Helping Students Work Through Mathematical Conjectures

In this thoughtful article in Teaching Children Mathematics, Amy Hillen and Tad Watanabe (Kennesaw State University) say that an important Common Core math reasoning skill is making conjectures and assessing them based on evidence. They suggest the following 60-minute lesson for elementary students:

• The teacher displays 1-9 on the board: 1 2 3 4 5 6 7 8 9
• The teacher picks the number 4 and writes it on another part of the board.
• The teacher calls on a student to pick another number – 8 is chosen.
• The teacher writes the 8 by the 4.
• “With these two numbers, which two-digit numbers can we make?” the teacher asks. 
• 48 and 84
• “Which is larger?” 84. The teacher writes the numbers using vertical notation:  84

      48

• The teacher asks students to subtract, and the answer is 36. 
• “Let’s try this with some other numbers.” This time, a student picks the first number, 7. The teacher picks the second number, 3 (making sure the difference is going to be 36).
• The teacher again sets up the subtraction problem:  73 – 37, and the answer is 36.
• “It’s the same!” students exclaim. “It’s always going to be 36!” 
• The teacher explains that this is a conjecture and writes it on the board – “If two different numerals are picked randomly to form 2 two-digit numbers, the difference will always be 36.” 
• “I wonder if this will always be true,” says the teacher. “How can we find this out?” Students suggest some ideas, and the teacher has students spend ten minutes working with a partner – just enough time for them to come up with only one or two possible combinations (students were given a template to set up the subtraction problems). 
• Students post their subtraction problems on the board.
• Everyone looks at the examples (duplicates are removed), and students notice that some problems have answers other than 36. Their initial conjecture was not true. 
• Now students are asked to work in pairs to revise their initial conjecture and formulate new conjectures based on the set of subtraction problems posted on the board. 
• Some students are boggled by the number of problems on the board and are prompted to sort them into groups with the same answer and asked, “What do you notice about the problems whose difference is 36?”
• Now the whole class discusses different conjectures. One student says that when the numbers chosen are next to each other, the answer is always 9. The teacher has the student clarify what “next to each other” means.
• Another student says that when the chosen numbers are separated by one number, the answer is always 18. 
• Now students work in pairs again to try to refine the conjecture. Students often notice that the difference will always be a “nine fact” – a multiple of nine; that all the problems that have the same difference involve numerals whose difference is the same; and the difference will always be a product of 9 and the difference between the two.
• Students are challenged to explain to a skeptic why these conjectures will always be true. The teacher asks, “How do we know that there is not a problem that we don’t have on the board whose difference is not a multiple of 9?” One approach is “proof by exhaustion” – trying all the possible problems. But is there another way? This might lead to organizing examples in a systematic way.
• The teacher concludes the lesson by pointing out the important reasoning-and-proving work they have done, the importance of perseverance when the initial conjecture proved false, and how that led to a more sophisticated finding. In fact, conjecturing and proving (and disproving) are the essence of doing mathematics.

To see a delightful video of a Japanese teacher teaching this lesson, go to (free registration):

http://www.globaledresources.com/user-info.php?id=curious-subtraction. 

“Mysterious Subtraction” by Amy Hillen and Tad Watanabe in Teaching Children Mathematics, December 2013/January 2014 (Vol. 20, #5 p. 294-301), http://bit.ly/1hz3128; the authors can be reached at ahillen@kennesaw.edu and twatanab@kennesaw.edu. 

From the Marshall Memo #516