Many students struggle with decimals – whether performing basic arithmetic operations, ordering and comparing decimal numbers, rounding with decimals, etc. One of the greatest misconceptions is that the decimal number system is unique. Too often, decimals are introduced as a new (i.e., different) and daunting topic, which sets up students to view decimals numbers as a separate system from the whole number system.
Using Digi-Block decimals right from the start, the students see that decimals are an extension of the number system they already know. They see for themselves that the same patterns hold true: each block opens to reveal ten of the next smaller block. This, of course, is why a solid mastery of operations with whole numbers is essential before the students see the decimal blocks. The decimals come to life when students recognize (dare I say, discover) these same patterns for themselves.
So just promise you won’t introduce the decimals prematurely, okay? Great. Thanks!
Alright, on to the crux of this post! How exactly do I introduce the decimal blocks? There are many ways, but these are my two favorite ways to start the lesson:
I set up what I call the “Progression” of blocks like this:
I point to the block-of-1000 and ask my students, “What happens when I unpack this block?” If they need prompting, I ask, “What’s the first thing I’d find if I open this block?”
The answer, of course, is 10 blocks-of-100. I provide the visual proof:
Then I ask, “What would I find if I unpacked a block-of-100?” The answer, of course, if is 10 blocks-of-10.
Next I ask, “What would happen if I unpacked a block-of-10?” The answer, of course, is 10 single blocks.
I ask my students to explain the pattern explicitly: each time they unpack a block, they find 10 of the next smallest inside.
Now we turn our focus to the single block. I ask, “What would happen if I opened a single block?”
This can either be a class discussion or an individual exercise. You can use this lesson plan and worksheet set as an aid, 1.3 Tenths and Hundredths
You’ll be amazed by answers you get. Here are some samples of student work that never fail to make me smile:
Aren’t these fabulous? From here, I either go to “Option 2,” or directly to “The Unveiling” (see below).
I give my students a division problem that I know will result in a decimal answer. Once again, it’s vitally important that the students have previous experience modeling division with whole numbers before jumping in with decimals. Even for students who are proficient in long division (and especially for those only proficient in the long division algorithm), modeling what happens during the process is very enlightening. Don’t skip these foundational lessons! Make sure your students have had practice modeling division with just whole numbers.
Here, we’ll model the problem 497 ÷ 4. I give them a word problem to ground this problem in reality. Each team of students starts with 497 blocks and 4 paper plates and goes about dividing the blocks evenly between the 4 plates.
When they’ve finished making their groups, they’re left with a rather unsatisfying answer: Each group has 124 blocks, but there’s one extra leftover block, a remainder of 1.
I play up this tension. “Oh! One leftover?! How are you going to share that one evenly?”
Now we’re ready to unveil the decimals!
The Unveiling: Behold the Decimals!
I let some uncomfortable tension build. I hold up a single block and say something like, “Boy, don’t you wish you could break this thing open?” and, “It certainly seems like the single block should be able to unpack.”
Then I ask an important question: “If I unpacked this block, how many smaller blocks do you think there would be?”
The students know the pattern. They answer ten.
Here, I start to ham it up. I say, “What if I told you I have a magic electric Exacto knife that gets this block unpacked?” Some students are skeptical, some are hopeful. With a big act, I take a single block, turn around and pretend to perform surgery on the block, all the while making over-the-top “magic electric Exacto knife” sounds. When I turn around, I’m holding 10 tenth blocks. The students go wild.
Very soon thereafter, something incredible happens. The students intuitively ask, “Do they get even smaller?” Or, “Can we open up one of those?” I love it – this always makes my teacher heart skip a beat. “What do you think?” I ask them. All around, there will be nods. “If I could open it up, how many pieces would there be?” It’s so intuitive to them that, of course, the answer is ten. I repeat the procedure converting a tenth block to 10 hundredths and voila! The kids are hooked.
If you taught with Option 2, let your students resolve their division problem with decimal blocks. Each group gives me their single remainder block, I “dissect” it for them, and hand them 10 tenths. When they’ve shared their tenths evenly, they will have two leftover tenths.
They give me these, and I give them 20 hundredths, which they divide evenly. The answer is much more satisfying now! Each plate has 124.25.
Some astute students will make the connection that each plate received ¼ of the original remaining block, which is represented by the decimal 0.25. To avoid cramming too much into one lesson, I usually don’t explicitly point out this equivalency (we’ll have more lessons to teach this concept later), but I love that the physical model makes the concept so visually clear.
Okay, cool, but are the decimal blocks really equivalent?
For a really powerful visual, I tape 10 tenths together like this and pass it around the room. If you’re dexterous, you can also tape together 10 hundredths. There are usually a few students who take on the challenge of constructing these models for me after school.
The students can compare a group of 10 tenths to a single block……… and a group of 10 hundredths to a tenth.
And of course, it won’t take long for students to ask if they can open the hundredth block. I ask students to draw what they think it would look like. And then to explain what they think would keep happening with the system.
Members Only Club
At the end of the lesson, I have my students take an oath that they won’t tell their younger siblings about the decimal blocks. This adds to the special aura of the lesson… the students feel like they’ve been initiated into an elite club. How empowering to realize they can extend their place value knowledge to infinitely small numbers! I balance this carefully, though: The key is that while the decimals feel special, they are part of the same familiar base-10 system, and the same patterns will hold true with the decimal blocks. In the coming year, the students will have dozens of lessons to prove this for themselves. Stay tuned!
For your next decimal lesson checkout the amazing introduction to repeating decimals