I’m really grateful for the deep bench I have on this blog, the readers who take the time to share with me the mathematical objects that intrigue them. Adam Poetzel, secondary math ed prof at the University of Illinois, sent along Orbeez, which is pretty aptly described by this commercial:

Basically, small things that grow big in water. The Orbeez website puts the volume increase at a factor of 100 while the instructional manual puts it at 150. Controversy! Which is right? Or are they both wrong?

I went to Toys R Us and bought a starter pack for $8.00.

I dunked ’em for a few hours and got this:

A few ideas here. Start informally. Move from the concrete to the abstract. The informal question is, “**how many times bigger is it, really**?” Ask the students to write down guesses. Write a few up on the board. Perhaps print that photo out and have them draw what they think “150 times the volume” would look like. (Am I alone in thinking this looks way

*way*smaller than 150 times bigger?)

Ask them what information they need to answer the question exactly. Put up this photo.

Here’s the math:

Okay, Orbeez, just watch yourself, that’s all I’m saying.

Ideally, you’ll move from the relatively laborious calculation of volume to the relatively simple comparison of the diameters using scale proportions. ie. if the large volume is really 150 times the small, then the large radius has to be at least 150^(1/3) = 5.3 times the small.

**The Goods**

The problem archive, including:

- the commercial,
- the manual,
- the website screenshot,
- before / after photo of Orbeez,
- before / after photo of Orbeez with ruler.
- Orbeez’ internal expansion measurements (given different water sources) [see this post].

**Addendum**

This is the rare WCYDWT investigation that would be even better with real stuff rather than all these digital replications of real stuff. Buy some Orbeez off Amazon. Let your students dunk their own Orbeez on day one. Perform the investigation on day two.

**2011 Jan 11**: Sharon Cohen, the brand manager of Orbeez, stops by to drop some knowledge on us all.

## 56 Comments

## jme

January 25, 2011 - 11:23 am -Being a stats guy, I’d just point out that the first thing I think of when I see this is that you’re not going to get the exact same expansion factor every time.

You did it once and got 109.6. Extensions into statistics:

(0) Was 150x right on average and this one was an outlier?

(1) Move from asking “What’s the expansion factor?” to “What’s the distribution of expansion factors?”

(2) How can we use the resulting distribution to decide whether this one case (109.6) was unusual or not?

## Dan Meyer

January 25, 2011 - 11:43 am -Yeah, good call. Let’s chalk those up to another benefit of students performing the investigation in class.

## Debbie

January 25, 2011 - 12:33 pm -Am I the only watching the advert and thinking, “but what would you do with them?”. Even the excitement of the Magic Machine actually took 70 minutes before the orbeez were ready to roll down the Magic Chute.

The maths problem is about 100 times more interesting than the product itself.

## Adam Gessaman

January 25, 2011 - 12:43 pm -I’m just a history teacher, so I’ve been largely observing the WCYDWT series and thinking about parallels in my subject area.

But I’d be curious to chart the Orbeez’s rate of expansion. Is it linear? Does it follow a predictable pattern?

These aren’t necessarily the most vital questions, but Dan’s reference to “a few hours” made me wonder how long it took to expand to twice, four times, and then to maximum size. A series of regular measurements, I suppose, might answer the question “how long must I wait before I can begin measuring?”

## Whitney

January 25, 2011 - 2:00 pm -I can’t immediately see how this would be useful, but it’d be nice to find a way to do a volume displacement here between the two different sized Orbeez.

If you had the rate of expansion, you could do it without sealing the Orbeez.

## jme

January 25, 2011 - 2:14 pm -@Whitney

Are they expanding by absorbing water?

Place a large number of them in a large graduated cylinder filled with (an excessive amount of) water. Video the expansion. Then you should be able to infer the (average) rate of expansion for one.

They are small enough that it probably would be tough to do with just one, but you’ll totally see the water level drop with lots of them (assuming they work by absorbing water, of course).

How’s that?

## louise

January 25, 2011 - 4:03 pm -The volume stays about the same – the water goes out of the runny water and into the polymer. Less water volume, more polymer volume, volume is about the same (actually may be slightly more, due to absorption mechanism).

Plastic grow creatures are actually easier to use – they are flatter. Students can trace around the “start” and measure the thickness. Then the only piece of homework i require all year- take it home ( yes, i will supply a cup if you do not have one in your house. I do not supply water, but I tell them they can use pop or pee if they have to) and soak for at least 24 hours. Bring it back in a plastic bag (supplied). You can get insects, dinosaurs and sea creatures.

The advantage is that students can measure multiple distances and see if they all have the same expansion ratio.

Applications include how much more chocolate you get in a regular M&M than an M&M mini (what should the trade value be?), and why don’t they make the giant M&Ms any more (how many of those in a bag?)? Or even, since it’s that sort of polymer, how much polymer do you think is in a newborn diaper? A 6-month-old diaper? A year-old diaper (the age of the child not the diaper)?

Finally, a measured chunk of dry ice in a latex glove is a lot of fun, and waves at students by the end of the day, if anchored correctly. Really helps to see the 1000X expansion from solid/liquid to gas.

## jme

January 25, 2011 - 4:18 pm -@louise –

Ah, of course, silly me. For some reason I had the image of a sponge stuck in my head.

Anyway, to address the original point, you can still estimate (average) volume change using a big vat of water and a bunch of pre-expanded and post-expanded ones, I suppose.

## Steve Phelps

January 25, 2011 - 4:28 pm -I am glad we will begin working with volume ideas next week, instead of having worked with volume ideas last week.

## Whitney

January 25, 2011 - 4:34 pm -@jme

Thanks!

