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GEOMETRY The Geometry course builds on Algebra I concepts and increases students’ knowledge of shapes and their properties through geometry-based applications, many of which are observable in aspects of everyday life. This knowledge helps develop visual and spatial sense and strong reasoning skills. The Geometry course requires students to make conjectures and to use reasoning to validate or negate these conjectures. The use of proofs and constructions is a valuable tool that enhances reasoning skills and enables students to better understand more complex mathematical concepts. Technology should be used to enhance students’ mathematical experience, not replace their reasoning abilities. Because of its importance, this Euclidean geometry course is required of all students receiving an Alabama High School Diploma. School systems may offer Geometry and Geometry A and Geometry B. Content standards 1, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 27, 31, 32, 33, 34, 35, 43, 44, 45, 46, 47, 48, 49, 50, 51, and 52 must be taught in the Geometry A course. Content standards 2, 12, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 28, 29, 30, 36, 37, 38, 39, 40, 41, and 42 must be taught in the Geometry B course. Students will: GEOMETRY Congruence Experiment with transformations in the plane. 1. Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment based on the undefined notions of point, line, distance along a line, and distance around a circular arc. [G-CO1] 2. Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). [GCO2] 3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. [G-CO3] 4. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. [G-CO4] 5. Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. [G-CO5] ACTIVITY: Use the TI-83/84 graphing calculators to graph functions and then practice editing the lists of ordered pairs to graph images which are transformations of these functions. A sheet of instructions for a simple activity involving transformations can be found below. All of the keystrokes for the calculator are given. The transformations can be extended to include composition of more than one transformation, etc. 1 Transformations on the TI-83/84 Graphing Calculators By Linda Bridges 1. Make a list of ordered pairs that will be used to graph a particular object. 2. Enter those ordered pairs into L1 and L2 on the calculator. Go to STAT and then ENTER. If there are numbers already in L1 and L2, clear them. To clear the lists, highlight upward to the name of the list (L1 or L2) and then hit CLEAR and ENTER. Once lists are cleared, enter the X-coordinates into L1 and the y-coordinates of the ordered pairs into L2. 3. To graph the ordered pairs, make sure all other graphs are turned off. To do this, hit the “y=” key on the first row and hit “clear” until all graphs have been cleared. Also, the stat plots must be turned off. To do this, hit “2nd” and “y=” and then scroll to number 4 “Plotsoff” and hit ENTER twice until it says “done”. Now you have all graphs cleared out and are ready to graph the ordered pairs. Hit “2nd” and “y=” again and then ENTER. Scroll over to “ON” and then press ENTER until “on” is highlighted. Under the “type”, choose the 2nd type of graph (scatterplot with connected dots) and then press enter until it is the only graph type highlighted. Under “xlist” type L1 and under “ylist”, type L2. To type L1, you must first hit the “2nd” key and then the “1” key. To type L2, you must first hit the “2nd” key and then the “2” key. Choose a “mark” to be used to graph the points. You may choose the square, the cross, or the dot. VERY IMPORTANT: Before attempting to look at the graph, hit the ZOOM key on the top row and then scroll down to number 9, “ZoomStat”, and press enter. You should now see the ordered pairs you placed in the lists graphed and the dots connected with line segments to form the picture. 4. To observe transformations, you must enter more data into lists 3 and 4. KEEP LISTS 1 and 2 AS THEY ARE. DO NOT DELETE OR CHANGE THESE TWO LISTS. TRANSLATIONS: Hit the “Stat” key and ENTER and then go to L3 and L4. If you need to clear these lists, highlight the list name at the top and then hit “clear” and “enter”. Then suppose you want to move everything to the right 2 places and down 4 places. Then you need to have “X+2” and “Y-4”. You do not have to change each individual ordered pair. GO to L3 and arrow upward to the list name and then press ENTER. You should now see the cursor at the bottom of your screen and it is waiting for you to give it a formula. Since you want to add 2 to your x values and these are in L1, type in “L1 +2” and then press ENTER. REMEMBER: To type L1, you must hit the “2nd” key and then the “1” key. After you have typed “L1+2” and pressed ENTER, you should immediately see L3 filled with the L1 values plus 2. Then go to