teaching stories

Volumes of Revolution & 3D Modeling

I had a conversation with a colleague a few years ago about volumes of revolution in Calculus. We were both a few years removed from our own Calculus courses in high school and college, and we were talking about how we thought about the concept visually.

For those that need a refresher, here is the idea behind a volume of revolution. Imagine you have a solid object that can be lined up with the x-axis so that its cross section looks like the image below. The object would have a pointy end at the origin (0,0) and a circular face located at x = 1. The closest real world object that fits this description is a Hershey’s Kiss.

The object is axially symmetric about the x-axis. If you were to cut the object with a knife so that the cut passes through the pointy end and the center of the flat face, the image at left would always be the cross section.

A volume of revolution is usually defined by an even simpler idea. Take a region of a graph and rotate it in a circle around some axis. The region at left is defined by rotating the area under the graph of y = x ^{2 around the x-axis.}

My colleague’s way of visualizing this idea started with the solid itself. Cut it into a series of discs, each of width dx , and then analyze a single differential disc to come up with an integral expression for the entire volume. This requires being able to visualize the entire solid first, and then see how it can be cut into discs.

I didn’t see it this way. I could visualize the solid usually, but to then mentally cut the solid into discs, and then construct a differential volume seemed to have one too many steps to make it simple. I focused on the step that made conceptual sense to me: start with a defined region and rotate it around an axis to create a solid. The differential strip of area we had been making underneath the graph since the first introduction of the definite integral was what I always visualized during integration. I could visualize taking that strip and rotating it around to form a disc, and using that concept for the differential volume. Then add up these discs through an integral to find the volume.

When I taught volume of revolution for the first time, I wanted to introduce it in a way that would emphasize how I had come to understand the concept. Granted, this assumes my way will work for the students, but so far it seems to be doing so pretty well.

Three dimensional computer modeling programs (Blender, Pro-E, Autodesk Inventor, etc) all have a function called ‘Revolve’ which is, by definition, how volumes of revolution are created. The idea is that you define a region, pick an axis, and then the software will create a 3D solid and display it. Having a copy of Pro-E from our FIRST Tech Challenge team, I was able to introduce the process with a series of demonstrations live with the software. Some examples:

The students immediately saw what was going on, and didn’t think much of the process. I could quickly make a sketch, revolve it, and then rotate the object around for students to see what it would look like if actually in front of them. We then proceeded to revolve strips under and between graphs to generate discs and washers. Writing the integrals was then a fairly simple process.

I think the difficulty that might come up with this type of problem is the visualizing step. Students must visualize the 3D shape in order to solve problems related to its volume. I think having this sort of tool available has made a big difference in my students seeing what it means to create a volume of revolution, which then leads to an easier time conceptualizing how to then find its volume using Calculus.

Party games & geometry definitions

Today’s geometry class started with a new random arrangement of student seats. It never fails to amaze me how the dynamics of the whole room change with a shuffle of student locations.

The lesson today was the first of our quadrilateral unit. Normally after tests, I don’t tend to have homework assignments, but I decided to make an exception with a simple assignment:

Create a single Geogebra file in which you construct and label all of the quadrilaterals given in the textbook: parallelogram, rhombus, square, kite, rectangle, trapezoid, and isosceles trapezoid.

This appealed to me because I really dislike lessons in which we go through definitions slowly as a group. I also knew that giving the students some independence in reviewing or learning the definitions of these quadrilaterals was a good thing. Sometimes they are a bit to reliant on me to give them all the information they need. For this assignment, students would need to understand the definitions of quadrilaterals in order to construct them, and that was a good enough for walking into class today.

The warm-up activity involved looking at unlabeled diagrams of quadrilaterals, naming them, and writing any characteristics they noticed about them from the diagrams:

Some had trouble with the term ‘characteristics’, but a peek down at the chart just below on the paper helped them figure it out:

Based on what they knew from the definitions before class, I had them complete this chart while talking to their new partner. There was lots of good conversation and careful use of language for each listed characteristic.

This led to the next thing that often serves as an important (though often boring) exercise: new vocabulary. I used one of my favorite activities that gets students focused on little details – each student received one of the following four charts. The chart is originally from p. 380 of the AMSCO Geometry textbook, and was digitally ruined using GIMP.

