Colleton County School District
Mathematics Curriculum
Subject: Geometry
Strand: I. Geometric Structure A. Axiomatic Systems
Standard: A1. Develop an awareness of the structure of a mathematical system, connecting definitions, postulates, logical reasoning, and theorems
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s): 1. Read and understand definitions
2. Describe the relationships between segments (midpoint, perpendicular, parallel, intersecting) and angles (bisector, vertical, supplementary, complementary)
3. Recognize and use postulates and theorems in reasoning
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. Draw diagrams to illustrate relationships between line segments and angles
2. Use patty paper to verify a midpoint, an angle bisector, and to show that vertical angles are congruent
Resources:
1. Geometry: An Integrated Approach (D.C. Heath) , Section 2.2, 2.3, 2.4, 2.6. Corresponding extra practice worksheets are also available
2. Merrill Geometry: Applications and Connections Sections 1-2, 1-4, 1-5, 1-6, 1-7, 1-8
3. Discovering Geometry (Key Curriculum) 2.3, 2.4
Assessment(s) Samples - PACT – Exit Exam
Draw a diagram: B is between A and C. Find the length of each segment.
AB = 2w + 5 AB = ______
BC = 3w – 7 BC = ______
AC = 18
Ans. w= 4
AB = 13
BC = 5
PSAT – SAT Correlations: Samples
R, S, and T are collinear. S is between R and T. RS = 2x + 1, ST = x – 1,
RT = 18. Solve for x, then determine the length of RS.
(A) 6 (B) 5 (C) 13 (D) 16 (E) 12
Ans: C
Vocabulary:
Postulate
Theorem
Sample Lessons (Addendum)
Strand: I. Geometric Structure A. Axiomatic Systems
Standard: A2. Recognize that the study of geometry was developed for a variety of purposes and has historical significance
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s):
1. Explore early uses of geometry by cultures such as the Egyptians, Greeks, Romans
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. Students will present short “historical moments in geometry” summarizing contributions of specific individuals
2. Students will build 3-D polyhedra and explain the historical significance
3. Use the Lightspan Web Trip Maker (see resources below) to create an online exploration for students to research 4-5 questions about the history of Geometry
Resources:
1. Encyclopedia and internet research
2. Geometry: An Integrated Approach (D.C. Heath)
pp. 17, 27, 41, 70, 160, 398, 434, 627
3. Merrill Geometry: Applications and Connections p. 29, 74, 127, 229, 245, 293, 421, 468, 179, 565, 579, 591
4. Lightspan Web Trip Maker http://www.lightspan.com
5. Discovering Geometry .7, 4.7, 6.5, 7.7
Assessment(s) Samples - PACT – Exit Exam
Column A Column B
(3a) + (4a) (5a)
(A) If the quantity in Column A is greater
(B) If the quantity in Column B is greater
(C) If the two quantities are equal
(D) If the relationship between the two quantities cannot be determined
Ans. C
PSAT – SAT Correlations: Samples
Column A Column B
The circumference of π
a circle having a radius
of
(A) If the quantity in Column A is greater
(B) If the quantity in Column B is greater
(C) If the two quantities are equal
(D) If the relationship between the two quantities cannot be determined
Ans. B
Vocabulary:
Archimedes
Pythagoras
The concept of p
Euclid
Platonic solids
M. C. Escher
Sample Lessons (Addendum)
Strand: I. Geometric Structure
B. Verification of Conjectures
Standard: B1. Explore attributes of geometric figures using constructions with straight-edge and compass, paper folding, and dynamic, interactive geometry software.
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s):
1. Basic constructions include
· Copy a segment
· Copy an angle
· Bisect a segment
· Bisect an angle
· Construct a line perpendicular to a given line from a point not on the line
· Construct a line perpendicular to a given line at a point on the line
2. Identify special segments in a triangle: median, altitude, perpendicular bisector
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. Use straight-edge and compass for constructions
2. Use patty paper for folding exercises
3. Use Geometer’s Sketchpad for computer-based constructions
Resources:
1. Geometry: An Integrated Approach (D.C. Heath) 1.7, 4.7, 2. 5.1 – 5.3
2. Merrill Geometry: Applications and Connections pp. 26, 31, 45-46, 57-58, and section 5-1
3. Geometer’s Sketchpad software--demo available http://www.keypress.com/sketchpad/sketchdemo.html
4. Discovering Geometry Chapter 3
Assessment(s) Samples - PACT – Exit Exam
In the given figure, use a protractor to draw a line m through point A perpendicular to segment AB. Give the measure of the smaller angle formed by the intersection of lines l and m.
