The Goods
Download the full archive [11.3 MB], including:
- Dan + Chris – Question
- Dan + Chris – Answer
- Dan + Chris + Annie – Question
- Dan + Chris + Annie – Answer
See, I had to do something about this problem:
This is one of those problems that wakes mathophobes up at night in a panicked, cold sweat. It’s so universally hated it’s a pop cultural cliché, a symbol of everything the layperson dislikes about math but can’t quite verbalize.
As math teachers, we step into the ring with this problem every year. In California you’re guaranteed at least one such problem on your summative student exam. But it’s a boutique problem. How much time can you really offer it in class? How will you treat it? There’s a fairly straightforward explicit formula for its solution:
Do you teach your students the formula and hope their memory serves? What kind of conceptual underpinning can you offer them without spending two weeks on it? In what ways can we improve this problem?
First, Fix The Visual
Since this problem represents itself as a real, no-fooling application of math to the world outside the math classroom, we owe it to our students to ask, “is this a good visual representation of that world?”
A: No. It’s clip art. We could upgrade the clip art to stock photography but both representations are decorative where we’d prefer something descriptive, a visual that is, itself, a useful site for analysis rather than mere drapery.
Here is that visual:
Bean Counting – Problem #1 from Dan Meyer on Vimeo.
This makeover was challenging. The task (whether painting a house or filling a cup of beans) is its own unit of measure. Think hours per house or minutes per cup. If the students manage to determine seconds per bean, you still have a math problem, but the task has changed significantly. So I sped up the tape to preserve that task. Also, the house-painting problem assumes identical houses and a constant rate of house-painting, which is the kind of unreality that exists only to serve a math problem. Here, though, we have used multimedia to inoculate that pseudocontext. Here, the glasses are identical every time and the actors are listening to a song in their earbuds that does keep them working at a constant rate, even when they’re working together.
Second, Fix The Motivation
Again, I hear you: who cares about two clowns filling a cup with beans. Try it, though. Play the video. Ask your students what questions they have. Ask them how many minutes they think it’ll take Dan and Chris together. Ask them to put down a number they know is too large and a number they know is too small. Write five of their guesses on the board.
Then move on to whatever else you had planned for the hour. Let us know how that goes.
Third, Teach
Lecture your way through the problem. Or, better, give your students a moment to reveal to you the tools they bring to the table. I’m only insisting here that when you make the brave (and, to the eyes of many students, ludicrous) claim that math has any application to the world outside the math classroom that you represent that world well and you get the motivation right. Your decisions past that will have a lot to do with your students, their developmental readiness, and your relationship to them.
But even if you lecture, don’t offer the formula. Build, instead, off their existing understanding of speed. This problem is nothing more than a strange unit for speed: percent per second. From there, what seemed complicated and dizzying becomes a straightforward application of rates: find Chris and Dan’s combined rate; divide into 100.
Fourth, Play The Answer
This is dramatic catharsis. I have no idea how to design a study to test for the effect of watching an answer rather than hearing it from the teacher’s mouth or reading it in an answer key but, anecdotally, it’s enormous.
Bean Counting – Answer #1 from Dan Meyer on Vimeo.
Fifth, Offer The Extension
Bean Counting – Problem #2 from Dan Meyer on Vimeo.
Your students’ reaction to this extensions is a meter stick for the effectiveness of your approach in step three. Will they take it in stride? Will they fall apart? If they have an inflexible understanding of the problem they may take this approach:
Which is understandable, but incorrect. (This is the explicit formula for three people.)
