## The Best Intro to Decimals Lesson You Can Teach Your Class

Oct22

The Best Intro to Decimals Lesson You Can Teach Your Class

A brief word before we begin…

Introducing decimal blocks is exciting.  However, it is essential that your students have ample exposure to the whole number 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 decimal 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, 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, using the top half of this worksheet.

You’ll be amazed by the answers you get.  Here are some samples of student work that never fail to make me smile:

See more student work here!

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…)

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.

## NCTM Denver Raffle Winners

Oct09

Digi-Block is pleased to announce the three raffle winners from the NCTM conference in Denver.  Each received a free classroom starter kit (a \$300 value).  Congratulations!

Tabitha Savage

K-4 Math Coach

Putnam County Schools, Tennessee

Kathleen Strange

K-5 Math Coach

Mount Diablo Unified School District, California

Zaia Thombre

Nashville, Tennessee

Swing by the Digi-Block booth at future conferences for your chance to win!

## Subtraction with Digi-Block vs Base-10 Blocks

Oct03

Watch to see how the same subtraction problems are done with Digi-Block and how they’re done with Base-10 blocks. If you’ve ever seen your students make mistakes while trading (with base-10 blocks), you’ll immediately love that there is no trading with Digi-Block! After you watch the 43 – 15 video, be sure to check out the 430 – 150 video. With Digi-Block, you see how these two problems are essentially the same, just shifted a power of ten (it’s like zooming in and out with a camera). With base-10 blocks, this connection is lost and it seems like an entirely new problem.

Here’s the 43 – 15 video:

Here’s the 430 – 150 video:

## How a 7 year old invented something better than the decimal point

Sep25

The decimal point baffles many children. They often don’t know what it’s for and what the numbers that follow it represent. It doesn’t have to be this way! The best success I’ve had teaching the decimal point has come from guiding my students to essentially inventing it themselves.

When we’re first teaching kids numbers, we’re only concerned with whole numbers (1, 2, 3, 4, etc.) so we can simply say that the number, or digit, that’s furthest to the right corresponds to the ones, which with Digi-Block is the smallest block. But what if we want to be able to express numbers that aren’t whole numbers, such as 20.75? Or 0.3? I can introduce the decimal blocks, but now there is no longer a “smallest” block, because the blocks could in theory always get smaller and smaller, so we can no longer use the simple rule that “the smallest block is the number at the right end.” We need something new!

The way I teach explores these ideas and shows the need for a new device for writing numbers. This engages and excites students. They might come up with a solution that works. Even if they don’t, they’ll now appreciate the solution and have a deeper understanding of why we’ve introduced something new.

First, you need to explore the idea that there’s stuff out there smaller than 1, but bigger than 0. When I teach with Digi-Block blocks, I ask kids, what do I find if I open up a block-of-10? They quickly respond, “10 ones!” I then hold up a one, or single, and ask, what do you think I’d find if I could open this block up too? They think about it and say, “10 smaller blocks inside?” And I say, “Yes, you’re right!” And then I show them the tenths blocks.

Start by writing a multi digit number down, such has “572”. Ask your student to read you the number. Ask him “How do you know which digit is the hundreds? Which digit is the tens? Which digit is the ones?” This question might leave him staring blankly back at you or you might have an interesting discussion about place value.

Regardless, the next step is to open up your student’s thinking. You’ll use different color ink for each power of ten. The idea here is that we can use color, not position, to determine the value of each digit. Tell him “I’m going to use purple for hundreds, blue for tens, and green for ones.” Now write the same digits on a piece of paper, but use green for the “2,” purple for the “5,” and blue for the “7.” Now ask your student to read you this number. If he deciphers the code correctly, he should realize it’s the same number as before: 572! Try a few more examples until he gets the hang of it.

Now ask him if he can think of a different way, maybe without color. I had one student who wrote numbers in different sizes to represent powers of ten. It looked like this:

As he tried to write more and more numbers, and longer numbers that required more than 3 digits, he decided this wasn’t very practical.

I then presented him with our problem: we have things smaller than 1. I bring the tenths blocks back out to show him what I mean. (It’s not important to name these yet, you’re just using them to show a unit smaller than the singles.)

I put out several blocks (a few hundreds, tens, ones, and tenths) and said, how do I write this down? I then took them away and said, or if you write a number down for me, a number that should use those little blocks (the tenths blocks), how will I know what to build?

After some time to think about it. I came back to the “572” we had written before. I asked which way do the digits mean bigger blocks? Which way do they mean smaller blocks? He quickly saw that the digits to the left were for larger blocks and the digits to the right were for smaller blocks. He jumped up and said that the digit for the smaller block must to the right of the one!

