The Coordinate Plane

The Coordinate Plane TEKS/TAKS: 1.a, 2.b, 4.a, 7.a Objective: You will graph ordered pairs on a coordinate plane. Used in: Locating items in publishing, archaeology, and aquatic explorations. Vocabulary: Coordinate plane, x-axis, y-axis, origin, quadrants, ordered pair, x-coordinate, y-coordinate Additional Reading: Textbook 1-1 pg. 6

Real World Application Aquatic Engineering The Dutch Delta Plan, which controls the flow of the Atlantic Ocean on the southwest coast of the Netherlands, is one of the greatest achievements of engineering. The system was built using a grid system of nylon mattresses with graded gravel and rocks to support concrete piers and steel gates. The horizontal axis is labeled with letters and the vertical axis with numbers. Suppose the first five piers were placed at 5B, 2K, 8D, 7A, and 12E. Sketch a graph showing the positions of the first five piers.

Challenge Problem Make a table of values to determine the five points that lie on the graph of y= x2-2x+6. Use the following values of x: 1, 2, 3, 4, and 5. Plot the five points. Do they appear to be collinear? Why or why not?

Challenge Homework Pgs. 10-11 #27, 29, 34, 37, 47

Points, Lines, and Planes TEKS/TAKS: 1.a, 1.b, 2.b, 4.a Objective: You will identify and model points, lines, planes, coplanar points, intersecting lines and planes, and solve problems by listing possibilities Used in: Representing real-life (tangible) objects Vocabulary: Planes, lines, points, space, possibilities Additional Reading: Textbook 1-2 pg. 12

Real World Application Anatomy Has anyone ever told you to stand up straight? If your posture is perfect, you should be able to draw a straight line from your ear to your ankle, running through your shoulder, hip, and knee. Study the posture of five of your friends or relatives. How many of them seem to have good posture according to the straight line rule? What percent of the people you observed have good posture?

Challenge Problem The Hawaiian game of lu-lu is played with four disks of volcanic stone. The face of each stone is marked with a series of dots. A player tosses the four disks and if they land all face-up, 10 points are scored, and the player tosses again. If any of the disks land facedown on the first toss, the players gets to toss those pieces again. The score is the total number of dots showing after the second toss. List the possible outcomes after the first toss.

Measuring Segments TEKS/TAKS: 1.a, 1.b, 2.a, 2.b, 4.a, 7.a, 7.c, 8.c Objective: You will find the distance between two points on a number line and between two points in a coordinate plane and use the Pythagorean Theorem to find the length of the hypotenuse of a right triangle. Used in: Segment measures are used in discovering characteristics of many geometric shapes. Vocabulary: Between, measure, Ruler Postulate, Pythagorean Theorem, distance formula Additional Reading: Textbook 1-4 pg. 28

Real World Application World Records On September 8, 1989, the British Royal Marines stretched a rope from the top of Blackpool Tower (416 feet high) in Lancashire, Great Britain, to a fixed point on the ground 1128 feet from the base of the tower. Then Sgt. Alan Heward and Cpl. Mick Heap of the Royal Marines, John Herbert of Blackpool Tower, and TV show hosts Cheryl Baker and Roy Castle slid down the rope establishing the greatest distance recorded in a rope slide. Draw a right triangle to represent this event. How far did they slide?

Challenge Problem Draw a figure that satisfies all of the following conditions: Points A, B, C, D, and E are collinear. Point A lies between points D and E. Point C is next to point A, and BD = DC.

Midpoints and Segment Congruence TEKS/TAKS: 1.a, 2.a, 2.b, 4.a, 7.a, 7.c Objective: You will find the midpoint of a segment, and complete proofs involving segment theorems. Used in: Finding the midpoint is often used to connect algebra to geometry. Vocabulary: Midpoint, segment bisector, theorems, proof, paragraph proof, informal proof Additional Reading: Textbook 1-5 pg. 36

Real World Application Transportation Interstate 70 passes through Kansas. Mile markers are used to name many of the exits. The exit for U.S. Route 283 North is Exit 128, and the exit to use U.S. Route 281 to Russell is Exit 184. The exit for Hays on U.S. Route 183 is 3 miles farther than halfway between Exits 128 and 184. What is the exit number for the Hays exit?

