ASSESSING CLICKER EXAMPLES VERSUS BOARD EXAMPLES IN...

ASSESSING CLICKER

EXAMPLES VERSUS BOARD

EXAMPLES IN CALCULUS

Kimberly A. Roth

Juniata College

1700 Moore Street

Huntingdon, PA 16652

[email protected]

ASSESSING CLICKER EXAMPLES

VERSUS BOARD EXAMPLES IN

CALCULUS

Abstract:

The combination of classroom voting system (clicker) questions and peer

instruction has been shown to increase student learning. While implementa-

tions in large lectures have been around for a while, mathematics has been

increasingly using clickers in classes of smaller size. In Fall 2008, I conducted

an experiment in order to measure the effect of clickers on exam performance.

The experiment was conducted with two sections of Calculus 1 using the same

examples either on the board or with clickers. By comparing only the lower-

performing students in the sections and alternating use of the clicker between

sections, I was able to obtain modestly significant results showing improved

exam performance for low performing students despite a small sample size.

Keywords: classroom response system, in class polling, peer instruc-

tion, ConcepTests, Calculus, exam performance

1 INTRODUCTION

Classroom voting systems(clickers) and peer instruction have been used

in the sciences, especially physics, for some time[6] and in mathematics

classrooms more recently. Initially they were often used in large lecture

classes, but they are now being used in smaller classes as well. Studies

have shown improvement in final exam scores [9], better performance

on conceptual questions [11], and that peer instruction is necessary to

make clickers more effective [9].

2

There are several articles detailing why one might want to try clickers

with peer instruction and discussions of how implementation works ([1],

[4] in general and [3] for math specifically). I was inspired to use clickers

by a talk by Craig Nelson at a Juniata faculty conference.

Juniata College is a small liberal arts college with a majority of stu-

dents from the local area in fairly rural central Pennsylvania. I currently

use clickers in my calculus classes and this study is of one semester’s

Calculus 1 students. Calculus 1 at Juniata College is taught in sections

that cap at thirty students and meet for four 55 minute sessions a week.

The course is taken mainly by science and math majors, along with a

few business majors and students from other majors fulfilling part of

their quantitative requirement. We used Hughes-Hallett, Gleason, Mc-

Callum et al., Calculus, Fourth Edition, [7] at the time the data was

collected, although we have since moved to the fifth edition. We use

Maple throughout the semester.

Implementation of clickers into the calculus course took more time

than I had initially expected. Clicker questions are often multiple choice

or True/ False, and so are structured differently than examples I do at

the board. I started with just adding warmup clicker questions, and then

slowly integrating them into my lectures. Figuring out which examples in

lecture were better served by clicker questions continues to be something

I consider every semester. I use clickers to determine their participation

grades, which accounts for five percent of the overall grade, and count

only whether or not the student answered the questions. There are many

other ways to incorporate clicker use into a student’s grade; I chose this

method to avoid having to choose which questions to grade.

Technically, initial setup was a bit time consuming with the original

clicker model. However, my college changed clicker types after my first

semester to a model which is easier to set up, although the software

has had some technical issues. Juniata has the students buy their own

clickers, and primarily uses the INTERWRITEPRS R© radio frequency

clickers. Most of my students have already had to purchase their clickers

for organic chemistry or physics, so I use whatever type of clicker is

chosen by those two departments.

3

For questions, I use mainly questions from the Hughes-Hallett, Glea-

son, McCallum et al., ConcepTests to accompany Calculus, Fourth Edi-

tion, copyright 2005 [8], a fifth edition is currently in use. In additions I

use questions from the Good question project [5], and some of my own

design. The number of resources for questions keeps increasing. Project

MathQuest at Carroll College keeps a good list of sources [10].

