Presentation on theme: "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,"— Presentation transcript:
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= x 2 -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?
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
"name": "Challenge Problem Draw BA and BC such that they are opposite rays and BE bisects
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?
"name": "Challenge Problem
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 E k = 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 Segment Relationships 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.
Verifying Angle Relationships 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.
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.
Challenge Problem Square dancing involves four couples. If each member of the square shakes hands with every other member of the square except his or her partner before the dance begins, what is the total number of handshakes?
Angles and Parallel Lines TEKS/TAKS: 2.a, 2.b, 3.d, 9.a Objective: You will use the properties of parallel lines to determine angle measure. Used in: Construction and interior decorating Vocabulary: None Additional Reading: Textbook 3-2 pg. 131
Real World Application Interior Decorating Walls in houses are never perfectly vertical. To hang wallpaper, a true vertical line must be established so the pattern looks nice. The paperhanger uses a plumb line, which is a piece of string with a weight at the bottom, to make the vertical line for the first piece of wallpaper. How can she be sure that all of the pieces of wallpaper are vertical if she does not use the plumb line again?
Challenge Problem In the figure below, explain why you can conclude that <1 is congruent to <4, but you cannot state that <3 is necessarily congruent to <2. C AB D 3 4 1 2
Slopes of Lines TEKS/TAKS: 1.a, 1.b, 2.b, 4.a, 7.a, 7.b, 9.a Objective: You will find the slopes of lines and use slope to identify parallel and perpendicular lines. Used in: Finding the distance between two points. Vocabulary: Slope, if and only if Additional Reading: Textbook 3-3 pg. 138
Real World Application Construction According to the building code in Crystal Lake, Illinois, the slope of a stairway cannot be steeper than 0.88. The stairs in Li-Chih’s home measure 11 inches deep and 7 inches high. Do the stairs in his home meet the code requirements? Explain your answer.
Challenge Problem A line contains the points at (-3, 6) and (1, 2). Using slope, write a convincing argument that the line intersects the x-axis at (3, 0). Graph the points to verify your conclusion.
Proving Lines Parallel TEKS/TAKS: 1.a, 1.b, 2.b, 4.a, 7.a, 7.b, 9.a Objective: You will recognize angle conditions that produce parallel lines and prove two lines are parallel based on given angle relationships. Used in: Construction and physics Vocabulary: None Additional Reading: Textbook 3-4 pg. 146
Real World Application Construction Carpenters use parallel lines in creating walls for construction projects. Copy the drawing below. Label your drawing and describe three different ways to guarantee that the wall studs are parallel.
Challenge Problem Suppose lines a, b, and c lie in the same plane and a||b and a||c. Draw a figure showing lines a, b, and c. Explain how you would prove that b||c.
Parallels and Distance TEKS/TAKS: 1.a, 2.a, 2.b, 4.a, 7.a, 7.b, 9.a Objective: You will recognize and use distance relationships among points, lines, and planes. Used in: Finding the distance between points and lines and between parallel lines and parallel planes. Vocabulary: Equidistant Additional Reading: Textbook 3-5 pg. 154
Real World Application Construction Dominique wants to install vertical boards to strengthen the handrail structure on her deck. How can she guarantee the vertical boards will be parallel?
Challenge Problem Find the distance between point P(6, -2) and the graph of line k whose equation is y = 7.
Spherical Geometry TEKS/TAKS: 1.a, 2.a, 2.b, 4.a, 7.a, 7.b, 9.a Objective: You will identify points, lines, and planes in spherical geometry and compare and contrast basic properties of plane and spherical geometry. Used in: Understanding the relationship of locations on the surface of Earth. Vocabulary: Plane Euclidean geometry, spherical geometry, non-Euclidean geometry Additional Reading: Textbook 3-6 pg. 163
Real World Application Space Travel According to Einstein’s Spherical Universe, geometric properties of space are similar to those on the surface of a sphere. Based on this model, what conclusion follows about the path of a spaceship along a straight line? Explain your reasoning.
