I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.


  1. Brian Cormier

    October 4, 2008 - 4:33 pm -

    For one thing, you can use it as a Do Now during a probability and counting unit. In Costa Rica, each license plate must be 6 digits. How many possibilities are there?(easy) If in California, each license plate is one digit, one letter, then 5 digits. How many possibile license plates are there?(tougher) In California, each license plate consists of 7 characters: 6 numbers and one letter. The letter can be placed in any position. How many possibile license plates are there?(toughest)

  2. Going the obvious route:

    How many unique license plates can Costa Rica give out compared to California?

    A little less obvious:

    Perhaps compare the number of registered vehicles in Costa Rica to California. How many unique license configurations are left to give out?

    Maybe with a little more digging you could even get your hands on how many new licenses are given out each year by each government. How long until they’re going to have to redesign their plate or numbering system?

  3. Certainly, looking at the sheer number of possibilities of each plate is a good mathematical problem.

    But then we need to ask ourselves, “so what?” And the so what here that I can see is why would California use that 1-letter system? What can be inferred about the number of cars in Costa Rica compared to that of California?

    Another state could be introduced. In Michigan, we recently went from a 3 numbers followed by 3 letters system to a 3 letters followed by 4 numbers system. What is the increase of number of plates here? Why would such a change be made?

  4. Extension on the above: why would CA need such a great number of plate combos? This is like using these plates as primary documents in social science.
    Set up how you make a comparison, the powers of observation needing to be at their strongest (Hocus Focus would work well for this, too).
    Talk about format and requirements.
    I’m sure there some kind of inductive/deductive reasoning lesson here.
    Talk about tone, how formatting communicates a message, what each plate says about that country: the dirt, the vehicle material, the colors, etc.
    On a related note, these plates are dying for a font study, in yearbook and journalism classes specifically.

  5. Written response journal: How do you feel about the proliferation of world-destroying automotive convenience, considering the overwhelming evidence that human activity in the form of hydrocarbon combustion leads directly to global warming while systematically oppressing native peoples and decimating their hopes of transition to a green, eco-tourist economy? Discuss your outlook on what the future holds for your generation in the context of the prison-industrial complex that typically produces license plates by virtually enslaving minority populations and simultaneously denying them the franchise over either crimes of poverty or falsified charges, without overlooking the pernicious effect of racial profiling, and the impact these forces have on democracy.

  6. I’m no math teacher but I might follow the above with something like this . . .

    Give them some real data to chart the percentage increase of cars per year in CA. Use this data to forecast future license numbers needed.

    At what point will CA need to change its license system?

    What should the new system be in order to handle the calculated rate of growth without having to change the system for X years into the future?

    –other extensions include using road and environmental data

    At what point would driving become too painful to continue?
    How many miles of new road would this kind of growth require? What would that cost? (are you taking inflation into account?)

    How many neutral carbon credits would CA need to buy to neutralize the output of all these vehicles? What would that cost? etc. etc.

    I agree with Todd that you could do a whole “personality” profile on areas based on the typography and design of the license plates.

  7. Which vehicle owner pays more for gas in a month?

    Interdisciplinary unit for Math, English and Social Studies.

    Write a short story for each picture. After researching the local economy and businesses of each region along with current gas prices, identify all variables involved in arriving at your answer. Story must match clues provided by the pictures (ie. sports car/longer commute/smaller gas tank vs truck/shorter commute/larger gas tank). Provide references for all information used. Distribute rubric prior to assignment to give students an idea of how much detail is expected.

    Can’t wait to see what others come up with. Great post Dan…

  8. I’ve seen the possible number combinations on license plats questions as an into activity in Saxon Math 76 (5th in Cali, 6 or 7th elsewhere). The photos make it MUCH better.

    As earlier commenters suggested, I would have the class figure out combinations. I would just start with the Costa Rican plate in the first lesson (remember, we’re talking upper elementary, they will need more scaffolding for this).

    You could do a Language Arts activity by having them create vanity plates, and limiting it to a certain number of positions, and/or # of numbers vs. letters to extend it.

  9. Finding myself less intrigued by the numerical gymnastics (although I see that as the must-do if the plates themselves are the key variable in the 2 images).

    Instead, I’m decidedly more curious about the contextual irony of the plates vs. how they are afixed to cars that seem to scream out: switched at birth.

    I’d also add: the rust won me at the snapshot hello.

