E2. Measurement
Specific Expectations
The Metric System
 capacity versus volume:
 a container with dimensions 10 cm × 10 cm × 10 cm (volume 1000 cm^{3}) has the capacity to hold 1 L of liquid:
 There is a relationship between volume and capacity.
 Volume can describe many aspects of the same object, so it is important to clarify “which volume” is being measured. For example, the volume of a cup could refer to:
 the volume of liquid the cup could hold (i.e., its capacity – see the note below);
 the volume of material needed to make the cup; or
 the volume of space needed if packing the cup in a box.
 Volume is measured in cubic units, and the measure represents the number of cubes needed to completely fill an object. Similar to units of area, a cubic unit is an amount of volume, and can come in any shape. Units of volume can be decomposed, rearranged, partitioned, and redistributed to better fill a volume and minimize gaps and overlaps.
 The rowandcolumn structure of an array that is the basis for indirectly measuring area (see Grade 4, E2.5) also helps structure the count of cubic units and is used to indirectly measure volume (see E2.7).
 Common metric units of volume include cubic centimetres (cm^{3}) and cubic metres (m^{3}). Common metric units of capacity are millilitres (mL), litres (L), and kilolitres (kL).
 There are relationships between metric units of capacity and volume: 1 mL of liquid occupies 1 cm^{3} of space, and a 1 L container has an interior volume of 1000 cm^{3}.
 The relationship between volume and capacity means that the volume of an object can be found using displacement: the amount of water displaced by an object (or the amount that the water rises) when it is submerged is equal to its volume. For example, if an object is dropped into 1 L of water, and the water level rises to 1.5 L, the change is 500 mL, which is equal to a volume of 500 cm^{3}.
Note
 Volume and capacity are not the same thing. An object will always take up space (volume), but it may not have capacity. A solid, for example, has volume but no capacity. The capacity of an object will also depend on its design; it may not be the same as the volume. For example, a vase may have a solid chunk of glass for its base, resulting in less capacity than volume.
 In reallife experiences, units of volume and capacity may be used interchangeably.
Provide students with a variety of clean, empty milk cartons (250 mL, 500 mL, 1 L, or 2 L), and discuss what the capacity measurement means (capacity describes the amount of milk that the carton can hold, which does not include the gable). Have them use centimetre cubes (cm^{3}) to find the volume of the space inside the carton by arranging the cubes inside the carton in rows and columns to form a layer and repeating that layer until the carton is filled. Discuss why there might be discrepancies in the measures (e.g., the container is not completely filled with milk; there may be gaps between the cubes), and use this to draw out the distinctions and relationships between these attributes. Have students use these relationships to identify benchmarks for volume and capacity (e.g., a bag of milk takes up approximately 1000 cm^{3} of space, which is the same as the base ten thousands block; 1 mL takes up the space of the base ten unit). Have students use their benchmarks to estimate the volume and capacity of a variety of things, such as everyday grocery items.
To reinforce the concept that 1 millilitre of liquid occupies 1 cm^{3} of space, have students use displacement to measure volume. For example, have them predict the displacement of water in a measuring cup when an object with a known volume (e.g., base ten rod or flat) is submerged in it (e.g., if a measuring cup had 50 mL of water and a rod was submerged in it, the measurement would now be 60 mL). Once the relationship and strategy are understood, have students use displacement to indirectly measure the volume of other objects.
Have students solve problems involving units of capacity and volume, such as the following:
 An Earth Day festival at the local park provides a water tank with drinkable water so that participants can refill their water bottles. The tank measures 100 cm × 100 cm × 100 cm and is full. If each participant has a water bottle with a capacity of 500 mL (i.e., 500 cm^{3}), how many empty water bottles can be filled?
E2.2
solve problems involving perimeter, area, and volume that require converting from one metric unit of measurement to another
 problems involving conversions from one metric unit to another:
 length: determining the amount of trim needed to go around a room when you have a scale drawing of the room with dimensions in centimetres
 area: determining the floor space of a room when you have a scale drawing of the room with dimensions in centimetres
 volume: determining how many 250 mL cups of water are needed to fill a fish tank measured in cubic centimetres or metres
 Multiplicative relationships exist when converting from one metric unit to another. The relationships differ when converting units of length, units of area, and units of volume.