In both problems, though, the formula obscures the fact that nothing more complicated than speed runs beneath this problem. I’d rather my students developed that understanding than a full set of what reader Bowen Kerins calls “single-use tools”:
High school math is filled with specific tools for one purpose only. Use this box to solve a word problem about people painting houses, but this other box for this other problem. Use FOIL to expand a binomial multiplied by another binomial, but don’t try it on a trinomial! It makes no sense, and contributes to students’ feelings that mathematics is a giant toolbox you either know or don’t know how to use. [via e-mail, with permission]
Jump back into that video I linked earlier. It’s a useful depiction of a locker room-full of students who understand math to be a giant toolbox you either know or don’t know how to use. Confronted with two numbers, they multiply, they add, they average. They’re just striking the two numbers against each other, looking for sparks, looking for a number they can live with. It’s impatient problem solving.
Then the mathemagician enters the scene, reveals the “simple formula,” and computes it.
Check out the look on Junior’s face. It’s like he’s seen a magic trick. He asks, “Are you sure?” and then takes the mathemagician’s word for it. Meanwhile, we’ve upgraded the representation of the problem and not one of our students has had to take our word on the answer. They just watch it.
Featured Comment
I passed out calculators for the bean counting problem, but made them give guesses (and back them up) first. Some couldn’t wait, and started crunching numbers.
The catharsis was definitely more potent in the bean counting video than the Little Big League video (“Wait, what’d he say?”). Once the time stopped at 4.5 minutes, students started with bragging.
“Ahhh! I TOLD yoouuuuu!”
49 Comments
Colin
April 10, 2011 - 3:23 pm -Maybe it’s because I’m an electrical engineer, but wouldn’t this be easiest to explain as a resistor network? It is identical and *immensely* practical unlike house painting & bean counting rates. It is entirely realizable too with a few resistors and a multimeter.
Dan + Chris + Annie act as three resistors in parallel and since they have different “resistance” to the same “potential” (the beans & plate are the same distance apart, if you will) then they will count beans at different rates just like resistors passing electrons at different rates (current) to the same voltage.
R = Ra*Rb/(Ra + Rb) is two resistors in parallel
R = 1/(1/Ra + 1/Rb + 1/Rc) is three resistors in parallel
What if Chris were twice as far from the beans & cup or there were two different houses to paint in a row? I don’t see that house painting nor bean counting really holds up to the generalization that resistor networks get you.
Dan Meyer
April 10, 2011 - 3:41 pm -So let’s run the cost/benefit analysis. Both examples accomplish the same mathematical objective. Yours is practical. Mine certainly isn’t. This is an Algebra 1 course, though, and your example seems to require prior knowledge that the students and probably their teachers don’t have. That’s a tough one.
Colin
April 10, 2011 - 4:15 pm -[What’s the syntax for quoting?]
Apriori knowledge is a good, valid point but how much knowledge? (I am extremely biased so I cannot answer that.)
Why do fire exits in gyms have multiple double sets of doors instead of just a single door? Why is it easier to blow through a normal straw vs. a coffee stirring straw? Why are hoses at a gas station smaller than fire truck hoses? Why does air flow out of a balloon faster than water (blow it up and let go without tying it)?
It’s all the same kind of thing and all those questions could segue into something more abstract like resistors (you can’t see resistance nor electrons) that gives you a more solid, repeatable demonstration. Heck, you could derive Ohm’s law by experimentation and then answer series & parallel equations. A couple AAA batteries, some resistors, and a multimeter and let the kids “loose.” With some pre-conceived resistor values it shouldn’t take more than 5 minutes to get a basic idea of series & parallel.
It seems like there’s something you could do there with hands-on work and all the questions above could be some kind of teaser to get into it *AND* answer the standard painting question without having to resort to rote memorization.
But I’m not a teacher so: IMHO.
Shawn Cornally
April 10, 2011 - 5:52 pm -The key here is the reaction you created in me, and I even KNEW you were trying to do it. It’s kind of like watching one of those illusionist specials from the nineties; I know they’re going to do some trick, and I’m hell-bent on trying to show you for the charlatan you are, yet I can’t help but get suckered into the act. (Not implying the D. Meyer is a charlatan, it’s just an analogy)
As we fight the good fight (for student engagement) I think all this magic-bullet-real-life talk should be thrown out the window. For surely bean counting is silly, but, if it makes you want to figure something out, you’ve done it. It crosses that activation energy barrier from teacher-coerced participation to willing-thoughtful participation.