L4 and scroll upward and highlight the list name and push ENTER. You now want to tell the calculator to take everything in the y list which is L2 and subtract 4. Thus, after you have highlighted L4 and hit ENTER, you should now see the cursor at the bottom. Type “L2-4” and hit ENTER and then list 4 should be filled will all your y values minus 4. To see the graph of these, go to “2nd” and “y=” and then scroll down to Plot2. DO NOT TURN PLOT ONE OFF___KEEP IT ON. Scroll to Plot 2 and press ENTER. Follow 2 the same instructions as before to turn it on and then choose the 2nd option under the type. Then you must type in L3 (“2nd” and “3”) for the x-list and then L4 (“2nd” and “4”) for the ylist. IMPORTANT: Choose a “mark” different from the one that you chose for the original figure so that you will be able to distinguish the new figure from the original figure when both are graphed together on the same screen. Then go to ZOOM and choose number 9 and then hit ENTER. You can see both graphs and how the second graph is shifted 2 to the right and down 4. You can do other translations in a similar manner. For instance, “L1-3” and “L2+5” placed as formulas into L3 and L4 would move the original graph 3 to the left and up 5. 5. REFLECTIONS about Y-axis. To see reflections, leave L1 and L2 as they are. DO NOT CHANGE THESE LISTS. To reflect about the y-axis, you must have (-X,Y) as the ordered pairs. Thus go to L3 and L4 and clear those out. Then highlight L3 name at the top and press ENTER. Type in negative sign and then L1 and press ENTER. Remember there is a difference between the negative sign next to the enter key and the subtraction operation symbol. You must type in “- L1”, since you want all your x-values to be the opposite sign. GO to L4 and highlight its name at the top and press ENTER and then type in “L2” and press ENTER since you want the y values to stay the same. Then go to “2nd” and “y=” and then go immediately to ZOOM and then #9 and press enter. (You have already told it to graph all four lists in two different statplots. You do not have to tell it to do that again). However, you must tell the calculator to ZOOMSTAT each time before you view another graph. 6. REFLECTIONS about the X-axis. To see reflections, leave L1 and L2 as they are. DO NOT CHANGE THESE LISTS. To reflect about the x-axis, you must have (X,-Y) as the ordered pairs. Thus go to L3 and L4 and clear those out. Then highlight L3 name at the top and press ENTER. Type in “L1” and press ENTER. GO to L4 and highlight its name at the top and press ENTER and then type in “negative sign and L2” and press ENTER since you want the y values to be the opposite of the original y values. Then go to “2nd” and “y=” and then go immediately to ZOOM and then #9 and press ENTER. (You have already told it to graph all four lists in two different statplots. You do not have to tell it to do that again). However, you must tell the calculator to ZOOMSTAT each time before you view another graph. 7. ROTATION of 180 degrees about the origin. If you clear L3 and L4 and then set L3 equal to the negative of L1 and set L4 equal to the negative of L2, you should be able to see a rotation of 180 degrees. 8. ROTATION OF 90 Degrees Counterclockwise about the origin. If you clear L3 and L4 and then set L3 equal to the negative of L2 and set L4 equal to L1, you should be able to see a rotation of 90 degrees counterclockwise. 9. DILATIONS: To see a dilation (stretching or shrinking) of the figure, try multiplying both L1 and L2 by a constant---either less than 1 or greater than 1---to see how the size of the object changes. This is a good exercise for talking about scale factors and similar figures. 3 OTE: I started out with a picture of a house, using these original ordered pairs. L1 L2 3 1 3 6 6 9 7 7 7 10 8 10 8 6 8 1 3 1 4 A quick overview of various transformations can be found at: http://www.regentsprep.org/Regents/math/algtrig/ATP9/funclesson1.htm Summarizes and illustrates: Reflections over the x-axis, -f(x); reflections over the y-axis, f(-x); translations, f(x+a) and f(x) + a; horizontal and vertical stretches or compressions, f(ax) or af(x); Additional information can be found at: http://www.regentsprep.org/Regents/math/algtrig/ATP9/reviewTranformations.htm Summarizes line reflections (in the y-axis, in the x-axis, in y = x, in y= -x); point reflections in the origin; rotations of 90 degrees, 180 degrees, and 270 degrees; dilations of scale factor k; and translations. The website also summarizes which transformations are isometries. A composition of transformations occurs when two or more transformations are combined to form a new transformation. Watch the order. Perform the transformation indicated on the right first, then do the transformation indicated by the one on the left. For instance, ry-axis o rx-axis = rorigin This implies that a reflection in the x-axis followed by a reflection in the y-axis is the same as a single transformation of a reflection in the origin. Likewise, rx-axis o T(2,3) indicates a translation of (x+2, y+3) followed by a reflection in the x-axis. A glide reflection is an example of a composition of transformations. A glide reflection is the composition of a glide (translation) and a reflection across a line parallel to the direction of the translation. A glide reflection is an isometry, since translations and reflections are both isometries. (The composition of isometries is an isometry). An isometry is a transformation in which the preimage and the image are congruent. Direct Isometry - orientation is preserved - the order of the lettering in the figure and the image are the same, either both clockwise or both counterclockwise. Opposite Isometry - orientation is not preserved - the order of the lettering is reversed, either clockwise becomes counterclockwise or counterclockwise becomes clockwise. The following website gives a concise review of all transformations and the isometry status of each—whether the transformation is an opposite isometry, direct isometry, or not an isometry. http://www.regentsprep.org/regents/math/geometry/GT5/properties.htm 5 ACTIVITY: Using masking or colored tape, divide the classroom floor into a coordinate plane, with the tape indicating where the X and Y axes are located. (Be sure to identify the orientation of these axes, by labeling the positive x-axis and the positive y-axis.) The students’ desks will then be divided up into the four quadrants. Students will be asked to identify who is their: reflection over the y-axis, reflection over the x-axis, point reflection in the origin, translation under a given rule, etc. Then a student will be asked to tell how they could be transformed to another student’s position, etc. ACTIVITY: Matching Exercise. Students will be given a card with a set of points whose coordinates are given. Then students will be given another card with a transformation activity on it—for instance, “dilation with a scale factor of 3” or “ reflection in the line y = x, translate (x-3, y+4), etc. Students will then identify their new points after the transformation has been performed. ACTIVITY: Use Geometer’s SketchPad software to experiment with transformations. Steps for using the software to investigate transformations can be found at http://mathbits.com/MathBits/GSP/ScratchTransformations.htm Understand congruence in terms of rigid motions. (Build on rigid motions as a familiar starting point for development of concept of geometric proof.) 6. Use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. [G-CO6] 7. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. [G-CO7] 8. Explain how the criteria for triangle congruence, angle-side-angle (ASA), side-angle-side (SAS), and side-sideside (SSS), follow from the definition of congruence in terms of rigid motions. [G-CO8] Congruence between two geometric objects can be defined as a rigid motion of the plane that maps one of the objects onto the other. ACTIVITY: Students will use straws, scissors, and rulers to cut 2 inch, 3 inch, and 4 inch segments. Then they will form a triangle by stringing these together with twine. Are all triangles formed by different groups the same? Identify congruent angles, congruent sides, etc. Use this knowledge of congruent figures to find missing side lengths and missing angle measures of figures known to be congruent. Then students will construct triangles with two and three inch sides and a 60 degree included angle. Are all the triangles formed just alike? Then they will do a similar exploration with ASA and AAS congruence conjectures to see if these conditions always produce congruent triangles. Then they will explore the AAA and the SSA options to see if these alway produce congruent triangles. (Of course the AAA and the SSA situations do not always produce congruent triangles. ) 6 Prove geometric theorems. (Focus on validity of underlying reasoning while using variety of ways of writing proofs.) 9. Prove theorems about lines and angles. Theorems include vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; and points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints. [G-CO9] 10. Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180º, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10] 11. Prove theorems about parallelograms. Theorems include opposite sides are congruent, opposite angles are congruent; the diagonals of a parallelogram bisect each other; and conversely, rectangles are parallelograms with congruent diagonals. [G-CO11] Make geometric constructions. (Formalize and explain processes.) 12. Make formal geometric constructions with a variety of tools and methods such as compass and straightedge, string, reflective devices, paper folding, and dynamic geometric software. Constructions include copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line. [G-CO12] 13. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. [G-CO13] Please note: If you do not know where the center of the circle is, you may draw two chords, segment AB and segment BC. Then construct the perpendicular bisector of each chord. They should meet at the center. To inscribe a regular hexagon in a circle, first locate the center of the circle. Make a point anywhere on the circle. This will be the first vertex of the hexagon. Set the compass on this point and set the width of the compass to be the distance between the first vertex and the center of the circle. Now, with the tip of the compass on the first vertex, mark an arc on the circle with the compass equal to the distance between the center of the circle and the first vertex. This will be the second vertex of the hexagon. Continue doing this until all 6 vertices have been located. Connect the points to form the 6 sides of the hexagon. (Since each side of a regular hexagon is equal to the distance from the center to any vertex, this construction simply sets the compass width to that radius, and then steps that length off around the circle to create the six vertices of the hexagon.) 7 To inscribe an equilateral triangle in a circle, start with an inscribed hexagon (constructed with the above instructions). Then draw the line segments indicated below to form an equilateral triangle. To inscribe a square in a circle, draw a diameter of the circle. Then construct another diameter perpendicular to the first one. Then connect vertices as shown in the diagram to make a square. 8 Similarity, Right Triangles, and Trigonometry Understand similarity in terms of similarity transformations. 14. Verify experimentally the properties of dilations given by a center and a scale factor. [G-SRT1] a. A dilation takes a line not passing through the center of the dilation to a parallel line and leaves a line passing through the center unchanged. [G-SRT1a] A dilation is a transformation that produces an image that is the same shape as the original, but is a different size. (A figure and its dilation are similar). The center of dilation is a fixed point in the plane about which all points are expanded or contracted. It is the only point that does not change in a dilation. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. [G-SRT1b] 15. Given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain using similarity transformations the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. [G-SRT2] 16. Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3] Prove theorems involving similarity. 17. Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [G-SRT4] To prove the Pythagorean Theorem using similarity, start with right triangle ABC with right angle at C. Draw an altitude from point C to the hypotenuse. Then it can be shown that Triangle ABC is similar to Triangle CBH and Triangle ABC is similar to Triangle ACH. Since Triangle ABC is similar to Triangle CBH, it follows that AB/CB = BC/BH and c/a =a/e and a2=ec. Since Triangle ABC is similar to Triangle ACH, it follows that AB/AC= AC/AH and c/b=b/d and b2=cd. Hence, a2+b2=ec+cd=c(e+d)=c(c)=c2. 9 18. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. [G-SRT5] Define trigonometric ratios and solve problems involving right triangles. 19. Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle leading to definitions of trigonometric ratios for acute angles. [G-SRT6] 20. Explain and use the relationship between the sine and cosine of complementary angles. [G-SRT7] 21. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.* [G-SRT8] Apply trigonometry to general triangles. 22. (+) Derive the formula A = ( 1/2)ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. [G-SRT9] 23. (+) Prove the Law of Sines and the Law of Cosines and use them to solve problems. [G-SRT10] 24. (+) Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). [G-SRT11] Circles Understand and apply theorems about circles. 25. Prove that all circles are similar. [G-C1] 26. Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle. [G-C2] 27. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. [G-C3] 28. (+) Construct a tangent line from a point outside a given circle to the circle. [G-C4] Find arc lengths and areas of sectors of circles. (Radian introduced only as unit of measure.) 29. Derive, using similarity, the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. [G-C5] Expressing Geometric Properties With Equations Translate between the geometric description and the equation for a conic section. 30. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. [G-GPE1] 10 Use coordinates to prove simple geometric theorems algebraically. (Include distance formula; relate to Pythagorean Theorem.) 31. Use coordinates to prove simple geometric theorems algebraically. [G-GPE4] Example: Prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, 3 ) lies on the circle centered at the origin and containing the point (0, 2). 32. Prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point). [G-GPE5] 33. Find the point on a directed line segment between two given points that partitions the segment in a given ratio. [G-GPE6] Label the unknown point as (x,y). Use the fact that all three points must be collinear and the slope between any 2 of the points is the same to write an equation relating x and y. Find the distance between the left endpoint and (x,y) and then the distance between the right endpoint and (x,y). Set the ratio of these two distances equal to the desired ratio. Solve for x and y to find the coordinates of the desired point. Consider the point P(xP , yP) that lies on the line segment AB joining points A(xA , yB) and B(xA , yB): Point P divides AB in the ratio k1 : k2 , and the coordinates of point P are given by these formulas: The blue formula is for three dimensional space, where you have 'z' coordinates as well as 'x' and 'y'. Source for above formula and information: http://www.teacherschoice.com.au/Maths_Library/Analytical%20Geometry/AnalGeom_3.htm 34. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.* [G-GPE7] 11 Use coordinates to prove simple geometric theorems algebraically. (ALABAMA) 35. Determine areas and perimeters of regular polygons, including inscribed or circumscribed polygons, given the coordinates of vertices or other characteristics. (ALABAMA) Geometric Measurement and Dimension Explain volume formulas and use them to solve problems. 36. Give an informal argument for the formulas for the circumference of a circle; area of a circle; and volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments. [GGMD1] Using a limit process, one can find the formula for the area of a circle by looking at the areas of regular polygons inscribed in the circle and determining what happens as the number of sides of the regular polygons continues to increase. Cavalieri’s Principle: 2-dimensional case: Suppose two regions in a plane are included between two parallel lines in that plane. If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3-dimensional case: Suppose two regions in three-space (solids) are included between two parallel planes. If every plane parallel to these two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. To see an explanation of how to use Cavalieri’s principle to find the volume of a sphere, look at this site: http://en.wikipedia.org/wiki/Cavalieri's_principle 37. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.* [G-GMD3] 38. Determine the relationship between surface areas of similar figures and volumes of similar figures. (ALABAMA) Visualize relationships between two-dimensional and three-dimensional objects. 39. Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify threedimensional objects generated by rotations of two-dimensional objects. [G-GMD4] Consider using computer software to generate 3-D solids of revolutions by rotating twodimensional objects, so that students can mentally visualize these cross sections more easily. 12 Modeling With Geometry Apply geometric concepts in modeling situations. 40. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).* [G-MG1] 41. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, British Thermal Units (BTUs) per cubic foot).* [G-MG2] 42. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost, working with typographic grid systems based on ratios).* [G-MG3] STATISTICS AND PROBABILITY Conditional Probability and the Rules of Probability Understand independence and conditional probability and use them to interpret data. (Link to data from simulations or experiments.) 43. Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B. [S-CP3] For two events (whether they are independent or not), P(A and B) = P(A)*P(B/A), where P(B/A) means probability of B given that A has already occurred. The formula can also be written: P(A and B)=P(B)*P(A/B), where P(A/B) means the probability of A given that B has already occurred. Solving for P(A/B) gives the formula, P(A/B)= P(A and B)/P(B). Likewise, P(B/A)=P(A and B)/P(A). If two events A and B are independent events, then P(A/B)=P(A) and likewise, P(B/A)=P(B). For instance, suppose you are drawing two cards from a standard 52-card deck. Suppose that you are replacing the first card before you draw the second card from the deck. In other words, you are drawing from a “full deck” each time. Then if you calculate the probability of drawing an ace on the first draw and then a king on the second draw, the second draw is independent of the first draw, since you are replacing the card after each draw. It really does not matter what you drew the first time. It does not affect the probability on the second draw. Therefore, the probability of drawing and ace and then a king = P(A and B) = P(A)*P(B)= 4/52 * 4/52. However, if you used the general formula