It focuses on reasoning about geometric figures and their properties. Students are expected to make conjectures, prove theorems, and find counterexamples to refute false claims. They are expected to use properties and theorems to construct geometric objects and to perform transformations in the coordinate plane. An optional enrichment unit extending the study of three-dimensional objects to the geometry of a sphere is included for study as time permits. The habits and tools of analysis and logical reasoning developed through studying geometric topics can and should be applied throughout mathematics. With this in mind, the closing unit in this course applies these tools to probability and probability distributions.

Appropriate use of technology is expected in all work. In Geometry, this includes employing technological tools to assist students in the formation and testing of conjectures and in creating diagrams, graphs, and data displays. Geometric constructions should be performed using geometric software as well as classical tools, and technology should be used to aid three-dimensional visualization. Testing with and without technological tools is recommended.

Appropriate use of technology is expected in all work in this Traditional Plus course sequence. In Algebra I this includes employing technological tools to assist students in the formation and testing of conjectures, creating graphs and data displays, and determining and assessing lines of fit for data. Testing with and without technological tools is recommended.

How a particular subject is taught influences not only the depth and retention of the content of a course but also the development of skills in inquiry, problem solving, and critical thinking. Every opportunity should be taken to link the concepts of Geometry to those encountered in middle school and Algebra I as well as to other disciplines. Students should be encouraged to be creative and innovative in their approach to problems, to be productive and persistent in seeking solutions, and to use multiple means to communicate their insights and understanding.

The Major Concepts below provide the focus for the Geometry course. They should be taught using a variety of methods and applications so that students attain a deep understanding of these concepts.

• Geometric Representations
• Reasoning in Geometric Situations
• Similarity, Congruence, and Right Triangle Trigonometry
• Circles
• Three-Dimensional Geometry
• Probability and Probability Distributions

Maintenance Concepts should have been taught previously and are important foundational concepts that will be applied in this Geometry course. Continued facility with and understanding of the Maintenance Concepts is essential for success in the Major Concepts defined for Geometry.

• Geometric Objects and Their Properties
• Length, Area, and Volume
• Graphing in the Coordinate Plane
• Rigid Motions in the Coordinate Plane
• Proportional Reasoning and Similarity
• Two- and Three-Dimensional Representations
• Elementary Data Analysis
• Simple Probability

A. Geometric Representations

Students who have experienced a rigorous middle school mathematics curriculum will be familiar with geometric objects and their measurements and properties. Rather than review these topics directly, this course begins the year with problems involving angles, polygons, circles, solids, length, area, and volume set in the context of probability. It extends the representational repertoire by introducing discrete graphs that can also serve as a vehicle for clarifying real-life situations.

Successful students will:

A1 Recognize probability problems that can be represented by geometric diagrams, on the number line, or in the coordinate plane; represent such situations geometrically and apply geometric properties of length or area to calculate the probabilities.

Example: What is the probability that three randomly chosen points on the plane are the vertices of an obtuse triangle?

A2 Use coordinates and algebraic techniques to interpret, represent, and verify geometric relationships.

Examples: Given the coordinates of the vertices of a quadrilateral, determine whether it is a parallelogram; given a line segment in the coordinate plane whose endpoints are known, determine its length, midpoint, and slope; given the coordinates of three vertices of a parallelogram, determine all possible coordinates for the fourth vertex.

1. Interpret and use locus definitions to generate two- and three-dimensional geometric objects.

   Examples: The locus of points in the plane equidistant from two fixed points is the perpendicular bisector of the line segment joining them; the parabola defined as the locus of points equidistant from the point (5, 1) and the line y = -5 is .

A3 Construct and interpret visual discrete graphs and charts to represent contextual situations.

1. Construct and interpret network graphs and use them to diagram social and organizational networks.

   A graph is a collection of points (nodes) and the lines (edges) that connect some subset of those points; a cycle on a graph is a closed loop created by a subset of edges. A directed graph is one with one-way arrows as edges.

   Examples: Determine the shortest route for recycling trucks; schedule when contestants play each other in a tournament; illustrate all possible travel routes that include four cities; interpret a directed graph to determine the result of a tournament.

2. Construct and interpret decision trees to represent the possible outcomes of independent events.

   A tree is a connected graph containing no closed loops (cycles).

   Examples: Classification of quadrilaterals, repeated tossing of a coin, possible outcomes of moves in a game.

3. Construct and interpret flow charts.

B. Reasoning in Geometric Situations

The geometric objects studied in a rigorous middle school curriculum provide a context for the development of deductive reasoning while also providing a context for students to explore properties, using inductive methods to propose and test conjectures. Students are expected to build on the understanding and formal language of reasoning introduced in Algebra I and use it to describe and justify geometric constructions and theorems.

