UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Relations Versus Functions/Domain and Range Station You will be given a ruler and graph paper. As a group, use our ruler to determine whether or not each relation below is a function. Beside each graph, write our answer and reasoning.. =. + = continued U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Relations Versus Functions/Domain and Range. = + How did ou use our ruler to determine whether each relation was a function?. Use our ruler and graph paper to sketch a function. Use the vertical line test to verif that it is a function. For the relations below, determine whether or not the are functions. Eplain our answer.. {(, ), (, ), (, ), (, )}. {(, ), (, ), (, )} U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Instruction Goal: To provide opportunities for students to develop concepts and skills related to creating and interpreting eponential graphs representing real-world situations Common Core State Standards F IF. F IF. Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a contet. Graph functions epressed smbolicall and show ke features of the graph, b hand in simple cases and using technolog for more complicated cases. d. (+) Graph rational functions, identifing zeros and asmptotes when suitable factorizations are available, and showing end behavior. e. Graph eponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. Student Activities Overview and Answer Ke Station Working with groups, students determine the -intercepts and solutions to eponential functions using their graphs. Then, students are given a pair of points and asked to determine the eponential function that passes through those points. Answers. - - - - - - - - - - - - - - - - - - - - -intercept: (, ) < < U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models. Instruction - - - - - - - - - - - - - - - - - - - -. (, ) - - - - - - - - - - - - - - - - - - - -. no -intercepts - - - - - - - - - - - - - - - - - - - - = U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Station Instruction Working with groups, students use calculators to evaluate and graph eponential functions. Answers. f ( ) =. - - - - - - - - - - - - - - - - - - - -. f ( ) =. - - - - - - - - - - - - - - - - - - - -. - - - - - - - - - - - - - - - - - - - - no -intercepts U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models. - - - - - - - - - - - - - - - - - - - - Instruction. no -intercepts - - - - - - - - - - - - - - - - - - - -. no -intercepts - - - - - - - - - - - - - - - - - - - - -intercept at (, ). The graph must cross the -ais, so the equation must include an addition or subtraction operation in addition to the eponential operation. U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Station Instruction Working with groups, students use eponential functions to calculate compound interest according tn to the formula r A = P + n. Answers. A = +. A = $.. A = +. A = $.. A = + A = $... A = +. A = $.. A = +. = $. A = +. = $. The account with % interest has the better ield since that account will ield approimatel $. and the account with the.% interest rate will ield approimatel $.. Station Students will be given an eponential function and asked to generate a table of values and the graph. Then students will eamine the equation, table of values, and graph for defining characteristics of eponential functions. Answers. Answers will var. See sample answer on the following page. U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Instruction f() / /. -intercepts: none; -intercept:.. Answers will var. Sample answer: variable in eponent. Answers will var. Sample answer: It grows quickl.. Answers will var. Sample answer: It rises to the right and levels off toward the left. Materials List/Setup Station colored pens or pencils Station graphing calculator; colored pens or pencils Station calculator Station none U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Discussion Guide Instruction To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to debrief the station activities. Prompts/Questions. What is an eponential function?. When does a function have an -intercept?. What is compound interest?. Wh can it be difficult to estimate compound interest?. How do ou determine if an equation is eponential?. What is the general shape of the graph of an eponential function? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses. An eponential function is a function in which the variable is in the eponent.. A function has an -intercept when its graph crosses the -ais.. Compound interest is interest that accumulates according to the total (principal plus interest) alread in the account, not just according to the principal.. The amount on which the percentage is based keeps changing.. An eponential equation has a variable in the eponent.. The general shape is a curve that etends toward infinit on one side and approaches the -ais on the other side. U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Possible Misunderstandings/Mistakes Incorrectl manipulating numbers, variables, or eponents Not understanding the laws of eponents Assuming that all functions have zeros Incorrectl calculating squares, cubes, etc., of integers between and Confusing a negative eponent with a fractional eponent Incorrectl using the eponent function of a calculator Incorrectl appling the formula of compound interest Instruction Not understanding the relationship between an eponential function and its graph Not generating the table of values correctl Plotting points incorrectl Miscalculating the - and -intercepts U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Station Work with our group to answer each question.. Graph =. Where is the -intercept? What are the roots of this function?. Graph =. Does this function have an -intercept? If so, estimate where it is. U- continued
