Geometry Modeling Activities for the Middle & High School Math Classroom

"Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards." Common Core Mathematics

Introduction:

Mission Statement

The purpose of this Wiki page is to provide a compilation of professional learning resources for teachers of middle
and high school mathematics and geometry courses. Our hope is to help current and future teachers find resources, such as lesson plans and activity ideas, that promote engaging math instruction related to geometry and modeling.

What is Geometry?

Geometry is the branch of mathematics that focuses on measurement and properties of shapes, including lines, points, angles, surfaces, solids, and more.

What is Modeling?

Mathematical modeling is the process of applying mathematical understanding to a system. Modeling with geometry applies geometric understanding to real world situations and problems.

Common Core State Standards: Apply Geometric Concepts in Modeling Situations

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

G-MG.3.Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Unfolding Geometry.
A team of teachers from the UK sought to liven up geometry instruction in their school. As a result, their team formed an innovative, authentic, and engaging geometry lesson that incorporates project-based learning, cooperative learning, inquiry, and real world problems. The activity promotes student engagement and interest in mathematics both in and outside of the classroom by contextualizing math work into possible career choices. The authors of this study are current and pre-service teachers who teamed up with teachers of other disciplines and outside sources such as community organizations and businesses.
Having briefly studied local award-winning buildings and architecture, students went on a field trip to meet with the professionals and interview them. The students' interests and conversations with the architects covered topics such as geometrical shapes' constructability, urban planning, computer modeling, and more. In the article, the authors designed a project-based learning tasks/activity in which students (ages 13-15) grappled with advanced topics such as various the stability of three-dimensional shapes like dodecahedrons (which students later hand-made out of paper origami), Euler's formula, the concept of perfect thirds, and more. At the end of the article, the authors offer suggestions and encouragement on how to adapt their lesson to fit your classroom, school, and community. Applicable Common Core State Standards:

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Reference: Leroux, H. & Santos, S. (November, 2009). Unfolding geometry. Mathematics Teaching, 216, 16-19.

This article focuses on accommodating students with learning disabilities in the geometry classroom; however, it also applies to elementary age students, at-risk students, and students with ADHD/ADD. The premise of the intervention/teaching strategy applies ideas from universal design for learning and the flipped classroom, which involves students watching teacher pre-made videos of procedures as well as watching videos of themselves performing the learning targets. The researchers found increased understanding, achievement, and retention of geometric ideas and properties.

Extensions: This idea/strategy does not have to be limited to students in special education; this technique could greatly benefit all students in the geometry/math classroom.

Applicable Common Core State Standards:

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Reference: Cihak, D.F. & Bowlin, T. (2009). Using video modeling via handheld computers to improve geometry skills for high school students with learning disabilities. Journal of Special Education Technology, 24(4), 17-29.

From the National Council of Teachers of Mathematics journal, The Mathematics Teacher, this activity builds connections between hands-on experimentation and geometric properties of volume. In the activity/lesson, the teacher-researchers use an inquiry based approach to engage students with making predictions, graphing results, and applying knowledge of geometric properties. This lesson could be applied to a range of grade levels from 8-12, as long as the instructor modifies, adapts, or identifies different focus areas depending on the grade level. Applicable Common Core State Standards:

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Reference: Gomez, E. & Wolfson, R.A. (March, 2012). Filling bottles with water. The Mathematics Teacher, 105(7), 542-548.

Applicable Common Core State Standards:Also from the National Council of Teachers of Mathematics, this journal article provides "project lessons in astronomical telescope design and construction to students in second-year algebra" (Siegel, et. al, 2008, p. 490). In exploring the ideas of telescope optics, students engage with the following math concepts: conic sections, parabolas, Euclidean proof, Cartesian coordinates, focal length, convergence, and vertices. Additionally, the teacher-researchers utilize teaching strategies and tools such as paper folding, mirrors, modeling, experimentation and investigation, and educational technology. Although the lesson plan is very topic specific (and required the teachers to have a certain amount of background knowledge related to astro-telescopes), the lesson is very detailed for anyone interested in the topic or looking to duplicate it: the authors include graphs, diagrams, and pictures of student work, along with the questions used to guide student exploration. Answers and student feedback are also provided. Overall, this is a very hands-on activity for students that also connects math to real world problems.

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Reference: Siegel, L.M., Dickinson, G., Hooper, E.J., & Daniels, M. (March, 2008). Teaching algebra and geometry concepts by modeling telescope optics. The Mathematics Teacher, 101(7), 490-497.

