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Sort and Classify 1 2 3 4 5 6 Write down as many ideas as you recall about objects that are congruent. Include thoughts on how you might prove that two objects are congruent. Write down as many ideas as you recall about objects that are similar. Include thoughts on how you might prove that two objects are similar. Use a compass to construct a segment congruent to the original. Use a compass to create a segment similar to the original. Write a complete sentence to describe how the compass can be used to accomplish these tasks. 7 Practice Set 1. Describe how your class sorted the shapes from the previous pages. 2. Use complete sentences to define congruence. 3. Write down the ideas about similar objects that surfaced in class discussion today. 4. Use complete sentences to describe how a compass can be used to create two congruent segments. Review 5.Use properties of equality to solve for x in each equation. x a. 5x 17 21 b. 8 3 6 c. 4x 12 9 x 11 6. Explain how someone can tell the difference between linear and exponential functions. If you can’t remember, look it up and write down your notes. 7. To what mathematics can you connect today’s task? Today’s task reminded me of … (topic and explanation). 8 Double the Logo Your assignment is to double the logo drawn below. Instructions: Take two rubber bands of equal length and tie them together so that there are equally-sized bands on either side of the knot. Pick and mark an anchor point somewhere on the paper. Pin the end of rubber band to the anchor point with your finger. On the opposite side of the other band, place a pen. Trace a new object while keeping the knot consistently on top of the figure you are trying to enlarge. Your task is to: 1. Follow directions as accurately as you can, but don’t worry if your pictures are wiggly. 2. Explain what in this procedure caused the shape to be twice the size of the original one. What happens if you choose a different anchor point? 9 3. How can you perform the same process if instead of rubber bands you used a ruler? Try it out. 4. List as many different things you notice that (a) stayed the same (b) changed in this process. 10 Additional Space for Double the Logo Questions and Notes 11 Before and After Before: Use wikisticks, a ruler, and protractor to measure segments, arcs, and angles of circle O. Notation: Record all measurements in the table on the next page. Create a point of dilation on the paper. After: Use rubber bands and/or a ruler to create dilated images with the following scale factors: 2, 5/2, and 4. Record all measurements in the table. 12 AB OE ED ACB Perimeter nD EOB Calculate Perimeter AB Scale Factor 2 5/2 4 List as many different things you notice that (a) stayed the same (b) changed in this process. 13 Practice Set 1. Explain the class definition of dilation. 2. Explain what happens to a segment upon dilation. 3. What is the circumference of a circle? 4. Define the number pi. Do not list digits. Review 5. Determine whether each set of data is linear, exponential, or neither. Explain how you know. n 0 1 2 3 f(n) 5 10 20 40 n 0 1 2 3 f(n) 1 4 9 16 6. Use properties of equality to solve for x. 9 x 18 9x 18 27w n 0 1 2 3 f(n) 21 19 17 15 tx 18 27w 7. To what mathematics can you connect today’s task? Today’s task reminded me of … (topic and explanation). 14 Find the center 1. Rectangle A′B′C′D′ is the image of rectangle ABCD under a certain dilation. Find the center of dilation P and the scale factor. Explain how you found it and why your method works. 2. The small rectangle A′B′C′D′ is the image of rectangle ABCD under a certain dilation. Find the center of dilation P. What can you say about the scale factor? 15 3. Triangle A′B′C′ has the same shape as triangle ABC, but can't possibly be the image of triangle ABC under any dilation. Explain why not. 4. Determine a sequence of transformations that will take the triangle ABC to triangle A′B′C′. 16 Notes for Finding the Center 17 As a group, answer the following three questions: What is a scale factor? How does the scale factor 2/5 compare with the scale factor 5/2? How does the scale factor 2/5 compare with the scale factor 4/10? What is a Rational Number? What is not a Rational Number? Notes on natural numbers, whole numbers, integers, real numbers. 18 Explain how to simplify each of the following operations on rational numbers. Properties of Equality do not apply here. Is the set of rational numbers closed over addition? Subtraction? Is the set of rational numbers closed over multiplication? Division? 19 Practice Set Write two sentences to compare your previous ideas of similarity with the ideas that we surfaced today. How do these ideas compare with the class definition of dilation? Draw a triangle A ' R ' T ' under the dilation with the given center and the scale factor: a. Center O, scale factor ½ b. Center T, scale factor ½ c. Center O, scale factor 2 Use the figures below to answer these questions: a. What is the scale factor that can be applied to figure B to get figure A? b. What is the scale factor that can be applied to figure A to get figure B? 20 21 22 Dilations on a Coordinate System A pool design originally is given as a quadrilateral with vertices A(1,1), B(3,5), C(6,8), D(0,4), The owner wants it 3 times larger. Create the diagram of the original design and the dilated design. Use coordinates to determine measurements and verify the scale. Notes 23 Create a design with at least five key points. List the coordinates of these points. Draw and describe the image after each operation as it compares with the original figure. 1. f ( x, y ) ( x 7, y 9) 2. f ( x, y ) ( y, x) 3. f ( x, y ) ( y, x) 4. f ( x, y ) (4 x, 4 y ) 5. f ( x, y ) ( x,3 y ) 24 Notes: Which of the previous transformations were congruent transformations? Which of the previous transformations were dilation transformations? Which of the previous transformations were neither congruent nor dilation transformations? Write two sentences to describe how you will know what type of transformation is in use? What is similarity? 25 From Here to There. List the coordinates of points A, B, C, D, and E. Cut out the shape below the grid and paste anywhere on the coordinate system. List the coordinates of points A’, B’, C’, D’, and E’. Trade with another student. Try to determine a sequence of transformations that map ABCDE to its image. Use proper notation. B 4 E C 2 A D -10 -5 5 10 15 -2 -4 -6 Cut me out: 26 Given a triangle ABC with vertices A(-1,2), B(0,3), and C(5,-2), create a matrix of coordinates for xB xC x transformation. Use the format A and call this matrix A. y y y B C A Perform each of the following operations. Use complete sentences to describe the transformation caused by the operation. 1 0 1. * A 0 1 3 0 2. * A 0 3 5 5 5 3. A 3 3 3 4. Create a matrix operation(s) that would reflect your triangle about the x-axis and dilate from the origin by a scale factor of 7. 27 Practice Set Name ________________________ Given and the scale factor, k, determine the coordinates of the dilated point, the center of dilation is the origin. . You may assume 1. 2. 3. Given and the dilated point , find the scale factor. (What did you multiply the coordinates of A by to get A’?). You may assume the center of dilation is the origin. 4. 5. 6. For the given shapes, draw the dilation, given the scale factor and center. 7. , center is 8. , center is 9. , center is http://middlemathccss.wordpress.com 28 10. Explain the effects of the matrix operation on a matrix A of coordinates describing a polygon. 0 1 1 0 * A 11. Perform the following operations. 3 2 7 5 3 2 7 5 3 2 7 5 3 2 7 5 12. Create and label a point of dilation, D. Dilate the image with a scale factor of 1.5. 29 30 As a group, recall… What are rational numbers? What are not rational numbers? Simplify each of the following and determine whether the number is rational or irrational. 8 81 18 1 Are there other sources/examples of irrational numbers besides the any found above? Extra practice 31 32 Notes from Going Round in Circles 33 Practice Set Write an equation of a circle for each center and radius or diameter measure given. 1. (0, 0) r = 2 2. (1, 1) r = 5 3. (-3, 2) d = 4 4. (-6, -1), d = 10 Find the coordinates of the center and the measure of the radius for each circle whose equation is given. 5. ( x 5)2 ( y 2)2 49 6. ( x 3)2 ( y 7)2 100 7. x 2 ( y 4)2 7 Simplify each expression. 