E2. Measurement
Specific Expectations
The Metric System
E2.1
use appropriate metric units to estimate and measure length, area, mass, and capacity
 reallife contexts for measurement:
 length:
 distance of a walk
 perimeter of a ball diamond, soccer field, or cricket field
 area:
 yard or garden
 floor space needed to accommodate an art exhibition
 mass:
 mass of a bunch of bananas in order to determine its cost
 mass of an animal in order to know the amount of medication required
 capacity:
 approximate size of a container to hold leftover food
 size of a container to hold soil for the school garden
 length:
 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 and 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. 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 English Canada, understanding the system reinforces the connection to place value:
Metric Prefix  kilounit  hectounit  decaunit  unit  deciunit  centiunit  milliunit 
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 
 Canada, as well as all but three countries in the world, has adopted the metric system as its official measurement system. It is also universally used by the scientific community because its standard prefixes make measurements and conversions easy to perform and understand. However, Canadians also commonly refer to the imperial system in daily life (gallons, quarts, tablespoons, teaspoons, pounds), and the Weights and Measures Act was officially amended in 1985 to allow Canadians to use a combination of metric and imperial units. The most appropriate unit is dependent on the context. Sometimes it is a metric unit (this is the emphasis in this expectation), sometimes it is an imperial unit, and sometimes it is personal nonstandard unit or benchmark.
 Although the size of a unit may change, the process for measuring an attribute remains the same. This is true whether using inches, centimetres, or hand spans.
Have students work in pairs to select one object in the classroom that is “very small” and one object in the classroom that is “very large”. They could identify the measurable attributes of these objects – do they have length? mass? capacity? area? – and decide what units should be used to measure these attributes. Support students in recognizing that the choice of units is based on several factors, not just the size of the object. After they choose an appropriate unit, have students use benchmarks to estimate the measurements of these attributes and then measure the different attributes of the object as accurately as a purpose requires, recognizing that some attributes may be too challenging to measure (although some students might find creative ways to approximate or research these measures). For example, students might decide to measure different attributes of a reading table or the teacher’s desk.
Provide a variety of measurement scenarios, for example:
 Have students estimate and measure the area of different spaces in the school (e.g., the library, the gym, their classroom, the basketball court, the school community garden).
 Have students estimate and measure the mass of everyday objects based on the mass of a benchmark object. For example, a bag of potatoes has a mass of 5 kg. Does the chair have more or less mass than the bag of potatoes?
 Have students estimate and measure the capacity of everyday objects based on the capacity of a benchmark container. For example, a container holds 1 litre. Can you think of a container that holds less than 1 L? How much less?
Support students in making connections to the science and technology curriculum, where they are investigating mass, volume, solids, liquids, and gasses.
E2.2
solve problems that involve converting larger metric units into smaller ones, and describe the base ten relationships among metric units
 types of conversions:
 kilometres to metres
 metres to centimetres
 centimetres to millimetres
 litres to millilitres
 kilograms to grams
 base ten relationships:
 1 cm is equivalent to 10 mm
 100 cm is equivalent to 1 m
 types of problems:
 determining the perimeter when not all measurements are in the same unit:
 comparing measurements expressed in different units:
 one container has a capacity of 1440 mL, and another container has a capacity of 2 L
 Conversions within the metric system rely on understanding the relative size of the metric units (see E2.1) and the multiplicative relationships in the placevalue system (see Number, B1.1).
 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.
 There is an inverse relationship between the size of a unit and the count of units: the smaller the unit, the greater the count. Remembering this principle is important for estimating whether a conversion will result in more or fewer units.
Note
 Although this expectation focuses on converting from larger to smaller units, it is important that students understand that conversions can also move from smaller to larger units using decimals. Exposure to decimal measurements is appropriate for Grade 5 students.
Have students discuss whether given statements, such as those below, are true or false. Have them 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 by 10, 100, or 1000. Have them estimate whether the result will be larger or smaller and describe how many times greater one unit is than the other.