Challenge Problem Explain why triangle ABC could not exist in plane Euclidean geometry. Could triangle ABC exist in non-Euclidean spherical geometry? Explain your reasoning. Include a drawing. A B C
Classifying Triangles TEKS/TAKS: 1.a, 1.b, 2.b, 4.a, 7.a, 7.c, 9.b Objective: You will identify the parts of triangles and classify triangles by their parts. Used in: Architecture and crafts Vocabulary: Triangle, polygon, sides, vertices, acute, obtuse, right triangle, equiangular, scalene, isosceles, equilateral Additional Reading: Textbook 4-1 pg. 180
Real World Application Architecture Consider a figure of the basic structure of the geodesic dome. How many equilateral triangles are in the figure. How many obtuse triangles are in the figure?
Challenge Problem The Pythagorean Theorem states that in a right triangle, the square of the measure of the hypotenuse is equal to the sum of the squares of the measures of the legs. Use patty paper to create several obtuse and acute triangles. Measure the sides of each. Square the measure of each side. Compare the square of the measure of the longest side to the sum of the squares of the measures of the other two sides. Make a conjecture about the relationship of the square of the measure of the longest side compared to the sum of the squares of the measures of the two shorter sides.
Measuring Angles in Triangles TEKS/TAKS: 1.a, 1.b, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b Objective: You will apply the Angle Sum Theorem and you will apply the Exterior Angle Theorem. Used in: Construction, design, and astronomy Vocabulary: Exterior angle, remote interior angle, flow proof Additional Reading: Textbook 4-2 pg. 189
Real World Application Astronomy Leo is a constellation that represents a lion. Three of the brighter stars in the constellation form a triangle LEO. If the angles have measures indicated in the figure at the right, find m
"name": "Real World Application Astronomy Leo is a constellation that represents a lion.",
"description": "Three of the brighter stars in the constellation form a triangle LEO. If the angles have measures indicated in the figure at the right, find m
Exploring Congruent Triangles TEKS/TAKS: 1.a, 1.b, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.a, 10.b Objective: You will name and label corresponding parts of congruent triangles. Used in: Crafts, arts, and construction Vocabulary: Congruent triangles, congruence transformation Additional Reading: Textbook 4-3 pg. 196
Real World Application Crafts Use graph paper to design a quilt using congruent triangles. Classify the triangles used by name and identify those that can be proved congruent.
Challenge Problem On graph paper, draw six congruent right scalene triangles. Cut out the triangles. Arrange the triangles so that congruent sides fit together. Try several different arrangements. How many different shapes with four sides can you make? How many different shapes with three sides can you make?
Proving Triangles Congruent TEKS/TAKS: 1.a, 1.b, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 7.a, 9.b, 10.a, 10.b Objective: You will use SSS, SAS, and ASA Postulates to test for triangle congruence. Used in: Determine if two triangles are congruent Vocabulary: CPCTC Additional Reading: Textbook 4-4 pg. 206
Real World Application Recreation Tapatan is a game played in the Phillipines on a square board as shown at the right. The players take turns placing each of their three pieces on a different point of intersection. After all the pieces have been played, the players take turns moving a piece along a line to another intersection. A piece cannot jump over another piece. A player who gets all his or her pieces in a straight line wins the game. Point E bisects all four line segments that pass through it. What can you say about triangle GHE and triangle CBE? Explain. What you can say about triangle AEG and triangle IEGA? Explain. What can you say about triangle ACI and triangle CAG? Explain. A D G BC F IH E in the middle
Challenge Problem Write a proof. Given:
"name": "Challenge Problem Write a proof.Given:
More Congruent Triangles TEKS/TAKS: 1.a, 1.b, 2.b, 3.b, 3.d, 3.e, 4.a, 9.b, 10.a, 10.b Objective: You will use the AAS Theorem to test triangle congruence, and to solve problems by eliminating possibilities. Used in: Broadcasting Vocabulary: Eliminate the possibilities Additional Reading: Textbook 4-5 pg. 214
Real World Application History It is said that Thales determined the distance from the shore to enemy Greek ships during an early war by sighting the angel to the ship from a point P on the shore, walking a distance to point Q, and then sighting the angle to the ship from that point. He then reproduced the angles on the other side of line PQ and continued these lines until they intersected. How did he determine the distance to the ship in this way? Why does this method work? PQ
Challenge Problem Can two triangles be proved congruent by AAA (Angle-Angle-Angle)? Justify your answer completely.
