Author: Catherine Michini
School/Organization:
Philadelphia High School for Girls
Year: 2020
Seminar: A Visual Approach to Learning Math
Grade Level: 912
Keywords: algeba, geometry, Math, NCTM, PSAT
School Subject(s): Math, Algebra, Geometry
Many high school students lack true understanding of surface area, volume, and the difference between the two measures. This curriculum unit uses handson and PowerPoint slides to increase spatial visualization skills that can increase comprehension of quantifying spatial measurement.
The curriculum is College Preparatory Mathematics Geometry, or CPM 2, used with their permission. I’ve added PowerPoint slides to assist teachers in presenting the topics of surface area and volume.
The unit consists of 6 lessons that range from spatial visualization skills to building polyhedral using gum drops and toothpicks. The technique for finding surface area is developed with the use of nets, which are introduced before the vocabulary and surface area lessons. The culminating activity, or theme problem is to find the surface area of a skyscraper in order to buy power washing supplies.
The lessons can be used together to present many of the surface area and volume topics, or individually. The lesson which introduces spatial visualization can be used completely out of the context of surface area and volume.
Download Unit: CatherineMichini.pdf
High schools have changed over the last century and with them, the teaching of mathematics. Since the beginning of the 20^{th} century, there have been debates over content and pedagogy. The debate over teaching Algebra and Geometry to all students or only to those who are college bound has seesawed with the decades and in some circles, is still happening now.
The progressivist’s camp believes that students should learn what they want and that their math courses should have direct practical applications, like making a budget or completing a tax return. An example of that would be shop math or consumer math. In fact, prominent education experts in the early 1900’s, Dr. William Heard Kilpatrick and Dr. Edward L. Thorndike argued that the study of algebra and geometry were of little value to boys and even less value, only 1%, to girls. Others, comprised of professionals and educators grounded in mathematics, see the value in the mental discipline required in Algebra and Geometry and the need for higher level math courses for everyone, for its applications and intrinsic value. (Klein, 2003)
In the 1980’s the National Council of Teachers of Mathematics (NCTM) published two documents with the power to reform the teaching of Mathematics. The release of An Agenda for Action and the Curriculum and Evaluation Standards for School Mathematics changed the way mathematics was taught. Communication, making connections and creative, critical thinking were emphasized and rote was deemphasized. (NCTM 1980,1989) The ground was laid for constructivism in mathematics classrooms! These documents were the forerunners of today’s Common Core State Standards.
The goal in our TIP course, “A Visual Approach to Learning Mathematics”, taught by Professor Robert Ghrist, was to improve our ability to motivate and communicate mathematical ideas and principles through the use of creative design, technical illustration, and animation. Professor Ghrist, who has his own YouTube channel, taught us basic skills in PowerPoint to assist us in creating lessons to help our students visualize mathematics, a key to many of the 8 Standards of Mathematical Practice published as part of the Common Core State Standards for Mathematics in 2010. Noted author, Stanford professor and You Cubed founder, Jo Boaler agrees with the importance of visualizing in mathematics. In her book Mathematical Mindsets Unleashing Students’ Potential Through Creative Math, Inspiring Messages and Innovative Teaching, she states “When we don’t ask students to think visually, we miss an incredible opportunity to increase their understanding. (Boaler, 2016, p. 63) In his Edutopia article, Jeremiah Ruesch, reminds us that “our jobs as educators is to set a stage that maximizes the amount of learning done by students, and teaching students mathematics in [a] visual way provides a powerful pathway for us to do our job well.” (Reusch, 2017) The purpose of this curriculum unit is to do just that for volume and surface area in Geometry, using College Preparatory Mathematics’ curriculum from CPM 2.
In 1999, the US Department of Education recommended 10 recommended mathematics programs. College Preparatory Mathematics (CPM) was one of the six exemplary programs listed. The books were student centered, the students worked in teams, the problems required discussion, and the homework was spiraled.
I use the CPM Geometry textbook and will base the lesson that follow on their unit entitled Spatial Visualization. As a teacher with many years of experience, it was refreshing to see that geometry and creativity abide together.
According to the 2015 TIMMS report, in Advanced Mathematics, the U.S. average percent correct in geometry was the lowest category, with 38% correct, compared to the Advanced Mathematics overall average of 44% correct.( nces.ed.gov, 2020) In NCTM’s 2000 publication, “Principles and Standards for School Mathematics, the authors state: “Students’ skills in visualizing and reasoning about spatial relation ships are fundamental in geometry. Some students may have difficulty finding the surface area of threedimensional shapes using twodimensional representations because they cannot visualize the unseen faces of the shapes. Experience with models of threedimensional shapes and their twodimensional “nets” is useful in such visualization. Students also need to examine, build, compose, and decompose complex two and threedimensional objects, which they can do with a variety of media, including paperandpencil sketches, geometric models, and dynamic geometry software.” (NCTM, 2000, p. 237)
Pierre van Hiele and his wife Dina developed a theory about difficulties students have in learning geometry. They named 5 levels of thinking in geometry and reasoned that a student could not reach subsequent levels without first understanding the basic level of visualization (Zalman, 1981). This ties in precisely with how my curriculum unit begins.