## gasstationwithoutpumps

January 25, 2011 - 5:12 pm -Before scrolling down, I tried measuring the (low-resolution) image in the blog. I got diameters of 16 pixels and 70 pixels, for a ratio of 4.38. Cubing that gave me a volume ratio of 84.

The 4/3 pi stuff is irrelevant, since the shape is not changing.

To get a ratio of 106, I’d need to have 15 pixels and 71 pixels, which is possible given the anti-aliased edges. It would be best, of course, to use the original higher-resolution image to do the measurements.

Of course, one could try doing the expansion inside a small box, in which case the shape would matter.

## Cathy

January 25, 2011 - 8:01 pm -My kids love these! Saw a YouTube video of a girl placing the dry Orbeez into water, vinegar, vinegar & baking soda, and boiling water to see which grew the largest. Kids could make predictions about the liquid, time required to grow to maximum volume, etc. and then carry out the corresponding experiments.

I also came across this video http://goo.gl/qIdql which would lead to lots more questions.

## RM

January 25, 2011 - 8:05 pm -I am new to posting here, but really enjoy your blog. Thanks for sharing such great ideas and getting me thinking about my teaching in many new ways.

I used a lesson similar to this last year with my 6th grade students. We were learning about percent change. I noticed that those dinosaurs that you put into water and they grow (like these: http://www.stevespanglerscience.com/product/1247) had on their packaging that it will grow 300%. Well, what better way than test it with your students.

They had to ask what we needed to test, how to measure, how to chart, and calculate the change from original to “grown” and back (as it states it will return to the original size).

I will be honest, the lesson was not a great success. They take much longer to grow than I predicted. I was fortunate enough to have some other work for the students to practice. I am hopeful that I can revamp and make this lesson a success this year.

## Dan Meyer

January 25, 2011 - 8:12 pm -Personally, I think these things look pretty boring, but I’m willing to admit a) I don’t know what kids like, and b) Cathy’s time-lapse video is pretty awesome.

Can we all get on the same page though and agree that this kid’s reaction is wildly,

wildlyover the top?## Adam Poetzel

January 26, 2011 - 11:50 am -The manual says Orbeez should be soaked for AT LEAST 2 to 3 hours. I actually let mine sit in the water for over a day and a half and the results were interesting. There was an obvious disparity in size between different Orbeez. I had a few actually grow to over 500 times the original volume! Some only grew a bit over 100 times their original volume.

I love jme’s idea here:

“(1) Move from asking “What’s the expansion factor?” to “What’s the distribution of expansion factors?””

Taking the problem from a straight geometry lesson to more of a statistics focus. Great!

I imagine their claim of 100 or 150 times the volume is based on the 2 to 3 hours of soaking….

## Brian Caine

January 26, 2011 - 3:08 pm -Reminds me of the good old decalcified egg and diffusion lab… soak an egg in vinegar overnight to remove the shell, then in various substances to change shape.

Couple of spin-offs:

1) after soaking in water, place into corn syrup and leave another set outside to compare subsequent decrease in size and rates of decrease. Messy…

2) have students determine whether density changes over between dehydrated, partially hydrated, and fully hydrated states.

As always good stuff!

## Brian Caine

January 26, 2011 - 3:10 pm -edit to above… “change shape” was about as poor a word choice as possible. Sorry.

## Sharon Cohen

January 30, 2011 - 9:00 am -Great to read this feedback–I’m the brand manager of Orbeez. The disparity (150/100) is based on the fact that growth depends on ionic content of the water–the purer the water the larger they grow. The very same Orbeez wll grow to a different size depending on the water purity. The number we chose ended up being a marketing decision (100 is a powerful figure) but we should have been consistent. It’s impossible to choose one accurate number.

Here are some stats to consider using different kinds of bottled water (we have averaged the results). The first number is after 5 hours of soaking, the second after a day and a half. They should reach 90 of full size after 4 hours, 100 percent after 8 hours.

Smart Water 13mm; 13.45mm

Dasani 13.62mm; 13.45mm

Fiji 10.2mm; 11.85 mm

Market Pantry 13.3mm; 14.3mm

Arrowhead 9.4mm; 10.68mm

Aquafina 13.4mm; 13.96

Help me out here–soneone do the math and tell me the most accurate number. And then–is is volume or size?

## Ginger

January 31, 2011 - 3:21 pm -Just curious–what happens as Orbeez dry out again? Do they return to their original size and shape? If so, how long does that take?

## Sharon Cohen

January 31, 2011 - 4:09 pm -Depends on storage conditions. If left in a single layer in heat, they can dry out and return to original size within a day. In a closed container out of the sun they can last months, even years. Many science, math and creative applications in this product. Also interesting to study their environmental uses–they were originally developed by the Department of Agriculture to reduce reduce the risk of drought. They are planted under golf course sod to reduce water usage. The clear ones are invisible in water. Colored ones reflect and transmit light in interesting ways…

## Phil

February 21, 2011 - 10:59 am -I love this discussion – there are so many multidisciplinary options available. I tried to convert it faithfully so it’s easy for others to find, to digest, and benefit from the ideas here. I’d love any suggestions or feedback on how to make it more useful.

## Matt Kardon

March 17, 2011 - 6:48 am -You could figure out the growth rate using some combination of statistics (linear regression) and calculus.

You can put one Orbee in water every 10 minutes for a few hours. Then take them all out at the same time.

Next, measure the volumes and plot volume vs time or radius vs time.

You can use different methods to estimate the growth function. Does the radius grow in a linear manner, or the volume, or the surface area? Maybe nothing is linear.

You may need to put in a few Orbeez every 10 minutes and take the mean volume to reduce variation.

## Bob Strickland

March 27, 2011 - 7:55 am -What happens to orbeez when you put them in boiling water?