There have been some interesting conversations about the process of multiplying and dividing by 10, 100 and 1000.
We started by asking the kids what they thought a decimal was. Here are a few of their responses:
A decimal is....
- less than a whole and represented by the decimal point in it
- like a fraction
- a fraction of an integer (number) which can be divided and multiplied to make a number
- another way of representing fractions and percentages e.g 25% = 0.25 of a whole
- part of a whole number
- a dot that indicates the number after it is less than one whole
- a decimal is a fraction of a whole number usually used in complex maths problems or as an alternative to percentages
- a decimal is a way to to show numbers smaller than a whole and a way to show exact numbers
So next we started to play with some numbers.
What happens when you multiply a decimal by 10?
Here's a few responses from the kids...
0.37 x 10 = 0.370Here is a very common mistake. We know that when we multiply by 10, you just put a zero on the end. This indicates that the student doesn't really understand what is going on with the process, that to increase by a factor of 10 we move the decimal point one position (since we have a base-10 number system). With whole numbers, this might involve adding a zero to the end of the number. With a decimal, you need to move the decimal point.
0.37 x 10 = 30.7This indicates that the idea of place value is still not fully developed. Why does the zero suddenly appear in the middle of the number? The student has remembered that something happens with a zero so slots one in where it doesn't belong.
0.37 x 10 = 0.47This is a bit more like addition than multiplication. An interesting way to solve the calculation but failing to grasp the idea that multiplying by 10 will increase a value tenfold.
The Importance of Context
Many of the issues that arose from these examples of errors and misunderstandings came from a lack of context. I had presented the "question" 0.37 x 10 in a purely symbolic form, without a background, context or even a diagram. I did this deliberately to see what would happen.
I found out....
Yes - we need to teach the skills.
But - the skills need to be presented in a meaningful way - a context that explains what is being asked.