The students had a good time filling in the missing information and conferring with each other to make sure they had it all. We then came up with some examples of consecutive vertices, angles, diagonals, and opposite sides.

From their work with the chart and using the new vocabulary whenever possible, we then did the following:

What information would you need in order to prove that a quadrilateral is… (use as much of the new vocabulary as possible!)

a square?

a rhombus?

a parallelogram?

a rectangle?

a trapezoid? (an isosceles trapezoid?)

a kite?

I was really pleased with how they did with this exercise – they really seemed to be interacting with the definitions and vocabulary well.

Finally, we arrived at the part that was the most fun. You know that annoying ice-breaker you sometimes are forced to do at professional development sessions where you wear something on your head and have to get the other attendees to tell you who you are?

I hate that activity. That usually means it’s perfect for my students:

Here are the quadrilaterals:
Quadrilaterals – Who-Am-I activity

The students were all smiles during the ten minutes or so we spent going through it – yes, I had one too! They were using the vocabulary we had developed during the day and were pretty creative in getting each other to guess the dog names as well.

In the end, I feel pretty good about how today’s set of activities went. The engagement level was pretty high and everyone did a good job of interacting with the definitions in a way that will hopefully lead to understanding as we start proving their properties in coming classes.

Turning random facts into logistics curves – ODE per day series continued.

I previously wrote about making sure that every class during our unit on differential equations starts with some differential equation they can see or feel in a concrete way.

During the last class, we investigated a draining tank using the video posted by Dan Meyer at his blog.

Today we did something different. I told them that I was doing an experiment with a simple task. They all needed to find the answers to some questions as quickly as possible:

When they found the answers, I wanted them to quickly throw a hand in the air to let me know. I told them to be honest – they didn’t know what I was doing with the information yet, so there really wasn’t a chance to skew it.

I then showed them the slide with the questions:

I also simultaneously started the following Python program. (UPDATE: Code is posted here.) This let me easily record any time a student raised his/her hand.

I then pasted the data directly into a Geogebra spreadsheet and graphed the data…

…and then fit a logistics curve to the data:

They had seen and heard the concept of learning/performance curves before, but it was really great to be able to develop one on the spot with the class. I was impressed with how good the data turned out. It was then neat to be able to show the differential equation that describes this type of phenomenon and solve it to get this type of function.

As is probably obvious, I only have ten students in this group. It would be really cool to try something like this with a bigger group and see if the data fits as nicely.

Building meaning for momentum from discussions, definitions, and data.

Today we started our next unit in physics with a ‘next time question’ from Paul Hewitt:

My reason for giving this was specifically because of the fact that we haven’t learned anything about it. I wanted the students to speak purely from their intuition. I asked them the following:

We aren’t quite ready to answer this by calculation, but I do want you to make a guess.

Will they move together faster than, slower than, or with the same speed as the ball?

Would your answer change if the ball bounced off Jocko instead of him catching it?

Student responses included:

We need to know if he bends backwards when he catches it, because that will affect it.
No matter how he does catch it, he will move slower. The larger mass will result in a smaller acceleration.
The clown has a non-conservative force, so the kinetic energy will decrease.

Interesting responses. We talked a bit about collisions and throws and catches of objects and what they ‘felt’ doing this with different objects. I introduced the idea that it might be nice to have a physics quantity that contains the direction and rate information of velocity, as well as the mass. I told them that physicists did, in fact, have such a quantity called momentum. They responded with a few non-physics related ways they had heard the term and described what it meant.

To figure things out about how momentum relates to collisions, I then had them analyze the three air track collision videos from the Doane Physics video library using Tracker. Their tasks were as follows:

Find the momentum of each cart before and after the collision for the video you are assigned. Calibration information is contained in the first frame of each video.
Find the total momentum of the system before and after the collision.
Find the total kinetic energy of the system before and after the collision.
What is thechange of the momentum of the system during the collision?
What is the change of the kinetic energy of the system during the collision?
Talk to your classmates and compare your answers for the three different videos.

It was pretty cool to see them jump in with Tracker and know how to analyze things without too much trouble. Fairly soon afterwards, we had some initial velocities and final velocities, and changes in momentum to compare.

I was, of course, leading them toward something with the change calculations.

We calculated the changes in momentum, which were non-zero. Were the magnitudes significant? A student suggested looking at the percent change compared to the initial momentum. For the first two videos, the loss was less than 1%, though for the third it was around 20%.