P(A and B) = P(A) * P(B/A), then you would have 4/52* probability of drawing a king given that you drew an ace on the first draw. However, it does not matter what you drew first; thus, you would still have 4/52*4/52. Now let’s consider that you are going to draw two cards and not put the first one back before drawing the second card. If we once again consider the probability of drawing an ace on the first draw and then a king on the second draw, these are not independent events. Thus, we must use the formula P(A and B) = P(A) * P(B/A)= 4/52 * 4/51, since the probability of drawing a king on the second draw GIVEN that you drew an ace on the first draw means that there are still 4 kings in the deck but only 51 cards. Use a randomized response activity with cards to illustrate how one can use conditional probability and a deck of cards to determine if someone is guilty of a wrongdoing. (Write-up of this activity follows). 13 44. Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. [S-CP4] Example: Collect data from a random sample of students in your school on their favorite subject among mathematics, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results. Observe the following example of a two-way frequency table. GRADE 10th 11th 12th TOTAL a. Math 40 15 6 61 FAVORITE SUBJECT Science 9 16 10 35 English 11 19 25 55 TOTAL 60 50 41 151 What is the probability of selecting at random a student who is a 10th grader and says his favorite subject is math? ANSWER: 40/151 14 b. What is the probability of selecting at random a student who says math is his favorite subject given that the student is a 10th grader? (The condition that the student is a 10th grader narrows our choices down to 60 students.) Thus the probability is 40/60= 2/3. c. Assuming the student chose math as his favorite subject, what is the probability that he is a 10th grader? The assumption that the student favors math narrows our choices down to 61 students. Thus the probability is 40/61. d. What is the probability that the student will favor science given that the student is in the 10th grade. The answer is 9/60 = 3/20. ARE TWO EVENTS INDEPENDENT? In the example above are grade level of student and choice of favorite subject independent events? In other words, is there a relationship between grade level in high school and favorite subjects? If there is a relationship, they are not independent events. If there is no significant relationship, the two events are independent. Use software to determine whether or not they are independent. TI-83/84 graphing calculators work well. Conduct a Chi-Square Test of Independence. To do this, enter the numbers, excluding totals, in each cell of a matrix A. In the example above, matrix A will be a 3X3 matrix. 40 9 11 A = 15 16 19 6 10 25 Enter matrix A into the calculator by pressing 2nd and Matrix and then highlighting A and scrolling over to EDIT and push ENTER. Since this is a 3X3 matrix, type in 3X3 on the top row and push enter. You are now ready to enter the elements in each cell of the matrix. Once finished, press 2nd and QUIT. Press the STAT key on the TI-83/84 calculator and then arrow over to TESTS. Scroll down and choose “Х2-Test” and press ENTER. Then next to Observed, type in A . Scroll down to CALCULATE and press ENTER. You will see the test statistics. Look at the p-value. The smaller the p-value, the more the test leans toward supporting the conclusion that there is a relationship between the two variables. Usually you want p to be less than 0.05 or in some cases, 0.01. In this example, p = 7.5575976E-7 or 0.000000756. Thus, there appears to be a relationship between grade in high school and choice of favorite subjects. In other words, they are NOT independent events. 45. Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. [S-CP5] Example: Compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer. Refer to information under #43 and #44. Example: 15 The probability of having a certain disease is 0.05. The probability of testing positive if you have the disease is 0.98. The probability of testing positive if you do not have the disease is 0.10. What is the probability that you have the disease if you test positive for it? A tree diagram is helpful, but it can be solved with a formula P(have the disease/ test positive) which is read “the probability that you have the disease given that you test positive” = P(have the disease and test positive) DIVIDED BY P(test positive)= (0.05)(.98 ) = 0.34 (0.05)(.98 ) (0.95) (.10) Use the rules of probability to compute probabilities of compound events in a uniform probability model. 46. Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model. [S-CP6] Refer to information under #43 and #44. Example: Suppose you roll a number cube with sides labeled 1, 2, 3, 4, 5, or 6. What is the probability that you roll an even number, given that the number rolled is greater than 3? Solution: Since you know that the number rolled is greater than three, it is given that it is either a 4, 5, or 6. Thus the probability of even is 2/3. The formula is: P(even/greater than 3) “read probability of rolling an even number, given that the number rolled is greater than 3” = P(even and greater than three) / P(greater than 3) = 2/6 = 2/3. 3/ 6 47. Apply the Addition Rule, P(A or B) = P(A) + P(B) - P(A and B), and interpret the answer in terms of the model. [S-CP7] The general rule of addition is P(A or B) = P(A) + P(B) - P(A and B), regardless of whether the two events are mutually exclusive or not. If two events are mutually exclusive (both events cannot occur at the same time), then the formula could be P(A or B) = P(A) + P(B). For instance, suppose you are going to draw a card from a standard 52-card deck. Consider the probability that the card is either a jack or a 5. These are mutually exclusive, since a card cannot be both a jack and a 5. Thus, the formula, P(A or B) = P(A) + P(B), could be used. Thus P(jack or 5)= P(jack) + P( a five)= 4/52 + 4/52 = 8/52. If we use the general formula, P(A or B) = P(A) + P(B) P(A and B), we would get the same answer, since P(A and B) = 0, since the events cannot both occur at the same time. For two mutually exclusive events, P(A and B) will always be 0. Thus P(jack or five) = P(jack) + P(5) - P(jack and 5)= 4/52 +4/52- 0 = 8/52. (Same answer as before). Now, let’s consider the probability of drawing one card from a standard 52 card deck and getting either a jack or a red card. These are not mutually exclusive events, since a card can be BOTH a jack and a red card—jack of hearts or jack of diamonds. Thus, we must use the formula, 16 P(A or B) = P(A) + P(B) - P(A and B)= P(jack or red)= P(jack) + P(red) – P(both jack and red)= 4/52 + 26/52 – 2/52 = 28/52. If you think about this, there are only 28 cards in the deck that are either a jack or a red card: Jack of spades, Jack of clubs, and 26 red cards (which will include the two red jacks. Thus, the general addition rule if P(A or B) = P(A) + P(B) - P(A and B). 48. (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model. [S-CP8] Refer to information under #43. 49. (+) Use permutations and combinations to compute probabilities of compound events and solve problems. [SCP9] A permutation of a set of objects is an arrangement of some or all of these objects in a specific order. The order of the objects is important. For example, FER is a different permutation than RFE, etc. A combination of a set of objects is a selection of some (or all) of these objects with no regard to order. Order is NOT important. The combinations FER and RFE are the same combination. Formulas: nPr = n! (n r )! and nCr= n! r!(n r )! Example: Find the probability that a committee of 10 students chosen from an organization consisting of 40 male and 35 female students will include 3 males and 7 females. First of all, how many committees of size 10 from a group of 75 can be chosen? ORDER IS NOT IMPORTANT, so use Combinations. Thus 75C10 = 8.289311064E11 or 828,931,106,400 This gives us the total number of possible committees. You may use the TI-83/84 calculators to find permutations and combinations. In the case of trying to find 75C10, you first type in 75 and then press the MATH key, scroll over to PRB and then down to nCr, press ENTER and then type in 10 and press ENTER for the answer. You will find permuations the same way, except you will choose nPr rather than nCr. We wish to compute the probability for having 3 males and 7 females on the committee. Again order is not important, so we use combinations. 40C3 * 35C7 = 9880*6,724,520= 66,430,257,600 We now find the probability by dividing the number of ways to have the desired committees divided by the total number of possible committees= (66,430,257,600)/(828,931,106,400) = 0.08 Example: In how many ways can a code sequence be formed if it must consist of 3 letters followed by 3 digits and none of the digits nor letters may repeat? Since we cannot repeat and order does make a difference, we are choosing 3 different letters from 26 and 3 different digits from 10, AND order is important. Thus, we use 26P3 for the letters and then 10P3 for the digits. The answer is 26P3*10P3 = 11,232,000. 17 PLEASE NOTE: If the letters and digits could have been repeated, we would not necessarily be choosing 3 distinct letters from 26, etc. Thus, we could not use permutations, since permutations and combinations require that the subset chosen has no repetitions. We would then have used the counting principle and the answer would have been 26*26*26*10*10*10 = 17,576,000 Using Probability to Make Decisions Use probability to evaluate outcomes of decisions. (Introductory; apply counting rules.) 50. (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). [S-MD6] 51. (+) Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). [S-MD7] 18