## Key Concepts

- Place value of the digits in the given number
- Place value blocks
- Using place value blocks to add and subtract decimals

### Representing a decimal number using place value blocks

**Place Value:**

Place value can be defined as the value represented by a digit in a number on the basis of its position in the number.

For example,

The 5 in 358 represents 5 tens or 50.

The 5 in 7005 represents 5 ones or 5.

The 5 in 46,5 represents 5 tenths or 5/10

**Note:**

The face value of the digit remains the same in any number, but its value depends on where it is in the number.

**Place value blocks:**

They are base 10 blocks.

The place value of a digit can be represented using base 10 blocks. Base-10 blocks are especially useful in providing the ways to physically represent the concepts of place value and addition, subtraction, multiplication, and division of numbers.

**For example**

Represent 452 using place value blocks.

While representing the decimal numbers, we will use the following base 10 blocks:

We use **flats** to represent the **whole** and **bars** to represent **tenths** and **cubes** to represent the **hundredths** part.

For example,

Represent 0.41 using place value blocks.

Here it clearly shows that there are 4 bars and 1 cube.

That is 4 tenths and one hundredths = 4/10 + 1/10

Let us practice a few.

- Write the digits of the decimal number 6 .23 under the correct column.

**Answer: **

- Write the decimal number that is represented by the base-10 blocks.

- If we think flat as one whole, write the decimal number represented by the base 10 blocks below.

- Form the decimal number represented by the yellow blocks.

### Addition of Decimals using Model

We know that,

These blocks can be used to add until hundredths.

**Example:** Add 3.22+ 5.94

**Step 1:** Consider 3.22

We have 3 ones, 2 tenths and 2 hundredths à place three flats, 2 bars and 2 cubes

**Step 2: **

Consider the second addend 5.94.

Here we can see that there are 5 ones, 9 tenths and 4 hundredths à Place 5 flats, 9 bars and 4 cubes.

**Step 3: ** Start adding ones, tenths and hundredths.

We have

That is, we have 8 ones, 11 tenths and 6 hundredths.

We know that 10 tenths make a whole flat. Hence, 11 tenths is nothing but one whole and one tenth.

That means we have 9 ones, 1 tenth and 6 hundredths.

- 3.22+ 5.94= 9.16

**Example 2:** 4.35+5.65

**Step 1:** Consider 4.35 à we have 4 ones, 3 tenths and 5 hundredths.

So, place 4 flats, 3 bars and 5 cubes

**Step 2:** Consider 5.65 à we have 5 ones, 6 bars and 5 hundredths.

So, place 5 flats, 6 bars and 5 cubes

**Step 3:**

Now start adding them up.

We can see 9 flats, 9 bars and 10 cubes.

When you observe the total, there are 10 cubes.

That means 10 hundredths (10/100), which is nothing but one tenth.

Now add this bar to the rest of the bars.

10 tenths = one whole

Adding this to the rest of the flats, overall we have ten flats.

That means 4.35+5.65= 10

### Subtraction of Decimals using Models

Subtraction always seems to be difficult but using base-10 blocks, we can make it simple.

**Example 1: **

Subtract 5.43- 1.29

**Step 1:** Consider 5.43

We can see 5 ones, 4 tenths and 3 hundredths.

Replacing them with base-10 blocks.

**Step 2:**

Now we need to subtract 1.29.

That is 1 ones, 2 tenths and 9 hundredths from the above.

**Step 3:**

If we observe cubes in the first number, we can find only 3.

But we need to subtract 9.

That means we need some regrouping in the first number.

That is, take one bar and separate the cubes.

Now the arrangement changes to,

**Step 4: **

From this, remove one flat, two bars and 9 cubes.

We are now left with 4 flats, 2 bars and 4 cubes.

Therefore, 5.43- 1.29 = 4.14

**Example 2: **

Subtract: 2.73-0.94

**Step 1:** Consider 2.73

We can see 2 ones, 7 tenths and 3 hundredths

That means we have to replace them with 2 flats, 7 bars and 3 cubes.

**Step 2:** Consider the second number 0.94.

That means there are 0 ones, 9 tenths and 4 hundredths.

That is 9 bars and 4 cubes.

To subtract this from the first one, let us start observing from cubes. We can take out 4 cubes from 3 cubes.

That means we need to regroup.

So, separate one bar into cubes.

Then 2.73 becomes

By removing 4 cubes we are left with 9 cubes.

Now observe the tenth place 0.94.

We have 9 tenths.

We can remove 9 bars from 6 bars. So need to regroup.

That is by converting 1 flat into 10 bars, we will have 16 bars.

Remove 9 bars from 16 bars, we get,

Therefore, 2.73-0.94 = 1.79

**Example 3:**

Without adding the decimals, tell if the sum of 0.43+0.35 is less than or greater than one? Explain.

The total number of bars we can find here is 7, which is less than 10 bars.

We know that 10 bars make one whole.

Hence, the sum of 0.43+0.35 is less than 1.

### Concept Map

### What have we learned

- Representing decimal number using base-10 blocks
- Adding decimals using base-10 blocks
- Subtracting decimals using base-10 blocks

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