Successful students will:

B1 Use the vocabulary of logic to describe geometric statements and the relationships among them.

1. Identify assumption, hypothesis, conclusion, converse, and contraposition for geometric statements.
2. Explain and illustrate the role of definitions, conjectures, theorems, proofs, and counterexamples in mathematical reasoning; use geometric examples to illustrate these concepts.

B2 Apply logic to assess the validity of geometric arguments.

1. Analyze the consequences of using alternative definitions for geometric objects.
2. Use geometric examples to demonstrate the effect that changing an assumption has on the validity of a conclusion.
3. Make, test, and confirm or refute geometric conjectures using a variety of methods.
4. Demonstrate through example or explanation how indirect reasoning can be used to establish a claim.
5. Present and analyze direct and indirect geometric proofs using paragraphs or two-column or flow-chart formats; explain how indirect reasoning can be used to establish a claim.

   Example: Explain why, if two lines are intersected by a third line in such a way as to make the corresponding angles, alternate interior angles, or alternate exterior angles congruent, then the two original lines must be parallel.

B3 Analyze, execute, explain, and apply simple geometric constructions.

1. Apply the properties of geometric figures and mathematical reasoning to perform and justify basic geometric constructions.
2. Perform and explain simple straightedge and compass constructions.

   Examples: Copy a line segment, an angle, and plane figures; bisect an angle; construct the midpoint and perpendicular bisector of a segment.

3. Use geometric computer or calculator packages to create and test conjectures about geometric properties or relationships.

C. Similarity, Congruence and Right Triangle Trigonometry

Building on the understanding of rates, ratios, and proportions developed in middle school and applied to number and algebraic situations in Algebra I, the geometric concepts of similarity and congruence are established in this course along with the conditions under which triangles, and later other polygons, are similar or congruent. Students are expected to prove theorems about the similarity of triangles or congruence of angles and triangles using various formats (paragraph, flow-chart, or two-column chart). Applications of similarity include origin-centered dilations in the coordinate plane and right triangle trigonometry.

Successful students will:

C1 Identify and apply conditions that are sufficient to guarantee similarity of triangles.

Informally, two geometric objects in the plane are similar if they have the same shape. More formally, having the same shape means that one figure can be mapped onto the other by means of rigid transformations and/or an origin-centered dilation.

1. Identify two triangles as similar if the ratios of the lengths of corresponding sides are equal (SSS criterion), if the ratios of the lengths of two pairs of corresponding sides and the measures of the corresponding angles between them are equal (SAS criterion), or if the measures of two pairs of corresponding angles are equal (AA criterion).
2. Apply the definition and characteristics of similarity to verify basic properties of angles and triangles and to perform using straightedge and compass or geometric software.
3. Identify the constant of proportionality and determine the measures of corresponding sides and angles for similar triangles.
4. Use similar triangles to demonstrate that the rate of change (slope) associated with any two points on a line is a constant.
5. Recognize, use, and explain why a line drawn inside a triangle parallel to one side forms a smaller triangle similar to the original one.

C2 Identify congruence as a special case of similarity; determine and apply conditions that guarantee congruence of triangles.

Informally, two figures in the plane are congruent if they have the same size and shape. More formally, having the same size and shape means that one figure can be mapped into the other by means of a sequence of rigid transformations.

1. 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).
2. 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 the measure of two pairs of corresponding angles and the length of the side that joins them are equal (ASA criterion).
3. Apply the definition and characteristics of congruence to make constructions, solve problems, and verify basic properties of angles and triangles.

   Examples: Verify that the bisector of the angle opposite the base of an isosceles triangle is the perpendicular bisector of the base; construct an isosceles triangle with a given base angle.

C3 Extend the concepts of similarity and congruence to other polygons in the plane.

1. Identify two polygons as similar if have the same number of sides and angles, if corresponding angles have the same measure, and if corresponding sides are proportional; identify two polygons as congruent if they are similar and their constant of proportionality equals 1.
2. Determine whether or not two polygons are similar.
3. Use examples to show that analogues of the SSS, SAS, and AA criteria for similarity of triangles do not work for polygons with more than three sides.

C4 Analyze, interpret, and represent origin-centered dilations

An origin-centered dilation with scale factor r maps every point (x, y) in the coordinate plane to the point (rx, ry).

1. Interpret and represent origin-centered dilations of objects in the coordinate plane.

   Example: In the following figure, triangle A’B’C’ with A’(9,3), B’(12,6), and C’(15,0) is the dilation of triangle ABC with A(3,1), B(4,2), and C(5,0). The scale factor for this dilation is 3.

