**High School Number Sense Lesson 24: Cubes (Memorization)**

Today I'm going to tell you a little secret about cubes. But first, let me say that those who write number sense tests love to test us on our knowledge of cubes--specifically

**from 1 to 15**. On high school tests this year, cubes (exponents--not solids) showed up

**9 times**within the first 24 problems; the median spot was

**question # 12**.

**Number Dojo Level: 119**

Questions about cubes come up in

**various forms**:

- As a straight memorization test (What is 13 cubed?)--version 1
- Or (What is 13 x 13 x 13?)--version 2
- As a disguised memorization test (What is 169 x 13?)
- As a cube root (What is the cube root of 2197?)--
*this will show up later this month as a Middle School lesson* - As a volume question for polyhedrons (What is the volume of a cube with an edge of 13?)--
*another separate topic*

This concept covers the first 3 variations; I'll save the last 2 for later. Now for:

**The Secret**

Perfect cubes (numbers with integers as their cube roots) can end in any digit from 0 - 9. In fact, if you list the perfect cubes in ascending order, the last digit will follow a specific pattern every time. See if you notice it:

^{3}=

**0**

1

^{3}=

**1**

2

^{3}=

**8**

3

^{3}= 2

**7**

4

^{3}= 6

**4**

5

^{3}= 12

**5**

6

^{3}= 21

**6**

7

^{3}= 34

**3**

8

^{3}= 51

**2**

9

^{3}= 72

**9**

10

^{3}= 100

**0**

11

^{3}= 133

**1**

12

^{3}= 172

**8**

13

^{3}= 219

**7**

14

^{3}= 274

**4**

15

^{3}= 337

**5**

16

^{3}= 409

**6**...

**It may not be obvious, but:**

- The units digit of consecutive perfect cubes is
**unique through the first 10**, and then it repeats in the same order. - Any integer ending in
**0, 1, 4, 5, 6, or 9**has a perfect cube with the**same last digit**as the original number. - Any integer ending in
**2**has a perfect cube ending in**8**, and vice versa. - Any integer ending in
**3**has a perfect cube ending in**7**, and vice versa. - So, if you
**switch 2 & 8**, and**3 & 7**, you can name the units digits in order (every time) by counting from 0 - 9 (with those substitutions).**0, 1, 8, 7, 4, 5, 6, 3, 2, 9...** **MATH IS BEAUTIFUL!**

**Use my flashcards to memorize these!**

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

cubespowers.pdf |

**Up Next for High School: RepDec.aaaa, RepDec.abab**