1. I don’t know too much about manipulatives. So, I may be misusing the term. What I’ve attempted to do here is associate each dimension with a color. Perhaps it will be easier to work with colors than with raw dimensions. What we’re going to do is to manipulate tiles that are similar to dominoes. First, we come up with a key that relates each dimension to a color. Here’s my sample: Foot Gallon Hour Inch Liter Mile Minute Ounce Quart Second Color Key Second, we define a bunch of standard conversion tiles. Think of each tile as being a fraction, with one color “over” the other. Note that each conversion tile has a reciprocal tile (e.g. Orange over Light Blue and Light Blue over Orange). 3600 Secs1 Hour 5280 Feet 1 Mile60 Minutes1 Hour3600 Secs1 Hour 5280 Feet 1 Mile60 Minutes1 Hour

2. 1 Foot12 Inches1 Quart 32 Ounces 1 Gallon4 Quarts1 Liter 33.8 Ounces 1 Gallon4 Quarts1 Quart 32 Ounces 1 Liter 33.8 Ounces 1 Foot12 Inches Okay, you get the idea. There are obviously dozens more I could do. The point would be to make sure that you have the right tiles for the domain of the dimensions used in your problem set. Now, let’s take Carolyn’s example. A car drives 60 miles in 50 minutes. What is its speed in miles per hour? My color key says that miles are lime green, and minutes are dark red. So, I make up my own tile to represent this. It is lime green over dark red, with the appropriate numbers written in: 50 Minutes60 Miles60 Secs1 Minute60 Secs1 Minute

3. I want to get to miles per hour, which would be lime green over light blue. So, here’s the game: You “play” tiles by lining them up from left to right. If you have a given color in the top half (numerator) of one tile and the bottom half (denominator) of the other half, then that color is cancelled out. When you string together enough tiles to net out at the color combination you want (in our example, lime green over light blue), you stop playing tiles. Then you look at the numbers on your tiles. Multiply the top row (numerators) to get your final numerator. Multiply you bottom row (denominators) to get your final denominator. At least I think it works. I just invented it today, so it hasn’t gone through thorough testing. Let’s continue with our example by playing a dark red over light blue tile: 50 Minutes60 Miles60 Minutes1 Hour The red in the bottom of the first tile cancels the red in the top of the second tile, so we are left with lime green over light blue, which is our target. So, let’s multiply our numerators and denominators: 50 Minutes60 Miles60 Minutes1 Hour = 3600 Miles 50 Hours = 72 Miles 1 Hour Yeah! It works. Okay, that was an easy example. Let’s try something a little harder…

4. A car drives for 10 minutes at 30 miles per hour. How many feet does it travel? Here, you’ve got to have a little understanding to get started. They give us two things: 10 minutes and 30 mph. We’ll represent each with a tile: 30 Miles 1 Hour 10 min Notice that I have no denominator dimension in the first tile. That’s what you have with a single value as opposed to a ratio. Now, I know this is going to get sloppy, so I want to introduce some “tally tiles.” These are little single-color tiles that we keep off to the side. We have one pile for the contents of our numerator and one for the denominator. Whenever we play a new tile, we add the corresponding tally tiles to our little piles. But--like in Old Maid, I guess—when we have a pair (a tally tile in the numerator pile is the same color as a tally tile in the denominator pile) we pull those paired tally tiles out of our piles. When our tally piles show the numerator and denominator we desire, we stop playing tiles. Well, maybe it will be clearer once we work through this example… We start with the setup below. Remember, our target answer is in feet, which are dark blue. Note that here we don’t want a rate (fraction); we just want a single-unit value, which will be a numerator only.

5. 30 Miles 1 Hour 10 min Tally Piles There was a tile back there that converted lime green to dark blue. Let’s play that. We want the one with lime green in our denominator to cancel the lime green that is already in our numerator. 30 Miles 1 Hour 10 min Tally Piles 5280 Feet 1 Mile Wait. We’ve got a “pair” in our tally piles, so that should be: 30 Miles 1 Hour 10 min Tally Piles 5280 Feet 1 Mile Now, I want to get the light blue out of our denominator. I see a tile that has light blue in the numerator and orange in the denominator. Let’s play that:

6. 30 Miles 1 Hour 10 min Tally Piles 5280 Feet 1 Mile Now, I want to get the red out of the numerator. I’ve got a tile that will do that: 3600 Secs1 Hour60 Secs1 Minute 30 Miles 1 Hour 10 min Tally Piles 5280 Feet 1 Mile3600 Secs1 Hour Check out that tally pile! We’ve got our target color. Now, we multiply the values in our numerators and denominators: Num = 10 * 30 * 5280 * 1 * 60 = 95040000 Denom = 1 * 1 * 3600 * 1 = 3600 So our answer is 9504000 / 3600 = 26400, and the units on that are dark blue…er…feet! Now, an astute observer might point out that our last two plays converted from light blue to orange, then from orange to red. In other words, we went from hours to seconds, then seconds to minutes. We could have saved a step by converting directly from hours to minutes (light blue to red). That would make our solution look like this:

7. 30 Miles 1 Hour 10 min Tally Piles 5280 Feet 1 Mile60 Minutes1 Hour Our tally pile still comes out right, but we’ve got different values to multiply across. Num = 10 * 30 * 5280 * 1 = 1584000 Denom = 1 * 1 * 60 * 1 = 60 So our answer is 1584000/ 60 = 26400, so we still got the same answer. This proves that you don’t have to find the optimal path to your answer. As long as you can make the dimensions get to your target, you will get the right answer. The dimensions are your guide. All right. I have no idea if this has been the least bit illuminating. Or has it increased the level of confusion? There is nothing inherently easier about manipulating colors than working with the units, themselves. But it is hard to know what is easier for another person to visualize and manipulate.