PSAT – SAT Correlations: Samples
Segment AB is the perpendicular bisector of segment GH.
(A) GAB
(B) ABH
(C) AHB
(D)HAB
(E) GAB
Ans. C
Vocabulary:
Bisect Midpoint
Perpendicular Median
Straight-edge Altitude
Compass Perpendicular Bisector
Sample Lessons (Addendum)
Strand: I. Geometric Structure
B. Verification of Conjectures
Standard: B2. Make and verify conjectures about angles, lines, polygon, circles, and three-dimensional figures, choosing form a variety of approaches such as coordinate, transformational, or axiomatic
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s):
1. Write different styles of proofs (paragraph, two-column, flow, coordinate)
2. Analyze which approach of proof should be used (coordinate, transformational, or axiomatic)
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. “Defend your actions” – students will give reasons of defense when presented with common problems such as being accused of cutting class, or driving too fast.
2. Make a connection between the above “defense” and a geometric proof
3. Make a connection with prior learning by asking students to justify the steps in solving an equation using the properties of equality for real numbers
Resources:
1. Geometry: An Integrated Approach (D.C. Heath) 3.4, 3.6, 4.3 – 4.5, 6.4, 9.2
2. Merrill Geometry: Applications and Connections sections 2-1, 2-3, 2-4, 2-6, 2-7 on 2-column proofs; pp. 129, 193, 240-241 on paragraph proofs, pp. 595-600 on coordinate proofs
3. Discovering Geometry Chapter 14
Assessment(s) Samples - PACT – Exit Exam
In the figure, the area of the rectangle is
(A) 63 (B) 36 (C) 27 (D) 24 (E) 18
PSAT – SAT Correlations: Samples
The figure above shows vertex P of a square PQRS, not shown. If side PQ is parallel to either the x- or the y-axis and PQRS has an area of 16, which of the following could NOT be the vertex Q of the square?
(A) (-1,7) (B) (-1,3) (C) (3,-1) (D) (3,7) (E) (7,3)
Ans A
Vocabulary:
Conjecture
Paragraph proof
Two-column proof
Flow proof
Coordinate proof
Indirect proof
Sample Lessons (Addendum)
Strand: I. Geometric Structure
C. Logical Reasoning and Proof
Standard: C1. Determine whether the converse of a conditional statement is true or false
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s):
1. Recognize and use conditional statements
2. Write the converse of a conditional statement and determine if it is true
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. Identify key words “if…, then…..”, “only if”, “if and only if”.
2. Sample conditional “If I live in Walterboro, then I live in SC.” (true)
Converse: “If I live in SC, then I live in Walterboro.” (False – I could live in Charleston, Round O, etc)
Resources:
1. Geometry: An Integrated Approach (D.C. Heath) 2.4
2. Geometry Addison - Wesley (199 edition)
3. Merrill Geometry: Applications and Connections section 2-2
4. Discovering Geometry Chapter 13
Assessment(s) Samples - PACT – Exit Exam
Given the conditional statement “If mA = , then A is acute.” Determine whether the converse is true or false. Verify your reasoning.
Ans. Converse: “If A is acute, then mA = ”
FALSE. A could have any measure less than 90.
PSAT – SAT Correlations: Samples
Assume the following statement is true: “If it is raining, then I will stay indoors.” Which of the following statements is true?
I. If I stay indoors, then it is raining.
II. If I do not stay indoors, then it is not raining.
III. If it is not raining, then I will go outside.
(A) I only (B) II only (C) III only
(D) I and II only (E) None
Ans. B
Vocabulary:
Conditional
Converse
Biconditional
Hypothesis
Conclusion
Counterexample
Sample Lessons (Addendum)
Strand: I. Geometric Structure
C. Logical Reasoning and Proof
Standard: C2. Use logical reasoning to draw conclusions about geometric figures from given assumptions
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s): The student will:
1. Form the negation and the contrapositive of a conditional statement.
2. Use the Law of Syllogism and the Law of Detachment to reason deductively
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. Conditional: “If the weather is good, then I will go swimming.”