Any chance you could do a tutorial on putting timers, counters, and layers on things in iMovie for schmucks like me that can’t afford the fancy software?
Dan Meyer
April 10, 2011 - 6:13 pm -No. Not because I don’t have time or interest or whatever but because iMovie can’t perform these tasks, even though they’re pretty trivial. And that sucks.
I’m a little scared to post this on the main floor of the blog, but if anyone ever wants me to do the grunt work of timers, counters, and layers for some video they have, I am more than happy to oblige. Holler at me.
David Taylor
April 10, 2011 - 11:02 pm -I like this, Dan.
I’m also wondering – how big is the glass? And how many beans are there?
Would it be a good idea to extend it, to allow pupils to have a go at it first. Maybe give them a glass, and the beans, and have them record their own times (for A, for B and for A and B) and maybe estimate their A and B times from their A and their B, and take it from there?
Christopher Danielson
April 11, 2011 - 5:40 am -Colin: Syntax for quoting is here: blockquote in html
Shawn: Dan did do an AfterEffects tutorial here. While I don’t have AfterEffects, I do have QuickTime 7 Pro ($30 upgrade from QuickTime) which will do layers and masking. Not timers directly, but a masked video of your smartphone’s timer will do the trick. An hour or so with QT Pro and its pdf manual and you’ll be up and running.
Colin again: Keep in mind that one of the objectives in Dan’s use of multimedia is easy in/easy out. It’s really easy for a teacher to never try to integrate with the physics teacher because “there is no common prep time” or “we don’t own the equipment” or… There are myriad excuses-many of them valid for not doing hands-on exploration of resistors in an Algebra I course. But what excuse is there for not trying bean counting? In today’s high school classrooms, the technology is generally available to show the videos that Dan made for all of us. There really is no excuse. So bean counting has the potential to make a real and immediate impact in a wide range of classrooms.
David: What would be added/lost by knowing these things? In the text of the post, Dan writes:
So in what ways will the task change if we know the number of beans in the glass? I’m not sure I know at first glance.
And finally Dan: Bravo!
Anyone who hasn’t clicked on the link to the stock photography example needs to in order to compare the profoundly different psychological effects of the video with the photograph.
I am not convinced that the percent per minute solution will be the primary one from students. I’ll give it a shot when I have the opportunity (and I’ll solve it myself) to test the hypothesis.
Dan Meyer
April 11, 2011 - 6:26 am -Hm. I don’t know. My reservation is that great pains were taken in the creation of the video to make sure that, mathematically, this problem works. The biggest pain was creating musical tracks to ensure that we maintained the same rate the entire time. My suspicion is that a) students won’t maintain a constant rate on their own and b) they will slow down when they work together. So you have to balance the extra student engagement against the fact that the math just isn’t going to work out. That’s a deal-breaker for me.
Shari
April 11, 2011 - 6:33 am -I have questions about context. These are questions that keep me challenged when trying to create good math applications.
I understand that no one works at a constant rate of speed and that each situation is different. I can look at the rates of speed runners achieve during different segments of a marathon. Throw in a mega hill, and that rate of speed will drop during that segment. But, it is still helpful information to the runner to know his/her average speed over the entire course.
The same is true in any business. The owner of the house painting business needs to know, on average, how much time it will take each of his employees to paint the average one-story house in order to take jobs, give fairly accurate dates of completion, and schedule his/her employees. This owner doesn’t calculate the exact surface area of every house.
To make this problem more applicable to a real world situation, can’t you just add more background?
Kelly is scheduling his employees for a week of house painting. He knows that Sally can paint an average 1-story house in 4 hours, while it takes John about 6 hours to paint roughly the same surface. If he schedules Sally and John to work together, how much time should Kelly estimate for the two to complete the painting of average 1-story houses?
