Secondary Mathematics Benchmarks Progressions, Grades 7–12: Geometry (G)

Geometry is an ancient mathematical endeavor. The roots of modern geometry were laid by Euclid more than twenty-two centuries ago. Since that time, geometry has been an integral part of mathematics and a common vehicle for teaching the critical skill of deductive reasoning. It offers a physical context in which students can develop and refine intuition, leading to the formulation and testing of hypotheses and ultimately resulting in the justification of arguments, both formally and informally. Geometry also describes changes in objects under such transformations as translation, rotation, reflection, and dilation. It helps students understand the structure of space and the nature of spatial relations. The measurement aspect of geometry provides a basis by which we quantify the world. Geometry is prerequisite for a broad range of activities and leads to methods for resolving practical problems; it can help find the best way to fit an oversized object through a door, aid in carpentry projects, or provide the basis for industrial tool design. Solving practical problems relies to some extent on approximate physical measurements but also rests on geometric properties that are exact in nature. The axioms and definitions of geometry assure us of such measures as the volume of a cube 1 unit on a side, the measure of an angle of an equilateral triangle, or the area of a circle with a given diameter. Grounded in such certainty, geometry provides an excellent medium for the development of students’ ability to reason and produce thoughtful, logical arguments.

G.A.1 Angles and triangles

a. Know the definitions and basic properties of angles and triangles in the plane and use them to solve problems.

• Know and apply the definitions and properties of complementary and supplementary angles.
• Know and apply the definitions and properties of interior and exterior angles.

Tasks related to this benchmark: Congruence Challenge, Regional Triangles

b. Know and prove basic theorems about angles and triangles.

• Know the triangle inequality and verify it through measurement.

  In words, the triangle inequality states that any side of a triangle is shorter than the sum of the other two sides; it can also be stated clearly in symbols: If a, b, and c are the lengths of three sides of a triangle, then a < b + c, b < a + c, and c < a + b.

• Verify that the sum of the measures of the interior angles of a triangle is 180°.
• Verify that each exterior angle of a triangle is equal to the sum of the opposite interior angles.
• Show that the sum of the interior angles of an n-sided convex polygon is (n – 2) × 180°.

  A common strategy is to decompose an n-sided convex polygon into n – 2 triangles.

• Explain why the sum of exterior angles of a convex polygon is 360°.

  A possible explanation is that a person walking completely around the perimeter of a convex polygon would have turned through the number of degrees in each external angle at each vertex and would have made one complete revolution when reaching the starting point again. So, the sum of exterior angles must be 360°.

G.A.2 Rigid motions in the plane

a. Represent and explain the effect of translations, rotations, and reflections of objects in the coordinate plane.

• Identify certain transformations (translations, rotations, and reflections) of objects in the plane as rigid motions and describe their characteristics.

  Translation, rotation, and reflection move a polygonal figure in the plane from one position to another without changing its linear or angular measurements, i.e., without altering its size or shape.

• Demonstrate the meaning and results of the translation, rotation, and reflection of an object through drawings and experiments.

b. Identify corresponding sides and angles between objects and their images after a rigid transformation.

Corresponding angles are those that lie between edges that correspond under the map showing that the triangles have the same size and shape.

c. Show how any rigid motion of a figure in the plane can be created by some combination of translations, rotations, and reflections.

G.A.3 Measurement

a. Make, record, and interpret measurements.

• Recognize that measurements of physical quantities must include the unit of measurement, that most measurements permit a variety of appropriate units, and that the numerical value of a measurement depends on the choice of unit, and apply these ideas when making such measurements.
• Recognize that real-world measurements are approximations; identify appropriate instruments and units for a given measurement situation, taking into account the precision of the measurement desired.
• Plan and carry out both direct and indirect measurements.

  Indirect measurements are those that are calculated based on actual recorded measurements.

b. Apply units of measure in expressions, equations, and problem situations.

