Math 12 - for 4 th year math students

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1 Math 12 - for 4 th year math students This portion of the entire unit should be completed in 6 days The students will utilize handout notes, measuring tapes, textbooks, and graphing calculators for all lessons in this unit. by Joseph Sedita

2 Overview This 7-day unit is taken from the UCSMP component of the Math 12 curriculum used in the City of Buffalo Schools. Topics include data collection, functions, modeling, interpretation of data, and analysis/evaluation of data. Students will have already learned about linear, quadratic and exponential functions in the 3 rd year course Advanced Algebra, so they should be familiar with notation and forms of the equations. Day Measure circumference of wrist, neck, and waist 1 Record measurements on board Using the calculator find linear regression models for Wrist vs. neck Neck vs. waist Wrist vs. waist Find the correlation coefficient for each bivariate set Interpret the meaning of the correlation coefficient based on graphs Predict the neck and waist measurements of Buffalo Bills lineman John Fina whose wrist measures 24.5 cm. Day Graph an exponential growth function, determine the domain and range 2-3 Graph the compound interest formula as a function of years Predict the amount in the account after a given number of years Predict the number years it would take to have a certain amount in the account Graph an exponential decay function, determine the domain and range Graph the future value of an annuity formula as a function of time Predict the number years it would take to have a certain amount in the annuity Day Given a set of data, find an exponential model algebraically 4 Find an exponential model using a graphing calculator Determine which is a better model linear or exponential Predict the value of the function for an independent variable value within the data set (interpolation) Predict the value of the function for an independent variable value outside the data set (extrapolation) Day Find the y-intercept of a quadratic function 5 Find the x-intercepts of a quadratic function using the quadratic formula Given a set of data, find a quadratic model algebraically Use the calculator to solve a system of equations Day Given a set of data, find a quadratic model using the calculator 6 Predict the value of the function for a given value Given a set of data find an appropriate model by examining the graphs of the data and any regression equations that may have been found

3 Lesson 1 Linear Models and Correlation Objectives Identify the properties of the regression line and correlation coefficient Find and interpret linear models Graph linear function Use scatterplots to draw conclusions about linear models Procedure 1. Hand out student notes 2. Have students work with a partner to measure the circumference of their wrists, neck, and waist and record the data on the chart on the class blackboard. 3. Have students input the data into three lists on their calculators 4. Make a scatterplot of the data in the first two lists Wrist vs. neck 5. Use the calculator to draw the regression lines for the scatterplot 6. Find the correlation coefficient for the regression line 7. Sketch the scatterplot and the regression line 8. Repeat the graphing process for Neck vs. waist and Wrist vs. waist 9. Predict the neck and waist size for Buffalo Bills tackle John Fina whose wrist measures 24.5 cm. Materials Measuring tapes Chart to record data (overhead or on board) Graphing calculators for students Graphing calculator with overhead Student textbooks Assignment Page , 16,17

4 Math 12 Notes With a given set of data we can: 1) graph the data on the calculator in a scatterplot 2) draw the line of best fit (regression line) or other regression equation 3) find the correlation coefficient 4) predict outcomes of data not contained in the set The average weight in kilograms for girls of various ages in the United States is given in the table below Age in years Weight in kg REMEMBER TO Clrlist BEFORE YOU BEGIN 1 TO GRAPH A SET OF DATA press STAT choose 1:Edit enter the data in the lists press ENTER after each piece of data x-values in one list y-values in another be sure that both lists have the same number of entries press 2nd Y= (STAT PLOT) press ENTER select On Type: scatterplot XList: L1 YList: L2 Mark: whichever you like (the + works best) press ZOOM 9 You should have a graph

5 2. TO FIND THE EQUATION FOR THE REGRESSION LINE press STAT select CALC select 4: LinReg (ax+b) you will want the calculator to graph this along with the data points you already have graphed with the cursor as in the picture at the right press 2 nd 1, 2 nd 2, VARS select Y-VARS, select 1: Function, select 1: Y 1 your screen should look like this press ENTER the equation is given in slope intercept form where a is slope and b is the y-intercept Press Y= to see that the regression equation is already in the function menu press GRAPH your line will be drawn in the scatterplot