Yes, I know there are many “abouts” and “averages” in there, but isn’t that how this works in real life? After doing this calculation, Kelly will likely add on a bit of buffer time based on problems that could occur, but the initial calculation will give Kelly a starting point for his scheduling.
Jason Dyer
April 11, 2011 - 7:23 am -I think the answer video might be a little too fast. It’s a.) hard to see the speed variation between the bean-counters; I was genuinely curious what it looked like and b.) kills any kind of suspense. I wouldn’t do real time, but maybe 60 seconds rather than 30?
Amy Zimmer
April 11, 2011 - 7:54 am -While I appreciate all the work that went into the bean counting and video and all the stuff I don’t know how to do, I agree with Shari, that bean counting is just not the same as house painting. Average rate is important to us runners, to contractors, to a motorist trying to get to work. I am going to stick to the average rate for doing real jobs and let my students make up a context that has some interest for them (as long as it is appropriate…I really don’t need any kegger/beer bong info thank you!)
And I do truly appreciate the mathematical model that I would gladly share too…many representations.
Keep up all the amazing work…
Yours, Amy
Dan Meyer
April 11, 2011 - 8:10 am -Always with the cost-benefit analysis. More realistic context (runners, painters, etc.) is inarguably a win. An inability to represent those contexts visually is arguably a loss. An inability to let students confirm the answer visually in those contexts is inarguably a loss. But as long as we’re constantly running these numbers and calculating the pros and cons, you have my blessing (FWIW) on whatever you decide.
R. Wright
April 11, 2011 - 1:21 pm -In regard to this type of problem (which I suppose is meant to serve as some kind of twisted justification for “covering” rational functions), I find the perhaps less-popular example of pipes that flow at different rates to be more captivating than anything having to do with human labor. Also, it’s conceivable that from there, one could, by natural analogy, introduce students to the basics of electrical resistors, which are an honest-to-goodness area of application of rational functions, as noted by Colin.
On a more general related note, isn’t the typical depth of coverage of rational functions in algebra classes pretty absurd? I find it difficult to mention anything beyond ratios of linear polynomials in good conscience, unless students happen to show curiosity in that direction.
Christopher Danielson
April 11, 2011 - 1:26 pm -Oops. Sorry Shawn. Here’s the link to Dan’s very helpful video masking tutorial.
Dan Meyer
April 11, 2011 - 1:33 pm -Look, I’ll do it. Don’t dare me. I’ll do it. I’ll buy a kiddie pool, fill it up with one hose, then get rid of the water and fill it again with another hose at a different rate, then get rid of the water and fill it again with both hoses. What do I do with the water, though, so I don’t get yelled at like I did that one time when I threw away 60,000 papers?
Don
April 11, 2011 - 2:32 pm -Rather than using beans why not stick with paint? Have this be a measure of time painting something.
The advantage over beans is that it’s got a reasonable real-life purpose – you want to split your pile of widgets that need painting between Dan and Chris and you’d like to give them the appropriate number they need.
(Do students give a damn about real life stuff like just-in-time manufacturing? I find process interesting so it would matter to me but maybe not to ‘normal’ folk…)
So anyway, if you want to split up the widgets so that they finish at the same time this actually is useful. Using the painting aspect seems to me like you could add some real-world secondary bits to this – what if Chris is way faster but ruins 1 in 30 widgets or uses twice as much paint? Chris charges three times as much per hour, is he worth it?
If you want to film something less expensive and permanent what about some sort of assembly? Nuts and bolts in quantity are pretty cheap. Or maybe Home Depot would let you film one of their employees putting together a BBQ grill.
Actually the grill might be a great example. Odds are you’d have to give Dan and Chris different workspaces if they were putting together grills and you want to move them in a pallet to close to each of them – think how pissed off Chris would be if being faster than Dan meant he had to go carry a heavy box over to his area every 10 minutes because he’d used up his pile already!
