# Building Functions - Thinking Ahead to Calculus

My ninth graders are working on building functions and modeling in the final unit of the year. There is plenty of good material out there for doing these tasks as a way to master the Common Core standards that describe these skills.

I had a sudden realization that a great source for these types of tasks might be my Calculus materials. Related rates, optimization, and applications of integrals in a Calculus course generally require students to write models of functions and then apply their differentiation or integration knowledge to arrive at a result. The first step in these questions usually involves writing a function, with subsequent question parts requiring Calculus methods to be applied to that function.

I dug into my resources for these topics and found that these questions might be excellent modeling tasks for the ninth grade students if I simply pull out the steps that require Calculus. Today's lesson using these adapted questions was really smooth, and felt good from a vertical planning standpoint.

I could be late to this party. My apologies if you realized this well before I did.

# #Teachers Coding - Bingo Cards

When I attended a Calculus AB workshop back in 2003, one of the nice takeaways was a huge binder of materials that could be used immediately with students. I ended up scanning much of those materials and taking the digital versions with me when I moved overseas.

One of these activities was called derivative bingo. This was a set of two sheets, one with a list of expressions, and another a 5 x 5 bingo card with the derivatives of those expressions. It was perfect to use after introducing derivative rules such as the product and quotient rules to develop proficiency.

It also wasn't as fun of an activity for two reasons. The first was that I only had one bingo card provided as part of the activity. Since everyone had the same card, everyone would really obtain five in a row after doing the same set of problems. Yes, I could have made different ones at some point in the past twelve years to resolve this problem, but I never thought about it with enough advance time to do so. The other reason was that the order of the list of derivatives was carefully designed so that you only obtained five in a row after doing most of the problems provided. Good for the purposes of getting students to do more practice, but definitely an attribute that hacks the entertainment value even more.

As you might expect, I wrote a computer tool to manage this. You can visit this site and see a sample card. Reload the page, and you'll generate a new one.

This turned into a nice little competition between groups of students, and I kept a tally of how many total rows had been matched by each group as they developed. The different cards led to some great conversations between students about their results:

I use the KaTEX rendering library to make the mathematical expressions look good. If you would like to edit the files for use with your own class, you can go to the GitHub repository here and download a zip file with all of the files. You'll find instructions there for changing the code to fit your needs. If those instructions don't make sense to you, let me know.

If you would just like a set of cards for the derivative practice activity that is ready for use with a class, that PDF is here: derivative-bingo-class-files

# Intermediate Value Theorem & Elevators

I've used the elevator analogy with the intermediate value theorem before, but only after talking students through the intermediate value theorem first. This time, I took them through the following thought experiment first:

## Step 1:

You enter the elevator on floor 2. You close your eyes and keep them closed until you arrive at floor 12, twenty seconds later.

Questions for discussion:

• At approximately what time was the elevator located at floor 7? How do you know? What assumptions are you making?
• Was there a time when the elevator was at floor 3? Floor 8? How do you know?
• Were you ever at floor 13? How do you know? Are you really sure?

## Step 2:

Another day, you again enter the elevator on floor 2. You again keep your eyes closed, but another person gets on from some floor other than floor 2. You keep your eyes closed. The other person leaves the elevator at some point. After 60 seconds, you are on floor 12, and you open your eyes.

Questions:

• Was there a time at which the elevator was at floor 7? How do you know?
• Was there a time at which the elevator was at floor 13? How do you know?
• What was the highest floor at which you can guarantee the elevator was located during the minute long trip? The lowest floor?

## Step 3

On yet another day, you are once again entering the elevator at floor 2 to go to floor 12. You close your eyes, same story as before. Another person gets on the elevator and leaves. This time, however, you open your eyes just long enough to see that the person leaves the elevator at floor 15. As before, the entire trip takes 60 seconds.

Questions:

• Was there a time at which the elevator was at floor 7?
• Was there a time at which the elevator was at floor 13? How do you know?
• Make a list of all of the floors that you can guarantee that the elevator could have stopped at during the 60 second trip.
• Can you guarantee that the elevator was never located at floor 17?

We then visited the driving principle to why we can do this thought experiment: why can we come to these conclusions without opening our eyes in the elevator? What is it about our experiences in elevators that makes this possible?

