Recently on the AP Calculus electronic discussion group the question came up, "What software do you use for generating graphs. For example, for a limits test. How can I generate a graph with removable discontinuities, jumps, etc. Is there a good software program out there for generating graphs?"

There were several replies of programs that use esoteric nomenclature or fancy tricks.

One person replied "I don't have one that works well with piecewise functions (most will handle continuous pw functions with some cajoling and using logical operators (<, <=, >, >=, and, or etc) with Boolean (ie 1 or 0) output, but typically go bonkers when you have discontinuities)... I was just creating a diagram for f(g(x)) where f and g are PW functions for a precal class, and everything I have has come up woefully short (the TI's won't even evaluate Y1(Y2(x)) if you've used the logical operator kluge to get pw functions).

So I'm also all ears if someone has a graphing program that will handle PW functions in a reasonable way."

The most reasonable way I have found to do piecewise functions is with the TI-Nspire. Here was my response with some additional pictures and explanation included:

The TI-Nspire Software may be what you are looking for.

It will do piecewise functions in a very natural way - just like they look in a textbook. There is a math template for piecewise functions. Just press the button (highlighted in red in the first picture on the left) or you can find it in the catalog. The nice thing about the catalog is that it gives a verbal identifier as you can see in the picture on the left.

The Nspire will also do composite functions. You can even change the name of a graph from f1(x) to f(x) and g(x) and let h(x)=f(g(x)). I tested it with a composition of piecewise functions and it worked great too. This was really easy to do. Here is what I did: Open a graph page, arrowed over and deleted the 1 in f1(x) so it would say "f(x)=". Then I press the math template button and filled in the blanks. Next I arrowed down and made it g(x). Finally, I arrowed down one more time and entered h(x)=f(g(x)).

Also with the TI-Nspire CAS you can do many implicit/conics by entering, for example, f1(x)=zeros(x^(2)+y^(2)-25,y)

The graphs look really clean (not large pixels) if you use View > Computer. CTRL J is the shortcut for getting screen shots.

You can download the software that comes free with new purchases of the TI-Nspire Touchpad at http://education.ti.com/educationportal/downloadcenter/SoftwareList.do?website=US&tabId=1&paneId=6002#10

You could also get the TI-Nspire or TI-Nspire CAS Teacher Software for free from a Learn & Earn program http://education.ti.com/html/nspire/learn-earn.html

Yesterday in Physics First and in my AP Calculus class we studied the relationship between position, velocity and acceleration with graphs. We collected data with CBR2 motion detectors. Students plugged these into their handhelds and I could also plug it into my computer and begin collecting data automatically with the TI-Nspire Software.

Notice that the labels on the axes were automatically changed from x and y to Dist(m) and Time(s). These labels can be manually changed of deleted entirely.

On a Data & Statistics page the labels are the variable names.

The students then analyzed the graphs with menu > Trace > Graph Trace. It was fun to have the students predict what the corresponding velocity-time graph would look like and use a dynamic Data & Statistics page to easily who what the calculated solution would be. This is easily done by clicking the left side of the screen. I . I also enjoy just pressing tab to quickly change the variable. The default of the motion detector is to measure the distance every 0.05 seconds. At the same time it populates two other columns of data: dc01.vel1 for velocity and dc01.acc1 is the acceleration for data collection trial 1.

This was a graph of a ball rolling up an incline. At first it increased its distance away from the motion detector and then it came back down. The little imperfection near the peak was when it moves a bit horizontally near the top of the incline. Usually I use a nice racquetball but this time a used a larger less perfectly spherical ball we got from this summer's public library reading program. As you can see the acceleration was a component of -9.8m/s^2. Students who have at least had geometry may be able to even figure out what the angle of the incline was. This ball on an incline experiment is outlined at the end of an Xtreme Calculus activity. There is a link to this TImath.com activity from this site.

From the first time I taught algebra, I've always said that there is now better way for students to develop an understanding of slope than to walk it out and immediately see what kind of graph they are making because of how fast they are moving.