6 3. TO PREDICT A VALUE NOT GIVEN BY THE DATA For example: We want to know what a 7 year old girl would be expected to weigh. press 2 nd MODE (QUIT) CLEAR this will put you back on the regular work screen input your value on the work screen press 7 STO X,T,θ,n ENTER press VARS select Y-VARS select 1:Function press ENTER select 1:Y1 this is the regression line you defined press ENTER press ENTER again to display the value calculated using your regression line So a girl 7 years old weighs kg

7 Lesson Day Exponential functions Objectives Identify the variables, domain, and range of exponential functions Describe properties of exponential functions Graph exponential functions Interpret properties of relations from graphs Procedure 1. Use a simple bacterial growth problem to begin such as: In a microbiology lab a lab technologist has isolated 10 bacteria in a dish. The biologist knows that number of bacteria in the dish doubles every hour. How many bacteria will be present in 5 hours? 2. Exponential equations are all of the form y = ab x. a = beginning amount b = growth factor x = # of growth periods 3. Have the students create a table of data with two column headings hours and number of bacteria. 4. Show that the equation y = 10(2) x is representative of the data 5. Graph the function 6. Have the students find a window that will allow them to see the number of bacteria that will be present in 24 hours, 48 hrs, 1 week, etc. 7. Do a radioactive decay problem such as: The half-life of a certain radioactive element is 40 days, if a scientist had 10 grams of the substance present initially how much would be present in 90 days? 8. Have the students find an exponential function that describes the situation 9. Have the students graph the compound interest formula as a function of time. Amount = P 1 + r nt when $1000 is compounded daily at 6%. n 10. Ask the students to find an appropriate window for this function 11. Ask the students how many years it would take to have $100, Ask the students how much will there be in 25, 50, 75 years. 13. Have the students graph the future value of an annuity formula when $50 is deposited each week at a rate of 7.5% APR. y=50((((1+(.075/52))^(52x)-1))/(.075/52)) how long will it take toaccumulate $1,000, Try the same formula with different interest rates and deposits to demonstrate to the students the value of long term investment Materials Graphing calculators for students Graphing calculator with overhead Student textbooks Assignment Lesson Master 2-4

8 Lesson Day 4 - Modeling Data Using Exponential functions Objectives To find and interpret exponential models To use scatterplots to draw conclusions about models for data Procedure 1. Hand out problem situations 2. Use the two data points in problem 1 to find an exponential model by solving a system of equations for the values of a and b in the general form of the exponential equation y = ab x 3. Graph the data points on the calculator and verify that the equation found above does indeed pass through the two points 4. Have the students make a scatterplot using the national debt data 5. Have the students find an exponential model for the data 6. Graph the model and the scatterplot and verify that the model fits 7. Have students find a linear model for the same data 8. Graph both models and the scatterplot and show that the exponential model more closely represents the data 9. Using the exponential model, find the amount of debt in Using the exponential model, predict the amount of debt in 2005 Materials Graphing calculators for students Graphing calculator with overhead Student textbooks Handout with problem situations Assignment Lesson Master 2-5

9 Math 12 Lesson 2-5 data 1. In a laboratory experiment on the growth of insects, there were 74 insects three days after the beginning of the experiment and 108 insects five days after the beginning of the experiment. The information is summarized in the table below. Days after number Beginning of insects a) Find an exponential model for the data algebraically. b) Find the number of insects at the beginning of the experiment. c) Predict the number of insects that will be present on the 8 th day. 2. The data below was obtained from the U.S. Office of Management and Budget. The data show that the national debt of the United States has been increasing significantly since Use your graphing calculator to find an exponential model of the form y = ab x to represent the data in the table. Year Amount of debt (in billions) a) Write the equation for your model. b) Use your model to estimate the debt in Since this value falls within the data we have this process is called interpolation. c) Use your model to estimate the debt in Since this value is beyond known values of the data, this process is called extrapolation.