Don
April 11, 2011 - 2:36 pm -Unrelated – it sure would be super if we could know when a thread we commented on got updated… http://wordpress.org/extend/plugins/subscribe-to-comments/
Dan Meyer
April 11, 2011 - 7:44 pm -See, we can’t use anything true to life because the math doesn’t work out in life. People bump into each other. They scrabble for nuts and bolts. Their rates don’t remain constant throughout an individual task nor are their average rates equivalent over the discrete tasks.
(This is, incidentally, a great argument for scrapping this lame, lame application of rational expressions. I’m working within the system I have, though.)
I have created an application problem – however unlikely to be found in the world outside the math classroom – where the math does work, where there is a pay-off for committing to the math.
David Taylor
April 12, 2011 - 12:12 am -Christopher: My idea was to have pupils do it first, and attempt to estimate the amount of time they took together in a similar way to how I approach Pythagoras’ Theorem – have them draw a right-angled triangle, give me the lengths of the two shorter sides, and give them the length of the longer side in return.
Dan: I appreciate that this will result in mathematical accuracy being lost, but it also allows for discussion regarding inaccuracies and the factors leading to them.
Laura
April 12, 2011 - 3:12 am -@ Jason:
I agree about the time…it happened a little too fast for me, too. I might be remembering wrong, but I vaguely recollect some other WCYDWT video being played at regular speed, then sped up or something? Maybe it’s the water tank video? Could that happen with this video?
Although I guess the purpose of the regular speed of the water tank was to motivate students to find out quicker (to use the math). Hm.
In any case, if no real time is used, then I agree that slowing down the video just a bit might be helpful. It was awesome though :) Loved it.
Nora
April 12, 2011 - 5:50 am -I decided to wait to post until I actually tried the videos with my class. The videos were great the way they were. My students were engaged much more than they would have been with a textbook problem or stock photos.
Here’s what happened:
I’m glad that I started the discussion, as Dan suggested, by asking for answers that were too low and too high. This lead the students to some insightful observations. Then I let them go in groups to find an answer.
After a while we discussed answers, showed the answer video and then solved the problem together.
Next, I showed the second problem with Annie and the kids were ready for it this time, but had trouble solving their equation.
The videos are fine. If it went too fast, no worries, I just showed it a second time. I liked the fact that the students couldn’t determine who was counting faster because that would have biased their answer for Annie. Believe it or not, students gave answers suggesting that Annie could count all those beans in under 2 minutes. Hmmm…
Dan, thanks again for an easy and productive day at work.
Rick Fletcher @TRFletcher
April 12, 2011 - 9:07 am -I’m a college professor of chemistry – and these freaking formulae. I had to be in Washington DC end of last week and I have a grad student who has been bugging me to stand in and teach my large lecture course – so here is the opportunity. I get back and first thing Monday morning one of my students who is feeling severely challenged by the course is in my office to show off that she now understands stoichiometry problems. I thought “Wonderful, great transition over a weekend but that is the way it works sometimes!” She shows me the “method” the grad student used in lecture and now she totally gets it.
You guessed it – the formula for three types of stoichiometry problems. I don’t teach from a formula. It’s all strategy and reasoning. ( I still have students every year wondering when I am going to ask them to memorize the periodic table – like their high school teacher did.)
I don’t have the time to start over with this student or all of the others. I’m really not sure how to handle this problem. Suggestions are welcome – but I am stunned how this kind of thing can happen so fast.
Dan Meyer
April 12, 2011 - 9:32 am -My concern with formula in my problem is that it sets my students up to be blindsided if the parameters of the problem change in even the most trivial ways. If a student came to me excited that she could successfully complete dozens and dozens of problems where two people paint a house, I could reveal the limits of her knowledge by changing the parameters. What, specifically, is your concern with formula in your case?
gasstationwithoutpumps
April 12, 2011 - 9:53 am -“He knows that Sally can paint an average 1-story house in 4 hours, while it takes John about 6 hours to paint roughly the same surface.”