2 9. 7 5 8. ( x 1)2 ( y 1)2 10 10. 4 1 7 9 11. 4 1 7 9 12. Choose a center of dilation and dilate the given figure using a scale factor of 1/3. 34 13. Label each of the following as linear, exponential, or neither. Explain how you know. a. f (n) 14 3n b. f (n) 90n2 1 c. f (n) 16 3n d. f (n) 11 .4(n 1) 2 e. f (n) 1 ( ) n 3 f. 4 x 8 y 17 14. Determine the median and the IQR for the following data set: 17, 15, 14, 15, 18, 11, 12 15. Determine the y-intercept for the following functions. a. y 15 3x b. y 16n 3 c. y 14 2 x 35 36 37 Cut 38 39 Practice Set Reduce each radical. Do write a decimal approximation. State whether each number is rational or irrational. a. 50 b. 88 c. 484 40 14. Use properties of equality to isolate the variable x. x 15 ax b c t x 8 h w 15. 41 Cross Sections Use clay to create several of each of the following shapes: cylinder, sphere, and cone. Use the dental floss to cut a planar cross-section of each figure. For each slice, describe in words how you cut the figure (be specific!) and sketch the planar cross-section. (The sketch should be only in two dimensions.) *Try several different angles to slice each shape. Cylinders Spheres Cones 42 History and Information: Conic Sections 43 Rabbit Farm Task Mary and Sam are going to start raising bunny rabbits to sell at Easter. They bought 24 feet of fence to create a rectangular enclosure. A. Draw the different enclosures Sam and Mary could make. Make sure to label the dimensions of each rectangle. B. Create multiple models to represent and organize the information from your rectangles. C. Sam and Mary want to sell a lot of rabbits at Easter. Which one of the enclosures would allow them to raise the most rabbits? Explain. 44 Planting a Garden Part I 1. John has an existing square garden in his backyard. He wants to increase the width by 4 feet and the length by 6 feet. A. What is the increase to the area of John’s garden? B. Write a rule to describe the area of John’s garden. x x 2. Bob has a garden with a fence around it. He wants to maximize the garden’s area to get the most out of the garden. He can’t afford to buy more fence. A. What can Bob change to maximize the garden area? B. Show and justify how your change created the largest possible garden. x+7 x+3 45 Planting a Garden Part II Describe the similarities and differences between Bob’s original garden and his new garden. We developed 3 rules to describe Bob’s garden: (x + 5 )2 (x + 5 ) (x + 5 ) x2 + 10x + 25 A. Show how these rules are the same and describe their differences. B. Relate the similarities and differences of the rules back to the area model of Bob’s garden. 46 Practice – More Gardens 1. James has a garden, shown below. What is the area of each part of the garden (ie: corn, peas,…)? x 3 x Potatoes Corn 4 Flowers Peas 2. Susan knows the area of three sections of her garden, what is the area of the missing piece? What are the dimensions of the entire garden? Area is measured in square feet. Pumpkins Squash x2 6x Green Beans Onions 36 3. Write an expression that represents James’ (#1) and Susan’s (#2) gardens in more than one form. 47 4. If the following rule x 2 8 x 15 is used to describe a garden, draw a model of the garden. Make sure to label the dimensions and areas of each piece. Write the expression for this garden in one other form. For each, Rewrite the expression in different forms. ( x 7)2 ( x 14) 2 ( x 4)2 3 ( x )2 2 In each circle equation, use multiplication to write the expressions in different forms. ( x 1)2 ( y 4)2 9 ( x 8)2 ( y 3)2 25 ( x 13)2 ( y 7)2 121 ( x 4)2 ( y 6)2 8 48 How would we go from general conic equation to standard form? Ex. Add what you need to create a perfect square trinomial to the following expressions. Then write in a different form. x2 + 24x + x2 + x2 -30x + x2 – 40x + x2 2 x 5 Show x 2 14 x 49 as the area of a square. + 49 1 x2 x 3 Show x 2 10 x 25 as the area of a square. 49 Build perfect squares and use properties of equality to transform the general conic equation of a circle into standard form. Then graph the circle. 1. x 2 8 x y 2 2 y 64 2. x 2 y 2 14 x 12 y 4 0 3. x 2 2 x y 2 55 10 y 50 Practice Set 1. Show x 2 24 x 144 as the area of a square. 2. Show x 2 30 x 225 as the area of a square. 3. Find the value of c that makes x 2 16 x c a perfect square trinomial. Then write the expression as a perfect square. 4. Find the value of c that makes x 2 11x c a perfect square trinomial. Then write the expression as a perfect square. 5. Build perfect squares and use properties of equality to transform the general conic equation of a circle into standard form. Then graph the circle. x 2 24 x y 2 6 y 137 y 2 24 y 2 x x 2 120 6. Simplify the following expressions. 7 4 3 15 2 9 16 2 8 1 3 1 4 4 2 51 7. Use complete sentences to describe what happens to the segments in figure during dilation. 8. What are similar figures? How do similar figures compare with congruent figures? 9. Determine the rate of change or slope in each of the following linear functions. f (n) 15 3n f (n) 1 6.5n f (n) 31n f (n) 14 f ( n) 7 n 3 10. Determine the median and IQR for the following data set. Create a box plot to represent this information. 31, 14, 2, 18, 15, 27, 19 11. Write several sentences to describe the different conic sections. 52 We are Similar... Can You Tell? Choose three integers between -5 and 5: a = b = c= Define a coordinate system on the grid below. Graph the circle with center (a, b) and radius c . Write your neighbor's numbers here: a = coordinate system. b= c= and graph their circle on the same Record your circle's equation in standard form: Record your neighbor's circle's equation: *Show that the two circles are similar. Are all circles similar? Explain your response. 53 Practice Set Graph the following circles. 1. ( x 1)2 ( y 3)2 25 2. ( x 4)2 ( y 4)2 121 3. ( x 3)2 y 2 25 4. x 2 ( y 2)2 81 Build perfect squares to create an equation of a circle in standard form. 5. x 2 12 x y 2 18 y 5 6. x 2 24 3 10 y y 2 7. 7 x 2 28x 7 y 2 14 y 35 0 8. Write several sentences to describe why all circles are similar. 9. Write several sentences to describe the conic sections. 54 10. Complete the following operations. 9 3 2 10 9 3 2 10 9 3 2 10 9 3 2 10 11. Create and label a point of dilation, D. Dilate the image with a scale factor of 1.5. 55 56 Investigate and Report Your group letter _____ I have an appropriate packet ______ Groups: A. Relationships between measures of arcs and central angles, inscribed angles. B. Relationships between measures of segments (chords, chord parts, secant segments, tangent segments) C. Relationships between measures of angles formed by secants, tangents, radii. Question to consider throughout the investigation: Do these relationships hold under dilations? Notes from groups A. Relationships between measures of arcs and central angles, inscribed angles. 57 Practice Set from Group A 1. What is a central angle? Use a diagram to clarify your explanation. 2. Describe the relationship between a central angle and the arc it intercepts. 3. Find the measure of the arc or central angle indicated. Assume that lines which appear to be diameters are actual diameters. 4. What is an inscribed angle? Use a diagram to clarify your explanation. 5. Describe the relationship between an inscribed angle and the arc it intercepts 58 6. Find the measure of the arc or angle indicated with a question mark. 7. Solve for x. 8. Draw a right triangle and circle such that the right angle is an inscribed angle and the hypotenuse is a diameter. 59 Notes from groups B. Relationships between measures of segments (chords, chord parts, secant segments, tangent segments) 60 Practice Set from Group B Refer to your notes on the previous page! 61 62 Notes from groups C. Relationships between measures of angles formed by secants, tangents, radii 63 Practice Set from Group C 64 65 15. Determine the missing length of each right triangle. 24 x x 7 8 25 10 3 x Label each missing length as rational or irrational. Explain your response. 66 67 Notes from Inscribing and Circumscribing Right Triangles 68 69 Practice Set 1. Describe key information that helped your class find the missing radii in the inscribed circles. 2. Determine the missing side in each triangle. a. leg length = 6 leg length = 8 hypotenuse length = ? b. leg length = 7 leg length = ? hypotenuse = 7 2 3. Determine the value of x in each figure. 