Analyzing Isosceles Triangles TEKS/TAKS: 1.a, 2.b, 4.a, 9.b, 10.b Objective: You will use the properties of isosceles and equilateral triangles. Used in: Design, carpentry, and navigation Vocabulary: None Additional Reading: Textbook 4-6 pg. 222
Real World Application Navigation Sau-Lim is the captain of a ship and he uses an instrument called a “pelorus” to measure the angle between the ship’s path and the line from the ship to a lighthouse. Sau-Lim finds the distance that the ship travels and the change in the measure of the angle with the lighthouse as the ship sails. When the angle with the lighthouse is twice that of the original angle, Sau-Lim knows that the ship is as far from the lighthouse as the ship has traveled since the lighthouse was first sighted. Why?
Challenge Problem Draw an isosceles triangle ABC with vertex angle at A. Find the midpoints of each side. Label the midpoint of AB point D, the midpoint BC point E, and the midpoint of AC point F. Draw triangle DEF. Name a pair of congruent triangles. Explain your reasoning. Name three isosceles triangles. Explain your reasoning.
Applying Congruent Triangles TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 4.a, 7.a, 7.b, 9.b, 10.b Objective: You will identify and use medians, altitudes, angle bisectors, and perpendicular bisectors in a triangle. Used in: Engineering, sports, and physics. Vocabulary: Perpendicular bisector, median, altitude, angle bisector Additional Reading: Textbook 5-1 pg. 238
Real World Application Physics Physicists often make calculations based on the center of gravity of an object. Follow the steps and answer the questions below to investigate the center of gravity of a triangle. a.Draw a large acute triangle that is not isosceles. b.Construct the three medians of the triangle. What do you notice? c.Construct three angle bisectors of the triangle. What do you notice? d.Construct three altitudes of the triangle. What do you notice? e.Construct the three perpendicular bisectors of the sides of the triangle. What do you notice? f.Cut out the triangles you made for parts a-e. Place the point where the segments intersect on the flat end of a pencil for each of the triangles. What do you observe? g.What changes would occur in the construction in parts b-e if the triangle were right or obtuse instead of acute?
Challenge Problem Draw any triangle ABC with median AD and altitude AE. Recall that the area of a triangle is one-half the product of the measures of the base and the altitude. What conclusion can you make about the relationship between the areas of triangle ABD and triangle ACD?
Right Triangles TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b Objective: You will recognize and use tests for congruence of right triangles. Used in: Trigonometry Vocabulary: Leg, hypotenuse Additional Reading: Textbook 5-2 pg. 245
Real World Application Construction A wooden bridge is being constructed over a stream in a park. The braces for the support posts came from the lumberyard already cut. The carpenter measures from the top of the support post to a point on the post to find where to attach the brace to the post. Explain why only one measurement must be made to ensure that all of the braces will be in the same relative position.
Challenge Problem In the figure below, m
"name": "Challenge Problem In the figure below, m
Indirect Proofs and Inequalities TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b Objective: You will use indirect reasoning and indirect proof to reach a conclusion, you will recognize and apply properties of inequalities to the measures of segments and angles, and you will solve problems by working backwards. Used in: Law and advertising Vocabulary: Indirect reasoning, indirect proof, working backward Additional Reading: Textbook 5-3 pg. 252
Real World Application Law The defense attorney said to the jury, “My client is not guilty. According to the police, the crime occurred on July 14 at 6:30 p.m. in Boston. I can prove that at that time my client was attending a business meeting in New York City. A verdict of not guilty is the only possible verdict.” Is this an example of indirect reasoning? Explain why or why not.