Painting time is far less than prep time for house painting, unless you are doing a Victorian house with a lot of fancy trim in multiple colors. Prep time for the outside of the house has more to do with the number and locations of windows and shrubbery than with area. It is not clear that the “painting” example can really be rescued as a real-world problem.
Zeno
April 12, 2011 - 11:13 am -What’s the first thing that comes to mind when you see a magic trick? Don’t you immediately wonder how the trick works? Isn’t that true of most people, kids included?
So if you show your students the “magic” formula for solving the two painter problem, won’t that naturally motivate an interest in learning why the formula works?
And if the students understand why the formula works, rather than just memorizing it, wouldn’t that help them in recovering it if memory fails, and in extending it correctly to additional painters?
Shari
April 12, 2011 - 11:58 am -I understand how motivating it is for students to visually see the problem and verify their work by seeing the solution. I think it’s a great approach. But, by setting the bar at visuals-only, are you not boxing your students in in another way?
I think it is important for students to experience the nebulousness of math, too. Sometimes there isn’t an exact answer. Sometimes there’s more than one answer. Sometimes there’s a range of answers that work depending on how you approach the problem.
In this example, you took an unrealistic situation and replaced it with another unrealistic situation. The only difference is the visuals. For me, it would be more important to create a realistic situation, even if that situation doesn’t have exact answers and exact rates. I’d want students to develop an understanding of less-then-exact real-world problems.
Jason Stein
April 12, 2011 - 12:04 pm -I have often seen this presented as a Common multiple problem. If A takes 4 hours to paint a house and B takes 6 hours to paint a house then in 24 hours they paint 10 houses. So it takes them 2.4 hours per house.
Laura
April 12, 2011 - 6:50 pm -@ Zeno
Zeno, I agree with you about the magic. And this extends to math at times. But I would argue that many kids (especially in middle to high school) do not see math in the same light. When they see something work, they often assume that the answer behind the “magic” of that formula lies beyond their grasp. Math is a wolverine to them, right?
This is why teachers face the challenge of students inflexibly clinging to formulae. The kids don’t understand a formula, and they don’t really want to, because they’re afraid that if they get their hands dirty it’ll just confuse them even more. I’ll admit that was me! I was a mathophobe (an English major), and in college I dreaded calculus. I had an engineering friend (God bless him) who would pull all-nighters with me before an exam as I was frantically trying to memorize and cram. If he tried to go in depth I remember frantically interrupting, “Whoa whoa! Don’t tell me too much or I might get confused!”
I teach 2nd grade, so my math expertise is like a droplet in all of your oceans, but I vividly remember the first time I began to love math: it was a math education course in grad school. The prof showed a visual representation of Pythagorean’s Theorem…simple stuff, right? No one had ever showed me how this worked. Once I saw that the random formula I had to memorize ten years ago actually MEANT something, THEN my appetite was whetted and I began to search for the secrets behind the “magic.”
I guess all this rambling to say that I don’t think what you’re positing is untrue, I just think it might be the exception rather than the rule. Maybe a formula first approach in Dan Meyer’s remedial algebra class in the spring (when he’s helped all his students to tame the wolverine) would be just as effective as what we see here. Maybe that kind of curiosity (having the same attitude toward math as we do magic) is what we hope to foster in our students…
Rick Fletcher @TRFletcher
April 12, 2011 - 8:22 pm -It’s the same problem as learning the formula in your case. The students need to be problem solvers – I can always throw a problem at them that they have not yet learned a formula answer and besides that, there are far too many different types of problems to learn enough formulae. The useful skill is to problem solve. My problem in this particular case is that the student feels accomplished and I have to find a way to gently burst the bubble that she might not have learned as much as she thinks and that it will take her a lot more time than she spent. These are the hazards of working with new teachers…
Bowen Kerins
April 12, 2011 - 9:58 pm -Rick, I think you hit in on the head here: there are very simple ways to extend a problem that move beyond formulas and require thoughtful analysis, so we should make the thoughtful analysis become part of the process in building the formulas… or even deciding if a problem type is important enough to deserve a formula. I wish more attention was paid to manipulating formulas once they’re seen as important — how many kids memorize r = d/t as a different formula from d = rt? Using and manipulating formulas is far more important a skill than memorizing them.