After using clickers in calculus for a few semesters, I wanted to inves-

tigate two questions. First, do clicker examples help students do better

on related exam questions than examples done on the board? Second,

do clickers perform better for graphical questions than other types of

questions? Juniata has a Scholarship of Teaching and Learning(SoTL)

that runs regular workshops on topics related to SoTL. In January of

2008, Cheryl McConnell who is in Accounting and Anita Salem who is

in Mathematics at Rockhurst University led a workshop which included

framing questions and methodologies and also provided individual con-

sultations. During my individual consultation Anita Salem gave me the

idea of alternating methods of examples as described in the experimental

design. This design was refined through discussion with David Drews

who was the leader of the SoTL group on campus at the time and during

a presentation and questions session at the biweekly SoTL brown bag

lunches.

2 EXPERIMENTAL DESIGN

In the fall of 2008 I taught two sections of Calculus 1, section 3 and

section 4, out of the four sections offered by my department. Each

section used clickers daily, always for a warm-up question and usually

for several questions during lecture. Warmup questions were frequently

a review from the previous day’s material or sometimes preview of the

current day’s. The students saw the warmup question when they arrived

in the classroom. I then took 10 to 15 minutes of homework questions.

After this, the students compared their answers with their neighbors

(pair and share) for peer instruction, and then gave a final answer for

the warmup questions, which we then discussed. Then, lecture occurred

4

for the rest of the 55 minute class with interspersed clicker questions.

The number of in-lecture questions varied from zero to five depending

on the topic. The questions during lecture followed the more traditional

format of answer, pair and share, answer again, and then discuss as a

whole. In addition to the clicker questions, during lecture I also had

the students answer other questions I wrote on the board by calling on

students who raised their hands and provided part or all of an answer

which I transcribed.

For each of the four in-class exams I picked two topics to test. In one

section I did clicker questions with board examples for the topic, and in

the other section I did only essentially identical board examples. The

sections were reversed for the other topic. The only difference in the

clicker and board questions was that the board questions did not have

multiple choice answers; the questions themselves were the same. On

the in-class exams there were questions on both of the topics. The final

had questions on all eight of the topics.

Each session had some clicker questions, so the sections could not tell

that I was replacing clicker questions with board examples. The days I

presented the topics were videotaped and I wrote journal entries about

the sessions. The students were also given a survey on the usefulness

and enjoyment of clickers at the end of the semester. I kept copies of

all of the individual student work for each topic on the exam, in case it

would be useful.

The topics were as follows, and after each topic I will give the ab-

breviation by which I will refer to it later. For exam 1 the topics were

identifying formulas of graphs of sine and cosine (trig) and checking con-

tinuity of functions (cont). For exam 2 the topics were making graphs of

f ′ from graphs of f (f ′)and finding tangent lines and determining if the

tangent line over or underestimated function values (tan line). Exam 3

had determining and classifying critical points by looking at the graph

of f ′ (crit pts) and basic related rates problems (rel rate). Exam 4 had

determining the graph of an antiderivative from the graph of the func-

tion (F ) and calculating left and right sums for a function (LHS/RHS).

in-class exams were given during class time and so the exam questions

5

from the clicker examples had to be slightly different. The final was

given to both sections at the same time and so there was only one final.

In topic selection, I tried to pick fairly specific topics like finding

the equation of the tangent line, as opposed to calculating a derivative.

The topics were picked by importance in the course and the number of

clicker questions I had available on the topic. Since I thought clickers

might show the most effect with graph-related questions in particular, I

always chose one graphical and one non-graphical topic per exam. Since

concepts often appear multiple times, I used the first occurrence of a

topic if possible. Concepts that occur multiple times affect how the final

scores are influenced.

I will illustrate the process. These examples were picked not because

they are the best examples of questions using clickers, but because they

had significant results. They are listed in order of ascending significance.

The full list of clicker / board examples and is available in the appendix.

For the tangent line example, the first clicker question was the fol-

lowing.

The equation of the tangent line to f(x) = −2ex at x = 0 is:

a) y = −2x− 2

b) y = −2(x− 0)

c) y = −2exx− 2

d) y = −2ex(x− 0)

This was followed by pair and share and discussion of the correct

answer, b, along with what was wrong with the incorrect answers and

then this follow-up question.

