Geometry and Measurement : Measurement
Key Concepts
Two-Dimensional Shapes
- The perimeter of complex figures can be determined by breaking down each part of the figure.
- Similarly, the area of complex figures can be determined by breaking down complex shapes into simpler, familiar ones.
- Changes in perimeter and area vary based on the dimension and the number of dimensions affected by scaling factors.
Three-Dimensional Objects
- Total surface area and volume can be determined by breaking down complex three-dimensional objects into simpler, familiar ones.
- The side-length relationship for right triangles (Pythagorean theorem) can be used to find unknown attributes of certain three-dimensional objects and complex solids.
- Changes in total surface area and volume vary based on the dimension and the number of dimensions affected by scaling factors.
- A pyramid’s volume is one-third that of a prism with the same base and height.
- A cone’s volume is one-third that of a cylinder with the same base and height.
Measurement Systems
- Measurements can be converted into equivalent units and expressed with varying degrees of precision.
- Measurement units indicate the attribute being measured (length, area, volume, capacity); for example, a square unit measures area, and a cubic unit measures volume.
Types of two-dimensional shapes and three-dimensional objects
This table provides an overview of the attributes and shapes that students work with from Grade 4 onwards. Two-dimensional shapes and three-dimensional objects are added from year to year.
| Grade Level | Two-Dimensional Shapes | Three-Dimensional Objects |
|---|---|---|
| 4 |
|
|
| 5 |
|
|
| 6 |
Quadrilaterals (area and perimeter):
Composite Polygons (area and perimeter) |
Prisms (total surface area) Pyramids (total surface area) |
| 7 |
Circles (area and circumference) |
Cylinders (total surface area and volume) Prisms (total volume) |
| 8 |
Composite two-dimensional shapes (solve problems involving area, perimeter, circumference) Applying the Pythagorean relationship to solve problems involving an unknown side length for a given right triangle |
Three-dimensional objects (solve problems involving total surface area and volume) |
| 9 |
Solve problems involving the side-length relationship for right triangles (Pythagorean theorem). The impact of changing one or more dimensions of a two-dimensional shape on the perimeter and the area. |
Solve problems involving the relationship between the volume of prisms and pyramids, and between the volume of cylinders and cones. The impact of changing one or more dimensions of a three-dimensional object on the total surface area and volume. |
Units of Measurement
The following table provides an overview of the units of measurement that students use from Grade 2 onwards. These units are progressively added from year to year.
| Grade Level | Metric System |
|---|---|
| 2 | cm, m |
| 3 |
mm, cm, m, km cm2, m2 |
| 4 | g, kg, mL, L |
| 5 |
Convert larger metric units into smaller ones (base ten relationships) Use appropriate metric units to estimate and measure length, area, mass, and capacity |
| 6 | Convert smaller units to larger ones and vice versa |
| 7 |
Apply relationships between mL and cm3 Convert from one metric unit of measurement to another (for example, mm3, cm3, m3) |
| 8 | Represent very large (mega, giga, tera) and very small (micro, nano, pico) metric units |
| 9 | Solve problems involving different units within a measurement system and between measurement systems |
There is a direct link between the International System of Units (SI), place value, and powers of ten. This table shows the connection, including the prefixes and symbols used. These prefixes are also used with other units of measure such as bytes, watts, pascals, and hertz.
| Prefix | Powers | Place Value | Length, Area, Volume | Capacity | Mass |
|---|---|---|---|---|---|
| Tera (T) | 1012 | Trillion | terametre | teralitre | teragram |
| Giga (G) | 109 | Billion | gigametre | gigalitre | gigagram |
| Mega (M) | 106 | Million | megametre | megalitre | megagram |
| Kilo (k) | 103 | Thousand | kilometre | kilolitre | kilogram |
| Hecto (h) | 102 | Hundred | hectometre | hectolitre | hectogram |
| Deca (da) | 101 | Ten | decametre | decalitre | decagram |
| 100 | Unit | metre (m) | litre (l) | gram (g) | |
| Deci (d) | 10−1 | Tenth | decimetre | decilitre | decigram |
| Centi (c) | 10−2 | Hundredth | centimetre | centilitre | centigram |
| Milli (m) | 10−3 | Thousandth | millimetre | millilitre | milligram |
| Micro (µ) | 10−6 | Millionth | micrometre | microlitre | microgram |
| Nano (n) | 10−9 | Billionth | nanometre | nanolitre | nanogram |
| Pico (p) | 10−12 | Trillionth | picometre | picolitre | picogram |
Reflection
- How can real-life measurement situations help students deeply understand and apply key measurement concepts?
- How can the use of measurement contexts enhance students’ conceptual understanding of operations and algebraic concepts?