After just wading into the basic idea of percent, I wrote this short story to help illustrate the benefits and power of using a standard reference frame to describe and compare fractional quantities. My number rockets made a few quick Oh Yeahs! at the question for the first scenario about whether or not to be worried, but that certainty gave way with a few groans at the implication of the second scenario. Thank goodness for learning disequilibrium.
I haven’t felt like taking out a loan, renting a pick-up truck, building an annex to my home, and buying a two year supply of cat food for my two cats. Making just one stop for cat food every two years is an attractive plan, but it comes with unreasonable associated costs. On the other hand, I don’t mind stopping every two weeks. That’s not a mad hassle and it’s not a budget buster. Two cases every two weeks has seemed just fine, but inevitably, demand sneaks up on and exceeds supply, and then crisis mode kicks in. I hate making a special trip for cat food at 9:50 PM just so the wild animals won’t starve to death before I can make it to the cat store.
At first I thought I could use my Reminder app and figure out some 2-case/3-case pattern every so many weeks, but the repeat menu is limited. So then I wondered how I could most efficiently buy cat food once only every two weeks and still ensure that my cats will never be hungry again.
Google Spreadsheet killed this problem. Number rockets can fiddle with different amounts, explore different initial conditions (Do we need an initial condition?), and let their rules play out over several months until they are satisfied with their solution. Is it simple? Is it efficient? Is it easy for a harried human to operate?
QUESTION: Two cats each eat one can of cat food twice every day. Cat food is sold in cases of 24 cans. What is the most efficient way to buy cat food once only every two weeks (because that’s an easy schedule to remember) and always be sure to have enough cat food on hand?
Cat Food Spreadsheet is here
Cat Food Graph x-axis: #weeks. y-axis: #cases of cat food.
click for Desmos view
Along with their number puzzles, Ken Ken produces lovely riddles (these arrive in weekly subscription emails). I have played with hijacking the graphic and flying it into a small field of Bloom questions, hoping to create opportunities for classroom conversation and fierce, standards-slaying critical thinking. I am keeping the redesigned, printable riddles in this Box.
I easily careened through Sam’s detour barricades, first by believing that my rich problem isn’t rich enough, and then by lacking a sense of personal ownership of my classroom, which was easier to do since I’m not yet a classroom teacher. (I’m certified and working as an edtech in a middle school, sometimes along with a dedicated seventh-grade resource room teacher.) I like this prompt because it has helped me realize that richness is in the mind of the beholder.
Our kids are wounded; they rarely or only barely get to gleam. Maybe a few school years spent mostly addressing insufficient skills development via flat, skills-based worksheeting and workscreening exercises have persistently dulled their appetite for mathematical curiosity and wonder, two important dispositions that are hard to develop in the absence of math success (see: vicious cycle). Recently, our number rockets have been struggling to understand area and perimeter. They have made progress, but dealing with the area of “chipped” rectangles is still a challenge for them. This activity aims to 1) promote analytical problem-solving skills and 2) focus thinking on the concept of area. It includes a kinesthetic activity because that’s important, helpful, and always inviting. Students will be given a chipped rectangle area task and a set of 10 rectangles cut out of grid-paper. They will begin by finding the component paper rectangles that fill the task shape’s area (for two of the tasks, there are two sets of component paper rectangles), and continue as guided.
I hear the questions because I have also asked some of them: Isn’t this too guided? Wouldn’t it be better, more open-ended, and potentially more empowering, to show less and allow students to discover processes and relationships by themselves? Yes and yes. But. Kids will not benefit from invitations that they are too scared to accept, or that they are too nervous to like. An image comes to mind: once in a Dallas, TX neighborhood, I saw a young child riding her small bicycle. It had only one training wheel, the other presumably having been removed after some time spent riding with both. Brilliant, I thought; a brilliant mix of guided and self-controlled exploration.
So, my intent is to guide our students to experiences of success with mathematical reasoning and to nudge them into developing the habits (mathematical practices) they will need if they will ever want to risk looking for mathematical structure, and maybe there find beauty.
Mow a huge lawn. Earn $75.
Knock down clothes line; shred grandmother’s robe. Pay $30.