   

2. Explain why the image under an origin-centered dilation is similar to the original figure.
3. Show that an origin-centered dilation maps a line to a line with the same slope and that dilations map parallel lines to parallel lines (lines passing through the origin remain unchanged and are parallel to themselves).

C5 Show how similarity of right triangles allows the trigonometric functions sine, cosine, and tangent to be properly defined as ratios of sides.

1. Know the definitions of sine, cosine, and tangent as ratios of sides in a right triangle and use trigonometry to calculate the length of sides, measure of angles, and area of a triangle.
2. Derive, interpret, and use the identity sin²θ + cos²θ = 1 for angles θ between 0° and 90°.

   This identity is a special representation of the Pythagorean theorem.

D. Circles

Once students are familiar with methods of proof and have mastered theorems associated with angles, triangles and other polygons, they are ready to turn their efforts to recognizing and establishing the relationships among the lines, angles, arcs, and areas associated with a circle. Application of these relationships to contextual situations as well as to diagrams is expected in this course.

Successful students will:

D1 Recognize and apply the definitions and the properties of a circle; verify relationships associated with a circle.

1. Know and apply the definitions of radius, diameter, chord, tangent, secant, and circumference of a circle.
2. Recognize and apply the fact that a tangent to a circle is perpendicular to the radius at the point of tangency.
3. Recognize, verify, and apply the relationships between central angles, inscribed angles, and circumscribed angles and the arcs they define.

   Example: Show that a triangle inscribed on the diameter of a circle is a right triangle.

4. Recognize, verify, and apply the relationships between inscribed and circumscribed angles of a circle and the arcs and segments they define.

   Example: Prove that if a radius of a circle is perpendicular to a chord of the circle, then it bisects the chord.

D2 Determine the length of line segments and arcs, the measure of angles, and the area of shapes that they define in complex geometric drawings.

Examples: Determine the amount of glass in a semi-circular transom; identify the coverage of an overlapping circular pattern of irrigation; determine the length of the line of sight on the earth’s surface.

D3 Relate the equation of a circle to its characteristics and its graph.

Examples: Determine the equation of a circle given its center and radius, and conversely, given the equation, determine its center and radius.

E. Three-Dimensional Geometry

It is important for students to develop sound spatial sense for the three-dimensional world in which we live. Volume and surface area of basic solids addressed earlier in middle school will be expanded here to include the calculation of slant height. Beyond the determination of volume and surface area, students should be able to identify the intersection of two or more planes or the figure formed when a solid object is cut by a plane (a cross-sectional slice). Visualization of a three-dimensional object should be extended from the interpretation of nets and multiple views, to the identification and analysis of a solid formed by rotating a line or curve around a given axis, a skill useful in architecture, carpentry, and the study of calculus. An optional topic in three-dimensional geometry extends the study of three dimensions to the exploration of geometry on a sphere, a non-Euclidean space where all lines intersect (that is where the parallel postulate from Euclidean geometry does not hold).

Successful students will:

E1 Determine surface area and volume of solids when slant height is not given; recognize and use relationships among volumes of common solids.

1. Determine the surface area and volume of spheres and of right prisms, pyramids, cylinders, and cones when slant height is not given.
2. Recognize and apply the 3:2:1 relationship between the volumes of circular cylinders, hemispheres, and cones of the same height and circular base.
3. Recognize that the volume of a pyramid is one-third the volume of a prism of the same base area and height and use this to solve problems involving such measurements.

E2 Analyze cross-sections of basic three-dimensional objects and identify the resulting shapes.

Example: Describe all possible results of the intersection of a plane with a cube, prism, pyramid, or sphere.

E3 Describe the characteristics of the three-dimensional object traced out when a one- or two-dimensional figure is rotated about an axis.

E4 Analyze all possible relationships among two or three planes in space and identify their intersections.

1. Know that two distinct planes will either be parallel or will intersect in a line.
2. Demonstrate that three distinct planes may be parallel; two of them may be parallel to each other and intersect with the third, resulting in two parallel lines; or none may be parallel, in which case the three planes intersect in a single point or a single line, or by pairs in three parallel lines.

E5 Recognize that there are geometries other than Euclidean geometry in which the parallel postulate is not true.

E6 Analyze and interpret geometry on a sphere. OPTIONAL ENRICHMENT UNIT

1. Know and apply the definition of a great circle.

   A great circle of a sphere is the circle formed by the intersection of the sphere with the plane defined by any two distinct, non-diametrically opposite points on the sphere and the center of the sphere.

   Example: Show that arcs of great circles subtending angles of 180 degrees or less provide shortest routes between points on the surface of a sphere.

