Primary Type: Lesson Plan
Status: Awaiting Review
This is a resource from CPALMS (www.cpalms.org) where all educators go for bright ideas!
Resource ID#: 49410
Three Dimensions Unfolded
Students will use nets of prisms to find the surface area of composite 3-D figures. Students will learn to identify the faces of 3-D figures that are
needed to find the surface areas, and those that are not needed.
Subject(s): Mathematics
Grade Level(s): 7
Intended Audience: Educators
Suggested Technology: Document Camera, Basic
Calculators, LCD Projector
Instructional Time: 1 Hour(s) 45 Minute(s)
Resource supports reading in content area: Yes
Freely Available: Yes
Keywords: surface area, prism, pyramid, three-dimensional figures, 3-D
Resource Collection: CPALMS Lesson Plan Development Initiative
ATTACHMENTS
TDU_Lesson_Assessment_revised.docx
TDU_Lesson_Assessment_Key_revised.docx
TDU_Sample_Net_revised.docx
TDU_Sample_Net_with_dimensions_labeled_revised.docx
TDU_Buildings_revised.docx
TDU_Red_Shet_Net_revised.docx
TDU_Practice_(r)_revised.docx
TDU_Practice_Key(r)_revised.docx
TDU_Formative_Assessment_(r)_revised.docx
TDU_Formative_Assessment_Key_(r2)_revised.docx
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 be able to find the surface area of prisms and pyramids.
Students will be able to solve real-world and mathematical problems involving finding the surface area of prisms and pyramids.
Prior Knowledge: What prior knowledge should students have for this lesson?
The following are standards that should be already mastered to ensure success in this lesson:
Understand that shapes in different categories (e.g., rhombuses, rectangles, and others) may share attributes (e.g., having four sides), and that the shared
attributes can define a larger category (e.g., quadrilaterals). Recognize rhombuses, rectangles, and squares as examples of quadrilaterals that do not belong to any
of these subcategories.
Students should be able to apply the area and perimeter formulas for rectangles in real-world and mathematical problems.
Classify two-dimensional figures based on the presence or absence of parallel or perpendicular lines , or the presence or absence of angles of a specified size.
Recognize right triangles as a category, and identify right triangles.
Classify two-dimensional figures in a hierachy based on properties.
page 1 of 5 Understand that attributes belonging to a category of two-dimensional figures also belong to all subcategories of that category.
tudents will be able to represent three-dimensional figures using nets made up of rectangles and triangles and use the nets to find the surface area of these
figures. Students will apply these techniques in the context of solving real-world and mathematical problems.
Guiding Questions: What are the guiding questions for this lesson?
What are the attributes of prisms and pyramids?
How can you use the properties of three-dimensional shapes to find the surface area?
How are the surface areas of prisms and pyramids with the same bases similar and different from each other?
What information is needed to find the surface area of different geometric figures?
What are some applications for finding surface area in the real world?
How does understanding what the net looks like help you find the surface area of a three-dimensional shape?
Teaching Phase: How will the teacher present the concept or skill to students?
The teacher will begin by presenting the students with the following question on the board as an opening activity:
What does "Surface Area" mean? (the total area of all the surfaces of a three-dimensional figure.)
What information is needed when finding the surface area of a prism or pyramid? (the dimensions of each face of the three-dimensional figure.)
The teacher should give the students a few minutes to write down their ideas on dry erase boards or paper and share with teammates.
Possible Student Answers: First you need to identify what two-dimensional shapes each of the faces of three-dimensional figures have (rectangular prism: 6
rectangles; triangular prism: two triangles, and three rectangles; rectangular pyramid: one rectangle and four triangles; triangular pyramid: four triangles.) After you
have identified what types of faces you have you can then find the area of each of the faces and add them together. For efficiency you can count the number of faces
that are duplicated and then find the area of just one of those and multiply it by the number of that shape.
It would be useful to have pictures or three-dimensional representations of the shapes to reference while students present their answers, or have students come up to
the board and draw example figures.
To start the lesson, the teacher should show the video clip that shows different buildings in New York City from: New York City
Buildings (http://www.videezy.com/urban/412-new-york-city-buildings; 19 seconds of ariel footage at the rooftop height passing several buildings. Graffiti appears on
some of the buildings, however, at 15 seconds, there is a large area of graffiti to reference.) The video clip is used with permission from Videezy.com
Say, "Some building owners noticed graffiti on the side of their building and wanted the graffiti painted over. Can the surface area of the graffiti be found if we know
the base area of the building and the height of the building?" (no, we would need to know how much of each side of the building is connected/covered by the next
building; we'd need to know the dimensions of the part of the building that is covered in graffiti only.) "How do you suppose the building owners know how much paint
they will need to cover up the graffiti on the buildings?" (find the surface area; measure the sides and find the area of each side.) If the students do not suggest that
they only need to cover the exposed surfaces, be sure to emphasize this. Point out subtracting the areas of the windows if applicable.
Ask the students why it would be important to know how much paint it would take before the painters start their work (so they have enough; so they know the cost.)