In “real life” there are very few memorized formulas. Those who frequently use a specific formula generally internalize it through repeated calculation and use; and those who use a lot of different formulas use books and websites.
I wish I knew good ways to help the student you mention: perhaps a different situation that leads to the same formula but isn’t obviously the same?
Thanks Dan for the mention! I expanded my comments on general-purpose tools and posted them to the fledgeling blog linked from this comment. These videos are excellent examples of problems that can be solved well without formulas, or even solved in the purpose of building formulas, with thinking and exploration at the forefront.
louise
April 14, 2011 - 7:24 pm -I was thinking I could have two or 3 students painting sidewalk areas with water. One student has a fat (3″) brush, one gets a 1/2″ brush. You can add in a third student.
Three groups, each group “paints” one sidewalk slab, and times it. The water has to cover the slab completely. Figure out how long it will take any pair to paint a slab if they work together, then try it.
Good for an entire hour outside. Sure, none of us really paint the sidewalk with water for a living, but it does translate to working together to reduce the task. Also, then I will move them on to washing my classroom windows…
Aaron B.
April 14, 2011 - 8:16 pm -Dan – looking forward to seeing/meeting you in Green Lake next month. Drinks are on me if you have any free time.
Another great idea that I can’t wait to use in class. It isn’t so much about the math but more about the engagement something like this creates. Even if someone wanted to go into the detailed mechanics of problems like these, you will get more mileage with an introduction like this. Thanks again for all you do.
luke hodge
April 16, 2011 - 1:35 pm -louise,
I like the painting idea, but don’t see the payoff for the hour spent outside. It seems like the kids would already have good intuition about a brush six times as wide painting six times as quickly. Unless the kids all brush at the same rate, which seems unlikely, the size of the brush is confounded with the brush rate – the math will not work out at all.
Seems like if you just had someone slowing counting off “one one thousand” with some people drawing or shading one square per second & others 1/2 or 3 or whatever, you would get better understanding in less time. It might be interesting to compare & predict the results for different teams racing to 20 squares: which team will get 20 squares first, how many squares will they win by, how much of a “head start” should the slow team get, etc.
Michael Paul Goldenberg
April 17, 2011 - 7:35 am -“He knows that Sally can paint an average 1-story house in 4 hours, while it takes John about 6 hours to paint roughly the same surface.”
Combined rate problems. Love ’em, though only when I’m doing test-prep tutoring, etc., since that’s pretty much the only place I worry about them.
I like the approach suggested above by Jason Stein, as I’ve not seen that one before.
That said, it’s always nice to have an intuitive approach to offer kids. First, recognize that many, many folks will look at this and answer “5 hours.” That makes for an interesting entry point for a lot of conversation about problem-solving, estimation, and common sense. Anyone who, pushed to think about, honestly believes that it takes two people working together longer than it takes the faster person to do it working alone may know something we don’t know about the job or two people in question, but probably s/he’s just not thinking about the situation, just jumping to the “obvious” solution of trying to average rates. Doesn’t usually work very well (see the classic problem from an SAT, c. 1979, with a woman driving to and from work along the same route at 30 and 40 mph, respectively, in 1 hour total time: most people will answer that the total distance driven must be 35 mph.)