 True or False: 5.6 cm = 0.56 mm
 True or False: 3400 km = 34 m
 True or False: 4 m = 4000 mm
 True or False: 500 L = 50 000 mL
 True or False: 3.4 kg = 34 g
Have students find interesting measurement statistics about animals or people (e.g., how far can a grasshopper jump? how much can a weightlifter lift?). Have students express these measurements in different metric units and discuss the impact of choosing that unit (e.g., if a skunk can spray 4.5 m, what is the effect of saying it can spray 450 cm or 4500 mm?). Also, have them apply proportional reasoning to solve problems such as: After 4 hops, a grasshopper is 1 m from its starting position. How many centimetres is the grasshopper from the starting position after 20 hops? As an extension, students could apply proportional reasoning and express these measurements using related but creative nonstandard units (e.g., an Olympic long jumper can jump more than 28 grasshopper hops).
Support students in making connections to the science and technology curriculum; for example, making models of organs or components of body systems.
Angles
E2.3
compare angles and determine their relative size by matching them and by measuring them using appropriate nonstandard units
 making a nonstandard unit to measure angles:
 creating differentsized sectors by repeatedly folding a paper circle in half and then cutting out the sectors:
 two times for 4 sectors
 three times for 8 sectors
 four times for 16 sectors
 creating differentsized sectors by repeatedly folding a paper circle in half and then cutting out the sectors:
 comparing angles:
 the angle of the yellow hexagon pattern block is equal to four of the acute angles of the tan rhombus, which will be called the tan wedge from here on:
 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.
 Angles are often difficult to transport and compare directly (i.e., by overlaying and matching one against another); therefore, angles are often compared indirectly by using a third angle to make the comparison:
 If the third angle can be adjusted and transported, it can be made to match the first angle and then be moved to the second angle to make the comparison directly. This involves the property of transitivity (if A equals C, and C is greater than B, then A is also greater than B).
 If the third angle is a smaller but fixed angle, it can operate as a unit that is iterated to produce a count. Copies of the third angle are fitted into the other two angles to produce a measurement. The unit count is compared to determine which angle is greater and how much greater.
 In the same way that units of length are used to measure length, and units of mass are used to measure mass, units of angle are used to measure angles. Any object with an angle can represent a unit of angle and be used to measure another angle.
Have students build an “anglecapturing tool” by joining two arms (paper, geo strips, cardboard) with some type of pivot (e.g., a paper fastener, or a nut and bolt). Demonstrate how they can adjust the arms of the tool to match and “capture” the size of an angle. Have them match and compare captured angles to any other angles. They can use their anglecapturing tool to record the size of their captured angles in their notebook; classify them as acute, right, obtuse, or straight; and order them from largest to smallest.
Have students use a tan rhombus pattern block as a nonstandard unit to measure the angles of the other pattern blocks. Support them as they notice how many tan wedges fit into an angle, and recognize, for example, that each angle on the green triangle pattern block is equivalent to two tan wedges, and that the angles on the orange square are equivalent to three tan wedges. Next, have them predict the number of tan wedges needed to measure and compare the angles of other shapes. Have them extend this learning by using their nonstandard tan wedge to measure the angles “captured” with their anglecapturing tool in Sample Task 1. Guide them to recognize that the use of a unit, like the tan wedges, enables a move from comparing angles (Sample Task 1) to measuring angles (Sample Task 2). Through the use of a unit, the question can change from “Which angle is bigger?” to “How big?” and “How much bigger?”.
Provide students with a cutout circle. Have them fold the circle in half until they have a very small sector. Have them use this sector as a nonstandard measuring tool to compare the sizes of angles for various shapes.
E2.4
explain how protractors work, use them to measure and construct angles up to 180°, and use benchmark angles to estimate the size of other angles
 making connections between nonstandard units and benchmark angles:
90° (QuarterTurn)  


180° (HalfTurn)  

 locating 30° on a protractor:
 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 180° are placed together, they form a straight line, as demonstrated on a 180° protractor.
 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 was 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. 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 use a protractor to make an accurate measurement (i.e., a count of degrees):
 align the vertex of the rays with the vertex of the protractor (i.e., the midpoint of the protractor where all the degree angles meet);
 align ray with the zero line on the protractor, similar to measuring from zero with a ruler;
 choose the scale that begins the count at zero and read the measurement where the ray crosses the number scale, i.e., if the rays open to the right, use the inner scale, and if the rays open to the left, use the outer scale.
 Being able to identify benchmark angles, such as 45°, 90°, 135°, and 180°, is helpful for estimating other angles.
Ask students how many tan wedges are needed to form a straight line. Guide them to make the connection that every straight line is 180°, so each wedge is 30°. Have them determine the angle measures of the other pattern block pieces in degrees.