   Since the earth is nearly spherical, this method is used to determine distance between distant points on the earth.

2. Use latitude, longitude, and great circles to solve problems relating to position, distance, and displacement on the earth’s surface.

   Displacement is the change in position of an object and takes into account both the distance and direction it has moved.

   Example: Given the latitudes and longitudes of two points on the surface of the Earth, find the distance between them along a great circle and the bearing from one point to the other.

   Bearing is the direction or angle from one point to the other relative to North = 0°. A bearing of N31°E means that the second point is 31° East of a line pointing due North of the first point.

3. Interpret various two-dimensional representations for the surface of a sphere, called projections (e.g., two-dimensional maps of the Earth), called projections, and explain their characteristics.

   Common projections are Mercator (and other cylindrical projections), Orthographic and Stereographic (and other Azimuthal projections), pseudo-cylindrical, and sinusoidal. Each projection has advantages for certain purposes and has its own limitations and drawbacks.

4. Describe geometry on a sphere as an example of a non-Euclidean geometry in which any two lines intersect.

   In spherical geometry, great circles are the counterpart of lines in Euclidean geometry. All great circles intersect. The angles between two great circles are the angles formed by the intersecting planes defined by the great circles.

   Examples: Show that on a sphere, parallel lines intersect—that is, the parallel postulate does not hold true in this context; identify and interpret the intersection of lines of latitude with lines of longitude on a globe; recognize that the sum of the degree measures of the interior angles of a triangle on a sphere is greater than 180°.

F. Probability and Probability Distributions

The reasoning that is so integral to this course can and should be applied to other areas of mathematics as well as to life experiences. One common type of reasoning is statistical reasoning, which has its grounding in probability distributions. Just as we opened this course by looking at geometric settings for probability, we close it with an introductory look at using probability to help make informed decisions.

Successful students will:

F1 Calculate and apply probabilities of compound events.

1. Employ Venn diagrams to summarize information concerning compound events.
2. Distinguish between dependent and independent events.
3. Use probability to interpret odds and risks and recognize common misconceptions.

   Examples: After a fair coin has come up heads four times in a row, explain why the probability of tails is still 50% in the next toss; analyze the risks associated with a particular accident, illness, or course of treatment; assess the odds of winning the lottery or being selected in a random drawing.

4. Show how a two-way frequency table can be used effectively to calculate and study relationships among probabilities for two events.

F2 Recognize and interpret probability distributions.

1. Identify and distinguish between discrete and continuous probability distributions.
2. Reason from empirical distributions of data to make assumptions about their underlying theoretical distributions.
3. Know and use the chief characteristics of the normal distribution

   The normal (or Gaussian) distribution is actually a family of mathematical functions that are symmetric in shape with scores more concentrated in the middle than in the tails. They are sometimes described as bell shaped. Normal distributions may have differing centers (means) and scale (standard deviation). The standard normal distribution is the normal distribution with a mean of zero and a standard deviation of one. In normal distributions, approximately 68% of the data lie within one standard deviation of the mean and 95% within two.

   Example: Demonstrate that the mean and standard deviation of a normal distribution can vary independently of each other (e.g., that two normal distributions with the same mean can have different standard deviations).

4. Identify common examples that fit the normal distribution (height, weight) and examples that do not (salaries, housing prices, size of cities) and explain the distinguishing characteristics of each.
5. Calculate and use the mean and standard deviation to describe characteristics of a distribution.
6. Understand how to calculate and interpret the expected value of a random variable having a discrete probability distribution.

F3 Apply probability to practical situations to make informed decisions.

Examples: Evaluate medical test results and treatment options, analyze risk in situations where anecdotal evidence is provided, interpret media reports and evaluate conclusions.

1. Communicate an understanding of the inverse relation of risk and return.
2. Explain the benefits of diversifying risk.

About the Benchmarks

Elementary (K–6) Strands and Grade Levels

Secondary (7–12) Strands

Secondary Model Course Sequences

Secondary Assessments and Tasks

Correlations to the Secondary Benchmarks

Supporting Resources

Home

Jump to:

A. Geometric Representations

B. Reasoning in Geometric Situations

C. Similarity, Congruence and Right Triangle Trigonometry

D. Circles

E. Three-Dimensional Geometry

F. Probability and Probability Distributions

Tasks Related to This Course

• Archeological Similarity
• Chinese Restaurant
• Common Differences
• Cones Launch
• Congruence Challenge
• Gamers
• How Odd . . .
• Is Your Score Normal?
• Neighborhood Park
• Regional Triangles
• Rock, Paper, Scissors
• Rolling for the Big One