E2. Measurement
Specific Expectations
The Metric System
E2.1
measure length, area, mass, and capacity using the appropriate metric units, and solve problems that require converting smaller units to larger ones and vice versa
- reasons for converting units:
- to make numbers smaller and mental calculations easier (e.g., converting 5000 cm to 50 m)
- to compare measurements expressed in different units (e.g., one container has a capacity of 1440 mL and another container has a capacity of 2 L)
- The choice of an appropriate unit depends on which attribute is being measured and the reason for measuring it.
- The attribute to be measured determines whether to choose a unit of length, area, mass, or capacity.
- The reason or context for measuring determines how accurate a measurement needs to be. Large units are used for broad, approximate measurements; small units are used for precise measurements and detailed work.
- When choosing the appropriate size of unit, it is helpful to know that the same set of metric prefixes applies to all attributes (except time) and describes the relationship between the units. Although not all metric prefixes are commonly used in English Canada, understanding the system reinforces the connection to place value:
| Metric Prefix | kilo-unit | hecto-unit | deca-unit | unit | deci-unit | centi-unit | milli-unit |
| Unit Value | 1000 units | 100 units | 10 units | 1 unit |  unit |
 unit |
 unit |
| Place Value | thousand | hundred | ten | one | one tenth | one hundredth | one thousandth |
- For any metric unit, the next largest unit (e.g., the unit to its left) is 10 times as great, and the next smallest unit (e.g., the unit immediately to its right) is one tenth as great. Both place value and the metric system use the same system of tens, so converting between units parallels multiplying or dividing by powers of 10 (e.g., by 10, 100, 1000). For example, since 1 m is
- There is an inverse relationship between the size of a unit and the count of units: larger units produce a smaller measure, and smaller units produce a larger measure. This principle is important for estimating whether a conversion will result in a larger or smaller count of units.
- Because both place value and the metric system are based on a system of tens, metric conversions can be visualized as a shifting of digits to the left or right of the decimal point a certain number of places. The amount of shift depends on the relative size of the units being converted. For example, since 1 km is 1000 times as long as 1 m, 28.5 km becomes 28 500 m when the digits shift three places to the left.
- Conversions are ratios, so the same tools that are useful for scaling and finding equivalent ratios are useful for unit conversions (e.g., double number lines, ratio tables, ratio boxes).
Ask students to share real-life contexts that involve linear measurements (e.g., distance between places); area (e.g., painting a wall); mass (e.g., weighing of fruit); and capacity (e.g., filling a drinking glass). For each of the suggested contexts, have students pose a problem that requires them to convert among units. For example: The distance between two communities is 13 907 m. How far is it in kilometres? Or: What is the area of a 2.43 m × 3.04 m wall in square centimetres?
Have students gather (from home, or images from online or newspapers) labels with measurements on them (mass, capacity, or linear dimensions). Have students convert these measures to a larger unit (millilitres to litres; grams to kilograms) and vice versa and discuss why a certain unit might be chosen over another.
To ensure that these conversions are sensible, guide students to recognize the base ten structure of the metric system and the clues that metric prefixes provide about the relative size of the units. Have them use place-value patterns and the multiplicative relationships between metric units to perform their calculations. To ensure that conversions are reasonable, guide students to predict and estimate the conversion first before carrying out the calculation, drawing on previous learning and their own experience, to remember that smaller units mean larger numbers.
Provide students with true or false statements, such as the ones below. Ask them to explain their rationale using their knowledge of metric prefixes and a unit’s size; their understanding of place value; and their mental math strategies for multiplying and dividing by 10, 100, or 1000.
True or false?
- 74 mm = 0.74 cm
- 350 mL = 35 L
- 1500 g = 1.5 kg
- 125 mL = 0.125 L
- 75 g = 7.5 kg
Pose problems involving measurement and rates. For example:
- For the community feast to celebrate the summer solstice and Indigenous Peoples Day, your family has volunteered to purchase the flour for bannock and bread making. If 10 kg of flour costs $9, how much does 100 g cost?
- If a 250 g box of granola is on sale for $2.50, and a 1.5 kg box of granola costs $16.00, which is the better value for money?
- If 1 L of paint covers up to 10 m2, how many millilitres is needed to cover 2 m2?
Have students recall metric relationships between grams and kilograms, and support them in using these relationships, and others that they notice in the problem, to make decisions. Discuss strategies for solving the problem, and guide them to notice how the “times 10” relationship between metric units and the place-value system makes it possible to do conversions mentally. Support students in using organizers such as a ratio table to highlight multiplicative relationships. For example, the ratio table below shows a way to record strategies for determining the cost of 100 g of flour if 10 kg costs $9:
Angles
E2.2
use a protractor to measure and construct angles up to 360°, and state the relationship between angles that are measured clockwise and those that are measured counterclockwise
- measuring angles in counterclockwise and clockwise directions:
- 90° angle, measured counterclockwise
- 270° angle, measured clockwise
- “If the black angle is 90°, then I know that the red one must be 270° because together they make a circle that adds up to 360°”:
- 180° angle measured counterclockwise and 180° angle measured clockwise:
- measuring a 30° angle clockwise using a circle protractor:
- measuring a 330° angle counterclockwise using a circle protractor:
- The lines (rays) that form an angle (i.e., the “arms” of an angle) meet at a vertex. The size of the angle is not affected by the length of its rays.
- Protractors, like rulers or any other measuring tool, replace the need to lay out and count individual physical units. The protractor repeats a unit so there are no gaps or overlaps and includes a scale to keep track of the unit count.
- A degree is a very small angle and is a standard unit for measuring angles. When 360° are placed together, they form a circle.