The confusion between surface area and volume is an especially prevalent issue in Geometry. I have found this to be true in my own classes and researchers have verified this issue. Allison Dorko and Natasha Speer, at the time, both from the University of Maine found that Calculus students often thought that by adding the area of the faces of a solid, they were incorporating a 3^{rd} dimension. (Dorko & Speer, 2013) In later research, they noted the significance that units of measure play in student understanding. 73% of the almost 200 Calculus students in their study made mistakes in labeling the units in the assigned area and volume tasks. (Dorko & Speer, 2015) This curriculum unit uses spatial visualization and manipulatives to introduce surface area and volume in order to strengthen student understanding. Barbara Kinach, an associate professor of mathematics education, states in her article in the Mathematics teacher “More emphasis on spatial reasoning is a way to increase meaning when students study Geometry.” (Kinach, 2012, p. 534)
In the PSAT and SAT, this topic in Geometry is in the category Problem Solving and Data Analysis. The students at the Philadelphia High School for Girls have not been scoring well in this category. As shown in the graphs below, the percentage of students performing well in this category, did not increase, even after taking Geometry.
Content Objectives
The purpose of my curriculum unit is to improve the student understanding of volume and surface area through visualization. I am hoping that the ability to construct and deconstruct a solid, fill it with cubes and measure the area of its faces, will strengthen a student’s concept of the topic and therefore improve student performance in Problem Solving and Data Analysis tasks.
I already use various websites to visualize mathematics in my classroom, such as Desmos, Which one Doesn’t Belong, Would you Rather, and Estimation 180. My students love the opportunity to have fun and do something they think is not math. I also use models in my classroom: cubes, algebra tiles, tissue boxes, etc., whatever I can find. That doesn’t work so well when learning is all virtual. I was especially excited to create PowerPoint slides to show how 2 dimensions can become 3 dimensions and to analyze 3D objects. Because one problem can take as many as 10 slides, each problem is its own PowerPoint. They are not all perfect, but I believe they get the idea across. Professor Ghrist told me not to let the perfect become the enemy of the good and I have followed his advice.
The unit begins by asking visual puzzlers, two and threedimensional geometric descriptions of situations and 3D interpretations of 2D sketches, including nets. The purpose of the first lesson is to develop, acknowledge and appreciate the value of spatial visualization with fun activities. One of the Do Now activities asks students to discuss ways children develop spatial and visualization skills and how often boys have different experiences than girls. With handheld video games as much a part of our current high school’s generation as TV was for me, one would think that the gender gap in spatial thinking would be reduced. In quite technical, neuroscientific detail, Li Yuan and his research colleagues at Shaanxi Normal University in Xi’an China, found that males outperform females in large and smallscale spatial abilities. (Yuan, 2019)
Polyhedra are introduced with a net and students are asked to build them from gum drops and toothpicks. Vocabulary “vertices” “edges” and “faces” are introduced, and students are asked to discover Euler’s formula. Polyhedra are named by the number of faces and bases. Students also use isometric grid paper to draw 3D solids. Volume is introduced as the number of cubes in a solid. Mat plans are also utilized to record a 3D solid. Surface area is introduced with the nets. After the surface area is found, the students build the polyhedron. Students will also find the slantheight of a pyramid. The unit culminates with a theme problem: finding the surface area of the Transamerica Pyramid in San Francisco, although I altered mine to use a local Philadelphia skyscraper.
This curriculum unit uses many strategies for visualization and cooperative learning. Many of these strategies I have learned from CPM training and others are specific to the exact topic.
When students work regularly in study teams, there are a few basic guidelines that CPM suggests teachers establish in their classrooms: Each member of the team is responsible for his/her own behavior, Each member of the team must be willing to help any other team member who asks for help, You should only ask the teacher for help when all team members have the same question, Use your team voice. Other CPM team strategies, which Jo Boaler also mentions in Mathematical Mindsets are to have students think about a problem by themselves and then share with their teams. CPM also recommends reserves the first few minutes a student and the team spends with a problem be without pencils in the “Teammates Consult” strategy. In this way, the team can clarify the problem and discuss possible solution methods without one student getting ahead of the others with their own approach.
In this curriculum unit students will be doing a number of hands on activities with extra materials most days. One of the roles a team member can serve is Resource Manager whose job is to get supplies for the team and make sure the team is cleaned up. The resource manager also is responsible for calling the teacher over for questions. You may want to switch team roles daily to vary student tasks. The other team roles are Recorder/Reporter, Facilitator, and Task Manager.
