MYP Standard Mathematics Volume Notes

1. Volume is about three-dimensional space: how much space an object occupies.
2. In IB MYP/Standard Mathematics, volume connects geometry (solids), unit conversion, and real applications such as capacity.

Volume Units Work Differently From Length Units

1. For length, each step (for example from m to cm) is a factor of 100.
2. For area, each step is squared.
3. For volume, each step is cubed.
4. That single idea explains most volume conversions.

Why Squared And Cubed Units Grow So Fast

1. If you scale a shape by a factor of $k$:
   1. Lengths scale by $k$
   2. Areas scale by $k^2$
   3. Volumes scale by $k^3$
2. So changing from meters to centimeters multiplies lengths by $100$, but multiplies volumes by $100^3=1{,}000{,}000$.

Cubes Make Unit Conversion Visual

A cube is the easiest solid for understanding volume because its volume is side$^3$.

Converting m$^3$ And cm$^3$

1. Since $1\text{ m}=100\text{ cm}$,
   1. a cube of side $1\text{ m}$ has volume $1\text{ m}^3$
   2. a cube of side $100\text{ cm}$ has volume $(100\text{ cm})^3=1{,}000{,}000\text{ cm}^3$
2. So, $$1\text{ m}^3=1{,}000{,}000\text{ cm}^3$$

Converting cm$^3$ And mm$^3$

1. Since $1\text{ cm}=10\text{ mm}$, $$1\text{ cm}^3=(10\text{ mm})^3=1000\text{ mm}^3.$$
2. So:
   1. cm$^3$ to mm$^3$: multiply by $1000$