E2. Measurement
Specific Expectations
The Metric System
E2.1
represent very large (mega, giga, tera) and very small (micro, nano, pico) metric units using models, base ten relationships, and exponential notation
 very large metric units:
 kilounit: 1 thousand units:
 kilogram: mass of an object
 kilometre: distances travelled on land and in the air
 megaunit: 1 million units:
 megahertz: frequency of electromagnetic radiation for broadcasting stations
 megapixels: picture resolution
 gigaunit: 1 billion (1 × 10^{9})^{ }units:
 gigametre: distance of planets from the Sun
 gigabit: bandwidth of a network link
 teraunit: 1 trillion (1 × 10^{12})^{ }units:
 terabyte: data storage
 terasecond: approximately 32 000 years
 kilounit: 1 thousand units:
 very small metric units:
 milliunit: 1 thousandth of a unit:
 millimetre: thickness of a card
 millilitre: volume in cooking and baking
 microunit: 1 millionth of a unit:
 micrometre: measure of microscopic objects
 microsecond: duration of a highspeed strobe flash
 nanounit: 1 billionth (1 × 10^{−9})^{ } of a unit:
 nanosecond: time for light to travel
 nanometre: length a fingernail grows in 1 s
 picounit: 1 trillionth (1 × 10^{−12})^{ }of a unit:
 picometre: measure of an atom
 picosecond: speed of lasers
 milliunit: 1 thousandth of a unit:
 Technology has enabled accurate measurements including very small and very large measures.
 All metric units are based on a system of tens, and the metric prefixes describe the relative size of a unit (see Grade 4, SE E2.2). Whereas units from kilo to milli are scaled by powers of 10, units beyond these are scaled by powers of 1000. Exponents are helpful for representing these relationships.
Metric Prefix  Meaning  Factor 
teraunit  1 trillion units  1 unit × 1 000 000 000 000 (10^{12}) 
gigaunit  1 billion units  1 unit × 1 000 000 000 (10^{9}) 
megaunit  1 million units  1 unit × 1 000 000 (10^{6}) 
kilounit  1 thousand units  1 unit × 1000 (10^{3}) 
unit  one unit  1 unit (10^{0}) 
milliunit  one thousandth of a unit  1 unit ÷ 1000 ( or 10^{−}^{3}) 
microunit  one millionth of a unit  1 unit ÷ 1 000 000 ( or 10^{−}^{6}) 
nanounit  one billionth of a unit  1 unit ÷ 1 000 000 000 ( or 10^{−}^{9}) 
picounit  one trillionth of a unit  1 unit ÷ 1 000 000 000 000 ( or 10^{−}^{12}) 
Show students a metre stick, stating that it is being used as a scaled model. Pose questions similar to the following:
 If the metre stick (1 m long) represents 1 km, what does 1 cm represent? What does 1 mm represent?
 If the metre stick represents a megametre (Mm), what does 1 cm represent? What does 1 mm represent?
 If the metre stick represents 1 mm, where would a micrometre (μm) be on the metre stick?
Similarly, use base ten blocks to model capacity units; consider the thousands block as a kilolitre (kL), a litre (L), or a millilitre (mL), and use that as a reference to visualize the other units.
Have students, in pairs, consider whether the statements below are true or false.
 Earth is about 13 megametres in diameter.
 The Sun is about 1.4 gigametres in diameter.
 Modern humans have been around for about 6 teraseconds.
 Light takes about a nanosecond to travel 30 centimetres.
 Human life expectancy is between 2 and 3 gigaseconds.
Have students describe these measurements using more common units, discuss strategies to convert them into very large or very small units, and verify the statements above.
Lines and Angles
E2.2
solve problems involving angle properties, including the properties of intersecting and parallel lines and of polygons
 angles formed by the transversal line intersecting two parallel lines:
 properties of angles and intersecting and parallel lines:
 supplementary angles (add up to 180°):
 complementary angles (add up to 90°):
 opposite angles are equal, so ∠a = ∠c and ∠f = ∠h:
 alternate angles (Zpattern) are equal, so ∠c = ∠e:
 corresponding angles (Fpattern) are equal, so ∠c = ∠g:
 cointerior angles (Cpattern) add up to 180°, so ∠d + ∠e = 180°:
 interior angles of polygons:
 triangle: sum of the angles is 180°
 quadrilateral: sum of the angles is 360°
 any polygon: sum of the angles is (n − 2) × 180, where n is the number of sides
 exterior angles:
 ∠ACD is an exterior angle of △ABC:
 ∠ACB + ∠ACD = 180°:
 Angles can be measured indirectly (calculated) by applying angle properties.
 If a larger angle is composed of two smaller angles, only two of the angles are needed to calculate the third.
 Angle properties can be used to determine unknown angles.