For a short while this summer, I’ve been working with a rising ninth-grader, Hat (not his real name), establishing and repairing his foundational math concepts and skills. For one of our episodes, I taught operations with negative numbers using an idea that I’d left undeveloped since I’d thought about it almost a year ago (during the New Blogger Initiation festival!). At the time, I wasn’t sure about Casino Zero’s working mechanism, mostly because I didn’t need one. Now I needed one.
Hat is an entrepreneurial number rocket with access to a riding lawn mower, making semi-mad stacks of cash this season. A simple story and board-game model provided all the control I needed. Here’s the set-up:
- The Game is played for pretend money but not with pretend money; it’s played with value chips.
- Chips are in two places: the World has chips (in the center of the game board) and the Player has chips.
- The Game starts with the World and the Player both having no money, but in the form of zero value, that is, as equal amounts of positive and negative value chips.
- Chip denominations are 5 and 25. (I would have also used 10 if I’d had a fourth color.) Those denominations come in both positive and negative, too. So, chips are +5, -5, +25, and -25, with positive and negative denoting increasing or decreasing wealth. That kind of “direction” of monetary value is not so strange.
- Positive and negative were used to denote one other direction: the flow of value. I suppressed the urge to talk about magnitude and vectors, but I did define positive flow as meaning “in the direction from World to Player” and negative flow as “in the direction from Player to World.” Hat agrees that this language captures the idea of chips either coming to him (getting paid) or going away from him (having to pay).
- The Game is played by rolling a die, moving a marker, and either earning money (a wealth-increasing event) or paying money (a wealth decreasing event), with plenty of ridiculousness along the way. Obviously, this game is easily adaptable to any story you want to make.
- Hat keeps a ledger with columns labeled Direction (In/Out), Number of Chips, Chip Value, Change in Wealth, and Total Wealth. He writes an expression to describe what happens for each play, then simplifies it to see the result. For example, you earn $75, so your ledger looks like +, 3, +25, +75, (new sum): . If you had to buy gas, your ledger might look like -, 8, +5, -40, (new sum): . An important supporting piece here is that the result of the operations must match the change in wealth, which is obvious because the game tile says what that change is and also because you can see your rising and falling fortunes for yourself in your pile of chips. (Hat sees that having equal stacks of opposite chips means having zero value and zero dollars.) With each event, Hat and I look at the outcome of the operations: (+) • (+) = (+), (–) • (+) = (–), and so on, building some sense about the “sign rules” for multiplication.
- Finally, the big mathematical moment happens! The World owes you $30 but it has none to give because there are no more +chips in the middle of the board. What can you do? If you go to the cashier right now and cash in your chips (for the pretend money), what will your wealth be? What about the wealth that the World owes you? What can you do with the chips in your hand that will increase the wealth you have in your hand, and so, the money you get from the cashier?
- Give away negative chips? Really? What would that look like? So, -, 6, -5, +30, or . Now, when you go to the cashier with fewer negative chips, you’ll receive more money; your wealth has increased: (–) • (–) = (+).
We only had time for one game and never got into some of the more interesting combinations of operations that might be required to create the necessary change in wealth. And for sure, not only would this have been more fun and more kindly chaotic with 3 or 4 players playing at 5 or 6 tables, but Hat and his game-mates would also have had some conversation, argument (I hope), and other social-learning energy to help drive the lesson forward. Having concrete experiences that illustrate abstract ideas helps secure those ideas. Even semi-authentic, not-exactly-real-world experiences can help students create cognitive pegs to hang math concepts on.
To add more support to the idea of negative factors making positive products, I drew Hat into Chris Adams’s great response to Dan Meyer’s call-for-tweets about A Negative Times a Negative. Hat’s gears audibly clicked into place when he played that movie in his head.
If you try a Casino Zero type game with your students, ping back how it went and add your ideas for making it better.
Pong in the City.
- Pong has been enlarged to be played on the side of the Cira Centre in Philadelphia. If the controller was enlarged as much as the video screen, how big would the controller be?
- Find a picture of something in the world that is as big as the controller would be and create a Photoshop file that shows it along with the Cira Centre building as if it was really there and you had taken that picture.
This becomes a compelling question only when my number rockets are eager to know: Who won the guessing game? Only when they are hungry.