# Multiplying Decimals

In Texas, multiplying decimals with products to the hundredths was added to the 5th grade curriculum with the last standards revision, and today I tackled it with some of our 5th graders. Shifting from multiplying whole numbers to multiplying decimals is a huge shift, so that means that the learning needs to be concrete.

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Time to bring out the base-10 blocks. As I was planning for my lesson, I thought through all the variations related to multiplying with decimals, and here’s what I came up with:

1. Whole number factor times mixed number factor (eg., 2 x 1.3)
2. Mixed number factor times mixed number factor (eg., 1.3 x 1.5)
3. Whole number factor times a decimal factor (eg., 2 x 0.8)
4. Mixed number factor times a decimal factor (eg., 1.3 x 0.4)
5. Decimal factor times a decimal factor (eg., 0.7 x 0.3)

think I got them all!

Before getting into the meat of the lesson, we had to cover some basics. First, we needed to establish the value of the base-10 blocks. When using base-10 blocks with whole numbers, the flat typically represents 100, the rod represents 10, and the cube represents 1. But when we shift to decimals, the materials take on new values. With decimals, the flat becomes the whole, meaning that it is now 1. That makes the rod one 10th, and the cube one 100th.

Next, I needed to make sure that the students understood how to make an area model to represent multiplication. So I asked them to use their base-10 blocks to create an area model showing 2 x 3. We labeled the sides and discussed that one side showed a length of 3 and the other showed 2.

Next up, I asked them to model 2 x 1.3. I really didn’t give them much more direction than that. I wanted to see what they would do with it. They struggled a little, but were actually very interested in the problem. I restated the problem as 2 groups of 1.3. That helped quite a bit and one of my pairs figured it out which, of course, led to a light bulb moment for the other group. We labeled the sides 2 and 1.3. I asked students to find the product, and they added the base-10 blocks and got 2.6. I pointed out that, as with whole numbers, the product was greater than either of the factors. So far so good.

Moving down the list, I asked them to try 2.3 x 1.5. Oh, BTW, with each new problem I asked them what had changed. They had to use vocabulary like factors, whole numbers, mixed numbers, and decimals to explain the differences in the problem types. That’s a huge advantage of small group instruction–you can really focus on getting the students to use precise mathematical language.

Okay, this was a trickier one. Again, I gave them very little direction. I just kept reminding them that one side had to show 2.3 and the other 1.5. They showed the 2.3 first, quite easily. When trying to show 1.5 on the other side, they initially had the rods placed vertically. I just rotated them so they were horizontal. Then I told them they had to fill it in to make it into a rectangle. They actually figured out on their own that they would need to use 100ths. Pretty cool! Once they had filled out their rectangle, I again asked them to find the product. Notice that this time they had to trade ten 10ths to make a 1. They found the correct solution of 3.45. I asked if the product was greater than or less than the factors. They said greater.

When we get ready to connect the manipulatives to the standard algorithm, they can draw their models and see why we place the decimal where we do in the product. So much better than just teaching tricks!

Finally, we were ready to move on to a decimal by a decimal. I really played up the drama, telling them with great flourish that next I was going to show them something that would blow their minds! Time to move to grid paper and attempt multiplying a decimal by another decimal. I wrote 0.7 x 0.4 on the whiteboard and asked what was different now. They told me that both factors were decimals. Perfect. I reminded them that one side needed to show 0.7 and the other 0.3. This I had to show them a little more directly. We used one color to color in 0.7 (red). We labeled that side to show that it had a length of 0.7. Then I showed them how to use the other color to shade 0.3 (blue), and we labeled it. I pointed out the area where the two colors overlap and explained that was the product of 0.7 x 0.3. They counted up the squares and got 0.21. Immediately, one student said, “Oh! 7 x 3 = 21!”.  I said Yes, but twenty-one what? And he told me twenty-one hundredths. Hmmmm, what do you notice about our product? Is it greater than or less than our factors? The students have just finished a unit on comparing decimals, so I was glad they could tell me that 0.21 was less than both 0.7 and 0.3. I reminded the students that we were taking something less than a whole (0.7) and we had less than a whole group of it (0.3). That’s a really hard concept to wrap your mind around.

It was definitely a whirlwind tour of multiplying decimals, but I’m glad they were able to see the connections between all the different models and notice that sometimes when we multiply the product is smaller than the factors. And I’m confident that the hands-on learning together with the small group setting really helped them own the learning. I’d love to hear your comments on multiplying decimals! I’m sure you’ve got great things to share!

If you are looking for some practice resources for multiplying decimals, check out this resource, which contains an I Have/Who Has? game and matching picture/expression cards. Now in both print and digital!

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1. Marlana says:

I love this! I struggle with teaching these models. I am going to try your way.

2. Shelly says:

Wow…. We were just working with kiddos today and this was so hopeful!!!! I know how to do it myself, but helping kids work through and understand more than just because I said so and put the decimal here.

Some of my kiddos made me feel like a super hero because they said you make this seem so easy!!! I love your other examples and can’t wait to share with the kids!!

3. Denise says:

I think using a concrete model helps students visualize the meaning of multiplying decimal values. We use models after students estimate products with decimals. Students are given task cards with two decimal factors and the digits of the product WITHOUT the decimal point. Students use estimation to determine where the decimal point belongs in the product. Example: 2.3 X 5.1 gives a product with the digits 1173. Students use estimation to place the decimal. They know the product is a bit more than 10, so there is only one logical place the decimal can go. Therefore, the product is 11.73. Then we move to the area model using base-ten blocks.

