Geometry Standards
MA19.GDA.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.
Knowledge
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.
Skills
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).
Understanding
Students understand that:
- rewriting radical expressions of rational and irrational numbers can help in recognizing geometric patterns.
Vocabulary
Rational numbers
Irrational numbers
Geometric Patterns
MA19.GDA.2
Use units as a way to understand problems and to guide the solution of multi-step problems.
MA19.GDA.2a
Choose and interpret units consistently in formulas.
MA19.GDA.2b
Choose and interpret the scale and the origin in graphs and data displays.
MA19.GDA.2c
Define appropriate quantities for the purpose of descriptive modeling.
MA19.GDA.2d
Choose a level of accuracy appropriate to limitations of measurements when reporting quantities.
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.
Vocabulary
| Units | Justify | Formulas | Accuracy | Area |
|---|---|---|---|---|
| Scales | Identify | Scale | Margin of error | Direct measurement |
| Descriptive modeling | Quantities | Consistency | Perimeter | |
| Interpret | Dimensional analysis | Precise | Volume |
MA19.GDA.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.
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.
Vocabulary
vertices
Function rules
linear equations
System of equations
MA19.GDA.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.
COS Examples
Example: Rearrange the formula for the area of a trapezoid to highlight one of the bases.
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.
Vocabulary
Literal equations
Variable
Constant
MA19.GDA.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.
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
Vocabulary
Graphically Finite solutions
Infinite solutions
MA19.GDA.6
Derive the equation of a circle of given center and radius using the Pythagorean Theorem.
MA19.GDA.6a
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.
MA19.GDA.6b
Derive the distance formula from the Pythagorean Theorem.
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)
Vocabulary
Pythagorean theorem
Radius
Translation
MA19.GDA.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.
COS Examples
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.
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.
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
MA19.GDA.8
Use technology to organize data, including very large data sets, into a useful and manageable structure.
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.
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.
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.
Vocabulary
Continuous data
Discrete data
quantitative
Categorical
line of best fit
Curve of best fit
Scatter plot
MA19.GDA.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.
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.
Vocabulary
Dot plots
Histograms
Box plots
Scatter plots
Univariate data
Bivariate data
MA19.GDA.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.
MA19.GDA.10a
Explain how standard deviation develops from mean absolute deviation.
MA19.GDA.10b
Calculate the standard deviation for a data set, using technology where appropriate.
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
Vocabulary
Center
Median
Mean
Spread
Interquartile range
Standard deviation
Absolute mean deviation
MA19.GDA.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.
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.
Vocabulary
Outliers
Center
Shape
Spread
Mean
Standard deviation
MA19.GDA.12
Represent data of two quantitative variables on a scatter plot, and describe how the variables are related.
MA19.GDA.12a
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.
MA19.GDA.12b
Use technology to find the least-squares line of best fit for two quantitative variables.
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.
Vocabulary
Quantitative variables
Scatter plot
Residuals
MA19.GDA.13
Compute (using technology) and interpret the correlation coefficient of a 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.
Vocabulary
Interpret
Correlation coefficient
linear relationship
MA19.GDA.14
Distinguish between correlation and 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
Vocabulary
Correlation
Causation
MA19.GDA.15
Evaluate possible solutions to real-life problems by developing linear models of contextual situations and using them to predict unknown values.
MA19.GDA.15a
Use the linear model to solve problems in the context of the given data.
MA19.GDA.15b
Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the context of the given data.
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).
Vocabulary
Quantitative variables
Scatter plot
Residuals
Slope
Rate of change
Intercepts
Constant
Ordered pairs
Horizontal lines
Vertical lines
MA19.GDA.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.
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.
Vocabulary
Dissection arguments
Cavalieri's Principle
Cylinder
Pyramid
Cone
Ratio
Circumference
Parallelogram
Limits
Conjecture
Cross-section
MA19.GDA.17
Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.
MA19.GDA.17a
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.
MA19.GDA.17b
Apply geometric concepts to find missing dimensions to solve surface area or volume problems.
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.
Vocabulary
- Dissection arguments
- Principle
- Cylinder
- Pyramid
- Cone
- Ratio
- Circumference
- Parallelogram
- Limits
- Conjecture
- Cross-section
- Surface Area
MA19.GDA.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.
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.
Vocabulary
Coordinates
vertices
perimeter
Area
Distance formula
Evaluate
Accuracy
MA19.GDA.19
Derive and apply the relationships between the lengths, perimeters, areas, and volumes of similar figures in relation to their scale factor.
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
Vocabulary
Derive
Apply
Scale Factor
Similar figures
Ratio of length
Ratio of perimeter
Ratio of area
Ratio of volume
MA19.GDA.20
Derive and apply the formula for the length of an arc and the formula for the area of a sector.
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.
Vocabulary
Similarity
Constant of proportionality
Sector
Arc
Derive
Arc length
Radian measure
Area of sector
Central angle
Dilation
MA19.GDA.21
Represent transformations and compositions of transformations in the plane (coordinate and otherwise) using tools such as tracing paper and geometry software.
