This blog serves two purposes: First, I just shared a lesson with the Polite Pirates (my class) that went so well that I want to share it with everyone. And, second, due to its complexity, several students need additional clarification. I am hoping that by writing this down, I can make clear how the business of bridge-building works.
Yesterday morning I dug out the colorful, connectable, plastic blocks I’d stored away in a bin under the counter. I told my students that they would be building bridges. Cheers rang out. They were to work as a team to construct a way for a car to travel from one pile of dictionaries to another. The Polite Pirates cheered with joy.
“There’s a catch, however. You have to buy the blocks!” I exclaimed. The class groaned. “Each one costs ¼ of a dollar. (I’ve been teaching fractions and mixed numbers.)”
“How much is that?!” a few students grumbled.
“You know how much a quarter of a dollar is,” I accused.
“Twenty-five cents?” a student clarified.
“Yes. You will work as a team of engineers. Use your Spiral Notebooks to keep track of the number of blocks. You’ll need to figure out the total cost of your bridge.
“Oh, one more thing: Your bridge is going to make you money. That’s right! Sure, it will cost you to build it, but once it is done, you can charge a toll for cars to use it. For every foot of bridge you are allowed to charge one dollar.
This is the slightly confusing part (one of them, anyway): If about twenty vehicles travel across your bridge per hour, how long will it take to make (dramatic pause) one thousand dollars?
Before setting them loose, I showed them that this problem was doable. (The looks on their faces were incredulous.) “Let’s say you build a 10 foot bridge. How much money can you charge to cross it?”
“Ten dollars.”
“Right, but that is $10 per car. If twenty cars travel across your bridge in one hour, how much money do you earn?”
Thinking… “Two hundred dollars,” a student offers.
“Good; You are correct. Where did that number come from?” I prompt. I want the class to know how to do these calculations.
“I multiplied ten by twenty.”
“Right. If your bridge makes $200 every hour, how long will it take to get to a thousand dollars?” Their minds were working, now!
“Five!” several students shouted in unison.
“Don’t forget that you need to use some of that money to pay for the building blocks that you used to construct the bridge,” I remind them. And, they’re off!
The Polite Pirates had a blast working together. I was impressed that they almost instantly formed the idea of making supports to hold up longer sections of bridge. In this way they could earn more toll money. Of course, they had to count the blocks that they used to hold up their bridge when tallying up the cost of building materials. I didn’t let them use anything other than the building blocks for construction.
Once the bridges got to be several feet long, every single student on the team was needed to hold the bridge in place as more supports and lengths were added. I overheard one third grader explaining to his partners that even though it would cost more to use extra blocks, they would make more money from tolls, because they could make their bridge longer. I reinforced this idea by sharing, “Sometimes you have to spend money to make money, folks!”
After a timer I had set went off, I had everyone stop building and count up the number of blocks that they had used. One team split up the task by divvying up the sections to be counted. They then added all the numbers together. Next, it was time to calculate the cost of all of those blocks. One team had used 355 blocks! How could they figure out 25￠ per block?
I reminded them that they already knew what a quarter of 100 was. “Separate the 300 from the 55,” I told them. “Each 100 would be how much?”
“Twenty-five dollars,” someone answered.
“That’s right. Now, how many twenty-fives do you have? We’re talking about 300 blocks.” I wrote 100÷4=25 on the board. When someone suggested that they needed 3 twenty-fives, I put X3 under the 25. “These might seem like really big numbers, but you already know what ‘three-quarters of a dollar is,” I prompt.
“Seventy-five cents!” a few blurt out.
“Not cents, though…” I can see the gears turning behind my students’ eyes. It feels like I can hear the steam coming from their ears. They even gasp with understanding.
“Seventy-five dollars.”
“Now, for the 55 other blocks. Is there a number close to 55 that is divisible by four,” sounded like Greek to them. I reworded my question in a more leading way. “Can 48 be evenly divided by 4?” This connected with their math facts. A student raised his hand.
After deciding that 48 blocks would cost $12, we tackled the leftovers. “What’s left?” I asked the Polite Pirates who were sitting so patiently on the carpet in the front of my classroom. Counting up from 48 to 55, we discovered there were seven blocks left. “How can we figure out the cost of these?”