• When necessary, convert measurements from one unit to another within the same system.

c. Use measures of weight, money, time, information, and temperature.

• Identify the name and definition of common units for each kind of measurement, e.g., kilobytes of computer memory.

d. Record measurements to reasonable degrees of precision, using fractions and decimals as appropriate.

A measurement context often defines a reasonable level of precision to which the result should be reported.

Example: The U.S. Census Bureau reported a national population of 299,894,924 on its Population Clock in mid-October of 2006. Saying that the U.S. population is 3 hundred million (3x10⁸) is accurate to the nearest million and exhibits to one-digit precision. Although by the end of that month the population had surpassed 3 hundred million, 3x10⁸ remained accurate to one-digit precision.

Task related to this benchmark: A Safe Load

G.A.4 Length, area, and volume

a. Identify and distinguish among measures of length, area, surface area, and volume.

• Identify and distinguish between the perimeter or circumference of a two-dimensional geometric figure and its area.
• Identify and distinguish between the surface area of a three-dimensional geometric object and its volume.

Task related to this benchmark: Neighborhood Park

b. Calculate the perimeter and area of triangles, quadrilaterals, and shapes that can be decomposed into triangles and quadrilaterals that do not overlap.

• Know and apply formulas for the area and perimeter of triangles and rectangles to derive similar formulas for parallelograms, rhombi, trapezoids, and kites.

Tasks related to this benchmark: Leo's Painting, Neighborhood Park

c. Determine the surface area of right prisms and pyramids whose base(s) and sides are composed of rectangles and triangles.

d. Know and apply formulas for the surface area of right circular cylinders, right circular cones, and spheres.

• Explain why the surface area of a right circular cylinder is a rectangle whose length is the circumference of the base of the cylinder and whose width is the height of the cylinder.

Task related to this benchmark: Cones Launch

e. Know and apply formulas for the volume of right prisms, right pyramids, right circular cylinders, right circular cones, and spheres.

Task related to this benchmark: Cones Launch

f. Estimate lengths, areas, surface areas, and volumes of irregular figures and objects.

G.B.1 Angles in the plane

a. Know and distinguish among the definitions and properties of vertical, adjacent, corresponding, and alternate interior angles.

• When a line intersects two parallel lines, identify pairs of angles that are vertical, corresponding, and alternate interior.

Task related to this benchmark: Congruence Challenge

b. Identify pairs of vertical angles and explain why they are congruent.

Task related to this benchmark: Congruence Challenge

c. Identify pairs of corresponding, alternate interior, and alternate external angles in a diagram where two parallel lines are cut by a transversal, and show that they are congruent.

Task related to this benchmark: Congruence Challenge

d. 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.

Task related to this benchmark: Congruence Challenge

e. Apply properties of lines and angles to perform basic geometric constructions.

Example: Using only a straight edge and compass, construct the perpendicular bisector of a given line segment and the bisector of a given angle.

G.B.2 Coordinates and slope

a. Represent and interpret points, lines, and two-dimensional geometric objects in a coordinate plane.

Task related to this benchmark: Neighborhood Park

b. Determine the area of polygons in the coordinate plane.

• Determine the area of quadrilaterals and triangles situated in a coordinate plane whose base/height/length/width are horizontal or vertical line segments.
• Determine the area of a polygon in the coordinate plane by encasing the figure in a rectangle and subtracting the area of extraneous parts.

  Example: Find the area of the shaded pentagon by subtracting the area of the five un-shaded triangles from the area of the rectangle shown.

  

c. Know how the word slope is used in common non-mathematical contexts, give physical examples of slope, and calculate slope for given examples.

• Interpret slope as a physical characteristic.

  Example: The slope of a ramp with horizontal length 12 feet and vertical rise 4 feet is 4/12 or 1/3.

• Find the slopes of physical objects (roads, roofs, ramps, stairs) and express the answers as a decimal, ratio, or percent.

d. Calculate the slope of a line in a coordinate plane.