- Since a degree is such a small unit, standard protractors often use a scale (typically in increments of 10) with markings to show the individual degrees. If every degree were labelled with a number, the protractor would need to be much larger.
- Protractors usually include a double scale to make it easier to count the degrees in angles that open clockwise and those that open counterclockwise. On a 180º protractor, the outer scale goes from 0° to 180° and reads from left to right whereas the inner scale goes from 0° to 180° and reads from right to left.
- To make an accurate measurement (i.e., a count of degrees) using a protractor:
- align the vertex of the ray with the vertex of the protractor (i.e., the midpoint of the protractor where all the degree angles meet);
- align one arm ray with the zero line of the protractor, similar to measuring from zero with a ruler;
- choose the scale that begins the count at zero, use the scale to count the degrees in the angle, and read the measurement where ray crosses the number scale – that is, if the rays open to the right, use the inner scale, and if the rays open to the left, use the outer scale.
- Many common protractors are semi-circular, meaning the scale only counts 180°. There are two strategies to measure or construct a reflex angle: measure the angle beyond the straight angle and add 180° to that amount or subtract the remaining angle from 360°.
Note
- Smaller angles may be added together to determine a larger angle. This is the additivity principle of measurement.
Show students a semi-circle and a full-circle protractor, and have students compare their similarities and differences. Guide students to notice the connection between the everyday use of the term “360” to describe a full circle and the idea that angles can extend to 360°. Have students make connections between reading the protractor clockwise and counterclockwise. Ask them to identify the angle measures in degrees for the straight angle and the right angle, and the range of angle measures for acute and obtuse angles. Have them use both types of protractors to measure angles greater than 180° (reflex angles), including those found in everyday life.
To apply measurement and the properties of quadrilaterals (see E1.1), have students measure the angles of convex quadrilaterals, such as a dart, and find the sum of their interior angles. As they measure the sum of angles in different quadrilaterals, guide students to notice that they all add to 360° (a fact students could also demonstrate by tearing the vertices from a quadrilateral and rearranging them into a 360° circle).
- Square:
- Parallelogram:
- Right Trapezoid:
Have students also measure the exterior angles of various quadrilaterals and support them in noticing that for quadrilaterals, the sum of an internal angle and its corresponding exterior angle is always 180°. This is information they will use in E2.3.
Have students measure the angles in a rotation, both clockwise and counterclockwise (see E1.4). As they rotate an object 270° and 90°, either by hand or using technology, discuss why the images end up at the same coordinates.
Have students measure the interior angles of various polygons, including those found in everyday life, and determine the sum of their angles. Discuss any similarities and differences.
Have students draw a design that includes a variety of angles, including at least one right angle, one acute angle, one obtuse angle, and one reflex angle. Have them measure each angle, write the measurements on a separate piece of paper, and then trade designs with a classmate. Support students in measuring the angles in each other’s designs and checking any measures that do not match. For discrepancies within a few degrees, draw out the notion that all measurements are approximate and that more precise measurements are sometimes limited by the tools used; for other discrepancies, have students check the scale they used or the accuracy of their calculations in determining a reflex angle.
E2.3
use the properties of supplementary angles, complementary angles, opposite angles, and interior and exterior angles to solve for unknown angle measures
- supplementary angles:
- pairs of angles that have a sum of 180°:
- complementary angles:
- pairs of angles that have a sum of 90°:
- opposite angles:
- the equal non-adjacent angles formed by two intersecting lines:
- interior angles in a triangle:
- ∠A, ∠B, and ∠C are inside △ABC:
- ∠A = 110°
- ∠B = 50°
- ∠C = ?°
- exterior angles:
- ∠ACD is an exterior angle of △ABC:
- Angles can be measured indirectly (calculated) by applying angle properties. Measuring angles indirectly is often quicker than measuring them directly and is the only choice if the location of an angle is impossible or impractical to measure.
- Smaller angles may be added together to determine a larger angle. This is the additivity principle of measurement.
- Angle properties can be used to determine unknown angles.
- A straight angle measures 180°: this property is used to determine the measurement of a supplementary angle and is applied when determining the exterior angles of a polygon.
- A right angle measures 90°: this property is used to determine the measurement of a complementary angle.
- Interior angles of quadrilaterals sum to 360°; this property is used to find an unknown angle in a quadrilateral.
- Interior angles of triangles sum to 180°; this property is used to find an unknown angle in a triangle.
- Angle properties can also be used to determine other unknown measures (e.g., the exterior angle measures of a polygon) or to explain why opposite angles are equal.
Have students apply the additivity principle and other angle properties to determine the angle measures of pattern blocks and to use different pattern blocks to draw benchmark angles of 30°, 45°, 60°, 90°, 120°, 135°, 180°, 270°, and 360°. For example, they might:
- identify the angles in the orange square as 90°
- deduce that if, 1 square + 3 tan rhombuses (acute angle) = 180°, then the tan rhombus must have an angle measure of 30°
- deduce that if the obtuse angles of 3 blue parallelograms make up 360°, then using division tells them each angle must be 120°
- combine the obtuse angles of the blue parallelogram (120° + 120°) with the acute angle of the tan rhombus (30°) to make a reflex angle of 270° and demonstrate how the same angle could have been created by subtracting the orange square (90°) from a full circle (360°) (see E2.2)
Ask students to draw a straight line and place a point on the line. Ask them to draw another line from that point at any angle. Ask them to estimate the angle measures, then use a protractor to confirm their measures. In small groups, have them compare their diagrams and their angle measures. Ask them to determine the sum of the two non-straight angles in each diagram and describe what they notice. Share that these angles are called supplementary angles.