Watch this 1-minute video of me demonstrating the directions for *The 1-10 Card Investigation*. I’ve found that demonstrating helps students understand exactly what they are supposed to accomplish. Then I give students time to work on the problem, either individually or in pairs.

There are several things I especially like about this investigation.

- It doesn’t depend on students having a particular mathematical skill that could make the problem inaccessible or inappropriate—they all know how to put numbers in order.
- No answer book is needed. When a student makes it work, they have the verification that their solution is correct.
- There are different ways to figure out how to arrange the cards in order to accomplish the task, including using trial and error, and learning from unsuccessful arrangements.
- Students typically learn fairly quickly that they need some way to keep track of their arrangements to learn from their unsuccessful tries, and it’s valuable experience for them to decide how to represent their thinking in a way that’s useful to them.

**Classroom Suggestions**

Here are ideas for using this activity in the classroom.

As Students Explore

After students have worked on the solution for 10 minutes or so, you might interrupt them and ask, “What have you noticed?” or “What have you tried so far?” Sharing ideas can help spark others’ thinking. After students have solved the problem, ask students to share their solution strategies in small groups or with the class. Discuss how various strategies are similar or different. Perhaps ask students to arrange the cards using someone else’s approach.

Extensions

For students who are interested in additional challenges, ask them to repeat with all of the cards in a suit, from Ace to King. Or try it moving two cards to the bottom of the deck each time, instead of one.

Providing Cards for All Students

When enough decks of playing cards aren’t available, I distribute five 3-by-5-inch cards to each student to cut in half and number from 1 to 10. This also has the benefit that students can take the cards home and share the investigation with their families.

For Younger Students

To make *The 1-10 Card Investigation* more accessible for younger students, have them try it with four cards, numbered from 1 to 4. Then, if they’re able and interested, extend to five cards, numbered from 1 to 5.

Post the Directions

If you’d like to post the directions for your students, here they are from *About Teaching Mathematics, Fourth Edition*, page 43.

This will be a great puzzle for the start of the year – accessible to everyone in the class, and can start the conversation about recording what you are doing, and persevering to get to a solution. And a chance for students to begin to explain what they did to solve it, and why. Everything you want in those first few puzzles.

This is a nice activity. They can also make the problem simpler by using fewer than 10 cards, to test some of their ideas.

Something related I’ve done is analyze “baby” shuffles, shuffling the deck with a small number of cards. Even with just 4 cards, you can have an interesting conversation:

(a) what is a “perfect” shuffle

(b) do all the cards change place in a perfect shuffle?

(c) what happens if you repeat a perfect shuffle?

It usually happens that students don’t recognize all the attributes of a shuffle and particularly miss whether the top half of the deck goes to my right hand or left hand. That sets up an easy sleight of hand where I show them one version several times, then do the other version and ask them to figure out what went wrong.

I love this puzzle. We found it in the book

Family Mathand played with it in our middle school math club. A fun challenge for all ages 🙂Marilyn,

I love this activity. I have done it with 5th and 6th graders many times. I usually start with 20 cards, but starting with 10 is definitely more accessible. The kids consistently love this puzzle.

I especially like it when students figure out a new way to solve this puzzle. I will often ask students who finish quickly: “Now that you have solved it, is there a more efficient strategy or approach you could have used?” There are LOTS of really cool ways to solve this!

This past year, a group of my students got really into this and started making their own extensions. Like you suggest, they tried placing 2 or 3 or 4 cards underneath. Others wanted to try with more than 20 cards! I saw many of them working on the bleachers during recess.

In order to try to keep up with my students, I wrote a program that solves this for any number of cards while placing any number of cards on the bottom. Let me know if you want me to post the program. I don’t want to give away all of the answers. I could also email you a link to keep it private.

Tyler

Thanks for your feedback. If you’re willing to share your program, send a link. Anyone who doesn’t want to see your solution doesn’t have to peek.

Here is a link to the code on a website that will run a program to figure out the order of the cards: Order of the Cards.

You can change how many cards (cardCount) or how many cards are skipped (skips) at the top of the code.

Click run when you have it set up the way you want. Let me know if you find any bugs!

Tyler

If there are N cards (1, 2, …, N) with M skipped each time: Can you predict where card K will appear in the solved ordering? (What would such a description even look like?)

MQ

mathwater.wordpress.com

Tyler,

What is the formula used?

Let me know if you can. Thanks!

This is a great activity! I can hardly wait to try it with my 1st Graders! Thank you 🙂

Also, I have been looking for your book “A Day Without Math” as (I lent out my copy and never got it back) I use it with both my 1st Graders and to teach my Pre-Service Teachers. However, I can’t seem to find it anywhere. Any suggestions? It’s such a fantastic teaching tool!

Thank you so very much for all you do and inspire!

Trish Simmons

Thanks for your comment, but I didn’t write “A Day Without Math.” I checked online and there’s a book titled “A Day with No Math” by another Marilyn. Maybe that’s the one? It sounds interesting and I’m interested in tracking it down.

Thank you for sharing this problem. It has a great entry point so I could try it myself with scratch paper. The feeling of accomplishment after getting it is awesome. This task really tests perseverance and organizing your thinking. I like to stump my friends on Facebook with these. I will definitely use this as a classroom activity for sure!!! Thanks. Makes me want to explore your book.

I am going to use this lesson right off the bat this year. This lesson is a great way to show how making mistakes are OK because we learn from them to grow and make better decisions. One of my goals is to create a culture in my classroom where risk taking is encouraged and mistakes are a positive instead of a negative. Thanks for this activity!

I did it by running the game backwards. Is that cheating?

Nah, not cheating. Just another problem-solving approach.

Hi, Marilyn-

I was so excited to see this challenge on your blog! At the beginning of each year, I go into the 4th grade classes and present this same problem. Students work to find solutions and post them on my office door, noting “Woo Hoo!” on their solutions. The teachers and I always enjoy seeing the level of interest and engagement as the students work through the task, particularly with those children who do not think of themselves as “good” math students. Thanks for the smile today!

Thank you so much for sharing. I tried it today with my 7th graders and they loved it. I introduced it as a warm up but I took 20 minutes of class because they really wanted to do it themselves.

I used to have the order written down and I haven’t used this for years. I can’t find the card order and have been working on it for about 20 minutes. Can someone give me the card order. Thanks

I’m working on a blog post to respond to your request. Stay tuned.

I solved your interesting 1-10 Card Investigation puzzle eight different ways. Have you seen my response on a recent email? I’d be happy to send it to you again if you did not get it.

Sorry, it didn’t reach me. Please post again.