Why Bother With Percent Anyway?

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.

bluestarcafe from Barry Lewis on Vimeo.


The Mathematics of Hungry Cats


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

Ken Ken + Bloom

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.

Seven Must-Have Graph Grooming Tips

A style guide for our number rockets as they learn what a well-groomed graph wears before it ever even thinks of stepping out of the house.

Known and Unknown Data Devils

Independence and Dependence: Examples of two kinds of stories that show up as 2-variable data analysis models in math instruction.

I thought I understood independent and dependent variables. Then I tried to explain it to myself so that I could help explain it to the 7th grade RTI3 students that I support. Then I fell into a philosophy hole of cause and effect. The kinds of basic data sets (e.g., size of stamp collection per year, inches grown over several years) that they were working with seemed to get pretty slippery when attempts were made to get them (the data) to hold still long enough to be able to determine the independent and dependent variables. What follows is my attempt to crawl out of that hole and back into the well-reasoned light of day, hopefully carrying with me a sackful of language that I can use with our students.

1. A Clear, Intuitively Available Case of Cause and Effect

The closer you get to a campfire*, the hotter it feels.
Suppose you have a tape measure and a thermometer.
You are some distance from a campfire, and as you move closer to it, the thermometer shows higher temperatures.
You record your distance from the fire at regular intervals.
At each interval, you record the temperature.
Does being closer to the fire cause the temperature to increase, or
Does an increase in temperature cause you to be closer to the fire?
Question (alternate)
Does your distance to the fire get shorter because the height of the thermometer column gets taller, or
Does the height of the thermometer column get taller because your distance to the fire gets shorter.
What are the two variables?
distance and temperature
Which variable changes because the other variable changes?
temperature changes because distance changes
Which variable depends on the other?
temperature depends on distance
Which is the dependent variable?
Temperature is the dependent variable because it changes depending on how close you are to the fire.
Which is the independent variable?
Distance is the independent variable because WE determine how it changes, rather than the temperature of the fire. How close you are to the fire depends on where you decide to stand, NOT on how hot it is where you stand.
*The campfire is stable and burns at about the same intensity while the data is being collected.

2. A Not Very Clear Case of Cause and Effect, or Maybe Even No Case At All

The temperature outside changes over a 24 hour period.*
Suppose you have a clock and a thermometer.
You go to the same spot outside and you notice that the temperature is different at different times of the day.
You record the time you go outside at regular intervals.
At each interval, you record the temperature.
Does the time of day you go outside with the thermometer cause the temperature to be what you measure, or
Does the temperature you measure cause you to go outside when you do.
Ummm, neither? There is no cause here. Temperature changes, but not because of when we measure it or the length of time between measurements. On one day, the temperature might drop 3 degrees in the hour between 2:00 PM and 3:00 PM; on another day, during the same interval at the same time of day, it might rise or it might stay the same. If there is no cause and effect, and if we want to talk about dependent and independent variables, then we need a different perspective on the variables in this story.

What are the two variables?
time and temperature

Do these variables change in predictable ways or in ways that WE control?
Time changes in a completely predictable way. If we record the first measurement of temperature at 8:00 AM, and then again at regular (evenly spaced) intervals of 15 minutes, we can predict with 100% certainty what time it will be when we record the fortieth measurement, but we can NOT be certain what the temperature will be at that time (at 9:45 PM).
We have chosen to measure temperature at regular intervals of time. We have NOT chosen to measure time at regular intervals of temperature (for example: what time is it at every temperature change of 2º?). We have freely selected and determined time intervals as the feature to compare unknown changes in temperature with.
We may have a hypothesis about how temperature will change over time on a given day, but the exact manner of change is unpredictable. We can not be certain of the exact temperature at 8:00 AM or 9:45 PM or at any other time until we actually measure it.
This story looks at how temperature changes compared with just one feature, or variable: time. We could have compared temperature with humidity, or proximity to the ocean, or the angle at which sunlight falls, or weather, or altitude, longitude, or latitude, or many other variables (probably along with time). We have chosen change in time to be the (sole) basis for comparison with change in temperature.

Which variable changes in ways that we understand and can predict with absolute certainty?
Question (alternate)
Which variable changes in some certain way regardless of how the other variable changes?

These new questions replace cause and effect with insulation. A variable is independent when its way of changing is fully known and when it is protected from any influence that might cause it to change differently. The independent variable will only change in some way that we know exactly, a way that we have chosen and which we control, regardless of what the dependent variable does. In other words, the independent variable only changes the way we say it will, or the way we know it will, whereas the dependent variable can change however it damn well pleases.

Does this mean that the dependent variable is a rebel, a lawless action, an event wholly random and uncontrolled? Maybe. It could mean that, unless we see a pattern in its behavior. We need the controlled, predictable, wholly known and independent variable as a background against which we can look at and compare the dependent variable. That is the only way we can ever notice pattern.

Which is the independent variable?
time, because we are certain about how it will change (its rate of change)
Which is the dependent variable?
temperature, because we are uncertain about how it will change (or if it even has a rate of change…that we can see)

Here, I  think, are criteria that I can ask our number rockets to evaluate: certainty, the independence of the variable that is above it all, and uncertainty, the whatever-it-is that the other variable depends on, the stuff we don’t know about yet, the mystery.

*This could just as easily be: number of stamps collected per year, number of socks lost in the laundry per month, number of watermelon seeds accidentally swallowed per summer, number of fish caught per day, number of home runs hit by bearded, left-handed hitters against right-handed pitchers in US stadiums located south of the 40th. parallel per season, etc. A dependent variable is caused to happen by the independent variable only when a pattern can be found. Until then, it’s the independent devil we know versus the dependent devil we don’t.

Then, when I finally went poking around, I found these:

Math Forum


Physical Science

Social Science

MTBoS Mission 1: How rich is rich?

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.

[Materials here: ten sets of constituent paper rectangles for the three rectangle problems.]

The Rs of Summer

Watch Reinventing Summer School to stop kids’ Learning Loss on PBS.

Wanting students to, first of all, show up for, then engage with and truly benefit from summer school, Rhode Island resorts to relevance and relationships. There’s rigor here, too, both in the field and back in the classroom. Anyway, once relevance and relationships are established, you can dial up the rigorometer to whatever reasonable values you like. (See technicians adjusting an early, analog rigorometer here.) Fall will come and these schools will not have the special funding that made these summer experiences possible, but minds have been changed, and there will be some teachers and students with new ideas needing to be hatched.

What’s lost?
So, is summer learning loss now Summer Learning Loss, or SLL, a worrisome addition to our list of education initialisms? I wonder though, if something is lost, if it doesn’t just fray or decay but straight up disappears, in just ten, warm weeks, could it be that it’s not learning getting lost so much as working memory winking out of existence? That’s a question that guides my reflection of my own practice, especially if I’m not teaching in a woods or on an ocean: What do I do to engage students with deep learning, the kind that knows its way around the hood, can read maps, can ask for and make use of directions, and can figure out how to get home.