He wrote a 4 digit number down, “4823,” but realized it looked like four thousand eight hundred twenty three.

I asked him, what can we do to let me know which digit is for which size block?
(Like how we used color before…)

He thought about it and then showed me his idea: he would circle the digit in the ones place. I thought this was fantastic! After some more thinking, I think this is actually better than the decimal point.

Circling the digit in the one’s place shows the relationship between the digits to the left and to the right of the one’s place much more clearly. Ten is one power of ten greater than 1, and one tenth is one power of ten less than 1. 100 is two powers of ten greater than 1 and one hundredth is two powers of ten less than 1, and so on. The decimal point obscures this and makes it look like the relationship between 1 and one tenth, 10 and one hundredth, and so on, should be emphasized…

Back to my student! I told him I thought his idea was great, but a long time ago, when people had to solve this problem, they came up with something a little different: the decimal point. I showed him how to write a number with a decimal point and then asked him to build it in blocks.

## Use Digi-Block to show Subtraction with Borrowing | 32 – 15 = 17

Sep16

The written algorithm of subtraction with borrowing can be pretty confusing to teach and to learn. What does crossing out the number mean? Carrying the one? Etc.

When you use Digi-Block, this operation becomes demystified. If you want to take 15 away from 32, you’ll quickly realize there aren’t enough singles (or ones) available to take away 5 ones. So you open up one of the tens. You now have one less blocks-of-10, this is the “crossing out.” When you open that block-of-10 you’re revealing 10 singles. That’s “carrying the one.” In essence, you’re adding 10 ones to whatever amount of ones you started with.

Want to see a different problem modeled out? Let us know in the comments!

## Life without Base-10 is a mess!

Aug14

Base-10 allows us to think about quantities large or small in an organized way. It helps us understand the magnitude of a number and lets us efficiently do operations.

Let’s look at what life might be like without Digi-Block blocks:

Can you guess how many blocks there are here?

Seriously, think of a guess! OK, now I’ll pack the blocks…

Much easier, right?

You could figure out that there are 123 blocks in the first image by individually counting each block. This isn’t a wrong approach, but it’s quite slow and tedious. Viewing each block as an individual unit makes it difficult to efficiently think about large quantities.

When you conceptualize numbers with a base-10 mindset, as in the second photo, you can see how ones units compose tens units, tens units compose hundreds units, and so on. Understanding this makes it much easier to visualize quantities. With the blocks packed up, allowing us to see this quantity in base-10, it takes just a quick glance to see that the quantity is 123 — composed of 1 block-of-100, 2 blocks-of-10, and 3 ones.

Aug14

## Beyond Base 10 Blocks: Numbers From the Real World

You don’t need a math textbook to help your kids study their math skills at home. If you have a newspaper or magazine, then you have access to a variety of math activities reinforcing all sorts of skills right at your fingertips.

### Number Search

To teach your young elementary school children to identify numbers, have them read through a newspaper and circle or highlight any number they see between 1 and 100. Have your child say the numbers aloud as he or she finds them. Then, make a list of all the numbers that you’ve found together on a page, and order them from smallest to largest.

This is also a great intro to having your children engage with age-appropriate current events.

### Counting Book

You and your child can make your own counting book using numbers and pictures found in a newspaper. Have your child cut out pictures from a newspaper and then compile a picture book. For example, for page one of your book, your child will need to find a picture from the newspaper with just one thing in it. Page two of your book should have a picture from the newspaper with two similar things in it, and so on.

At the bottom of each page of the book, have your child write the number of items in the picture and the name of the items. Once the book is complete, have your child read the book to you and other friends and family.

### Sorting and Organizing: Scavenger Hunt

For this activity, you and your child will need a newspaper/magazine, a pair of scissors, paper, and glue. Search the pictures for examples of cylinders, cubes, and other geometric shapes. Cut out pictures and glue them on a piece of paper to make a book of geometric shapes. Group the like shapes together and have your child write the name of the shape at the bottom of each page.

Aug14

## Beyond Base 10 Blocks: Fun with Numbers

How many times will your kids ask, “Are we there yet?” while traveling? Kids hate being cooped up in a vehicle for long. Rather than rely on those TVs in the backseat, parents can help entertain their kids while on the road and reinforce their math skills at the same time. With these sure-to-please activities, your child will have tons of fun on the road without even realizing that he or she is learning.

