The Winding Game

Lynne and her assistants set up 12 chairs in a circle. On the chairs, they have taped the pictures of the animals that the students drew in their groups. Each chair has a different animal, and the animals are arranged according to the circle of the zodiac. Lynne sends the students back to their desks and asks them to bring their books when they gather around the chairs. The class quiets when Lynne raises her hand. She directs Stasha to read the setup for practicing for the Winding Game (see Figure 5.4). Lynne points out that she has prepared cards for numbers from 1 through 70 instead of through 144. Some teams will have six players instead of five, and the game will have six rounds, meaning that one person in the teams of five will have to take two turns.

FIGURE 5.4 The Chinese Zodiac and the Winding Game

Source: From The Algebra Project Inc. Transition Curriculum, Unit 2, Lesson 5, p. 57; with permission.

Lynne then describes the game to the class:

Now I’m going to ask you to watch very carefully as I demonstrate the game. I want to see if you can figure it out. I am going to pick a number from the pile and put it in my pocket after I look at it. I want you to guess my number. Now you’ll have to watch very carefully. I’m going to walk around the circle and then sit down. The thing that will help you to know my number is division form A. That will help you. And remember to watch very closely.

Starting at the monkey chair, Lynne walks around the circle two times and then sits down on the chair with the dog drawing on it. “Now raise your hand if you think you can tell me what my number is—the number in my pocket.” Lynne makes no comment or change of expression as the students share their guesses—2, 22, 24, 2, 26, 7, 2, 26. Some students have no guess. Lynne offers a hint, “I am the dividend.” Several students exclaim, “Oh, I think I get it.” When Lynne asks once more and a student guesses 22, Lynne tells the class that she is going to do the walk again and that they should watch closely. As Lynne walks, she reminds the students, “My number is the dividend.” Lynne completes her second walk, and more students’ hands shoot up with confidence to guess Lynne’s number. Without taking any guesses, Lynne asks Steven to model the game.

Lynne tells the class, “I’m going to give Steven his number, and he’ll model the game.” Ivan tells Lynne that he is not certain how Steven will know how many times to walk around the circle. Lynne suggests that dividing by twelve may give him a hint and that he should watch very carefully during this next round. While Steven is getting his number, student-talk, such as “I get it” . . . “I don’t get it,” can be heard throughout the room.

Lynne positions Steven in front of the monkey chair and tells the class that he is ready. Steven signals to the class that he is starting and walks twice around the circle, stopping at the pig. This time there are more “Oh, I get it” replies filling the room than before. Lynne calls on Marcus, who announces, “27!” Acknowledging this correct answer, Lynne asks the students to write Steven’s number in division form A. Katherine reads hers: “27 = 2 x 12 + 3.” Lynne writes this equation on the board and asks the students to verify that this is the equation they also wrote in their notebooks.

Lynne returns to Stasha and asks her to read the instructions for practicing the Winding Game (see Figure 5.5). Before moving on, Lynne asks, “What is the mathematical name for the winding number, the number that tells you how many complete revolutions to go around?” Lynne presses the class for answers until she is certain that students recognize the winding number as the quotient. Then she poses, “What’s another word for residue?” Estelle volunteers, “The remainder.” “So what do you call the number you were given?” asks Lynne. Diego remembers, “The dividend. And the twelve is the number we’re dividing by.”

Next Lynne directs the students to look at the recording chart for the Winding Game (see Figure 5.6) and gives them an example of how they will fill it out when they play the game. For example, “If Jasmine is the first member of the team to play, you write her name on the first space under ‘Winder.’ Now let’s establish the order of the teams.” There are four teams in the room, so Lynne assigns each a spot on the practice sheet signifying the order in which they will play the game. When students watch the players complete their trips around the circle, they write the number of winds in the “W” column on the sheet, the residue or remainder number in its column, and the equation in division form A in the corresponding space on the practice sheet. Students can also write the zodiac sign in the column designated for it.