Estimates for y = −2ex using the tangent line to y = −2ex at

x = 0 are:

a) overestimates

b) underestimates

c) depends on what you are estimating

The followup was followed by pair and share and discussion of the

correct answer, b.

6

In the section that got the board example,section 4, I did the same

tangent line question and then the followup without the multiple choice

answers. The examples were done on the board. I posted the problem

and asked the students to respond by raising their hands to questions

about the steps, such as ”What is the first step?”, and ”What do you

do next?”.

The following question was asked on the final.

Find the equation of the tangent line to y = f(x) = sin (2x)+1

at x = 0. Make sure to show your work!

Does the tangent line over or underestimate values of f(x)

near x = 0? Why?

For the trigonometry (trig) example these were the clicker questions

from [8]:

Figure 1. Reprinted with permission of John Wiley & Sons, Inc.

7



8

The graphs were drawn on the board for the board examples and the

same questions were asked of them without the multiple choice options.

The third question was followed up with “What is the equation for the

graph?”, although I forgot to do this in the clicker section that semester.

On the final this question appeared. The graph is from [7].

For the graph below:


a. What is the period?

b. What is the amplitude?

c. Find a possible equation for the graph.

For the determining continuity (cont) examples, these were the clicker

questions from [8].

True or False: The function is continuous on the given in-

terval.

f(x) = 1x−2 on [0, 3]


terval.

f(x) = 1x−2 on [−1, 0]


terval.

f(x) = etan (θ) on [−π2 ,π2 ]

When using the clicker questions, I have some students state why

they picked their answer. These questions were written on the board for

9

the board example, asking if they were continuous and why or why not,

without the True or False portion.

This question occurred on the first exam in the board example sec-

tion, which was section 3 for this topic.

Is the function f(x) =√x

x2−7x+6 continuous on the following

intervals. Justify your answer!

a. [−3, 0]

b. [2, 3]

c. [0, 3]

This question occurred on the first exam in the clicker section, which

was section 4 for this topic.


x2−5x+6 continuous on the following

intervals. Justify your answer!

a. [−1, 1]

b. [0, 1]

c. [1, 3]

The sections were quite different from each other in makeup and

behavior. Section three had 26 students, 9 of whom were high school

students, 2 international first year students, 24 domestic first year stu-

dents, and 1 junior. Section four had 14 students, 4 of whom were high

school students, 4 international first year students, 5 domestic first year

students, and one sophomore. In this particular semester, the number

of high school students was higher than usual; normally there are 1 or

2 high school students per section. It is also worth noting that the high

school students in the different sections were from different high schools.

Section 3 was a talkative class and had an unusual student who was a

substantial discipline problem. Section 4 was much more quiet, in fact

they might be the most quiet class I have ever taught. In overall final

grades, section three was a stronger group of students. Note that the

sample size is quite small here.

10

method trig(3) cont(4) f ′(4) tan line(3) crit pts (3) rel rate(4) F (3) LHS/RHS(4)

clicker 11.5 13.3 13.2 11.5 11.0 6.9 11.4 10.6

board 12.7 11.8 13.1 11.2 8.9 6.6 10.3 11.7

Table 1. Table consists of means by section for regular exam questions for

all students. The number after topic is section of the clicker example.

3 RESULTS

For my first look at the data, I looked at the means of the scores on the

exam questions, as we see in Tables 1 and 2 (all scored out of 15 points

except for the related rates, which was out of 10). Most means for clickers

were higher, but none of them were statistically significant. However I

noticed that every section had a number of high performing students

who earned nearly all the points on every question. These students were

doing well, no matter what the method of example delivery was.

So I decided to split each section into a higher performing group

and a lower performing group and look mostly at the low performance

group. To split the sections, I took the portion of the third exam score

that was unrelated to the questions analyzed in the clicker project, found

the overall median and then split the sections into the higher perform-

ing above the median group and the lower performing less than or equal

to the median group. The third exam was used because in analyzing

previous years’ exam data, the third exam was the strongest predictor

of final grade for an in-class exam. I did not use the final exam grade,

because by the final two of the weaker students had dropped the class.