• Explain how this relates to slope as a physical characteristic.

  The measurement of slope in a physical context does not include any inherent concept of positive or negative direction, as is the case with slope in a coordinate plane.

e. Interpret and describe the slope of parallel and perpendicular lines in a coordinate plane.

• Show that the calculated slope of a line in a coordinate plane is the same no matter which two distinct points on the line one uses to calculate the slope.
• Demonstrate why two non-vertical lines in a coordinate plane are parallel if and only if they have the same slope and perpendicular if and only if the product of their slopes is –1.
• Use coordinate geometry to determine the perpendicular bisector of a line segment.

G.B.3 Pythagorean theorem

a. Interpret and prove the Pythagorean theorem and its converse.

The most common proofs employ geometric dissection.

b. Determine distances between points in the Cartesian coordinate plane.

• Relate the Pythagorean theorem to this process.

c. Apply the Pythagorean theorem and its converse to solve problems.

Task related to this benchmark: Archeological Similarity

G.B.4 Circles

a. Identify and explain the relationships among the radius, diameter, circumference, and area of a circle.

• Identify the relationship between the circumference of a circle and its radius or diameter as a direct proportion.
• Identify the relationship between the area of a circle and the square of its radius or the square of its diameter as a direct proportion.
• Explain why the area of a circle equals its radius times one-half of its circumference.

  One way to "see" this is to slice a circle into many small pie-shaped pieces and then match them head to toe. The result will be a bumpy rectangular shape whose height is approximately the radius of the circle and whose width is approximately half the circumference of the circle.

Task related to this benchmark: Satellite

b. Show that for any circle, the ratio of the circumference to the diameter is the same as the ratio of the area to the square of the radius and that these ratios are the same for different circles.

• Identify the constant ratio A/r² = ½Cr/r² = C/2r = C/d as the number π and show that although the rational numbers 3.14, or are often used to approximate π, they are not the actual values of the irrational number π.
• Show that the area of a unit circle (one whose radius is 1) is π.
• Identify and describe methods for approximating π.

  Examples: Archimedes constructed a paired sequence of regular polygons with increasing numbers of sides that inscribed and circumscribed a unit circle and used the areas of the polygons to determine upper and lower bounds for the successive approximations for the area of the circle—hence of π; the Buffon Needle problem provides a method of approximating π using geometric probability—

  Note: New computers and software are often tested by calculating π to billions of places.

c. Know and apply formulas for the circumference and area of a circle.

• Determine the perimeter and area of a semicircle and a quarter-circle.

Tasks related to this benchmark: Cones Launch, Satellite

G.B.5 Scaling, dilation, and dimension

a. Analyze and represent the effects of multiplying the linear dimensions of an object in the plane or in space by a constant scale factor, r.

Example: Multiplying the lengths of the sides of a polygon by r results in a polygon having the same shape as the original.

Mathematically, having the same shape means that the image will have the same number of angles and sides as the pre-image, that all angles will preserve their measure, and that corresponding sides will be proportional.

• Use ratios and proportional reasoning to apply a scale factor to a geometric object, a drawing, a three-dimensional space, or a model, and analyze the effect.

Task related to this benchmark: Neighborhood Park

b. Describe the effect of a scale factor r on length, area, and volume.

• Explain why triangles in the plane with corresponding sides having a scale factor r have areas related by a factor of r² and tetrahedrons in space with corresponding sides having a scale factor r have volumes related by a factor of r³.

  Since any polygon can be decomposed into a finite number of triangles, it follows that any two similar polygons with scale factor r have areas related by a factor of r². A similar argument works for polyhedral figures in space.

• Extend the concept of scale factor to relate the length, area, and volume of other figures and objects.

  Examples: Explain how a change in the length of radius affects the area of a circle and the volume of a sphere; compare the metabolic rate of a man with that of someone twice his size (the metabolic rate of the human body is proportional to the body mass raised to the ¾ power).