1. Jorey Beamesderfer says:

I like this too. I teach 6th grade and may try to get them to estimate prior to using the models.

1. Ann Zisser says:

I have taught these methods before, but I always want to tell them the โwhyโ. How do you explain to them why the model changes to that overlap model when multiplying two decimal numbers like 0.7 x 0.3?

1. Donna Boucher says:

Yes, it’s tricky! You are finding less than a whole group of less than a whole, if that makes sense. Think of this progression. 2 x 3 is 2 groups of 3, or 6. 1/2 times 3 is 1/2 a group of 3 (or 3 groups of 1/2), or 1 1/2. 0.7 x 0.3 is 7 tenths of a group of 3 tenths. I find it helps to use the “groups of” verbiage and connect it back to whole numbers.

4. Angela says:

How long did you work with a small group on this?

1. Donna Boucher says:

This lesson, which was an introductory first-teach lesson, was about 25 minutes long.

5. Stephanie says:

This is so helpful! I never learned multiplying decimals visually; I just learned how to adjust the decimal point. This really helps create a clear picture of what is actually happening when multiplying two decimals results in a smaller number. That can be very confusing to explain to kids, so this activity is a great way to prove it to them. I think this would be helpful for middle school students as well who need to practice their multiplication skills!

6. Rhonda says:

Help please! We are working with this model but are not sure what happens when we work with hundredths. It breaks down for us. Do you have an example of that? Rhonda

1. Donna Boucher says:

Right, as far as I know, it won’t work with hundredths times hundredths. At that point, they would probably be using the standard algorithm, but their work with concrete and pictorial models will help them better understand the decimal placement.

1. Rhonda says:

Thanks so much. I do love the concept and I think it helped the students. I admire all of your work and I know I am one of your biggest fans! Thanks for all you do. Rhonda

7. Alice says:

Thanks for this! I have been using these manipulatives to teach decimals but I love how your explanation is set out so clearly! Going to do this lesson today:)

8. Beverly Thompson says:

I am not seeing the share buttons for your blog anymore. I have always been able to share with my fellow teachers. Am I looking in the wrong place?

1. Donna Boucher says:

If you just want to share a particular post, the icons are at the bottom of the post, right before the comments. There are also social share buttons at the top to visit my Facebook, Twitter, etc. Let me know if you still don’t see them!!

9. Nicole Schuler says:

The models can get so confusing to the kids. I noticed my advanced classes struggle with the models more than my regular classes because they just want to do the math. But we spent FOREVER on models this year. But if was at the beginning of the year. By the time it was STAAR review, they had forgotten most of it.
I like the 2 groups of 1.3, saying it that way. Thanks for shantung!

1. Donna Boucher says:

Retention is always a problem, Nicole, and that’s so frustrating. If you’re using a workshop approach, consider having a previous skill activity as one of your workstations to provide that spiraling review throughout the year.

10. A.T. says:

Thank you for this post!

Hoo!! so helpful! then I can help my boy throat this! txs

12. Nina says:

I’m teaching 4th grade math for the first time after teaching 1st and 2nd for 10 ten years! I loved using base ten blocks in this way. What should the kids know how to do before teaching a lesson like this?

13. Edna says:

1. Donna Boucher says:

We usually represent the multiplication of a decimal by a decimal by shading the grids, as shown in the post. It’s difficult to do with base ten blocks, because you’re looking for the area where they overlap.

14. Shelby Sommerville says:

15. Sara says:

How would you model Mixed number factor times a decimal factor (eg., 1.3 x 0.4)? Thank you!

16. Vanessa Hudson says:

How do you explain that when multiplying to decimals (2.3 x 3.6) that you have to fill in the the rest of the area model with hundredths? I don’t want to say that the hundredths are making it a rectangle. Is there an explanation? Thank you.

1. Donna Boucher says:

I like for them to experience it first and then have the discussion of why it works, if that makes sense. So they create the rectangle, and then I can ask, “Why did we need to use hundredths to complete the rectangle?”

17. Natalie Simon says:

This is so cool! I’m very new to teaching, but although I have seen these manipulatives used before to describe decimals, I have not seen them used to represent decimal multiplication – this is very visually intuitive and elegant!

18. Susie says:

I would love you to do a post on manipulatives and decimal division, also a 5th grade standard I believe.

1. Donna Boucher says:

Great suggestion! Thanks!

19. Saul Schultheis-Gerrys says:

I notice you show the calculation with the traditional algorithm. Why not use the area model or partial product method. (Or FOIL?)

1. Donna Boucher says:

Yes, this post focused primarily on using manipulatives and then making a connection to the standard algorithm. My thinking was that by the time students are multiplying and dividing decimals they have had plenty of experiences with strategies such as the area model and partial products to build an understanding of the algorithm in their work with whole numbers. I went back to base-10 blocks since decimals operate in a way that’s so different from whole numbers. But since the standard algorithm works pretty much the same way for whole numbers and decimals, I left out the other strategies. Of course, if students aren’t secure with the standard algorithm, then definitely include work with the area model and partial products. Great point! Thank you.