**Middle School Number Sense Lesson 39: Estimating Square Roots**

Last month I introduced the topic of cube roots, which dealt mostly with memorization. Today let's look at estimating square roots. By definition, a square root

**R**of a number

**N**is the number that, when multiplied by itself, equals the original number

**N**. This concept appeared

**13 times**this year in middle school--as early as question # 30 and as late as question # 70, with a median placement at

**question # 40**.

**Number Dojo Level: 203**

First let me say that I am no genius when it comes to HTML, so I will have to settle on using the

**√**sign today or "sqrt" or "the square root of"--instead of placing the whole number under the radicand. Please forgive my ignorance. But I will teach you how I solve these problems.

**How to Solve:**

- Break the number into
**pairs**, starting at the right. If the number has an odd number of digits, its first number by itself is its own "pair." Your answer to the square root problem will have as many digits as the resulting pairs. (In other words, for a 5- or 6-digit number, there are 3 pairs, so your answer will have 3 digits). - Look at the first pair (or digit). Let's call this
**P**. Think of the largest perfect square that is**less than or equal to**that pair (and let's call it**L**). Take the square root of this number and write your result down. This will be the first digit of your answer. - Now think of the smallest perfect square that is
**greater than**the first pair (and let's call it**G**). In your mind, place these numbers in order:**L < P < G**. On a number line, think of where that P lands (between L & G). If it is about halfway, your 2nd digit will be**4, 5, or 6**. If P is pretty close to L, your 2nd digit will be**0, 1, 2, or 3**. If P is pretty close to G, your 2nd digit will be**7, 8, or 9**. Write down your 2nd digit. - Write a 0 for each remaining pair in the original number.

*Confused? Let's look at some examples!*

**Example 1: * √3474**

- Break the number into pairs:
**34**and**74**. Your answer will be 2 digits. - Look at the first pair:
**34**. Think of the largest perfect square less than or equal to 34, which is 25. Write down its square root:**5**. This is your first digit. - Now think of the smallest perfect square greater than 34, which is 36. 25 < 34 < 36, and 34 is pretty close to 36, so the 2nd digit will be 7, 8, or 9. Let's use
**8**; write this down. - Your answer is
**58**, because there are were 2 pairs in the original number. - The actual answer is about
**58.94**, and the (+/- 5%) range is**between 56 and 61**. You are safely in range.

**Example 2: * √44378**

- Break the number into pairs (starting at the right):
**4 43 78**. Notice the first 4 is by itself. Your answer will be 3 digits. - Look at the first pair:
**4**. Think of the largest perfect square less than or equal to 4, which is 4. Write down its square root:**2**. This is your first digit. - Now think of the smallest perfect square greater than 4, which is 9. 4 =< 4 < 9, and 4 is at the beginning of that range, so the 2nd digit will be 0, 1, or 2. Let's use
**1**; write this down. - There is one pair left, so write down a
**0**. Your answer is**210**. - The actual answer is about
**210.66**, and the (+/- 5%) range is**between 201 and 221**. You are safely in range.

**Example 3: * √443780**

- Break the number into pairs:
**44 37 80**. Your answer will be 3 digits. - Look at the first pair:
**44**. Think of the largest perfect square less than or equal to 44, which is 36. Write down its square root:**6**. This is your first digit. - Now think of the smallest perfect square greater than 44, which is 49. 36 < 44 < 49, and 44 is toward the middle-end of that range, so the 2nd digit will be 7, 8, or 9. Let's use
**7**; write this down. - There is one pair left, so write down a
**0**. Your answer is**670**. - The actual answer is about
**666.17**, and the (+/- 5%) range is**between 633 and 699**. You are safely in range.

**Example 4:**

*** √(251 x 479)**

- I personally prefer to break this up into two pieces and then multiply:
**√251 x √479**. - √251 is almost √256, which is
**16**. √479 is almost √484, which is**22**. - 16 x 22 = 32 x 11 (using
**Double/Half**) =**352**. - The actual answer is about 346.74, and the (+/- 5%) range is
**between 330 and 364**. You are safely in range.

**Example 4 (method 2):**

*** √(251 x 479)**

- Using
**OrderOfOper2**, simplify within the parentheses. (251 x 479) is about (250 x 480), which is 120,000. - √120000; break the number into pairs:
**12 00 00**. Your answer will be 3 digits. - Look at the first pair:
**12**. Think of the largest perfect square less than or equal to 12, which is 9. Write down its square root:**3**. This is your first digit. - Now think of the smallest perfect square greater than 12, which is 16. 9 < 12 < 16, and 12 is toward the middle of that range, so the 2nd digit should be 4, 5, or 6. Let's use
**4**; write this down. - There is one pair left, so write down a
**0**. Your answer is**340**. - The actual answer is about 346.74, and the (+/- 5%) range is
**between 330 and 364**. You are safely in range.

**Example 5: * √184 x √215**

- When the two numbers are close like this (within about 20% of each other), simply take the average of the two and write that down. The average of 184 & 215 is 199.5; write down
**199**. - The actual answer is about 198.90, and the (+/- 5%) range is
**between 189 and 208**. You are safely in range.

**Example 6: * √3421000**

- Break the number into pairs:
**3 42 10 00**. Your answer will be 4 digits. - Now look at the first 2 pairs together as a 3-digit number. Think of the largest perfect square less than or equal to 342, which is 324. Write down its square root:
**18**. These are your first 2 digits. - (This step is not actually necessary, but I'll do it for illustration purposes). Now think of the smallest perfect square greater than 342, which is 361. 324 < 342 < 361, and 342 is almost exactly in the middle of that range, so the next digit should be
**5**. Write this down. - There is one pair left, so write down a
**0**. Your answer is**1850**. - The actual answer is about 1849.59, and the (+/- 5%) range is
**between 1758 & 1942**. You are well within range. (I said Step 3 wasn't necessary because you could've simply added two 0's to the 18 to get 1800, which you will notice is still within range).

**Here's a free worksheet to help you practice EstSquareRoot:**

estsquareroot.pdf |

**Up Next for Middle School: EstMult%**