compare, estimate, and determine measurements in various contexts
represent very large (mega, giga, tera) and very small (micro, nano, pico) metric units using models, base ten relationships, and exponential notation
Show students a metre stick, stating that it is being used as a scaled model. Pose questions similar to the following:
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.
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.
solve problems involving angle properties, including the properties of intersecting and parallel lines and of polygons
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 missing-angle and missing-length puzzles, such as the one below.
Have students invent their own missing-measurement 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.
solve problems involving the perimeter, circumference, area, volume, and surface area of composite two-dimensional shapes and three-dimensional objects, using appropriate formulas
Have students solve problems involving composite, non-routine 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 eye-catching, 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.
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
To support students in recognizing and understanding the Pythagorean relationship, guide them through experiences such as the following:
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: