Elementary students really enjoy learning based on tangible experience. When they can touch, move, and mix concrete objects, the learning process feels like a game, and this entertainment-type cognition is more natural for such young explorers. On the other hand, education should educate, and by providing students with toys or too much stimulation teachers may distract from learning. Thanks to research based on comprehensive observations, started as long ago as the 19th century, scientists, methodologists, and teachers developed and proved a positive impact of manipulatives — a palpable way to model multiple representations of math concepts. Manipulatives provide students with engaging physical experience without switching attention to gameplay.

In the Happy Numbers curriculum, there are several basic models that include manipulative mechanics. Up to Grade 2, students practice with manipulatives based on real objects, such as fruit, birds, etc. and become familiar with Base-10 Blocks, a more abstract visual model in which concrete objects are replaced by cubes. But when it comes to working with multi-digit numbers, Base-10 Blocks become inconvenient and obsolete. Therefore, Happy Numbers upgrades the pictorial representation of these numbers to the PVC, or Place Value Chart, in which students compose numbers using Place Value Disks.

It’s important to recognize that the origin of math can be traced back to commerce. Place Value Disks, which look similar to coins, help to foster a conceptual understanding of this historical basis of math. In addition, students indirectly learn how to handle money, a helpful skill for real life.

Now let’s take a closer look at the application of PV Disks!

## Have a Whack at Place Value Disks

Place Value Disks are pretty intuitive, as all manipulatives should be. Here’s how PV Disks are introduced in Happy Numbers.

To see the full exercise, follow this link.

Happy Numbers wants students to develop procedural fluency. That’s why they need to not only distinguish the disk’s value but also easily recognize the place value of each digit and the column corresponding to it in the table.

To see the full exercise, follow this link.

In these introductory exercises, instant feedback provided by Happy Numbers software helps students to quickly understand the model. From here on out, PV Disks become a very handy interactive visual support. Their usage ranges from *immediate assistance*, as shown below…

To see the full exercise, follow this link.

…to *scaffolding for conceptual understanding*.

Let’s now go over some important concepts that benefit from use of the PVC model.

## 1. Trading

Place Value Disks can be “exchanged” in the same way as money in our everyday life. The relation between disks of different values is simple, but very important. For example:

– 10 ones disks represent the same amount as 1 tens disk,

– 10 tens disks represent the same amount as 1 hundreds disk,

etc. It means that 10 ones disks can be *replaced *by 1 tens disk and vice versa, which is, in other words, *trading *(a.k.a. renaming, regrouping, and carrying over).

Engaging experience with the PV Disks helps students to grasp and apply the concept of trading. They start with simple exercises that require adding disks or taking them away:

Students have already added a tens disk to the model of 491. The group of 10 tens disks turns into 1 hundreds disk when dragged to the hundreds column.

To see the full exercise, follow this link.

Trading works in both directions. Click the Play button to watch a subtraction-related example of trading a tens disk for 10 ones disks.

Students arrive at this screen after realizing that there are not enough ones disks to give the monkey.

To see the full exercise, follow this link.

This is just the beginning: trading is included in all four of the operations below.

## 2. Column Addition and Subtraction of Whole Numbers

Column addition/subtraction is based on:

– Working place by place, right to left

– Trading between the places when needed

Place Value Disks illustrate both operations and help relate the model to the column addition/subtraction record.

Let’s walk through an example step-by-step. Here, both addends are represented by Place Value Disks.

First of all, students need to model addition of ones by combining the ones disks:

Then they trade 10 ones disks for 1 tens disk:

…and record the two-digit sum of ones in the ones and tens columns respectively.

To see the full exercise, follow this link.

With the help of PV Disks, the addition of numbers with a transition across tens or hundreds becomes much easier to grasp.

## 3. Multiplication of Multi-Digit by Single-Digit Whole Numbers

Multiplication by a single-digit number is one of two skills that forms the base of the Standard Algorithm for multiplication (the second is multiplication by 10, 100, 1,000, etc.).

Modeling multiplication by a single-digit number is very close to modeling Column Addition. The similarity is no surprise since both operations are place-by-place procedures with the trade connecting neighboring places. And multiplication is repeated addition, after all!

For example, here’s the number 124 represented by disks:

As students did before with Column Addition, they start modeling the multiplication of 124 x 3 by “multiplying” the ones disks:

Then they trade 10 ones disks for 1 tens disk:

Students work on the tens next. They:

1) “multiply” the original tens disks

2) combine the result with the tens disk they got from the trade:

To see the full exercise, follow this link.

Combining the disks in the tens column completes modeling of the second step — working with the tens of number 124. The result represents the tens digit of the product 124 x 3.

The visual model shown above displays the principle of working with any digit of a multi-digit number when multiplying by a *single-*digit number.

## 4. Division of Multi-Digit by Single-Digit Whole Numbers

Since division is the inverse of multiplication, it can also be shown using PV Disks. This type of visual modeling helps students grasp the basics of division in general and the Long Division method in particular.

Let’s look at an example. To represent division of 75 ÷ 3, Happy Numbers shows the dividend with disks and the divisor as three empty boxes below.

Students model division by distributing disks evenly into the boxes.

Like addition, subtraction, and multiplication, division by a single-digit number requires place-by-place work. However, division requires work from the left to the right, unlike the other three operations. So, students start division with the leftmost place:

Students see that the partial quotient for tens is 2, the leftover is 1, and a trade is needed to move to the next step:

After the trade, students divide ones and find the next partial quotient. Again, they model the division by evenly distributing ones disks:

Here’s the result!

To see the full exercise, follow this link.

Modeling with PV Disks truly helps students recognize that:

– Division can be done step by step, from left to right

– Each step of the division process consists of finding a partial quotient

– There may or may not be a remainder at any step in the process

– Any remainder from the previous step should be traded and contributes to the number which students will divide in the next step

These statements *remain valid for division by a two- or more-digit number*. The PVC model is important since there is hardly any other model that represents these four statements with multi-digit division.

## 5. Decimals

With the experience of modeling whole numbers and operations with them, students are ready to extend their modeling with PV Disks to decimals. Introduction to it starts with the ten*th*s and hundred*th*s. So, the disk model gets new ten*th*s and hundred*th*s disks.

To see the full exercise, follow this link.

The ratio between the disks remains the same:

– 10 hundredths disks represents the same amount as 1 tenths disk

– 10 tenths disks represent the same amount as 1 ones disk

It’s no wonder that the PV Disks modeling of decimal operations echoes that of the whole number operations.

The following screenshots show already familiar cases of trading:

Trading when subtracting decimals.

To see the full exercise, follow this link.

Trading when multiplying decimals.

To see the full exercise, follow this link.

At the same time, modeling of some decimal-specific situations helps students gain insight into them, as shown below.

The first example is multiplying a decimal by a whole number. One way to do that is to ignore the decimal point, multiply the numbers as whole numbers, and then identify the right place for the decimal point. PV Disk modeling supports the conceptual understanding of this strategy:

Students model two types of multiplication separately, one with the decimal point and one without it. This experience helps them figure out that the digits of the two methods are the same.

To see the full exercise, follow this link.

The second example introduces a new concept. In decimal division, it’s possible to divide to one or more places beyond the rightmost digit of the dividend. For example, students model division of 8.9 ÷ 2:

First two steps: division of ones and tenths.

Due to the equivalence of 8.9 and 8.90, it’s possible to add the hundredths column to the chart:

Then, students trade one tenths disk for ten hundredths disks and continue the division:

To see the full exercise, follow this link.

So, in decimal division, it’s possible to continue dividing the “remainder”. This contrasts with whole number division, which requires the quotient and remainder to be whole numbers.

## When Math Is Taking Shape

The emphasis of mathematical learning has shifted from memorizing solution patterns to building meaning behind these patterns. That’s why interactive models are the way to better understanding of the standard algorithms. They contribute to the student’s engagement and grasp of the concept. With PV Disks, students receive a comprehensive visual representation, which helps them master operations with whole numbers and decimals and make the next step in developing abstract thinking.

Join Happy Numbers to enhance your math curriculum with the best pedagogy!