    This texture-esque bit would lead me down a conversational road with my students of discussing the shared element of intuited authenticity throughout our worlds, as well as how often incongruity sparks intrigue.

    As a writer or as a human being, the context which houses the variable shapes the ultimate meaning(s).

  10. Mathematically, it is screaming combinatorics, and I’d pick on the same mathematical content as the majority. The only wrinkle I’ve got to add is, what’s the ultimate potential of this system – number plates world-wide. If we allow up to seven characters, then the answer is between 50 and 100 billion (depending on what we do with O, I, blank).

    Still, I think the *content* has been dealt with by now. How do we think this would be best used instructionally? Should it be done during a specific part of the syllabus, or as an ‘unattached’ project? Would there be any use presenting to students in the same way Dan’s done for us, and asking “what Maths can we do here?”

    What other resources should we use in the same lesson as these number plates? Would you use these as the lead-in to a topic, or otherwise, what scaffolding would you put in place during previous lessons?

  11. This I think would be a great way of discussing problem that seems simple at first but, is in fact in depth and would require much thought.

    First of course start with some basic permutations of character sets and see how many plates would be possible in each.

    Then throw a curve and add in readability requirements. This would eliminate confusing characters such as I and 1. What would this do to

    After this you can introduce the problem of license plates spelling inappropriate words. To fix most of this problem just remove vowels and vowel lookalikes and you can no longer spell a word. You still have the problem of acronyms and slang terms. the only way to completely remove this issue is to have each letter isolated to where they can’t form meaningful strings of characters. This is probably why there is only one letter on that California plate.

    This can be a fun class experiment add in the requirement that it the plates need to memorable. This is where you can summarize the theory of clumping in memory retention. So you can now create diffrent plate styles. for example is three letters, three numbers and a space between better then 7 characters.

    I actually think this would be a great demonstration of how complex many real world problems are. It would fit into one of those “simple” problems you give at the beginning of class but the students find out the problems is not a simple as it looks.

  12. Hmm…tough one for me, but in English, I see some early work in comparison/contrast possible here. It could lead to a discussion about how each country views its central parts (CR has a Central American identification, but US is broke up into specific states).

  13. Though I am a math/science guy, here’s a journaling topic for the english crowd, or a “think about this” for the tech-geeks.

    Notice that the two license plates use different fonts. Some countries (areas in Netherlands) have begun to utilize different fonts based upon the readability of the fonts to OCR cameras. You know the ones that take a picture of your license plates, then kindly mail you a ticket for running a red light, or speeding.

    “What is your opinion of using cameras to capture speeders, and possible flaws with the system, such as illegible fonts on license plates?”

    As a physics teacher, I guess the appropriate discussion for my class would be to set up a hypothetical speed limit, and give a frame rate to my students. Then have my students calculate the displacement of the vehicle at given frame rate if the car were speeding. Expanding on this, how about the percent decrease in the font appearance at the relative distances.

  14. I’m skipping the math teacher response and going along the lines of Todd, Jethro, and Christian.

    Racial profiling is a huge issue for my students. We know, but rarely discuss, that your license plate number can increase the chances of being pulled over. (The first two numbers on the plate indicate which county a car is registered in. It happens that my county is the reservation.)

    Anecdote Break

    My class window overlooks the parking lot. Last winter, a student wanted to know which one was my car.

    “That blue one.”

    “No way. How’d you get rez plates?”

    “Well, I live here…”

    My Ethics/Social Studies/Life Skills/stretch for English plan for these pictures is to start off brainstorming about what we know about the people in each car. Spin into discussion of profiling and stereotyping. When could it be beneficial? When is it harmful? What have you experienced?

  15. I’m surprised nobody has mentioned this (maybe dan doesn’t have too many readers in CA) but that second plate is for commercial vehicles (pick-up trucks and SUVs included) only. A standard passenger car has the plate structure # LLL ###, giving many more permutations.

    At some point in the early 80s, California switched from a ### LLL structure to its current one. Why?

    When I bought my first (and only) new car in 1996, my license plate was 3SUB728. What is the newest plate that you have seen? How many cars have received new plates since then? How many cars have been newly registered per year since then? per month? per day? per hour? How does this compare to Costa Rica?

  16. How bout this: If all license plates in the world used the same format and registration numbers were mutually exclusive, what information would you need to suggest a viable format?