55 10 x 2x-17 8 9 4. Use properties of equality to determine the value of the variable. g 4( x 1) 2 x 7 7 g 11 3 5. Write a function that describes each situation. a. You started the day with a 450 calorie breakfast. Each minute of your hike burned 2 calories. b. The trout population in your neighborhood pond was recorded at 36, increasing at a rate of 2% per year. 70 The Arc de Triomphe roundabouts in Paris, France… on a coordinate system! Write the equation of the highlighted traffic circle. Does this traffic circle have any y-intercepts? Does this traffic circle have any x-intercepts? If the circle has any intercepts, determine the locations. Write the equation of a second traffic circle in the image. If this second traffic circle has any x or y intercepts, determine the locations. Do these traffic circles intersect? Explain your answer. Describe how the use of a coordinate system could help in descriptions and calculations of this traffic scene. 71 Consider the circle ( x 4)2 y 2 9 . Graph. Does this circle have any y-intercepts? Does this circle have any x-intercepts? If the circle has any intercepts, use more than one method to find them. Consider the circle ( x 4)2 ( y 3)2 16 . Graph. Does this circle have any y-intercepts? Does this circle have any x-intercepts? If the circle has any intercepts, use more than one method to find them. 72 Consider the circle ( x 5)2 ( y 5)2 4 . Graph. Does this circle have any y-intercepts? Does this circle have any x-intercepts? If the circle has any intercepts, use more than one method to find them. Notes from finding intercepts of circles 73 74 Practice Set 1. Consider the circle ( x 4)2 ( y 2)2 9 . Graph. Does this circle have any y-intercepts? Does this circle have any x-intercepts? If the circle has any intercepts, use more than one method to find them. 2. Consider the circle ( x 5)2 ( y 1)2 25 . Graph. Does this circle have any y-intercepts? Does this circle have any x-intercepts? If the circle has any intercepts, use more than one method to find them. 3. Consider the circle ( x 1)2 ( y 3)2 16 . Graph. Does this circle have any y-intercepts? Does this circle have any x-intercepts? If the circle has any intercepts, use more than one method to find them. 75 4. Consider the circle ( x 6)2 ( y 7)2 121 . Graph. Does this circle have any y-intercepts? Does this circle have any x-intercepts? If the circle has any intercepts, use more than one method to find them. 5. Determine XY. 6. The mB 56 and mAC 82 . Determine the measure of AD . 7. Simplify the following expressions. 8 1 8 1 9 4 9 4 8 1 9 4 8. Write a situation or story that could be described with the following function. f (n) 30 10.50n 76 Revisiting intersections of a plane and a cone Use the conic definition and the distance formula to determine the equation of the parabola. 77 How would this direction change the equation? Examples 78 Given a parabola with vertex at (0,0) and focus at (0,2), sketch the parabola and write the equation. Given a parabola with vertex at (0,0) and focus at (0,-3), sketch the parabola and write the equation. Given a parabola with vertex at (0,0) and directrix at y = -5, sketch the parabola and write the equation. Given a parabola with vertex at (0,0) and directrix at x = 8, sketch the parabola and write the equation. 79 Practice Set Write the equation of the parabola given the following information. Sketch each. 1. Vertex (0,0) and focus (0,4). 2. Vertex (0,0) and focus (2,0). 3. Vertex (0,0) and focus (0,-6). 4. Vertex (0,0) and focus (-7,0). 5. Vertex (0,0) and directrix x = -5. 6. Vertex (0,0) and directrix y = 2. 7. Write the definition of a parabola. 8. What are the reflective properties of a parabola? 9. Given two plans, which would you choose in order to gain the most money in the first ten years? a. Beginning with $400, the account increases $100 per year. b. Beginning with $400, the account increases at a rate of 5% per year. 80 Warm-Up Together Tell whether the parabola opens up, down, left, or right. How can you tell? x 2 8 y x 2 12 y y 2 16 x y 2 24 x Graph the equation. Identify the focus and the directrix. y 2 16 x x 2 12 y 4x y2 0 x 2 y2 x2 9 y 4 y 2 32 x 0 81 Write the equation of the parabola with the given focus, vertex at (0,0). Focus (0, -1) Focus (5,0) A reflector for a satellite dish is parabolic in cross section, with the receiver at the focus. The reflector is 1 foot deep and 20 feet wide from rim to rim. How far is the receiver from the vertex of the parabolic reflector? Write down some ideas on how the equation of our parabola would change if we did not select the origin (0,0) as the location of the vertex? 82 Name the center of the circle. x 2 y 2 25 ( x 6)2 ( y 1)2 121 In the previous two equations, where did you see the center of the circle? How might this idea be applied to the equations of parabolas? Sketch a parabola with vertex at (4,5) and focus at (4,8). Write the equation of the parabola. Sketch a parabola with vertex at (4,5) and focus at (0,5). Write the equation of the parabola. Determine the vertex, focus, and directrix of the parabola. Draw a sketch to assist you. ( x 9)2 12( y 2) 83 Practice Set 1. How does the equation of a parabola change if the vertex moves from (0,0) to another location? 2. Write the equation of a parabola with the vertex at the origin and focus at (0, 4.5). 3. Write the equation of a parabola with the vertex at the origin and the directrix at y = 5.25. 4. Write the equation of a parabola with the vertex at (-5,8) and focus at (-15, 8). 5. Write the equation of a parabola with the vertex at (-8,-9) and directrix at x = 0. 6. Find the x and y intercepts of the circle given by the equation ( x 5)2 ( y 11)2 49 . 7. Determine the missing number to build a perfect square trinomial. x 2 14 x ? Then write as a perfect square. 8. Build perfect squares to write the equation of the circle in standard form. x2 4x y 2 6 y 9 0 84 9. Determine a point of dilation and dilate the figure with a scale factor of 7/2. 10. Use properties of equality to solve for c. 11. Simplify the following expressions. 3 9 3 9 5 10 5 10 9(c 10) 2c 14 3 9 5 10 85 Describe the following equations. Rewrite the expressions in different forms. ( x 4)2 3( y 2) ( y 1)2 12( x 11) x 2 4( y 9) Determine the vertex, focus, and directrix and sketch the graph of the parabola. x 2 4 x 8 y 28 y2 4x 8 y How does the equation of a parabola compare to the equation of a circle? 86 We are Similar... Can You Tell? Choose three numbers between -5 and 5: a = b= c= Graph a parabola with vertex (a, b) and a coefficient c (hint: c = 4p in the equation). Do the same with your neighbor’s choices for a,b,and c. Record your parabola's equation: Record your neighbor's parabola's equation: Show that the two parabolas are similar. 87 Notes from Are We Similar? 88 Practice Set 1. Explain how you know that all parabolas are similar. 2. Compare the equations of parabolas and circles. 3. Find the vertex, focus and directrix of the parabola. Sketch. x 2 24 y 12 ( x 4) 2 12( y 2) x 2 8 x 2 y 10 x 2 10 x 32 y 1 0 4. Write the equation for the parabola with vertex at (1, 0) and directrix at x = -5. 5. Write the equation for the parabola with focus at (-4,0) and directrix at x = 2. 6. Describe how to slice a cone to get a parabolic cross-section. 89 7. Sketch the circle. Use algebra to find the x and y intercepts. ( x 4)2 ( y 3)2 25 8. Describe how to slice a cone to get a circular cross-section. 9. Describe the effects of dilation on a line segment. 10. Compare linear and exponential functions. 11. Explain which properties are used in the following example of finding the value of x. 8( x 3) 1 2 x 17 8 x 24 1 2 x 17 8 x 2 x 40 6 x 40 20 x 3 90 What is a function? Which of the following are functions? Make a scatterplot of each. What is the vertical line test? Write function equations for as many of the above data sets as you can. 91 Are all parabolas graphs of functions? Consider the area from this toothpick pattern. Use multiple representations to describe the area as it changes with each step. 92 Consider the total number of dots in each step of the pattern. Use multiple representations to describe the total number of dots as it changes with each step. In general, 93 Tally the total number of segments that can be drawn among n points, no three of which are collinear. Begin with two points. 