The structure of this post comes from Dan Meyer’s Three Acts of A Mathematical Story.

# Act 1

**Play the teaser video.**

**Have students write down questions they have about the video.**Ask them to be silent during this part so that other student’s questions don’t affect their own. By this point, I’m sure that 95% of them will have the same question:

**How long does it burn?**Ask them to write down their guess to the answer to this question (once again silently). Record their guesses on the board; this tends to buy them into the process. There are reluctant guessers in every class who are afraid of being wrong and this is an important lesson for them to learn.

# Act 2

Offer some resources for the students to work on.

All videos used the following stop-motion settings:

Two photos per minute.

Video compiled with 10 frames per second.

Small candle start mass: 13g

Small candle end mass: 5g

Medium candle start mass: 51g

Medium candle end mass: 5g

Large candle start mass: 287g

Large candle end mass: 30g

Using the stop-motion settings students first need to calculate how long the small and medium candles burned, in order to make a linear regression of **candle burn mass** vs.** candle burn time**.

# Act 3

Answer video:

Have them calculate the final burn time of the large candle. Talk about possible error points (do color, fragrance, or other factors affect burn time?)

Maybe(?) show the following graph:

Extension Questions:

- If I want a candle to burn for 100 hours, how large does it have to be?
- If I have a candle with 3 wicks that has a mass of 150g, how long will it burn?
- (more?)

As always **any** and **all **feedback is extremely appreciated by me. Please let me know what you think. I design this stuff in a vacuum and I want to know how to make it better. Thanks, Dan.

Hi Dan.

Nice exercise and worthy of Dan Meyers.

I remember doing a similar exercise as an introduction to a science class when I was in Grade 9. The emphasis of that exercise was to record as many observations about the burning candle as one possibly could. Most people make about 50 distinct observations.

While your emphasis is on math, I can see how the two exercises can be combined leading to both more science and math questions. For instance, how deep does the melted wax get before it starts dripping? Or, how long does it take before the black smoke starts eddying or helicing? And, of course, obvious questions such as how hot are the different colors of the flame?

Great idea for a math exercise. It forces kids to confront the big picture before getting into more detail-orientated math.

Shawn

One observation: In your second extension question you ask how long a candle of known mass burns if it has three rather than one wick. Do the wicks always act independently (linearly, that is a third of the time it would take one wick to burn) or by some other means due to the fact that there are three of them (is this as simple as any combined work question, like two painters paint a house)? I don’t know. I think you would have to test and film this yourself before you ask such a question. But wouldn’t it be interesting to do so?

Question for you, Dan: in what ways did you feel constrained by the three-act framework? What good bits did you have to cut from your lesson in order to make it fit? In what ways, then, was the framework useful?

I’m pretty sure you’re the only person alive who’s taken my framework seriously enough to use it, so I’m very curious about your experience.

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I love it, but I’ve got to play devil’s advocate for a bit. Just for fun.

What was done in order to determine that it was a linear function, or did the class just assume that it would be linear?

Also, it looks like it worked out OK here, but I have a tough time extrapolating by that much. I actually make a pretty big deal in my class about being REALLY careful or else you come up with really absurd numbers sometimes. I found it helpful to give my kids some data points that looked linear but were actually logarithmic so they could see what happened when they made a huge jump. Maybe that is next for you?

@Shawn

That’s an excellent question re: triple wicks. I’d bet that it continued to be linear because really the wax is just a fuel can, and three wicks would burn three times as fast. But, I don’t want to make an

assout ofuandme, so I’m going to go back to Target and see if they have the same brand of candle in different sizes (hopefully with a multi-wick option too).@Dan Meyer

Benefits of the 3 Act framework:

* It makes it much easier to see the raw material and what math can be done

* The 3 Act framework makes it easier to see how it would go in a class. There are times when I’ve thrown interesting material at a class and it didn’t stick because of too much or too little information provided. The framework allows for a focus on

whatshould be provided to the students, and possibly more importantly,whenit should be provided.Constraints of the 3 Act framework:

* It might have a lack of flexibility. If the students wanted to measure something different other than the question I thought they would ask, then I may not have the information necessary to scratch their itch. This constraint is why the twitter tag #anyqs is helpful because it can provide some insurance on what question will be. It allows you to be better prepared for the anticipated question(s).

@sarcasymptote

You’re right, of course. The extrapolation is a hefty one, I’ll see if Target has other sizes to make this less of a leap.

As far as the linear nature of the candle sizes vs. burn lengths goes, I’d have to find a way for the students to understand how a candle works. The students would find them a bit mysterious. I made a guess at the linearity because I related the wax mass to a fuel can, so of course it would be linear. My guess was on. A nice followup to this might be some data of a similar experiment, (ice melting?) that looks linear for small samples, but is something else.

Lovely. Great production value.

I’m curious: why do all your videos start about 10:00?

On a note of more consequence, I wonder how you chose mass as the information to give, rather than dimensions.

Love this. Very nice job. Here’s what I thought about as I watched:

Is there a difference in the thickness of the wicks in the various candles?

The diameter of the candle vs change in mass. Larger candles don’t melt out to the edge. What is the size of the heat affected zone? Do long skinny candles burn at the same rate as short fat ones with equal mass?

I see 3 wicks as 3 candles with a shared fuel source and 3 overlapping heat affected zones.

I start my MAT program in a couple weeks to get my masters degree and initial licensure. I’m loving all this. Thanks!

Oh…one more thing…If I use the first two data points for linear extrapolation, don’t I way underpredict burn time? The goodness-of-fit shown in your graph depends on there being three known points. I’m curious how this affects classroom dynamics.

@Christopher They start at 10:00 because I set up the candles to burn when I went to bed, and the stop-motion software is on my iphone. So that big sucker took 10 or so nights to finally burn down.

I chose mass instead of volume (that was my original thought), because it would have been hard to measure the volume of the candle wax left over at the end. I could have remelted the end bits into a different shape that was easy to find the volume of, or could have used the Archimedes method to find the volume. Seemed easier and more direct to measure the mass. Though I would certainly have this type of discussion with the class during this activity.

And yes, the students will under predict the burn time by using the first two data points. That’s okay with me. The packaging says the burn time should be 40 hours. It ends up being 60ish.

But I do need more data points to make this linear extrapolation better. Working on it.

@Jim

Working on it, gonna get some more data points. Though I’d bet that the size/shape doesn’t have much to do with the actual burn rate of the candle. It might affect how much candle is left at the end. I’m going to stick with candles from the same source, so that the wicks and wax types are the same.

Have you considered cutting the candle to get different lengths? Wonder if that would work–worried about the wick holder at the bottom maybe being an issue if it is not melted into the wax.

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