Reasoning about Decimal Operations

Written by Donna Boucher

Donna has been a teacher, math instructional coach, interventionist, and curriculum coordinator. A frequent speaker at state and national conferences, she shares her love for math with a worldwide audience through her website, Math Coach’s Corner. Donna is also the co-author of Guided Math Workshop.

The Common Core State Standards (CCSS) now have 5th graders computing with decimals. CCSS 5.NBT.7 reads:

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.

Well, that’s certainly a mouthful! This post is going to focus on place value strategies, or what I call number sense.

This post contains affiliate links, which simply means that when you use my link and purchase a product, I receive a small commission. There is no additional cost to you, and I only link to books and products that I personally use and recommend.

First, computing with decimals is essentially the same as computing with whole numbers. Don’t believe me?  Multiply 12.1 x 34, 121 x 34, and 121 x .34  What do you get? All three products contain exactly the same digits-4114. The only difference is place value, or in other words, the placement of the decimal point. We traditionally teach students to count the places after the decimal point in the factors and move that many places from the right of the product and insert the decimal point. That certainly works, but it’s just a “trick”, and tricks can lead to unreasonable answers.

John Van de Walle suggests that instruction on computation with decimals must start with estimating. If students can accurately estimate products and quotients, they are more likely to correctly place the decimal point. Look at the card shown below. The digits in the product of the two numbers are shown below the multiplication problem. Without multiplying, decide where the decimal point should go.

Did you place the decimal after the 28? Why? How would you justify your answer? 
 
With this next problem, you are placing the decimal points in the factors. It’s trickier because there could be more than one right answer (don’t you just love that!).
What solution or solutions did you come up with? 18 x 14.5 would work, but so would 1.8 x 145. Could you explain your reasoning on both those answers?
Not surprisingly, the same process works with division. Look at the card below. Where would you place the decimal in the quotient? Why?

So, I hope that helps some. Remember, you always want your kiddos to understand the math they’re doing. I think this little activity is very powerful and really encourages deep understanding.

Click here to download a set of cards like the ones above that you can use in your classroom. And, please, if you teach math, don’t start the year without a copy of the Van de Walle book for your grade level.

multiplying decimals

 

18 Comments

  1. Diane Hubacz

    I am so addicted to your blog! I especially love that you are from Texas ( I teach in The Woodlands!). Check out my post today about Math Analogies!

    I have nominated your blog for the Leibster Award! Come by my blog to grab it and pass it on!

    Diane
    Teaching with Moxie

    Reply
    • Donna Boucher

      The Woodlands? Well, howdy neighbor. Ha ha. I’m following your blog now, too!

      Reply
  2. Viv

    Thank you so much for the great cards! I love Van de Walle’s work.

    Reply
    • Donna Boucher

      Van de Walle is the man! 🙂

      Reply
  3. Karen Greenberg

    This is great. Thank you! I, too, love anything where the students have to explain why they are doing something. I can’t wait to use these in my classroom.

    Reply
    • Donna Boucher

      You’re very welcome, Karen. Thanks for stopping by!

      Reply
  4. Fontenot's Firebreathers

    I just found this blog by a link in Pintrest! I love it! I am always working on growing my math skills. I move to 4th grade from 5th this next school year! I am always looking for new ideas! Thanks for the great site!

    Reply
    • Donna Boucher

      Gotta love Pinterest! Good luck in your new assignment.

      Reply
  5. Anonymous

    Just found this via Pinterest – I’m a UK teacher and I think this will be a great extension task for some of my Year 4s. Thanks ever so! 🙂

    Reply
    • Donna Boucher

      Wow, kinda cool! Love international sharing! 🙂

      Reply
  6. Anonymous

    Thank you. I really like your ideas and it is refreshing to know others math teachers out there truly get the math. Van De Walle is a valuable resource!

    Reply
  7. Nancy

    When I first read about this in Van de Walle’s book, I was so excited! It’s so much easier and makes more sense than counting and moving the decimal. But once you move to both numbers less than 1, it’s a little more difficult. Any suggestions on how this applies to decimals that are in the hundredths? Like 0.03 x 0.12

    Reply
    • Donna Boucher

      Nancy, check out this post on multiplying decimals!

      Reply
      • Nancy

        Thanks – I think using those models for decimals is great.

        Reply
  8. Tiffany

    I hate this math. I can’t help my 5th grader because you don’t make this easy for parents to explain to kids. I’m really good at math, and I now have to google things in order to help him with his homework. This is horrible

    Reply
    • Donna Boucher

      I agree that it is very different from the way we learned math. I can see why you find it frustrating.

      Reply
  9. Krystal L. Smith

    Such a great post! I remember reading Van de Walle’s book in graduate school, and being upset at how my instructor expected us to learn and then teach math. I am glad I have a growth mindset mentality now because I am much more open to what I have read over a decade ago, and how and why I need to teach math differently than the way I was taught.

    Reply
  10. Sylvia Parker

    Thank you! I loved how you used reasoning to show why this works. While knowing a trick can be helpful, in the long term, it does not help with mathematical understanding. I’m off to rethink my lessons; in a good way. 🙂

    Reply

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