I need your help. I’ve been thinking of ways to better connect decimal multiplication with fraction multiplication, especially in terms of justifying moving the decimal point. Sixth graders may recall counting the number of decimal places, but do they really know what they are counting and why they are counting it?

The following is work in progress and I would appreciate your input. How can my attempt at creating curiosity be improved upon? Are the examples sufficient or should they be revised? What am I missing?

I’m anticipating students will say, “Count the decimal places,” so my follow up is this question.

I don’t think students will connect fraction multiplication with decimal place value so my next move will be to ask them to compare the two expressions.

Some students may recognize the two expressions are equivalent, but will they make the place value connection?

I prepared a second example that “culminates” with this slide.

Since the above problem has numbers in the ones place, some students may recognize 2 x 2 = 4, so it would be reasonable to place the decimal between the 5 and 6. I would certainly accept that, but my point is to discuss powers of ten. Yet I need to be careful. My sixth graders haven’t been exposed to negative exponents. Would it be appropriate to describe it as powers of tenths? Is this they best way to address the mathematics? I’ll likely begin with establishing a pattern before I present these slides.

I’m also thinking of creating a similar lesson which connects dividing decimals with dividing fractions using the common denominator method. The students are familiar with the common denominator method, but I wonder if it will be too messy?

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If you have been teaching multiplication using area, then this problem lends itself well to that. When you multiply 2/10 by 8/10 you have a grid of 16 colored areas out of 100, which would then be 16/100ths. If you connect that back to your 16 answer and do several of those, the students should figure out the “two decimal places” reasoning.

Thanks, Alisa. As I mentioned in a previous comment students are familiar with area models, but I think it would be best to add those visual representations along side the abstract fraction and decimal representations. I can’t tell you how much time I’ve spent thinking about this. Anticipating what students will say, attempting to provide clarity, and now you’ve given me even more to think about. I appreciate you taking the time to comment.

In my mathematical reasoning course, they taught us to use the phrase “we need to reunitize”, and banned “move the decimal”. I agree with the method that Alisha Carter posted about, if you have practiced multiplying with an area model, this is easier to see.

Students have practiced with area models but after reading yours and other the comments thus far I’m compelled to create a slide or two that includes the representational area model along with the abstract fraction and decimal. That may make a better connection while minimizing confusion. Also, re-unitize is a new vocabulary word for me. It provides mathematical clarity. I’ll be introducing it. Thank you for giving me something to think about.

I am so glad to see you writing about this. I see people writing about this every so often but never chime in. I am not a math teacher; I am more like a student who is still trying to figure things out.

I know that the decimal moves one way or the other, but I can never remember which way it moves. Now, I know that some teachers really dislike thinking about his as “moving the decimal,” but hear me out!

First (risking derision) I think that it’s okay to think about this process as moving the decimal. It took a long time to realize that the direction that I move the decimal shouldn’t be a memory action, but, rather, an action that is governed by estimation.All the memory needs to tell me is that 0.2 x 0.8 is that the decimal should be moved two places, so it’s either 16.00 or 0.16. How do I know which one?

Thinking about this, I know that 2 x 8 is 16, so 0.2 x 0.8 can’t be 16. I might be tempted to think the answer is 1.60, but then further thought tell me that 0.2 x 8 equals 1.6, which makes sense because there is only one decimal place in the question and in the answer. So this leaves me with showing two decimal places, as in the question, 0.16.

The point is that I am using my number sense to figure out which way to move the decimal.

I am okay with talking about ” moving the decimal over” :for this reason: In proofs that mathematicians use when looking at infinite series, the way that they multiply 0.99999999 by 10 is to MOVE TO DECIMAL OVER one place. This is absolutely common practice in upper mathematics, so deriding this method in elementary mathematics makes no sense to me. I hope that from my explanation above it’s clear that I do believe that developing a number sense needs to come first,

In your second example shwoing 2.45 x 2.3, you mention you would accept thinking about the answer as something close to 5 as being reasonable (as opposed to 56) but you want to reveal the math behind the decimals, which is understandable. But the way you have it in your slides, to me, invites confusion because there are so many different numbers to look at. If you want to simplify this to make it more palatable I would recommend using repeating numbers. Take a look at this mathologer video https://www.youtube.com/watch?v=SDtFBSjNmm0 which, at the 9:30″ mark, shows what you are thinking about, and shows it in a clear and accessible way.

Thank you so much for taking the time to write such thoughtful comments. I wholeheartedly agree that number sense comes first. Too often students are in a procedural mode and don’t stop to think if their answer is reasonable. I also appreciate the feedback on decluttering the slides. That will be my first order of business and it sounds like the link you provided will help me do that. Again, thanks for the feedback.

Mary, thank you for your post on this. If I can offer one more suggestion and leave it up to others to determine if this is a reasonable way to teach this.

What if you start by writing…

Ten x ten = ______

Hopefully all can agree = hundred and have a short discussion on modeling “groups of. ” Ten groups of ten is one hundred.

Then write

Tenth x tenth = _______

Now model or discuss I have a tenth groups of a tenth, what is that? (This is really clunky to say here, I can’t think of a better way at this second, but the area model or a visual can help here)

Hundredth.

Now ask them write it with numbers replacing the words.

10 x 10 = 100

.1 x .1 = .01

Then ask is this the same every time?

Hundred x ten = thousand

100 x 10 = 1000

Hundredth x tenth = thousandth

.01 x x .1= .001

What do you notice? What do you wonder?

Now what is

8 tenths x 2 tenths ? It is going to be hundredths, but how many? 16 hundredths

.8 x .2 = .16

I believe that most students miss the concept of 0.8 being 8 there when they just say “zero point eight” but if they keep the units in mind it could make more sense?

Would love some critique and pushback on this.

First of all, I like your reference to “I have a tenth groups of a tenth, what is that?” It invites discussion. What I noticed in your approach is the lack of reference to physical fractions (i.e. 2/10). That’s neither good or bad, just an observation. My intent was twofold 1) connect it back to fractions (tenths, hundredths, etc), then demonstrate the “movement” on a place value chart. I didn’t mention that in my post, but showing a chart would add a necessary visual representation. And 2) another reason why I wanted to use fractions was I thought this would be an opportunity for students to experience fractions as something that is actually helpful and not something to fear.

I hope this discussion continues. For me, the best part of reading a blog post is the comments. Thanks.

I knew there were good reasons for my choosing to teach early childhood. I have no idea of what you are doing, or its point. 🙂

Hi Norah, All I’m trying to do is make sense of a “fancy” pattern! On a related, but different note, I recently became aware of this organization. Are you familiar with http://earlymath.erikson.edu/

Thanks for that, Mary. I wasn’t aware of the organization before. I appreciate your letting me know.