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models. Graph =. Does this function have an -intercept? If so, estimate where it is.. An eponential function passes through the points (, ) and (, ). What is the function? Graph our answer. U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Station Using a calculator, work with our group to graph each function and evaluate the function at the given value.. f ( ) =. f ( ) =. f ( ) =. + f ( ) = continued U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models. Graph =. If there is an -intercept, what is it?. Graph =. If there is an -intercept, what is it? continued U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models. Graph = ( ). If there is an -intercept, what is it?. Graph = + ( ). If there is an -intercept, what is it?. For an eponential function to have an -intercept, what must be true of the equation? U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Station The formula for compound interest is r A = P + n, where A is the final total (principal plus interest), P is the initial amount (principal), r is the interest rate, t is the amount of time in ears, and n is the number of times the interest compounds per ear. Work with our group to set up and then solve each equation. Round answers to the nearest penn. tn. An account with an initial balance of $, has interest of.% that compounds quarterl over four ears. What is the balance at the end of the fourth ear?. An account with an initial balance of $, has interest of.% that compounds monthl over two ears. What is the balance at the end of the second ear?. An account with an initial balance of $ has interest of % that compounds ever other month over five ears. What is the balance at the end of the fifth ear?. An account with an initial balance of $, has interest of.% that compounds monthl over three ears. What is the balance at the end of the third ear?. If ou have $, to invest for two ears, which account has the better ield: an account that compounds quarterl at %, or one that compounds monthl at.%? U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models Station You will work with an eponential function at this station. Use the eponential function below for the following problems. f() =. Create a table of values for our function. f(). Find the - and -intercepts.. Graph our function below. continued U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Comparing Eponential Models. Looking at the equation, what are some defining characteristics of an eponential function?. Looking at the table of values, what are some defining characteristics of an eponential function s table of values?. Looking at the graph, what are some defining characteristics of an eponential function s graph? U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Interpreting Eponential Functions Station Instruction Working with groups, students determine properties of the graphs of eponential functions. Answers. a. (, ) b. > if b >, and < if b < c. no d. =. a. all real numbers b. > c. = d. (, ) Station Working with groups, students determine the end behavior of eponential functions. Students use their observations to determine based on the formula whether a formula represents eponential growth or deca. Answers. a. approaches b. grows without bound. a. grows without bound b. approaches. a. decreases without bound b. approaches. Eponential deca; the function approaches as becomes infinitel larger.. Eponential growth; the function approaches infinit or grows without bound as becomes infinitel larger. U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Interpreting Eponential Functions Station Instruction Student pairs graph eponential functions, checking their work with a graphing calculator. Answers.. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -.. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - Materials List/Setup Station graphing calculator Station graphing calculator Station graphing calculator Station graphing calculator; graph paper U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Interpreting Eponential Functions Discussion Guide Instruction To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to debrief the station activities. Prompts/Questions. What is an eponent?. What is an eponential function?. What are the differences among the graphs of eponential functions when the base is negative, when the base is a fraction, and when the base is a negative fraction? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses. An eponent is a number that tells the number of times the base is to be multiplied b itself.. An eponential function is a function in which the variable is in the eponent.. A base that is negative will be reflected over the -ais, and a base that is a fraction will be reflected over the -ais. If the base is a negative fraction, then it will be reflected over both the - and -aes. Possible Misunderstandings/Mistakes Incorrectl manipulating numbers, variables, or eponents Not understanding the laws of eponents Assuming that all functions have zeros Assuming that an eponential function has a vertical asmptote as it tends toward unbounded growth Not understanding the difference between growth and deca Incorrectl calculating squares, cubes, etc., of integers between and Confusing a negative eponent with a fractional eponent U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Interpreting Eponential Functions Station Work with our partner to evaluate each epression. Show our work.. f () = f () =. f () = f () = ( ) = ( ) = f f. f ( ) = f () = f f () = =. f () = ( ) = ( ) = ( ) = f f f. f () = f () = ( ) = f. f () = ( ) = ( ) = f f U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Interpreting Eponential Functions Station Consider the graph of the given function. Work with our group to answer each question. Show all our work.. = b a. Where is the -intercept? b. What is the range? c. Does the function have an zeros? If so, where are the? d. Does the function have an asmptotes? If so, where?. = a. What is the domain? b. What is the range? c. Does the function have an asmptotes? If so, where? d. Does the function have an zeros? If so, where are the? U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Interpreting Eponential Functions Station Work with a group to answer each question. Show all our work.. =. a. What is the end behavior as approaches infinit? b. What is the end behavior as approaches negative infinit?. = a. What is the end behavior as approaches infinit? b. What is the end behavior as approaches negative infinit?. = ( ) a. What is the end behavior as approaches infinit? b. What is the end behavior as approaches negative infinit?. = Does this function represent eponential growth or deca? Eplain.. = Does this function represent eponential growth or deca? Eplain. U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Sequences Discussion Guide Instruction To support students in reflecting on the activities and to gather some formative information about student learning, use the following prompts to facilitate a class discussion to debrief the station activities. Prompts/Questions. What is an arithmetic sequence?. What is a series?. Is a sequence finite or infinite?. What is a geometric sequence?. How is a geometric sequence different from an arithmetic sequence?. How could a geometric sequence be related to eponential functions? Think, Pair, Share Have students jot down their own responses to questions, then discuss with a partner (who was not in their station group), and then discuss as a whole class. Suggested Appropriate Responses. An arithmetic sequence is an ordered group of numbers separated b a common difference.. A series is the partial sum of a sequence.. A sequence can be bounded (finite) or infinite.. A geometric sequence is an ordered set of numbers that increase or decrease b a common ratio, r.. An arithmetic sequence is an ordered set of numbers that increase or decrease at a common difference, d. The terms in an arithmetic sequence are defined b addition or subtraction; the terms in a geometric sequence increase or decrease b a common factor.. To find the terms of a geometric sequence, we use an eponential calculation. U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Sequences Station Work with our group to answer each question. Show all our work. Use the calculator if ou need help. For problems, let a = t and d = t.. What is a n?. What is a?. What is the sum of the first terms in the sequence? For problems, let a = and d =.. What is a n?. What is a?. What is S? Answer the following questions about sequences.. Look at the sequence,,,,... What is S?. Look at the sequence,,,,,,... Is it arithmetical? Eplain.. Think of the sequence of positive odd integers. What is the th term of that sequence?. What is S of the sequence of positive odd integers? U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Sequences Station Working with our group, graph each geometric sequence as an eponential function..,,,,....,,,,... continued U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Sequences.,,,,,....,,,,... continued U-
UNIT LINEAR AND EXPONENTIAL RELATIONSHIPS Station Activities Set : Sequences.,,,,... U-
UNIT REASONING WITH EQUATIONS Station Activities Set : Solving Sstems b Graphing Station At this station, ou will find four inde cards with the following linear sstems of equations written on them: = + = = ; ; = + = + = ; = + + = Work together to match each sstem of linear equations with the appropriate graph below. Write the appropriate sstem of linear equations beside each graph... continued U-