At the end of James Nickels explanation of projectile motion he wrote

"Such is one of the amazing revelations that mathematics gives us about the many nuances of the physics of motion, motion governed by the wisdom of God in Christ (Colossians 1:15-17; 2:1-3). These covenantal laws are treasures that man discovers by investigating the creation order (Proverbs 25:2). Classical physics, founded by men like Galileo, Kepler, and Newton, is a sequence of one fascinating revelation after another. This essay is only one example of these quantitative wonders."

## Monday, August 30, 2010

## Saturday, August 28, 2010

### Why I Nspire? Part 2 - software

I wonder how many of my students' parents read Good House Keeping. It was neat to see the positive review from the experts at the Good Housekeeping Research Institute lab for the kind of technology we are using in my classroom. Also of interest, that I've been meaning to link to, is a news release about the TI-Nspire Touchpad that came out about the time I was in San Diego for the large annual math teacher conference NCTM. Finally, here are answers to the top 10 questions parents ask about graphing calculators.

So back to the topic of ...

Why do I use the TI-Nspire CAS Touchpad in my classroom? Why would a parent want to get the TI-Nspire or TI-Nspire CAS for their child?

2. the SOFTWARE - When you buy a typical graphing calculator, that is all you get - a handheld device that is nice and durable (I wouldn't recommend subjecting it to extreme frigid temperatures). The batteries last a lot longer than a laptop. But if you get the new TI-Nspire, you also get the student software included. A few months ago you would have to buy it separately, retail price close to $100, and I suppose it is still available. But NOW it comes with the software. As TI's website says, it is a 3-for-1.

There are many schools now that are using laptops. I've heard of schools were the underclassmen are getting tablet PCs. The Nspire Student Software would be a perfect fit for this situation because they can use the emulator-like software and then when it comes time to take tests like the AP, PSAT, SAT, etc, then they are familiar and comfortable with the handheld Nspire that works just like the software.

More ways to distribute files

There are a lot of great activities that are ready to use that are available on sites like ti-mathnspired.com and timath.com. Teachers today also have ways to post them to secure school websites. Our school uses edline. I've also heard of schools that use RenWeb and Angel. When students check to see what the assignment is they can also click the tns (Nspire software file) that I have posted. If they are using one of the computers they installed the software on then it will open right up. In addition to the password protected website, our school also has a website where I have posted activities.

Collecting files

These files that students do can be collected or turned in for a grade via email or by sending the files to their handheld. No longer does a student need another piece of software (like TI-Connect for the TI-83/84 or TI-Computer Link for the Nspire). The TI-Nspire Student Software or Teacher Software makes the process of transferring files to a handheld connected to their computer as easy and drag and drop. Files sent to their handheld can be collected from students with TI-Connect-to-Class or with the TI-Nspire Navigator.

Pedagogical benefits

It has been my experience that students with the software explore mathematics more. This is huge! It is the dream of teachers that students would engage with the subject on their own. This additional opportunity to use mathematical technology empowers students to wonder 'what if' and then actually investigate it. I think this is particularly true for the TI-Nspire CAS (computer algebra system - for a comparison to the numeric Nspire check out this link).

It is easier for students to navigator the menu options and 'mouse around' with their computer. With the latest TI-Nspire 2.1 operating system larger files like those found on Math Nspired open so much faster and response with the Touchpad has improved as well, but when you use a computer to explore math and science topics with TI-Nspire technology then everything is extremely fluid and quick to respond.

I also think there are a number of pedagogical benefits of color. Students who use the software enjoy mathematics in color.

Sure there are reasons to use the handheld:

(i) It is useful on high stakes tests where a computer isn't allowed - you wouldn't want students looking things up on the internet or 'phone(text)-a-friend' to find an answer.

(ii) The handheld is useful for easily grading and keeping track of formative assessment when the teacher uses Navigator.

(iii) The handhelds are great for portable data collection with Vernier probes. You can go outside in the snow and use temperature probes or walk around in the hallway to learn about slope with the CBR2. It is nice not to be constrained to a computer.

Another feature of the software that students are likely to use include screen shots for lab reports. Stat

students will enjoy copy pasting data and easily porting it to their handheld. The software enables students to maximize the benefits of multiple representations of split screens. Currently I will not likely often use the computer view on the software for presentations in class or at conferences, but you get a lot more real estate with the smaller font of the software's computer view.