A student proposed the possibility that the change should be zero if no momentum is lost during the transfer. Comments were made about how that made sense in the context of our previous unit on energy – things feeling right when all of a quantity can be accounted for.

I then did a little pushing (since we were almost out of time) about what this might mean about total initial momentum and total final momentum. I also gave them definitions for elastic and inelastic collisions. I then assigned them a couple simple questions that I wanted them to figure out if we can say that the change in total momentum before and after is zero:

Then it was time for Calculus.

______________________________________

I don’t usually like giving students information. I don’t like giving it away without some sense of where it comes from. I also like when students can discover quantities without equation definitions. Sometimes though, the simplicity of an idea like momentum and its power can come from taking the calculation itself as a tool that can be used to analyze a situation.

In previous classes, I have given the definition, shown situations in which momentum is conserved, and then asked students to use this idea of momentum conservation with their math skills to find unknown quantities. I really liked this alternate approach today of using momentum itself to analyze a situation and then have the idea of conservation come out of discussion. I think its potential for ‘stickiness’ in the minds of students is much greater this way.

Experiencing an ODE per day

I don’t like how applications of math are presented as a “special topic” once the theoretical has been understood. There are, admittedly, some aspects of concepts that are more thoroughly understood with background knowledge. I subscribe to an approach that bounces between applied and theoretical whenever possible.

This especially applies to differential equations. I tell my calculus students the story of my time at college when I took a course in differential equations. I spent a lot of time trying to understand how the processes of solving differential equations actually worked. It was the same way I studied multi-variable calculus the semester before, lectures for which I found fairly intuitive. The lectures for differential equations, on the other hand, were extremely technical and involved processes that were not clearly motivated by, well, anything from my professor.

This was also a time when my tolerance for pure mathematics was fairly low. As an engineering student, I needed to have an application nearby in order to push through the theoretical, otherwise my internal ‘what’s the point’ light would start flashing and I would tune out. There was very little of this in the lectures, but I pushed for understanding in my completion of problem sets and studying for the first exam.

My grade on the first exam was a 73. I was shocked. I also decided to give in to the suggestions of the sophomore students that had taken it before, who said just to memorize it all. I didn’t like it, but I did it, and my grade subsequently shot up. It was not until I took courses in system design, heat transfer, dynamics, and control systems, when I saw how differential equations actually worked and could work to understand much of the theory behind them.

Footnote: This entry is not in any way going to be an indictment of my university mathematics education (which was on the whole fantastic), a commentary on the perils of testing (which I do like discussing), or on how pure mathematics is not a rich course of study (which I further believe is NOT the case after teaching for several years and actually doing recreational mathematics on my own and with students). I don’t care to be boxed into any of those categories – the point is coming, I promise.

Last year was my first time teaching Calculus. Knowing how powerful differential equations are, I had prepared a full day where we spent looking at various differential equations and how they are used to model real phenomena from a bunch of different fields. What I found, however, was that my Calculus students reacted in much the same way as my other math students would when they sensed a day of word problems was ahead. There had to be a better way.

Here’s the plan:

Every day, I will show some physical situation with changes that can be modeled using a differential equation. No simulation allowed unless I can also show an actual apparatus that the students can visibly see, feel, hear, or otherwise sense changing over time, distance, etc. There’s something really powerful about generating data real-time – especially when it is related to something students can sense themselves.

My progress thus far:

Day 1: Newton’s Law of Cooling

I started class by asking a student to bring a mug of hot water from the water dispenser around the corner. When he returned, I tossed in a temperature sensor that I had connected to a National Instruments myDAQ board, and without much other commentary, started some review of antiderivatives. Close to the end, I stopped the LabVIEW program from running, and showed the resulting lovely graph of Temperature vs. time.

This resulted in all sorts of questions and discussions- when was the temperature changing the fastest with respect to time? What would happen to the derivative of temperature with respect to time as time went on? What is the physical meaning of this?

One of the students noted that this happened because the temperature of the cup was higher than the temperature of the room. This started a mini-discussion about situations where the temperature of the cup would rise. This all motivated the idea behind Newton’s law of cooling beautifully.

