**Definition**

**Base 10 blocks are a set of four different types of blocks that, when used together, can help you to see what a number looks like and understand its value.** Not just simple mathematical operations such as addition and subtraction, base 10 blocks can be used to help understand the concepts such as multiplication, division, volume, perimeter, and area. Units rods, flats and cubes together make the base 10 blocks.

**Types of Base 10 blocks**

There are four types of base 10 blocks, namely,

- Units
- Rods
- Flats
- Cubes

Let us learn about them one by one.

**Units**

Units in the base 10 blocks are used to represent the ones. In other words they represent the numeral 1. Below is a general representation of units in base 10 blocks –

**Rods**

Rods in the base 10 blocks are used to represent the tens. In other words they represent the numeral 10. Below is a general representation of rods in base 10 blocks –

**Flats**

Flats in the base 10 blocks are used to represent the hundreds. In other words they represent the numeral 100. Below is a general representation of flats in base 10 blocks –

**Cubes**

Cubes in the base 10 blocks are used to represent the thousands. In other words they represent the numeral 1000. Below is a general representation of cubes in base 10 blocks –

**Addition and Subtraction using Base 10 Blocks**

In order to understand addition and subtraction of numbers using base 10 blocks it is important to recall the place value system of numbers.

**What is Place Value System?**

Place value is the basis of our entire number system. This is the system in which the *position* of a digit in a number determines its value. The place value of a digit in a number is the value it holds to be at the place in the number. Therefore, the number 65,471 is different from 17,645 because the digits are in different positions.

**Importance of place value in addition and subtraction**

Place value has an important role to play in defining the algorithm for addition as well as subtraction. We know that the place value is the basis of our entire number system. This is the system in which the *position* of a digit in a number determines its value. The place value of a digit in a number is the value it holds to be at the place in the number. Therefore, the number 65,471 is different from 17,645 because the digits are in different positions.

In the place value system, the numbers are written by using the digits 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9 with each symbol getting a value depending on the place it occupies in the place value chart. For example, three million two hundred seventy eight thousand nine hundred forty-seven represents a collection of 3 million, 278 thousand, 9 hundred, 4 tens and 7 units. Therefore, it is written by putting 3 in the millions place, 2 in hundreds thousand’s place, seven in ten thousand places, 8 in the thousands place, 9 in the hundreds place, 4 in ten’s place and 7 in the units place. We use the base ten number system to write numbers with each place having a value; ones, tens, hundreds, thousands, and so on.

Let us understand this by an example.

**Example**

Write down the place of each digit in the number 845

**Solution**

We have been given the number 845. Let us obtain the place value of each digit of the given number.

The place value of 5 = 5 × 1 = 5

The place value of 4 = 4 × 10 = 40

The place value of 8 is 8 × 100 = 800.

**Place Value and Face Value of a Digit in a Numeral**

The place value and the face value of a digit in a number can be defined as –

**Place Value –** The place value of a digit in a number is the value it holds to be at the place in the number.

**Face Value** – The face value of a digit in a number is the digit itself.

Now, we know that the place value of a digit depends on its position, whereas the face value does not depend on its position. For example, in the number six thousand eight hundred forty seven, i.e. 6847, the face value of 7 is 7. Similarly, the face values of 4, 8 an d6 are also, 4, 8 and 6 respectively. However, the digit –

7 has the place value 7 x 1 = 7, since it is in the units place

4 has the place value 4 x 10 = 40, since it is in the tens place

8 has the place value 8 x 100 = 800, since it is in the hundreds place

6 has the place value 6 x 1000 = 6000, since it is in the thousands place

In the expanded form, the number 6847 will be written as

6847 = 6 x 1000 + 8 x 100 + 4 x 10 + 7

It is evident from this that a number is the sum of the place values of all its digits.

Also, **Place value of a Digit = Face Value x Position Value**

**However, it is important to note here that the place value of 0 is 0 itself, wherever it may be.**

**Using the Place Value system for learning through base 10 blocks**

The following points need to be considered for using the place value system for learning through base 10 blocks –

- It is important to remember how a unit cube defines it as a “unit” or “one.”
- Next, we put ten unit cubes side by side and place a ten rod next to it. Here it is important to observe that they are of the same length. These rods are defined as a “ten.”
- Next, we place ten ten rods side by side. This makes a square with 100 unit cubes. We define the flat as a “hundred.”
- Now we stack ten hundred flats together. Here we define the large block as a “thousand.”
- The ten thousand blocks are stacked to illustrate ten thousand.
- Lastly, we measure off or try to imagine the size of a hundred thousand flats and a million cube.

**Addition and Subtraction by regrouping using base 10 Blocks**

Let us recall how we add or subtract numbers using regrouping. Let us consider two numbers 5 and 16. How will we add these two numbers? Let us find out.

5 = 0 tens + 5 ones

16 = 1 tens + 6 ones.

Therefore,

5 + 16 = 0 tens + 5 ones

+ 1 tens + 6 ones

———————-

1 tens + 11 ones

———————-

Now, 11 ones = 1 tens + 1 ones

Therefore, we have 5 + 16 = 1 tens +1 tens + 1 ones = 2 tens + 1 ones.

Hence, 5 + 16 = 21

Let us consider another example

**Example**

Let us consider two numbers 28 and 16. How will we add these two numbers? Let us find out.

**Solution**

We will follow the following steps to add these numbers.