Challenge Problem Mr. Mendez was checking on the date he had attended a four-day conference a year ago. The page in his record book was torn and all that remained of the date for the meeting was “ber 31”. What was the month of the first day of the conference?
The Triangle Inequality TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b Objective: You will apply the Triangle Inequality Theorem. Used in: Carpentry and mathematics history Vocabulary: Triangle Inequality Theorem Additional Reading: Textbook 5-5 pg. 267
Real World Application Carpentry Salina is building stairs and wants to nail a brace at the base of each stair as shown in the figure. The brace attaches at the bottom of the rise and anywhere along the tread. The stairs have an 18-cm rise and a 26-cm tread. There is a pile of braces 5 centimeters, 20 centimeters, 24 centimeters, and 45 centimeters long that Salina can use. Which of the lengths can she sue as a brace?
Challenge Problem Is it true that the difference between any two sides of a triangle is less than the third side? Explain your reasoning?
The Triangle Inequality TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 9.b, 10.b Objective: You will apply the SAS Inequality and the SSS Inequality. Used in: Physical therapy and biology Vocabulary: SAS, SSS Additional Reading: Textbook 5-6 pg. 273
Real World Application Physics The discovery of the lever allowed ancient people to accomplish great tasks like building Stonehenge and the Egyptian pyramids. A lever multiplies the force applied to an object. One example of a lever is the nutcracker. Use the SAS or SSS Inequality to explain how to operate the nutcracker. fulcrum force load
Challenge Problem Suppose that plane F bisects AC at B for a point D in F, DC > DA. What can you conclude about the relationship between AC and plane F? AB D C F
Parallelograms TEKS/TAKS: 1.a, 2.b, 4.a, 5.a, 7.a, 7.b, 9.b Objective: You will recognize and apply the properties of a parallelogram, and you will find the probability of an event. Used in: Transportation and interior design Vocabulary: Quadrilaterals, parallelogram, diagonals, probability Additional Reading: Textbook 6-1 pg. 291
Real World Application Language In a commercial for CompuServe Computer Discount House, the announcer says that he thought a parallelogram was a “telegram for gymnasts at the Olympics.” Look up the suffix “gram” in a dictionary. Then make a conjecture about why a parallelogram is named as it is.
Challenge Problem Consider parallelogram RSTV. As the measure of angle R decreases, what must happen to the measure of angle V? What is the maximum measure for angle V? Explain your reasoning.
Tests for Parallelograms TEKS/TAKS: 1.a, 1.b, 2.b, 4.a, 5.a, 7.a, 7.b, 9.b, 10.a Objective: You will recognize and apply the conditions that ensure a quadrilateral is a parallelogram and you will identify and use subgoals in writing proofs. Used in: Engineering and arts Vocabulary: Identifying subgoals Additional Reading: Textbook 6-2 pg. 298
Real World Application Drafting Before computer drawing programs became available, blueprints for buildings or mechanical parts were drawn by hand. One of the tools drafters used, a parallel ruler, is shown above. Holding one of the bars in place and moving the other allowed the drafter to draw a line parallel to the first in many positions on the page. Why does the parallel ruler guarantee that the second line will be parallel to the first?
Challenge Problem Ellen claims she has invented a new geometry theorem: “A diagonal of a parallelogram bisects its angles”. She gives the following proof: Given: parallelogram MATH with diagonal MT Prove: MT bisects
"name": "Challenge Problem Ellen claims she has invented a new geometry theorem: A diagonal of a parallelogram bisects its angles .",
"description": "She gives the following proof: Given: parallelogram MATH with diagonal MT Prove: MT bisects
Squares and Rhombi TEKS/TAKS: 1.a, 2.a, 2.b, 3.b, 3.c, 3.d, 3.e, 4.a, 5.a, 7.b, 9.b Objective: You will recognize and apply the properties of squares and rhombi. Used in: Art and construction Vocabulary: Rhombus, rhombi, square Additional Reading: Textbook 6-4 pg. 313
Real World Application Construction The opening for the reinforced doors of a storage shed is shaped like a square. Identify the quadrilaterals that make up the doors. Explain why you think the doors are shaped as they are.