UNIT REASONING WITH EQUATIONS Station Activities Set : Solving Sstems b Graphing... What strateg did ou use to match the sstems of linear equations with the appropriate graph? U-
UNIT DESCRIPTIVE STATISTICS Station Activities Set : Displaing and Interpreting Data. es. Answers will var but should be in the form = m + b.. Answers will var. Instruction. BMI Hours. No. There are too man other factors involved (such as activit level and diet). There seems to be a correlation, but we can t prove causation.. Yes; (, ) and (,.) Station Working with groups, students analze a data set to find a linear relationship between variables. Students use a calculator to conduct linear regression. Answers. Grade. es. Answers will var. Hours U-
UNIT DESCRIPTIVE STATISTICS Station Activities Set : Displaing and Interpreting Data. (, ). =. +.. Yes. The linear relationship is ver close. Instruction. We can t prove causation from this data. As the outlier shows, some people ma not stud because the alread know the material well. However, the data suggests that the more time spent studing, the higher the test grade will be. Materials List/Setup Station Station Station Station graph paper; nine inde cards with the following numbers written on them:,,,,,,,, graph paper; ruler calculator; colored pens or pencils; graph paper calculator; colored pens or pencils U-
UNIT DESCRIPTIVE STATISTICS Station Activities Set : Displaing and Interpreting Data Station Work with our group to answer each question about the data set. Use the calculator to calculate medians and create our graphs. A class wants to find out if there is a correlation between the number of hours studied and grades on the midterm eam. The students log their hours and their grades, as follows. Studing (hours) Grade Studing (hours) Grade. Enter the numbers into our calculator to graph the results on a scatter plot. Sketch our plot below. Grade Hours continued U-
UNIT DESCRIPTIVE STATISTICS Station Activities Set : Line of Best Fit Instruction Goal: To provide opportunities for students to develop concepts and skills related to creating and analzing scatter plots and lines of best fit to represent a real-world situation Common Core State Standards S ID. S ID. Represent data on two quantitative variables on a scatter plot, and describe how the variables are related. a. Fit a function to the data; use functions fitted to data to solve problems in the contet of the data. Use given functions or choose a function suggested b the contet. Emphasize linear, quadratic, and eponential models. b. Informall assess the fit of a function b plotting and analzing residuals. c. Fit a linear function for a scatter plot that suggests a linear association. Interpret the slope (rate of change) and the intercept (constant term) of a linear model in the contet of the data. Student Activities Overview and Answer Ke Station Students will be given graph paper and a ruler to help them create a scatter plot. Then the will analze the scatter plot to determine the correlation between the data and describe the slope. Answers. Graph: Math Test Score Hours studied The graph is a scatter plot. Test score (%). The test scores increase as the amount of time she studies for each test increases.. Positive correlation; the longer she studied for the test, the higher her test score; positive slope because the line increases from left to right. U-
UNIT CONGRUENCE, PROOF, AND CONSTRUCTIONS Station Activities Set : Corresponding Parts, Transformations, and Proof Answers Instruction. (, ), (, ), (, ); (, ), (, ); (, ). Yes, because the size and shape remained the same.. A and B, because corresponding sides and angles are congruent.. Yes, because the size and shape remained the same.. The have the same size and shape. Station Students will be given four inde cards with the following written on them: SSS; SAS; ASA; AAS. Students will work together to match the inde cards to real-world eamples of SSS, SAS, ASA, and AAS. Then the will eplain how SSS, SAS, ASA, and AAS relate to congruent triangles. Answers. ASA. AAS. SSS. SAS. Answers will var.. side-side-side; side-angle-side; angle-side-angle; angle-angle-side. These are was to prove two triangles are congruent. Materials List/Setup Station Station ruler; protractor graph paper; ruler; push pins; rubber bands Station graph paper; ruler; cardboard triangle created from a triangle with vertices (, ), (, ), and (, ) in the coordinate plane Station four inde cards with the following written on them: SSS; SAS; ASA; AAS U-
UNIT CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES Station Activities Set : Parallel Lines, Slopes, and Equations Station At this station, ou will find rulers. Use these to help ou determine whether or not the following lines are parallel. Look at the graph below.. Are these lines parallel?. Eplain two was ou can tell lines are parallel.. What is the shortest distance between these two lines?. Draw a line that is parallel to the given line below.. How do ou know our line is parallel? U-