94 95 Practice Set Find the first and second differences for each set of data. Use these differences to model each of the following data sets with a quadratic formula. 1. X 0 1 2 3 4 2. 3. X 0 1 2 3 4 Y -4 3 14 29 48 5. X 0 1 2 3 4 Y .25 .75 3.25 7.75 14.25 7. X 0 1 2 3 4 X 0 1 2 3 4 Y 3 -4 -15 -30 -49 4. Y 1 5.5 12 20.5 31 X 0 1 2 3 4 Y -7 -2 9 26 49 Y 5 3 3 5 9 X 1 2 3 4 5 Y 9 32 69 120 185 X 1 2 3 4 5 Y -4 -6 -6 -4 0 6. 8. 96 97 98 Notes from Tile Table 99 Consider the total number of toothpicks in the figures below. Use multiple formats to describe the pattern. Two students looked at the total number of toothpicks as well. Their thoughts are diagramed below. In what ways do the following expressions represent the figures below them? 100 Practice Set Instructions: analyze each table. Classify the pattern as linear or quadratic. Find the rate of change, and write a function for each. 1. 1 -3 2 -9 3 -15 4 -21 5 -27 6 -33 7 -39 3 30 4 42 5 56 6 72 7 90 8 110 0 -3 1 -7 2 -7 3 -3 4 5 5 17 6 33 2. 3. 101 -2 -3 -1 .5 0 4 1 7.5 2 11 3 14.5 4 18 -3 -28 -2 -11 -1 0 0 5 1 4 2 -3 3 -16 -2 17 -1 11 0 5 1 -1 2 -7 3 -13 4 -9 5. 6. 102 7. In #3, the values in the output column start out decreasing, and then they change to increasing (you may want to make a graph to verify this). Could that ever happen with a table of values for a linear pattern? Justify your answer. 8. Find the vertex, focus, and directrix of the parabola. Sketch. ( y 3)2 56 x 20 2 x 2 2 x 24 y 7 0 9. Are all parabolas graphs of functions? Explain your response. 10. Are all parabolas similar? Explain your response. 11. What transformations result in congruent figures? 12. What transformations result in similar figures? 103 In the game “Angry Birds”, little birds are launched at targets. The path of the bird flying through the air is a parabola. Use the width of a bird as your scale. Find the equation of the parabola. 104 All Angry Birds that are launched move at the same acceleration. The way to select a different target is by changing the angle of the launch. Choose your next target, select 3 points that will define the parabola that leads to the target, then find the equation of the parabola. 105 You made it to Lake Powell. It is probably hot and you are waiting around for your parents to take care of everything, so put on your swimsuit and go jump off the marina (make sure the water is deep enough or just pretend to jump). The path of your dive can be modeled by the equation y = -16t2 + 8t Fill in the table: Now explain the meaning of the “t” and “y” values. Not sure? Perhaps a graph would help. Create a graph using the points. When do you hit the water? How do you know that? What is the highest you jump? At what time are you at the highest point of your jump? At what time interval are you going up? At what time interval are you going down? 106 After you have had a chance to anchor the houseboat in deeper water, do the following problems. Assuming the deck of the houseboat is 10 feet above the surface of the water and you jump from the deck. Your dive can be modeled by the following equation. y = -16t2 + 6t + 10 Create a table using appropriate values for t. When did you hit the water? According to your table, what is the highest point of your jump and at what time were you at the highest point? Is this time half-way between the beginning time and the time you hit the water? Why or why not? Find the height at t = 3.75 How can you use this information to find the actual time that you reached the highest point of your dive? 107 Cliff diving at Lake Powell is prohibited; however, if you dived off a cliff that is 30 feet tall, the dive might be modeled by the equation y = -16t2 +4t + 30. Once again, create a table using appropriate values for t. What does t = 0 represent? What does the height at t = 0 represent? What does “t” refer to at the time ”y” = 0? Find the height at t = .125 and use this information to find the maximum height and the time at which it will occur. Now suppose the crazy person in your group decides to jump off the 50 foot cliff, write an equation that might model his/her dive. Construct a graph of a parabola that opens down. Now construct several horizontal lines intersecting the graph at several locations. How can the intersection points of each horizontal line with the parabola be used to find the location of the maximum value of the parabola? 108 With your group, brainstorm and take notes on the benefits of each of the following quadratic formats. ( x 7)2 4( y 8) f (n) 2n2 12n 2 5x 2 10 x 8 y 4 0 ( x 5)2 2( y 3) f ( x) 3x 2 4 x 5 3x 2 8 x 5 y 6 0 109 Practice Set Write each of the following quadratic equations in a form that is easy to see the coordinates of the vertex. Sketch the circle. Use algebra to find the x and y intercepts. 7. ( x 10)2 ( y 1)2 36 8. ( x 7)2 ( y 4)2 81 9. Compare the equations of circles and parabolas. 10. Compare the rates of change of linear, exponential, and quadratic functions. 110 Although the playing surface of a football or soccer field appears to be flat, its surface is actually shaped like a parabola so that rain runs off to either side. The cross section of a field with synthetic turf can be modeled by where x and y are measured in feet. From this equation, find the width of the field. What is the maximum height of the field’s surface? Considering the maximum height in the middle of the field, how much higher is the field at the vertex than at the sidelines (lowest point). Calculate the slope from the vertex to the sideline (slope of the secant line). Do you think this slope would be detectable by the players on a field? 111 A duck dives under water and its path is described by the quadratic function y = 2x2 -4x, where y represents the depth of the duck in meters and x represents the time in seconds. An object is launched at 19.6 meters per second (m/s) from a 58.8-meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = –4.9t2 + 19.6t + 58.8, where s is in meters. When does the object strike the ground? 112 Consider the functions below. Rewrite the equations so that you can more easily find the roots. Find the roots. a. f ( x) x 2 40 x 113 b. f ( x) x 2 6 x 1 c. y 100 x 2 1500 x 113 114 Practice Set Find the intercepts for the appropriate variable. In each equation, the “y” has already been set to 0. 115 9. Find the vertex, focus, and directrix of the parabola. Sketch. y 2 36 x 6 y 10. Find the differences and write a function that describes the data. X 0 1 2 3 4 Y 9 32 69 120 185 11. Determine the missing term to build a perfect square trinomial. Then write each as perfect square. x 2 36 x ? x 2 ? 144 12. Compare the equations of circles and parabolas. 13. Describe the result of a dilation on a rectangle. 14. Simplify the following expressions. 1 3 11 22 1 3 11 22 1 3 11 22 116 Quickly sketch the graphs of the following functions. 1. Graph y = x2 – 4 2. Graph y = x2 + 4 Where are the roots on each graph? When you look in the mirror what do you see? a) Is it real? b) Is it imaginary? c) What is it? d) What does the image see when it looks in the mirror? e) Is it real? Imaginary? f) What applications does the image have in the real world? The roots are not real, but there are images that have applications in the real world, Describe a way to reflect the parabola without real roots in such a way that the image has roots. Circle the roots on the graph of the parabola or the roots of the image as needed. 117 Refer to functions 1 and 2, and use algebra to solve. In your group, discuss whether or not your results correspond to the roots you have circled. What is i? What are Complex Numbers? Simplify and write each as a complex number in the form a + bi. 9 7 + 4i 25 2 – 6i 2 5 + (-3)i 118 The geometry of Complex Numbers Plot 5 + 2i on an Argand diagram (on the complex plane). Calculate the distance from 0 + 0i to 5 + 2i on the complex plane. Modulus 119 Plot -4i on the complex plane. Determine 4i . Plot, as accurately as you can, the complex number 3 3i . Determine the modulus. Find the modulus of any complex number, a + bi. Plot the complex number 3 + 4i on the complex plane. Calculate 3 4i . Give the coordinates of 7 other points which are the same distance from 0 + 0i. Plot these numbers on the same complex plane. 120 When adding two complex numbers, what is the geometric change? 121 Practice Set 15. Simplify and write the following as complex numbers, a + bi. 49 6 12 73 122 16. What is the geometric effect of adding two complex numbers on the complex plane? 