Another hook to the lesson would be to introduce students to the works of the artists Christo and Jeanne-Claude at http://www.christojeanneclaude.net/ where
they can see works of the artists' projects that involve covering large landmarks and objects with material.
One of their projects that has been in progress since the 1970's is "The Mastaba," the world's largest sculpture in the United Arab Emirates. The sculpture is to be
constructed of oil barrels to appear as a mosaic, however, from a distance, the sculpture appears to take the form of a trapezoidal prism. It will stand 492 feet tall and
have a base of 984 feet x 738 feet. Ask the students, "How would you figure out how much paint it would take, if the artists wanted to paint the faces of the prism a
different color than the other faces?" (find the areas of the trapezoids, then find the areas of the rectangular sides and the top. Make sure the students understand
that the bottom would not be needed if the task was painting.) "How many surfaces would need to be covered?" (five, the 2 trapezoidal bases {front and back}, the
two sides, and the top.) Many other questions can be developed based on this sculpture and others from these artists.
The teacher will display the images from the document called Buildings (see attachment.) It shows 4 different buildings from around the world that are composed of
composite shapes. The teacher should elicit from the students what shapes make up each building. The teacher should randomly choose a student to answer
Next, the teacher will activate prior knowledge to ensure that students have enough information to complete the guided practice effectively.
The teacher should ask, "How can you find the surface area of an object that has different shaped sides?"
Possible Student Answer: If you make a net that represents the object, you can see all the different shapes at one time and be able to find the areas of each shape
one at a time, then add them together; If you know the net of the shape, then you can see what types of faces you have, and the number of faces that are duplicated;
It makes it easy to organize your work, because you can label each side neatly; In a rectangular prism you will have opposite parallel rectangular faces with equivalent
areas (top & bottom, right side & left side, front & back); In a rectangular prism there may be more than one set of opposite parallel faces being equivalent, but it
depends on the shape; In a triangular prism you have one set of opposite parallel sides that are triangles, and then three rectangles that may or may not be
equivalent depending on the type of triangles you have (equilateral, isosceles, or scalene); In a rectangular pyramid you will have one rectangle, and four triangles
with at least the opposite pairs being equivalent to each other because the base of the triangle is the base of the rectangle as well; In a triangular pyramid you will
have four triangles, but they may or may not be equivalent depending on the type of triangles you have (equilateral, isosceles, or scalene).
If students are struggling to come up with this information, ask them about each three-dimensional figure individually. It would be useful to have students come up to
the board and sketch example nets to help explain their answers.
The teacher should ask a student to provide a review of how to find the area of rectangles and triangles - which are needed to find the surface area of prisms and
pyramids. (Area of a Rectangle = base x height; Area of a Triangle: ½ x base x height.)
Display this information on the board for students to reference.
Display a rectangular prism on the board (document camera with the attached document or sketched on the board.)
page 2 of 5 Ask the students to find the surface area. The teacher should observe the different methods the students use to attack the problem. Look for deconstructing the prism,
calculations, algorithms. After students have had sufficient time to formulate a response, ask for solutions. If the majority of the students are successful, then the
teacher should proceed to the guided practice phase of this lesson.
If the students have arrived at multiple solutions, without acknowledging a correct solution, the teacher should display the Sample Net and ask a student to come and
label the net with the appropriate dimensions. The teacher may wish to hand out the Sample Net with Dimensions sheet that is pre-labeled with dimensions to
students with visual-spatial deficits.
Ask the students to show the work needed to find the area of each face of the prism. Ask them if they see anything 'notable' about all six areas. (They should note that
three of the areas are repeated.) Ask a student to explain why. (They should respond that the front and back, the top and bottom, and the left and right sides are
congruent.)
The teacher should introduce the symbol used to represent congruent measures (hash mark/congruency mark.) The teacher should demonstrate the marks on a
figure such as a rectangle and emphasize that only sides marked with the same mark can be considered congruent. Some polygons may contain more than one set of
congruent parts, for example:
If the students need to see the concrete example, the teacher should have a pre-cut sample net that can be folded into a prism, and the students can see that the
faces with the same areas are the same size and also parallel to each other.
Guided Practice: What activities or exercises will the students complete with teacher guidance?
The guided practice activity involves students using nets to represent the "Red Shed on the South Bank" building that is constructed of a large rectangular prism and
topped with four congruent square prisms.
The teacher will refer back to the image of the "Red Shed on the South Bank." An anecdote should be told that the city would like to paint the shed a different color
and needs to know how much paint to purchase, as the budget to do this is limited. The teacher can take a vote for a new color to engage the students and get them
vested in the task.
In order to find out how much paint will be needed, the city needs to determine how much of the building will be painted. Ask the students how to find how much will
be painted. Possible answers include: find the area of each prism then add; find the area of all the exposed surfaces by finding the areas of each face, then subtract
the bottom of the large prism and the bottoms of each of the four smaller prisms.
**The teacher should ensure all students understand that NOT all surfaces of all five prisms will be counted, as the surfaces where smaller prisms meet the larger
prism and the bottom of the larger prism will not be exposed.