- We know that 28 = 2 tens + 8 ones and 16 = 1 tens + 6 ones
- Now, 8 ones + 6 ones = 14 ones. As the sum of the digits at the one’s place exceeds 9, you must carry ones into tens.
- 14 ones = 10 ones + 4 ones = 1 tens + 4 ones
- Write 4 under ones column and carry
- Add the tens
- 1 ten ( that was carried over ) + 2 tens + 1 ten = 4 tens.
- Thus 28 + 16 = 44

Now, how to add the numbers using Base 10 blocks?

Suppose we want to add 28 and 34. How will we represent this addition? Below is the graphical representation of this addition.

Let us take another example.

Suppose we wish to add 75 and 48

The following would be the addition of these two numbers using base 10 blocks –

**Multiplication and Division of Numbers using Base 10 Blocks**

We know that the process of finding out the product between two or more numbers is called multiplication. The result thus obtained is called the **product**. Suppose, you bought 6 pens on one day and 6 pens on the next day. Total pens you bought are now 2 times 6 or 6 + 6 = 12. This can also be written as 2 x 6 = 12 the process of multiplication involves the following steps

- First, we write the multiplicand and the multiplier in columns.
- First, multiply the number at the one’s place of the multiplier with all the numbers of the multiplicand and write them horizontally.
- Make sure you write numbers from right to left and each number below the corresponding place value of the multiplicand.
- Now, move to the next line.
- Place a 0 at the one’s place of this line.
- Now, look for the digit at the ten’s place of the multiplier. Multiply the number at the ten’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked 0.
- Again move to the next line.
- Place a 0 at the one’s as well as ten’s place of this line.
- Now, look for the digit at the hundred’s place of the multiplier. Multiply the number at the hundred’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked the two zeros.
- Continue in this manner by adding an extra zero in each line until you have reached the end of the multiplier
- Add the numbers vertically according to their place values.
- The number so obtained is your result.

**Let us understand this by an example**

For example, Multiply 132 by 13

Solution

- First, we write the multiplicand and the multiplier in columns.

2. First multiply the number at the one’s place of the multiplier with all the numbers of the multiplicand and write them horizontally.

3. Place a 0 at the one’s place of the next line

4. Now, look for the digit at the ten’s place of the multiplier. Multiply the number at the ten’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked 0.

5. There is no more number in the multiplicand. Now, add the numbers vertically according to their place values.

6. The final answer is 1716. Hence 132 x 13 = 1716

Let us see another example where we 3 digits in the multiplicand.

**For example, Multiply 364 by 123**

**Solution**

1. First we write the multiplicand and the multiplier in columns

**2.**** **First multiply the number at the one’s place of the multiplier with all the numbers of the multiplicand and write them horizontally.

3. Place a 0 at the one’s place of the next line

4. Now, look for the digit at the ten’s place of the multiplier. Multiply the number at the ten’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked 0.

5. Place a 0 at the one’s as well as ten’s place of the next line.

6. Now, look for the digit at the hundred’s place of the multiplier. Multiply the number at the hundred’s place of the multiplier with all the numbers of the multiplicand and write them horizontally in the line where you had marked the two zeros.

7. There is no more number in the multiplicand. Now, add the numbers vertically according to their place values.

8. Hence the final product is 44,772. We can say that 364 x 123 = 44772

Now, if we wish to do the same using Base 10 Blocks, how would we do that? Suppose we wish to multiply 26 and 14. Following would be the multiplication of these two numbers using base 10 blocks –

Similarly, Division can be done as repeated subtraction or through building and analysing rectangular arrays. This can be done by letting the cube, flat, rod, and unit represent 1, 0.1, 0.01, and 0.001, respectively.

**Advantages of using Base 10 Blocks**

The following are the advantages of using base 10 blocks for various operations in mathematics

- A major advantage is that helps in the visualisation of the addition, subtraction, place value, counting, and number sense thereby aiding in understanding the various mathematical concepts.
- Practicing operations through Base 10 blocks helps in the cognitive development as a whole.

**Challenges while using Base 10 Blocks**

Although base 10 blocks serve as an excellent tool to understand the various operations in mathematics, it has its share of challenges as well. Some of the challenges which we might face during the use of base 10 blocks are –

- At time we tend to see the tens rod as a unit of ten which makes it seem as a single unit. That means that when we are counting 2 ten sticks and 3 ones, we can mistakenly count it as “ 5 ” instead of “ 23 ”.
- Regrouping using base 10 blocks is an area that it prone to errors.
- Another challenge I that when using base-ten blocks, we can begin by separating the two addends into different piles, but once composed, there is no distinguishing between the two addends. This can further lead to errors in the addition of numbers using base 10 blocks.

**Key Facts and Summary**

- Base 10 blocks are a set of four different types of blocks that, when used together, can help you to see what a number looks like and understand its value.
- Units in the base 10 blocks are used to represent the ones. In other words they represent the numeral 1.
- Rods in the base 10 blocks are used to represent the tens. In other words they represent the numeral 10.
- Flats in the base 10 blocks are used to represent the hundreds. In other words they represent the numeral 100.
- Cubes in the base 10 blocks are used to represent the thousands. In other words they represent the numeral 1000.
- The place value of a digit in a number is the value it holds to be at the place in the number.
- The face value of a digit in a number is the digit itself.
- At time we tend to see the tens rod as a unit of ten which makes it seem as a single unit. That means that when we are counting 2 ten sticks and 3 ones, we can mistakenly count it as “ 5 ” instead of “ 23 ”.
- Regrouping using base 10 blocks is an area that it prone to errors.
- Another challenge I that when using base-ten blocks, we can begin by separating the two addends into different piles, but once composed, there is no distinguishing between the two addends. This can further lead to errors in the addition of numbers using base 10 blocks.

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