Task related to this benchmark: Neighborhood Park

c. Recognize and use relationships among volumes of common solids.

• Recognize and apply the 3:2:1 relationship between the volumes of circular cylinders, hemispheres, and cones of the same height and circular base.
• 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.

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

A dilation centered at the origin with scale factor r maps the point (x, y) to the point (rx, ry).

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.

• Analyze the effect on an object when it is subjected to an origin-centered dilation.

  The image under an origin-centered dilation will have the same shape as the original object.

• Show that a dilation maps a line to a line with the same slope and that dilations map parallel lines to parallel lines (except for those passing through the origin, which do not change).

Task related to this benchmark: Neighborhood Park

G.B.6 Similarity and congruence

a. Interpret the definition and characteristics of similarity for triangles in the plane.

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.

• Know that two triangles are similar if their corresponding angles have the same measure.
• Know that the ratio formed by dividing the lengths of corresponding sides of similar triangles is a constant, often called the constant of proportionality, and determine this constant for given similar triangles.

Task related to this benchmark: Archeological Similarity

b. Apply similarity in practical situations.

• Calculate the measures of corresponding parts of similar figures.
• Use the concepts of similarity to create and interpret scale drawings.

Task related to this benchmark: Archeological Similarity

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

• 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 two pairs of corresponding angles are congruent (AA criterion).
• Apply the SSS, SAS, and AA criteria to verify whether or not two triangles are similar.
• Apply the SSS, SAS, and AA criteria to construct a triangle similar to a given triangle using straightedge and compass or geometric software.
• Identify the constant of proportionality and determine the measures of corresponding sides and angles for similar triangles.
• Use similar triangles to demonstrate that the rate of change (slope) associated with any two points on a line is a constant.
• Recognize that a line drawn inside a triangle parallel to one side forms a smaller triangle similar to the original one; explain why this is true and apply this knowledge.

Task related to this benchmark: Archeological Similarity

d. Explain why congruence is 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.

• 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 two pairs of corresponding angles and their included sides are all equal (ASA criterion).

Task related to this benchmark: Congruence Challenge

e. Apply the definition and characteristics of congruence to make constructions, solve problems, and verify basic properties of angles and triangles.

• Verify that two triangles are congruent if they are formed by drawing a diagonal of a parallelogram or by bisecting the vertex angle of an isosceles triangle.
• Verify that the bisector of the angle opposite the base of an isosceles triangle is the perpendicular bisector of the base.
• Verify that the base angles of an isosceles triangle are equal.

Task related to this benchmark: Congruence Challenge

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

A closed plane figure is called a polygon if all of its edges are line segments, every vertex is the endpoint of two sides, and no two sides cross each other.

• 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.
• Determine whether or not two polygons are similar.
• 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.

G.B.7 Visual representations

a. Relate a net, top-view, or side-view to a three-dimensional object that it might represent.

Sufficient information must be given so that "hidden sections" are well-defined.

• Visualize and be able to reproduce solids and surfaces in three-dimensional space when given two-dimensional representations (e.g., nets, multiple views).
• Interpret the relative position and size of objects shown in a perspective drawing.
• Visualize, describe, and identify three-dimensional shapes in different orientations.

Task related to this benchmark: Cones Launch

b. Draw two-dimensional representations of three-dimensional objects by hand and using software.

• Sketch two-dimensional representations of basic three-dimensional objects such as cubes, spheres, pyramids, and cones.
• Create a net, top-view, or side-view of a three-dimensional object by hand or using software.

c. Visualize, describe, or sketch the cross-section of a solid cut by a plane that is parallel or perpendicular to a side or axis of symmetry of the solid.

G.B.8 Geometric constructions

a. Carry out and explain simple straightedge and compass constructions.