  17. Also of note is the emissions testing sticker, which brings up both math and science questions.

    Why is testing the emissions of vehicles important? When is the cutoff for when vehicles need to be tested every year? Who decided this cutoff?

  18. If I went the English route (maybe mix in some stats) I’d use the craigslist robbery guy and mix it with similar looking plate permutations, common car types and colors and a dyslexic witness. Then I’d dip into how inaccurate eye witness accounts are – all as a set up for court room writing prompt (heavy on the fact vs. opinion side of things)

    So basically they’d be brainstorming how to pull off a bank robbery and then defend themselves.

  19. I was going to point out that the California plate was a truck plate.

    Obvious questions (at or about the 7th-8th grade level):

    – What is the expected ratio of trucks to passenger cars, based on the license plate numbering schemes (0A00000 vs 0AAA000)?

    – Assuming that the truck plates start with 0A00000, how many plates were issued before that one?

    – Why doesn’t the Costa Rica license plate have any letters?

    – When Costa Rica runs out of numbers, will they be better off with adding another digit to their plates, or making one of the digits be a letter?

    – Why does the California plate have an Arizona frame?

  20. My kids are younger, so I’d work on basic facts with them:

    (1) I’ll throw out a number (“sixteen!”) and then you race to make that number using the digits on the license plate and any operations you care to add.

    (2) I’ll give you a number (e.g., 16) and you see how many different ways you can make that number using the digits on the license plate and any operations you care to add. (Qs for thought: what makes one plate better than another? Does it matter what number I choose?)

    (3) There are only nine types of license plate in the world. To find out which you’re looking at, cast out the nines. (practice adding single digit numbers, and homework is to find someone on your street who has one of each)

    If I had older kids, I’d observe that both the plates have duplicated digits and ask them whether that’s likely or unlikely? How would we verify/calculate that? Are the odds different (and if different, better or worse) if we’re looking for “any two digits the same” vs “any two consecutive digits the same”?

  21. This conversation is on the verge of a Lesson Study. With some focused organization and implementation, this could turn into a model for the future of teaching and learning about student learning. Kudos to Dan for a fantastic idea!

    As for what I can do with this. . .

    I am currently teaching Linear Programming in my Advanced Algebra class. It would be neat to see the costs in making each plate, the price the DMV charges to acquire the plate and perhaps do some cost benifit analysis of adding another character to the plate. At what point would the plate need to get longer or double stacked? How many plate numbers are recycled (if any) back into the mix because the car they belong to no longer exists. What information can you get from looking at a motorcycle plate, or a trailer plate, etc.? Based on seeing just one plate, can you give an accurate estimate of the number of registered cars in a state or country( +- 10, +- 100, +-1000, +- 10000, . . .)

    It would be neat if this conversation would create a lesson, all of us teach that lesson and then come back and discuss the lesson and try to improve it. It could be something as simple as a daily warm up or a full blown lesson.

    What a neat idea for a forum like this!

  22. It would be interesting to take these plates, and other samples and have the kids figure out how many possibilities there are (that’s been stated), see how many drivers there are (that’s been stated), but then compare those neighbors to comparative economics. Does the number of possible plates reflect how wealthy a nation is?

  23. I liked the “What’s your problem” post. Too few idealists, futurists, and every other kind of ist (i.e. Technology Special-ISTs) forget what life is like in a real classroom.

  24. Am I the only guy who’d be asking if we could track down the owners of the respective vehicles? Then ask about what other numbers are public information?

  25. I’d simply ask kids to brainstorm what the two have/might have in common…and what is/might be different about them… then have them choose one or more of those commonalities or contrasts to write a script for the story of these two plates. Purely for creative writing and flexible thinking…but then I am a humanities/gifted teacher type.

    I also would have used Nat’s idea above but made it a broader number sense/algebra game for kids to take with them everywhere: each time you pull up behind a plate at an intersection, formulate a number sentence by adding symbols anywhere amid the letters and numbers on that plate to create a true statement:

    6+95-2=99 That one was too easy.

  26. I actually used this concept to teach my students about combination earlier in the school year.

    Instead of a still or drawing I took my classes out into the teacher parking lot and we walk around from car to car picking on the uglier cars, and not accidentally, skipping my car altogether…that s until one of the students managed to find it.

    This concept above all others stayed with them because it brought it home for them….it made the math real.

    We also did this with phone numbers, area codes, exchanges, gmail passwords, etc.

    Awesome post by the way. I’ll try not to be so late on the next “lesson study”