When this question was met with blank stares, I quickly drew seven (very ugly) squares on the dry erase board. I drew a hasty circle around the first four. “Each of these cost 25￠. How much money is four quarters?” Lights blinked on in every students’ eyes.
“One dollar…!”
“And…” I prompted for the cost of the remaining three unaccounted for blocks.
“A dollar and 75￠,” a student finished the thought.
“So, these seven cost $1.75, the 48 blocks cost $12, and the 300 blocks cost $25 X 3,” I summed up our calculations thus far. “What will this team need to do next? Tell the person next to you.” Then I sent everyone back to their seats to work out the costs of their bridges. I told them to figure out the prices independently. Then compare your work with your teammates. In this way you can double-check your math accuracy.
I walked around the room, helping students with their division facts. There was a wonderful hum of productive struggle. Some individuals figured out that they would have to add more than one additional hour on to their original answer in order to account for the cost of the bridge. Most found that only one hour would do the trick.
Now that all of the math was done, it was time to write about it. I had posted a question in the Polite Pirates’ Google classroom: “How long will it take for your engineering firm to make one thousand dollars?”
They could work with partners and discuss their writing with their team, but each student was responsible for producing their own explanation of what they did. I left the math that I’d shown them on the board, so they could copy it into their notes, or just write about what we figured out together.
One of the things I like most about using Google classroom on iPads is how easy it is to use the “Speak to Text” feature. It’s true, you have to teach and practice rereading and editing your text, in order for students to use this effectively, but it speeds up typing entire paragraphs. It also helps facilitate a more “Discourse” -style text. When my students purely type, they are less likely to include opening statements, and they will leave out key details. Through the process of “Telling” their iPads what they did, you get a more structured description. And, because it is easy and fast to do, students don’t have a problem including more details.
I hope you enjoyed hearing/reading about this lesson that my students experienced this week. If you have ideas on ways to improve or modify it, let me know in the comments. Thanks, and take care.
The only reason that I knew about this interaction was the hurt child came over and told me about it. It almost seemed like she wasn’t sure if she should be upset or not. As she told me her story, others, overhearing the tale, chimed in with their two-cents. This additional commentary seemed to spin the story into a negative frenzy, so that I put an end to the retelling in order to address the class. 
What the boy did when he pointed out the way a little girl looked different from himself was a natural thing to do. We humans can’t help but categorize things (Vanderbilt, 2016). We have genres for music, books, movies, art, dance; You name it! It’s much easier to find things when they are labeled. When I’m in the mood to read a mystery, it’s much faster and easier to go to that section of the bookstore, rather than sift through every single shelf. But, what about those books that are very funny true stories about a person’s life that happen to also be highly mysterious? Now what? Often times, we simply create a new category. 
Tom Vanderbilt (2016) goes a step further, suggesting that the more we like something, the more we want to categorize. A person who enjoys candy will separate a bag of treats into several groups of like sweets. How many times has the book-lover rearranged his or her library, recategorizing titles? I have grouped books that I read close in chronology, only to separate them after reading new texts. 
I came up with an activity to help reinforce this concept in my classroom. I call it “Building Bridges”. We have a snack/play time in the afternoons, when we give ourselves a brain break. I instituted a “constructive” play rule, where kids use items that can be connected or built upon in order to make things, rather than just mess around. We have Legos, Lincoln Logs, Tinker Toys, 30+ year-old Robotix toys from my childhood, and we also use the colorful interlocking blocks that came with our math curriculum. Because we have the most of this last medium, I chose to have the whole class use the colorful blocks in a contest. (The competition makes it more exciting.) Groups of kids–I not only let the kids pick their own groups, I didn’t limit the size of groups–arranged and connected blocks to form bridges that spanned distances between desks. I told them that the bridge had to be able to hold and transport a Thymio robot–This is a little robot that students can program to drive. The robot was the trick. Students worked to make their bridges wide enough, thick enough, and strong enough for the robot to use. Rather than drop robots on the floor during trial runs, we measured the weight with a simple scale. The robot was equal in weight to a half-full pencil box.
The Captain of Class 