All of the 8 Standards of Mathematical Practice can be used during lessons in this unit. Students will make sense of problems and persevere. This is especially applicable in the surface area problems and the theme problem. Students will reason abstractly and quantitatively. The best example of this is SV104 where students have to decide if a 5foot stick can fit in a cube that is 3ft on each side. Students will construct viable arguments and critique the reasoning of others. When students are discussing their answers to the spatial visualization questions, these skills will definitely be put into practice. Students will model with mathematics using all of the handson activities. The theme problem is the “cherry on top” for this practice. Students will use appropriate tools strategically, especially while measuring the nets of their 3D solids to find the surface area. Students will also attend to precision with this activity. Students are asked to look for and make use of structure when generalizing a rule for the slant height or the length of a 3D diagonal. Students will look for and express regularity in repeated reasoning when discovering Euler’s Formula for the number of vertices, faces and edges of a cube in SV79.
Other teaching strategies utilized in the unit are cooperative assessments, jigsaw, using academic vocabulary, and applying skills to a realworld problem. Of course, the most consistent and important strategy used in this unit is using visualization; a most important skill for the growing world of online learning.
Lesson 1: Spatial Visualization
Learning Objective: At the end of this lesson, students will be able to visualize and 2dimensional and 3dimensional geometric representations of given situations. They will be able to describe the geometric representations using academic vocabulary and/or labeled drawings. Students will also be able to interpret 2dimensional drawings as representing real world objects.
Materials:
Procedures:
Notes: Lesson 1 can take 1 or 2 class periods.
Lesson 2 – Isometric Drawings
Learning Objective: At the end of this lesson, students will be able to draw a 3D cube on 2D isometric grid paper. Students will be able to copy and draw on isometric grid paper, a 3D solid made up of stacks of cubes. Students will be able to use a MAT plan to create a drawing on isometric grid paper. Students will be able to find the volume of any 3D solid given its MAT plan or its drawing on isometric grid paper.
Materials:
Procedures:
Notes:
Lesson 3 – Building Polyhedra
Learning Objectives: At the end of this lesson, students will be able to use polyhedral vocabulary to accurately find the number of vertices, edges and faces of a given polyhedron. Students will discover and use Euler’s Polyhedral Formula to check the accuracy of a polyhedron’s number of faces, vertices and edges. Students will also be able to identify and name polyhedral by their number of sides or base.
Materials:

Procedures:
Notes:
Lesson 4 – Building Polyhedra, Volume
Learning Objectives: At the end of this lesson, students will be able to list similarities and difference between pyramids and prisms. Students will also be able identify the base of any polyhedra and name them accordingly. Students will continue to find the volume of prisms, for this lesson using the formula Volume = (Area of Base)(Height of the Prism). Students will find and accurately draw pyramids and prisms using dotted segments for hidden edges and open circles for hidden vertices.
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Procedures:
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Lesson 5 – Surface Area
Learning Objectives: At the end of this lesson, students will be able find the surface area of a polyhedron given its net after first measuring the appropriate dimensions. Students will use the handson experience to find the surface area of any given pyramid or prism drawing by first analyzing the faces of the polyhedron, then finding the area of each face, and finally, finding the sum of the areas.
Materials:
Procedures:
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Lesson 6 – Threedimensional diagonal and Theme Problem.
Lesson Objectives: At the end of this lesson, students will be able to find the slant heights of the lateral faces of a pyramid and a 3dimensional diagonal for a rectangular prism. Students will also be able to solve a realworld problem utilizing surface area.
Materials:
Procedures:
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Alex, J. K., & Mammen, K. J. (2016). Lessons Learnt from Employing van Hiele Theory Based Instruction in Senior Secondary School Geometry Classrooms. EURASIA Journal of Mathematics, Science and Technology Education, 12(8). doi: 10.12973/eurasia.2016.1228a
Boaler, J., & Dweck, C. (2016). Chapter 5/Rich Mathematical Tasks. In Mathematical mindsets: Unleashing students’ potential through creative math, inspiring messages and innovative teaching. San Francisco, CA: JosseyBass & Pfeiffer Imprints.
© CPM Educational Program. All rights reserved. Used with permission. CPM Math 2 (Geometry) CPM Educational Program 9498 Little Rapids Way, Elk Grove, CA 95758
Dorko, Allison, and Natasha M. Speer. “Calculus Students’ Understanding of Volume.” Investigations in Mathematics Learning, vol. 6, no. 2, 2013, pp. 48–68., doi:10.1080/24727466.2013.11790332.