Challenge Problem Point C lies on AB such that AC = ¼ AB. If the endpoints of AB are A(8, 12) and B (-4, 0), find the coordinates of C.

Exploring Angles TEKS/TAKS: 1.a, 1.b, 2.a, 2.b, 4.a, 5.a Objective: You will identify and classify angles, use the Angle Addition Postulate, find the measures of angles, and identify and use congruent angles and the bisector of an angle. Used in: Astronomy, construction, art, and engineering Vocabulary: Angle, opposite rays, sides, vertex, interior, exterior, degrees, measure, Protractor Postulate, congruent, angle bisector Additional Reading: Textbook 1-6 pg. 44

Real World Application Entertainment John Trudeau used angles to help him design The Flintstones pinball machine. The angle that the machine tilts, or the pitch of the machine, determines the difficult of the game. The angles at which the flipper will hit the ball are considered when the ramps, loops, and targets of the game are placed. Mr. Trudeau recommends that The Flintstones machine be installed with a pitch of 6° to 7°. Is this an acute, right, straight, or obtuse angle?

Challenge Problem Draw BA and BC such that they are opposite rays and BE bisects <ABD. If m<ABE = 6x + 2 and m<DBE = 8x – 14, find m<ABE.

Angle Relationships TEKS/TAKS: 1.a, 2.a, 2.b, 4.a, 9.a Objective: You will identify and use adjacent, vertical, complementary, supplementary, and linear pairs of angles, and perpendicular lines, and to determine what information can and cannot be assumed from a diagram. Used in: Geology, sports, and construction Vocabulary: Perpendicular lines and adjacent, vertical, supplementary, and complementary angles, linear pair. Additional Reading: Textbook 1-7 pg. 53

Real World Application Sports In the 1994 Winter Olympic Games, Espen Bredesen of Norway claimed the gold medal in the 90-meter ski jump. When a skier completes a jump, he or she tries to make the angle between his or her body and the front of his or her skis as small as possible. If Espen is aligned so that the front of his skis make a 15° angle with his body, what angle is formed by the tail of the skis and his body?

Challenge Problem <R and <S are complementary angles, and <U and <V are also complementary angles. If m<R = y – 2, m<S = 2x + 3, m<U = 2x – y, and m<V = x – 1, find the values of x, y, m<R, m<S, m<U, and m<V.

Inductive Reasoning and Conjecturing TEKS/TAKS: 1.a, 2.b, 3.d, 4.a Objective: You will make conjectures based on inductive reasoning. Used in: Law, higher level mathematics and science, research Vocabulary: Inductive reasoning, conjecture, counterexample Additional Reading: Textbook 2-1 pg. 70

Real World Application Billiards Consider a carom billiard table with a length of 6 feet and a width of 3 feet. Suppose you start in the upper left-hand corner and shoot the ball at a 45° angle. Use graph paper to trace the path of the ball. Make a conjecture about shooting the ball from any corner of a table this size at a 45° angle.

Challenge Problem Determine if the conjecture below is true or false. Explain your answer and give a counterexample if it is a false conjecture. Given: x is an integer. Conjecture: -x is negative.

If-Then Statements and Postulates TEKS/TAKS: 1.a, 2.b, 3.a, 3.b, 3.c, 3.d, 4.a Objective: You will write statements in if-then form, you will write the converse, inverse, and contrapositive and you will identify and use basic postulates about points, lines, and planes. Used in: Understanding if-then statements helps determine the validity of conclusions. Vocabulary: If-then statements, conditional statements, hypothesis, conclusion, converse, inverse, contrapositive, negation, Venn diagram Additional Reading: Textbook 2-2 pg. 76

Real World Application Biology Use a Venn diagram to illustrate the following conditional about the animal kingdom. “If an animal is a butterfly, then it is an arthropod.”

