Multimodal Geometry Project
Learning Module
- Creator(s):
This work is for a sophomore level Geometry class. In their study of Geometry in prior chapters, the focus has been on two-dimensional shapes. Students have learned about the following: the special properties of triangles, parallelograms, regular polygons and circles. Students have learned to describe and analyze these shapes and the properties that are associated with each. For example, students have tools to find an interior angle of a regular hexagon, to calculate the length of the hypotenuse of a right triangle, and to measure the perimeter of a triangle or the area of a circle.
In this work, we will turn the focus to learn about three-dimensional shapes (called solids), such as cubes and cylinders. Prior to this, students will have studied rectangular prisms and cubes. In this set of modules, students will focus on cylinders and prism. Students learn several ways to represent three-dimensional solids and develop methods to measure their volumes and surface areas. The use of multimodalities will be used throughout the modules to engage students in a variety of ways.
Multimodality is a way of meaning making that is vastly different from traditional teaching models (Cope, 2016) Text, image, sound and various other media are used simultaneously to make meaning. In the video below, the elements of multimodality is explained further.
Education at Illinois. (2016, April 19). 8.1 Representation, Communication and Design [Video]. YouTube. https://youtu.be/zJ3UVat1Llk
As you heard in the video, the three components are: representation, communication and design. Dr. Cope also states that literacies are about creating a message. This message must be appropriate for the context and medium (Cope & Kalantzis, 2009).
The 6 updates listed below should be done in sequential order and can be a part of any sophomore level high school course. Each update would correlate with one 50 minute class period. The sequence of information follows the College Preparatory Mathematics curriculum (www.cpm.org) that is used in my current school. I chose this topic because I am looking for a way to incorporate new media literacies, multimodalites and peer reviews into my current curriculum in order to make this portion of the chapter more meaningful and useful for my students.
Throughout this unit, standards will be assessed from the Common Core State Standards and each module is scheduled to take one day. These are the standards:
G-GMD.1: Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone.
G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve 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)
Additionally, students will be using the Math Practice Standards below:
CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
CCSS.MATH.PRACTICE.MP4 Model with mathematics.
CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
CCSS.MATH.PRACTICE.MP6 Attend to precision.
CCSS.MATH.PRACTICE.MP7 Look for and make use of structure.
CCSS.MATH.PRACTICE.MP8 Look for and express regularity in repeated reasoning.
The culminating project will allow students the opportunity to use their new knowledge to become meaning makers as they construct a structure using information learned. In this project, students will research components that make buildings strong. They will then test the strength by placing weights on top of their building until it collapses.
Lesson Objective: To pre-assess key vocabulary terms that will be used throughout the unit.
The goal of this first module is to pre-assess key vocabulary terms that will be used throughout the unit. Please visit the Padlet linked HERE and add anything you know about the terms. You can include text, images, videos, podcasts, websites, etc. The goal here is to take a look at these terms and to see what you know before we begin the new material. Be sure to scroll to the right to see the entire list. The words on the Padlet are below:
Task: After you have added to the Padlet, take a look at what others have added and comment below with any connections, similarities, or differences you notice. Additionally, please take this pre-unit survey so that I can assess your background knowledge. It is not for a grade and is simply a formative assessment that will help me guide the learning throughout the unit. Survey can be found here: https://docs.google.com/forms/d/1_pZBlOcH6vDb8zi3LX6R0qY5dkwqCBx9BXrp7kI-3gY/edit
Content Standard: 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)
Math Practice Standard: CCSS.MATH.PRACTICE.MP4 Model with mathematics.
Lesson Objective: To pre-assess key vocabulary terms that will be used throughout the unit.
The goal of this first module is to pre-assess key vocabulary terms that will be used throughout the unit. Students will be asked to do a carousel with the following words:
A carousel can be done in class or virtually. In class, this would look like having the words written on large poster paper around the room. Students walk around the room and write or draw anything that comes to mind on each poster under the word. Virtually, you could do a Padlet where students type in their answers or add images to the online platform. This gives the students an opportunity to engage in a multimodal way by including media elements such as images, podcasts, videos, etc. This gives you, the teacher, and opportunity to formatively assess students' knowledge of the important terms. Take note of what the understandings and the misconceptions are be sure to incorporate these into future teaching as appropriate.
