E2. Measurement
Specific Expectations
The Metric System
E2.1
explain the relationships between grams and kilograms as metric units of mass, and between litres and millilitres as metric units of capacity, and use benchmarks for these units to estimate mass and capacity
- mass:
- 1 kilogram (kg) is equivalent to 1000 grams (g)
- benchmarks:
- 1 gram: a paper clip, a leaf
- 25 grams: a spoon
- 50 grams: 10 nickels, a small chicken egg
- 500 grams: a brick of butter, 3 large apples
- 1 kilogram: a small watermelon
- 5 kilograms: a bowling ball, an average-size pumpkin
- 10 kilograms: a bag of flour
- capacity:
- 1 litre (L) is equivalent to 1000 millilitres (mL)
- benchmarks:
- 1 millilitre: 20 drops of water
- 5 millilitres: 1 teaspoon
- 15 millilitres: 1 tablespoon
- 200 millilitres: a juice box
- 350 millilitres: a mug
- 1 litre: a milk carton
- Millilitres and litres are standard metric units of capacity. Grams and kilograms are metric units of mass:
- 1 kilogram (kg) is equivalent to 1000 grams (g).
- 1 litre (L) is equivalent to 1000 millilitres (mL).
- 1 millilitre (mL) of water has a mass of 1 gram (g).
- 1 millilitre (mL) of liquid occupies the space of a 1 centimetre (cm cube.
- Although standard and non-standard units are equally accurate for measuring (provided the measurement itself is carried out accurately), standard units allow people to communicate distances and lengths in ways that are consistently understood.
- The metric system is universally used among scientists because it uses standard prefixes for measurements and conversions. Metric units are the standard unit for all but three countries in the world.
Note
- Canada officially adopted the metric system in 1970, through the Weights and Measures Act. This Act was amended in 1985 to allow Canadians to use a combination of metric and imperial units (called “Canadian” units in the Weights and Measures Act). In addition to metric units, other commonly used units of capacity are gallons, quarts, cups, tablespoons, and teaspoons; other commonly used units of mass are ounces, pounds, and tons. Measuring with imperial units follows the same process as measuring with metric and non-standard units. Only the units and the measuring tools differ. Imperial units are the typical units used in construction and the trades. Students in elementary grades learn to work with metric units first.
Show students a balance scale with markings for grams and kilograms. Provide them with various items to measure in grams using the scale. Have them identify benchmark items for 25 g, 50 g, and 500 g. Have them predict how many of these benchmark items would be needed for 1 kg, then measure to verify their predictions. Through this task, support students in developing an understanding that 1000 g is equivalent to 1 kg.
As a home connection, encourage students to visit a grocery store to find fruits and vegetables of different masses, and use these to create personal benchmarks. Have them use their benchmarks to estimate whether a mass is closer to 500 g, 1 kg, 5 kg, or 10 kg and measure to confirm their prediction. Support students in recognizing that the relationship between grams and millilitres is also helpful in developing benchmarks.
Have students use metric measuring cups, measuring scoops, beakers, and graduated cylinders to measure the capacity of containers using millilitres; a combination of litres and millilitres; and litres only, rounding the capacity to the nearest whole litre. (Note: In Grade 4, it is not an expectation for students to write with decimals to thousandths, so they could express the measure by saying 1 L plus 250 mL, or a little more than 1 L, and so on.) Have them find objects with benchmark capacities (1 L, 500 mL, 10 mL, and 1 mL) and use these to estimate. Have them use the relationship between a centicube and 1 mL to connect the size of the place-value blocks to their capacity (e.g., the thousands cube could hold 1 L of liquid).
Show students that 1 cup is approximately 250 mL (benchmark). Confirm that 4 cups is approximately 1000 mL or 1 L. Provide them with various items that have capacities from 50 mL to 4 L. Ask students to use these benchmarks to establish other benchmark measures, then use them to estimate and measure.
E2.2
use metric prefixes to describe the relative size of different metric units, and choose appropriate units and tools to measure length, mass, and capacity
Metric Prefix | kilo | hecto | deca | unit | deci | centi | milli |
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 |
The metric system continues beyond the chart above, in both directions, to describe even larger and smaller amounts. This is covered in Grade 8.
- The metric system parallels the base ten number system. One system can reinforce and help with visualizing the other system.
- The same set of metric prefixes is used for all attributes (except time) and describes the relationship between the units. For any given unit, the next largest unit is 10 times its size, and the next smallest unit is one tenth its size.
Note
- Although not all metric prefixes are commonly used in Canada, understanding the system reinforces the connection to place value.
Show students the metric prefixes from kilo- to milli- and discuss the relationship between them and the place-value system. Ask students to share their experiences with measuring using other types of units (e.g., a horse is typically measured in hands; diamonds are measured in carats).