This video shows educator Suney Park using geometric modeling to teach her 6th Grade students about the connections between and interpretations of area and perimeter. In the video, Suney sets up a challenge for them: they must seat 22 guests at a table fore a feats. What dimension table will allow them to fit the most food? How about the least amount of food? The video demonstrated students working in groups and learning through exploration. Throughout the video, Suney explains her lesson plan and her approach to engaging her students in the subject. She also clarifies the connections to the Common Core Standards.

Applicable Standards:

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

This video follows the teaching of Eeva Reeder and a real-world project she assigns her students. After realizing that they could not articulate what they had been learning in class, she developed a project that would force them to realize what they did and did not understand. The several layers of the project are laid out in the video. This type and style of project would be a useful resource for teaching because it engages the students, promotes and requires deep learning as well as collaboration, and it connects what they are learning in class to the real world.

Applicable Standards:

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

In this resource, Dan Meyer presents a problem: will the meatballs fit in the sauce without causing it to overflow? The problem is presented in an interactive way with a format that allows for exploration of the problem in a technology-equipped classroom. The activity is well suited for group discussion, and can be presented to an entire class. It provides all of the necessary details to formulate an answer, and reveals the correct answer as you progress through the activity. Once the preliminary problem is solved, further questions are presented to stimulate conversation and better understanding. Dan also presents a series of other problems in a similar format that can also be used in the classroom.

Applicable Standards:

G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

G-MG.2. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).

Relevance:

Geometry is quite relevant in our world today, and modeling it is used in all sorts of fields. This website has a lot of great resources that show the different fields in which geometry is applied and modeling is relevant. So, when your students ask you why they need to learn things like this, you can have plenty of answers!

Page created/updated by Allie Calderon, Jim Kaiser, and Lydia Wright.

## Geometry Modeling Activities for the Middle & High School Math Classroom

"Modeling is best interpreted not as a collection of isolated topics but rather in relation to other standards. Making mathematical models is a Standard for Mathematical Practice, and specific modeling standards appear throughout the high school standards."Common Core Mathematics## Introduction:

The purpose of this Wiki page is to provide a compilation of professional learning resources for teachers of middleMission Statementand high school mathematics and geometry courses. Our hope is to help current and future teachers find resources, such as lesson plans and activity ideas, that promote engaging math instruction related to geometry and modeling.

Geometry is the branch of mathematics that focuses on measurement and properties of shapes, including lines, points, angles, surfaces, solids, and more.What is Geometry?

Mathematical modeling is the process of applying mathematical understanding to a system. Modeling with geometry applies geometric understanding to real world situations and problems.What is Modeling?## Common Core State Standards: Apply Geometric Concepts in Modeling Situations

G-MG.1.Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).G-MG.2.Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).G-MG.3.Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).## Common Core Standards Pages:

High School GeometryModeling StandardsModeling With Geometry## Lesson & Activity Resources:

Unfolding Geometry.A team of teachers from the UK sought to liven up geometry instruction in their school. As a result, their team formed an innovative, authentic, and engaging geometry lesson that incorporates project-based learning, cooperative learning, inquiry, and real world problems. The activity promotes student engagement and interest in mathematics both in and outside of the classroom by contextualizing math work into possible career choices. The authors of this study are current and pre-service teachers who teamed up with teachers of other disciplines and outside sources such as community organizations and businesses.

Having briefly studied local award-winning buildings and architecture, students went on a field trip to meet with the professionals and interview them. The students' interests and conversations with the architects covered topics such as geometrical shapes' constructability, urban planning, computer modeling, and more. In the article, the authors designed a project-based learning tasks/activity in which students (ages 13-15) grappled with advanced topics such as various the stability of three-dimensional shapes like dodecahedrons (which students later hand-made out of paper origami), Euler's formula, the concept of perfect thirds, and more. At the end of the article, the authors offer suggestions and encouragement on how to adapt their lesson to fit your classroom, school, and community.

Applicable Common Core State Standards:

- G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
- G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Reference: Leroux, H. & Santos, S. (November, 2009). Unfolding geometry.Mathematics Teaching,216, 16-19.## Using Video Modeling via Handheld Computers to Improve Geometry Skills for High School Students with Learning Disabilities

This article focuses on accommodating students with learning disabilities in the geometry classroom; however, it also applies to elementary age students, at-risk students, and students with ADHD/ADD. The premise of the intervention/teaching strategy applies ideas from universal design for learning and the flipped classroom, which involves students watching teacher pre-made videos of procedures as well as watching videos of themselves performing the learning targets. The researchers found increased understanding, achievement, and retention of geometric ideas and properties.- Extensions: This idea/strategy does not have to be limited to students in special education; this technique could greatly benefit all students in the geometry/math classroom.