Challenge Problem Use a ruler to draw a quadrilateral with perpendicular diagonals that is not a rhombus.
Kites TEKS/TAKS: 2.a, 2.b, 9.b Objective: You will recognize and apply the properties of kites. Used in: Art and construction Vocabulary: Kite Additional Reading: Textbook 6-4B pg. 320
Real World Application Construction Draw AC in kite ABCD. Use a protractor to measure the angles formed by the intersection of AC and BD. Measure the interior angles of kite ABCD. Are any congruent?
Challenge Problem Given kite ABCD, label the intersection of AC and BD as point E. Find the lengths of AE, BE, CE, and DE. How are they related?
Challenge Homework Pg. 320 Write at least five original conjectures about kites. Justify your conjectures.
Trapezoids TEKS/TAKS: 1.a, 2.b, 3.c, 3.e, 4.a, 5.a, 7.a Objective: You will recognize and apply the properties of trapezoids. Used in: Sailing and engineering Vocabulary: Trapezoid, bases, legs, base angles, isosceles trapezoid Additional Reading: Textbook 6-5 pg. 321
Real World Application Architecture Tray ceilings make a room appear to be larger than it actually is. The trays are make using wood or drywall panels. What type of quadrilaterals are used to make the tray ceiling shown above?
Challenge Problem Draw a trapezoid with two right angles. Label the bases and the legs. Try to draw an isosceles trapezoid with two right angles. Is it possible? Explain why or why not.
Using Proportions TEKS/TAKS: 1.a, 2.b, 3.e, 4.a, 5.a, 9.b, 11.b Objective: You will recognize and use ratios and proportions and you will apply properties of proportions. Used in: Movie props, literature, and meteorology Vocabulary: Ratios, proportion, cross products, means, extremes Additional Reading: Textbook 7-1 pg. 338
Real World Application Cartography The scale on a map indicates that 1.5 centimeters represents 200 miles. If the distance on the map between Norfolk, Virginia, and Atlanta, Georgia, measures 2.4 centimeters, how many miles apart are the cities?
Challenge Problem At each gas stop on a recent trip, Keshia recorded the distance she traveled since the last stop and the amount of gas she purchased for her midsize car. Find the average number of miles per gallon. StopsDistance (mi)Gas (gal) 135113 227511 3362.512.5 437212
Exploring Similar Polygons TEKS/TAKS: 1.a, 1.b, 2.b, 3.e, 4.a, 5.a, 7.b, 9.b, 11.b Objective: You will identify similar figures and solve problems involving similar figures. Used in: Cartography, gardening, and construction work Vocabulary: Similar figures, dilation Additional Reading: Textbook 7-2 pg. 346
Real World Application Construction On the floor plan for a new house, one inch represents 18 feet. If the living room is 3/4 inch by 5/8 inch, what are the dimensions?
Challenge Problem Use what you know about slope and distance to show that two triangles are similar. Triangle ABC has vertices A(0,0), B(12,0), and C(6,9). Triangle DEF has vertices D(18,0), E(26,0), and F(22,6).
Identifying Similar Triangles TEKS/TAKS: 1.a, 1.b, 2.b, 3.e, 4.a, 5.a, 9.b, 11.b Objective: You will identify similar triangles and use similar triangles to solve problems. Used in: Surveying, forestry Vocabulary: None Additional Reading: Textbook 7-3 pg. 354
Real World Application Surveying (pg. 360) Mr. Cardona uses a carpenter’s square, an instrument used to draw right angles, to find the distance across a stream. He puts the square on top of a pole that is high enough to sight along OL to point P across the river. Then he sights along ON to point M. If MK is 2.5 feet and OK = 5.5 feet, find the distance KP across the stream.
