Introducing Digit Disguises with a small game

Because [reasons], my game Digit Disguises has been on my mind recently, and reading the original blog post from 2019, I suddenly realised I had never shared my ideas on how to introduce the game to a whole class at once. This blog post fixes that. To keep in the spirit of it, I have not put a link to the original blog post about the game yet, and will introduce it here the way I would in a class. It reads like a recipe, but of course you as the teacher can make your own professional decisions about what to do. I just find that approximately this plan works and want to share it.

  1. Tell the class that you’re going to be playing a game called Digit Disguises in small groups, but first we’ll play a smaller version as a class.
  2. Choose one volunteer to be the One Who Knows by whatever method you like best. You might like to choose someone who would get a boost from succeeding publicly. In my experience, even someone not confident with algebra will totally understand what to do as the One Who Knows, and will get a kick out of being the one with the secret knowledge.
  3. Give the One Who Knows a sheet with A, B, C, D and spaces next to them. Show everyone what the sheet looks like before you hand it over to the One Who Knows. There is a printable version of the sheet at this link.
  4. Ask the One Who Knows to secretly write the numbers 1, 2, 3, 4 next to the letters A, B, C, D in any order they like, but maybe mix them up a bit. Tell them to show nobody, not even you.
    (I have chosen the numbers 1, 2, 3, 4 for a reason. I deliberately did not include 0 because I think it’s  good for people to have the success of come up with the strategies around 0 for themselves later during a real game. Also I need four different numbers so that there is just enough to have to tease apart the logic in this game.)
  5. Tell everyone that One Who Knows has disguised the numbers 1, 2, 3, 4 as letters and their goal is to collectively figure out which number is disguised as which letter.
    The way they’re going to do this is to ask the One Who Knows to do a calculation using two different letters and +, -, × or ÷. Then the One Who Knows will tell us all what letter is the answer.
    It’s important that the One Who Knows never says a number, only a letter.
  6. To start them off, ask the One Who Knows for A+B, and remind them that they are not allowed to say a number out loud. There’s a 2/3 chance that the answer won’t be one of the letters, and they’ll ask you what to do. You can say they’ve already done it: if it’s not one of the letters, then just say it’s not one of the letters.
    (If it doesn’t come up here, you can deal with the “not a letter” thing whenever it first comes up.)
  7. Regardless of what happens with that first question, write down the question and the response on the board/screen/document camera, and then ask the class if they can say what that means for which number is disguised as which letter.
  8. Now ask the class to suggest things to ask, and get the One Who Knows to respond. Write the questions and the answers up as you go. Each time, ask everyone what the response means for what you know about which number is disguised as which letter. You can also ask the person who suggested the calculation why they suggested what they did.
  9. Note: There’s no need to push them too hard on the “what does it mean”. Just one idea is enough each time. However I do think it’s important to make sure they realise at some point that a response of “not a letter” is not actually failure but can give them information.
  10. Note: It is very likely for someone to suggest early on something like A÷A. Tell the One Who Knows not to answer and ask the person why they suggested that, to which they are likely to say that the answer is 1, and then they’ll know what letter 1 is. Celebrate their thinking, because it’s a really important thing about numbers that they’re using there. But then say that actually the rules of the game say you have to use two different letters. In fact, if nobody does suggest a move like A÷A, then it might be worth asking at some point why the rules say you have to use two different letters, and then someone will possibly suggest this as a reason.
  11. At some point, you will have figured out which number is disguised as which letter, and at that point, you should ask ask the class if they are sure. It’s a good idea to ask someone to go through the reasoning so far that got you to the point you are at, and then ask again if they’re sure. When they are, ask the One Who Knows to reveal the sheet and tell everyone if the class is right.
  12. Celebrate the win, and thank the One Who Knows for their help.
  13. Now it’s time to play the game yourselves, but there will be some differences…
    • This time, you won’t just be disguising 1 to 4, but the digits from 0 to 9. How might that be different?
    • This time, instead of everyone and the One Who Knows, it will be two teams. Both teams will disguise the digits as letters, and both will take turns ask questions to figure out the other team’s disguises. How might that be different?
    • This time, when you think you know the other team’s letters and numbers, you can ask them all by saying which letter is which number, but you’ll only get one chance. If you’re right for all of them, you’ll win. If you’re wrong for any of them, you’ll lose. How might that be different?
  14. Split them into teams and put the teams into pairs by whatever method you like best.
  15. Give them the Digit Disguises game board handout.
  16. Remind them of the rules.
    • Take turns asking questions.
    • Your questions are calculations with two different letters.
    • Respond with the letter that the answer is disguised as, or say “not a letter”.
    • Nobody ever says a number …
    • …until the very last turn when you guess them all.
    • Feel free to write down whatever you want as you go to help you figure it out.
    • The handout for the game has all the instructions if you forget.
  17. Let them play and circulate to hear their awesome thinking and point it out.
  18. Stop them after everyone has had a few turns but before anyone has finished the game to have a discussion about how the game is going so far, and to highlight some great thinking you’ve seen. It’s important to remember you don’t have to wait until the end of the game to get some good learning out!
  19. Whenever you choose to end, have a debrief. You may want to ask the class if they have any questions about the game they want to investigate or discuss. Some possible questions are listed in the original Digit Disguises blog post. But basically it’s up to you and your class what you do with the game now.

I hope this is helpful to you. It will certainly help future me when I next introduce Digit Disguises to a group.

PS: Special thanks go to the participants in my 2021 MASA conference session who lived through the first version of this, and to Michaela Epstein, who I discussed it with while originally planning that conference session.

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