Courses of Study : Mathematics

Students:

• Rewrite given expressions involving radicals.
• -Simplify expressions involving radicals.
• Use the operations of addition, subtraction, multiplication, and division, with radicals within expressions.

Students know:

• Order of operations, Algebraic properties, Number sense.
• Computation with whole numbers and integers.
• Radicals.
• Rational and irrational numbers.
• Measuring length and finding perimeter and area of rectangles and squares.
• Volume and capacity.
• Rewrite radical expressions.
• Pythagorean theorem.

Students are able to:

• Simplify radicals and justify simplification of radicals using visual representations.
• 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).

Students understand that:

• rewriting radical expressions of rational and irrational numbers can help in recognizing geometric patterns.

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.

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).

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.

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.

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.

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.

Students:

• Given the equations to a set of lines.
• Use simultaneous linear equations. to produce the coordinates of the vertices of a polygon.

Students know:

• Substitution, Elimination, and Graphing methods to solve simultaneous linear equations.

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.

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.

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.

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.

Students know:

• Properties of equality and inequality

Students are able to:

• Accurately rearrange equations or inequalities to produce equivalent forms for use in resolving situations of interest.

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.

Learning Objectives:
GEO.4.1: Accurately rearrange equations and inequalities to produce equivalent forms for use in resolving situations of interest.

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.

Students know:

• Appropriate methods to find ordered pairs that satisfy an equation.
• Techniques to graph the collection of ordered pairs to form a line

Students are able to:

• Accurately find ordered pairs that satisfy the equation.
• Accurately graph the ordered pairs and form a line

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

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.

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.

Students know:

• Key features of a circle.
• The Pythagorean Theorem, Midpoint Formula, Distance Formula.

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.

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)

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.

Students:

• Given both univariate and bivariate data that suggest a linear association.
• Use mathematical and statistical reasoning to draw conclusions and asses risk.

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.

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.

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.

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).

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.

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.

Students are able to:

• recognize patterns, trends, clusters, and gaps in the organized data.

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.

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.

Students know:

• Techniques for constructing dot plots, histograms, scatter plots and box plots from a set of data.

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.

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.

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.

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.

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.

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.

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

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.

Students:

• Given multiple data sets, recognize and explain the differences in shape, center, and spread, including effects of outliers on mean and standard deviation.

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.

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.

Students understand that:

• Differences in the shape, center, and spread of data sets can result from various causes, including outliers and clustering.

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.

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.

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.

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.

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.

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.

Students:
Given a data set of two quantitative variables,

• Use technology to compute and interpret the correlation coefficient of a linear relationship.

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.

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.

Students understand that:

• using technology to graph some data and look at the regression line that technology can generate for a scatter plot.

Learning Objectives:
GEO.13.1: Define mean, standard deviation, population, sample, and correlation coefficient.
GEO.13.2: Calculate the correlation coefficient.

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.

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.

Students are able to:

• distinguish between correlation and causation

Learning Objectives:
GEO.14.1: Define correlation and causation.

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.

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.

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.

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).

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.

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.

Students know:

• Techniques to find the area and perimeter of parallelograms.
• Techniques to find the area of circles or polygons.

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.

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.

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.

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.

• Dissection arguments

Principle

Cylinder

Pyramid

Cone

Ratio

Circumference

Parallelogram

Limits

Conjecture

Cross-section

Surface Area

Students know:

• Techniques to find the area and perimeter of parallelograms, Techniques to find the area of circles or polygons

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.

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.

Learning Objectives:
GEO.17.1: Define Cavalieri's principle, circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone; oblique, radius, diameter, height, and 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.

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.

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.

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.

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.

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.

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 .

Students know:

• Scale factors of similar figures.
• The ratio of lengths, perimeter, areas, and volumes of similar figures.
• Similar figures.

Students are able to:

• Find the scale factor of any given set of similar figures.
• Find the ratios of perimeter, area, and volume

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

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.

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.

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.

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.

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.

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.

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.

Students know:

• Characteristics of transformations (translations, rotations, reflections, and dilations).
• Methods for representing transformations.
• Characteristics of functions.
• Conventions of functions with mapping notation.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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).

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

Students know:

• Use technology and other tools to discover patterns and relationships in figures.
• Use patterns. relationships and properties to construct figures.

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.

Learning Objectives:
GEO.29.1: Construct a copy of a segment, angle, bisection of a segment, 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.

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.

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).

Students are able to:

• Use known and developed definitions and logical connections to develop new definitions.

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.

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.

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.

Students know:

• Requirements for a mathematical proof.
• Techniques for presenting a proof of geometric theorems.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

Students know:

• Techniques to find measures of geometric shapes.
• Properties of geometric shapes.

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.

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.

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.

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.

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).

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.

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.

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.

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

Students know:

• Properties of geometric shapes.
• Characteristics of a mathematical model.
• How to apply the Mathematical Modeling Cycle to solve design problems.

Students are able to:

• Accurately model and solve a design problem.
• Justify how their model is an accurate representation of the given situation.

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.

Learning Objectives:
GEO.38.1: Define density, area, and 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.