Multiply Before Divide

Let’s ponder for a second.  How would you teach students how to solve this problem:

2.4 x 1.3

Would you multiply 24 x 13 and then count decimal places?  In years past, this is the method I taught students.  No reason to count decimal places other than it will give the correct placement of the decimal point in any problem.  I anticipated this strategy being taught to the group of rising 6th graders I’ve had the privilege to work with this summer.

In our previous session, we made sense of the standard algorithm.  I planned to transition next to dividing decimals using the standard algorithm.  As I consulted with Van de Walle on how to make this concept make sense for students, he gave this advice.

With this, I changed my plan to address multiplying decimals before dividing decimals.  In thinking forward, I knew I wanted the kiddos to apply the place value understanding we discussed with the standard algorithm as they divided decimals.  As I considered Van de Walle’s advice and what would be the most effective way to make these connections, I decided to focus on the relationship between whole numbers and decimals.  This might be easier for me to show you my thoughts rather than type it out:

Disregard the fact it says, “Multiplying Decimals in Context using the Standard Algorithm”. 🙂

I wanted the group to think about the relationship between the whole numbers and the decimals, so I begin with the Same, Different strategy.  They noticed the obvious like, same digits, some have decimals while others do not and 130 has a zero while 13 does not.  This was the perfect setup for what I wanted to highlight for them.

Let’s consider 13 x 17 = 221 and 13 x 1.7 and the relationship between 17 and 1.7.  1.7 is ten times smaller than 17, therefore the product of 13 and 1.7 will be ten times smaller.  What about 13 x 17 and 1.3 x 1.7?  1.3 is ten times smaller than 13 and 1.7 is ten times smaller than 17.  This makes the factors one hundred times smaller than the original whole number factors.  Which will in turn make the product one hundred times smaller than the original product of 221.

After conducting a think aloud, I assigned some parallel problems for practice to the group.  With the parallel problems, they worked in partner pairs answering 3 questions.  Each partner had a different problem, but both problems had the same answer.  This helped generate conversations around their thinking.

The biggest struggle was determining what the product would be when it is ten times, one hundred times or one thousand times smaller than the original.

In 4th grade, students learn as digits shift to the left, they become ten times larger.  Or as the standard states, the digit is ten times what it represented in the place to the right.  4.NBT.1 Recognize that in a multi-digit whole number, a digit in one place represents ten times what it represents in the place to its right. For example, recognize that 700 ÷ 70 = 10 by applying concepts of place value and division.

In 5th grade, students learn as digits shift to the left, they become ten times larger; and as digits shift to the right the become ten times smaller.  5.NBT.1 Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.

I have yet to find in the standards where it states the decimal point moves.  THAT’S BECAUSE THE DECIMAL NEVER MOVES!  IT IS ALWAYS BETWEEN THE ONES AND TENTHS PLACES.  Now remember, students’ understanding of multiplying decimals will support their understanding of dividing decimals.

The images above and below shows the shifting of digits as it becomes ten times smaller and one hundred times smaller.  Images similar to this helps to reinforce the two standards learned in 4th and 5th grade and debunks the idea of the decimal point moving. (Oh and I kind of told them if their teachers says the decimal point moves not to believe them.). I may have said it once or twice before, but if you are teaching students the decimal point moves, you are undermining the place value understanding emphasized 1st through 5th grade.

The group asked if I would give them a problem in which they could practice by first solving the multiplication problem of whole numbers and then determine the products of related problems.

In the long run, they will understand why counting decimal places of factors will match the decimal places in the product.  This will help them understand division problems such as 9.8 divided by 0.3, which I will share in my next post.

Thanks for reading.

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