Negation: “If the weather is not good, then I will not go swimming.”
Contrapositive: “If I do not go swimming, then the weather is not good.”
2. Law of Syllogism example:
Given: “If Ryan gets a B on this test, then he will pass the course.
If he passes the course, then he will get a scholarship.”
Conclusion: “If Ryan gets a B on this test, then he will get a scholarship.”
3. Law of Detachment example:
Given: “If you live in Walterboro, then you live in Colleton County.
My teacher lives in Walterboro.”
Conclusion: “My teacher lives in Colleton County.”
Resources:
1. Geometry: An Integrated Approach (D.C. Heath) 3.3, 3.4, Extra practice
2. Merrill Geometry: Applications and Connections section 2-3
3. Logic Puzzles
4. Discovering Geometry Chapter 13
Assessment(s) Samples - PACT – Exit Exam
Assume the conditional statements are true. Write the conclusion.
“If the stereo is on, then the volume is loud.
If the volume is loud, then the neighbors will complain.”
Ans. “If the stereo is on, then the neighbors will complain.”
PSAT – SAT Correlations: Samples
If , , and are coplanar and is perpendicular to and is perpendicular to , which of the following statements are true?
I. and are skew
II. is parallel to
III.
(A) I only (B) II only (C) III only (D) I and II only
(E) I and III only
Ans B
Vocabulary:
Negation
Contrapositive
Law of Syllogism
Law of Detachment
Sample Lessons (Addendum)
Strand: I. Geometric Structure
C. Logical Reasoning and Proof
Standard: C3. Construct and judge the validity of a logical argument consisting of a set of premises and a conclusion
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s):
Decide whether a statement follows from given information.
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. Use syllogisms by Lewis Carroll (Heath 3.3) – students could write a biography of this writer/mathematician
2. Determine if a statement is sometimes, always, or never true.
Examples:
Given: Points A, B, and C lie on a segment
Conjecture #1: A, B, and C are collinear (always true)
Conjecture #2: B is between A and C (sometimes true)
Resources:
1. Geometry: An Integrated Approach (D.C. Heath) 3.3,
P. 125, 128,131, 251
2. Logic puzzles (included in resource kit)
3. Merrill Geometry: Applications and Connections chapter 2
4. Discovering Geometry Chapter 13
Assessment(s) Samples - PACT – Exit Exam
If each of the 3 distinct points A, B, and C are the same distance from point D, which of the following could be true?
I. A, B, C, and D are the four vertices of a square
II. A, B, C, and D lie on the circumference of a circle.
III. A, B, C, and D lie on the circumference of a circle whose center is D
(A) I only (B) II only (C) III only (D) II and III only
(E) I, II, and III
Ans C
PSAT – SAT Correlations: Samples
In the figure, if m1 = m2, which of the following
statements are true?
I. m3 = m2
II. is parallel to
III.
(A) I only (B) II only (C) III only (D) I and II only
(E) II and III only
Ans D
Vocabulary:
quantifier
counterexample
Sample Lessons (Addendum)
Strand: I. Geometric Structure
C. Logical Reasoning and Proof
Standard: C4. Use inductive reasoning to formulate a conjecture
Taught By: ( designated by quarter, may or may not be in final form)
Objective(s):
Identify and use inductive reasoning
Performance Descriptors ( may be combined into the objective category)
Instructional Strategies (samples)
1. Site everyday examples – “fried chicken on Thursday”, “movie on Friday”
2. Determine the next three numbers in a pattern based on inductive reasoning; for example, the Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13,... (Students could investigate the historical significance of this famous number pattern.
3. Determine the number of diagonals from one vertex of a convex polygon
4. Investigate number patterns in Pascal’s Triangle
5. Investigate growth patterns based on concrete representations; for example,
6. A Mobius strip has only one surface. Investigate other characteristics by coloring, drawing lines, and cutting. (See Merrill Enrichment resources)
Resources:
1. Geometry: An Integrated Approach (D.C. Heath) P. 57, 273
3. Merrill Geometry: Applications and Connections sections 2-1, 6-2, p. 658
4. Discovering Geometry Chapter 1
Assessment(s) Samples - PACT – Exit Exam