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Understanding Place Value of Decimal Numbers to the Thousandths

Hi, I’m Tim Hudson. I’m the Curriculum Director at Dreambox Learning and I want to thank you for taking a few moments to look at how Dreambox develops place value understanding in students. We’ll take a look at a few of our tools and just want to briefly talk about what place value is and why it’s so important to kids. It’s critical to understand place value if you want to have any chance of future success in Mathematics. You have no chance really of understanding the computational algorithms in the upper elementary grades if you don’t understand place value.

So, in Dreambox, as we know -all great teachers know that developing mathematical understanding of place value in the early grades requires that we make coherent connections and have consistent learning progressions from the early grades, place value content with whole numbers, all the way through decimals later because if you don’t understand place value of whole numbers, place value of decimals is going to be nearly impossible to really understand.

Now, place value is more than just knowing that the number 49 is four tens and nine ones. That’s sort of the simplified version, the one digit for every place, but a place value is far more than that. You consider with decimals that .25 is not just two tenths and five hundredths. We also in a lot of meaningful context need to think of .25 as twenty-five hundredths. And if we lock kids into that sort of one-digit-per-place idea, we’re sort of keeping their understanding of place value numbers shallower than it needs to be for long term success. 49 is also one ten and 39 ones or two tens and 29 ones. It’s kind of like how when we talk about time. Sometimes we say, well, this meeting will last 90 minutes or we might say this meeting will last an hour and 30 minutes or an hour and a half. They’re all the same duration of time. It’s just a difference in how we’ve grouped and how we’ve worked with the sort of place value of time. So let’s take a look at some of Dreambox’s tools and how we really developed understandings deeply in the minds of young children.

One of the challenges with mathematics software is we can’t have actual physical Manipulatives for kids to pick up in their hands and we know that’s a challenge. But at Dreambox, we’ve developed Virtual Manipulatives that give very compelling digital context and Digital Manipulatives, Virtual Manipulatives for students to make sense of the mathematical ideas. This first mathematical Virtual Manipulative that we’re going to show you for place value is our packing work space. What we do is you’ll see we have 49 loose items and students need to complete this table in terms of how many tens there are and how many ones there are. So, a student doing this lesson will be given something like two tens and 29 ones and they probably notice that, “Oh 49, well, that’s four tens and nine ones.” That’s the sort of simplified place value version with one digit per place. The student when they’re looking at this particular problem might also recognize, “Oh, if I have 49 loose items there aren’t any boxes of tens.” But for these next couple of answers as you can see, the student needs to bring up the work space, the packing workspace.

This is the concrete Digital Manipulative that students can interact with to help them make sense of place values. So, you’ll see the student packs a box so that’s one box that they’ve packed and at that point they realize, “Oh, that’s 39 ones that are left over to go with that one box of ten.” Then the student packs a second box, thinks about it, packs a third box, thinks about it, and then decides, “You know what? They don’t need the workspace anymore.” And at that point they can answer that last question that 49 is also equivalent to three tens and 19 ones.

As students interact with these Virtual Manipulatives, you can actually see what they’re thinking; actually see how they’re trying to work out the deep understanding of place value. So, here’s an example where now we’ve moved up into the hundreds, and a student is working with 452 loose items. So, a lot of students notice right off the bat “Well, that’s four hundreds, five tens, and two ones.” But, the student then isn’t a 100% sure where to go next so the student’s able to pack a case of 100 and at that point they see that, “Well, that’s one group of 100, no tens, and I’m left with 352 loose items.” Well, that’s another valid answer. That is a valid way to think about 452. The students have figured out the rest of the answers, can pack a case, pack two cases, pack three cases, pack a fourth case to bring about a 400 with 52 loose items, and can pack a box of one, bringing them to another answer of four hundreds, one ten, and 42 loose items, 42 ones. Once students get into the an immediate grade and start working with decimals, Dreambox lessons and Virtual Manipulatives build upon the great place value learning that kids acquired and understood in the early grades of elementary school.

Here is our “Decimal dials” as we call them. This is a Virtual Manipulative to help students understand the place value ideas inherent in decimals. In this question, kids are asked to build the number 548 using these dials. These dials operate very similar to the hands on a clock where when you adjust the minute’s hand, the hour’s hand moves in a very related fashion. So, as the students adjust the hundreds dial and moves it up to five hundreds, then moves over to the tens dial, and moves it up to four tens, you can actually see as can the student, the hundreds dial moving from five up to six because as you’ve moved from 500 to 540, you are getting closer to 600. Then the student can adjust the last one’s dial to eight. These dials were patterned after work done in Cathy Fosnot and Bill Jacob’s unit from context for learning called “The Mystery of the Meter.” They’re highly engaging and they really help build deep understanding of decimals for students.

The decimal dials as you can see from that previous example with whole numbers, they transition beautifully into working with decimal numbers. Here a student uses the same dials, but now the place values have changed so, the student instead of building a whole number, needs to construct 82.2. And we can see the student trying some things with the tens dial, moving that around kind of experimenting, seeing how that relates and impacts the other dials as well. And then working with the tens dial to see how many spins of the tens dial it’s going to take to move the other dials kind of like spinning the second hand on the clock to hopefully move the minutes and the hours. It’s going to take a lot of spins, kids realize this. So, ultimately the student decides to basically move the tens dial up to just past eight. That’s kind of like 8.2 tens and then the student adjusts the tenths dial to two and gets the answer correct.

This Dreambox Virtual Manipulative, these decimal dials are highly engaging for kids, and we actually create critical thinking opportunities for kids to really think deeply about decimals and about place value when we break one of the dials. Here a student is asked to build 7.3, but the tens dial is broken so, the student has to build 7.3 using only the ones dial and the hundreds dial. So, you’ll see the students spin the ones dial to seven and then use three spins of the hundreds dial in order to build 7.30. In this way, students are understanding that three tenths is the same thing as 30 hundreds.

Thanks for taking a few moments to learn about how Dreambox is Virtual Manipulatives doing an incredible job at helping students understand place value deeply from the very early grades on up through decimals. Thanks.

Tim Hudson, PhD
Curriculum Director
Dreambox Learning