Primary Type: Lesson Plan
Status: Published
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 28562
Sweet Surface Area
In this lesson, students will explore the relationship between volume and surface area through real world problem solving. They will work with a
partner as they are in charge with the task of finding the least expensive packaging (smallest surface area) for a given number of caramels
(volume). Students will justify their packaging strategy in a group discussion.
Subject(s): Mathematics
Grade Level(s): 7
Intended Audience: Educators
Suggested Technology: Interactive Whiteboard,
Overhead Projector
Instructional Time: 2 Hour(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: Surface area, volume, rectangular prism, cube
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
Surface Area Caramel Challenge.docx
sweet surface area student completed example.pdf
LESSON CONTENT
Lesson Plan Template: General Lesson Plan
Learning Objectives: What should students know and be able to do as a result of this lesson?
Students will work with math manipulatives to understand the relationship between volume and surface area.
They will determine that rectangular prisms comprised of cubes will have a constant volume but may have varying surface areas. The variety of the surface areas will
depend on the factors of the volume and the different ways students can arrange cubes within a given rectangular prism.
Prior Knowledge: What prior knowledge should students have for this lesson?
For this lesson to be successful, students need to have had experience with finding the surface area of rectangular prisms and cubes; this should not be a student's
first experience with surface area.
This prior experience can be either with using a ruler to measure the area of six sides of a rectangular prism and adding them together to find the total surface
area or with a formula for surface area.
Students also need to understand what volume is, as well as the difference between volume and surface area.
It will also be helpful if students have a working knowledge of factoring. If students can find the factors for a given number, then it will be easier for them to create
different arrangements of their cubes within the rectangular prism.
Guiding Questions: What are the guiding questions for this lesson?
Can rectangular prisms with the same volume have different surface areas?
Will a rectangular prism with a large volume automatically have a large surface area?
Is there a constant relationship or pattern between volume and surface area?
page 1 of 4 Teaching Phase: How will the teacher present the concept or skill to students?
1. Arrange students so that they are working with a partner. If students have a math partner or a peer buddy, then allow them to work together. If not, arrange
partners ahead of time so that students will be in successful pairings.
2. Give each pair 12 cubes.
3. Tell students to put the cubes in a straight line on their desks. This will create a long, skinny rectangular prism.
4. Give students time to think, pair, share as you review volume by asking, "What is the volume of your prism?" Students should respond 12 cubic inches because
there are 12 cubes in the prism.
5. Now give students time to think, pair, share and record their answers as you ask, "What is the surface area of your prism?"
6. Circulate and check calculations. It may help students to create a chart to keep track of surface area so that they do not miss any of the six sides of the prism.
Here is an example:
Front - 12 square inches
Back - 12 square inches
Top - 12 square inches
Bottom - 12 square inches
Side 1 - 1 square inch
Side 2 - 1 square inch
*Keep in mind that the front, back, etc. will depend on the perspective of the student. Just make sure they keep their perspectives constant throughout the lesson.
Total Surface Area - 50 square inches
1. Have a pair come up to the board and share their calculations.
2. Ask, "Are there any other arrangements you can make with your 12 cubes to still have a rectangular prism? If you can find any, arrange your cubes and calculate
your new surface area."
3. Circulate and check calculations. Students may arrange their cubes in a 6 by 2 or 2 by 6 prism (which will have the same surface area) or a 3 by 4 or 4 by 3 prism
(which will have the same surface area). The 6 by 2 prism will have a surface area of 40 square inches and the 3 by 4 prism will have a surface area of 38 square
inches.
4. Ask students to discuss why they think a prism with a constant volume of 12 cubes can have different surface areas. Students should respond that the different
arrangement of cubes allows for some prisms to be packed tighter than others; the tighter packing leads to a smaller surface area.
5. Now ask students to think about packs of gum. Ask, "How are packs of gum or candy usually packaged?" Students should respond that they are usually made as
small as possible; the sticks of gum are arranged as close together as possible to cut down on packaging.
6. If your school allows, bring in different packs of gum to show students how many pieces of gum are packed into a small, tight space. Ask students, "Why do you
think companies choose to package gum in this way?" Students should respond that less packaging costs the company less money; the smaller the surface area of
the package, the less money the company has to spend on excess packaging.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
Tell students they will work with their partners to find as many possible surface areas for two given volumes.