My students were primed to bring up continuity given that they worked through the concepts during the previous class. That said, there were quite a few lights that went on when I asked what it would be like to ride in a discontinuous elevator. Skipping floors, feeling the elevator move upwards and then arriving at a floor lower than where we started, or arriving at different floors just from closing or opening the doors.

Once we were comfortable with this, I threw the standard vocabulary of the intermediate value theorem:

Suppose f(x) has a maximum value M and a minimum value L over an interval [a,b]. There exists a value c in [a,b] such that L≤f(c)≤M as long as...

...and I left it there, hanging in the air until a student filled the silence with the condition of continuity over [a,b]. This was also a great time to introduce the idea of an existence theorem - it tells you that a mathematical object exists, and might give you some information on where to find it, but won't definitively tell you exactly where it is located. Fun stuff.

We then talked about other examples of functions that are or are not continuous. Students brought up crashing into a wall after moving at a non-zero velocity. I also have this group of students the following period for physics, so I brought up what the velocity versus time graph actually looks at when you zoom in to the time of impact. (I like that this wasn't a cognitive stretch for them given their experience zooming in on data on their calculators and graphs from Logger Pro.) The student that brought this up quickly argued himself back from saying that this was truly discontinuous.

This was a fun activity, and I'm glad I went through it. The concept of IVT is fairly intuitive, but we often present it in a way that doesn't emphasize why it is special. In previous years, I started with the graph of a polynomial function bouncing up and down, asked students for the max/minimum value, and then asked them to identify whether they could do this for any value in the range between the maximum and minimum. They could, but never really saw the point of why that was special. Forcing them to imagine closing their eyes, limiting the information available to them, and then seeing how far they could take that limited knowledge made a difference in how this felt on the teaching end. I've seen some pretty good responses on my assessments of this concept as well, so it seems to have done some good for the students as well. (Phew!)

# 2012-2013 Year In Review – Learning Standards

This is the second post reflecting on this past year and I what I did with my students.

My first post is located here. I wrote about this year being the first time I went with standards based grading. One of the most important aspects of this process was creating the learning standards that focused the work of each unit.

### What did I do?

I set out to create learning standards for each unit of my courses: Geometry, Advanced Algebra (not my title - this was an Algebra 2 sans trig), Calculus, and Physics. While I wanted to be able to do this for the entire semester at the beginning of the semester, I ended up doing it unit by unit due to time constraints. The content of my courses didn't change relative to what I had done in previous years though, so it was more of a matter of deciding what themes existed in the content that could be distilled into standards. This involved some combination of concepts into one to prevent the situation of having too many. In some ways, this was a neat exercise to see that two separate concepts really weren't that different. For example, seeing absolute value equations and inequalities as the same standard led to both a presentation and an assessment process that emphasized the common application of the absolute value definition to both situations.

### What worked:

• The most powerful payoff in creating the standards came at the end of the semester. Students were used to referring to the standards and knew that they were the first place to look for what they needed to study. Students would often ask for a review sheet for the entire semester. Having the standards document available made it easy to ask the students to find problems relating to each standard. This enabled them to then make their own review sheet and ask directed questions related to the standards they did not understand.
• The standards focus on what students should be able to do. I tried to keep this focus so that students could simultaneously recognize the connection between the content (definitions, theorems, problem types) and what I would ask them to do with that content. My courses don't involve much recall of facts and instead focus on applying concepts in a number of different situations. The standards helped me show that I valued this application.
• Writing problems and assessing students was always in the context of the standards. I could give big picture, open-ended problems that required a bit more synthesis on the part of students than before. I could require that students write, read, and look up information needed for a problem and be creative in their presentation as they felt was appropriate. My focus was on seeing how well their work presented and demonstrated proficiency on these standards. They got experience and got feedback on their work (misspelling words in student videos was one) but my focus was on their understanding.
• The number standards per unit was limited to 4-6 each...eventually. I quickly realized that 7 was on the edge of being too many, but had trouble cutting them down in some cases. In particular, I had trouble doing this with the differentiation unit in Calculus. To make it so that the unit wasn't any more important than the others, each standard for that unit was weighted 80%, a fact that turned out not to be very important to students.

