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]]>Common Core Standards Covered in this Lesson Plan:

CCSS.Math.Content.3.OA.A.1 | CCSS.Math.Content.3.OA.D.9 | CCSS.Math.Content.3.NBT.A.3 |

CCSS.Math.Content.3.NBT.A.3 | CCSS.Math.Content.3.MD.C.7d | CCSS.Math.Content.4.OA.C.5 |

CCSS.Math.Content.4.NBT.A.1 | CCSS.Math.Content.4.NBT.B.5 | CCSS.Math.Content.5.NBT.A.1 |

CCSS.Math.Content.5.NBT.A.2 | CCSS.Math.Content.5.NF.B.5a | CCSS.Math.Content.6.RP.A.1 |

We all know the “trick” of multiplying by ten. Just “add a zero,” right?

10 x 23 = 230.

But let’s not have our students rely on a trick alone. Today, let’s examine why this works.

Here, of course, is one group of 23 blocks (2 blocks-of-10 and 3 single blocks).

And here we have our 10 groups of 23, all lined up on the classroom floor.

There are several ways we could pack up the blocks to determine how many blocks there are all together. Today, I want to be strategic about how our class packs up the blocks so we can really examine what happens when we multiply by 10.

So… I help them see we have something convenient right before our eyes. There’s a special, efficient way we could pack them up:

Sometimes, all it takes is setting out some empty holders above our project. The students can envision each column of ten blocks sliding straight up to make a bigger block.

Each distinct column of 10 blocks is packed together…

Once packed, we can verify our answer: 230 (2 block-of-100 and 3 blocks-of-10).

But wait! Doesn’t this look awfully familiar?

I like to pause here so my students can really grapple with this. They realize that what they’re discovering is actually something they’ve known for a long time – that by definition 10 groups ARE 1 of the next sized unit! It’s almost too obvious. But, as with most things in mathematics, the elegance is in the simplicity.

Your students may not have the language to say this quite so specifically, but think of what incredible discoveries they’re making!

Now that’s powerful learning.

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Fun Variation:

Sometimes, for a special twist, at the beginning of the lesson, I split a classroom into three groups. Each group models a slightly different problem, such as:

Group 1: 9 x 23

Group 2: 10 x 23

Group 3: 11 x 23

When each group has laid out the repeated rows of 23 blocks, we pause and briefly discuss the visual similarities to each group’s set-up.

Then, each group packs up their blocks and shows the class their answer:

Group 1: 207

Group 2: 230

Group 3: 253

This is a great way to prompt the observation that there’s something memorable to the look of Group 2’s blocks. In fact, by seeing the three problems side-by-side, there’s a distinct and fun “a-ha” moment: there really is something uniquely special about multiplying by 10.

Want to learn more? Check out Lesson 3.6 in Operations with Whole Numbers and Decimals.

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A quick side note here. Perhaps your students will initially want to pack up the blocks like this:

Of course, this wouldn’t be wrong. In fact, it is vital that students be given plenty of opportunity to pack the blocks in whatever manner they like, to explore different possible packing patterns, and to reach the stage where they’re organically hungry to hear about why one method might be more strategic than another in certain scenarios. If they haven’t had enough free exploration time, forcing this method may feel overly rigid (or, even worse, it may convey the message that there is a single formulaic way to arrive at a correct answer).

If you haven’t already, start with some of these fantastic multiplication lessons with your students:

3.4 Record 1-Digit x 2-Digit Problems

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]]>Common Core Standards Covered in this Lesson Plan:

CCSS.Math.Content.3.OA.A.2 | CCSS.Math.Content.3.OA.A.3 | CCSS.Math.Content.3.OA.D.9 |

CCSS.Math.Content.3.NF.A.1 | CCSS.Math.Content.3.G.A.2 | CCSS.Math.Content.4.OA.C.5 |

CCSS.Math.Content.4.NBT.A.1 | CCSS.Math.Content.4.NBT.B.6 | CCSS.Math.Content.4.NF.C.6 |

CCSS.Math.Content.5.NBT.A.1 | CCSS.Math.Content.5.NBT.A.2 | CCSS.Math.Content.6.RP.A.1 |

A brief word before we begin our decimal lesson plan…

Introducing decimal blocks is exciting. The most important thing, however, is that your students have ample exposure to the whole number Digi Blocks prior to the lessons with decimals.

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:

Option 1:

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).

Option 2:

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!

Lesson Extension: Repeating Decimals

I definitely wouldn’t start with this lesson, but for a future lesson, here’s another favorite spin-off.

Give the students a division problem with an answer with a repeating decimal. For instance:

658 ÷ 3 (the answer will be 219.333333333 or 219.3 ̅)

The students start with 658 blocks and 3 paper plates.

When they divvy up the blocks, each plate gets 219 blocks, and we have 1 leftover.

The leftover single block “unpacks” to 10 tenths, and each plate gets 3 of the tenths, with 1 tenth leftover.

The leftover tenth block “unpacks” to 10 hundredths, and each plate gets 3 of the hundredths with 1 hundredth leftover.

The students can explain what has happened and what would continue to happen infinitely. They tell me what a repeating decimal is!

Similarly, 100 ÷ 3 is a fun problem. Some advanced students are able first to explain/draw/write about what will happen with the blocks. Then they can physically model the problem with blocks to check their answer.

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We’re excited to announce that Digi Make 10 is **free** for the month of April. Download now!. This special release is in honor of the National Council of Teachers of Mathematics conference in New Orleans.

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