Dorko, Allison, and Natasha Speer. “Calculus Students’ Understanding of Area and Volume Units.” Investigations in Mathematics Learning, vol. 8, no. 1, 2015, pp. 23–46., doi:10.1080/24727466.2015.11790346.
Kinach, and Barbara M. “Fostering Spatial vs. Metric Understanding in Geometry.” Mathematics Teacher, National Council of Teachers of Mathematics. 1906 Association Drive, Reston, VA 201911502. Tel: 8002357566; Tel: 7036203702; Fax: 7034762970; eMail: Orders@Nctm.org; Web Site: Http://Www.nctm.org/Publications/, 29 Feb. 2012, eric.ed.gov/?id=EJ981870.
Klein, D. (n.d.). A Brief History of American K12 Mathematics Education. Retrieved from http://www.csun.edu/~vcmth00m/AHistory.html
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston, VA: National Council of Teachers of Mathematics.
National Council of Teachers of Mathematics. (1989). Curriculum and Evaluation Standards for School Mathematics. Reston, VA:NCTM
National Council of Teachers of Mathematics (NCTM). An Agenda for Action. Reston, Va.: NCTM, 1980.
Ruesch, J. (2017, September 08). The Power of Visualization in Math. Retrieved from https://www.edutopia.org/article/powervisualizationmath
“Standards for Mathematical Practice.” Standards for Mathematical Practice  Common Core State Standards Initiative, 2010, www.corestandards.org/Math/Practice/.
U.S. Performance on the 2015 TIMSS Advanced Mathematics … (n.d.). Retrieved from https://nces.ed.gov/pubs2020/2020051.pdf
Yuan, L., Kong, F., Luo, Y., Zeng, S., Lan, J., & You, X. (2019). Gender Differences in LargeScale and SmallScale Spatial Ability: A Systematic Review Based on Behavioral and Neuroimaging Research. Frontiers in Behavioral Neuroscience, 13. doi: 10.3389/fnbeh.2019.00128
Zalman. (1981, November 30). Van Hiele Levels and Achievement in Secondary School Geometry. CDASSG Project. Retrieved from https://eric.ed.gov/?id=ED220288
Additional Resources
Bouck, Emily, Sara Flanagan, and Mary Bouck. “Learning Area and Perimeter with Virtual Manipulatives.” The Journal of computers in mathematics and science teaching. 34.4 (2015): n. pag. Print.
Obara, S. (2009). Where does the formula come from? students investigating total surface areas of a pyramid and cone using models and technology. Australian Mathematics Teacher, 65(1), 2533.
A website with many interactive math lessons, including surface area and volume. http://www.shodor.org/interactivate/lessons/SurfaceAreaAndVolume/
Academic Standards
From the Common Core State Standards for Mathematics:
An understanding of the attributes and relationships of geometric objects can be applied in diverse contexts—interpreting a schematic drawing, estimating the amount of wood needed to frame a sloping roof, rendering computer graphics, or designing a sewing pattern for the most efficient use of material.
During high school, students begin to formalize their geometry experiences from elementary and middle school, using more precise definitions and developing careful proofs.
Geometric measurement and dimension GGMD
Explain volume formulas and use them to solve problems
Visualize relationships between twodimensional and three dimensional objects
modeling with Geometry GMG
Apply geometric concepts in modeling situations
From PA Common Core Standards:
CC.2.3.HS.A.13
Analyze relationships between two‐dimensional and three‐dimensional objects.
CC.2.3.HS.A.3 Verify and apply geometric theorems as they relate
to geometric figures.
CC.2.3.8.A.1
Apply the concepts of volume of cylinders, cones, and spheres to solve real‐world and mathematical problems.
CC.2.3.HS.A.12
Explain volume formulas and use them to solve problems.
CC.2.3.HS.A.14 Apply geometric concepts to model and solve real‐ world problems.
From PA Geometry Assessment Anchors and Eligible Content
G.1.2.1.1 Identify and/or use properties of triangles
G.1.2.1.2 Identify and/or use properties of quadrilaterals
G.1.2.1.3 Identify and/or use properties of isosceles and equilateral triangles
G.1.2.1.4 Identify and/or use properties of regular polygons
G.1.2.1.5 Identify and/or use properties of pyramids and prisms
G.2.2.3.1 Describe how a change in the linear dimension of a figure affects its perimeter, circumference, and area (e.g. How does changing the length of the radius of a circle affect the circumference of the circle?).
G.2.3.1.1 Calculate the surface area of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.
G.2.3.1.2 Calculate the volume of prisms, cylinders, cones, pyramids, and/or spheres. Formulas are provided on a reference sheet.
G.2.3.1.3 Find the measurement of a missing length given the surface area or volume.
G.2.3.2.1 Describe how a change in the linear dimension of a figure affects its surface area or volume (e.g. How does changing the length of the circumference of the edge of a cube affect the volume of the cube?).