17. How many different complex numbers exist that have modulus = 5? Explain your response. 18. What is i? 19. Write an equation for the conic section. Circle with center at (-3,1) and radius 2. Parabola with vertex at (3,-2) and focus at (5,-2). 20. Determine the value of x. x x 7 3 110 12 123 What is the geometric effect of multiplying a complex number by a real number? 124 Plot the number 3 + 4i on the complex plane. Determine the modulus. a. Using the distributive property, multiply i(3 + 4i). Simplify the result and plot on the same complex plane. b. Multiply i(result of part a). Simplify the result and plot on the same complex plane. c. multiply i(result of part b). Simplify the result and plot on the same complex plane. d. Describe the geometric result of multiplying a complex number by i. e. Describe each of the following geometrically on the complex plane. i2 i3 i4 125 Algebra of complex numbers (1+3i)(7-5i) (5 + 3i)(5 – 3i) Can you create an example where the sum of two complex numbers was a real number? Can you create an example where the product of two complex numbers was purely imaginary? (real part = 0) 126 electrical current equation V=I•Z V is voltage, I is current, Z is impedance Z=V/I we can also separate the current and voltage using complex notation Z = V + Ii Definitions: Voltage – the difference in electrical charge between two points in a circuit, units are volts Current – a flow of electric charge, units are amperes or amps Impedance – the opposition to current flow in AC circuits. In DC voltages, the term resistance is used. Impedance is simply the measure of how the flow of electrons is resisted. Units are ohms. Problems The impedance in one part of a circuit is 4 + 12i ohms. The impedance in another part of the circuit is 3 – 7i ohms. What is the total impedance in the circuit? IT = I1 + I2 = IT = The current in a circuit is 8+3i amps. The impedance is 1 – 4i ohms. What is the voltage? I = 8 + 3i amps Z = 1 – 4i ohms V=I•Z=I•Z V= 127 Practice Set 128 129 Consider the following two complex numbers. -4 +5i -4 -5i Graph each on the complex plane and determine the moduli. Add the two complex numbers together and describe the result. Multiply the two complex numbers together and describe the result. Conjugates What is the conjugate of 3 – 3i? What is the sum of 3 – 3i and its conjugate? What is the product of 3 -3i and its conjugate? 130 How could the conjugate help with division of complex numbers? The voltage in a circuit is 45 + 10i volts. The impedance is 3 + 4i ohms. What is the current? V = 45 + 10i volts Z = 3 + 4i ohms V=I•Z I=V/Z 131 Practice Set 1. Write the conjugate of the complex number 6 + 2i. 2. Describe the result of adding a complex number and its conjugate together. 3. Describe the result of multiplying a complex number and its conjugate together. 4. Use conjugates and multiply by one to simplify the following: 5. Graph the equation. ( y 4) 2 8( x 1) ( x 1) 2 ( y 2) 2 1 6. Describe how you would determine x and y-intercepts of a graph, given the equation. 132 Deriving a formula Find the x-intercepts of the following quadratics. x 2 8 x 15 y 3x 2 18 x 2 y 4 x 2 bx 1 y ax 2 2 x 1 y 133 y ax 2 bx c The standard form of a quadratic function is given as f (n) ax 2 bx c . Why would we find the roots to this function? Practice using the quadratic formula to solve. The “y” variable has already been set to zero. 134 Practice Set 1. Find roots using the quadratic formula. The “y” variable has already been set to zero. 2. Determine whether each represents the equation of a parabola or a circle. Then build perfect squares to help you graph. x 2 y 2 6 x 4 y 12 0 y 2 2 x 20 y 94 0 135 3. Use properties of equality to find the x-intercepts. f ( x) 25 7 x 4 x 3.5 y 11 4. Create a histogram. Determine which measure of center and spread would be most appropriate (mean and standard deviation or median and IQR). Explain your response and find those measures. 5.3, 4.7, 6.0, 3.5, 5.1, 4.4, 8.1, 61 5. Use the circle properties to determine the value of x. x x 130 4 3 136 137 Practice Factoring. 138 If g h j 0 , what do you know? How could your work with the Polynomial Puzzler and the Zero Product Property help to solve quadratic functions? x2 – 16x + 64 = 0 2x2 + 7x= 15 – y y+ x2 = 6x What do you notice about all the roots on this page? 139 Extra practice 140 141 How does the factoring method compare with other methods of finding roots? When would you choose to use each method? Give an example. 142 Practice Set Solve by factoring Simplify the following. Do not write a calculator approximation. 11. 60 12. 140 13. 90 143 Write a function for the following situations. 14. Doris begins her business with a $150 loan. Every gadget she sells, she makes a $3.50 profit. 15. Randall bought two beta fish for his tank. Surprisingly, after three months, Randall had twenty beta fish in his tank. After another three months, Randall needed room for 200 beta fish. x y 0 12 16. 1 20 2 30 3 42 4 56 17. Use complete sentences to describe congruent figures. 18. Use complete sentences to describe similar figures. 19. Determine the distance between the points (-4, 11) and (2, 10). 20. Determine the length of the hypotenuse of a right triangle, if the lengths of the legs are 14 and 9. 144 Extra factoring practice Describe your comfort level with factoring. 145 Recall the good old days of linear functions… f ( x) 4 1x What is the y-intercept or starting amount? What is the change per step or slope? Calculate the x-intercept. Graph the linear function on the graph paper. f ( x) 6 1x What is the y-intercept or starting amount? What is the change per step or slope? Calculate the x-intercept. Graph the linear function on the same coordinate system. 146 We will use these linear functions to build a new function by multiplying them together. y (4 1x)(6 1x) . Distribute. What kind of function is this new function? Determine the y-intercept. Build a perfect square to determine the vertex of the parabola. Graph the parabola on the same coordinate system as the two linear functions. What relationship do you see between the graph of the linear functions and the new parabola? … and the x-intercepts? … and the y-intercepts? 147 148 Given two roots, could you determine the quadratic function? Ex. Given x = 3, x = -7, determine a quadratic function. Ex. Given x = 6 and x = -9, determine a quadratic function. Ex. Given x = 5 and x = -1, determine a quadratic function. Ex. Given x = 2 2 and x = , determine a quadratic function. 5 7 Ex. Given x = 3i and x = -3i, determine a quadratic function. 149 Given factors, what are the roots? Given roots, what are the factors? The function? 150 Given roots, what are the factors? The function? Sketch the function. a. 4i and -4i b. 3i and -3i c. 2+5i and 2-5i d. 5+7i and 5-7i Factor over the complex number system x 2 16 x 2 10 x 34 151 152 Practice Set 1. For each problem, draw a sketch of the function. Given factors, what are the roots? 2. Given roots, what are the factors? The function? 153 3. Expand the expression (x + 3)(x – 5i)(x + 5i) in these two ways: A. [(x + 3)(x – 5i)](x + 5i) B. (x + 3)[(x – 5i)(x + 5i)] Compare and contrast the methods. 4. Given roots, what are the factors? The function? A. 7i and -7i B. 3 + 9i and 3 – 9i 5. Write the quadratic formula. 6. Use the quadratic formula to solve x 2 10 x 16 0 7. Simplify. (3 + i)(7 – 5i) 154 Alternative notation: rational exponents Rewrite each of the following (from rational exponents to radicals or vice versa). 1 1 Ex. 8 3 Ex. 5 b Ex. m 6 Ex. w 1 Ex. 49 2 3 Ex. 243 5 = Ex. 25 1 2 2 Ex. 32 5 155 Practice Set Solve 156 157 Complete your dominoes. Arrange your completed dominoes to ends that correctly match. 