### What needs work:

• The vocabulary of the standards needs to be more precise and clearly communicated. I tried (and didn't always succeed) to make it possible for a student to read a standard and understand what they had to be able to do. I realize now, looking back over them all, that I use certain words over and over again but have never specifically said what it means. What does it mean to 'apply' a concept? What about 'relate' a definition? These explanations don't need to be in the standards themselves, but it is important that they be somewhere and be explained in some way so students can better understand them.
• Example problems and references for each standard would be helpful in communicating their content. I wrote about this in my last post. Students generally understood the standards, but wanted specific problems that they were sure related to a particular standard.
• Some of the specific content needs to be adjusted. This was my first year being much more deliberate in following the Modeling Physics curriculum. I haven't, unfortunately, been able to attend a training workshop that would probably help me understand how to implement the curriculum more effectively. The unbalanced force unit was crammed in at the end of the first semester and worked through in a fairly superficial way. Not good, Weinberg.
• Standards for non-content related skills need to be worked in to the scheme. I wanted to have some standards for year or semester long skills standards. For example, unit 5 in Geometry included a standard (not listed in my document below) on creating a presenting a multimedia proof. This was to provide students opportunities to learn to create a video in which they clearly communicate the steps and content of a geometric proof. They could create their video, submit it to me, and get feedback to make it better over time. I also would love to include some programming or computational thinking standards as well that students can work on long term. These standards need to be communicated and cultivated over a long period of time. They will otherwise be just like the others in terms of the rush at the end of the semester. I'll think about these this summer.

You can see my standards in this Google document:
2012-2013 - Learning Standards

I'd love to hear your comments on these standards or on the post - comment away please!

# Volumes of Revolution - Using This Stuff.

As an activity before our spring break, the Calculus class put its knowledge of finding volumes of revolution to, well, find volumes of things. It was easy to find different containers to use for this - a sample:

We used Geogebra to place points and model the profile of the containers using polynomials. There were many rich discussions about wise placement of points and which polynomials make more sense to use. One involved the subtle differences between these two profiles and what they meant for the resulting volume through calculus methods:

The task was to predict the volume and then use flasks and graduated cylinders to accurately measure the volume. Lowest error wins. I was happy though that by the end, nobody really cared about 'winning'. They were motivated themselves to theorize why their calculated answer was above or below, and then adjust their model to test their theories and see how their answer changes.

As usual, I have editorial reflections:

• If I had students calculating the volume by hand by integration every time, they would have been much more reluctant to adjust their answers and figure out why the discrepancies existed. Integration within Geogebra was key to this being successful. Technology greases the rails of mathematical experimentation in a way that nothing else does.
• There were a few many lessons that needed to happen along the way as the students worked. They figured out that the images had to be scaled to match the dimensions in Geogebra to the actual dimensions of the object. They figured out that measurements were necessary to make this work. The task demanded that the mathematical tools be developed, so I showed them what they needed to do as needed. It would have been a lot more boring and algorithmic if I had done all of the presentation work up front, and then they just followed steps.
• There were many opportunities for reinforcing the fundamentals of the Calculus concepts through the activity. This is a tangible example of application - the actual volume is either close to the calculated volume or not - there's a great deal more meaning built up here that solidifies the abstraction of volume of revolution. There were several 'aha' moments and I saw them happen. That felt great.

I feel sorry for the way spreadsheets are used most of the time in school. They are usually used as nothing more than a waypoint on the way to a chart or graph, inevitably with one of its data sets labeled 'Series 1'. The most powerful uses of spreadsheets come from how they provide ways to organize and calculate easily.

I've observed a couple things about the problem solving process among students in both math and science.

• Physics students see the step of writing out all of the information as an arbitrary requirement of physics teachers, not necessarily as part of the solution process. As a result, it is often one of the first steps to disappear.
• In math, students solving non-routine problems like Three Act problems often have calculations scrawled all over the place. Even they are written in an organized way, in the event that a calculation is made incorrectly, any sets of calculations that are made must be made again. This can be infuriating to students that might be marginally interested in finding an answer in the first place.
• Showing calculations in a hand written document is easy - doing so in a document that is to be shared electronically is more difficult. There are also different times when you want to see how the calculation was made, and other times that you want to see the results. These are often presented in different parts of a report (body vs. appendix) but in a digital document, this isn't entirely necessary.