Clarifying with an example, Lynne asks the students what would be the wind if the number were 26. Several students respond, “2”; others reply, “I still don’t get it.” Lynne directs the students

Instructions for Practicing the Winding Game

• 1. A player picks a number out of a hat. The player doesn't tell the number to anyone except the teacher. Suppose the number 38 is picked.
• 2. The player goes to the strategy desk to figure out how many laps she or he will have to go around the circle and how many chairs will be left over. Having picked the number 38, the player could just walk around the circle counting chairs until getting to the number 38. Or she or he could use a strategy to wind around the circle (walking quickly without counting the chairs).
• 3. The player always starts at the monkey (zero) chair and goes around until getting to the correct chair. By starting at the monkey chair and walking around the circle, the player gets to the number 38 by counting each chair. Or she or he winds around the circle, passing the monkey chair three times, counts two more chairs, and sits down in the second chair.
• 4. Each student records the winding number (how many times the student passes the monkey chair) and the residue number (the number on the chair where the student sits after the last time passing the monkey chair). In our example, the winding number is 3. The residue number is 2.
• 5. Each student uses division form A to write a division equation that represents his or her observation of the player's turn.

Division Form A Dividend = (Quotient x Divisor) + Remainder

38 = (3 x 12) + 2

FIGURE 5.5 Winding Game Instructions

Source: From The Algebra Project Inc. Transition Curriculum, Unit 2, Lesson 5, p. 68; with permission.

FIGURE 5.6 Recording Chart for the Winding Game

Source: The Algebra Project Inc. Transition Curriculum, Unit 2, Lesson 5, p. 70; with permission.

to decide within their groups who will go first, second, and so on. Sensing that some students are a bit confused, Lynne tells the class:

This is just a practice game; it’s not for points. Although we will be playing games for points during this term, this game isn’t one of those kinds. You will be able to ask your team for help

in figuring out the wind number, the equation, and so forth. You will get clearer about this as it goes along. And one other thing—one sheet that will help you is the sheet of equivalence classes that you made out in your groups earlier. If, for example, Marcus happens to pull the number 42, you can tell by looking at the sheet that he’s going to wind three times and that he’ll sit on chair 6, right?

Lynne: OK, Marcus, I guess you’re first. Pick a number from the envelope. Everybody needs to

be very quiet while Marcus is doing this so you can really concentrate. Now, what’s the one you always start at?

Students: The monkey.

Marcus draws a number and backs away from the group so he can read the number in private. He walks up to the monkey chair and announces to the class, “I’m starting.” Marcus walks around the circle six times and drops into the buffalo chair.

Lynne: Let’s see what we have. What was Marcus’s wind? Tymesha?

Tymesha: Six.

Lynne: The residue?

Diego: Five.

Lynne: And what zodiac sign is that? Anthony?

Anthony: The buffalo.

Lynne: Now let’s hear the equation. We need it in division form A, remember.

Stasha: That would be 77 = 6 x 12 + 5.

Lynne: If you didn’t get that equation, write it in now Your turn will come tomorrow Let me

leave you with this: the way I wound is not the only way to wind around the circle. With more practice, you might find more efficient ways to wind around the circle.

As the students are putting away their notebooks and getting ready to leave Lynne’s class for the day, Nathan is overheard saying, “I know another fun way.” Michelle inquires, “What is your way?” “To get 77, I’d go around 10 times, then go backward four times, then figure out how many chairs to walk—like five chairs—so I’d end up at the right one. That would really confuse the class! But I think I’d be right!” declares Nathan grinning.

Yvelyne’s visit to Lynne’s class at Young Achievers comes to an end but not before several students wave a happy farewell to her as they exit the classroom. “Come back to see us, Dr. McCarthy.” A little stunned, she replies, “Thanks for making me part of your class. I’d love to come back.” And as I turn off the tape, I find myself saying, “I’d love to visit, too.”