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:
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:
- 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!