CASIO Education Workbook Series ACTIVITY SAMPLES with the CASIO fx-9750GII WHAT’S INSIDE: • Algebra I: Fun with Functions • Algebra II: Saving for a Rainy Day • Geometry: Quite the Quadrilateral • Statistcs: The Cost of Car Insurance • Pre-Calculus: Area of a Triangle and Circular Sector • Calculus: A Graphical Look at Continuity
Fun With Functions Teacher Notes Topic: Functions and Function Notation NCTM Standard: • Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules. • Relate and compare different forms of representation for a relationship. • Identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations.
Fun With Functions “How To” The following will demonstrate how to store a value for a variable, input a function to generate a table, and enter data into a table to construct a graph with the Casio fx-9750 GII. To store a value for a variable: 1. To store 5 for the variable x, input: 5bfl. To input a function and generate a table: 1. From the Main Menu, highlight the TABLE icon and press l or 5. 2. To enter a function such as 3x – 1, input: 3f-1l. 3.
2. Press d then u(TABL) to display the table. 3. To display a corresponding y-value for a specific x-value, highlight any x-value and enter the desired value. To display the corresponding y-value when x = 18, input 18l. Note: You do not need to change the tables settings; you could just enter all given x-values and create a custom table. To enter data into a list and graph the data: 1. From the Main Menu, highlight the STAT icon and press l or 2. 2. In List 1, input: 1l2l3l. 3.
Fun With Functions Activity Functions help establish various types of numeric patterns, based upon whether those functions are linear, quadratic, cubic, etc. Building a strong foundation in Algebra includes a comprehensive study of linear functions. Functions are a rule used to calculate values. Functions are written using a specific notation called function notation. Each function has an independent and a dependent variable. The independent variable is the value you get to choose or control.
Questions 1. Given the function f(x) = 3x + 7, evaluate the function at f(4). _________________________________________________________________ 2. Given the function f(x) = -2x – 5, evaluate the function where -5 < x < -1. (Let x = set of integers) _________________________________________________________________ 3. Given the function f(x) = 2x2 + 6x – 5, evaluate the function at f(-2). _________________________________________________________________ 4.
6. Enter the data into the calculator and determine the linear function. List 1 List 2 0 8 2 5 6 3 2 0 _________________________________________________________________ 7. Lauren works as a babysitter to earn some extra money. She charges her customers seven dollars an hour. Write a function to determine the amount of money Lauren will earn if she works x hours.
Solutions 1. f(4) = 19 2. f(-5) = 5, f(-4) = 3, f(-3) = 1, f(-2) = -1, f(-1) = -3 3. f(-2)= -9 4. The function is: y = 3x – 1 or f(x) = 3x – 1 5.
6. This relation fails the vertical line test as evident by the two coordinates directly above each other. Since the relation fails the vertical line test, the relation is not a function. 7. The function is y = 7x or f(x) = 7x. If Lauren works four hours, she will earn $28. If Lauren works seven hours, she will earn $49. If Lauren works 11 hours, she will earn $77. 8. The function is y = .07(x - 50) + 30 or f(x) = .07(x - 50). For 50 text messages per month, you will spend $30.
Saving for a Rainy Day Teacher Notes Topic Area: Patterns and Functions – Algebraic Thinking NCTM Standard: • Understand patterns, relations, and functions by interpreting representations of functions of two variables using symbolic algebra to explain mathematical relationships. Objective Given a set of formulas, the student will be able to use the GRAPH Menu and G-Solve Function to solve problems involving the investment of money.
Saving for a Rainy Day “How To” The following will demonstrate how to enter a given formula into the GRAPH module the Casio fx-9750GII, graph the data, and use G-Solve to find x- and yvalues. Example Formulas: ⎛1+ r⎞ A = P⎜ ⎟ ⎝ n ⎠ A = nP (1+r)n n where n = 4, P = 100, and r = x where n = x, P = 1000, and r = 0.05 Steps for using the GRAPH Menu: 1. From the Main Menu, press 3 for the GRAPH icon. 2. Enter the first formula into Y1: by entering: 1000(1+fz4)^4l. 3.
Steps for using G-Solve: 1. Press Ly(G-Solv), u( > )and q(Y-CAL). 2. To find the y-value when x = 0.1, input the following: .1l. 3. Press Ly(G-Solv), u( > ) and w(X-CAL). 4. To find the x-value when y = 2000, input the following: 2000l.
