Illuminations: Hay Bale Farmer

# Hay Bale Farmer

 In this lesson, students will use dimensions of round and square hay bales to calculate and compare volumes. They also calculate unit prices to determine which hay bale is the better value. Finally, students explore how to fit round and square bales into a barn to maximize volume, and decide which type of hale bale is the best choice.

### Learning Objectives

 Students will: Calculate and compare volumes of different solid figures Calculate and compare unit rates Verbally and graphically describe arrangements of rectangular prisms and cylinders in a given space Calculate percent of a given value

### Materials

 Pictures of hay bales (optional) Cylindrical and rectangular prism blocks (optional) Hay Bale Farmer Activity Sheet Solution – Question 3 Overhead

### Instructional Plan

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

1. 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 baleVolume of square bale ≈ 113 ft39 ft3 ≈ 12.6

2. The quantities needed for 1 year are:
• 1,778 square bales

or

• 142 round bales

Note: These values are rounded up since you can only buy whole bales.

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.

3. 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.
4. 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.
5. 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.

### Questions for Students

 Why did we round up when answering the questions in this activity? [You can only purchase whole bales, so while we may calculate that we need 13.2 bales, we cannot buy 0.2 bales. Therefore, we must buy 14 full bales of hay.]

### Assessment Options

 Have students write a journal entry, using information on their activity sheet to provide a mathematically based argument for purchasing one type of hay bale.

### Extensions

 Students can bring in their own data values by going outside and actually measuring hay bales, or by using the current price for round and square bales. Students can calculate the number of round bales that would fit in a barn that has different dimensions or a different shape than the one use in the activity sheet. Since rectangular bales fit better in a rectangular barn, would round bales fit better in a round barn?

### Teacher Reflection

 Did students struggle with understanding the different types of hay bales? How could you introduce farming to them? Was students’ level of enthusiasm and involvement high or low? Why? Were students able to personally connect to the information? How could you adapt the lesson for higher or lower achievers? Did students understand how different shapes can lead to different answers? What else could you have done to emphasize this point?

### NCTM Standards and Expectations

 Measurement 6-8Understand, select, and use units of appropriate size and type to measure angles, perimeter, area, surface area, and volume. Develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders. Solve simple problems involving rates and derived measurements for such attributes as velocity and density. Number & Operations 6-8Understand and use ratios and proportions to represent quantitative relationships. Work flexibly with fractions, decimals, and percents to solve problems.
 This lesson was prepared by Katie Hendrickson as part of the Illuminations Summer Institute.

1 period

### NCTM Resources

 More and Better Mathematics for All Students
 © 2000 National Council of Teachers of Mathematics Use of this Web site constitutes acceptance of the Terms of Use The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring mathematics learning of the highest quality for all students. The views expressed or implied, unless otherwise noted, should not be interpreted as official positions of the Council.