In Metres  Visual Representation  In Centimetres  
Length (one dimension) 
1 metre 
1 m 

Area (two dimensions) 
1 square metre 
1 m^{2}= 100 cm × 100 cm 

Volume (three dimensions) 
1 cubic metre 
1 m^{3}= 100 cm × 100 cm × 100 cm 
Note
 The goal is not to memorize these relationships as formulas, but to have the tools to visually represent and understand the relationships, and to use them to calculate unit conversions.
 The relationships between square centimetres and square metres, and between cubic centimetres and cubic metres, are ratios.
 Since 1 m = 100 cm, then 5 m = 500 cm.
 Since 1 m^{2} = 10 000 cm^{2}, then 5 m^{2} = 50 000 cm^{2}.
 Since 1 m^{3} = 1 000 000 cm^{3}, then 5 m^{3} = 5 million cm^{3}.
 Since 1 mL has a volume of 1 cm^{3} (see E2.1), by extension, there are 1 000 000 millilitres in 1 m^{3}.
 In reallife experiences, units of volume and units of capacity may be used interchangeably, and conversions among these units may be necessary.
Have students make a square that is 1 m × 1 m (1 m^{2}) out of newspaper, newsprint, craft paper, or wrapping paper. Ask them to measure the perimeter of the square in metres, centimetres, and millimetres. Ask them to describe the relationship between:
 the perimeter measured in metres and the perimeter measured in centimetres
 the perimeter measured in metres and the perimeter measured in millimetres
 the perimeter measured in centimetres and the perimeter measured in millimetres
Pose questions that require students to use these relationships to solve problems. For example, if the perimeter of a rectangular garden is 6 m, what are the dimensions of the garden in centimetres if its length is twice its width? Have students use the 1 m^{2} paper square from Sample Task 1 to visualize and answer the following questions:
 How many 1 cm × 1 cm squares will fit in this 1 m × 1 m square?
 How many 1 dm × 1 dm squares will fit in this 1 m × 1 m square?
 How can you change the 1 m × 1 m square to make a triangle that has the same area? How does the perimeter change when you change the shape?
Have students describe these area relationships and use diagrams and ratio tables to convert, for example, 8.25 m^{2} to square centimetres.
Have students make the faces of a cube that is 1 m × 1 m × 1 m by taking six 1 m^{2} paper squares from Sample Task 1 and taping them to a skeleton of a cube made with metre sticks. Have them determine how many centimetre cubes would be needed to fill this 1 m^{3} cube. Then, have students use this relationship to solve problems such as: How many 250 mL measuring cups are needed to fill a tank that is 1 m × 2 m × 3 m?
Have students solve problems involving units of capacity and volume (see E2.1), such as the following:
 Suppose this class is going on a hike with Grade 7 classes from other schools. There is a water tank with drinkable water halfway through the hike so students can refill their water bottles. The water tank holds 1 m³ of water, and each student’s water bottle has a capacity of 500 mL (i.e., 500 cm^{3}). How many students will be able to fill up their water bottles before the water tank needs to be refilled?
Support students in recognizing the relationship between cubic centimetres and millilitres and in using drawings to visualize the situation.
Circles
E2.3
use the relationships between the radius, diameter, and circumference of a circle to explain the formula for finding the circumference and to solve related problems
 relationship between radius, diameter, and circumference:
 radius is half the diameter:
 strategy to show the relationship between circumference and diameter:
 measure the circumference and diameter of objects of various sizes using a string or measuring tape (e.g., hula hoop, washer, ring from a ring toss)
 calculate the ratio between the circumference and diameter of each object to two decimal places (e.g., circumference is 47 cm; diameter is 15 cm; ratio is 47 : 15 or about 3.13 : 1)
 For some shapes and some attributes, length measurements can be used to calculate other measurements. This is true for the circumference of a circle. Indirectly measuring the circumference of a circle is quicker and more accurate than measuring it directly (e.g., with a string).
