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 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.
Stephen Anderson