Saving for a Rainy Day Activity Investing for the future is usually the last thing on a person’s mind when they are just entering the workforce. Paying bills, buying groceries, and purchasing a home are usually at the top of the list. However, putting money away in some form of savings should be the number one priority of every budget. Social security and retirement plans are often not enough to allow a person to continue to afford their current lifestyle.
5. What would be the future value of an annuity if $10,000 is invested yearly for 5 years at 4% APR? _________________________________________________________________ 6. What would be the future value of an annuity if $10,000 is invested yearly for 5 years at 10% APR? _________________________________________________________________ 7. For a principal amount of $10,000, what is the difference between the amount of income earned at 2.
12. Calculate the future value of an investment of $100 a year, at 5% APR, for 10 years. _________________________________________________________________ 13. Calculate the future value of an investment of $100 a year, at 5% APR, for 30 years. _________________________________________________________________ 14. Calculate the difference between question 12 and question 13. _________________________________________________________________ 15.
19. What is the difference between the investment for 10 years and the investment at 30 years? _________________________________________________________________ 20. What is the benefit of starting to invest early? _________________________________________________________________ _________________________________________________________________ Extension 1. Credit cards charge interest, however, that interest is compounded daily.
Solutions 1. $5256.33 2. $5416.32 3. $6105.10 4. $52,563.28 5. $54,163.22 6. $61,051.00 7. $8,487.72 8. 1.9 years 9. 3.8 years 10. 5.5 years 11. Answers will depend on student experience 12. $1,257.79 13. $6,643.88 14. $5,386.09 15. $6,288.95 16. $33,219.42 17. $26,930.47 18. $627.45 19. $566.66 20. Answers will vary depending on student experience but should involve the fact that the longer money is invested, the more money you will have. Extension Solution 1.
Quite the Quadrilateral Teacher Notes Topic Area: Properties of Parallelograms NCTM Standards: • Use geometric ideas to solve problems in, and gain insights into, other disciplines and other areas of interest such as art and architecture. • Use Cartesian coordinates and other coordinate systems, such as navigational, polar, or spherical systems, to analyze geometric situations. • Investigate conjectures and solve problems involving two- and threedimensional objects represented with Cartesian coordinates.
Quite the Quadrilateral “How-To” The following will demonstrate how to enter a set of coordinates into two lists using the Statistics mode of the Casio fx-9750GII. After the list is set up, you will find the slope of a line containing the points, save the equation in the Graph mode, and find the intersection of two lines. You will then find the length of a segment. Line segment AB has endpoints at (−5, -2) and (3, 6) and segment CD has endpoints at (−6, 4) and (3, −7).
To graph the two equations and find the intersection: 1. From the Main Menu, highlight the Graph icon and press l or press 3. 2. To graph the two equations, highlight each equation and press q (Sel) to turn the function on; when the equal signs are highlighted, you know the equation is selected. Then press u(Draw). 3. While viewing the graph, press y(G-Solv) y(ISCT) to find the intersection of the two equations. 4. The coordinates are displayed at the bottom of the screen.
Quite the Quadrilateral Activity The parallelogram is a special quadrilateral with special properties that is used in a variety of areas, especially in design. In this activity, we will explore the properties and then solve some problems using those properties. Questions The diagram at the right shows Quad ABCD. By definition, a parallelogram is a quadrilateral with both pairs of opposite sides parallel. 1. (6, 6) C (-2, 4) B Find the equation of a line that contains the following points: a.
7. Find the length for the following segments to the nearest tenth. a. AB : _______________________________________________________ b. 8. DC : ________________________________________________________ What can you conclude about the opposite of a parallelogram? _________________________________________________________________ Draw the two diagonals for the figure. We are now going to look at their properties in relation to quadrilaterals. 9. Find the equation for the following segments. a.
16. What conclusion can be made about angles in a parallelogram? _________________________________________________________________ One method of demonstrating vector addition is by creating a parallelogram. The sum is the coordinates of the fourth vertex of the parallelogram. 17. Given the diagram below and using the properties of parallelograms, find the sum of v1 and v2. _________________________________________________________________ 18. The magnitude of a vector is equal to its length.
Solutions: 1. a. y = 0.25x + 4.5 b. y = 0.25x − 0.75 2. The slope of each line is 0.25. 3. a. y = 2x + 8; b. y = 2x − 6 4. The slope of each line is 2. 5. The opposite sides are parallel. 6. a. b. 7. a. b. 8. (6 − −2)2 + (6 − 4)2 = 8.2 (3 − −5)2 + (0 − −2)2 = 8.2 (− 2 − −5)2 + (4 − −2)2 = 6.7 (6 − 3)2 + (6 − 0)2 = 6.7 Opposite sides of the parallelogram are equal.