This probably occurs because the third exam contains the applied opti-

mization problems and related rates, two of the most difficult topics for

students. The median split made my sample even smaller, 13 students

from section 3 and 8 from section 4 in the lower performing group. So

there were 13 students from section 3 and 6 students from section 4 in

the higher performing group.

Examining the higher performing group in Tables 3 and 4 you can see

the clicker does not appear to have much effect. Sometimes the clicker

11


clicker 13.2 13.9 11.9 11.5 13.5 6.7 12.1 10.3

board 11.2 13.5 11.7 9.9 12.3 5.8 11.3 9.1

Table 2. Table consists of means by section for final exam questions for all

students. The number after topic is section of the clicker example.


clicker 12.8 13.8 13.5 13.0 11.9 8 11.5 12.8

board 12.8 13.0 14.3 13.8 8.9 8.2 11.8 11.3


higher performing students. The number after topic is section of the clicker

example.

example yields a higher exam problem average, sometimes it does not.

It is useful to note that section 3 earned higher overall grades in the

course. Section 3 has greater than or equal averages on exam questions

for 10 out of 16 questions.

Considering only the lower performing students in Table 5 and 6,

the differences between means with two sided t-tests for the continuity

question on the in-class exam (p=.096), the trigonometry question on

the final (p=.091) and the tangent line question on the final (p=.081)

were statistically significant with α = .1. If the t-test were one sided

the results would be less than .05, but since I do not want to assume

that clickers always improve scores I did not use a one sided test. With

the sample sizes as small as they are I have chosen to consider α = .1


clicker 14.1 14.2 11.7 11.8 14.3 6.8 11.7 9.5

board 13.1 14.1 13.3 11.8 13.2 6.9 13.8 11.7

Table 4. Table consists of means by section for final exam questions for higher

performing students. The number after topic is section of the clicker example.

12

Two sided t-tests for difference in Means between Clicker and Board.question clicker N board N p-value df

trig(3) 10.2 13 11.5 8 .379 17

cont(4) 13.3 8 10.5 13 .096 17

f ′(4) 13.0 8 11.9 13 .477 18

tan line(3) 10 13 9.2 8 .696 15

crit pts (3) 10.1 13 8.9 8 .554 12

rel rate(4) 6 8 4.9 13 .494 17

F (3) 11.2 11 9.13 8 .365 15

LHS/RHS(4) 10.9 8 11 11 .483 15


lower performing students. The number after topic is section of the clicker

example. Bolded results are significant. α = .1

Two sided t-tests for difference in Means between Clicker and Board.question clicker N board N p-value df

trig(3) 12 12 9.4 7 .091 9

cont(4) 13.7 7 12.9 12 .487 16

f ′(4) 12.1 7 9.9 12 .302 14

tan line(3) 11.2 12 8.3 7 .081 11

crit pts (3) 12.6 12 11.6 7 .535 10

rel rate(4) 6.6 7 4.7 12 .143 13

F (3) 11.6 12 10.6 7 .593 13

LHS/RHS(4) 8.8 7 8.8 12 .982 11

Table 6. Table consists of means by section for final exam questions for lower

performing students. The number after topic is section of the clicker example.

. Bolded results are significant. α = .1

13

significant, even though .05 would be more common. Looking at the

means themselves, most clicker exams led to a higher, but not signifi-

cantly higher average, except for the trig example on the in-class exam,

the F graph on both the in-class and the final, and the LHS/RHS ques-

tion. Note that the N values change slightly throughout the semester

due to students who dropped the class and one fourth exam I failed to

photocopy.

4 CONCLUSIONS AND FURTHER QUESTIONS

Overall, statistically it looks like the data so far indicate that clicker ex-

amples with peer instruction may improve exam performance in specific

cases for lower performing students, although not as significantly as we

would hope. However, in order to solidify this conclusion I would need

to take more data. Clickers may improve performance on exams in a

small class setting, but even if they do not, they do not seem to hurt

performance on exams. It does not appear that they have this effect

particularly for graphical questions as I had originally hypothesized.

