**High School Number Sense Lesson 57: Dividing and then Multiplying Squares of Numbers with the Same Ratio**

Today's lesson looks and sounds harder than it is. As far as I can tell, this concept just started showing up only recently: in the 2014-15 school year. It appeared

**12 times**last year, with a median placement at

**question # 55**.

**Number Dojo Level: TBD**

To recognize this concept, look for something like this:

**(2a)**

^{2}÷ (a)^{2}x (a/2)^{2}- Three terms with two operators,
- Division followed by multiplication,
- A consistent ratio between terms (usually 1/2), and
- The same exponent on each term.

**How to Solve:**

- Make sure all 4 of the above conditions exist.
- Ignore the 1st & 3rd terms, and calculate ONLY the 2nd term.
- That's it. You're done. Move along. Pretty cool, right?

**Example 1 (the long way):**

**12**

^{2}÷ 6^{2}x 3^{2}= ___1. 12

^{2}= 144. 6

^{2}= 36. 3

^{2}= 9.

2. 144 ÷ 36 = 4. 4 x 9 =

**36**.

Example 1 (the short way):

Example 1 (the short way):

**12**

^{2}÷ 6^{2}x 3^{2}= ___1. Notice the pattern of 3 terms, 2 operators, division then multiplication, each term being 1/2 the previous, and all numbers being squared.

2. Execute the 2nd term. 6

^{2}=

**36**.

**Example 2:**

**(1/3)**

^{2}÷ (1/6)^{2}x (1/12)^{2}= ___1. Notice the pattern of 3 terms, 2 operators, division then multiplication, each term being 1/2 the previous, and all numbers being squared.

2. Execute the 2nd term. (1/6)

^{2}=

**1/36**.

**Example 3:**

**(2.4)**

^{2}÷ (1.2)^{2}x (0.6)^{2}= ___1. Notice the pattern of 3 terms, 2 operators, division then multiplication, each term being 1/2 the previous, and all numbers being squared.

2. Execute the 2nd term. (1.2)

^{2}=

**1.44**.

**Example 4:**

**32**

^{2}÷ 8^{2}x 2^{2}= ___1. This skill works with any ratio, as long as it's consistent across all terms. Notice the pattern of 3 terms, 2 operators, division then multiplication, each term being

**1/4**the previous, and all numbers being squared.

2. Execute the 2nd term. 8

^{2}=

**64**.

**Example 5:**

**12**

^{3}÷ 6^{3}x 3^{3}= ___1. Notice the pattern of 3 terms, 2 operators, division then multiplication, each term being 1/2 the previous, and all numbers being

**cubed**.

2. Execute the 2nd term. 6

^{3}=

**216**.

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

divmultsquare.pdf |

**Up Next for High School: FOIL3by3digit**