 The distance from any point on a circle to its centre is always the same. This distance is a circle’s radius (r).
 The diameter (d) of a circle is the longest distance from one side of a circle to another. The diameter will always pass through the centre of the circle, so it is the same as two radii (r). This relationship can be expressed symbolically as d = 2r or r = d ÷ 2.
 The perimeter of a circle – its distance around – is called its circumference (C). The circumference of a circle is a little more than three times the length of the diameter, or a little more than six times the length of the radius. This ratio is constant and is described using the Greek symbol π (spelled pi and pronounced like pie). This relationship can be expressed symbolically as C = πd or, equivalently, as C = π2r.
 Pi (π) is equal to , which is a very important relationship, used in many math and physics formulas. It is approximately equal to 3.14159, but it is an irrational number, which means that it can never be exactly calculated: the decimals never end, and it cannot be represented precisely as a fraction.
Note
 The number of decimals used to express π depends on the level of precision needed. Commonly, π is approximated as 3.14 (or in fraction form as ). However, sometimes “a little more than 3” is a sufficient estimate; at other times, such as when astrophysicists calculate the circumference of the observable universe, π must be calculated to 39 digits. Some scientific applications round π to hundreds of digits, and mathematicians, using supercomputers, have calculated π to trillions of digits.
To develop an understanding of the relationship between the radius, diameter, and circumference of a circle, have students use string or tape measures (paper or fabric versions) to measure several everyday circular objects (e.g., cups, cans, hula hoops). For each circle, have students measure the circumference and the diameter and determine the radius.
Have students organize their measurements in a table with columns to record radius, diameter, circumference (2πr or dπ), and the ratio of circumference to diameter. Ask them to write equivalent ratios in the form n:1. Have them compare their ratios, and then discuss this special ratio, known as pi (π), as a class. Have them find out more about pi and share their learning with the class.
Have students solve problems that involve circumference, diameter, or radius, such as:
 If a bicycle tire has a radius of 35 cm, how many kilometres will it travel if it makes 2000 revolutions?
 If you travel 10 km on this bike, how many times will the wheel turn?
Note: When bicycling, the distance a tire travels when it revolves once is equivalent to its circumference.
Have students solve these problems using approximations for pi, such as 3.14, then compare these results to those obtained using technology that uses pi (e.g., a calculator app that includes pi).
E2.4
construct circles when given the radius, diameter, or circumference
 constructing a circle using a compass:
 set the distance between the compass arms to the length of the radius:
 Compasses are often used to construct circles by hand. Another strategy is to attach a pencil to the end of a string. The radius must be known when using a compass or string to construct a circle.
 The relationships between the radius, diameter, and circumference may be used to determine the radius of a circle when its diameter or circumference is known.
 Circles of given measurements can also be constructed using technology. Dynamic geometry applications show how changes to the radius, the diameter, or the circumference affect the others.
Note
 The diameter (d) is twice the radius (r) (i.e., d = 2r), and the radius is half the diameter (r = d).
 The circumference is a little more than three times (and a very little over 3.14 times) the length of the diameter (C = πd).
 If one of the three measurements is known – the circumference, the radius, or the diameter – the other two can be measured indirectly by calculating.
Have students construct circles given the radius, given the diameter, and given the circumference.
Have students estimate the unknown measurements for the following circles, then construct them to verify:
 What are the circumference and the diameter of a circle with a radius of 8 cm?
 What are the circumference and the radius of a circle with a diameter of 9 cm?
 What are the radius and the diameter of a circle with a circumference of approximately 40 cm?
Have students describe the calculations they used for the estimate and the construction. Have them compare different strategies for constructing circles (e.g., compass, technology).
