G: Geometry with Data Analysis

Geometry with Data Analysis

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Transformations, congruence, and similarity are proved using rigid motions and similarity transformations. Circle theorems, coordinate geometry, conic sections, and formal constructions make up the geometry core. Trigonometric ratios, volume formulas, and statistical displays with regression complete the course.

G.1Extend understanding of irrational and rational numbers by rewriting expressions involving radicals, including addition, subtraction, multiplication, and division, in order to recognize geometric patterns.G.10Use 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.G.11Interpret 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.G.12Represent data of two quantitative variables on a scatter plot, and describe how the variables are related.G.13Compute (using technology) and interpret the correlation coefficient of a linear relationship.G.14Distinguish between correlation and causation.G.15Evaluate possible solutions to real-life problems by developing linear models of contextual situations and using them to predict unknown values.G.16Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.G.17Model and solve problems using surface area and volume of solids, including composite solids and solids with portions removed.G.18Given 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.G.19Derive and apply the relationships between the lengths, perimeters, areas, and volumes of similar figures in relation to their scale factor.G.2Use units as a way to understand problems and to guide the solution of multi-step problems.G.20Derive and apply the formula for the length of an arc and the formula for the area of a sector.G.21Represent transformations and compositions of transformations in the plane (coordinate and otherwise) using tools such as tracing paper and geometry software.G.22Explore rotations, reflections, and translations using graph paper, tracing paper, and geometry software.G.23Develop definitions of rotation, reflection, and translation in terms of angles, circles, perpendicular lines, parallel lines, and line segments.G.24Define 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.G.25Verify criteria for showing triangles are congruent using a sequence of rigid motions that map one triangle to another.G.26Verify experimentally the properties of dilations given by a center and a scale factor.G.27Given 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.G.28Verify criteria for showing triangles are similar using a similarity transformation (sequence of rigid motions and dilations) that maps one triangle to another.G.29Find patterns and relationships in figures including lines, triangles, quadrilaterals, and circles, using technology and other tools.G.3Find 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.G.30Develop 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.G.31Justify 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.G.32Use coordinates to prove simple geometric theorems algebraically.G.33Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems.G.34Use congruence and similarity criteria for triangles to solve problems in real-world contexts.G.35Discover and apply relationships in similar right triangles.G.36Use geometric shapes, their measures, and their properties to model objects and use those models to solve problems.G.37Investigate 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.G.38Use the mathematical modeling cycle involving geometric methods to solve design problems.G.4Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations.G.5Verify 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.G.6Derive the equation of a circle of given center and radius using the Pythagorean Theorem.G.7Use 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.G.8Use technology to organize data, including very large data sets, into a useful and manageable structure.G.9Represent 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.
Example Problems
Simplify:
You have found the following ages (in years) of all 4 rhinos at your local zoo:


What is the standard deviation of the age of the rhinos at your zoo? Round your answers to the nearest tenth.
What is the mean of the following numbers?
Zara studied the age (in years) and resale value (in thousand dollars) of a random sample of 40 used cars. Here is computer output on the sample data:

Regression: resale value vs. age
PredictorCoefSE Coef
Constant18.70.9
Age-1.850.31

Assume that all conditions for inference have been met.

Write the expression for the test statistic for testing the null hypothesis that the population slope in this setting is .
Leona collected data on the battery life (in hours) and price (in dollars) of a random sample of mobile phones. Here is computer output from a least-squares regression analysis on her sample:

PredictorCoefSE Coef
Constant191.31294.318
Battery life6.5563.788




Assume that all conditions for inference have been met.

Write the expression for the test statistic for testing the null hypothesis that the population slope in this setting is 0.
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