 A straight angle measures 180°; this is used to determine the measure of a supplementary angle.
 A right angle measures 90°; this is used to determine the measure of a complementary angle.
 The interior angles of triangles sum to 180°, the interior angles of quadrilaterals sum to 360°, the interior angles of pentagons sum to 540°, and the interior angles of nsided polygons sum to (n − 2) × 180. The angle properties of a polygon can be used to determine the measure of a missing angle.
 The properties above can be used to determine unknown angles when a line (transversal) intersects two parallel lines:

Note
 The aim of this expectation is not to memorize these angle theorems or the terms, but to use spatial reasoning and known angles to determine unknown angles.
 Smaller angles may be added together to determine a larger angle. This is the additivity principle of measurement.
 If two shapes are similar, their corresponding angles are equal (see E1.3). Recognizing similarity between shapes (e.g., by ensuring that the corresponding side lengths of a shape are proportional) can help to identify their corresponding angles.
Have students draw two parallel lines and a transversal (a line that passes through both of the others). Have them use a protractor to measure the angles formed and describe the relationships between them. Give students a diagram of parallel lines with a transversal and one angle measure indicated, not drawn to scale. Have them use the relationships to determine the other angle measures and then redraw the diagram to scale.
Present students with a diagram similar to the one below. Ask them to solve for the missing angle measurements if, for example, ∠b is equal to 51°. Support students in using known angles to justify their solutions:
Have students use angle properties as well as their understanding of scale drawings, dilations, and similarity (e.g., proportional side lengths; congruent angles), to solve missingangle and missinglength puzzles, such as the one below.
Have students invent their own missingmeasurement puzzles, incorporating the various angle properties. Challenge them to make puzzles with the minimum number of given angle measurements needed to determine the unknown angle measures. Have them exchange their puzzles with others in the class and discuss their strategies for determining the unknown angle measures.
Length, Area, and Volume
E2.3
solve problems involving the perimeter, circumference, area, volume, and surface area of composite twodimensional shapes and threedimensional objects, using appropriate formulas
 composite twodimensional shapes:
 composite threedimensional shapes:
 Twodimensional shapes and threedimensional objects can be decomposed into measurable parts.
 The attributes of length (including distance, perimeter, and circumference), area (including surface area), volume, capacity, and mass all have the property of additivity. Measures of parts can be combined to determine the measure of the whole.
 For some attributes and for some shapes, relationships exist that can be expressed as formulas. To apply these formulas to composite shapes and objects, the shapes and objects are decomposed into parts that have known formulas. For example, an Lshaped area could be decomposed into two rectangles, and the smaller areas added together to calculate the whole. (Note: This does not hold true for its perimeter.)
 Applying formulas to realworld contexts requires judgement and thoughtfulness. For example, to apply the formula for the area of a rectangle to a garden:
 determine whether the garden is rectangular;
 determine whether the garden is close enough to a rectangle that, for the needs of the moment, the formula can be applied;
 if not rectangular, determine whether the garden can be broken into smaller rectangles (e.g., if it is an Lshaped garden) and the areas combined;
 if decomposition results in other shapes, apply appropriate area formulas or draw a scale drawing on a grid and approximate the count of squares.
 Known length formulas at this grade level include:
 Perimeter = side + side + side + ...
 Diameter = 2 × radius or 2r
 Circumference = π × diameter or πd
 Known area formulas at this grade level include:
 Area of a rectangle = base × height
 Area of a parallelogram = base × height
 Area of a triangle = (base × height)
 Area of a trapezoid = (base 1 + base 2) × height (or its equivalent)
 Area of a circle = π × radius × radius or πr^{2}
 Known volume formulas at this grade level include:
 Volume of a prism = (area of the base) × height
 Volume of a cylinder = (area of the base) × height
Have students solve problems involving composite, nonroutine shapes. For example, have students find the distance around a racetrack. Support them in recognizing that this shape can be decomposed into a rectangle and two semicircles (which make up one circle), and have them identify how the distance across the track relates to the diameter of the semicircles. Have them combine the measurements to find the total. In a similar way, have them determine the area of the grass in the middle of the track by adding together the areas of its parts.
Have students build a structure that is composed of at least two different prisms or includes a cylinder. Tell them that the exposed surface of their structure is to be painted, so they need to determine the surface area to know how much paint they need.
Tell students they are playing the role of a marketer who must design a container for a given object. The container is to be appealing and eyecatching, made from prisms, cylinders, or a combination of both, and have a capacity of 3 L.
Have students make a prototype of their container. Note that the choice of object influences the containers students make. As a culmination, students could display their containers in an exhibit. Have them prepare a poster to accompany their container with information about what it is designed to hold, the material it is to be made from, the shapes that inspired it, and measurements such as surface area and volume. Students could also be challenged to create packaging with minimal environmental impact.