For my teaching, this means I will keep using clickers. Besides po-

tential educational advantages to using them, I enjoy using them in class

and the students report on their end of class survey that they enjoy them

as well.

Remaining questions: Is there a class size too small for clickers to be

effective? I had the unusual experience to have a class of 7 students in

Fall 2009 and while they still answered and enjoyed the clicker questions,

I had difficulty getting them to pair and share. I think the class was

too small for the sense of anonymity when discussing questions to kick

in. It would be interesting to study the effectiveness of classroom voting

across sizes of classes.

Is there a type of question for which clickers are most helpful? I

went into the experiment thinking that the graphical questions were the

most useful for clickers. Certainly the clickers allow me to do more

graphical examples in shorter time, since I do not have to draw the

graphs. The results are mixed on whether graphical examples work

14

better. The trigonometry example did show significant improvement,

but the derivative graphs, finding critical points from the graph of the

derivative, and antiderivative graphs did not.

This experimental design is portable and could be reused for a num-

ber of settings where larger numbers of sections are not available for

study. In fact, if one year’s data was not large enough to provide ad-

equate sample size, it would be possible to repeat the experiment in

a section in another semester, as long as the exam questions could be

reused. I considered that option for the Fall of 2009, but then had a

class of only seven students which was too small for my purposes.

Clicker questions were shown to improve exam performance in some

cases for lower performing students. I think it would be interesting if

there would be further study about which types of questions or which

types of topics have the greatest effect.

ACKNOWLEDGEMENTS

I was inspired to use clickers by a talk by Craig Nelson at a Juniata

faculty conference. The idea for this experiment came from Anita Salem

at a SoTL workshop at Juniata. Derek Bruff keeps the most detailed

reference list for clicker literature([2]) and I have greatly appreciated its

existence. Thanks to David Drews, Kathy Westcott, Phil Dunwoody,

Lynn Cockett, Kimberly Burch and the members of the Scholarship of

Teaching and Learning Group at Juniata College for their advice and

encouragement. Thanks especially to David for telling me that you can

never collect too much data. Thank you to the referees for their useful

comments about both style and substance. Thank to John Wiley & Sons

for allowing me to reprint their images.

SoTL projects at Juniata are partially supported by a grant from the

Teagle Foundation.

REFERENCES

[1] Bruff, D. (2009). Teaching with classroom response systems: Creat-

ing active learning environments. San Francisco: Jossey-Bass.

15

[2] Bruff, D. Classroom Response System (”Clickers”) Bibliogra-

phy http://www.vanderbilt.edu/ cft/resources/teaching

resources/technology/crs biblio.htm. Accessed 14 June 2010.

[3] Cline, K., Zullo, H., & Parker, M. (2007). Teaching with classroom

voting. Mathematical Association of America FOCUS, 27(5), 22-23.

[4] Duncan, D. (2005). Clickers in the classroom: How to enhance sci-

ence teaching using classroom response systems. San Francisco: Pear-

son Education.

[5] The Cornell Good Questions Project

http://www.math.cornell.edu/ GoodQuestions/, Accessed 14

June 2010.

[6] Crouch, C. H., & Mazur, E. (2001). Peer instruction: Ten years of

experience and results. American Journal of Physics, 69(9), 970-977.

[7] Hughes-Hallett, D., Gleason, A., McCallum, W., et al. (2005). Cal-

culus:Single Variable. John Wiley & Sons Inc. 4th edition.

[8] Hughes-Hallett D., Gleason A., McCallum W. et al.(2005), Con-

cepTests to accompany Calculus, Fourth Edition. John Wiley & Son.

[9] Miller, R.L., Santana-Vega, E., & Terrell, M.S. (2006). Can good

questions and peer discussion improve calculus instruction?, PRIMUS,

16(3).

[10] Project Math QUEST at Carroll College,

http://mathquest.carroll.edu/resources.html. Accessed 14

June 2010.

.