E2.5
show the relationships between the radius, diameter, and area of a circle, and use these relationships to explain the formula for measuring the area of a circle and to solve related problems
 strategy to explain the relationship between radius and area of a circle:
 divide a circle into sectors:
 cut out the sectors and form a “parallelogram”:
 determine the area of the “parallelogram”, making connections to the radius and circumference of the circle
 A relationship exists between the area of a circle (A) and its radius. This relationship can be expressed symbolically as A = πr^{2}. The relationship between the radius and the diameter (r = d) means that if either the radius or the diameter is known, the area can be measured indirectly, without the need to count square tiles. If the area is known, then the radius and the diameter can be determined using this relationship.
Note
 Visual proofs use the formulas for the area of a parallelogram or the area of a triangle to demonstrate the logic of the circle formula for area.
To support visualizing the relationship between the area of a circle and its radius, have students find the area of a circle by reshaping (decomposing and recomposing) it into a “parallelogram”. To do this, have them cut a circle into 8 or 16 sectors and rearrange the slices to form an approximate parallelogram (the more sectors, the closer the result is to a parallelogram):
Guide students to notice that the base of the parallelogram is half the circumference of the circle ( or = π × r) and that the height of the parallelogram is the radius of the circle (r).
Support students in recognizing that in this case, the area of the parallelogram (b × h) can be thought of as π × r × r = πr^{2}.
Have students do research on the Internet to find other visual proofs or explanations of this formula and share them with the class.
For a more concrete experience, have students construct a circle with a given diameter or radius on grid paper, count the squares and partial squares to estimate the area, and compare their findings against the formula.
Have students use the formula for the area of a circle to solve reallife problems, such as:
 A roundabout is being constructed to improve the flow of traffic. How much land is needed if the roundabout is to have a 75 m diameter?
 Determine if it is a better deal to buy 2 large pizzas or 3 medium pizzas (or any other combination) at a local pizza shop. Ask students to research the size and prices of these pizzas, then support them to develop a plan to compare the areas of the pizzas and their price per square centimetre.
 Which would cover a greater area:
 six circular sprinklers with a maximum radius of 3 m each, or
 two circular sprinklers with a maximum radius of 5 m each?
Have students explain their thinking.
Volume and Surface Area
E2.6
represent cylinders as nets and determine their surface area by adding the areas of their parts
 net of a cylinder with two closed ends:
 net of a cylinder with one closed end:
 net of a cylinder with two open ends:
 Area is additive: partial areas can be added together to find a whole area. Finding the surface area of a cylinder is an application of the property of additivity.
 Nets help to visualize the twodimensional shapes that make up a threedimensional object such as a cylinder. Cylinders, for their bases, have two parallel, congruent faces (see E1.1 for a full list of the properties of cylinders) that are joined by a rectangle (to produce a right cylinder) or a nonrectangular parallelogram (to produce an oblique cylinder).
 In reallife contexts, cylinders can have two closed bases (e.g., closed tin cans), one closed and one open base (e.g., cylindrical pencil holders), or two open bases (e.g., pipes; paper towel rolls).
Note
 Visualizing the net for a cylinder – imagining it in the “mind’s eye” – involves identifying the shapes that form its faces and recognizing how the dimensions of the cylinder relate to the dimensions of the different faces.
Show students a closed cylinder. Tell them that you would like them to make a net for the cylinder. Ask them what they need to know. Now have them construct the net. Then, ask them to find the surface area of the cylinder using their net and the formulas for the area of a rectangle and the area of a circle.
Have students identify and compare the surface area of cylinders in real life that are closed on both ends, closed on one end, and open (no bases). Guide students to reformulate their formula to find the surface area of a cylinder to account for each of the three variations.
 showing the relationship between the area of the base and the volume of a prism:
 stack several congruent rectangles to build layers with the same base
 notice that the volume of the prism is always the area of its base times its height:
 showing the relationship between the area of the base and the volume of a cylinder:
 stacking several congruent circles to build layers with the same base
 tilting the tower, resulting in an oblique cylinder with the same volume (this can also be demonstrated using a prism)
 noticing that the volume of the cylinder is always the area of its base times its height:
 Volume is measured in cubic units, and the measure represents the number of cubes needed to completely fill an object. Similar to units of area, a cubic unit is an amount of volume, and can come in any shape. Units of volume can be decomposed, rearranged, partitioned, and redistributed to better fill a volume and minimize gaps and overlaps (see E2.1).
