GEOMETRY

 

 

This course will provide the students with experience that deepens the understanding of two and three-dimensional objects and their properties.  Deductive and inductive reasoning, as well as investigative strategies in drawing conclusions are stressed.  Properties and relationships of geometric figures will include the study of points, lines, angles, planes, polygons with a special emphasis on quadrilaterals, triangles and right triangles, circles, polyhedra, and other solids.  An understanding of proof and logic will be developed.  Use of graphing calculators and computer drawing programs will be used.

 

Notes:

(1)     Prerequisite: Completion of Algebra I with a C or better

(2)     A Core 40 and AHD course with standards defined

 

Recommended Textbook: Glencoe Mathematics: Geometry by Glencoe/McGraw-Hill

 

 

 

prepared Spring 2004

 

By Susan Place

 

 

 OAK HILL UNITED SCHOOL CORPORATION


Course:  Geometry

 

Unit:  1

 

Textbook References:  Chapters 1 - 3

 

Suggested Time Frame:  9 weeks

 

Content Outline:

      Points

      Lines

      Planes

      Angles

      Distance

      Midpoint

      Logic

      Reasoning

      Proofs

      Conditional Statements

      Parallel Lines

      Perpendicular Lines

      Transversal

      Slope

      Equations of Lines

 

Standards to be Emphasized: 

Standard 1
Points, Lines, Angles, and Planes

Students find lengths and midpoints of lines. They describe and use parallel and perpendicular lines. They find slopes and equations of lines.

G.1.1         Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems.
Example: Find the length and midpoint of the line joining the points A (3, 8) and B (9, 0).

G.1.2         Construct congruent segments and angles, angle bisectors, and parallel and      perpendicular lines using a straight edge and compass, explaining and justifying the process used.
Example: Construct the perpendicular bisector of a given line segment, justifying each step of the process.    

G.1.3         Understand and use the relationships between special pairs of angles formed by parallel lines and transversals.
Example: In the diagram, the lines k and l are parallel. What is the measure of angle x? Explain your answer.

 

G.1.4         Use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines.
Example: Find an equation of a line perpendicular to y = 4x 2.

Standard 8
Mathematical Reasoning and Problem Solving

Students use a variety of strategies to solve problems.

G.8.1         Use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards.
Example: How far does the tip of the minute hand of a clock move in 20 minutes if the tip is 4 inches from the center of the clock?

G.8.2        Decide whether a solution is reasonable in the context of the original situation.
                 Example: John says the answer to the problem in the first example is 12 inches. Is his
                 answer reasonable? Why or why not?    

Students develop and evaluate mathematical arguments and proofs.

G.8.3         Make conjectures about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture.
Example: Calculate the ratios of side lengths in several different-sized triangles with angles of 90, 50, and 40. What do you notice about the ratios? How might you prove that your observation is true (or show that it is false)?

G.8.4         Write and interpret statements of the form if then and if and only if.
Example: Decide whether this statement is true: If today is Sunday, then we have school tomorrow.

G.8.5         State, use, and examine the validity of the converse, inverse, and contrapositive of if then statements.
Example: In the last example, write the converse of the statement.

G.8.6         Identify and give examples of undefined terms, axioms, and theorems, and inductive and deductive proofs.
Example: Do you prove axioms from theorems or theorems from axioms?

G.8.7         Construct logical arguments, judge their validity, and give counterexamples to disprove statements.
Example: Find an example to show that triangles with two sides and one angle equal are not necessarily congruent.

G.8.8         Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, and two-column and indirect proofs.
Example: In triangle LMN, LM = LN. Prove that
LMN @ LNM.

G.8.9         Perform basic constructions, describing and justifying the procedures used.         

                  Distinguish between constructing and drawing geometric figures.
                  Example: Construct a line parallel to a given line through a given point not
                  on the line, explaining and justifying each step.

 

Key Vocabulary: 

point                                                    conclusion

line                                                      parallel

plane                                                   perpendicular

ray                                                       proof

angle                                                   theorem

acute angle                                        postulate

obtuse angle                                      midpoint

right angle                                          bisector

straight angle                                     compass

vertical angle                                     transversal

linear pair                                           alternate interior angle

complementary angle                       alternate exterior angle

supplementary angle                        corresponding angle

collinear                                              consecutive interior angle

coplanar                                             skew lines

congruent                                           slope

concave                                              converse

convex                                                contrapositive

hypothesis                                          inductive and deductive reasoning

 

   

Suggestions for Incorporating Technology (for students): 

Geometer's Sketchpad

5 Minute Checks on Overhead

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

 

Suggestions for Incorporating Technology (for teachers):

Geometer's Sketchpad

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

TestCheck

Worksheet Builder

 

Suggestions for Differentiated Instruction: 

        Illustrate points, lines, planes, and angles in nature

        Use meterstick to find distance between points or objects

        List angle relationships

        Choose a traffic sign and describe the polygon

        Brainstorm conjectures and counterexamples in nature

        Use index cards to explain converse, inverse, and contrapositve statements

        Use index cards for proofs

        Illustrate alternate interior, alternate exterior, corresponding, and consecutive interior angles by marking a transversal and two parallel lines on the floor

        Compare slopes of parallel and perpendicular lines using note cards

 

 

Notes . . .