[11] Pilzer, S. 2001. Peer Instruction in Physics and Mathematics.

Primus. 11 (2): 185-192.

16

BIOGRAPHICAL SKETCH

Kimberly Roth is an Associate Professor of Mathematics at Juniata

College. She earned her B.A. in mathematics with a minor in computer

science from Oberlin College. She spent the year after her B. A. studying

at the Budapest Semesters in Mathematics. Her Ph.D. work was at

Penn State under the supervision of Greg Swiatek. Her field is complex

dynamics. She enjoys working on Scholarship of Teaching and Learning

projects and teaching both mathematics and statistics. Her hobbies

include singing in the Juniata College Choral Union, knitting, and wheel

thrown pottery.

17

5 APPENDIX

The following are the topics tested in this experiment. Included are

the reasons for topic choice, clicker questions for the topic, and exam

questions for the topic. Questions that had significant results are marked

significant. As explained further in the article the examples done on the

board were the same as the clicker items without the multiple choice

options and most had followup discussion of all of the options not just

the correct ones. All of graphs except for the first two for graphing

the derivative are from Hughes-Hallett [7] as well as some of the exam

problems themselves, as I usually put one verbatim homework question

on each exam.

5.1 Identifying formulas for graphs of sine and cosine (trig)

5.1.1 Reason for topic choice

I chose this topic because the students are often not as good at going

from a graph to a formula rather than the other way around. This was

the graphical topic of exam 1. Clickers were used in section 3 and board

example was used in section 4.

18

5.1.2 Clicker questions for this topic from [8]


19



20

The third question was followed up with what is the equation, although

I forgot to do this in the clicker section that semester.

5.1.3 Exam 1 question for section 3










21



5.1.5 Final exam question for both sections(significant)






5.2 Checking continuity of functions (cont)


This topic was chosen mainly because there were a fair number of clicker

questions available. This was the topic with the least interesting exam

question in my opinion. Clickers were used in section 4 and board ex-

ample was used in section 3.


True or False: The function is continuous on the given interval.

f(x) = 1x−2 on [0, 3]


f(x) = 1x−2 on [−1, 0]

22


f(x) = etan (θ) on [−π2 ,π2 ]

5.2.3 Exam 1 question for section 3(significant)


x2−7x+6 continuous on the following intervals.

Justify your answer!

a. [−3, 0]

b. [2, 3]

c. [0, 3]

5.2.4 Exam 1 question for section 4(significant)


x2−5x+6 continuous on the following intervals.


a. [−1, 1]

b. [0, 1]

c. [1, 3]

5.2.5 Final exam question for both sections

Is the function f(x) = 1√2x−5

continuous on the following intervals.


a. [0, 3]

b. [2.5, 4]

c. [3, 6]

5.3 Making graphs of f ′ from graphs of f (f ′))


This is a graphical topic and important to the course. It was also chosen

since this is a topic for which there are many clicker questions. Clickers

were used in section 4 and board example was used in section 3.

23



24



25



Given the following graph of f(x)

26

a. On the same axes, sketch the graph of the derivative of f(x).

b. Identify the interval or intervals (approximately) where f ′′(x) < 0.

How can you tell?




b. Identify the interval or intervals (approximately) where f ′′(x) > 0.

How can you tell?




27


b. Identify the interval or intervals (approximately) where f ′′(x) > 0.

How can you tell?

5.4 Finding tangent lines and determining if the tangent line

over or underestimated function values (tan line)


This is an important topic that repeats several times within the course.

It is also one that is often difficult for students. Clickers were used in

section 3 and board example was used in section 4.

5.4.2 Clicker questions for this topic

1. The equation of the tangent line to f(x) = −2ex at x = 0 is:

a) y = −2x− 2

b) y = −2(x− 0)

c) y = −2exx− 2

d) y = −2ex(x− 0)

2. Estimates for y = −2ex using the tangent line to y = −2ex at x = 0

are:

a) overestimates

b) underestimates

c) depends on what you are estimating


Find the equation of the tangent line to y =√x at x = 4. Make sure to

show your work!