158 159 160 161 162 163 Notes from Cutting Corners 164 Compare and sort each set of data. a. b. c. d. e. f. g. f (n) 13 2n h. f (n) 2(n 1)2 11 i. f (n) 5 2n 165 How will you decide what model (linear, exponential, regression) fits real data best? For each of the following, use technology to graph the data. Experiment to determine which regression model fits best. Write down your chosen model (equation) and explain your choice. Use your model to respond to the prediction prompt. A. The table below lists the total estimated numbers of AIDS cases, by year of diagnosis from 1999 to 2003 in the United States (Source: US Dept. of Health and Human Services, Centers for Disease Control and Prevention, HIV/AIDS Surveillance, 2003.) Predict the number of cases in the U.S. in the year 2010. Year AIDS Cases 1999 41,356 2000 41,267 2001 40,833 2002 41,289 2003 43,171 B. As a chambered nautilus grows, its chambers get larger and larger. The relationship between the volume and consecutive chambers usually follows the same pattern. The volumes v (in cubic centimeters) of nine consecutive chambers of a chambered nautilus are given in the table. The chambers are numbered from 0 to 8, with 0 being the first and smallest chamber. Chamber, n Volume (cm3), v 0 0 .787 1 0.837 2 0.889 3 0.945 4 1.005 5 1.068 6 1.135 7 1.207 8 1.283 166 C. The following data shows the age and average daily energy requirements for male children and teens. 1 1110 2 1300 5 1800 11 2500 14 2800 17 3000 D. On Tuesday, May 10, 2005, 17 year-old Adi Alifuddin Hussin won the boys’ shot-putt gold medal for the fourth consecutive year. His winning throw was 16.43 meters. A shot-putter throws a ball at an inclination of 45° to the horizontal. The following data represent approximate heights for a ball thrown by a shot-putter as it travels a distance of x meters horizontally. What would be the height of the ball if it travels 80 meters? Distance (m) 7 20 33 47 60 67 Height (m) 8 15 24 26 24 21 E. The concentration (in milligrams per liter) of a medication in a patient’s blood as time passes is given by the data in the following table: Time Concentration (Hours) (mg/l) What is the concentration of medicine in the blood after 1.5 hours have 0 0 passed? 0.5 78.1 1 99.8 2 50.1 2.5 15.6 167 F. The median income of U.S. families for the period 1970-1998. Predict the median income in 2015. Year 1970 1980 1985 1990 1995 1998 Income 9867 21,023 27,735 35,353 40,611 46,737 G. At 1821 feet tall, the CN Tower in Toronto, Ontario, is the world’s tallest self-supporting structure. [Note: This information is taken from College Algebra: A Graphing Approach by Larson, Hostetler, & Edwards (Third Edition), page 202.] Suppose you are standing in the observation deck on top of the tower and you drop a penny from there and watch it fall to the ground. The table below shows the penny’s distance from the ground after various periods of time (in seconds) have passed. Where is the penny located after falling for a total of 10.5 seconds? Time Distance (seconds) (feet) 0 1821 2 1757 4 1565 6 1245 8 797 10 221 H. The table below lists the number of Americans (in thousands) who are expected to be over 100 years old for selected years. [Source: US Census Bureau.] How many Americans will be over 100 years old in the year 2008? Number Year (thousands) 1994 50 1996 56 1998 65 2000 75 2002 94 2004 110 Cases found: http://www.algebralab.org/Word/Word.aspx?file=Algebra_QuadraticRegression.xml 168 Population Trends in China As the new U.S. Ambassador to China, you have been given the country’s population data from 1950 to 1995. You have been asked to prepare and explain a mathematical model that represents this growth for the State Department. Use multiple formats in your presentation. The following table shows the population of China from 1950 to 1995. Year 1950 Population 554.8 in Millions 1955 609.0 1960 657.5 1965 729.2 1970 830.7 1975 927.8 1980 998.9 1985 1990 1995 1070.0 1155.3 1220.5 169 Your intern forgot to give you the additional data on population trends in China. Does this new data affect your choice in models? Explain why or why not. Year 1983 Population 1030.1 in Millions 1992 1171.7 1997 1236.3 2000 1267.4 2003 1292.3 2005 1307.6 2008 1327.7 170 Practice Set Granite School District For each pattern, 1-6, write the function. Use your function to predict the 50th term in each sequence. (For the picture patterns, a numerical and verbal representation is sufficient). 7. Rewrite the quadratic to determine the vertex. x 2 8 x 10 y 171 8. Simplify. 6i – 3i 9. Simplify. 90 10. Simplify. i i2 i3 i4 11. What is the relationship between the measure of a central angle of a circle and its intercepted arc? 12. What is relationship between the measure of an inscribed angle of a circle and its intercepted arc? 13. Draw a diagram of a 90 degree inscribed angle. What do you know about its intercepted arc? 14. Describe the effects of dilation upon a line segment. 15. Rewrite as a radical. 3 a7 172 After some hard work, you have earned a season ticket in the student section of the college football games. And you are part of each half-time show! Each game, you are given a color card, which can be used to make designs (as shown below). In honor of the Month of Mathematics, the design will model y ( x 18) 2 , using a blue and orange color scheme. Graph the student section design. Y represents the row number and x represents the seat number. 173 Write a rule for row 9 of the design. How does that rule given instructions for each seat color? Write a rule for row 25 of the design. How does that rule given instructions for each seat color? 174 Graph each of the following. A. y > -x2 - 6x – 7 B. y > x2 – 3x + 2 175 For each of the following you are given a rule. Create a number line that can describe the rule in greater detail. A. x2 + x > 6 B. x2 + x 2 176 Practice Set Graph each of the following. 177 For each of the following you are given a rule. Create a number line that can describe the rule in greater detail. 7. x 2 6 x 8 0 8. 4 x 2 16 0 Write whether the following situations could be modeled with linear, exponential, or quadratic functions. 9. A set of data whose first difference changes, but whose second difference is constant. 10. A set of data whose first difference is constant. 11. A set of data multiplied by a common ratio each term. Answer the questions based on the two-way frequency table given. Show your work for each. green eyes not green eyes female male 7 4 18 19 12. Of all females, what percent do not have green eyes? 13. Of all green-eyed people, what percent are females? 14. Of all people, what percent are male? 178 Conics: Ellipses Describe how an ellipse results from the intersection of a plane and the cones. Describe similarities and differences between an ellipse and a circle. Describe similarities and differences between a parabola and an ellipse. Ellipse derives from the Greek term, elleipsis, which means "falling short". Why do you think this term was used to describe this figure? 179 Notes on ellipses 180 Equation of an ellipse centered at the origin. Find the vertices and foci of the ellipse and sketch its graph. x2 y 2 1 100 25 Find the vertices and foci of the ellipse and sketch its graph. x2 y 2 1 9 64 181 Write the equation of an ellipse with foci at ( 2, 0) and vertices at (5, 0) . The point in a lunar orbit nearest the surface of the moon is called perilune and the point farthest from the surface is called apolune. The Apollo 11 spacecraft was placed in an elliptical lunar orbit with perilune altitude 100km and apolune altitude 314 km (above the moon). Find an equation of this ellipse if the radius of the moon is 1728 km and the center of the moon is at one focus. 182 Practice Set 183 5. Use complete sentences to compare ellipses with circles. 6. Use complete sentences to compare ellipses with parabolas. 7. Use properties of equality to determine the value of x. 14 x 3 6 2 17 1 5 x 8. Write a function for each of the following data set. n f(n) n f(n) 0 1 2 3 4 5 3 12 48 192 768 3072 0 1 2 3 4 5 3 4.1 5.2 6.3 7.4 8.5 184 How will the equation change if the center of an ellipse shifts away from (0,0)? Write the equation of an ellipse with foci at (0,2) and (0,6) and vertices at (0,0) and (0,8). Find the vertices and foci of the ellipse and sketch its graph. ( x 3)2 ( y 2)2 1 5 100 185 Find the vertices and foci of the ellipse and sketch its graph. 9 x 2 18x 4 y 2 27 0 x2 2 y 2 6x 4 y 7 0 186 Practice Set Identify the center, vertices, and foci of each. Graph each equation. 187 Write the standard equation for the ellipses with the given characteristics. 5. Foci (5, 0) and (-5, 0). Vertices (9, 0) and (-9, 0). 6. Vertices (5, 0) and (-5, 0). Co-vertices (0,4) and (0, -4). 7. Choose a center of dilation and dilate the figure with a scale factor of 1.5. 