Lynne begins the lesson, which uses information about the Chinese zodiac, by directing students to answer the warm-up questions written on the board:
Task 1: The year 2002 is a year of the horse. How do you know?
Task 2: Use division form A, which states that Dividend = Quotient x Divisor + Remainder, to write an equation for 2002.
Task 3: What will be the next year of the horse? Explain/show
Each student has a placemat (used frequently in Chinese restaurants) on which is a drawing of the Chinese zodiac showing the 12 animals of the zodiac in symbol form (See Figure 5.1). The symbols are arranged in a large circle on the mat, and, with each animal, there is a list of corresponding years in the 20th century of their occurrence in the calendar (see Figure 5.2).
After a couple of minutes, Lynne asks for a volunteer to share the answer to the first question. Steven comes to the board and, holding his mat so that his classmates can see the circle, points to the symbol of the horse. He reads the dates of the years of the horse (see Figure 5.2) and then writes on the board 1930, 1942, 1954, 1966, 1978, and 1990. He then explains, “I looked at it and noticed that if I added 12 to 1930, I got 1942. Every time I added 12 to a year, I got the next year.” He takes the chalk and connects pairs of years, writing “+12” with each pair. “So I added 12 to 1990 and came up with 2002.”
FIGURE 5.1 The Chinese Zodiac
FIGURE 5.2 Year Charts: Twelve animals rule a particular year in rotation. In addition to lending characteristics to people born during that year, the animal also influences the quality of a particular year. So according to Chinese astrology, knowing which animal rules a future year can help you plan in advance for the trials and triumphs that that year may bring.
The following dialogue reveals the various ways in which students begin to develop strategies to solve the problem:
Stasha: I have a different way
Lynne: Come to the front of the class and show us.
Stasha: I looked at the year of the horse and saw 1990, so I counted all the way around the
circle, 1991, 1992, and 1993, until I got to the year of the horse again. That’s when I got to 2002. (Stasha demonstrates using herplacemat.)
Lynne: Now we’ve found two methods. Is there another one?
Samantha: I looked at the latest date, the largest number (1999), and the rabbit. Then I went to the tiger (1998). That was less, so I went forward and counted around until I got to 2002, which ended up on the horse.
Lynne: So I see a counting-on strategy, a similar strategy to what Stasha used. But, Samantha,
it was a little less counting than what Stasha did because you started at 1999 and she started at 1990. Any other way?
As Marcus comes forward, Lynne comments, “These brains are warm today!”
Marcus: OK, first I took 2002. (He writes 2002 in large numbers on the board.) Then I divided it by 12. I divided it by 12 because in our book we had divided other years by 12.
(Note: In an earlier lesson, when students first explored the Chinese zodiac and looked for patterns in the years, they were encouraged to try dividing the years by 12.) Marcus shows his division in a step-by-step process, mentioning each use of multiplication and subtraction along the way In the end, he points to his answer of 166 remainder 10.
Marcus: So the remainder of 10 tells me that it is the year of the horse because that’s what we did before.
Marcus points to a chart made by the students when they were figuring out remainders of each year on the placemat divided by 12. Other years of the horse, when divided by 12, gave remainders of 10.
Lynne: Will any year divided by 12, where the remainder is 10, be a year of the horse?
Does anybody know whether that’s true or not?
Wanda: Yes. It’s true because it says on the chart what the year of the horse is when the
remainder is 10.
Lynne: So if I got a remainder of 7, what year would it be?
Several students: Rabbit.
Lynne: Good. Ivan, you’ve got another way?
Ivan writes “166 r 10” on the board. Before going on, Lynne asks him to explain where he got these numbers. Ivan tells her that he did the same thing that Marcus did. Next on the board, he multiplies 166 times 12, showing each step along the way Pointing to 1992, he adds the remainder of 10 and then gets to 2002.
Lynne: Thank you, Ivan. This is something I see Tymesha do a lot. It’s called a check of the work.
You figured it out like Marcus and then showed us how to check your answer. You didn’t just figure it out; you made certain. Great. How about someone for the second warm-up on the board? That’s writing the equation in division form A. Courtney?
Courtney writes “12 = 166 x 2002 + 10” on the board and steps aside. Lynne, recognizing Courtney’s error and wanting her to look at the reasonableness of her answer, asks Courtney to explain what she wrote.
Courtney: I wrote the equation in division form A.
Lynne: Yes, that’s division form A. Tell me about your equation.
Courtney reads her equation from left to right. Lynne invites her to rework the division problem that she solved and adds, “I hear some classmates say they have some problem with what you wrote for your equation.” Then Lynne turns to the class and remarks, “But maybe Courtney will figure this out.” Courtney proceeds to write her long division calculation on the board (2002 - 12), gets an answer of 166 r 10, and then labels each number in the equation: 2002 = divisor; 10 = remainder; 166 = quotient; 12 = dividend.
Dividend Quotient Divisor Remainder
12 = 166 x 2002 + 10
Noticing that Courtney has confused which numbers are the dividend and the divisor, Lynne asks her to look at her long division calculation, label each number in it, and explain her labels as she’s doing it. As Courtney starts to label 12 as the dividend, she realizes her mistake and says, “Oops!” She switches the 12 and the 2002 in her equation and is satisfied that her labels are correct.
Dividend Quotient Divisor Remainder
2002 = 166 x 12 + 10
Lynne: Now I know what all the confusion was about in your first answer.
(Moving on, Lynne returns to helping students discover patterns.)
Lynne: So when’s the next year of the horse, and how do you know that?
Lynne: When’s the next? When will it cycle around again so we’ll have the next time to say,
“Happy New Year, it’s the year of the horse.”
Lynne: And how do we know that?
Anthony: It goes around by 12; I added 12.
Moving her hand around in a broad circle to dramatize, Lynne says, “Yes, it’s a cycle that goes around in 12’s.”