Introduce students to the lesson through a discussion about hay bales. If students are unfamiliar with hay bales, bring in some pictures and discuss the shapes along with the common uses for hay. Use the following questions as a guide:
- What types of hay bales exist?
[Round and square. Be sure to mention to students that although these terms are mathematically inaccurate, you will use them during the lesson because they are the standard terms used in practice.]
- What shape are the square bales?
[rectangular prisms]
- What shape are the round bales?
[cylinders]
If students do not have any background knowledge of hay bales, an alternative problem could be finding the volume of soda cans versus juice boxes, and how many of each can fit into a large cardboard box. You could also compare round oatmeal canisters to rectangular cereal boxes. Ask students to name similar products that are sold in different-shaped containers and how they think the distributors choose the shapes.
Tell students that they are going to investigate which type of hay bale is the better deal mathematically. Students in some communities may already have some information about their family’s preferred hay bale, and they may volunteer their opinion and the contributing factors to the choice. If they seem eager to contribute, allow them to make predictions, such as which takes more space and which costs more. Remind them that a lot of factors go into deciding which type to purchase, but today they will just be looking at two factors.
Distribute the Hay Bale Farmer activity sheet to each student. Read the introduction and ensure that students understand the diagrams. Put students into groups of 3 or 4 of mixed ability, and have them begin working on answering the questions. You may choose to make cylindrical and rectangular prism blocks available to help students reason through the problems and draw their diagrams.
As students work, circulate among the groups and provide guidance as needed. If you want to keep all students on pace with the rest of the class, you can have students answer one question at a time and discuss their solutions as a class. Otherwise, wait until all students are finished to discuss the answers. Have students share their answers and draw diagrams on the board. Three-dimensional drawings are difficult for many students, so stress the importance of communicating the mathematics over creating an accurate diagram in the problem.
Solutions
- You would need to purchase approximately 12.6 square bales. If students are stuck, you may want to suggest that they find the volume of each.
Volume of round bale = 
× r2 × h = × (3)2 × 4 ≈ 113 ft3
Volume of square bale = l × w × h = 3 × 2 × 11/2 = 9 ft3
Volume of round bale
Volume of square bale |
≈ |
113 ft3
9 ft3 |
≈ 12.6 |
- The quantities needed for 1 year are:
To determine which is the better value, students could find the unit rate of dollars per ft3:
- Square bales cost $0.31/ft3.
- Round bales cost $0.18/ft3.
They could also multiply the cost per bale times the number of bales needed:
- Square bales cost $4,889.50 for a 1-year supply.
- Round bales cost $2,840 for a 1-year supply.
Either way, the round bales are far more economical.
- The barn will fit 1,728 square bales or 108 round bales. Students should draw 2 diagrams, as shown on the Solution – Question 3 overhead. The square bale diagram should show 18 bales of hay fitting along the length, 12 along the width, and stacked 8 high. The round bale diagram should show 6 bales of hay along the length and 9 along the width, stacked 2 high. Square bales fit a greater volume inside the barn.
- 34 bales will have to be stored outside. To find this solution, take the difference of 16,000 and the volume that can be stored in the barn, then divide by the volume of one bale and round up. Since 34 round bales of hay have a volume of approximately 3,845.3 ft3, the 10% loss will amount to about 384.5 ft3, which is the equivalent of 3.4 round bales.
- Student responses may vary. While the round bales cost less by volume, they are harder to store. The square bales can be stored more compactly so less hay needs to be stored outside, which results in less wasted hay due to mold.