 

 

 

 

 

Course:  GEOMETRY

 

Unit:  2

 

Textbook References:  Chapters 4 - 6

 

Suggested Time Frame:  9 weeks

 

Content Outline:

      Triangle Relationships

      Congruent Triangles

      Proportions and Similarities

      Right Triangles

      SSS

      SAS

      ASA

      AAS

      HL

      Bisectors

      Medians

      Altitudes

      Geometric Mean

      Pythagorean Theorem

      Trigonometry

 

Standards to be Emphasized:

Standard 4
Triangles

Students identify and describe types of triangles. They identify and draw altitudes, medians, and angle bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They apply inequality theorems.

G.4.1         Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular.
Example: Use a drawing program to create examples of right, acute, obtuse, scalene, isosceles, equilateral, and equiangular triangles. Identify and describe the attributes of each triangle.

 G.4.2         Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors.
Example: Draw several triangles. Construct their angle bisectors. What do you notice?

G.4.3         Construct triangles congruent to given triangles.
Example: Construct a triangle given the lengths of two sides and the measure of the angle between the two sides.

G.4.4         Use properties of congruent and similar triangles to solve problems involving lengths and areas.
Example: Of two similar triangles, the second has sides half the length of the first. The area of the first triangle is 20 cm2. What is the area of the second?

   G.4.5         Prove and apply theorems involving segments divided proportionally.
Example: In triangle ABC,  is parallel to . What is the length of ?

 G.4.6         Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles.
Example: In the last example, prove that triangles ABC and APQ are similar.

G.4.7         Find and use measures of sides, perimeters, and areas of triangles. Relate these measures to each other using formulas.
Example: The gable end of a house is a triangle 20 feet long and 13 feet high. Find its area.

G.4.8         Prove, understand, and apply the inequality theorems: triangle inequality,                   inequality in one triangle, and the hinge theorem. 
Example: Can you draw a triangle with sides of length 7 cm, 4 cm, and 15 cm?

G.4.9        Use coordinate geometry to prove properties of triangles such as regularity,                 congruence, and similarity.
Example: Draw a triangle with vertices at (1, 3), (2, 5), and (6, 1). Draw another triangle with vertices at (-3, -1), (-2, 1), and (2, -3). Are these triangles the same shape and size?

 

Standard 5
Right Triangles

Students prove the Pythagorean Theorem and use it to solve problems. They define and apply the trigonometric relations sine, cosine, and tangent.

G.5.1         Prove and use the Pythagorean Theorem.
Example: On each side of a right triangle, draw a square with that side of the triangle as one side of the square. Find the areas of the three squares. What relationship is there between the areas?

G.5.2         State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.
Example: In triangle ABC with right angle at C, draw the altitude  from C to . Name all similar triangles in the diagram. Use these similar triangles to prove the Pythagorean Theorem.

G.5.3         Use special right triangles (30 - 60 and 45 - 45) to solve problems.
Example: An isosceles right triangle has one short side of 6 cm. Find the lengths of the other two sides.

G.5.4         Define and use the trigonometric functions (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles.
Example: In triangle ABC, tan A = 1/5 . Find sin A and cot A.

G.5.5         Know and use the relationship sin2 x + cos2 x = 1.
Example: Show that, in a right triangle, sin2 x + cos2 x = 1 is an example of the Pythagorean Theorem.

G.5.6         Solve word problems involving right triangles.
Example: The force of gravity pulling an object down a hill is its weight multiplied by the sine of the angle of elevation of the hill. What is the force on a 3,000-pound car on a hill with a 1 in 5 grade? (A grade of 1 in 5 means that the hill rises one unit for every five horizontal units.)