To support students to better understand protractors, build on the work they did in E1.4, Sample Task 2 by taping together several tan wedges to make a semicircular protractor (e.g., six tan wedges = a straight angle). To keep track of the count of wedges, have students create a scale that counts the wedges (e.g., from 0 to 6), and, since angles can open in any direction, have them number the wedges in both directions. This simulates the double scale on a standard protractor.
To make a transparent nonstandard protractor (e.g., one where it is easier to recognize the angle beneath the protractor), have students fold a circular piece of wax paper into sectors and number the scale with a marker. Discuss whether the numbers should be placed on the lines or the spaces (they can think about a ruler for a comparison), and have them use their nonstandard protractors to describe how much bigger one angle is than another. In consolidating the learning from the task, guide students to compare their nonstandard protractors to standard semicircular protractors and note similarities and differences. Guide students to notice that degrees are simply really small sectors and that, instead of 6 tan wedges fitting into a straight line, it takes 180° to fill that space.
To further understand how a protractor works, have students compare different types of standard protractors (semicircular variations; angle rulers; circular protractors) and explain similarities and differences. Have them use different protractors to measure the angles of reallife and relevant objects and estimate to verify that the measurements are reasonable.
Have students randomly select an acute angle measure. Have them sketch that angle. Next, have them use the protractor to determine how close their drawing is to the actual angle. Repeat this activity for obtuse angles.
Area
E2.5
use the area relationships among rectangles, parallelograms, and triangles to develop the formulas for the area of a parallelogram and the area of a triangle, and solve related problems
 rectangles, triangles, and parallelograms with the same bases and heights:
 For some shapes and some attributes, length measurements can be used to calculate other measurements. This is true for the area of rectangles, parallelograms, and triangles. Indirectly measuring the area of these shapes is more accurate than measuring them directly (i.e., by laying out and counting square units and partial units).
 The spatial relationships between rectangles, parallelograms, and triangles can be used to determine area formulas. The array is an important model for visualizing these relationships.
 The area (A) of any rectangle can be indirectly measured by multiplying the length of its base (b) by the length of its height (h) and can be represented symbolically as A = b × h. It can also be determined by multiplying the rectangle's length (l) by its width (w), represented symbolically by A = l × w (see Grade 4, E2.6). This formula can be used to generate formulas for the area of other shapes.
 Any parallelogram can be rearranged (composed) into a rectangle with the same area. For all parallelograms it is true that:
 the areas of the parallelogram and its rearrangement as a rectangle are equal;
 the base lengths of the parallelogram and its rearrangement as a rectangle are equal;
 the heights of the parallelogram and its rearrangement as a rectangle are equal;
 this means that, the area of any parallelogram can be indirectly measured, like a rectangle, by multiplying the length of its base by the length of its height;
 this relationship can be represented symbolically using the formula A = b × h, where A represents Area, b represents base, and h represents height.
 Any triangle can be doubled to create a parallelogram (i.e., by rotating a triangle around the midpoint of a side). Any parallelogram can be divided into two congruent triangles. Half of a parallelogram is a triangle.
 For all triangles it is true that:
 the base lengths of the triangle and the parallelogram formed by rotating a copy of the triangle are equal;
 the heights of the triangle and the parallelogram are equal;
 the area of the parallelogram is double that of the triangle, and the area of the triangle is half that of the parallelogram;
 therefore, since A = b × h for a parallelogram, the area (A) of a triangle can be determined by multiplying the length of its base (b) by the length of its height (h) and dividing by 2;
 this relationship can be represented using the formula A = (b × h) ÷ 2. Because multiplying by one half is the same as dividing by 2, it can also be represented as A = (b × h).
Note
 Any side of a rectangle, parallelogram, or triangle can be its base, and each base has a corresponding height.
Have students compare the area of a rectangle to the area of several parallelograms drawn on grid paper, all with the same base length and height (although very different side lengths or “slant lengths”). Support students as they visualize and make predictions about the various areas. Encourage them to check their predictions by counting the squares in each of the shapes and by decomposing the shapes and rearranging the parts. Guide them to recognize that the area of a rectangle is the same as the area of a parallelogram with the same base and height. Reinforce that, regardless of how the parts of the parallelograms were rearranged, their areas are unchanged (conservation of area).