Here's my model for how a spreadsheet can address some of these issues:

Why I like it:

• The student puts all of the given information at the top. This information may be important or used for subsequent calculations, or not. It minimally has all of the information used to solve a problem in one place.
• The coloring scheme makes clear what is given and what is being being calculated.
• The units column is a constant reminder that numbers usually have units. In my template, this column is left justified so that the units appear immediately to the right of the numerical column.
• Many students aren't comfortable exploring a concept algebraically. By making calculations that might be useful easy to make and well organized, this sets students up for a more playful approach to figuring things out.
• Showing work is easy in a spreadsheet - look at the formulas. Depending on your own expectations, you might ask for more or less detail in the description column.

Some caveats:

• A hand calculation should be done by someone to confirm the numbers generated by the spreadsheet are what they should be. This could be a set of test data provided by the teacher, or part of the initial exploration of a concept. Confirming that a calculation is being done correctly is an important step of trusting (but verifying, to quote Reagan for some reason) the computer to make the calculations so that attention can be focused on figuring out what the numbers mean.
• It does take a bit of time to teach how to enter a formula into a spreadsheet. Don't turn it into a lecture about absolute or relative addressing, or about rows and columns and which is which - this will come with practice. Show how numbers in scientific notation look, and demonstrate how to get a value placed in another cell. Get straight into making calculations happen among your students and in a way that is immediately relevant to what you are trying to do. Then change a given value, and watch the students nod when all of the values in the sheet change immediately.
• Building off of what I just said, don't jump to a spreadsheet for a situation just to do it. The structure and order should justify itself. Big numbers, nasty numbers, lots of calculations, or lots of given information to keep track of are the minimum for establishing this from the start as a tool to help do other things, not an end in and of itself.
• Do not NOT

## NOT

hand your students a spreadsheet that calculates everything for them. If a student wants to make a spreadsheet for a particular type of calculation, that's great. That's the student recognizing that such a tool would be useful, and making the effort to do this. If you hand them a calculator for one specific application, it perpetuates the idea among students that they have to wait for someone else that knows better than them to give them the tool to use. Students should have the ability to make their own utilities, and this is one way to do it.

### Example from class yesterday:

We are exploring the way Newton's Law of Gravitation is used. I asked students to calculate the force of gravity from different planets in the solar system pulling on a 65 kilogram person on Earth, with Wolfram Alpha as the source of data. Each of them used a scientific or graphing calculator to calculate their numbers, with the numbers they used written by hand (without units) on their papers with minimal consistency. They grumbled about the sizes of the numbers. When noticeable differences arose in magnitude between different students, they checked each other until they were satisfied.

I then showed them how to take the pieces of data they found and put them in the spreadsheet in the way I described above. In red, I highlighted the calculation for the magnitude of the force for an object on Earth, and then asked a student to give me her data. This was the value she calculated! I was quickly able to confirm the values that the other students also had made.

I then had them calculate the weight of an object on Earth's surface using Newton's law of gravitation. This sent them again on a search for data on Earth's vital statistics. They were surprised to see that this value was really close to the accepted value for g = $9.8;m/s^2$. I then asked them in their spreadsheet how they might figure out the acceleration due to gravity based on what they already knew. Most were able to figure out without prompting that dividing by the 65 kilogram mass got them there. I then had them use that idea and Newton's Law of Gravitation to figure out how to obtain the acceleration due to gravity at a given distance from the mass center of a planet. I then had them use the spreadsheet model on their own to calculate the acceleration due to gravity on a couple of different planets, and it went really well.

The focus from that point on was on figuring out what those numbers meant relative to Earth. Often with these types of problems, students will calculate and be done with it. These left them a bit curious about each other's answers (gravity on Jupiter compared to the Moon) and opened up the possibilities for subsequent lessons. I'll write more about how I have grown to view spreadsheets as indispensable computing tools in the classroom in the future. A pure computational tool is the lowest level on the totem pole of applications of computers for learning mathematics or science, but it's a great entry point for students to see what can be done with it.

Files:

Centripetal Acceleration of the Moon - a comparison we used two days ago to suggest how a 1/r^2 relationship might exist for gravity and the moon.

