gr8_packaging

//It is expected that students will://
 * **Prescribed Learning Outcomes**

A5 solve problems that involve rates, ratios, and proportional reasoning [C, CN, PS, R] C2 draw and construct nets for 3-D objects [C, CN, PS, V] C3 determine the surface area of to solve problems [C, CN, PS, R, V] C4 develop and apply formulas for determining the volume of right prisms and right cylinders [C, CN, PS, R, V] ||
 * right rectangular prisms
 * right triangular prisms
 * right cylinders
 * **Planning for Assessment** || **Assessment Strategies** ||
 * Provide groups of students with paper, scissors, and tape, challenging them to create a rectangular prism, a triangular prism, and a cylinder. In writing or through dialogue, have students reflect on how the surface of a 3-D object can be created using only 2-D shapes.

Provide each student with a prism (ie. a wooden block) and have them draw each of the faces.

Have each student measure the necessary sides of their prism and label the side lengths in their diagram.

Have each student calculate the surface area of each prism by finding the area of each face and then determining the sum.

Using words and a diagram, each student is asked how they could combine faces into larger ones to reduce the number of area calculations needing to be performed.

Students are presented with the task of designing a package for a food product. With or without the use of computer software, each student designs one package in each shape of a rectangular, triangular, and circular prism, each one fitting within a bounding rectangular prism. (Note that "cylinder" is a special name for a circular prism, and it is beneficial for student understanding to identify that a cylinder is also a prism)

Using available measurement tools, students calculate the surface area of their package by creating a table of measurements outlining the calculation of each face's area.

Given a price rate per area of package material have students calculate the price of the packaging for each prism || Verify that students are able to identify all of the faces of a given prism.

Through discussion or student writing, assess students' ability to explain the relationship between the area of 2-D shapes and the surface area of 3-D objects.

Check student calculations to assess students' ability to describe and apply strategies for determining the surface area of a given object.

Assess students' abilities to apply a rate to solve a problem. ||
 * With or without the use of computer software, have students create prisms from faces by extending or "extruding" the face. (Faces should minimally include circles, rectangles, and triangles, but using regular polygons and irregular shapes will help develop the conceptual understanding of a prism, particularly in relation to volume.)

Provide students with some building blocks and ask them to create some rectangular prisms with dimensions (length by width by height) such as the following: a) 4 blocks by 5 blocks by 1 block b) 4 blocks by 5 blocks by 2 blocks c) 4 blocks by 5 blocks by 3 blocks d) 3 blocks by 2 blocks by 1 block e) 3 blocks by 2 blocks by 2 blocks f) 3 blocks by 2 blocks by 4 blocks

Then ask the students to answer the following questions: 1) How many blocks does it take to make each prism? 2) What is the area (in terms of blocks squared) of the top of each prism? 3) What is the relationship between the area of the base of each prism, the height of each prism and the number of blocks required to make each prism. 4) If the number of blocks required to make each prism represents the volume of the prism determine a formula for calculating the volume of a rectangular prism. Discuss whether or not their formula would apply to prisms with other bases. 5) What would happen to the volume if two of the dimensions were switched with each other?

Have students use the formula they developed to calculate the volume of each of their packages  and include each calculation in their table of measurements.

Students should determine a cost per volume ratio for each package to determine the most economical shape for their package.

If using 3D modeling software, students can design the packaging for each face of a package using computer graphics software, and superimpose the images onto the 3D model.

As an extension for more advanced students, investigation can be made into adjusting the dimensions of a given prism while keeping volume constant to maximize the volume to surface area ratio. || Verify that students are able to explain the relationship between the area of the base of a prism and its volume, and to generalize a rule for determining volume.

Ensure that students are able to recognize that the orientation of a prism does not affect its volume.

Collect students' calculations and assess their ability to apply a formula to determine the volume of a prism.

Assess students' ability to apply a conceptual understanding of volume to solve a problem.

Verify that students are able to determine and apply ratios to solve a problem. ||
 * Have students create prototype models of their packages by designing a net for each. Student use the exact measurements from their designs to draw out each face.

Present students with a variety of nets, including atypical arrangements for the three prisms covered, but also pyramids, regular polyhedra, and other solids. Without assembling the net, have them describe in words what the object would look like, and select a corresponding shape from a wide assortment of objects such as wooden blocks or polyhedral dice.

Have students verify their predictions by assembling the nets. || Verify that students are able to create nets for a given triangular, rectangular, or circular prism.

Assess students' abilities to predict 3-D objects that can be created from a given net and verify their prediction by constructing a 3-D object from the net.

Check that students are able to recognize that different nets can generate the same object. ||