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.

 Hay Bale Farmer Activity Sheet 

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 cylinders 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 = π × r² × h = π × (3)² × 4 ≈ 113 ft³ 

   volume of square bale = l × w × h = 3 × 2 × 1¹/₂ = 9 ft³ 

   

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.
    Solution – Question 3 Overhead 
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.

Assessment Options 

1. 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.
2. Use the Hay Bale Farmer Activity Sheet as a form of assessment.

Extensions 

Questions for Students 

1. 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.]

2. How do you determine the unit rate of dollars per ft³?

[You create a ratio, where units are in the denominator, and dollars are in the numerator. Then, you divide both the numerator and denominator by the denominator.]

Teacher Reflection 

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.

NCTM Standards and Expectations

• Work flexibly with fractions, decimals, and percents to solve problems.

• Understand and use ratios and proportions to represent quantitative relationship.

• Understand, 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.

Common Core State Standards – Mathematics

Grade 6, Ratio & Proportion

• CCSS.Math.Content.6.RP.A.1
  Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, ''The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.'' ''For every vote candidate A received, candidate C received nearly three votes.''

Grade 6, Ratio & Proportion

• CCSS.Math.Content.6.RP.A.2
  Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship. For example, ''This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.'' ''We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.''

Grade 6, Geometry

• CCSS.Math.Content.6.G.A.2
  Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = b h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.

Grade 7, Ratio & Proportion

• CCSS.Math.Content.7.RP.A.1
  Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction (1/2)/(1/4) miles per hour, equivalently 2 miles per hour.

Grade 7, The Number System

• CCSS.Math.Content.7.NS.A.3
  Solve real-world and mathematical problems involving the four operations with rational numbers.

Grade 7, Expression/Equation

• CCSS.Math.Content.7.EE.B.3
  Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

Grade 8, Geometry

• CCSS.Math.Content.8.G.C.9
  Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
  Make sense of problems and persevere in solving them.

• CCSS.Math.Practice.MP4
  Model with mathematics.

• CCSS.Math.Practice.MP5
  Use appropriate tools strategically.

• CCSS.Math.Practice.MP7
  Look for and make use of structure.