1. First, let the students work with 20 cubes (a prism with a volume of 20 cubic inches). As students work, ask them to find the smallest surface area for the 20 cubes.
The tightest packed arrangement is a 4 by 5 or 5 by 4 prism with a surface area of 58 square inches.
2. When students have solved the first problem, let them work with 19 cubes (a prism with a volume of 19 cubic inches). Before they begin working, ask them to
predict if this prism will have a smaller surface area than the one with 20 cubes. Most students will predict that the prism with a volume of 19 will have a smaller
surface area because the volume is smaller. However, since 19 is a prime number, it can only be arranged in a 19 by 1 or 1 by 19 prism. Either way, the surface
area is 78 square inches which is much greater than the prism with a volume of 20 and surface area of 58 square inches.
3. Discuss these findings as a class by asking students to now predict if they think a prism with a volume of 17 or 18 could have the smallest surface area. Students
should understand that 18 has more possible factors than 17 so it will have a smaller surface area. If students do not understand this from the discussion about 19
and 20, then allow them time work out the volumes and surface areas with 17 and 18.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
*Complete the independent practice before the closure.
Have students complete the SurfaceAreaCaramelChallenge.docx worksheet with their partners. The worksheet consists of a challenge to find the best way (smallest
surface area) to package a given number of caramels. Continue to use the one inch cubes as a model for the caramels because the cubes are easier to package than
caramels (Some candy caramels will not be perfect cubes due to melting, compacting, etc.).
READ THE WORKSHEET CAREFULLY BEFORE GIVING IT TO PAIRS; STUDENTS NEED TO BE CHECKED BY THE TEACHER AT CERTAIN POINTS. THEY WILL ALSO
NEED FOIL FOR THEIR FINAL SOLUTIONS.
It is up to the teacher to determine how many cubes to give each group to package; this is a wonderful opportunity for differentiated instruction. Struggling groups
may only need to work with 8 cubes, while more advanced groups may work with 50. I recommend using the numbers 8, 9, 10, 14, 15, 16, 21, 22, 25, 27, 30, 33,
36, 40, 45, 49 and 50 for the volumes (number of cubes), but you can use any numbers. You may want to allow students to choose their number from the given
numbers so they take ownership in receiving a "fair" number.
You can also have more than one group with the same number so that pairs can check each other's calculations during the closure. Here is an example of a student
completed worksheet from a pair of students with 16 cubes: SurfaceAreaCaramelChallenge.docx
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
*Complete the independent practice before the closure.
1. Arrange the entire class to sit in a circle or in a manner conducive to sharing ideas.
2. Have students sit next to their partners and hold their foil-wrapped packages of caramel (or cubes representing caramel). Remember, the objective of the
worksheet was for them to take a given volume and package the cubes into an arrangement with the smallest possible surface area.
3. Go around the circle and allow each group to share their volume and smallest surface area they found for that volume.
4. Record student answers on the board so that the class has a record of each others' work.
page 2 of 4 5. As students share, remind them to justify why and how they decided on the smallest surface area for their volume.
6. Encourage audience members to ask questions such as, "Were there any other ways you could have packaged your cubes?" or "Your volume was 30 and I know 3
times 10 is 30. Why did you choose an arrangement of 5 by 6 instead of 3 by 10?" This should continue to reinforce the idea that large volumes can be arranged to
have small surface areas, depending on the factors of the given volume.
7. Ask students if they notice anything about the volumes that are square numbers. For volumes of 16, 25, 36, and 49, the smallest surface areas should all result in
rectangular prisms that are also cubes.
To help students organize this new information from this lesson, keep their worksheets in a folder or math notebook. Understanding this relationship between volume
and surface area for rectangular prisms will help them as they begin to explore the volume and surface area for objects that are not rectangular prisms.
Summative Assessment
The teacher will determine if the students have reached the learning targets for this lesson when students are able to complete the SURFACE AREA CARAMEL
CHALLENGE worksheet and the related activities.
When students work through this activity, they will be able to find multiple surfaces areas for a given number of cubes (volume). They will solve a real world problem
by finding the smallest surface area possible for their given number of cubes.