9. a. y = 0.36x + 3.81 b. y = −0.8x + 2.4 10. (0.5, 2) 11. a. b. 12. a. b. (0.5 − −5)2 + (2 − −2)2 = 6.8 (0.5 − 6)2 + (2 − 6)2 = 6.8 (0.5 − −2)2 + (2 − 4)2 (0.5 − −5)2 + (2 − 4)2 = 3.2 = 3.2 13. The diagonals bisect each other. 14. a. b. 15. ∠ ABC ≅ ∠ CDA & ∠ BAD ≅ ∠ DCB 16 Opposite angles of a parallelogram are equal. 17. v1, + v2 = 9,8 18. a. v1 = 2 2 + 6 2 = 6.32 units b. v2 = 7 2 + 2 2 = 7.28 units c.
The Cost of Car Insurance Teacher Notes Topic: Data Analysis and Probability NCTM Standard(s) • For univariate measurement data, be able to display the distribution, describe its shape, and select and calculate summary statistics. Objective: Given a set of data, the student will be able to enter data into the statistics menu of the Casio 9750 GII, graph the data using a median box-and-whisker graph, and calculate the measures of central tendency.
The Cost of Car Insurance “How-To” The following will demonstrate how to enter a set of data into the Casio fx-9750GII, graph the data using a Box and Whisker Plot and find important information from the graph. Scores on the First Math Test 55 60 75 80 90 65 75 60 50 80 70 95 100 Scores on the Second Math Test 75 90 85 60 95 85 80 To enter the data from the table in the problem: 1. From the Main Menu, highlight the STAT icon and press l or 2. 2.
4. Press w(Box) for a box-and-whisker plot. 5. Make sure that the XList is List 1 and a Frequency of 1. If not, scroll down and press q to select a frequency of 1. 6. Press d, then q(GPH1) to view the graph. 7. Pressing q will display the statistical data from the list. To graph multiple sets of data: 1. Press d to go back one screen. 2. Press u(SET) and w(GPH2) to set the type of graph for StatGraph 2. 3. Press w(Box) for a box-and-whisker plot, then press N to change the XList to List 2. 4.
To perform a 2 variable statistic analysis of the data: 1. d twice until you are at the main stat screen. 2. Press w(CALC), then w(2VAR) for a two-variable analysis. 3. Scroll down to see the data.
The Cost of Car Insurance Activity For many years, actuaries have kept track of the driving records of car insurance policy holders. These statistics compare males and females and those under or above 21 years old. This data is used to determine the amount paid for car insurance premiums. In this activity, you will compare the cost of car insurance premiums that resulted from the analysis of this data.
5. 6. Use your Casio 9750GII to graph a box and whisker for each of the age and gender groups. Draw a sketch of each graph. Be sure to label the interquartile values for each age and gender group. Female < 21 Female ≥ 21 Male < 21 Male ≥ 21 What is the range of costs for car insurance for females over 21 years old? _________________________________________________________________ What is the mean cost? _________________________________________________________________ 7.
9. 10.
Solutions 1. Range = $2381 - $1773 = $608 Mean = $2035.40 2. Range = $3291 - $2459 = $8732 Mean = $2870.80 3. $2870.80 - $2035.40 = $835.40 4. Answers will vary 5.
6. Range = $1748 - $1129 = $619 Mean = $1456.60 7. Range = $2439 - $1477 = $962 Mean = $1847.80 8. $1847.80 – $1456.60 = $391.20 9. Male, 17 years old: Female, 18 years old: Male, 76 years old: Female, 35 years old: 10.
Area of a Triangle and Circular Sector Teacher Notes Topic Area: Trigonometric Applications NCTM standards: • Develop fluency in operations with real numbers using technology for morecomplicated cases. • Understand functions by interpreting representations of functions. Objective To calculate the area of a triangle and a circular sector using trigonometry. Getting Started In this activity, the student will learn how to calculate the area of a triangle and a circular sector using trigonometry.
Prior to using this activity: • Students should understand the difference between right triangles and nonright triangles. • Students should know the difference between degrees and radians as angle measurements. Ways students can provide evidence of learning: • Students will be able to calculate the area of any triangle. • Students will be able to calculate the area of a circular sector.
Area of a Triangle and Circular Sector “How-To” The following will demonstrate how to enter the data into the Casio fx-9750GII and interpret the results. To set the calculator to calculate in degrees: 1. From the Main Menu, highlight the RUN•MAT icon and press l or 1. 2. Make sure the calculator is in degree mode by pressing Lp, move the cursor down to Angle. Press q(Deg) to change it into degrees, thend.