One nice way to estimate combined work problems is to imagine twins who work at the same rate. Most folks can glean that two Sallys here will take 2 hours working together and that two John’s will take 3 hours working together. So one Sally and one John together had better take somewhere between those numbers, not between 4 and 6. Will it be EXACTLY the average of two Sallys and two Johns? Probably not, given the above-mentioned driving problem, but 2.5 hours is a hell of a lot better ballpark figure than 5. The latter is impossible unless we assume that two people together will screw around or get in one another’s way SO much that the combination of their efforts makes things take longer than the faster one working alone would manage – not out of the question, of course, but probably not a grand idea if you like getting a good score on tests.
In fact, the exact answer is 1/(1/4 + 1/6) = 12/5 hours or 2.4 hours. We were quite close given these numbers, only off by 6 minutes. Will our luck hold out for more complicated problems?
Hmm. Let’s add a third person, Ted, who alone can do the job in 2 hours working alone. Adding and dividing by 3 suggests that together the trio would take 4 hours, which for the same reasons as above we’ll dismiss. And using the same ballpark approach, we’d get 2/3 hour for three Teds, 4/3 hours for three Sallys and 2 hours for three Johns. Adding these and dividing by 3 gives 4/3 hours for the three working together.
The exact answer is 1/(1/2 + 1/4 + 1/6) = 12/11 hours. Our estimate was off by 8/33 of an hour, which is less than 16 minutes.
Obviously, the numbers we use have something to say about the closeness of the estimate we’ll get with this method. I could be fun to explore more cases to see if we can get a handle on how the number of people and the disparities amongst their individual rates affects accuracy of this method to arrive at an accurate estimate. Lots of “weighted average” things going on here, which is something I don’t think we get kids to ponder and work with enough.
Brendan Murphy
April 29, 2011 - 3:35 pm -I just wonder why not build a couple of Lego machines to build stack or destroy something. http://www.youtube.com/watch?v=de4xdOVVROQ
Pete Capewell
July 5, 2011 - 1:06 pm -‘Be discrete’ seems to be the key to making the concept concrete. It’s the move from continuous painting to counting something that scaffolds learners’ thinking toward a combined rate calculation.
What if we re-shot the house-painting video as a group of people *tiling* a kitchen wall at a constant rate per square metre? The advantage of Dan’s video is that learners can see a time to complete a concrete, integer-valued outcome. A tiling video would preserve authenticity to the original (albeit hackneyed) problem, but do so in a way that allowed the clever use of a click-track to preserve the rhythm of each performer achieving a unit of progress.
Still, as it is, it’s an old problem presented in a wonderfully new and simple way. Thanks, you’re an inspiration.
Tim Erickson
July 13, 2011 - 8:35 pm -@Shawn: in the Ancient Years, there was an iMovie plugIn (I can’t find it on my disk!) that let you put time codes in. A non-exhaustive search for such a thing for modern iMovie found this:
http://www.imovieplugins.com/plugs/timer.html
I bet there are even better ones, but this one is at least cheap.
Tim
Paul Wolf
December 7, 2011 - 1:25 pm -So if anyone wants to know how this would go over in an Algebra 2 class of relatively advanced ability, it works great.
I put up a slide with the original “textbook” problem, and we talked about it for a bit, then we did the first video, which was kind of slow going. Trying to get the kids to justify why we can add the rates was tough, but we got a great argument as to why we shouldn’t average them. Then we launched into the sequel, which they jumped into right away, and if they didn’t get the right answer, they at least got its reciprocal divided by 100.
By then the textbook problem was an afterthought.
Matt Vaudrey
March 21, 2012 - 12:41 pm -I did this today with Algebra I, GATE and General, students. I found that showing the Little Big League clip first got them thinking (mostly because it has attractive integer operators).
I passed out calculators for the bean counting problem, but made them give guesses (and back them up) first. Some couldn’t wait, and started crunching numbers.
The catharsis was definitely more potent in the bean counting video than the Little Big League video (“Wait, what’d he say?”). Once the time stopped at 4.5 minutes, students started with bragging.
“Ahhh! I TOLD yoouuuuu!”