I am excited about so many improved features of the software that came about with the 2.0 OS.

The release notes do a nice job at explaining what new.

So back to the topic of ...

Why do I use the TI-Nspire CAS Touchpad in my classroom? Why would a parent want to get the TI-Nspire or TI-Nspire CAS for their child?

2. the SOFTWARE - When you buy a typical graphing calculator, that is all you get - a handheld device that is nice and durable (I wouldn't recommend subjecting it to extreme frigid temperatures). The batteries last a lot longer than a laptop. But if you get the new TI-Nspire, you also get the student software included. A few months ago you would have to buy it separately, retail price close to $100, and I suppose it is still available. But NOW it comes with the software. As TI's website says, it is a 3-for-1.

There are many schools now that are using laptops. I've heard of schools were the underclassmen are getting tablet PCs. The Nspire Student Software would be a perfect fit for this situation because they can use the emulator-like software and then when it comes time to take tests like the AP, PSAT, SAT, etc, then they are familiar and comfortable with the handheld Nspire that works just like the software.

More ways to distribute files

There are a lot of great activities that are ready to use that are available on sites like ti-mathnspired.com and timath.com. Teachers today also have ways to post them to secure school websites. Our school uses edline. I've also heard of schools that use RenWeb and Angel. When students check to see what the assignment is they can also click the tns (Nspire software file) that I have posted. If they are using one of the computers they installed the software on then it will open right up. In addition to the password protected website, our school also has a website where I have posted activities.

Collecting files

These files that students do can be collected or turned in for a grade via email or by sending the files to their handheld. No longer does a student need another piece of software (like TI-Connect for the TI-83/84 or TI-Computer Link for the Nspire). The TI-Nspire Student Software or Teacher Software makes the process of transferring files to a handheld connected to their computer as easy and drag and drop. Files sent to their handheld can be collected from students with TI-Connect-to-Class or with the TI-Nspire Navigator.

Pedagogical benefits

It has been my experience that students with the software explore mathematics more. This is huge! It is the dream of teachers that students would engage with the subject on their own. This additional opportunity to use mathematical technology empowers students to wonder 'what if' and then actually investigate it. I think this is particularly true for the TI-Nspire CAS (computer algebra system - for a comparison to the numeric Nspire check out this link).

It is easier for students to navigator the menu options and 'mouse around' with their computer. With the latest TI-Nspire 2.1 operating system larger files like those found on Math Nspired open so much faster and response with the Touchpad has improved as well, but when you use a computer to explore math and science topics with TI-Nspire technology then everything is extremely fluid and quick to respond.

I also think there are a number of pedagogical benefits of color. Students who use the software enjoy mathematics in color.

Sure there are reasons to use the handheld:

(i) It is useful on high stakes tests where a computer isn't allowed - you wouldn't want students looking things up on the internet or 'phone(text)-a-friend' to find an answer.

(ii) The handheld is useful for easily grading and keeping track of formative assessment when the teacher uses Navigator.

(iii) The handhelds are great for portable data collection with Vernier probes. You can go outside in the snow and use temperature probes or walk around in the hallway to learn about slope with the CBR2. It is nice not to be constrained to a computer.

Another feature of the software that students are likely to use include screen shots for lab reports. Stat

students will enjoy copy pasting data and easily porting it to their handheld. The software enables students to maximize the benefits of multiple representations of split screens. Currently I will not likely often use the computer view on the software for presentations in class or at conferences, but you get a lot more real estate with the smaller font of the software's computer view.

I am excited about so many improved features of the software that came about with the 2.0 OS.

The release notes do a nice job at explaining what new.

## Monday, August 2, 2010

### Why I Nspire? Part 1

TI-Nspire technology is the most intelligently designed and mathematically enjoyable technology ever developed for the classroom. It has been an important part of my math and science classes for the past 3 years. This will be the first in a series of blog entries that will help to explain why I made the switch to this 21st century technology. So, by request, here is the first installment. Enjoy (even if classroom technology isn't your thing, it may be interesting to see what is out there today ... and why). [Part 2: The software, Part 3: Exam acceptance, Part 4: Math Print, Part 5: The complete summary list]

Why do I use the TI-Nspire CAS Touchpad in my classroom? Why would a parent want to get the TI-Nspire or TI-Nspire CAS for their child? Or even more generally, why should middle school through college teachers and students use graphing calculator technology frequently?