The students will each be provided with a piece of copy paper. Based on the picture of the Red Shed, students should sketch what they believe the net would look like.
This is a carry over from MACC.6.G.1.4 where students are asked to represent three-dimensional figures with nets. The teacher will circulate looking for appropriate
representations. The largest prism should be a rectangular prism. The four towers should be represented each by a square pyramid where the square bases are
proportionate to how they will be placed on the corners of the larger prism, and all four rectangular faces should be congruent.
The students will be provided with the Red Shed Nets copies (1 set per group), rulers, scissors, tape, and calculators (optional.) The teacher should provide written
directions on the board as well as tell the students that they are going to cut the nets, form the prisms, and calculate the exposed surface area to change the 'red'
shed to the selected color, so they will need to calculate the square centimeters that will be painted.
Each student should calculate the surface area individually on his/her own dry erase board and then compare the work within his/her groups. This provides an
opportunity to practice Mathematical Practice 3 - Construct viable arguments and critique the reasoning of others.
Bring the groups together and ask a student explain how the group found their solution. The student should come to the front of the class to explain, using the
document camera to show work, or work the problems out on the board. If the work is correct, the teacher should ensure that all students understand the process. If
the work is incorrect, ask another student to provide some appropriate error analysis.
Independent Practice: What activities or exercises will students complete to reinforce the concepts and skills developed in the
lesson?
Distribute the the independent practice (Practice Sheet).
page 3 of 5 While students are working, circulate throughout the room to monitor students' written responses and later conversations with their partner, intervene with guiding
questions and clarifications, challenge them to be persistent, and encourage them to revise their work, as needed.
After students have had sufficient time to answer the problems, assign partners. Ask students to share their results with their partners and revise their work, if
needed.
Call on students to display, share, and explain their answers.
The teacher will ask, "Which shape has the greater surface area?"
Possible student answers: The surface area of the second pyramid is greater than the surface area of the first pyramid. The first pyramid has a surface area of 96
square centimeters, and the second pyramid has a surface area of 129 square centimeters.
The teacher may choose to have students project their solutions using the document camera to show correct solutions.
The teacher asks, "Why do you think the second shape has a greater surface area, when they both had the same volume?"
Possible student answer: The second pyramid was much taller than the first pyramid, which caused the triangles to have greater areas.
Closure: How will the teacher assist students in organizing the knowledge gained in the lesson?
Have a whole class discussion about the contents of this lesson, including but not limited to the questions below.
The teacher asks:
How do you find the surface area of a three-dimensional object?
What information do you need?
How might you check your work?
When might someone calculate surface area in the real world? (When you want to paint a room or building; when you want to wrap a present in wrapping paper;
when you need to find out how much material will be needed to cover something.)
Summative Assessment
The teacher will administer the Lesson Assessment (see attachments), when students are ready.
The teacher should use the responses from the students to gauge how to proceed with the next lesson based on their understanding.
Formative Assessment
Students will be given a prior knowledge assessment, or pre-test, the day or two before the lesson is implemented. See Formative Assessment in the Attachments
section. This will allow the teacher to determine whether the students have the prerequisite knowledge to be successful in this lesson and determine the level of
support individuals will require during the lesson.
See the Independent Practice for embedded Formative Assessment.
Feedback to Students
Students will be given feedback by their peers as they work together to find solutions to the guided practice questions.
The teacher should provide constant feedback through group conferences as the students work with their groups, and during the individual practice as needed. At
the end of the lesson students will participate in a discussion about the individual practice problems.
ACCOMMODATIONS & RECOMMENDATIONS
Accommodations:
Students should have access to three-dimensional figures that represent the problems being discussed.
Students can be provided with nets to cut out and fold to create the shapes being discussed.
Dictionaries and glossaries or translations should be available to English Language Learners to clarify any unfamiliar vocabulary.
Extensions:
Students can use their knowledge of finding the surface area of three-dimensional figures to apply it to finding a missing dimension when given the surface area and
all other dimensions.
Suggested Technology: Document Camera, Basic Calculators, LCD Projector
Special Materials Needed:
Teacher Materials:
Buildings document (shown from the computer to projector, or printed and projected from document camera)
Keys: Formative Assessment, Practice, Lesson Assessment
page 4 of 5 Copies per group:
Red Shed Nets
Copies per student:
2 sheets of copy paper per student for activity
Formative Assessment worksheet
Sample Net document
As needed: Sample Net with dimensions labeled
Practice Worksheet
Lesson Assessment worksheet
Student Materials:
Basic Calculators
Rulers
Tape or Glue
Scissors
Dry erase boards/markers/erasers (optional)
Additional Information/Instructions
By Author/Submitter
This resource is likely to support student engagement using the following the Mathematical Practices:
MAFS.K12.MP.3.1- Construct a viable argument and critique the reasoning of others.
MAFS.K12.MP.4.1 - Model with Mathematics
SOURCE AND ACCESS INFORMATION
Name of Author/Source: Anonymously Submitted
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 5 of 5