 

Key Vocabulary:

right triangle                                                               centroid

acute triangle                                                             circumcenter

obtuse triangle                                                           incenter

Scalene triangle                                                        orthocenter

Isosceles triangle                                                      angle bisector

equilateral triangle                                                    perpendicular bisector

equiangular triangle                                                  ratio

base angle                                                                 proportion

included side                                                             similar

included angle                                                           cross products

remote interior angle                                                midsegment

altitude                                                                        geometric mean

median                                                                       Pythagorean triple

Trigonometric functions sine, cosine, tangent

 

 

 

Suggestions for Incorporating Technology (for students): 

Geometer's Sketchpad

5 Minute Checks on Overhead

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

Scientific Calculator

 

Suggestions for Incorporating Technology (for teachers):

Geometer's Sketchpad

Scientific Calculator

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

TestCheck

Worksheet Builder

 

Suggestions for Differentiated Instruction:

        Illustrate different types of triangles in nature

        Show that the angles of a triangle total 180 degrees by cutting the angles and arranging them in a straight line

        Students create notecards illustrating the different ways to prove triangles are congruent

        Explore circumcenter, centroid, orthocenter, and incenter by folding paper triangles

        Summarize a proof

        Compare and contrast theorems

        Students can find the height of a wall by using a shadow and proportions

        Explore the lengths of sides in special right triangles by paper folding

 

 

Notes . . .

  

 

Course:  Geometry

 

Unit:  3

 

Textbook References:  Chapters 7 - 9

 

Suggested Time Frame: 9 weeks

 

Content Outline:

      Quadrilaterals

      Parallelograms

      Rectangle

      Square

      Rhombus

      Trapezoid

      Kite

      Polygons

      Transformations

      Rotation

      Reflection

      Translation

      Vector

      Circles

      Arcs

      Chords

      Secant

      Tangent

 

Standards to be Emphasized:

Standard 2
Polygons

Students identify and describe polygons and measure interior and exterior angles. They use congruence, similarity, symmetry, tessellations, and transformations. They find measures of sides, perimeters, and areas.

G.2.1         Identify and describe convex, concave, and regular polygons.
Example: Draw a regular hexagon. Is it convex or concave? Explain your answer.

G.2.2        Find measures of interior and exterior angles of polygons, justifying the method used.
Example: Calculate the measure of one interior angle of a regular octagon. 
Explain your method.

G.2.3        Use properties of congruent and similar polygons to solve problems.
Example: Divide a regular hexagon into triangles by joining the center to each vertex. Show that these triangles are all the same size and shape and find the sizes of the interior angles of the hexagon.

G.2.4         Apply transformations (slides, flips, turns, expansions, and contractions) to polygons to determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips, and turns are congruent to the original shape.
Example: Use a drawing program to create regular hexagons, regular octagons, and regular pentagons. Under the drawings, describe which of the polygons would be best for tiling a rectangular floor. Explain your reasoning.

G.2.6        Use coordinate geometry to prove properties of polygons such as regularity,                 congruence, and similarity.
Example: Do these four points form a square: (2, 1), (6, 2), (5, 6), (1, 5)?

Standard 3
Quadrilaterals

Students identify and describe simple quadrilaterals. They use congruence and similarity. They find measures of sides, perimeters, and areas.

G.3.1       Describe, classify, and understand relationships among the quadrilateral square, rectangle, rhombus, parallelogram, trapezoid, and kite.
Example: Use a drawing program to create a square, rectangle, rhombus, parallelogram, trapezoid, and kite. Judge which of the quadrilaterals has perpendicular diagonals and draw those diagonals in the figures. Give a convincing argument that your judgment is correct.

G.3.4       Use coordinate geometry to prove properties of quadrilaterals, such as regularity, congruence, and similarity.
Example: Is rectangle ABCD with vertices at (0, 0), (4, 0), (4, 2), (0, 2) congruent to rectangle PQRS with vertices at (-2, -1), (2, -1), (2, 1), (-2, 1)?

Standard 6
Circles

Students define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths, and areas. They prove theorems about circles. They find equations of circles.


G.6.1         Find the center of a given circle. Construct the circle that passes through three given points (not in a straight line).
Example: Given a circle, find its center by drawing the perpendicular bisectors of two chords.

G.6.2         Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and tangent.
Example: What is the angle between a tangent to a circle and the radius at the point where the tangent meets the circle?

G.6.3         Prove theorems related to circles.
Example: Prove that an inscribed angle in a circle is half the measure of the central angle with the same arc.

G.6.4         Construct tangents to circles and circumscribe and inscribe circles.
Example: Draw an acute triangle and construct the circumscribed circle.

G.6.5         Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents).
Example: Find the measure of angle ABC in the diagram below.

 

 G.6.6         Define and identify congruent and concentric circles.
Example: Are circles with the same center always the same shape? Are they always the same size?

G.6.7         Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these measures to solve problems.
Example: Which will give you more: three 6-inch pizzas or two 8-inch pizzas? Explain your answer.

G.6.8         Find the equation of a circle in the coordinate plane in terms of its center and            radius.
Example: Find the equation of the circle with radius 10 and center (6, -3).