Have students bring products or pictures of products from home that have labels with capacity, mass, or length dimensions (e.g., a 325 mL mustard container; 454 g of butter; a 1.8 kg bag of dried cranberries; 7.62 m of aluminum foil). Have them think about the relationships between different metric units and pose questions related to the measures (e.g., “I have 1 L of mustard. About how many 325 mL containers can I fill?”; “I have 1.8 kg of cranberries. How many 100 g bags of cranberries can I fill?”).
To foster an appreciation of different perspectives from around the world, have students share places where they have been, have lived, or come from that use metric units not typically seen in Canada. As they hear about, for example, a 50 centilitre (cL) bottle of juice, discuss whether it would be more or less than 50 mL. They might want to explore this through their own research.
Have students colour-code a metre stick to show decimetres. Ask them to measure different lengths using millimetres, centimetres, decimetres, and metres (in Grade 4, students write with decimals to tenths, so they could express the measure by saying 1 m plus 100 mm, or 1.1 m, or 11 dm). Support students in recognize “times 10” relationships on the metre stick and in making connections to the place-value system.
Time
E2.3
solve problems involving elapsed time by applying the relationships between different units of time
- using a timeline to solve problems:
- determining the time elapsed from 8:45 a.m. to 3:15 p.m. (6 hours and 30 minutes):
- Elapsed time describes how much time has passed between two times or dates. Clocks and calendars are used to measure and/or calculate elapsed time.
- Addition, subtraction, and different counting strategies can be used to calculate the difference between two dates or times. Open number lines (timelines) can be used to track the multiple steps and different units used to determine elapsed time.
Note
- Elapsed-time problems often involve moving between different units of time. This requires an understanding of the relationships between units of time (years, months, weeks, days, hours, minutes, seconds), including an understanding of a.m. and p.m. as conventions to convert the 24-hour clock into a 12-hour clock.
Support students in using open number lines to keep track of and calculate elapsed time. Have students share their strategies and discuss the different approaches. Have them use timelines to track elapsed-time problems involving both dates and times.
Have students use open number lines to keep track of and calculate elapsed time. For example:
- If a train leaves at 6:30 a.m. and arrives at 5:58 p.m., how long is the ride?
- It’s a 4-hour drive to a friend’s home and we want to arrive at 1:00 p.m. What time should we leave?
- Your friend tells you that 24 days from today a new movie is coming out. What date will the movie be released?
Use bus or train schedules to create elapsed-time problems involving the 24-hour clock (e.g., “How long does it take for the bus to get from point A to point B?”). Have students move between the 24-hour clock and a.m.–p.m. conventions, and structure problems to increase in complexity, such as:
- when only one unit changes (e.g., from 08:15 to 12:15 or from 07:45 to 9:45)
- when both units change (e.g., from 08:15 to 11:45)
- when both units change and move from a.m. to p.m. (e.g., 8:45 a.m. to 1:15 p.m.)
- when the start and end times do not fall on the quarter hour (e.g., 08:34 to 13:29)
Have students pose and solve elapsed-time problems that involve using schedules based on a 24-hour clock (e.g., bus or train schedules). For example, if a train leaves at 06:30 and arrives at 23:15, how long is the train ride?
Vary the structure of elapsed-time problems (i.e., what is known and unknown), including the following:
- The elapsed time is unknown, and the start and end times are known (e.g., If a train leaves at 6:30 a.m. and arrives at 5:58 p.m., how long is the ride?).
- The start time is unknown, and the elapsed and end times are known (e.g., It is a 4-hour drive to a friend’s house. If we want to arrive at 1:00 p.m., what time should we leave?).
- The end time is unknown, and the start and elapsed times are known (e.g., Your friend tells you that 34 days from today is his dance recital. When is his dance recital?).
Angles
Right Angle | Straight Angle |
Acute Angle | Obtuse Angle |
- The rays that form an angle (i.e., the “arms” of an angle) meet at a vertex. The size of an angle is not affected by the length of its arms.
- A right angle is a quarter turn, and it is sometimes called a “square angle” because all angles of a square (or rectangle) are right. If two lines meet at a right angle, the lines are perpendicular.
- Angles can be compared by overlaying one angle on another and matching them. A turn greater than a right angle is an obtuse angle. A turn less than a right angle is an acute angle. A half-turn, where the arms of the angle create a straight line, is a straight angle.
- Right angles measure exactly 90°, a fact that will be addressed formally in Grade 5.
Have students use their arms to model a right angle, an obtuse angle, an acute angle, and a straight angle. As well, have them turn their bodies to face different directions, showing right, straight, acute, and obtuse angles as amounts of turn. To connect with their learning about the properties of light in science and technology lessons, have them describe the angle of a ray of light reflected off a mirror.