Day 2: Newton’s 2nd Law

The physics students weren’t impressed by this one. Part of the homework assignment from the previous day was to research and post information on the class wiki about a differential equation that (genuinely) held some meaning or interest for them. A couple of the students independently put on Newton’s 2nd, and I accepted it since they did it in slightly different ways. I then showed the students this apparatus (again, not a surprise to the physics students).

This time though, the focus was on the dynamics of an object on a spring. Giving the mass a nudge downward starting it oscillating nicely.

This led us to figure out what the forces acting on it were, namely gravity, the spring force, and possibly friction. This led to the differential equation form of Newton’s 2nd law. I did make available a Processing sketch I put together that contained the differential equation so they could see that this really was what governed the motion of the object.

We didn’t talk too much about the specifics of the program, as lines of computer code thrown at students tend to result in glassy eyes fairly quickly without proper preparation. We will look at programming again later on in the year though, so I’m not too upset that we didn’t talk about the details.

Future topics?

My hope is to include some lights attached to capacitors and resistors to show an RC circuit, a draining tank of water, deflection of a cantilevered beam, maybe even monitoring an oxygen sensor with a candle in a closed container. Part of me also wants to do a bar heated at one end, maybe a bit trickier since it is a partial differential equation, but I think it might also serve to get students thinking about how temperature might vary as functions of time and position. I don’t know what else, but I’m excited about the possibilities.

What are your favorite demonstrations of differential equations in action?

My tutor’s name is Geogebra CAS.

When I first started teaching, I learned that the best thing to have students do after factoring a trinomial was to have the students check by multiplying out the binomials. At the time, it naively made total sense – students don’t even need me to be there to practice! They can do this on their own while sitting on the subway or waiting for the bus – whatever dead time they have. The students that need to practice factoring can do as much of this as they need until they can factor with some degree of automaticity.

Some (not all) students took my advice. Of those that did, I often saw stuff like this:

x² – 4 = (x – 2)(x – 2)
= x² – 2x + x(2) + 4 = x² – 4

This was a worse situation than how we started – not only were they factoring incorrectly, but their inability to multiply binomials was giving them the false idea that they were doing a good job of factoring! This frustrated me to no end – even if I did give students time during class to practice and develop these skills, what could I tell them to do to improve outside of class? One colleague considered stopping giving homework because he saw it repeatedly reinforcing student errors. I didn’t go that far, but I did start grading homework to try to find mistakes.

The missing piece for these students is the lack of useful and correct feedback. Most of them learned the procedures, but made arithmetic or careless errors such as leaving out terms when simplifying. Without any correct data to make decisions on, these students were just going through a procedure and generating incorrect results, and using the incorrect results to validate an incorrect procedure. If they had a way to generate correct feedback, this experience would stop being worthless and instead become a useful method for developing student skills!

This is where CAS systems come in – Wolfram Alpha is nice, but Geogebra CAS is even better because of speed. I worked with a student that needed practice both in simplifying polynomial expressions and factoring polynomials completely. This is what I had him do while he sat with me:

Make up a pair of binomials of the form (x – 5)(4x – 5), multiply them, and then find the quadratic and linear coefficients. When you are ready, use the Simplify[] command to check your answer.
Make up a product of polynomials of the form 4x^2(x+5)(2x-5) . Multiply it out all the way on paper, and then check your result using the Simplify[] command.

After this step, we talked about how he could do this on his own and check his work. While we were sitting there, he made mistakes, but was able to catch them himself. He was the source of the problems, and was able to check and see if his final answers were correct. We then moved on to factoring practice:

Write out 15 products of binomials (3x-1)(x+5). For some of them, add a monomial factor. Include a couple sum and difference polynomials as well. Multiply any three of them out manually and check using Simplify[].
Use Geogebra to multiply any ten of the the rest of them and write down the resulting polynomials on a separate sheet of paper.
Eat dinner, watch TV, or something that has nothing to do with factoring.
Return to the paper and factor the ten polynomials you wrote down completely. Use Factor[] to check and make sure your final answers match what Geogebra produces – if there are differences, check to see if you have actually factored completely or not. Make a note of any repeat mistakes.

There is a whole lot of extra busy work involved in this process, but part of that is because it’s easy to factor a polynomial that you just generated moments before if you still remember the factors. For some students, this won’t matter, but it helps ensure that the exercises generates are actually useful. This student was on fire during class today, even though we were looking at a different topic entirely. I should have asked him directly whether this is the case, but perhaps the boost of confidence going through this process gave him is part of the reason. I also really like that this method allows the student to simultaneously work on multiplying polynomials and factoring them. My method beforehand would have been to stick to multiplying, then factoring, and then mix them up – there’s no reason to do this.