 


Key Vocabulary:

diagonal                                                         circle                                       isometry

trapezoid                                                        center                                     major arc

Isosceles trapezoid                                       circumference                        minor arc

rectangle                                                        chord

square                                                            circumscribed

rhombus                                                         inscribed

kite                                                                  radius

rotation                                                           secant

translation                                                      tangent

reflection                                                        diameter                                                        

transformation                                               semicircle

vector                                                              central angle

glide reflection                                               tessellation

 

Suggestions for Incorporating Technology (for students):

Geometer's Sketchpad

5 Minute Checks on Overhead

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

 

Suggestions for Incorporating Technology (for teachers):

Geometers Sketchpad

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

TestCheck

Worksheet Builder

 

Suggestions for Differentiated Instruction:

        Illustrate angle bisectors and diagonals with paper polygons

        Compose a picture using right triangles

        List similarities between a rectangle, square, and rhombus

        Use string to show congruent diagonals intersect at their midpoints

        Name examples of trapezoids in the real world

        Name examples of reflections in nature

        Explore the idea of tessellations with regular and irregular polygons

        Apply knowledge of circles to a 16 inch pizza

 

Notes . . .

 

 

 

 

 

 

 

Course:  Geometry

 

Unit:  4

 

Textbook References:  Chapters 10 - 13

 

Suggested Time Frame: 9 weeks

 

Content Outline:

      Areas of Polygons

      Areas of Circles

      Surface Area

      Prisms

      Cylinders

      Pyramids

      Cones

      Spheres

      Volume

      Congruent Solids

      Similar Solids

 

Standards to be Emphasized:

 

G.2.5         Find and use measures of sides, perimeters, and areas of polygons. Relate these measures to each other using formulas.
Example: A rectangle of area 360 square yards is ten times as long as it is wide. Find its length and width.

G.3.2         Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas.
Example: Of two similar rectangles, the second has sides three times the length of the first. How many times larger in area is the second rectangle?

G.3.3         Find and use measures of sides, perimeters, and areas of quadrilaterals.                 Relate these measures to each other using formulas.
Example: A section of roof is a trapezoid with length 4 m at the ridge and 6 m at the gutter. The shortest distance from ridge to gutter is 3 m. Construct a model using a drawing program, showing how to find the area of this section of roof.

   

Polyhedra and Other Solids

Students describe and make polyhedra and other solids. They describe relationships and symmetries, and use congruence and similarity.

G.7.1         Describe and make regular and nonregular polyhedra.
Example: Is a cube a regular polyhedron? Explain why or why not.

G.7.2         Describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given polyhedron.
Example: Make a net for a tetrahedron out of poster board and fold it up to make the tetrahedron.

G.7.3         Describe relationships between the faces, edges, and vertices of polyhedra.
Example: Count the sides, edges, and corners of a square-based pyramid. How are these numbers related?

G.7.4         Describe symmetries of geometric solids.
Example: Describe the rotation and reflection symmetries of a square-based pyramid.

G.7.5         Describe sets of points on spheres: chords, tangents, and great circles.
Example: On Earth, is the equator a great circle?

G.7.6         Identify and know properties of congruent and similar solids.
Example: Explain how the surface area and volume of similar cylinders are related.

G.7.7         Find and use measures of sides, volumes of solids, and surface areas of solids. Relate these measures to each other using formulas.
Example: An ice cube is dropped into a glass that is roughly a right cylinder with a 6 cm diameter. The water level rises 1 mm. What is the volume of the ice cube?

 

Key Vocabulary:

polyhedra                                                       great circle

apothem                                                         hemisphere

sector                                                             oblique

face                                                                 slant height

edge                                                               sphere

vertices                                                           surface area

cone                                                                volume

prism                                                              congruent solids

pyramid                                                          similar solids

cylinder

 


Suggestions for Incorporating Technology (for students):

Geometer's Sketchpad

5 Minute Checks on Overhead

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

 

Suggestions for Incorporating Technology (for teachers):

Geometer's Sketchpad

GeomPASS: Tutorial Plus

MindJogger Videoquizzes

TestCheck

Worksheet Builder

 

Suggestions for Differentiated Instruction:

        Draw tangents using a CD and a ruler

        Work with partners to the apothem by drawing a polygon and using the radius of the circumscribed circle

        Illustrate irregular shapes by taking pictures of real life objects

        Explain nets by cutting out shapes and folding them together

        Find the surface area of a paper towel roll or toilet paper roll

        Illustrate an oblique cylinder using a stack poker chips

        Find examples of cylinders, cones, pyramids, and spheres

 

Notes . . .