### Guess the Number

This game is similar to ‘I Spy,’ except numbers are used. Tell your child to think of a number between 1-100. You will try to guess your child’s number by asking questions such as, “Is it greater than 20?” “Is it an odd number?” Your child can only answer “yes” or “no.” After you guess your child’s number, you pick a number and let your child try to guess by asking similar questions.

### Find the Numbers

On a piece of paper, have your child make a chart listing numbers 1-50. Like a Bingo game of sorts, whenever they spot a number on a road sign, license plate, billboard, etc., have your child make an X on that number. The first to find all 50 numbers is the winner!

There are many other games involving math that you can make up while you’re on the road. If you keep your kids busy doing math, you won’t hear “Are we there yet?” nearly as often.

Aug14

## Beyond Base 10 Blocks: Understanding Money

When should you start teaching your child about money? How young is too young for an allowance? At what age should your child be able to solve math problems involving money? Parents and math teachers cannot agree on the answers to these questions. It all depends on your child. If he or she expresses an interest in learning about money, here are some tips and activities to help your child understand the concept of money.

Ages 6-10   For children ages 6-10, here are some basic concepts about money that you can reinforce at home.

• Identify Money: Make sure that your child can name each coin and their values.
• Make Change: Make sure that your child knows how much money to present at a sales counter when paying for an item. Also, make sure your child knows when to expect change back, and how much.
• Manage an Allowance: Make sure your child knows how to handle an allowance. This may require teaching your child to save up money for desired items.

Ages 11-13              As your child gets older, his or her understanding of money should increase. Here are some tips for children ages 11-13.

• Set Savings Goals: Encourage your child to set spending goals instead of just making impulsive purchases. For example, have your child save spending money for an upcoming family vacation or a class trip.
• Open a Savings Account: Allow your child to empty his or her piggy bank, count the money, and open a big-kid savings account at a local bank.
• Donate: Your child is at the right age to begin donating to charities. Your child should learn about worthy causes and donate a small portion of his or her money.
• Learn to Shop: Allow your child to spend his or her own money at a store as he or she sees fit. This will quickly teach your child about the value of money and merchandise, as well as the value of sales!

Ages 14-18   By the time your child reaches this age group, he or she should have a good understanding of the value of money. It is still very important to reinforce money concepts even at this age because it will not be long before your child is out in the real world on his or her own. Instilling responsible spending is critical. Here are some tips to help children ages 14-18 learn more about the value of money.

• Save for College: Teach your child about the different options for paying for college. Talk to your child about scholarships, student loans, and Pell grants.
• Get a Job: The quickest way for your child to learn the value of a dollar is for him or her to actually get a job. Your child will soon see how hard it is to earn money.
• Learn to Budget: Help your child create a budget to allocate money for all of the things he or she needs.

These tips for teaching the different age groups are not set in stone. Your child may be ready to learn these concepts at an earlier age, or may not be ready as soon as you think. Talk to your child and find out what he or she already knows about money and move ahead accordingly.

## Activity for Parents and Teachers to Teach the Size and Distances of Planets

Aug14

Practically all kids are interested in the planets and outer space. You can encourage their interests in astronomy while teaching them math and science at the same time with these simple activities for older elementary students. These activities will also help reinforce critical thinking skills.

Before beginning, make sure your child has some basic background information such as the names and order of the planets in our solar system. Showing your child a model or illustration would be helpful. Explain to your child that most illustrations show the planets in relative size. Discuss with your child the meaning of relative size, showing that the pictures depict how big the planets are when compared to each other and the sun. Ask your child to identify the smallest planet and the largest planet.

If your child understands the sizes and ratios of planets fairly easily, challenge them a little bit more by explaining the astronomical unit (AU). (An AU is a simplified number used to describe a planet’s distance from the sun. An AU is equal to the average distance from Earth to the sun, approximately 149,600,000 kilometers.) Help your child draw the following conclusions:

• Planets farther away from the sun than Earth have an AU greater than 1.
• Planets closer to the sun than Earth have an AU less than 1.

Teachers can emphasize the distance of the planets from the sun by taking a class outside to a large area. Different students could each represent a planet by taking the following number of steps away from an object representing the sun:

• Mercury = 1 step from the sun
• Venus = 2 steps from the sun
• Earth = 2.5 steps from the sun
• Mars = 4 steps from the sun
• Jupiter = 13 steps from the sun
• Saturn = 24 steps from the sun
• Uranus = 49 steps from the sun
• Neptune = 76 steps from the sun

After portraying the model, the teacher should ask students to describe what they observed about the distances of the planets from the sun and from each other. The teacher should point out the math connection of this activity, and thus explain how closely related math, science, and outer space actually are.

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