria for triangles to assist in solving the problem.
• Justify solutions and critique the solutions of others.

• Given a geometric figure, establish and justify relationships in the figure through the use of congruence and similarity criteria for triangles
Teacher Vocabulary:
• Congruence and similarity criteria for triangles
Knowledge:
Students know:
• Criteria for congruent (SAS, ASA, AAS, SSS) and similar (AA) triangles and transformation criteria.
• Techniques to apply criteria of congruent and similar triangles for solving a contextual problem.
• Techniques for applying rigid motions and dilations to solve congruence and similarity problems in real-world contexts.
Skills:
Students are able to:
• Accurately solve a contextual problem by applying the criteria of congruent and similar triangles.
• Provide justification for the solution process.
• Analyze the solutions of others and explain why their solutions are valid or invalid.
• Justify relationships in geometric figures through the use of congruent and similar triangles.
Understanding:
Students understand that:
• Congruence and similarity criteria for triangles may be used to find solutions of contextual problems.
• Relationships in geometric figures may be proven through the use of congruent and similar triangles.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.34.1: Develop an equation from given information to prove congruence or similarity.
GEO.34.2: Illustrate congruence and similarity in geometric figures.
GEO.34.3: Apply proportional reasoning to real world scenarios.
GEO.34.4: Solve proportions.

Prior Knowledge Skills:
• Analyze an image and its dilation to determine if the two figures are similar.
• Identify similar figures.
• Define similar.
• Identify congruent figures.
• Identify attributes of two-dimensional figures.
• Compare rotations to translations.
• Compare reflections to rotations.
• Compare translations to reflections.
• Define congruent and sequence.
• Apply the rule of proportional relationship to real world context.
• Recognize similar triangles.
• Define similar triangles, intercept, slope, vertical, horizontal, and origin.
• Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
• Analyze the graph to determine the rate of change.
• Generate the slope of a line using given ordered pairs.
• Graph a function given the slope-intercept form of an equation.
• Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
• Graph a linear equation given the slope-intercept form of an equation.
• Recognize that two sets of points with the same slope may have different y-intercepts.
• Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
• Recall that for a relationship to be proportional, the graph must pass through the origin.
• Demonstrate how to graph on a Cartesian plane.
• Recall that for a relationship to be proportional, both variables must start at zero.
• Identify the unit rate of two quantities.
• Recall how to write a ratio of two quantities as a fraction.
• Recall equivalent ratios and origin on a coordinate (Cartesian) plane.
• Define proportional, independent variable, dependent variable, unit rate.
• Identify proportional relationships.
• Locate/use scale on a map.
• Define scale, scale drawings, length, area, and geometric figures.
• Use a table or graph to determine whether two quantities are proportional.
• Define equivalent ratios and origin.
• Define unit rate, proportions, area, length, and ratio.
• Recognize polygons. M. 6.3.4: Restate real world problems or mathematical problems. M. 6.3.3: Calculate unit rate or rate by using ratios or proportions. M. 6.3.2: Create a ratio or proportion from a given word problem, diagram, table, or equation. M. 6.3.1: Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table. M. 6.3.16: Form a ratio. M. 6.3.15: Solve a proportion using part over whole equals percent over 100. M. 6.3.14: Identify a proportion from given information. M. 6.3.13: Calculate a proportion for missing information. M. 6.3.10: Create a proportion or ratio from a given word problem.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 5 Learning Activities: 3 Lesson Plans: 1 Classroom Resources: 1
35. Discover and apply relationships in similar right triangles.

a. Derive and apply the constant ratios of the sides in special right triangles (45o-45o-90o and 30o-60o-90o).

b. Use similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.

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

d. Demonstrate the converse of the Pythagorean Theorem.

e. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.
Unpacked Content Evidence Of Student Attainment:
Students:
• Given a collection of right triangles, discover and apply relationships in similar right triangles.
• Derive and apply the ratios of the sides of the original triangles to the ratios of the sides of the similar triangles.
• Communicate observations made about changes (or no change) to such ratios as the length of the side opposite an angle to the hypotenuse, or the side opposite the angle to the side adjacent, as the size of the angle changes or in the case of similar triangles, remains the same.
Summarize these observations by defining the six trigonometric ratios.
• Explain why the two smallest angles must be complements.
• Compare the side ratios of opposite/hypotenuse and adjacent/hypotenuse for each of these angles and discuss conclusions.

