Post from Marilyn’s Blog: The 1–10 Card Investigation

by Math Solutions Professional Learning Team, June 03rd, 2016

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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.

  1. 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.
  2. No answer book is needed. When a student makes it work, they have the verification that their solution is correct.
  3. 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.
  4. 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.

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.

q card: Arrange a set of ten cards, numbered 1 to 10, facedown so that the following occurs: 1. When you turn over the top card, it should be a 1. Place it faceup on the table. 2. Move the next card to the bottom of the deck, keeping it facedown. 3. When you turn over the third card, it should be a 2. Place it faceup on the table. 4. Move the card to the bottom of the deck, keeping it facedown. 5. Continuous this way, turning over the card, placing it faceup on the table, and moving the next card to the bottom of the deck. 6. When you're done, all the cards on the table should be faceup in order from 1 to 10.


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