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Using GeoGebra 3D & Augmented Reality to Build and Test 3D Mathematical Models

Explore and create

Explore and create : Creation lab

Thursday, December 3, 12:30–1:20 pm PST (Pacific Standard Time)

Timothy Brzezinski  
Learn how to use GeoGebra 3D Calculator to construct models of 3D objects using concepts students learn in a variety of math curricula. We will use augmented reality on our phones and tablets to test the accuracy of our models by superimposing our virtual constructions over the real objects.

Audience: Teachers, Teacher education/higher ed faculty, Professional developers
Skill level: Beginner
Attendee devices: Devices required
Attendee device specification: Smartphone: Android, iOS
Laptop: Chromebook, Mac, PC
Tablet: Android, iOS
Participant accounts, software and other materials: GeoGebra 3D Calculator (iOS) download for iPad/iPhone:

GeoGebra 3D Calculator (Google/Android):

Google Play Services for AR (Google/Android ONLY):

Interested participants should go to and quickly create a free GeoGebra account.

Topic: Augmented, mixed & virtual reality
Grade level: 9-12
Subject area: Math
ISTE Standards: For Educators:
  • Use technology to create, adapt and personalize learning experiences that foster independent learning and accommodate learner differences and needs.
  • Design authentic learning activities that align with content area standards and use digital tools and resources to maximize active, deep learning.
  • Explore and apply instructional design principles to create innovative digital learning environments that engage and support learning.
Disclosure: The submitter of this session has been supported by a company whose product is being included in the session

Proposal summary

Purpose & objective

Teachers will be able to
1) Use GeoGebra 3D Calculator to construct mathematical models of everyday, real-life objects (cubes, rectangular prisms, spheres, etc)
2) Save their constructions they create to their GeoGebra profile page.
3) Open up their constructions in GeoGebra 3D Calculator app on their smartphones and/or tablets.
4) Activate augmented reality within GeoGebra 3D calculator app on their phone to explore their models from inside and out.
5) Use augmented reality to TEST the accuracy of their mathematical models by superimposing these 3D virtual models over real objects that will be placed in their environment. The more accurate their model, the more this virtual models will want to it will stay locked-on the real model.

This workshop is targeted towards upper elementary, middle, and high school, & post-secondary mathematics teachers and administrators. In this workshop we will also complete steps (1) - (5) through using the tools of GeoGebra 3D Calculator mathematically model a Toblerone candy bar (triangular prism whose base is equilateral). In doing so, we will experience how concepts (in several mathematics curricula (algebra through precalculus) that students typically learn in 2D can be applied and explored in the 3D (real-world) realm.


Order & description of activities:
1) Download GeoGebra 3D Calculator app on our devices (from AppStore or Google Play), and quickly create a GeoGebra account online (or sign in to our GeoGebra account if a user has one already) (10 min)
2) Open up GeoGebra 3D Calculator on our phones. Construct basic solids both in the 3D AND augmented reality realm. Explore these surfaces in augmented reality. (5 min)
3) Open up GeoGebra 3D Calculator on our computers and Chromebooks. We will together, collectively as a group, discuss how we can use coordinate geometry and transformational geometry to effectively construct a perfect model of a Toblerone candy bar (whose triangular edge length = 3.5 cm and whose height = 21 cm). Through this discussion, teachers will see how many geometry and algebra concepts their students typically learn in 2D can easily be applied in 3D. (20 min)
4) Save our constructions we made in (3) above. We will then open up our constructions on our phones & tablets and use augmented reality to test the accuracy of our models. We will also be able to create quick screencasts of these. (10 min)
5) We will re-build this Toblerone now through using an algebraic functions approach. Since the absolute value function looks like the letter "V", we will transform an absolute value surface function to create this Toblerone a different way. (20 min)
6) We will then explore how easy it is to create a surface of revolution in GeoGebra 3D Calculator (rotating a segment 360-degrees about an axis to create a cylinder or cereal bowl). Then, we will switch gears to Algebra 1 and construct a linear function to rotate to create a perfect model of a cereal bowl (that will be in the room). We will then test our models in augmented reality (20 min).
7) Wrap up (5 min)

These times are for a 90-min session. If ISTE wants this to be a 60-min session insrtead, eliminate all of (5) and I will quickly demo (6) in 10 min.

Supporting research

The Common Core State Standards (CCSS) Initiative is a document that influences many statewide curricula in the United States. Other similar documents can be found within other countries. The CCSS deems "Modeling with Mathematics" as one of the 8 key standards for effective practice in teaching mathematics. This strand is the main idea for using augmented reality within this workshop.

Another key standard in the CCSS is "Attend to Precision". Here, augmented reality helps students determine how precise or accurate their virtual 3D models are.

Another key standard in the CCSS is "Construct Viable Arguments and Critique Reasoning of Others". When we collaborate as a group during this workshop, we will be doing this.

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Timothy Brzezinski, GeoGebra

Tim Brzezinski is a full time Independent Mathematics Education Consultant, Mathematics Teacher, and Accredited GeoGebra Trainer and Author. He is a former full time high school mathematics teacher w/15 years experience and part time university mathematics instructor w/15 years experience. Mr. Brzezinski facilitates engaging GeoGebra professional development workshops for mathematics teachers and administrators worldwide. In these, he demonstrates how GeoGebra can serve as an powerful platform that effectively fosters active, student-centered, discovery-based learning, meaningful remediation, and ongoing formative assessment. Mr. Brzezinski also authors interactive GeoGebra resources for publishers, EdTech companies, and mathematics teachers worldwide.

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