Given a contextual situation involving right triangles,
• Create a drawing to model the situation.
• Find the missing sides and/or angles using trigonometric ratios.
• Find the missing sides using the Pythagorean Theorem.
• Use the above information to interpret results in the context of the situation, including finding the areas of regular polygons.
Teacher Vocabulary:
• Side ratios
• Trigonometric ratios
• Sine
• Cosine
• Tangent
• Secant
• Cosecant
• Cotangent
• Complementary angles converse
Knowledge:
Students know:
• Techniques to construct similar triangles.
• Properties of similar triangles.
• Methods for finding sine and cosine ratios in a right triangle (e.g., use of triangle properties: similarity. Pythagorean Theorem. isosceles and equilateral characteristics for 45-45-90 and 30-60-90 triangles and technology for others).
• Methods of using the trigonometric ratios to solve for sides or angles in a right triangle.
• The Pythagorean Theorem and its use in solving for unknown parts of a right triangle.
Skills:
Students are able to:
• Accurately find the side ratios of triangles.
• Explain and justify relationships between the side ratios of a right triangle and the angles of a right triangle.
Understanding:
Students understand that:
• The ratios of the sides of right triangles are dependent on the size of the angles of the triangle.
• The sine of an angle is equal to the cosine of the complement of the angle.
• Switching between using a given angle or its complement and between sine or cosine ratios may be used when solving contextual problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.35.1: Define trigonometric (sine, cosine and tangent) ratios for acute angles, complementary angles, and Pythagorean Theorem.
GEO.35.2: Simplify, multiply, and divide radicals.
GEO.35.3:Discuss the relationship between sine and cosine angles within a triangle.
GEO.35.4: Solve equations using trigonometric ratios.
GEO.35.5: Apply properties of similarity to demonstrate the trigonometric ratios of right triangles.
GEO.35.6: Use Pythagorean Theorem to find the missing side of a right triangle.
GEO.35.7: Create an equation using the given information of a right triangle.
GEO.35.8: Identify the parts of a right triangle.
Examples: legs, hypotenuse, right angle.

Prior Knowledge Skills:
• Demonstrate how to find square roots.
• Identify right triangles.
• Solve problems using the Pythagorean Theorem.
• Recognize ordered pairs (x, y).
• Recall how to name points on a Cartesian plane using ordered pairs.
• Identify right triangles.
• Solve problems using the Pythagorean Theorem.
• Discuss strategies for solving real-world and mathematical problems.
• Recognize examples of right triangles.
• Define a right angle, Pythagorean Theorem, converse, and proof.
• Apply properties to find missing angle measures.
• Identify a transversal.
• Identify exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, and corresponding angles.
• Identify attributes of triangles.
• Define exterior angles, interior angles, vertical angles, adjacent angles, alternate interior angles, alternate exterior angles, corresponding angles, and transversal.
• Identify right angles and straight angles.
• Identify all types of angles.
• Identify proportional relationships.
• Locate/use scale on a map.
• Define scale, scale drawings, length, area, and geometric figures.
• Recall how to find unit rates using ratios.
• Define unit rate, proportions, area, length, and ratio.
• Analyze the area of other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes.
• Define area, special quadrilaterals, right triangles, and polygons.
• Recognize and demonstrate that two right triangles make a rectangle.
• Select manipulatives to demonstrate how to compose and decompose triangles and other shapes.
• Explain how to find the area for rectangles.
• Demonstrate how the area of a rectangle is equal to the sum of the area of two equal right triangles.
• Apply area formulas to solve real-world mathematical problems.
• Recognize polygons.
• Restate real world problems or mathematical problems.
• Calculate unit rate or rate by using ratios or proportions.
• Create a ratio or proportion from a given word problem, diagram, table, or equation.
• Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
• Form a ratio.
• Solve a proportion using part over whole equals percent over 100.
• Identify a proportion from given information.
• Calculate a proportion for missing information.
• Create a proportion or ratio from a given word problem.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 6 Learning Activities: 3 Classroom Resources: 3
36. Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a real-world object,
• Select an appropriate geometric shape to model the object.
• Provide a description of the object through the measures and properties of the geometric shape which is modeling the object.
• Explain and justify the model which was selected.
Teacher Vocabulary:
• Model
Knowledge:
Students know:
• Techniques to find measures of geometric shapes.
• Properties of geometric shapes.
Skills:
Students are able to:
• Model a real-world object through the use of a geometric shape.
• Justify the model by connecting its measures and properties to the object.
Understanding:
Students understand that:
• Geometric shapes may be used to model real-world objects.
• Attributes of geometric figures help us identify the figures and find their measures. therefore, matching these figures to real world objects allows the application of geometric techniques to real world problems.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.36.1: Estimate the dimensions of a given object.
GEO.36.2: Discuss the properties of a given object.
GEO.36.3: Identify the relationship of geometric representations to real-life objects.

Prior Knowledge Skills:
• Recognize attributes of geometric shapes.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.

 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 3 Classroom Resources: 3
37. Investigate and apply relationships among inscribed angles, radii, and chords, including but not limited to: the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Unpacked Content Evidence Of Student Attainment:
Students:
Given circles with two points on the circle,
• Compare the measures of the angles (with and without technology) formed by creating radii to the given points, creating chords from a third point on the circle to the given points, and creating tangents from a third point outside the circle to the given points, and conjecture about possible relationships among the angles.
• Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle is one half the central angle cutting off the same arc, and the circumscribed angle cutting off that arc is supplementary to the central angle relating all three).