Challenge Problem Is it possible that triangle ABC is not similar to triangle RST and that triangle RST is not similar to triangle EFG, but that triangle ABC is similar to triangle EFG? Explain.
Parallel Lines and Proportional Parts TEKS/TAKS: 1.a, 2.a, 2.b, 3.e, 4.a, 5.a, 7.a, 11.b, 11.c Objective: You will use proportional parts of triangles to solve problems and to divide a segment into congruent parts. Used in: History and navigation Vocabulary: None Additional Reading: Textbook 7-4 pg. 362
Real World Application Construction Hai was building a large open stairway and used wood strips as decoration along the inside wall. If the strips were spaced along the bottom as shown in the diagram above, at what distance should he attach the top of the strips if the strips are to be parallel? 8 ft x y z 2.5 ft2 ft1.5 ft
Challenge Problem Draw any quadrilateral ABCD and connect the midpoints E, F, G, H of the sides in order. Determine what kind of figure EFGH will be. Prove your claim.
Parts of Similar Triangles TEKS/TAKS: 1.a, 2.b, 3.b, 3.c, 3.e, 4.a, 5.a Objective: You will recognize and use the proportional relationships of corresponding perimeters, altitudes, angle bisectors, and medians of similar triangles. Used in: Photography, design, and art Vocabulary: None Additional Reading: Textbook 7-5 pg. 370
Real World Application Design Julian had a picture 18 centimeters by 24 centimeters that he wanted enlarged by 30% and then have the inside of the frame edges with navy blue piping. The store only had 110 centimeters of navy blue piping in stock. Will this be enough piping to fit on the inside edge of the frame? Explain.
Challenge Problem Consider two rectangular prisms shown at the right. What ratios are necessary to determine whether they are similar? If the first prism has dimensions that are three times as large as the second, will the volume also have a ratio of 1 to 3? Explain why or why not. s h w a b c
Fractals and Self-Similarity TEKS/TAKS: 1.a, 2.b, 4.a, 5.b, 11.a, 11.b, 11.d Objective: You will recognize and describe characteristics of fractals and will solve problems by solving a simpler problem. Used in: Nature and art Vocabulary: Sierpinski triangle, fractal, self-similar, strictly self-similar, solve a simpler problem Additional Reading: Textbook 7-6 pg. 378
Real World Application Solve a Simpler Problem Look at the diagonals in Pascal’s triangle. Find the sum of the first 25 numbers in the outside diagonal. Find the sum of the first 50 numbers in the second diagonal.
Challenge Problem The fractal in the pictures at the right is the space-filling Hilbert curve. Define the iterative process used to generate the curve. Why do you think it is called “space filling”?
Geometric Mean and the Pythagorean Theorem TEKS/TAKS: 1.a, 1.b, 2.b, 3.e, 4.a, 5.c, 8.c, 11.c Objective: You will find the geometric mean between two numbers, solve problems involving relationships between parts of a triangle and the altitude to its hypotenuse, and use the Pythagorean Theorem and its converse. Used in: Architecture Vocabulary: Geometric mean, Pythagorean triple Additional Reading: Textbook 8-1 pg. 397
Real World Application Motion Pictures In the movie The Wizard of Oz, the Scarecrow is looking for a brain. When the Wizard presents him with a Doctor of Thinkology degree, the Scarecrow immediately announces “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.” Do you agree with the “Scarecrow Theorem”? Explain.
Challenge Problem Draw an acute triangle and an obtuse triangle. In the acute triangle, draw an altitude to the longest side. In the obtuse triangle, draw an altitude from the obtuse angle. In either case, are the triangles formed by the altitude similar to the original? Explain.