MA19.GDA.21a
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.
MA19.GDA.21b
Compare transformations which preserve distance and angle measure to those that do not.
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.
Vocabulary
Transformation
Reflection
Translation
Rotation
Dilation
Isometry
Composition
Horizontal stretch
Vertical stretch
Horizontal shrink
Vertical shrink
Clockwise
Counterclockwise
Symmetry
Preimage
Image
MA19.GDA.22
Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.
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.
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
MA19.GDA.22a
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.
MA19.GDA.22b
Specify a sequence of rotations, reflections, or translations that will carry a given figure onto another.
MA19.GDA.22c
Draw figures with different types of symmetries and describe their attributes.
MA19.GDA.23
Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments.
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.
Vocabulary
Transformation
Reflection
Translation
Rotation
Dilation
Isometry
Composition
Clockwise
Counterclockwise
Preimage
Image
MA19.GDA.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.
COS Examples
Example: $\Delta ABC$ is congruent to $\Delta XYZ$ since a reflection followed by a translation maps $\Delta ABC$ onto $\Delta XYZ$.
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.
Vocabulary
Rigid motions
Congruence
MA19.GDA.25
Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another.
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).
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)
MA19.GDA.25a
Verify that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.
MA19.GDA.25b
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).
COS Examples
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.
MA19.GDA.26
Verify experimentally the properties of dilations given by a center and a 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.
Vocabulary
Dilations
Center
Scale factor
MA19.GDA.26a
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.
MA19.GDA.26b
Verify that the dilation of a line segment is longer or shorter in the ratio given by the scale factor.
MA19.GDA.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.
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.
Vocabulary
Similarity transformation
Similarity
Proportionality
Corresponding pairs of angles
Corresponding pairs of sides
Rigid Motion
MA19.GDA.28
Verify criteria for showing triangles are similar using a similarity transformation (sequence of rigid motions and dilations) that maps one triangle to another.
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.
Vocabulary
Similarity transformation
Similarity
Proportionality
Corresponding pairs of angles
Corresponding pairs of sides
Similarity criteria for triangles
Rigid Motion
MA19.GDA.28a
Verify that two triangles are similar if and only if corresponding pairs of sides are proportional and corresponding pairs of angles are congruent.
MA19.GDA.28b
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).
COS Examples
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.
MA19.GDA.29
Find patterns and relationships in figures including lines, triangles, quadrilaterals, and circles, using technology and other tools.
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.
Vocabulary
Conjectures
Construct
Congruent
Compass
Straightedge
MA19.GDA.29a
Construct figures, using technology and other tools, in order to make and test conjectures about their properties.
MA19.GDA.29b
Identify different sets of properties necessary to define and construct figures.
MA19.GDA.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.
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.
Vocabulary
Same side interior angle
Consecutive interior angle
Vertical angles
Linear pair
Adjacent angles
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
MA19.GDA.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.
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.
Vocabulary
Same side interior angle
Consecutive interior angle
Vertical angles
Linear pair
Adjacent angles
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
MA19.GDA.31a
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.
MA19.GDA.31b
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 $180^{\circ}$; 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.
MA19.GDA.31c
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.
COS Examples
Example: Prove that rectangles are parallelograms with congruent diagonals.
MA19.GDA.32
Use coordinates to prove simple geometric theorems algebraically.
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.
Vocabulary
Simple geometric theorems
Simple geometric figures
MA19.GDA.33
Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.
COS Examples
Example: Find the equation of a line parallel or perpendicular to a given line that passes through a given point.
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.
Vocabulary
Parallel lines
Perpendicular lines
Slope
Slope triangle
MA19.GDA.34
Use congruence and similarity criteria for triangles to solve problems in real-world contexts.
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.
Vocabulary
Congruence and similarity criteria for triangles
MA19.GDA.35
Discover and apply relationships in similar right triangles.
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.
Vocabulary
Side ratios
Trigonometric ratios
Sine
Cosine
Tangent
Secant
Cosecant
Cotangent
Complementary anglesconverse
MA19.GDA.35a
Derive and apply the constant ratios of the sides in special right triangles ($45^{\circ} -45^{\circ} -90^{\circ}$ and $30^{\circ} -60^{\circ} -90^{\circ}$).
MA19.GDA.35b
Use similarity to explore and define basic trigonometric ratios, including sine ratio, cosine ratio, and tangent ratio.
MA19.GDA.35c
Explain and use the relationship between the sine and cosine of complementary angles.
MA19.GDA.35d
Demonstrate the converse of the Pythagorean Theorem.
MA19.GDA.35e
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems, including finding areas of regular polygons.
][
MA19.GDA.36
Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.
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.
Vocabulary
Model
MA19.GDA.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.
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.
Vocabulary
Central angles
Inscribed angles
Circumscribed angles
Chord
Circumscribed
Tangent
Perpendicular arc
MA19.GDA.38
Use the mathematical modeling cycle involving geometric methods to solve design problems.
COS Examples
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.
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.
Vocabulary
Geometric methods
Design problems
Typographic grid system
Density