Computer algebra has been around for a while. The reason I think it’s now to the point where it can be transformational is that it’s easily accessible, easy to use, and almost instant. This idea of using technology (and particularly Geogebra) to help students develop their pencil and paper skills is one that really excites me. I’m excited to see if it works with the students that came in a bit behind but are willing to put in the time to catch up. I don’t want my class time to be spent learning algorithms – that defeats my strong belief that we should focus on teaching mathematical thinking, modeling, and problem formulation instead of algorithms. That said, students do need to be able to develop their skills, and this offers a personalized way to help them do so on their own.

Impressing the parents with Wolfram Alpha…it’s for your own good, kids.

I received a few emails from parents recently wondering how to help their children get better in math. Parents often apologize for not being strong at math themselves, and the students, in my case all teenagers, have trouble communicating with parents about, well, a lot of things, let alone math. Creating a genuine way for children and parents to communicate with each other about math has always been difficult. Thankfully, tools like Wolfram Alpha can come to the rescue.

Here is the advice I gave one parent this week whose child is learning to factor quadratic trinomials:

Think of four numbers, keep them between 1 and 8. For example, 2, 1, 5, 7.

You can then write them like this: (2x+1)(5x+7) or make some negative: (2x – 1)(5x+7).

Go to Wolfram Alpha, and in the main input bar, type what you wrote, as shown below:

A page will load with a part that looks like this, a bit of the way down the page:

Give the top one (10x^2+9x-7) to him, and say to factor it. A groan at this point is natural. And then he will remember how to do it. The final answer should be the same as what you entered into the website. You can come up with new numbers and do this as much as you want – it will only make him stronger. If he has trouble, make the 1st and 3rd numbers you pick be 1, and it will simplify it a bit.

Yes, it will result in at least some expression of teenage ‘come-on-mom/dad-ery’. But that’s probably going to happen anyway, right?

How good is your model? Angry Birds edition

With Algebra 2 this week, I decided it was time to get on the Angry Birds wagon. I didn’t even mention exactly what we were going to do with it – the day before, the students found the above image in the class directory on the school server, and were immediately intrigued. This was short lived when they learned they weren’t going to find out what it would be used for until the day after.

To maximize the time spent actually mathematical modeling, I used the video Frank Noschese posted on his blog for all students. They could pick any of the three birds and do the following:

Part A:
Birds are launched at 6, 13, and 22 seconds in the video. Let’s call each one Bird A, Bird B, and Bird C.
• Take a screenshot of any of the complete paths of birds A, B, or C.
• Import the picture into Geogebra. Create the most accurate model you can for the bird you selected. What is the equation that models the path? Does it match that of your neighbors?

Part B:
• Go back to the video and the part in the video for the bird that you picked. Move forward to a frame shortly after the bird is launched, take a screenshot, and put it again into Geogebra. Can you create a model that hits the landing point you found before using only the white dots that show only the beginning of the path?

If not, find the earliest possible time at which you can do this. Post a screenshot of your model and the equations for the models you came up with for both Part A and Part B.

My hope is not to just use the excitement of using Angry Birds in class to motivate knowing how to model using quadratic functions. That seems a bit too much like a gimmick. The most interesting and realistic use (and ultimately the most powerful capability of any model) of this source of data is to come up with as accurate of a prediction of the behavior of the trajectory as is possible using minimal information. It’s easy to come up with a quadratic model that matches the entire path after the fact. Could they do this only twenty frames after launch? Ten?

The students quickly started seeing how wildly the parabola changes shape when the points being used to model the parabola are all close together. This made obvious the importance of collecting data over a range of values in creating a model – the students caught on pretty quickly to this fact.

I think Angry Birds served as a cool “something different” for the class and has a lot of potential in a math class, as it does in physics. I am hoping to use this as a springboard to have students understand the power of models and ultimately choose something to model that allows them to predict a phenomenon that is of some importance to their own adolescent worlds. I don’t exactly know what this might be, and I have some suggestions for students to make if they are unable to come up with anything, but this tends to be one of those ideas that eventually results in a few students doing some very original work. Given my interest in ultimately getting students to participate in the Google Science Fair, I think this is just the thing to push them in the right direction of making their own investigation.