Given circles with chords from a point on the circle to the endpoints of a diameter,
• Find the measure of the angles (with and without technology), conjecture about and explain possible relationships.
• Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle on a diameter is a right angle).

Given a circle with a tangent and radius intersecting at a point on the circle,
• Find the measure of the angle at the intersection point (with and without technology), conjecture about and explain possible relationships.
• Use logical reasoning to justify (or deny) the conjectures (in particular justify that the radius of a circle is perpendicular to the tangent where the radius intersects the circle.
Teacher Vocabulary:
• Central angles
• Inscribed angles
• Circumscribed angles
• Chord
• Circumscribed
• Tangent
• Perpendicular arc
Knowledge:
Students know:
• Definitions and characteristics of central, inscribed, and circumscribed angles in a circle.
• Techniques to find measures of angles including using technology (dynamic geometry software).
Skills:
Students are able to:
• Explain and justify possible relationships among central, inscribed, and circumscribed angles sharing intersection points on the circle.
• Accurately find measures of angles (including using technology (dynamic geometry software)) formed from inscribed angles, radii, chords, central angles, circumscribed angles, and tangents.
Understanding:
Students understand that:
• Relationships that exist among inscribed angles, radii, and chords may be used to find the measures of other angles when appropriate conditions are given.
• Identifying and justifying relationships exist in geometric figures.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.37.1: Define inscribed angles, central angles, circumscribed angles, radius, chord, tangent, secant, and diameter.
GEO.37.2: Define inscribed and circumscribed circle of a triangle.
GEO.37.3: Apply knowledge of arcs, angles and chords to solve circle related problems.
GEO.37.4: Determine angle values for all angles formed in the exterior, interior and on the circle.
GEO.37.5: Determine lengths of intersecting chords and secants.
GEO.37.6: Discuss the relationship among inscribed angles, radii, and chords.
GEO.37.7: Illustrate inscribed and circumscribed circles of a triangle and quadrilaterals inscribed in a circle.
GEO.37.8: Illustrate radii, chords, diameters, tangents to curve, central, inscribed, and circumscribed angles.

Prior Knowledge Skills:
• Identify parts of a circle.
• Recall how to find circumference of a circle.
• Recall the meaning of a radius and diameter.
• Identify all types of angles.
• Recognize the attributes of a circle.
• Identify and label parts of a circle.
• Define diameter, radius, circumference, area of a circle, and formula.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.

Experiencing the mathematical modeling cycle in problems involving geometric concepts, from the simplification of the real problem through the solving of the simplified problem, the interpretation of its solution, and the checking of the solution's feasibility, introduces geometric techniques, tools, and points of view that are valuable to problem-solving.
 Mathematics (2019) Grade(s): 9 - 12 Geometry with Data Analysis All Resources: 2 Classroom Resources: 2
38. Use the mathematical modeling cycle involving geometric methods to solve design problems.

Examples: Design an object or structure to satisfy physical constraints or minimize cost; work with typographic grid systems based on ratios; apply concepts of density based on area and volume.
Unpacked Content Evidence Of Student Attainment:
Students:
Given a contextual situation involving design problems,
• Create a geometric method to model the situation and solve the problem.
• Explain and justify the model which was created to solve the problem.i

Note: Mathematical Modeling Cycle can be found in the Appendix of the COS document
Teacher Vocabulary:
• Geometric methods
• Design problems
• Typographic grid system
• Density
Knowledge:
Students know:
• Properties of geometric shapes.
• Characteristics of a mathematical model.
• How to apply the Mathematical Modeling Cycle to solve design problems.
Skills:
Students are able to:
• Accurately model and solve a design problem.
• Justify how their model is an accurate representation of the given situation.
Understanding:
Students understand that:
• Design problems may be modeled with geometric methods.
• Geometric models may have physical constraints.
• Models represent the mathematical core of a situation without extraneous information, for the benefit in a problem solving situation.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
GEO.38.1: Define density, area, volume.
GEO.38.2: Illustrate a design conflict (e.g., draw a chair and a desk where the chair will not fit under the desk).
GEO.38.3: Discuss the relationship between units in each modeling situation.
GEO.38.4: Calculate density (D), mass (m) or volume (V) using the formula, D = m/V.
GEO.38.5: Recognize appropriate units for various situations.

Prior Knowledge Skills:
• Define volume.
• Derive the formulas for the volume of a cone, cylinder, and sphere.
• Calculate the volume of three-dimensional figures.
• Solve real-world problems using the volume formulas for three-dimensional figures.

Alabama Alternate Achievement Standards
AAS Standard:
M.G.AAS.10.36 Use geometric shapes to describe real-world objects.