Special Right Triangles TEKS/TAKS: 1.a, 2.b, 4.a, 5.c, 11.c Objective: You will use the properties of 45-45-90 and 30-60-90 triangles. Used in: Sports, city planning, and landscaping Vocabulary: None Additional Reading: Textbook 8-2 pg. 405
Real World Application City Planning Granmichele is a city with a population of about 15,000 people in northern Sicily. The city has a hexagonal design as shown by the aerial view at the right. Consider all sides of the hexagons to be congruent. Suppose the perpendicular distance from the center of the city to each side of the one hexagon is 0.5 mile. Find the perimeter of that hexagon.
Challenge Problem Two parallel lines are cut by a transversal. The transversal makes a 120° angle with one of the parallel lines. A line bisects the 120° angle. Another line bisects the consecutive interior angle. Draw the figure and show that the triangle formed by the two angle bisectors and the transversal is a 30-60-90 triangle.
Ratios in Right Triangles TEKS/TAKS: 1.a, 2.b, 4.a, 5.c, 9.b, 11.c Objective: You will find trigonometric ratios using right triangles and you will solve problems using trigonometric ratios. Used in: Aviation, medicine, and astronomy Vocabulary: Trigonometry, trigonometric ratio, sine, cosine, tangent Additional Reading: Textbook 8-3 pg. 412
Real World Application Medicine A patient is being treated with radiotherapy for a tumor that is behind a vital organ. In order to prevent damage to the organ, the radiologist must angle the rays to the tumor. If the tumor is 6.3 cm below the skin and the rays enter the body 9.8 cm to the right of the tumor, find the angle the rays should enter the body to hit the tumor. Skin Organ Tumor 9.8 cm
Challenge Problem Draw a circle in Q1 of a coordinate plane with a radius of 1 unit. Each division on the x- and y-axes is 0.2 unit. Right triangles are formed by drawing a vertical line from the point on the x-axis to the circle, the connecting that point with the origin. Use the Pythagorean Theorem and trigonometry to complete the table of values for each triangle. Length of hypotenuse X-valueY-valueSin OCos O 10.2 10.4 10.6 10.8
Angles of Elevation and Depression TEKS/TAKS: 1.a, 2.b, 4.a, 11.c Objective: You will use trigonometry to solve problems involving angles of elevation or depression. Used in: Aerospace, architecture, and meteorology Vocabulary: Angle of depression, angle of elevation Additional Reading: Textbook 8-4 pg. 420
Real World Application Literature “In the Adventures of Sherlock Holmes: The Adventures of the Musgrave Ritual”, Sherlock Holmes uses trigonometry to solve the mystery. To find a treasure, he must determine where the end of the shadow of an elm tree was located at a certain time of day. Unfortunately, the elm had been cut down, but Mr. Musgrave remembers that his tutor required him to calculate the height of the tree as part of his trigonometry class. Mr. Musgrave tells Sherlock Holmes that the tree was exactly 64 feet. Sherlock needs to find the length of the shadow at a time of day when the shadow from an oak tree is a certain length. The angle of elevation of the sun at this time of day is 33.7°. What was the length of the shadow of the elm?
Challenge Problem Imagine that a fly and an ant are in one corner of a rectangular box. The end of the box is 4 inches by 6 inches, and the diagonal across the bottom of the box makes an angle of 21.8° with the longer edge of the box. There is food in the corner opposite the insects. What is the shortest distance the fly must fly to get to the food? What is the shortest distance the ant must crawl to get to the food? 4 in 6 in Food Insects 21.8°
Exploring Circles TEKS/TAKS: 1.a, 1.b, 2.b, 4.a, 7.a, 9.b, 9.c Objective: You will identify and use parts of circles and solve problems involving the circumference of a circle. Used in: Surveying, sports, and space travel Vocabulary: Circle, center, radius, chord, diameter, circumference, pi Additional Reading: Textbook 9-1 pg. 446
Real World Application Culture About one thousand years ago, wooden poles formed a gigantic circle 125 meters across in southern Illinois. The structure was a giant solar calendar that kept track of the seasons and the movement of the sun for Native North Americans. What was the circumference of this structure?