10 Lesson Day 5 - Quadratic Functions Objectives To identify the variables, domain and range of quadratic functions To describe properties of quadratic functions Graph quadratic functions Find a quadratic model for a set of data algebraically Procedure 1. The standard form of the quadratic equation is ax 2 + bx + c = 0 2. Have students find the y-intercept of the quadratic y = 3x 2 - x Have students find the x-intercepts (roots) of y = 3x 2 - x 14 using the quadratic formula 4. Hand out Quadratic notes including program for the TI Link the calculator to transfer the program QUADFORM to one student. Now have the students transfer the program until everyone has it. 6. Have students find the roots of the following y = 7x 2 + x + 5, y = 21x 2-23x 20 using their new program 7. Demonstrate how to find a quadratic model for a set of data using three points and solving a system of equations use example 3 on page Remind the students that they learned a method of solving systems of equations using matricies in Advanced Algebra. Materials Graphing calculators for students Graphing calculator with overhead Student textbooks Handout with program and example 3 from page 123 Assignment Homework Handout 2-6

11 Math 12 Quadratic Notes Program QUADFORM finds the roots of a quadratic equation ClrHome Disp "OTHER ROOT IS" Disp "INPUT" Disp F Frac Prompt A,B,C Goto 2 B^2-4ACD Lbl 1 Disp "DISCRIMINANT=" Disp "IMAGINARY ROOTS" Disp D B/(2A)R Pause ( abs(d))/(2a)i If D<0 Disp "REAL PART IS" Goto 1 Disp R Frac ( B+ (D))/(2A)E Disp "IMAG. PART IS" ( B- (D))/(2A)F Disp I Frac Disp "ONE ROOT IS" Lbl 2 Disp E Frac Example 3 on page 123 Day Price We know the standard form of a quadratic is y = ax 2 + bx + c. Using three points from the data set given (0, 350), (1, 340), and (2, 320), we can write a system of equations where we substitute the x and y values of these points into the standard form of the equation and get three equations with three unknowns a, b, c. 350 = a(0 2 ) + b(0) + c 350 = c (1) 340 = a(1 2 ) + b(1) + c 340 = a + b + c (2) 320 = a(2 2 ) + b(2) + c 320 = 4a + 2b + c (3) since c = 350 from (1) we substitute c into (2) and (3) to obtain -10 = a + b (4) -30 = 4a + 2b (5) solving for a and b we obtain a = -5 b = -5 and c = 350 thus the quadratic model is y = -5x 2 5x + 350

12 Lesson Day 6 Finding Quadratic Models Objectives Graph quadratic functions Find a quadratic model for a set of data using the graphing calculator Procedure 1. Have students work in groups of 2 or 3 to do the in class activity on page 119 of the text. 2. Demonstrate the procedure for finding the quadratic model using the calculator. 3. Have students find a quadratic model for the data given on Supplemental Problem Materials Graphing calculators for students Graphing calculator with overhead Student textbooks Homework Handout on back of Lesson Master 2-6 Assignment Front side of Lesson Master 2-6 with Extra Homework Problem on back

13 Math 12 Supplemental problem The data in the table below shows the average weight gain of three pigs on a hog farm. Twenty-four pigs were each given a dietary supplement in the form of pellets. The pigs were randomly selected to receive a daily dosage of pellets ranging from 0 to 7 pellets, the average weight gain of the three pigs receiving the same dosage is summarized in the table below. No of pellets (daily dosage) Percentage of weight gain Make a scatterplot of the data in the table Find an appropriate model for the data (linear, exponential or quadratic). Determine the best dosage to give to the hogs based on the data.

14 Problem to copy on the back of Lesson Master 2-6 Based on tests made by the Bureau of Public Roads, below are the distances (in feet) it takes to stop a car in minimum time under emergency conditions. Reaction time is considered to be.75 seconds. Speed (mph) Stopping Distance (feet) Make a scatterplot of the data. Find an appropriate equation to model this data. Write the equation. Using your model fill in the expected stopping distances for each speed listed in the table below. Speed (mph) Stopping Distance (feet)

15 Materials and Equipment TI-83 or TI83PLUS graphing calculators for students TI-83 or TI83PLUS graphing calculator with overhead projection unit 5 10 fabric or plastic measuring tapes textbooks UCSMP Functions, Statisics and Trigonometry, 2 nd Edition Lesson Masters from UCSMP Functions, Statisics and Trigonometry, 2 nd Edition Handouts as indicted Tennis Ball Resources UCSMP Functions, Statisics and Trigonometry, 2 nd Edition. Addison Wesley, Chapter 2 pp Functions, Statisics and Trigonometry, 2 nd Edition, Lesson Master Supplement. Addison Wesley, Physics. John Wiley& Sons,1989. Improve Your Scuba: Enriched Air Diver Manual. International PADI, Inc., 1995.