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?
- What shape are the round bales?
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
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
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
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.
- You would need to purchase approximately 12.6 square bales. If
students are stuck, you may want to suggest that they find the volume
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
- 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
Solution – Question 3 Overhead
- 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.
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.]
- 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
- 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
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
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
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
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
Solve real-world and mathematical problems involving the four operations with rational numbers.
Grade 7, Expression/Equation
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
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
Make sense of problems and persevere in solving them.
Model with mathematics.
Use appropriate tools strategically.
Look for and make use of structure.