Applicable Common Core State Standards:- G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Reference: Cihak, D.F. & Bowlin, T. (2009). Using video modeling via handheld computers to improve geometry skills for high school students with learning disabilities.Journal of Special Education Technology, 24(4), 17-29.## Filling Bottles with Water

From the National Council of Teachers of Mathematics journal,The Mathematics Teacher, this activity builds connections between hands-on experimentation and geometric properties of volume. In the activity/lesson, the teacher-researchers use an inquiry based approach to engage students with making predictions, graphing results, and applying knowledge of geometric properties. This lesson could be applied to a range of grade levels from 8-12, as long as the instructor modifies, adapts, or identifies different focus areas depending on the grade level.Applicable Common Core State Standards:

- G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).

Reference: Gomez, E. & Wolfson, R.A. (March, 2012). Filling bottles with water.The Mathematics Teacher, 105(7), 542-548.## Teaching Algebra and Geometry Concepts by Modeling Telescope Optics

Applicable Common Core State Standards:Also from the National Council of Teachers of Mathematics, this journal article provides "project lessons in astronomical telescope design and construction to students in second-year algebra" (Siegel, et. al, 2008, p. 490). In exploring the ideas of telescope optics, students engage with the following math concepts: conic sections, parabolas, Euclidean proof, Cartesian coordinates, focal length, convergence, and vertices. Additionally, the teacher-researchers utilize teaching strategies and tools such as paper folding, mirrors, modeling, experimentation and investigation, and educational technology. Although the lesson plan is very topic specific (and required the teachers to have a certain amount of background knowledge related to astro-telescopes), the lesson is very detailed for anyone interested in the topic or looking to duplicate it: the authors include graphs, diagrams, and pictures of student work, along with the questions used to guide student exploration. Answers and student feedback are also provided. Overall, this is a very hands-on activity for students that also connects math to real world problems.- G-MG.1. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder).
- G-MG.3. Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).

Reference: Siegel, L.M., Dickinson, G., Hooper, E.J., & Daniels, M. (March, 2008). Teaching algebra and geometry concepts by modeling telescope optics.The Mathematics Teacher, 101(7), 490-497.## A Video Resource Showing a Lesson that Engages Students in Geometry Modeling in the Classroom

This video shows educator Suney Park using geometric modeling to teach her 6th Grade students about the connections between and interpretations of area and perimeter. In the video, Suney sets up a challenge for them: they must seat 22 guests at a table fore a feats. What dimension table will allow them to fit the most food? How about the least amount of food? The video demonstrated students working in groups and learning through exploration. Throughout the video, Suney explains her lesson plan and her approach to engaging her students in the subject. She also clarifies the connections to the Common Core Standards.Applicable Standards:

## A Video Showing a Lesson that Follows a Real-World Project where Students Apply their in-Class Learning

This video follows the teaching of Eeva Reeder and a real-world project she assigns her students. After realizing that they could not articulate what they had been learning in class, she developed a project that would force them to realize what they did and did not understand. The several layers of the project are laid out in the video. This type and style of project would be a useful resource for teaching because it engages the students, promotes and requires deep learning as well as collaboration, and it connects what they are learning in class to the real world.Applicable Standards:

## A Video Resource Presenting Geometric Modeling using Meatballs, and Other Resources by Dan Meyer

In this resource, Dan Meyer presents a problem: will the meatballs fit in the sauce without causing it to overflow? The problem is presented in an interactive way with a format that allows for exploration of the problem in a technology-equipped classroom. The activity is well suited for group discussion, and can be presented to an entire class. It provides all of the necessary details to formulate an answer, and reveals the correct answer as you progress through the activity. Once the preliminary problem is solved, further questions are presented to stimulate conversation and better understanding. Dan also presents a series of other problems in a similar format that can also be used in the classroom.Applicable Standards:

## Relevance:

Geometry is quite relevant in our world today, and modeling it is used in all sorts of fields. This website has a lot of great resources that show the different fields in which geometry is applied and modeling is relevant. So, when your students ask you why they need to learn things like this, you can have plenty of answers!Page created/updated by Allie Calderon, Jim Kaiser, and Lydia Wright.

## Other Wiki Resources:

Modeling Story ProblemsModeling Geometry Activities and Games