1. RESEARCH shows it to be beneficial (and is not harmful to basic arithmetic skills) to all aspects of mathematical learning.

Independent research firms and major universities have done studies to learn about the effectiveness of graphing calculators on math achievement. Their findings?

-- When teachers incorporate graphing calculators into curriculum more often, and when students have more access to graphing calculators, math achievement increases (1,2).

-- Students who receive instruction using graphing calculators perform as well or significantly better on key math skills(3). Students using graphing calculators during class instruction did as well or better on conceptual, problem solving and operational math skills than those who did not use graphing calculators.

-- Students using graphing calculators during instruction -- but not during assessment -- performed as well or better in all five math skills areas(3), indicating that student math skills did not suffer without calculator use specifically during assessment.

-- Students who use graphing calculators have significantly better attitudes toward math than those who do not use graphing calculators(3).

(1) "Impact of Handheld Graphing Calculator Use on Student Achievement in Beginning Algebra," Heller Research Associates, January 2006

(2) "Effectiveness of Graphing Calculators in K-12 Mathematics Achievement," Empirical Education Incorporated (EEI), November 2005

(3) "A Meta-Analysis of the Effects of Calculators on Students' Achievement and Attitude Levels in Precollege Mathematics Classes," Aimee Ellington, November 2003

Update: But what about specific research about the TI-Nspire? Since I first wrote this blog entry more research has been done. You can now find specific experimental research, case studies, qualitative research, and the research basis for the TI-Nspire. There is also research specifically for CAS (computer algebra system). Click the above links over the words to read more.

The following is a quote from another website [with some comments embedded]: "In fact, a substantial body of educational research now indicates that graphing calculators can have significant benefits for mathematics learning. Research shows that students using graphing calculators develop flexible strategies for problem solving and a deeper appreciation of mathematical meaning than students who do not use graphing calculators (Ellington, 2003; Khoju, Jaciw, and Miller, 2005). In addition, students who use graphing calculators are better able to understand variables and functions

[function notation is promoted by the TI-Nspire better than any other device I know. With the TI-81/82/83/84 you would graph y1, y2, but with the Nspire you graph or define f1(x), or even g(x), etc.],

solve algebra problems in applied contexts [There are a large number of these types of problems available on TI-math.com and coming soon there are some great NASA questions and activities with the TI-Nspire],

interpret graphs, and move among varied representations [multiple representation is the hallmark of the TI-Nspire] —that is from graphs to tables to equations—than students who do not have access to the technology....

Research from the National Assessment of Education Progress (NAEP) has consistently shown that at the eighth grade level frequent use of calculators is associated with greater mathematics achievement. Moreover, research shows that teachers and students who used graphing calculators most frequently learned the most. On the NAEP assessment, 8th graders whose teachers reported that calculators were used almost every day scored the highest (NCES, 2001). Similarly, in examining an implementation that aligned graphing technology with a comprehensive math curriculum, Heller found that daily use of graphing calculators is more effective than infrequent use (Heller, 2005).

[My 8th grade daughter has been enjoying her TI-Nspire CAS for some time now. It is true that our state end-of-course assessment for algebra doesn't permit the use of CAS, but the TI-Nspire is permitted. The TI-Nspire CAS is allowed on the PSAT, SAT, and AP exams. With the wealth of Math Nspired activities available (and more coming soon) that tackle the tough-to-teach and tough-to-learn topics, I'm certain this research will be even more true today. These activities are easy to use by teachers who do not have just about any experience with the Nspire.]Research shows that the association between frequent graphing calculator use and high achievement holds true for a wide variety of grade levels, socio-economic backgrounds, geographic locations, and mathematical topics (National Center for Education Statistics, 2001; Ellington, 2003)."