188 Hyperbolas Describe how a hyperbola results from the intersection of a plane and the cones. Describe similarities and differences between a hyperbola and a circle. Describe similarities and differences between a parabola and a hyperbola. Hyperbola derives from the Greek term which means "over-thrown" or "excessive". Why do you think this term was used to describe this figure? What might hyperbolic geometry (non-Euclidean geometry) have to do with hyperbolas? 189 Notes on hyperbolas 190 How does the hyperbola equation compare to other conic equations? Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. x2 y 2 1 144 25 Write the equation of a hyperbola with foci at (0, 3) and vertices at (0, 1) . 191 How will the equation change if center changes? Identify the center, vertices, and foci. Then sketch the graph. ( y 1)2 ( x 1) 2 1 9 16 Write the equation of a hyperbola with foci at (1,3) and (7,3) and vertices at (2,3) and (6,3). Sketch the graph. 192 Find the vertices, foci, and asymptotes of the hyperbola and sketch its graph. 16 x 2 9 y 2 64 x 90 y 305 0 y 2 x2 4 0 193 Practice Set 194 5. Describe the relationship between the lengths of secant segments in the picture below. b c d a 6. Using point Q as the center of dilation, dilate the triangle by a scale factor of 2. Then a scale factor of 3. Q 195 Conic Sections, Extra Fun 196 Describe triangle ABC, including measurements of side lengths and angles. Using A as the point of dilation, dilate the triangle by a scale factor of 3/2. Describe your procedure. Draw and describe the result. Using A as the point of dilation, dilate the triangle by a scale factor of 2. Describe your procedure. Draw and describe the result. 197 Draw any triangle with one vertex at the center of the circles. Using the center as the point of dilation, dilate your triangle by a scale factor of 1/2. Describe your procedure. Draw and describe the result. Repeat using a scale factor of 3. Use one color to highlight all parallel segments. Use other colors to highlight congruent angles (same color, same measure). 198 Use one color to highlight all parallel segments. Use other colors to highlight congruent angles (same color, same measure). Explain your work. C D B A What is the relationship between AD and BC? If two parallel lines are cut by a transversal… 199 Extra Practice Identify the angle pairs below. Determine the unknown angle measure. Solve for x. 200 Eratosthenes estimated the Earth’s circumference by using the fact that the Sun’s rays are parallel. He chose a day when the Sun shone exactly down a vertical well in Syene at noon. On that day, he measured the angle the Sun’s rays made with a vertical stick in Alexandria at noon. Use one color to highlight all parallel segments in the diagram. Use other colors to highlight congruent angles (same color, same measure). Explain Eratosthenes method. 1 of a circle. At the time, the distance from Syene to 50 Alexandria was believed to be 575 miles. What was Eratosthenes estimate of the circumference of the earth? Eratosthenes determined that m2 201 Consider the following diagrams of parabolic dishes. Use one color to highlight all parallel segments in the diagram. Use other colors to highlight congruent angles (same color, same measure). Justify your responses. For each problem, determine the value of x and/or y. Justify your responses. 202 Practice Set 203 20. Simplify the following. Do not give a calculator approximation. 80 21. Simplify the following. 6 21 7 24 8 6 21 7 24 40a 2 6 21 7 24 22. Solve by graphing. y x 2 7 x 12 23. Graph the following circle. ( x 5)2 ( y 3)2 36 Use algebra to find any x and y intercepts. 204 For each of the following, color parallel segments, color congruent angles (same measure, same color). Then find the missing measure. 205 206 207 208 209 Practice Set 7. 210 8. 9. Answer questions based on the following two-way frequency table. muppet other puppet one-person 15 20 multiple-person 2 7 a. Of all other puppets, what percent are one-person? b. Of all one-person creatures, what percent are muppets? c. Of all multiple-person creatures, what percent are other puppets? 10. Solve for x. 10x+6 104 211 Forming logical arguments Fill out the Sudoku and provide justification as you go. 212 THE DIGIT PLACE GAME This game is the number version of the commercially available logic game called Master Mind. OBJECTIVE: The object of the game is to figure out a secret number using as few guesses as possible. RULES: One person thinks of a two-digit number. (We’ll use a three-digit number later.) The other players try to figure out what the secret number is by using some very specific clues. No digits are repeated within the secret number and no number begins with 0. A player makes a guess and the person with the secret number responds with two values. The first value (digit) stands for the number of correct digits in the guess. The second value (place) stands for the number of correct digits in the guess, which are in the correct place. The players continue to make guesses/conjectures and receive responses until they know what the secret number is. YOU SHOULD NOT SAY THE NUMBER; YOU ONLY NEED TO KNOW IT. PURPOSE: Remember that our purpose is to give logical arguments to support conjectures, so be prepared to explain why you think your conjectures are reasonable. Use the chart provided to determine the missing number. Write a paragraph explaining your reasoning. 1. Guess 73 34 67 52 Digit Correct 1 0 1 1 Place Correct 0 0 1 1 Digit Correct 0 1 1 0 1 Place Correct 0 1 1 0 0 2. Guess 95 46 43 12 87 3. Pick a 3-digit number and play the Number Game with a partner. 213 UNO Rules *Try to use all your cards. Green 3 is showing. Your hand consists of Blue 3, Yellow 10, Blue 10, Red 3. Show you can play the Yellow 10. 214 As a group, use acceptable formats to prove that the sum of all the angles in a triangle is 180 . Helpful diagram: Helpful hints: Color parallel segments. Color congruent angles. 215 Practice Set Fill out the Sudoku puzzle and provide justification as you go. Uno proof Given: Blue 4 is showing. Your hand of cards consists of Green 8, Red 6, Blue 8, Green 4, Red 9, Yellow 6, Blue 9, Yellow 10. Prove: You can play the Yellow 10. Play a game or two of minesweeper with a friend or family member. 216 Paper Fold: On one side of the paper, label two edges as 8 1/2 inches and two edges as 11 inches. On the opposite side of the paper, draw a happy face, a tree, or some other simple drawing. Fold the paper along one diagonal of the rectangle and cut/tear along the diagonal. Determine the length of the diagonal and record it along the hypotenuse of either or both right triangles. Take one of the two right triangles and set it aside. Take the other congruent right triangle and fold a perpendicular to the hypotenuse that passes through the vertex of the right angle. Cut or tear the triangle along this fold. Overlap the three triangles as shown. Color all congruent angles (same measure, same color). Color all parallel segments. Determine the lengths of all sides of all three triangles. Write the dimensions on the triangles. Determine the area of each triangle. Do the squares of the legs of each right triangle sum to the square of the hypotenuse? Explain why or why not. How do the ratios of the sides of the three triangles relate to the ratios of their areas? Reassemble your paper for reference. 217 Extra Practice Similar Triangles 218 219 Practice Set 220 You will be measuring the lengths of the sides of your triangle and comparing those lengths. You will then tell the class about your measurements and your comparisons. Brainstorm about what you need to accomplish these tasks and make our class discussion clear. Make a sketch of your group triangle. Triangle Number _______ *Note: do not write on your cardboard triangle, please. Record the measurements of the lengths of sides of your group triangle. Compare the lengths of sides of your group triangle. Use the notation and methods from the class discussion. Comparison 1 Comparison 2 Comparison 3 *Write your group comparisons on three note cards for display. *Use poster putty to hang your triangle on the front board. Do not hang your note cards. 221 Class volunteer “sorter” _____________________________ Describe how _____________________ sorted the triangles on the front board. Send a group member to the board to post your comparison note cards adjacent to your triangle. Describe the comparison note cards for triangles in the same category. Color your angle. Label each side as hypotenuse, adjacent leg, or opposite leg. Label and set up ratios of sides. Using the vertex of your angle as the point of dilation, draw a dilated triangle with scale factor 3. Include lengths on each segment. Label and set up ratios of sides for the dilated triangle. 