I’ve been a math educator for more than fifty years, and I thought that by now I’d probably heard every possible answer to problems I’d given students. But “120 and 30/5” for the answer to 12.6 × 10 was new to me. This answer was given by a fifth-grade boy at Malcolm X Elementary School in Berkeley, California. My colleague Ruth Cossey and I were visiting his class along with the students from Ruth’s math methods course at Mills College in Oakland, California. Ruth and I were part of the Math Solutions team that developed the Math Reasoning Inventory (you can read more about the MRI below). Ruth had prepared her Mills students to use the tool, and we blitzed the class, breaking up into pairs and interviewing all of the fifth graders. Then, after school, we met together with the teacher and the principal to talk about what we had learned. It was a terrific professional learning experience.
From developing the MRI, Ruth and I knew that the two most common incorrect answers students give for 12.6 × 10 are 120.6 and 12.60. (Of 7,881 students interviewed nationwide, 51 percent answered incorrectly, and these two answers represent 39% of the wrong answers given.) To hear a sampling of how students reasoned, watch these four video clips of students who answered incorrectly.
In contrast, the following video clip of a student who got the correct answer of 126 shows an explanation that revealed the student’s understanding of the distributive property.
We didn’t videotape the boy who gave the answer of 120 and 30/5, but the students from Mills College who interviewed him wrote down his explanation:
First I multiplied 12 x 10 and got 120.
Then I changed .6 to the fraction 3/5.
I multiplied 10 times 3/5, and that’s 30/5.
So the answer is 120 and 30/5.
In our after-school discussion, we first talked about whether “120 and 30/5” is a correct answer. Some were sure it was wrong, others felt it was right, and others weren’t sure. (Yes, the answer is correct, but it’s certainly not conventional. One way to think about it is that 30/5 is equal to 6, and 120 + 6 is 126.) Trying to make sense of the boy’s explanation reminded me that understanding the math we teach is essential for understanding how students reason. (On reflection, I think it would have been a good discussion to present the fifth graders with two possible correct answers—120 and 30/5, and 126—and talk about why they were equivalent.)
Try the following with students who are studying about decimals. Write 12.6 × 10 on the board and ask students to figure out the answer in their heads. After a few moments, have students share their answers and how they reasoned with a partner, and then lead a class discussion. Here are questions that may be useful for the discussion:
- How would you explain to someone why 12.60 can’t be the correct answer?
[Since 12.6 and 12.60 are equal, 12.60 can’t be 10 times 12.6.]
- How could you use addition to figure out the answer?
[Add 12.6 ten times: 12.6 + 12.6 + 12.6 …]
- Why doesn’t the rule for adding a zero when you multiply a number by 10 work here?
- Why is moving the decimal point one space to the right the same as multiplying by 10?
Then write on the board the fifth grader’s answer: 120 and 30/5. Tell the students that this is an unusual way to express the answer but that it’s mathematically correct. Use the think–pair–share instructional strategy to engage students in thinking about this possible solution.