Have students find right angles and perpendicular lines in objects around the classroom or school by using the corner of a sheet of paper as a “right-angle tool” to make comparisons directly. Have them also look for objects with angles that are greater than or smaller than a right angle and classify these as acute or obtuse. Have them add examples of straight angles to the list and discuss how a straight line is also a straight angle. Connect to learning from E1.1 by using the property of right angles to distinguish rectangles from other shapes, such as parallelograms and rhombuses.
Area
E2.5
use the row and column structure of an array to measure the areas of rectangles and to show that the area of any rectangle can be found by multiplying its side lengths
- constructing rectangles with an area of 12 square units:
- tiling a rectangle using square units:
- To measure the area of a rectangle, it must be completely covered by units of area (square units), without gaps or overlaps. The alignment of square units produces the rows and columns of an array, with the same number of units in each row.
- The array replaces the need to count individual units and makes it possible to calculate an area.
- Both the number of units in each row and the number of units in a column can be determined from the length of the rectangle’s sides.
- Thinking about a row or a column as “a group that is repeated” (unitized) connects the array to multiplication: the base of a rectangle corresponds to the number of squares in a row and the height of a rectangle corresponds to the number of squares in a column.
- Multiplying the base of a rectangle by its height is a way to indirectly measure the area of a rectangle, meaning it is no longer necessary to count all the individual square units that cover a rectangle’s surface.
Note
- Many students do not immediately recognize the row-and-column structure of an array; instead, it appears as a random scattering of squares, or a “spiral” of squares that goes around the outside towards the centre. Recognizing an array’s structure requires careful attention and instruction.
Provide students with 24 square tiles and ask them to build different rectangles. Have them state the dimensions of each rectangle. Support students in making connections between the dimensions of their rectangles, the row and column structure of an array, and the area of their rectangles (24 square tiles); for example, the base tells the number of columns and the height tells the number of rows. Repeat for other quantities of square tiles.
Provide students with paper squares and outlines of rectangles – some of which require the paper squares to be cut in half to fill the rectangle. Ask them to arrange the paper squares in rows and columns to fill in the rectangle, and tell them that they can cut the paper squares if they need to. Ask them to make connections between the dimensions of their rectangles, the row and column structure of an array, and the areas of their rectangles.
Provide students with rectangles sectioned into rows and columns, with part of each rectangle obscured or erased. Have students predict the area of each rectangle, then ask them to verify their predictions by filling in the missing lines to complete the row and column structure. Support them in using multiplication facts to determine the areas of the rectangles.
E2.6
apply the formula for the area of a rectangle to find the unknown measurement when given two of the three
- formulas for the area of a rectangle:
A = l × w A represents area. l represents length. w represents width. |
A = b × h A represents area. b represents base. h represents height. |
- The formula for finding the area of a rectangle can be generalized to describe the relationship between a rectangle’s side lengths and its area: (Area = base × height).
- Both multiplication and division can be used to solve problems involving the area of a rectangle.
- Multiplication is used to determine the unknown area when the base and height of a rectangle are given (Area = base × height).
- Division is used to determine either the length of the base or the length of the height when the total area is given (Area ÷ base = height; Area ÷ height = base).
- Either side length can be considered the base or the height of a rectangle.
- An area measurement needs to include both the number of units and the size of the units. Standard metric units of area are the square centimetre (cm^{2}) and the square metre (m^{2}). If a surface is completely covered by 18 square centimetres, the area of that surface is 18 cm^{2}. If a surface is completely covered by 18 unit squares, the area of that surface is 18 square units.
Note
- The area of a rectangle is used to determine the area formulas for other polygons. Using “base” and “height” rather than “length” and “width” builds a unifying foundation for work in Grade 5 involving the area formulas for triangles and parallelograms.
Pose problems where students are tasked to find an unknown area, base, or height when two of the three measurements are given. Have students brainstorm real-life and relevant spaces and objects that are rectangular (e.g., the floor area of the community centre), and use these contexts in posing and solving problems. For example:
- The front cover on a journal measures 96 cm^{2}. What are the possible dimensions of the journal? Which dimensions would you prefer if it was your journal and why?
- A gymnasium measures 28 m by 9 m. If tiles are 1 m^{2}, how many tiles are needed to cover the floor? (Multiplication names the unknown area.)
- The area of a garden is 24 m^{2}. What could its dimensions be? Which would you recommend? If you knew that one side was 6 m long, how long would the other side be? (Division names an unknown side length.)
For each problem, have students identify the multiplication and division sentences they could write and highlight the relationship between multiplication, division, and the measurement of area. Ensure that students recognize that a measurement needs both a number and a unit, and support them in choosing the appropriate unit for area and length.