Challenge Problem Use the Triangle Inequality Theorem to show diameter SA is the longest chord in circle P. That is, write a paragraph proof that shows SA > KR. (Hint: Draw PK and PR). SPA K R
Angles and Arcs TEKS/TAKS: 1.a, 2.b, 4.a, 5.a, 8.b, 9.b, 9.c Objective: You will recognize major arcs, minor arcs, semicircles, and central angles, you will find measures of arcs and central angles, and you will solve problems by making circle graphs. Used in: Data display, clocks, statistics Vocabulary: Central angle, arc, minor arc, major arc, semicircle, adjacent arcs, arc length, concentric circles, similar circles, congruent circles, congruent arcs Additional Reading: Textbook 9-2 pg. 452
Real World Application Clocks The Floral Clock in Frankfort, Kentucky, has a diameter of 34 feet. The hands on the clock form a central angle with the circular timepiece. Suppose the hour hand is on the 10 (A) and the minute hand is on the 2 (C) and the vertex is B. Find the measure of the central angle ABC. Find the arc length of the minor arc.
Challenge Problem Draw a diagram to explain how it is possible for two central angles to be congruent, yet their corresponding minor arcs are NOT congruent.
Arcs and Chords TEKS/TAKS: 1.a, 2.a, 2.b, 4.a, 5.a, 9.b, 9.c Objective: You will recognize and use relationships among arcs, chords, and diameters. Used in: Geology Vocabulary: Arc of the chord, inscribed polygon Additional Reading: Textbook 9-3 pg. 459
Real World Application Road Signs Jodi wants to create a new yield sign that inscribes the yellow isosceles triangle in a circle. Draw an isosceles triangle and explain how Jodi could use the theorems from this lesson to find the center of the circle that contains the yellow yield sign.
Challenge Problem Draw a circle R and choose a point P on the circle. Now draw six chords with P as one endpoint. Label the other endpoints A, B, C, D, E and F. Make a conjecture as to how the lengths of each chord are related to their distances from the center of the circle. Explain your reasoning.
Inscribed Angles TEKS/TAKS: 1.a, 2.a, 2.b, 4.a, 5.a, 9.c Objective: You will recognize and find measures of inscribed angles and apply properties of inscribed figures. Used in: Civil engineering and carpentry Vocabulary: Inscribed angle, intercepted arc Additional Reading: Textbook 9-4 pg. 466
Real World Application Civil Engineering A civil engineer uses the formula l = (2*pi*r*m)/360 to calculate the length l of a curve for a road, given a radius r and a central angle measure m. Find the length of the road from A to B in the diagram at the right. Round to the nearest foot. 70 ft B A 220 ft l
Challenge Problem Can an isosceles trapezoid be inscribed in a circle? Write a brief paragraph explaining your reasoning.
Tangents TEKS/TAKS: 5.a, 9.b, 9.c Objective: You will recognize tangents and use properties of tangents. Used in: Astronomy and aerospace Vocabulary: Tangent, point of tangency, interior, exterior, common internal tangent, common external tangent, tangent segments Additional Reading: Textbook 9-5 pg. 475
Real World Application Olympics At the 1996 Summer Olympics, Anthony Washington of Aurora, Colorado, finished fourth in the discus event with a throw of 65.42 meters. If he wound up in a circular pattern to throw the discus, along a tangent line (path) to the circle, use the figure to find the radius of the circle. x x 65.42 m 67.13 m
Challenge Problem A unit circle is a circle with a radius of 1. In the figure, circle O is a unit circle with QR tangent to circle O at R. Use the right triangle trigonometry definition of tangent to find tan theta. Q R1 1
Tangents TEKS/TAKS: 5.a, 9.b, 9.c Objective: You will recognize tangents and use properties of tangents. Used in: Astronomy and aerospace Vocabulary: Tangent, point of tangency, interior, exterior, common internal tangent, common external tangent, tangent segments Additional Reading: Textbook 9-5 pg. 475