compare, estimate, and determine measurements in various contexts
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
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?
Pose problems involving measurement and rates. For example:
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:
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
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).
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.
use the properties of supplementary angles, complementary angles, opposite angles, and interior and exterior angles to solve for unknown angle measures
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:
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.