Added to all of these general things about graphing calculators, consider also what the TI-Nspire Navigator brings to education. Many schools are considering 'clickers for the classroom' or student response systems. I'm sure there is a lot of research that has been done to show the benefits of the enhanced engagement of students gained with clickers. These clicker systems are typically expensive and limited in their use. However, I've found that for a comparable price, the TI-Nspire Navigator and TI-Nspire handheld are more robust than any clicker system. This Navigator system is versatile at improving documented formative assessment through quick polls were students can enter mathematics in the way that it naturally looks, automatic screen captures, and sending and receiving files. It has enabled me to get feedback from a quiet class and get 100% involvement in the class where one student would often blurt an answer before others had time to think.

References:

Dick, Thomas P. (1996). Much More than a Toy. Graphing Calculators in Secondary school Calculus. In P. Gómez and B. Waits (Eds.), Roles of Calculators in the Classroom pp 31-46). Una Empresa Docente.

Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students' achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education. 34(5), 433-463.

Heller, J. L., Curtis, D. A., Jaffe, R., & Verboncoeur, C. J. (2005). Impact of handheld graphing calculator use on student achievement in algebra 1: Heller Research Associates.

Khoju, M., Jaciw, A., & Miller, G. I. (2005). Effectiveness of graphing calculators in K-12 mathematics achievement: A systematic review. Palo Alto, CA: Empirical Education, Inc.

National Center for Education Statistics. (2001). The nation’s report card: Mathematics 2000. (No. NCES 2001-571). Washington DC: U.S. Department of Education.

UPDATE April 14, 2011: Thanks to a recent discussion on the Nspire Google Group, I've been made aware of additional research with findings available online. These sites include a detailed mixed method long term study in Scotland and is available at http://www.abdn.ac.uk/education/documents/UofA_Report_Mar10.pdf . Another pdf files summarizing the finding of teachers in a manner that is approachable to other teachers.

The lead researcher "observed that teachers changed their teaching methods dramatically when using TI-Nspire, both the way they taught each topic and the way they teach in general. They found that the use of the Npires together with a move towards an investigative approach with more opportunity for questioning and discussion led to a deepening in students' understanding, an increased pace of learning and an increase in motivation and engagement across all ability

levels." A brief summary is available here.

Why do I use the TI-Nspire CAS Touchpad in my classroom? Why would a parent want to get the TI-Nspire or TI-Nspire CAS for their child? Or even more generally, why should middle school through college teachers and students use graphing calculator technology frequently?

1. RESEARCH shows it to be beneficial (and is not harmful to basic arithmetic skills) to all aspects of mathematical learning.

Independent research firms and major universities have done studies to learn about the effectiveness of graphing calculators on math achievement. Their findings?

-- When teachers incorporate graphing calculators into curriculum more often, and when students have more access to graphing calculators, math achievement increases (1,2).

-- Students who receive instruction using graphing calculators perform as well or significantly better on key math skills(3). Students using graphing calculators during class instruction did as well or better on conceptual, problem solving and operational math skills than those who did not use graphing calculators.

-- Students using graphing calculators during instruction -- but not during assessment -- performed as well or better in all five math skills areas(3), indicating that student math skills did not suffer without calculator use specifically during assessment.

-- Students who use graphing calculators have significantly better attitudes toward math than those who do not use graphing calculators(3).

(1) "Impact of Handheld Graphing Calculator Use on Student Achievement in Beginning Algebra," Heller Research Associates, January 2006

(2) "Effectiveness of Graphing Calculators in K-12 Mathematics Achievement," Empirical Education Incorporated (EEI), November 2005

(3) "A Meta-Analysis of the Effects of Calculators on Students' Achievement and Attitude Levels in Precollege Mathematics Classes," Aimee Ellington, November 2003

Update: But what about specific research about the TI-Nspire? Since I first wrote this blog entry more research has been done. You can now find specific experimental research, case studies, qualitative research, and the research basis for the TI-Nspire. There is also research specifically for CAS (computer algebra system). Click the above links over the words to read more.