3. Enter 10 for M and 25 for V, and then highlight K. 4. Press u(SOLV). 6. Kinetic energy for this example is 3125 kg• 7. To use the Solver to find m , input values for K and V, highlight M and press u to find the value for M. Use this same method to find V. m .
Area of a Triangle and Circular Sector Activity Introduction In this activity, you will learn how to calculate the area of a triangle and a circular sector using trigonometry. The area of a triangle is defined to be one-half of the product of the lengths of the two sides (a, b) and the sine of angle (C) included between those two sides.
2. 3. Find the area of the following triangles for the given values using Heron’s formula and the Solve feature. a. a = 5, b = 7, c = 10 area = __________ b. a = 10, b = 8, c = 6 area = __________ c. a = 6, b = 4, c = 8 area = __________ Find the area of the following circular sectors for the given values using the formula in the Solve feature. a. r = 8, θ = π 5 b. r = 10, θ = c.
Solutions 1. a. area = 10.0376 units2 b. area = 18.7789 units2 c. area = 7.7135 units2 2. a. area = 16.2481 units2 b. area = 24 units2 c. area = 11.6190 units2 3. Note that angles are given in radians for this question. changed over to radian mode before proceeding. Calculator must be a. area = 20.1062 units2 b. area = 104.7198 units2 c. area = 34.
A Graphical Look At Continuity Teacher Notes Topic: Continuity NCTM Standard • Organize and consolidate their mathematical thinking through communication; communicate their mathematical thinking coherently and clearly to peers, teacher, and others. Objectives The student will be able to develop a visual understanding of how limits and continuity relate and be able to understand and communicate what it means for a function to be continuous at a point.
A Graphical Look At Continuity “How-To” The following will demonstrate how to graph a function, graph a split-defined function and examine its behavior on the CASIO fx-9750GII. Explore the behavior of the function f(x) = −x2 + 3x − 5. To display a graph of the function: 1. From the Main Menu, highlight the GRAPH icon and press l or press 3. 2. To delete any previous equations, highlight the equation and press w(DEL)q(Yes.) 3. Enter the equation in Y1 by pressing nfs+3f-5l. 4.
To find the vertex of the graph: 1. Press Ly(G-Solv)w(MAX). The coordinates of the vertex will be displayed at the bottom of the screen. [Note: w(MAX) was pressed since the vertex of this parabola is the highest, or maximum, point. If the graph of the parabola opened up, the vertex would be the lowest, or minimum, point and you would have chosen e(MIN).] ⎧− x2 + 3x − 5, x < 1.5 ⎪ Explore the behavior of the split-defined function g(x) = ⎨− 4, x = 1.5 . ⎪− x2 + 3x − 5, x > 1.
A Graphical Look At Continuity Activity This activity will have you explore the concept of continuity at a point. It will also allow you to discover that simply having a limit at a point will not guarantee that the function is also continuous. Using the CASIO fx-9750GII, you will be working in pairs or small groups. Questions Explore the behavior of the function f(x) = x2 − x − 6 around the vertex. 1. Graph the function using the initial view window, changing Ymin to −9.3 and Ymax to 9.
5. Explore the behavior of the following split-defined function: ⎧x 2 − x − 6, x < 0.5 ⎪ g(x) = ⎨− 6, x = 0.5 ⎪x 2 − x − 6, x > 0.5 ⎩ Use the same viewing window as before. Record what you see below. 6. What does the value of lim g( x ) appear to be? x → 0.5 _________________________________________________________________ 7. How does it compare to the lim f ( x ) ? x →0.
9. Find the following limits: a. b. c. 10. lim g(x) ____________________ lim g(x) ____________________ lim g(x) ____________________ x → 0.5 + x → 0.5 − x → 0.5 Find g(0.5). How does this compare to your answer for lim g( x ) ? x →0.5 _________________________________________________________________ _________________________________________________________________ _________________________________________________________________ 11.
Solutions 1. 2. (0.5, −6.25) 3. Nothing unusual should be seen. The function is continuous. 4. The limit is -6.25, the vertical value of the vertex. Answer will vary; care should be taken to point out that simply tracing to a value is not confirmation enough and can be tricky. Direct substitution is a valid explanation. A good answer might also include a mention of “passing through” or even a mention of continuity. 5. [Note: The discontinuity will not be immediately apparent.] 6. The limit is −6.25.
11. Answers will vary; a good answer will include the fact that the function has a gap or a hole or a jump at the point of discontinuity. The idea is to have the students begin to think about the fact that simply having a limit does not guarantee the continuity of a function.