Challenge Problem Consider the conditional, “If two angles are adjacent, they are not both acute.” Write the converse of the contrapositive of the inverse of the conditional. Explain how the result is related to the original conditional.

Deductive Reasoning TEKS/TAKS: 1.a, 2.b, 3.c Objective: You will use the Law of Detachment and the Law of Syllogism in deductive reasoning and you will solve problems looking for a pattern. Used in: You can use deductive reasoning to reach logical conclusions. Vocabulary: Law of Detachment, deductive reasoning, Law of Syllogism Additional Reading: Textbook 2-3 pg. 85

Real World Application Airline Safety The statement below is posted in airports throughout the U.S. Provide information necessary to illustrate logical reasoning using the Law of Detachment with this if-then statement. Attention All Travelers If any unknown person attempts to give you any items including luggage to transport on your flight, do not accept it and notify airline personnel immediately.

Challenge Problem Use deductive reasoning laws to write a true conclusion using all of the following three statements. Explain all steps used to arrive at your conclusion; remember, if a conditional is true then its contrapositive is true. If a person is baby, then the person is not logical. If a person can manage a crocodile, then that person is not despised. If a person is not logical, then the person is despised.

Using Proof in Algebra TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a Objective: You will use properties of equality in algebraic and geometric proofs. Used in: Forensic science, law, higher-level mathematics and science Vocabulary: Proof, two-column proof, properties of equality, properties of segments Additional Reading: Textbook 2-4 pg. 92

Real World Application Physics Kinetic energy is the energy of motion. The formula for kinetic energy is Ek = h · f + W, where h represents the work function of the material being used. Solve this formula for f and justify each step.

Challenge Problem What are some of the similarities and differences between the Transitive Property of Equality and the Transitive Property of Congruent Segments? Give an example of each property using segments and angles.

Verifying SegmentRelationships TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a Objective: You will complete proofs involving segment theorems. Used in: Geography Vocabulary: Reflexive, symmetric, and transitive Additional Reading: Textbook 2-5 pg. 100

Real World Application Measurement Some rulers have centimeters on one edge and inches on the other edge. About how long in centimeters is a segment that is 6 inches long? Are the two segments congruent? Explain.

Challenge Problem Draw and complete the proof. Given: PS is congruent to RQ. M is the midpoint of PS. M is the mipoint of RQ. Prove: PM is congruent to RM.

Challenge Homework Pgs. 104-106 #21, 27, 29, 33, 35, 43

Verifying AngleRelationships TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a Objective: You will complete proofs involving angle theorems. Used in: Art, nature, and architecture Vocabulary: Illusion Additional Reading: Textbook 2-6 pg. 107

Real World Application Architecture The Leaning Tower of Pisa in Italy makes an angle with the ground of about 84° on one side. If you look at the building as a ray and the ground as a line, then the angles that the tower forms with the ground form a linear pair. Find the measure of the other angle that the tower makes with the ground.

Challenge Problem Given <5 and <A are complementary. <6 and <A are complementary. m<5 = 2x + 2 and m<6 = x + 32. Find the m<5 and m<6.

Parallel Lines and Transversals TEKS/TAKS: 1.a, 2.b, 3.b, 3.d, 3.e, 4.a, 9.a Objective: You will solve problems by drawing a diagram, you will identify relationships between two lines or between two planes, and you will name angles formed by a pair of lines and a transversal. Used in: Architecture, agriculture, and air travel Vocabulary: Drawing a diagram, skew lines, transversal, interior, exterior, alternate exterior, consecutive interior, corresponding Additional Reading: Textbook 3-1 pg. 124

Real World Application Music The word “parallel” is used in music to describe songs moving consistently by the same intervals such as harmony with parallel voices. Find at least two additional uses of the word “parallel” in other school subjects such as history, electronics, computer science, or English.

The Coordinate Plane