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

Task related to this benchmark: Chinese Restaurant

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

Task related to this benchmark: Chinese Restaurant

G.C.1 Geometry of a circle

a. Know and apply the definitions and properties of a circle and the radius, diameter, chord, tangent, secant, and circumference of a circle.

b. Recognize and apply the fact that a tangent to a circle is perpendicular to the radius at the point of tangency.

Task related to this benchmark: Satellite

c. Recognize, verify, and apply statements about the relationships between central angles, inscribed angles, and the circumference arcs they define.

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

Task related to this benchmark: Cones Launch, Regional Triangles

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

An interior angle of a circle is an angle defined by two chords of the circle that intersect in the interior of the circle. An exterior angle of a circle is an angle defined by two chords of the circle that, when extended, intersect outside the circle.

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

Task related to this benchmark: Satellite

e. Determine the length of line segments and arcs, the size 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.

Tasks related to this benchmark: Regional Triangles, Satellite

G.C.2 Axioms, theorems, and proofs in geometry

a. Use geometric examples to illustrate the relationships among undefined terms, axioms/postulates, definitions, theorems, and various methods of reasoning.

• Analyze the consequences of using alternative definitions for geometric objects.
• Use geometric examples to demonstrate the effect that changing an assumption has on the validity of a conclusion.

Task related to this benchmark: Congruence Challenge

b. Present and analyze direct and indirect geometric proofs using paragraphs or two-column or flow-chart formats.

Task related to this benchmark: Regional Triangles

c. 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; find an equation of a circle given its center and radius and conversely, given an equation of a circle, find its center and radius.

d. 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 y = (x - 5)² - 2; the locus of points in space equidistant from a fixed point is a sphere.

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

G.D.1 Triangle trigonometry

a. 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.

Task related to this benchmark: Satellite

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

Task related to this benchmark: Satellite

c. 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.

G.D.2 Three-dimensional geometry

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

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

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

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

• Know that two distinct planes will either be parallel or will intersect in a line.
• 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, a single line, or by pairs in three parallel lines.

G.E.1 Spherical geometry

a. 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.

• 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.

b. 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.

• 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.

c. Interpret various two-dimensional representations for the surface of a sphere (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.

d. Describe geometry on a sphere as an example of a non-Euclidean geometry.

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

• 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°.

G.E.2 Vectors

a. Use vectors to represent quantities that have both magnitude and direction.

b. Add and subtract vectors, find their dot product, and multiply a vector by a scalar; interpret the results.

c. Use vectors to describe lines in two- and three-dimensional Euclidean space.

d. Use vectors and their operations to represent situations and solve problems.

e. Use vectors to represent motions of objects in two and three dimensions.

f. Apply parametric methods to represent motion of objects.

G.E.3 Conic sections

a. Develop and represent conic sections from basic properties.

• Know and apply the definitions for an ellipse and a hyperbola.

  An ellipse is the locus of all points on the plane the sum of whose distances to two given points, called the foci, is constant. A hyperbola is the locus of all points on the plane the difference of whose distances to two given points, called the foci, is constant.

• Identify a parabola, circle, ellipse, or hyperbola from its equation or key characteristics.

  Example: Know that a conic section whose eccentricity is 1 is a circle; identify x = -4y² + 8y + 15 as a parabola that opens to the left and has an axis of symmetry parallel to the x-axis.

• Derive the equations of circles and parabolas and of ellipses and hyperbolas with axes parallel to the coordinate axes and centered at the origin.
• Describe the effect that changes in the parameters of a particular conic section have on its graph.
• Explain how the key characteristics of standard algebraic forms of ellipses and hyperbolas are related to their graphical characteristics and translate between algebraic and graphical representations.

b. Describe how the intersection of a plane with a cone can form a circle, an ellipse, a parabola, or a hyperbola depending on the orientation of the plane with respect to the axis of the cone.

c. Apply conic sections in modeling real-world phenomena.

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