# Differentiation Rules - Making it Interactive

I always struggle during the days spent going over differentiation rules. The mathematician in me says the students need to see where the rules come from so that they aren't just a recipe. On the other hand, I see students glazing over a bit with notation and getting lost in the midst of the overall goal: how do we find shortcuts for finding derivative functions outside of using the limit definition every time?

I have also tried going through the derivations in class and having them just watch and see the progression on their own, without copying things down. Some compulsively copied despite my repeated requests not to do so - I think it was a situation of seeing copying notes down as an alternative to really digging in to what was actually going on. It's mindless to copy down notes, a great alternative to actually going through the steps of understanding.

Last year I made videos of the derivations and asked students to watch them outside of class in a one-off attempt at flipping. That didn't work - students said they watched but 'didn't get it', so my attempt to quiz them when they arrived in class was a bust.

This is my compromise this year: for finding the derivative of a constant, a constant times a function, and the power rule, students will be guided through what has essentially my lesson plan for previous lessons. Sums of functions, products, and quotients will be given first as applications of the limit rules, but the details of getting from the start to the finish will be kept as an exercise for later.

See my handout for today here:
03 - CW - Differentiation Rules

Thank you to Patrick Honner and Dan Anderson for their comments pushing me on this.

# Why SBG is blowing my mind right now.

I am buzzing right now about my decision to move to Standards Based Grading for this year. The first unit of Calculus was spent doing a quick review of linear functions and characteristics of other functions, and then explored the ideas of limits, instantaneous rate of change, and the area under curves - some of the big ideas in Calculus. One of my standards reads "I can find the limit of a function in indeterminate form at a point using graphical or numerical methods."

A student had been marked proficient on BlueHarvest on four out of the five, but the limit one held her back. After some conversations in class and a couple assessments on the idea, she still hadn't really shown that she understood the process of figuring out a limit this way. She had shown that she understood that the function was undefined on the quiz, but wasn't sure how to go about finding the value.

We have since moved on in class to evaluating limits algebraically using limit rules, and something must have clicked. This is what she sent me this morning:

Getting things like this that have a clear explanation of ideas (on top of production value) is amazing - it's the students choosing a way to demonstrate that they understand something! I love it - I have given students opportunities to show me that they understand things in the past through quiz retakes and one-on-one interviews about concepts, but it never quite took off until this year when their grade is actually assessed through standards, not Quiz 1, Exam 1.

I can determine the perimeter and area of complex figures made up of rectangles/ triangles/ circles/ and sections of circles.

...followed by an explanation of how to find the area of the figure. Where did she get this problem? She made it up.

I am in the process right now of grading unit exams that students took earlier in the week, and found that the philosophy of these exams under SBG has changed substantially. I no longer have to worry about putting on a problem that is difficult and penalizing students for not making progress on it - as long as the problem assesses the standards in some way, any other work or insight I get into their understanding in what they try is a bonus. I don't have to worry about partial credit - I can give students feedback in words and comments, not points.

One last anecdote - a student had pretty much shown me she was proficient on all of the Algebra 2 standards, and we had a pretty extensive conversation through BlueHarvest discussing the details and her demonstrating her algebraic skills. I was waiting until the exam to mark her proficient since I wanted to see how student performance on the exam was different from performance beforehand. I called time on the exam, and she started tearing up.

I told her this exam wasn't worth the tears - she wanted to do well, and was worried that she hadn't shown what she was capable of doing. I told her this was just another opportunity to show me that she was proficient - a longer opportunity than others - but another one nonetheless. If she messed up a concept on the test from stress, she could demonstrate it again later. She calmed down and left with a smile on her face.

Oh, and I should add that her test is looking fantastic.

I still have students that are struggling. I still have students that haven't gone above and beyond to demonstrate proficiency, and that I have to bug in order to figure out what they know. The fact that SBG has allowed some students to really shine and use their talents, relaxed others in the face of assessment anxiety, and has kept other things constant, convinces me that this is a really good thing, well worth the investment of time. I know I'm just preaching to the SBG crowd as I say this, but it feels good to see the payback coming so quickly after the beginning of the year.

