Volume of Irregularly Shaped Prisms

In this activity students will develop the idea that the volume of a right prism of ANY shape is just the area of the base times the height. To do this they will construct their own irregularly shaped prisms from snap cubes and then use the volume that they can determine by counting the cubes to develop the relationship. There is some practice at the end that can be done via worksheet or Desmos.

• Grade 7 -  determine, through investigation using a variety of tools and strategies (e.g., decomposing right prisms; stacking congruent layers of concrete materials to form a right prism), the relationship between the height, the area of the base, and the volume of right prisms with simple polygonal bases (e.g., parallelograms, trapezoids), and generalize to develop the formula (i.e., Volume = area of base x height).
• Grade 8 - determine, through investigation using a variety of tools and strategies (e.g., generalizing from the volume relationship for right prisms, and verifying using the capacity of thin-walled cylindrical containers), the relationship between the area of the base and height and the volume of a cylinder, and generalize to develop the formula (i.e.,Volume = area of base x height). Note: though there isn't a direct connection to the cylinder here, since a cylinder is a prism (circular prism), it will adhere to this relationship. 

• 2cm snap cubes (each student or pair of students will need up to 60)
• photocopy of the grid handout for each student or pair of students. Note that when you copy or print this out, make sure that the check box "fit" is selected to maximize the size of the grid making the squares the same size as the snap cubes. You may want to verify this before printing a bunch out for the class
• If you are doing this virtually, then you might want to have them actually build their shapes using this virtual manipulative.

1. Students are first given a blank grid and asked to trace two non rectangular shapes of different areas (no diagonals) using the grid lines.
2. Once they have their shapes they should determine the area (in units²) of each shape and enter that in their table. 
3. Students then take the cubes and connect them to create a 3D shape to match the profile of their drawn shapes (1 unit high).If you are doing this virtually, then you might want to have them actually build their shapes using this virtual manipulative.
4. Students can then determine the volume of each shape (in units³) by counting the cubes used and enter those values in the table
5. Next increase the height of each shape by one unit, determine the volume by counting the cubes and enter those values in the table.
6. Repeat step 5 so that the shapes are now 3 units high. 
7. Students are then to look for a connection between the height, area of the shape (base) and the volume and use that relationship to predict the volume of a shape that is 6 units in height (without building it). Students will hopefully see that the volume can be found by multiplying the area of the base times the height. 
8. Generalize the relationship (Volume = area of the base x height) and have students complete this worksheet to practice. For a Desmos version, click here.   

• Grid Handout (PDF, Google Drawing)
• Worksheet (PDF, Google Drawing, Desmos)
• Table (PDF, Google Doc)

Did you use this activity? Do you have a way to make it better? Did you find a mistake? If so tell us in the comment section. Thanks

Volume of a Pyramid

In grade 9 we are supposed to develop the formula for the volume of a pyramid via investigation. That is, by doing some sort of procedure (not algebraically). If you are lucky your schools might have a set (or several sets) of these volumetric solids so you can pour liquids from one to another to compare but if not, here is a cheap way to do the same thing (at least for right prisms). For an idea of how to do it via water and the volumetric solids, check out this post from Tap Into Teen Minds.

• MPM1D, MFM1P - develop, through investigation (e.g., using concrete materials), the formulas for the volume of a pyramid, a cone, and a sphere (e.g., use three-dimensional figures to show that the volume of a pyramid [or cone] is 1/3 the volume of a prism [or cylinder] with the same base and height, and therefore that V_pyramid= V_prism/3

• Each student or group needs six of the large nets to make their rectangular prism and three of the small nets to make the cube. See below to download the nets.
• Note that each of these shapes is a pyramid, just non symmetrical.

Note that these instructions are for students. You may just wish to do this as a demo.

1. Have the students construct the nine shapes using the nets (6 large and 3 small). 
2. Taking the six large shapes, construct three symmetrical pyramids. Note the dimensions (size of base and height).
3. Using the same six shapes, create a rectangular prism. Note the dimensions (size of base and height).
4. How do the height and base of the prism compare to the one of the symmetrical pyramids? 
5. How do the volumes of each pyramid compare to that of the prism?
6. Taking the three small shapes, note the type of solid and its dimensions (size of base and height)
7. Using the same three shapes construct a cube. Note the dimensions (size of base and height)
8. How do the height and base of the cube compare to each non symmetrical pyramid? 
9. How do the volumes of each non symmetrical pyramid compare to that of the cube?
10. If you had any prism and a pyramid with the same base and height, how would the volume of the pyramid compare to the prism?

As an added feature you may wish to construct a figure that folds in and out from one shape to several shapes as seen in the videos below.

• A rectangular prism in six parts (pdf) (doc) Answers (pdf) (doc)
• A cube in three parts (pdf) (doc) Answers (pdf) (doc)

Did you use this activity? Do you have a way to make it better? If so tell us in the comment section. Thanks