**What Geometry and Statistics Standards and Skills should you master for the Praxis 5161 Exam?**

Since the Praxis 5161 Math Credential enables one teach High School Math, it's an extremely profitable exercise to scrutinize the Math content standards applicable for High School Math teachers in your state. These are expectations for students that every current and prospective Math teacher ought to be familiar with.

The following content standards apply for Geometry and Statistics, in general.

The following content standards apply for Geometry and Statistics, in general.

__GEOMETRY__**:**The geometry skills and concepts developed in this discipline are useful to all. Aside from learning these skills and concepts, candidates will develop their ability to construct formal, logical arguments and proofs in geometric settings and problems.**Candidates shall**- Demonstrate understanding by identifying and giving examples of undefined terms, axioms, theorems, and inductive and deductive reasoning.
- Write geometric proofs, including proofs by contradiction.
- Construct and judge the validity of a logical argument and give counterexamples to disprove a statement.
- Prove basic theorems involving congruence and similarity.
- Prove that triangles are congruent or similar, and they are able to use the concept of corresponding parts of congruent triangles.
- Know and are able to use the triangle inequality theorem.
- Prove and use theorems involving the properties of parallel lines cut by a transversal, the properties of quadrilaterals, and the properties of circles.
- Know, derive, and solve problems involving the perimeter, circumference, area, volume, lateral area, and surface area of common geometric figures.
- Compute the volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres; and commit to memory the formulas for prisms, pyramids, and cylinders.
- Compute areas of polygons, including rectangles, scalene triangles, equilateral triangles, rhombi, parallelograms, and trapezoids.
- Determine how changes in dimensions affect the perimeter, area, and volume of common geometric figures and solids.
- Find and use measures of sides and of interior and exterior angles of triangles and polygons to classify figures and solve problems.
- Prove relationships between angles in polygons by using properties of complementary, supplementary, vertical, and exterior angles.
- Prove the Pythagorean theorem.
- Use the Pythagorean theorem to determine distance and find missing lengths of sides of right triangles.
- Perform basic constructions with a straightedge and compass, such as angle bisectors, perpendicular bisectors, and the line parallel to a given line through a point off the line.
- Prove theorems by using coordinate geometry, including the midpoint of a line segment, the distance formula, and various forms of equations of lines and circles.
- Know the definitions of the basic trigonometric functions defined by the angles of a right triangle. They also know and are able to use elementary relationships between them.
- Use trigonometric functions to solve for an unknown length of a side of a right triangle, given an angle and a length of a side.
- Know and are able to use angle and side relationships in problems with special right triangles, such as 30°, 60°, and 90° triangles and 45°, 45°, and 90° triangles.
- Prove and solve problems regarding relationships among chords, secants, tangents, inscribed angles, and inscribed and circumscribed polygons of circles.
- Know the effect of rigid motions on figures in the coordinate plane and space, including rotations, translations, and reflections.
- Derive and apply the Law of Sines and Cosines.

**Geometry Checklist of Skills:**You must:- Know and apply the distance formula.
- Know and apply the Midpoint formula.

- Know and apply symmetric of graphs with respect to the axis, y axis, origin, line y = x, line y = -x.
- Know the point-slope form, slope intercept form of equation of lines.
- Know the condition for parallel and perpendicular lines using slopes.
- Know properties of isosceles, equilateral, right triangles.
- Know and apply the Triangle Congruence Theorems: SSS, ASA, AAS, SAS, HL.
- Prove geometric theorems and propositions in Statement- Reason form as well as in coordinate form.
- Provide basic geometric theorems using Indirect Proof ( Proof By Contradiction).
- Know and apply properties of perpendicular and angle bisectors.
- Know and apply theorems concerning concurrency of perpendicular bisectors, angle bisectors and medians of a triangle relating to acute, obtuse and right triangles.
- Know and apply theorems on triangle inequalities.
- Know and apply properties of interior and exterior angles of convex polygons.
- Know and apply properties of parallelograms, squares, kites, rectangles, rhombus relating to their sides, angles and diagonals.
- Know formulae of area of basic quadrilaterals.
- Be able to identify transformations as isometrics.
- Be able to perform isometric transformations of points in the coordinate plane: Rotations about the origin as well as about the other points: Reflection about lines x = a, y = b as well as y=mx + b: translations.
- Know and apply the Similarity Theorems: AA, SAS, SSS.
- Apply concepts of proportion and similarity in word problems relating to geometric situations.
- Be able to perform dilations of objects in the coordinating plane.
- Solve Right Triangle Trigonometry: Determine missing sides and angles for a triangle.
- Know properties of 30°-60°-90°∆s and 45°-45°-90° ∆s.
- Use Trigonometric Ratios to determine coordinates of points lying on the vertices of regular polygons.
- Know properties of circles relating to tangent, chords, secants and angle of intersection.
- Be able to determine the areas and perimeters of regular polygons using trigonometry.
- Know properties of perimeters and areas of similar polygons.
- Be able to determine angle measures, areas and arc lengths of sectors in circles.
- Be able to solve basic problems relating to geometric probability.
- Know the Pythagorean Theorem.
- Prove geometric theorems and propositions in Statements – Reasons form well as in coordinate form.
- Prove basic geometric theorems using Indirect Proof (i.e. Proof By Contradiction).
- Know formulae concerning surface area and volume of 3-dimensional solids: Prism, Cylinder, Cone, Pyramid and Sphere.
- Know properties of surface area and volume for similar solids.
- Calculate areas and volumes of simple and complex 3-dimensional solids.
- Know Euler's Theorem relating to the faces, sides and vertices of polyhedrons.
- Be able to Prove and Apply the Law of Sines and Cosines.

- Be able to Prove Theorems on parallel lines: Consecutive Interior Angle Theorem, Alternate Interior Angle Theorem, Alternate Exterior Angle Theorem, Consecutive Interior Angle Theorem Converse, Alternate Interior Angle Theorem Converse, Alternate Exterior Angle Converse; that two lines are parallel using Converse Theorems
- Be able to Prove Triangle Sum Theorem, Exterior Angle Theorem.
- Be able to Prove and Apply the theorems of Triangle Congruence to prove congruence of angles and sides
- Be able to Prove the Base Angles Theorem and its Converse for Isosceles triangles.
- Be able to Prove the Perpendicular Bisectors Theorem and its converse.
- Be able to Prove the Angle Bisector Theorem and its converse.
- Be able to Prove the Mid-Segment Theorem.
- Know Triangle Inequalities related to sides and angles of a triangle.
- Be able to Prove Theorems relating to properties of Parallelogram: Opposite sides are congruent, Opposite angles are congruent, Consecutive angles are supplementary, Diagonals bisect each other.
- Be able to Prove that a Quadrilateral is a parallelogram: Converse Theorems of the Parallelogram Theorems
- Know and Apply the Properties of Rhombus, Rectangle, Square, Kite and Trapezoid to their aspects viz. Diagonals, sides and angles.
- Prove and Apply the Pythagorean Theorem and Properties of Special Right Triangles (30°-60°-90° ∆s and 45°-45°-90° ∆s).

- Be able to Prove Circle Theorems: Perpendicularity of Radius to the circle at the point of tangency; Tangents from external point being congruent;

- Be able to Prove the Inscribed Angle Theorem.
- Know Theorems about the chords of circles.
- Prove Theorems about inscribed polygons: Triangle inscribed in a semicircle is a right triangle; Quadrilateral inscribed in a circle has its opposite angles supplementary.

- Prove the Sum of Interior Angle Sum Theorem of a Convex Polygon

__STATISTICS__**:**This discipline is a technical and in-depth extension of probability and statistics.**Candidates shall****Describe Univariate Data:**- Calculate, interpret and apply the properties of the Measures of Central Tendency and dispersion of a data set.
- Determine the mean, median mode, standard deviation and variance of a data-set in myriad formats: table, histogram, frequency or probability distributions
- Calculate percentiles, given a data-set, and determine the X-value, given a percentile
- Be familiar with common displays of data: pie-charts, bar-graphs, histograms, dot-plots, box-plots, stem-and-leaf plots
- Know how the various Numerical Summaries – as Mean, Median, s.d., variance – are changed when the data is transformed ie. a constant is added / multipled to each datum;
- Know which measure(s) to use to describe the data, depending on the skewness of the distribution.

**Describe Bivariate Data:**- Know the difference between Correlation and Regression;
- Calculate and interpret the Correlation Coefficient,
*r*, and the Least-Squares Regression Line (LSRL) by hand, and by calculator; - Know the properties of the Correlation Coefficient and the LSRL
- Possess a broad comprehension of the derivation of the LSRL (know how the line came about)
- Compute the coefficients of the LSRL based on numerical summaries (mean and s.d. of the x- and y-variables, and the correlation coefficient, r)
- Make predictions using the LSRL.
- Understand the effects of outliers on correlation and the LSRL

**Methods of Generate Data:**- Possess a broad understanding of different ways of Producing Data and the Advantages (Benefits) and Disadvantages (Constraints) pertaining to each: Census; Surveys, Observation Studies and Experiments
- Describe and implement the different types of Sampling methods (Simple Random, Stratified, Systematic, Cluster, etc)
- Know the different kinds of Biases in Sampling (Selection Bias; Response Bias; Non-Response Bias);
- Understand the means to control Sampling Variability

**Apply Principles of Probability:**- Apply the basic Rules of Probability;
- Identify Dependent and Independent Events, Mutually Exclusive and Non-Disjoint Events
- Determine the probabilities of simple and compound events
- Calculate Conditional probability using the formula and a Tree Diagram or Probability table

**Describe Common Discrete and Continuous Distributions:**- Have a general knowledge of different kinds of distributions and their underlying assumptions and properties: Discrete (Binomial) and Continuous (Uniform, Normal);
- Be able to identify common distributions pertaining to real-life situations
- Be able to determine probabilities and percentiles pertaining to the Binomial, Normal and Uniform Distributions
- Calculate the Mean and standard deviation of a Binomial distribution
- Understand the notion of a continuous random variable and can interpret the probability of an outcome as the area of a region under the graph of the probability density function associated with the random variable.

**Statistical Inference:**- Be familiar with the notions of a statistic of a distribution of values, of the sampling distribution of a statistic, and of the variability of a statistic.
- Apply the Central Limit Theorem: know basic facts about the relation between the mean and the standard deviation of a sampling distribution and the mean and the standard deviation of the population distribution relating to sample means and sample proportions.
- Determine confidence intervals for a simple random sample from a normal distribution of data and determine the sample size required for a desired margin of error.
- Perform simple tests of hypotheses for a sample proportion and sample mean for a simple random sample from a normal distribution.
- Perform Tests of Hypothesis for the Chi-square Goodness of Fit and the Test of Independence of 2 categorical variables;
- Interpret the P-value from Tests of Hypotheses and draw suitable conclusions.
- Calculate and interpret Confidence Intervals for the true proportion and true mean based on a random sample from a normal distribution

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