The following is a quote from another website [with some comments embedded]: "In fact, a substantial body of educational research now indicates that graphing calculators can have significant benefits for mathematics learning. Research shows that students using graphing calculators develop flexible strategies for problem solving and a deeper appreciation of mathematical meaning than students who do not use graphing calculators (Ellington, 2003; Khoju, Jaciw, and Miller, 2005). In addition, students who use graphing calculators are better able to understand variables and functions

[function notation is promoted by the TI-Nspire better than any other device I know. With the TI-81/82/83/84 you would graph y1, y2, but with the Nspire you graph or define f1(x), or even g(x), etc.],

solve algebra problems in applied contexts [There are a large number of these types of problems available on TI-math.com and coming soon there are some great NASA questions and activities with the TI-Nspire],

interpret graphs, and move among varied representations [multiple representation is the hallmark of the TI-Nspire] —that is from graphs to tables to equations—than students who do not have access to the technology....

Research from the National Assessment of Education Progress (NAEP) has consistently shown that at the eighth grade level frequent use of calculators is associated with greater mathematics achievement. Moreover, research shows that teachers and students who used graphing calculators most frequently learned the most. On the NAEP assessment, 8th graders whose teachers reported that calculators were used almost every day scored the highest (NCES, 2001). Similarly, in examining an implementation that aligned graphing technology with a comprehensive math curriculum, Heller found that daily use of graphing calculators is more effective than infrequent use (Heller, 2005).

[My 8th grade daughter has been enjoying her TI-Nspire CAS for some time now. It is true that our state end-of-course assessment for algebra doesn't permit the use of CAS, but the TI-Nspire is permitted. The TI-Nspire CAS is allowed on the PSAT, SAT, and AP exams. With the wealth of Math Nspired activities available (and more coming soon) that tackle the tough-to-teach and tough-to-learn topics, I'm certain this research will be even more true today. These activities are easy to use by teachers who do not have just about any experience with the Nspire.]Research shows that the association between frequent graphing calculator use and high achievement holds true for a wide variety of grade levels, socio-economic backgrounds, geographic locations, and mathematical topics (National Center for Education Statistics, 2001; Ellington, 2003)."

Added to all of these general things about graphing calculators, consider also what the TI-Nspire Navigator brings to education. Many schools are considering 'clickers for the classroom' or student response systems. I'm sure there is a lot of research that has been done to show the benefits of the enhanced engagement of students gained with clickers. These clicker systems are typically expensive and limited in their use. However, I've found that for a comparable price, the TI-Nspire Navigator and TI-Nspire handheld are more robust than any clicker system. This Navigator system is versatile at improving documented formative assessment through quick polls were students can enter mathematics in the way that it naturally looks, automatic screen captures, and sending and receiving files. It has enabled me to get feedback from a quiet class and get 100% involvement in the class where one student would often blurt an answer before others had time to think.

References:

Dick, Thomas P. (1996). Much More than a Toy. Graphing Calculators in Secondary school Calculus. In P. Gómez and B. Waits (Eds.), Roles of Calculators in the Classroom pp 31-46). Una Empresa Docente.

Ellington, A. J. (2003). A meta-analysis of the effects of calculators on students' achievement and attitude levels in precollege mathematics classes. Journal for Research in Mathematics Education. 34(5), 433-463.

Heller, J. L., Curtis, D. A., Jaffe, R., & Verboncoeur, C. J. (2005). Impact of handheld graphing calculator use on student achievement in algebra 1: Heller Research Associates.

Khoju, M., Jaciw, A., & Miller, G. I. (2005). Effectiveness of graphing calculators in K-12 mathematics achievement: A systematic review. Palo Alto, CA: Empirical Education, Inc.

National Center for Education Statistics. (2001). The nation’s report card: Mathematics 2000. (No. NCES 2001-571). Washington DC: U.S. Department of Education.

UPDATE April 14, 2011: Thanks to a recent discussion on the Nspire Google Group, I've been made aware of additional research with findings available online. These sites include a detailed mixed method long term study in Scotland and is available at http://www.abdn.ac.uk/

The lead researcher "observed that teachers changed their teaching methods dramatically when using TI-Nspire, both the way they taught each topic and the way they teach in general. They found that the use of the Npires together with a move towards an investigative approach with more opportunity for questioning and discussion led to a deepening in students' understanding, an increased pace of learning and an increase in motivation and engagement across all ability

levels." A brief summary is available here.

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