Does the tangent line over or underestimate values of f(x) near x =

4? Why?

28


Find the equation of the tangent line to y = 1x at x = −1. Make sure to

show your work!


−1? Why?

5.4.5 Final exam question for both sections (significant)

Find the equation of the tangent line to y = f(x) = sin (2x)+1 at x = 0.

Make sure to show your work!


0? Why?

5.5 Determining and classifying critical points by looking at

the graph of f ′ (crit pts)


These questions are represented by a sequence of clicker questions. They

require students to understand the derivative tests for classifying critical

points well. Clickers were used in section 3 and board example was used

in section 4.

29

5.5.2 Clicker questions for this topic


30



31


Given the graph of the derivative function f ′ below.


a. Find the x−coordinates of the critical points of f .

b. For each critical point, indicate if it is a local maximum or local min-

imum or neither. Justify your answers!

c. Find the x− coordinate of the points on inflection. How do you know

they are points of inflection?



32















33

5.6 Basic related rates problems (rel rate)


Related rates are difficult for students, even when given the equation.

Clickers were used in section 4 and board example was used in section

3.

5.6.2 Clicker question for topic


5.6.3 Exam 3 question for section 3 from [7]

A spherical cell is growing at a constant rate of 400 µm3 per day (µm is

micrometers a measurement of length) At what rate is its radius increas-

ing when the radius is 10 µm? The volume V of a sphere with radius r

is V = 43πr

3.

34

5.6.4 Exam 3 question for section 4 from [7]

A spherical balloon is inflated so that its radius is increasing at a constant

rate of 1 cm per second. At what rate is the volume changing when its

radius is 5 cm? The volume V of a sphere with radius r is V = 43πr

3.

5.6.5 Final exam question for both sections from [7]

Gasoline is pouring into a cylindrical tank of radius 3 feet. When the

depth of gasoline is 4 feet, the depth is increasing at .2 ft/sec. How fast

is the volume of the gasoline changing at that instant? Note: Volume,

V of a cylinder is V = πr2h where r is the radius and h is the height.

5.7 Determining the graph of an antiderivative from the graph

of the function (F )


Just like graphing the derivative given the function, this is a graphical

topic and important to the course. It was also chosen since this is a

topic for which there are many clicker questions. Clickers were used in

section 4 and board example was used in section 3.

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Graph two antiderivatives of the function given below on the same axes.








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5.8 Calculating left and right sums for a function (LHS/RHS)


This is a topic that is repeated frequently at the beginning of integral

calculus. It is however not one I had often tested in the past due to

length and time it takes to grade. Clickers were used in section 3 and

similar board example was used in section 4.

5.8.2 Clicker question for this topic from [8]


This is followed up with questions about which is the LHS and what is

wrong with the other options.


Use the table for the decreasing function f(t) to compute the following.t 0 2 4 6 8

f(t) 15 13 12 10 7

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a. A right hand sum estimating∫ 8

0f(t) dt with n = 2. Show your work!

b. A left hand sum estimating∫ 8


c. For each estimate have you over or under estimated∫ 8

0f(t) dt ? How

do you know?


Use the table for the increasing function f(t) to compute the following.t 0 3 6 9 12

f(t) 4 7 8 10 15

a. A right hand sum estimating∫ 1

02f(t) dt with n = 2. Show your

work!

b. A left hand sum estimating∫ 1


c. For each estimate have you over or under estimated∫ 1

02f(t) dt ?

How do you know?


Use the table for the function f(t) to compute the following.t 0 1 2 3 4 5 6

f(t) 5 -7 4 -3 2 -1 -4

a. A left hand sum estimating∫ 6

0f(t) dt with ∆t = 3. Show your work!

b. A right hand sum estimating∫ 6

0f(t) dt with ∆t = 2. Show your work!

c. Can you tell if your estimates are under or over estimates∫ 6

0f(t) dt ?

Why?

ASSESSING CLICKER EXAMPLES VERSUS BOARD EXAMPLES IN...

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