Math is everywhere! – fractals on the Franz Josef glacier

One of the stops on our New Zealand adventure was at the Franz Josef glacier on the West coast. We went on the full day hike which gave us plenty of time to explore the various ice formations on the glacier under the careful eye of our guide. Along the way up the glacier, I took the following series of pictures:

All of these were taken on the way up the glacier. Can you tell in what order I took them? If you’re like my students (and a few others I have shown these to), you will likely be incorrect.

I realized as I was walking that this might be because of the idea of self-similarity, a characteristic of fractals in which small parts are similar to the whole. When I showed this set of pictures to my geometry class, I then showed them a great video video zooming in on the Mandelbrot fractal to show them what this meant.

The formations in the ice and the sizes of the rocks broken off my the glacier contributed to the overall effect. Here is another shot looking down the face of the glacier in which you can see four different groups of people for a size comparison:

The cooler thing than seeing this in the first place was discovering that it’s a real phenomenon! There are some papers out there discussing the fact that the grain size distribution of glacial till (the soil, sand, and rocks broken off by the glacier) is consistent throughout a striking range of magnitudes. The following chart is from Principles of Glacier Mechanics by Roger Leb. Hooke:

In case you are interested in exploring these pictures more, here are the full size ones in the same A-B-C-D order from above:

Oh, and in case you are wondering, the correct order is B-C-A-D.

Bugs on your windshield – An introduction to definite integrals

Considering how tired I was this morning on the first day back to school, I could only imagine how the students might be feeling. Today was the first day of our definite integrals unit in Calculus, and I decided to start off class today nice and easy with the following question:

Suppose I pay you to clean the classroom according to the following plan. I’ll give you $400 for the
first hour, $200 for the second, $100 for the third, and so on. If it takes you 6 hours to clean the room,
how much do you make?

They joked about the silliness of the plan and what they would do given this opportunity. Then they got down figuring out the solution. They were a bit rusty and many assumed there was something complicated going on, so some started recalling geometric series and writing functions involving 2^x. These students quickly gave in to peer pressure and just calculated the total directly. It’s always interesting to see how more experienced students decide not to take the simplest route (though in high school, I think I was one of them.)

The other warm-up question I gave for the day was the following:

The images below represent the windshield of the bus during one of Mr. Weinberg’s trips in New Zealand.
The windshield initially had no bugs on it.

The students were a bit annoyed at having to do this, but they got a much needed review of approximating derivatives. Most students used a central difference, with only a couple using just a forward or backward difference. The fact that they did both was really useful during discussions later on about using left, right, and midpoint calculations for integrals.

As tends to occur with my students, especially at this point in the year when they know most of my ideas don’t come out of nowhere, they demanded to see some of the pictures I took. I was, of course, happy to oblige:

I was able to show them a few more actual scenic pictures, which kept things light as they needed to be before diving into the tedium of calculating areas under curves manually.

The rest of the lesson went great and was essentially unchanged from last year, with the exception of using the following data table instead of a table of velocity vs. time:

Originally I was going to start the lesson with this, but added the second warm-up activity when I thought it might seem a bit too contrived to just throw a table like this at them without any feasible way of actually generating it. I also gave the warning that though the values in this table was made up (though some thought that it seemed completely in character for me to actually take the pictures every hour for the purposes of Calculus), it would be possible to generate such a table using the procedure they followed in the warm-up question.

We talked about how we might estimate the total number of bugs during each two hour interval if we knew the rate and assumed that rate was constant. The left hand and right hand sums came straight out of this. A couple students immediately thought about averaging together the two rates to do midpoint, and later on that led very nicely to a visual discovery of the trapezoidal rule. When we looked at what this process then meant graphically, most students seemed to find the overall concept pretty simple.

The mechanics of doing a left/right/midpoint sum with a function initially appeared more complicated, but having them set up the calculation using a table to organize the values (as with the smash rate table) made a big difference.

Overall, I think the students last year got along fine with the more traditional introduction finding displacement from a table of velocity vs. time data. They got the concept fine, as did the students this year when I showed them how it was really the same as what we did. I think it made a difference to be able to introduce the topic in a more quirky way that grabbed their attention slightly more than something that was just plain easy to understand.

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