Decomposing 2-Digit Numbers - First Grade Mathematics

In mathematics, "decompose" means to break something apart into smaller, simpler pieces. Just like you can take apart a toy castle built of blocks to see how each tower was made, you can decompose any two-digit number to see the tens and ones hidden inside it. Decomposing numbers gives you immense power because it lets you understand what each digit in a number is truly worth. In this chapter, you will learn how to break two-digit numbers into tens and ones, write numbers in expanded form, and explore multiple ways to decompose the same number.

Standard Form vs. Expanded Form

Every two-digit number can be written in two different ways: standard form and expanded form.

Standard Form vs. Expanded Form for 65:

Standard Form:     65     (The regular numeral)
                   /                  /  Tens Part:       60   5   Ones Part

Expanded Form:   60 + 5 = 65 (Shows the value of each part!)

Standard Form

Standard form is the everyday way we write numbers using digits, like 65. The position of each digit tells its place value: the 6 is in the tens place, and the 5 is in the ones place.

Expanded Form

Expanded form writes out the actual value of each digit connected by an addition sign: 60 + 5. It expands the number so you can see that the 6 actually means sixty, and the 5 means five.

Using a Place-Value Chart

A place-value chart provides labeled columns that keep tens and ones separated cleanly.

Place-Value Chart for Various Numbers:

Number | Tens | Ones | Expanded Form
-------+------+------+---------------
  27   |  2   |  7   | 20 + 7 = 27
  40   |  4   |  0   | 40 + 0 = 40
  83   |  8   |  3   | 80 + 3 = 83
  99   |  9   |  9   | 90 + 9 = 99

The Crucial Zero in Decade Numbers

Look at 40 in the chart. There are 4 tens and 0 ones. Expanded form is 40 + 0. The digit 0 must be written in the ones place to hold the space; otherwise, the 4 would look like four single units!

Decomposing Numbers in Flexible Ways

While decomposing into tens and ones is standard, numbers can also be broken apart in other creative ways.

Different Ways to Decompose 25:
Way 1 (Standard):  2 tens and 5 ones     (20 + 5)
Way 2 (Flexible):  1 ten and 15 ones     (10 + 15)
Way 3 (All Ones):  0 tens and 25 ones    (25)
All three ways equal 25!

Unbundling a Ten

Imagine you have 2 rods and 5 units (25). If you take one of the rods and unbundle it back into 10 loose units, you now have 1 rod and 15 units. You still have 25 total cubes! This flexible thinking will be your secret weapon when you learn subtraction with regrouping.

Practice Exercises

Exercise 1: Decompose the number 74 into tens and ones. How many tens are there, and how many ones are there? Write 74 in expanded form.

Exercise 2: Write the standard two-digit numeral that matches this expanded form: 50 + 9.

Exercise 3: Complete this place-value puzzle: 3 tens and 12 ones. What total number does this represent?

Exercise 4: Write the expanded form for the number 90. Explain why the number zero is needed.

Solutions and Explanations

Solution to Exercise 1: In the number 74, there are 7 tens and 4 ones. In expanded form, it is written as 70 + 4 = 74.

Solution to Exercise 2: 50 + 9 equals the standard two-digit numeral 59.

Solution to Exercise 3: 3 tens equals 30. 12 ones equals 12. Adding 30 + 12 gives 42. (You can also think of 12 ones as 1 ten and 2 ones; combined with the 3 tens, that makes 4 tens and 2 ones, which is 42).

Solution to Exercise 4: The expanded form for 90 is 90 + 0 = 90. The zero is necessary to show that there are zero single units. Without the zero holding the ones place, the digit 9 would represent 9 ones instead of 9 tens.