E2.4
describe the Pythagorean relationship using various geometric models, and apply the theorem to solve problems involving an unknown side length for a given right triangle
 geometric models:
 The properties of a right triangle can be used to find an unknown side length. The longest side of a right triangle is always opposite the 90° angle and it is called the hypotenuse.
 Given any right triangle, the length of the hypotenuse squared is equal to the sum of the squares of the lengths of the other two sides. This is known as the Pythagorean relationship. The length of the hypotenuse squared and the length of each of the sides squared can be represented as three squares formed with the three sides of the triangle, and then visualizing the sum of the areas of the two smaller squares as being equivalent to the area of the larger square.
 The Pythagorean theorem expresses this relationship symbolically: a^{2} + b^{2 }= c^{2}, where c is the length of the hypotenuse and a and b are the lengths of the other two sides of the triangle. For example:
 if side a is 3 units long, then a square constructed on this side has an area of 3^{2} or 9 square units;
 if side b is 4 units long, then a square constructed on this side has an area of 4^{2} or 16 square units;
 if the square on side c is equal to the combined areas of the squares on sides a and b, then the square on side c must have an area of 25 square units (9 square units + 16 square units);
 if the area of the square on side c is 25 square units, then the length of c must be or 5 units.
 The inverse relationship between addition and subtraction means that the Pythagorean theorem can be used to find any length on a right triangle (e.g., c^{2} − b^{2 }= a^{2}; c^{2} − a^{2 }= b^{2}).
 The Pythagorean theorem is used to indirectly measure lengths that would be impractical or impossible to measure directly. For example, the theorem is used extensively in construction, architecture, and navigation, and extensions of the theorem are used to measure distances in space.
Note
 Dynamic geometric software can provide students with opportunities to expand their thinking about the Pythagorean relationship related to the areas of shapes other than squares (e.g., three semicircles whose diameters are the sides of a triangle.)
 The properties of a square can be used to find its side length or area. The side length of a square is equal to the square root of its area (see also B1.3).
To support students in recognizing and understanding the Pythagorean relationship, guide them through experiences such as the following:
 Have students draw a right triangle on grid paper. Have them draw a square on each side of the triangle and determine the area of each one. Note that for the square on the hypotenuse, they will need to consider how partial squares can be rearranged to make six full squares in each of the triangles. Have them use the area of a square to find the side lengths of the triangle (i.e., the side length = the square root of the area of the square) and explain how drawing a square on a line segment can help find its length.
 To establish the relationship between the areas of the three squares on the triangle, have students cover the square on the hypotenuse with the squares from the other two side lengths (i.e., by cutting up the squares and rearranging the parts). Have them share their findings and verify that this relationship holds true for each right triangle constructed by their classmates. Support students in verifying that, while the relationship is true for right triangles, it does not hold true for nonright triangles. Consolidate the learning by generalizing the relationship for right triangles as a theorem: a^{2} + b^{2} = c^{2}.
 To demonstrate that the Pythagorean theorem applies to all right triangles, have students investigate visual proofs of the theorem. Have them search the Internet for dynamic models of the Pythagorean theorem. When they find one that they consider interesting and convincing, have them share it with their peers and explain why it is a visual proof for the theorem.
 Provide students with many opportunities to apply the Pythagorean theorem, particularly in contexts that show its relevance in realworld situations. For example, to connect the learning to work with coding, have them program a robot to construct a right triangle with two given side lengths. Draw on the inverse relationship between addition and subtraction to show how the theorem can find any unknown side length of a right triangle. Use a partwhole model to make this relationship visible.
One application for the Pythagorean theorem is carpentry and construction. Have students recognize that each diagonal in a rectangle creates two right triangles and that each of the two diagonals in a door frame should, therefore, form two congruent right triangles. Discuss how the Pythagorean relationship could be used to test whether the door frame is “square”, and support them in recognizing that, if the Pythagorean relationships applies to the two triangles in the door, then the angles in the door must be “right”. Have them apply their thinking to measure doors around the school (their base, their height, and their diagonal) and draw scaled models to prove or disprove that the doors are “square”.
Assign problems that require students to use the Pythagorean relationship. For example:
 Your class has decided to participate in a program called “Operation Spring Cleaning”. Each student will volunteer their time to assist an elderly neighbour with spring cleaning. You will be helping your neighbour to clean out her basement.
 While cleaning out your neighbour's basement office, you come across a large bulletin board that measures 1.8 m by 1.2 m. You think it would be perfect for your bedroom to help you organize your school assignments. Your neighbour says you are welcome to have it, but you can’t take it up the stairway in case it marks her newly painted walls. There is an office window that measures 1 m by 0.75 m. Will the bulletin board fit through it?