This just happened in our fifth grade class a couple of weeks ago.
We were talking about Math in Focus Workbook 5A Chapter 2 Whole Number Multiplication and Division (if you have the book, it’s on page 49). The goal is for student to know that 340 divided by 20 yields the same answer as 34 divided by 2, which is 17. Everyone was happy with that.
The tricky part came in when there was a remainder. In Question 2, 630 divided by 60.
Student A: “By following the previous example, we know that 630 divided by 60 is the same as 63 divided by 6. Ten 6s goes into 60, which gives a remainder of 3. The answer is 10 R 3.“
Student B: “No. Ten 60 goes into 600, so the remainder should be 30, not 3!”
Why is there a difference? This puzzled the class for a long time.
The explanation is the remainder of 3 is out of a division by 6, while the remainder of 30 is out of a division by 60. While the quotient is the same, the definition of the remainder is dependent on the divisor. Without specifying the divisor, the concept of a remainder is ambiguous. Another way to look at this is the ratio between remainder and divisor, which is 0.5 in both cases.
For more related resources, please refer to our Multiplication and Division page.