The Story of Measurement: Units, Systems and π

Welcome to Geometry and Measurement

This unit is about the shapes and sizes of real things: circles and triangles in designs, the units we measure with, how shapes change when we stretch them, right triangles, and the volume of pyramids and cones.

Remember from Grade 8: you used metric prefixes from pico to tera, the Pythagorean theorem, and the area, surface area and volume of composite shapes made from prisms and cylinders. We will build on all of that.

We start with a question that sounds simple: where do units come from?

A hook: how tall is that horse?

At a fall fair you read that a horse is 15.2 hands tall. That is not 15 and two-tenths. It means 15 hands and 2 inches. A hand is exactly 4 inches, so the horse is 15 × 4 + 2 = 62 inches at the shoulder.

The hand is a very old unit that started with the width of a real hand. Many units began this way, with body parts, steps, or things people could see every day.

What is a measurement system?

A measurement system is a family of units that work together. It has:

• base units for each kind of quantity (length, mass, time, capacity),
• smaller and larger units, and
• rules that connect them.

The metric system (its modern form is called SI, the International System of Units) is decimal: every unit is 10, 100, 1000 ... times another. 1 km = 1000 m, 1 m = 100 cm.

The imperial system uses other numbers: 12 inches = 1 foot, 3 feet = 1 yard, 1760 yards = 1 mile, 16 ounces = 1 pound. These come from history, not from one plan.

Canada began switching to metric in the 1970s, but you still meet imperial units: people give their height in feet and inches, lumber is sold as "2 by 4s", and many recipes use cups and teaspoons.

Measurement in different communities

Every culture measures, and many knowledge systems measure using the body, the land and the sky.

• Body-based measuring. Traditionally, a qajaq (an Inuit kayak) is built to fit the person who will paddle it, so the builder takes measurements from that person's body, such as their arm span. The boat is "custom-sized" instead of using one standard size.
• Time from the moon and seasons. Many First Nations communities describe the year by moons. For example, many Anishinaabe communities describe 13 moons in a year, each named for what is happening on the land at that time.
• Distance as travel time. Many people, in many cultures, describe distance by how long it takes to travel, which also tells you about the terrain and the season.

When you research an Indigenous measurement practice, use sources from the community itself: a First Nation, Métis or Inuit organization's own website, a cultural centre, or an Elder or Knowledge Keeper sharing with permission. Knowledge belongs to communities, and practices differ from one Nation to another.

The story of a geometric concept: π

For any circle, circumference ÷ diameter is the same number, π ≈ 3.14159. People have known this for thousands of years, but finding its value took clever geometry.

• A Babylonian clay tablet uses 3 1/8 = 3.125.
• An ancient Egyptian text, the Rhind papyrus, uses a method equal to (16/9)² ≈ 3.16.
• Around 250 BCE, the Greek mathematician Archimedes trapped π between two polygons: one drawn inside a circle and one drawn outside it.