222 Create a right triangle and color in the angle of focus. Provide appropriate labels. Sine: means “pocket”, mistranslation from Sanskrit “chord” Cosine: “complement” of sine. Tangent: “to touch” The ratio comparing the opposite leg to the hypotenuse of a right triangle is called the sine because... The ratio comparing the adjacent leg to the hypotenuse of a right triangle is called the cosine because... The ratio comparing the opposite leg length to the adjacent leg length of a right triangle is called the tangent because... 223 Extra Practice 224 Practice Set 225 11. Use algebra to determine any x and y intercepts. y 14 2 x ( x 2) 2 ( y 3) 2 4 ( y 3) 4( x 2) 2 12. Describe how to slice a cone (or double-cone) to get the following cross-sections. Circle Parabola Ellipse Hyperbola 226 The County building department has given you a scale drawing of the new ramp to be constructed outside of Farley’s Place. Color the angle of interest. Label the sides as hypotenuse, adjacent leg, opposite leg. Label and set up ratios of sides. Upon construction at Farley’s Place, the ramp has passed inspection. However, the length of the ramp surface is 4.7 meters. Determine the other dimensions of the ramp. 227 The calculator screen reports that sin17 .292. Explain how this value could be useful. Use the information below to draw at least two possible triangles. cos60 .5 tan 45 1 Angle of elevation. Angle of depression. 228 Practice Set 229 230 Determine the height of the flag pole in front of school (without measuring the height!). 231 Unit Circle 232 Radian Measure Convert to radian measure. 120 330 225 Convert from radian measure to degrees. 7 6 4 3 6 233 Draw an angle in standard position on the unit circle. Make it large enough that you can take notes. Color your angle. Label your sides as hypotenuse, adjacent leg, opposite leg. Include any lengths that you know, without taking measurements. Label and set up ratios of side lengths. Set up the Pythagorean theorem for this triangle. Given sin 4 , find the value of cos . 5 234 Determine the value of . 235 Reciprocal Definitions Cosecant Secant Cotangent Variations on the Pythagorean Identity 236 Practice Set 1. Convert from degrees to radian measure. 315 150 2. Convert from radian measure to degrees. 7 4 4 Given each of the following, define the other five trig ratios. Also determine the value of . 237 4. Write out the Pythagorean Identities. 5. Use algebra to determine the x and y intercepts. y 5 2 x 10 ( x 13) 2 y 2 100 6. Describe the relationship between a central angle of a circle and its intercepted arc. Draw a diagram. 7. Choose a center of dilation and dilate the figure with a scale factor of 2. 238 On the grids below, and using your ruler, draw two different right triangles. Label each of the angles A, B, and C so that angle A and B are acute angles, and angle C is 90˚. Measure the sides and label them as well. Triangle #1: Triangle #2: What relationship do angles A and B have? What name do we give special angles like this? Find the sine, cosine, and tangent for both triangles for both angles A and B. What relationship do you notice between trigonometric ratios of the complementary angles? Compare your results (remember that your triangles may be different than other group members). Organize your data below. 239 Use your understanding of the relationship between trig ratios of complementary angles to complete the following problems. (Note that these measurements are not accurate, so using your calculator will NOT help you!) 1. Given the triangle and trig ratios below, find sin(β) and cos(β). sin(α) = ⅓ cos(α) = ¼ β α 2. Suppose that sin(19˚) = 0.2 and cos(19˚) = 0.98. Find sin(71˚) and cos(71˚). 3. Suppose that sin(θ) = 0.35 and cos(θ) = 0.35. Find θ. Prove that the relationship between trigonometric ratios of the complementary angles holds true for the triangle below. A c b B a C Write a statement that generalizes the relationship between the sine and cosine ratios of complementary angles. Did you notice any other interesting things about trig ratios that you’d like share? 240 Proving Identities General strategies Begin with the more complicated expression If no other move suggests itself, convert the entire expression into sines and cosines Combine fractions by getting a common denominator Examples. Prove the identity. 1. (x – 1)(x + 2) – (x + 1)(x – 2) = 2x 2. 1 1 2 x x 2 2x 3. (cosx)(tanx + sinxcotx) = sinx + cos2x 4. 1 tan 2 x sec2 x 2 2 sin x cos x 241 242 Practice Set Use your definitions and theorems to prove the identities. 1. cot x csc x cos x 2. cot x sin x cos x 3. tan x sin x sec x 4. tan x cos x sin x 5. cot x cos x csc x 6. sin x sec x tan x 7. tan x csc x sec x 8. sec x (1 cos x ) 1 sec x 9. sin x (1 csc x ) sin x 1 10. tan x (1 cot x ) 1 tan x 11. cos x (sec x 1) cos x 1 12. csc y (sin y 1) 1 csc y 243 13. cot z (1 tan z ) cot z 1 14. sin y tan 2 y cot 3 y cos y 15. sin2 x sec2 x sec2 x 1 16. (1 tan 2 x ) cos 2 x 1 17. (1 tan y ) 2 sec 2 y 2 tan y 18. (cos x sin x ) 2 1 2 sin x cos x 19. (sin y cos y )(sin y cos y ) 1 2 cos 2 y 20. tan 2 x 1 1 sec 2 x cot 2 x 1 244 Ignoring the Obvious but Incorrect Use your unit circle to assist in the following exercises. 1. Let u 180 and v 90 . Find sin (u + v). Find sin(u) + sin(v) Describe your findings. Test your idea with a few other pairs of angle measures. 2. Let u 0 and v 360 . Find cos(u + v) Find cos(u) + cos(v) Describe your findings. Test your idea with a few other pairs of angle measures. 3. Find your own values of u and v that will confirm that tan(u v) tan(u ) tan(v) . 245 Two right triangles have been arranged such that the two angles in focus, alpha and beta, are adjacent. Assume only what you have been given on the diagram. Label the side lengths and determine sin( ) and cos( ) 1 246 Two right triangles have been arranged such that the two angles in focus, alpha and beta, overlap. Assume only what you have been given on the diagram. Label the side lengths and determine sin( ) and cos( ) 1 - 247 How could we use our identities and definitions to determine the values of tan(α + β) and tan(α - β). 248 Practice Set Name ______________________________ Rewrite each of the following as a sum or difference of two angles. Use 30 , 45 ,60 ,90 (and their equivalent radian measures) if possible. Prove the identities. 249 250 Plot 3 + 4i on an Argand diagram. Find the modulus of this complex number. Polar Form of a complex number 251 There and Back Again Find the polar form of the complex number 1. 3i 2. -2i 3. 2 + 2i 4. 4 – 7i Write the complex number in rectangular form. 5. 3(cos30 i sin 30) 6. 8(cos 210 i sin 210) 7. 5(cos 60 i sin 60) 8. 5(cos i sin ) 4 4 252 Practice Set Name ___________________________ 253 254 255 Notes on Shape Statements 256 257 Notes from Card Set A 258 259 260 261 262 Notes for Apple Pi 263 Consider a circle with radius 3. Dilate the circle by scale factors of 2, 3, 5, 7, and 10, using the center of the circle as the point of dilation. Create a sketch of these dilations. Use different representations to describe the area of each circle. Attempt to generalize any patterns. 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 Practice Set Name ________________________________ Calculate the area of each of the following figures. 281 282 A designer has given you a floor plan of your new yurt home. Determine the area of the design. (Ignore wall thickness). 283 Notes from Yurt Area 284 Given the scale for the design below, determine the actual area of the floor plan. 285 Determine the area of the regular pentagon. If figure is dilated with a scale of 3, what will the area of the dilated pentagon be? 286 287 288 289 290 291 Notes from Fill ‘er Up 292 293 294 295 Notes from Popcorn Anyone? 296 297 298 299 300 301 302 303 304 305 Notes from Propane Tanks 306 307 308 309 Notes from Area and Volume 310 Probabilities Involving Area A point is selected at random in the square. Calculate the probability that it lies in the triangle MCN. Determine the probability that a spinner lands on section 3? Dilate the above spinner by a scale factor of 2. Determine the probability that a spinner lands on section 3? Determine the probability of the spinner landing in region A. 311