# A sample of my direct instruction videos.

As I have previously mentioned, I am really excited to be creating Udacity style videos as resources for students in my classes. With my VideoPress upgrade in effect, I have included some of the two minute videos I put together for Calculus class tomorrow introducing limits. We have already spent some time exploring the concept of limits by graphing and evaluating numerically, but these videos are the start of a more formal treatment of evaluating limits algebraically.

I am very interested in feedback, so let me know what you think in the comments.

# Winning the battle over Python programming

Two stories to share after this week's activities with students about programming. I have posted previously about my interest in making Python a fundamental part of my classes this year, and so I am finding ways to include it when it makes sense to do so.

I have a couple of students that are bridging the gap between Algebra 2 and Precalculus with an independent study that I get to design. The tentative title of the course for their transcript is 'Fundamentals of Mathematical Thinking' and the overall goal is to get these students a chance to develop their fundamental skills to be successful in later classes. I see it as an opportunity to really dig in to some cool mathematical ideas and get them to, well, dig into the fundamentals of mathematical thinking. I don't plan too much emphasis on the algorithms (though we will spend some time working on skills in algebra, polynomial manipulation, functions, and other crucial topics where they are weak). Looking at a situation, exploring the way different variables might be used to model that situation, and then really digging in to abstract the variables into a model.

We are starting with what I think is the most fundamental application of this: sequences and series. Even simpler, the first task I gave the students was to look at the number of bricks in the rows of a triangular tower and use Python to add up the bricks in each row. This started as a couple of exercises getting to know Python's syntax. They are then taking programs I wrote to model this problem and adjusting them to find other sums, including the sum of even and odd numbers. One student that completed this task was intrigued that the sum of the latter consisted of perfect squares, but we didn't explore it any further at this point.

I then gave this student a bunch of sequences. His task was simple: model each one in Python and generate the given terms. This is a standard exercise for Algebra & Precalc students by hand, but I figured that if he could do this with Python, clearly he was able to figure out the pattern. I showed him how to write fractions using string concatenation (e.g. 1/3 = 1 + "/" + 3) which enabled him to develop the harmonic series. Today he figured out Fibonacci and a couple other new ones. It was really fascinating to see him mess around think deeply about the patterns associated with each one. I did tap him slightly in the right direction with Fibonacci, but I have otherwise been hands off. I am also having him write about his work to give him opportunities to work on his writing too. When he feels comfortable sharing it (and I have already warned him that this is the plan), I will post links to his work here.

The other new thing was in Calculus. I have shortened my review of Pre-Calculus concepts substantially, and have made the first unit a survey of limits, rate of change, and definite integrals. Most of this has required technology to explore local linearity and difference quotients. On Thursday, I introduced using rectangular sums to find area - they were otherwise stuck on counting boxes, and I could tell they felt it was like baby math. They really didn't know any other way.

In showing them rectangular sums, we had some pretty good discussions about overestimating and underestimating. The students had conversations about how rough the approximation with only 3 - 5 rectangles gave for area under a parabola. A couple of them figured out how to use more rectangles. I told them I was going to write a program to do this while they were sitting and working. I created this program and talked them through how it works. They thought it was too complicated to be worth the time, but I think they did understand the basic idea. I then changed the value of N and asked them what they thought that meant. They got it right the first time. I then pushed the value to higher and higher values of N and they immediately saw that it was approaching a limit. Game, set, match.

Today I had the AP students together working on another definite integral activity that focused on the trapezoidal rule. I showed them the code again and gave them the line that calculates area. It wasn't too much of a stretch for them to work their way to adjusting the program to work for the Trapezoidal rule. We ran out of time to discuss comparisons between the two programs, but they stayed late after class and into their lunch getting it working on their own computers and playing a bit. Here is what we came up with.

The big battle I see is two-fold.

• Help students not be intimidated by the idea of writing a program to do repetitive calculations.
• Give students opportunities to see it as necessary and productive to use a computer to solve a problem.

Sometimes these battles are the same, other times they are different. By using the built-in version of Python on their Macs, I have already started seeing them run commands and use text editors to create scripts without too much trouble. That's the first battle. My plan is to give lots of examples supporting the second one in the beginning, and slowly push the burden of writing these programs on to the students as time goes by and they become more comfortable with the idea. So far I am feeling pretty good about it - stay tuned.