 Indirectly measuring the volume of shapes is often quicker and more accurate than measuring volume directly (i.e., by laying out and stacking cubes).
 All prisms and cylinders have two congruent bases that are parallel to each other (see E1.1). This means that, at any height in a prism or cylinder, a “slice” could be made, and – as long as the slice is parallel to the base – the crosssection that is created will have congruent faces. From top to bottom, the area of any slice, or layer, is consistent, and it is always equal to the area of the base. This geometric property of prisms and cylinders forms the basis for the formula for calculating their volume.
 Right prisms and right cylinders have bases that are perpendicular to their sides.
 The rowandcolumn structure of an array, which helps structure the count of square units for area (see Grade 4, E2.5), also helps structure the count of cubic units and is used to indirectly measure volume.
 A unit is repeated to produce the given length (a row).
 A row is repeated to produce the given area of the base (a layer).
 A layer is repeated to produce the given height (the volume).
 The area of the base determines how many cubes can be placed on the base, which forms a single unit – a layer of cubes. The height of the prism determines how many layers of cubes it takes to fill the volume. Therefore, the formula for finding the volume of a rectangular prism is area of the base × height.
Note
 The same is true for any prism or cylinder: the area of the base determines how many cubes can be placed on its base, and the height determines how many layers of cubes it takes to fill the volume. This means that the formula for finding the volume of any cylinder or prism is area of the base × height.
Have students use pattern blocks to model the relationship between the volume of a prism and the area of its base. Have them stack pattern blocks to build a red trapezoidbased prism, a green trianglebased prism, a yellow hexagonbased prism, and a blue rhombusbased prism, all with the same height. As they do so, draw out the idea that every layer in the prism has the same area, and, by extension, every layer also has the same volume.
To support comparisons of the areas and volumes of the different pattern block prisms, ask questions such as the following:
 How do the volumes of these prisms compare, and how do you know?
 Which dimensions remained constant, and which ones changed?
 How would the volume change if you made pattern block prisms of different heights?
 If you needed to find the volume of a pattern block prism (or any prism) and could only request two measurements, which ones would you select and why?
Support students in generalizing a volume formula that would work with any pattern block prism (i.e., the area of the base times the height), and explain why this formula could be applied to any prism, including nonright prisms. (For different prisms, use examples generated from nets in E1.1, Sample Task 1.)
Guide students to consider whether this formula would also apply to cylinders. As students explain their reasoning, ensure that they recognize that, for both prisms and cylinders (and unlike pyramids), the area remains constant throughout their entire height. Have them inspect a variety of cylinders and prisms and identify the relevant dimensions needed to determine the volume of each, as follows:
 for prisms, the length and width dimensions needed to determine the area of the base, and the height
 for cylinders, the radius or diameter needed to determine the area of the base, and the height
After they have developed an understanding of the relationships between the volume, the area of the base, and the height of a prism or cylinder, have students apply these relationships to solve various problems, such as:
 What is the volume of a squarebased prism whose square base has a side length of 4 cm and whose height is 10 cm?
Guide students to recognize that if they have two of (a) the volume, (b) the area of the base, and (c) the height, they can multiply or divide to determine the unknown measurement. Support students in recognizing that the inverse relationship between multiplication and division enables them to use the formula to find more than just a missing volume. For example, they might consider the following:
 A cylinder has a diameter of 10 cm. What might its volume be? Give the radius, the area of the base, and the height for four possible cylinders. How do you know your answers are reasonable?
 A triangular prism has a volume of 10 m^{3}. What might its dimensions be? Give the dimensions for the base and the height, the total area of the base, and the height for four possible triangular prisms. How could you use these calculations to find a rectangular prism with a volume of 10 m^{3}?
Have students represent these situations with a diagram to show what is known and unknown, then use a chart to organize the various possibilities. Use these types of problems to help consolidate work with area, volume, and circles, and to reinforce the inverse relationship between multiplication and division.