Welcome to Mathnasium’s Math Tricks series. Today we are squaring numbers ending in 0 or 5 by observing patterns and applying our number sense.

To square a number, we multiply it by itself. Sometimes, this multiplication might be difficult to compute, but let’s observe a few patterns that might help us out — see if you and your child can spot them!

90^{2} = 8,100

110^{2} = 12,100

45^{2 }= 2,025

115^{2} = 13,225

Did you notice that the square of a number ending in 0, such as 90, ends in 00? How about that the beginning of the final answer is the square of the digits in front of 0 in the original number? In the case of 90, 9^{2} = 81.

Did you also notice that the square of a number ending in 5, such as 45, ends in 25? And that the beginning of the final answer is equal to the digits in front of the 5 multiplied by those same digits increased by 1? The square of 45 begins with the digits 20 because 4 multiplied by (4+1) is equal to 4×5 = 20. And this happens all the time!

Follow the examples below to observe more patterns and use them to find the square of each number.

# Example 1: Find 120^{2}.

** Step 1: Identify the number-sense pattern for squaring numbers.**

Since 120 ends with a 0, we turn to the first number-sense trick.

** Step 2: Write down two 0s for the end of the final answer.**

_____ 00.

** Step 3: Square the digits before the 0.**

We square the digits 12: 12^{2} = 12 x 12 = 144.

** Step 4: Put the answer to Step 3 in front of the 00 in the final answer.**

144 00.

** Answer: **120 squared is equal to 14,400.

# Example 2: Find 125^{2}.

** Step 1: Identify the number-sense pattern for squaring numbers.**

Since 125 ends with a 5, we turn to the second number-sense trick.

** Step 2: Write down 25 for the end of the final answer.**

_____ 25.

** Step 3: Increase the digits in front of the 5 by 1.**

We increase the digits 12 by 1: 12 + 1 = 13.

** Step 4: Multiply the original digits in front of the 5 by the digits from Step 3.**

We multiply 12 x 13 = 156.

** Step 5: Put the answer to Step 4 in front of the 25 in the final answer.**

156 25.

** Answer: **125 squared is equal to 15,625.

Now, with this strategy, you are ready to square numbers ending in 0 or 5 using the mental math and number sense. Click here for more practice problems, then check your answers here.

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