By evaluating the completed SURFACE AREA CARAMEL CHALLENGE worksheet, the teacher will be able to measure the impact of this lesson on student learning.
Students should be able to give evidence of the understanding that rectangular prisms with the same volume may have a variety of surface areas depending of the
arrangement of cubes within the prism.
Formative Assessment
For this lesson to be successful, students need to have experience with finding the surface area of rectangular prisms and cubes; this should not be a student's first
experience with surface area. Prior experience can be either with using a ruler to measure the area of six sides of a rectangular prism and adding them together to
find the total surface area or with a formula for surface area.
For a formative assessment:
1. Have 15 boxes of different sizes (a tissue box, cereal box, bandage box, etc.) on a table in the room.
2. Allow students to work with a partner to choose a box and find its surface area in inches by using a ruler.
3. Pairs need to record their calculations on paper. Each group can share a piece of paper.
4. Repeat this with another box.
If students pairs cannot find the surface area of two different boxes, then this lesson is not appropriate because they will be creating their own rectangular prisms with
unique surface areas.
Remediation should include working with rectangular prisms of different sizes and practicing to accurately determine the surface areas of the boxes.
Feedback to Students
Students will receive constant feedback during the lesson as they work with one inch cubes manipulatives. If students do not use the one inch cubes to create regular
rectangular prisms (prisms with no leftover cubes), then their partner or the teacher needs to assist the student immediately.
The teacher should circulate as student groups work through trial and error, packing their cubes and make sure the first package is arranged correctly before students
move on to the second or third package model.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Throughout the guided practice and surface area caramel challenge, students will be working with partners.
The teacher should carefully select partners so that all students will have the opportunity for success.
If students who usually need additional accommodations have a peer that they are used to working with, then make sure that person is their partner.
If necessary, allow extra time to complete the independent practice worksheet for students who need time accommodations.
Also, the teacher can differentiate the volume (number of cubes) each pair works with depending on the needs of the students in the group.
Extensions:
If students understand the complexity of the relationship between volume and surface area immediately, then they may need more challenging extensions.
Instead of giving students volume and having them find the surface area, it is much more difficult to give students the surface area and have them find potential
arrangements of cubes for the volume.
Students may also work with irregular prisms instead of rectangular prisms.
Suggested Technology: Interactive Whiteboard, Overhead Projector
Special Materials Needed:
One inch cubes (The number will vary by your class; if you have 24 students in 12 pairs in your class and everyone works with a volume of 20, then you need at
least 240 cubes.)
rulers
foil
gum packs and/or caramels (Both optional depending on your school rules.)
Further Recommendations:
I recommend using cubes throughout the lesson to represent caramels. Caramels are not usually perfect cubes due to melting or compression within their
packaging.
I also strongly recommend students work with one inch by one inch cubes; centimeter cubes are too small for them to manipulate and arrange successfully. DO
page 3 of 4 NOT use "pop cubes" or cubes that connect. The connections can skew the validity of the surface area calculations.
Be sure to attend to school rules about food in the classroom. Real caramels are not necessary for this lesson. If you cannot bring packages of gum to school, show
pictures of the packaging instead.
Some students may immediately understand that a 1 by 8 prism and 8 by 1 prism will have the same surface area. Some may not. If students do not understand
this, allow them to work through their problems without explicitly telling them about this relationship; it is more powerful if they discover it for themselves.
This lesson only covers the standard in terms of cubes and regular rectangular prisms. To fully cover this standard further instruction involving two- and threedimensional objects composed of triangles, quadrilaterals, and polygons is needed. I recommend doing this lesson first because students usually have an easier time
understanding surface area and volume with cubes and rectangular prisms first. Once they understand the concepts covered in this lesson, instruction can be
continued with other two- and three-dimensional objects.
SOURCE AND ACCESS INFORMATION
Contributed by: Megan Crombie
Name of Author/Source: Megan Crombie
District/Organization of Contributor(s): Wakulla
Is this Resource freely Available? Yes
Access Privileges: Public
License: CPALMS License - no distribution - non commercial
Related Standards
Name
Description
Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional
objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
MAFS.7.G.2.6:
Remarks/Examples:
Examples of Opportunities for In-Depth Focus
Work toward meeting this standard draws together grades 3–6 work with geometric measurement.
page 4 of 4