Here is a brief introduction to the platform of Padlet, if needed:
Padlet. (2017, February 02). Introduction to Padlet. [Video]. Retrieved from https://www.youtube.com/watch?v=U3P5QySmLeU
Additionally, students will take a pre-unit survey in the form of an Anticipation Guide. Collect the data and use that to make adjustments as needed in the following modules.
Lesson Objective: Students will learn what a 3D solid is and how to create one using an online tool.
Today, the focus is on three-dimensional solids. The objects you encounter every day are three-dimensional. To understand the shapes that you encounter daily, you will need to learn more about how three-dimensional shapes, called solids, can be created, described, and measured. An example of a solid is below.
When we see a solid such as this, we assume that blocks cannot hover in midair. That is, if a block is on the second level, assume that it has a block below it to prop it up. Use the e-tool linked HERE to recreate the solid above in a virtual space. Is there one more than one arrangement of blocks that could result in image above? If so, create more. For each arrangement you make on the e-tool, take a screenshot and upload in the task below.
Task: Take a screenshot of the solid(s) you created in the e-tool and upload in the comment section. Make two comments on your peers' pictures. In your comments, include whether you think their reasoning is valid. If you think they made a mistake, comment on ways to improve their figure.
Content Standard: 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)
Math Practice Standard: CCSS.MATH.PRACTICE.MP5 Use appropriate tools strategically.
Lesson Objective: Students will learn what a 3D solid is and how to create one using an online tool.
This module introduces students to the study of three-dimensional solids. Begin by explaining that all of the geometric focus in this course so far has been on two-dimensional figures and now students will begin to focus on three-dimensional figures. Review the Padlet/Carousel from the previous day and point out some key parts of the column of "Three-Dimensional". Expect some questions regarding the difference between two-dimensional and three-dimensional.
The focus on this lesson is to understand the shape of the solid and to begin foundational work with volume and surface area. As such, students will not be asked to draw the complete solid by hand, but to explore with the online software tool linked in the student section.
Lesson Objective: Students will learn how to represent three-dimensional solids using side views and a mat plan. Also, students will be introduced to volume as a form of measurement.
Thinking back to the solids we learned about in Module 2, we can use a mat plan to show how the blocks are arranged in the solid. Take a look at the Padlet from yesterday to better understand a mat plan. An example of a mat plan is below:
The number in each square represents the number of the blocks stacked in that location if you are looking from above. For example, in the right-hand corner, the solid is only 1 block tall, so there is a “1” in the corresponding corner of its mat plan. Verify that the solid you submitted in Task 1 matches the solid represented in the mat plan above.
Mat plans can help us find the volume of a shape very easily. Watch this brief video below on the definition of volume and how the cubes we have been working with play an important role in volume:
NUMBEROCK. (2015, July 28). Volume Song. [Video]. https://www.youtube.com/watch?v=LZxXUb9iAZc&vl=en
You can also read more about the definition of volume here: https://www.splashlearn.com/math-vocabulary/geometry/volume
The video and online reading tells us that the volume of any shape can be found by adding up each cubic unit. In a mat plan, the amount of cubes are listed out, so to find the total volume, you simply add the numbers together. In the mat plan above, the shape would have a volume of 13 cubic units.
Three-dimensional solid can also be represented by its side and top views. These views are two-dimensional and are visible when looking directly at the solid from the top, front and right hand side. The two images below show the different views.
Using the e-tool HERE, build the shape below and then explore the top, right and front views. After your exploration, complete the task below.
Task: After learning about the solids, mat plans and side/top views, take this assessment and submit to your teacher. The survey can be accessed by clicking HERE.
Content Standard: G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Math Practice Standard: CCSS.MATH.PRACTICE.MP1 Make sense of problems and persevere in solving them.
Lesson Objective: Students will learn how to represent three-dimensional solids using side views and a mat plan. Also, students will be introduced to volume as a form of measurement.
The first problem introduces the mat plan as one way to represent a three-dimensional solid. This problem also introduces volume as the number of cubes used to create the three-dimensional object. Allow students time to use the online software to create the shape.
The second problem introduces side and top views as another way to represent a three-dimensional solid. Allow students time to use the online software to create the left, right, and top views as instructed. As students are building, be sure to circulate and check for potential misunderstanding such as not including all the blocks, or not connecting the height of the block with the side views.
Their task is a survey that you can use as a formative assessment. Review the answers and be prepared to clear up any misunderstanding in the next module.
Lesson Objective: Students will be introduced to prisms. Students will also learn how to find the surface area of a solid.
Today you will continue to study three-dimensional solids and will practice representing a solid using a mat plan and its side and top views. As you work today, you will learn about a special set of solids called prisms and will study how to find the surface area and volume of a prism.
WARM UP: We will start with a warm-up from the previous module. The top, front and right hand views are shown below. Use the e-tool HERE and create a solid that matches these views. Find the volume. Once you have found the volume, check your answer with your neighbor.
Thinking back to the solid we have built in the prior modules, imagine what would happen if you dropped the entire solid in a bucket of paint. What parts of the solid would be covered? The space that would be covered is called the surface area. Watch the video below to see another example of what the surface area looks like in a few real life applications. Volume is also explained for comparison.
GreeneMiddleSchool. (2012, October 28). Volume vs. Surface Area. [Video]. https://www.youtube.com/watch?v=PsJd-BOBNGY
You can read more about surface area here: https://www.mathopenref.com/surface-area.html
The volume and surface area can be found for any shapes. For this module, we will focus on prisms. There are many types of prisms, and our focus right now is on the general shape and how to apply our three-dimensional building to find volume and surface area. Take a minute to watch this video on what prisms are and how they are created. Remember, the names of each are not quite as important and understanding how they are created.
KASAMathematics. (2010, December 19). [Video]. https://www.youtube.com/watch?v=TM004YsdvLQ
Task 1: Make your own figure of a solid and post the top, right and side views. From the views posted choose one and see if you can build the solid based on the given information. Post your picture. The creator of the initial views should check and see if the submission is correct. Offer feedback to your peers in the form of a comment under their submission.
Task 2: Read the question about Paul's tower below and comment with your answers. Also answer the following question in your comment: In general, what is the relationship between the area of the base of a prism, its height, and its volume?
Content Standard: 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)
Math Practice Standard: CCSS.MATH.PRACTICE.MP3 Construct viable arguments and critique the reasoning of others.
Lesson Objective: Students will be introduced to prisms. Students will also learn how to find the surface area of a solid.
As you introduce today’s lesson, give feedback on yesterday's survey. The new concepts build on the information from yesterday, so it's important that student errors are cleared up.
In the first problem, students are given front, right and top views and are asked to build the shape of the solid. This activity is challenging and students should use their e-tool to help them as needed. Important: We want the students to create a solid with the LEAST number of cubes, so continue to challenge them to use less.
Surface area is introduced in the next problem, when a solid is dropped into a bucket of paint. Refer back to the Padlet and draw connections from the students pre-assessment to the current idea of paint. A common misconception here is that the bottom is not included, so be on the lookout for that.
Once students have had a chance to learn about surface area, we start to move towards volume and surface area of prisms. The term prism is introduced and students use a variety of representations to find the volume and surface area. The general idea is that the volume is the area of the base (the number of cubes in each layer) multiplied by the height of the tower (number of layers). As for surface area, students may stick to counting the faces of each cube, but that is not going to work for very large shapes, so encourage students to find another method. We are looking for them to multiply the perimeter of the base by the height of the tower.
Lesson objective: Students will practice finding the surface area and volume of cylinders.
In prior modules, you investigated volume, surface area, and special three-dimensional solids called prisms. Today you will explore different ways to find the volume and surface area of related solid called a cylinder. Since theses shapes are not made up of cubes, we must find a general way to find the needed measurements. Watch the video below to find the volume and surface area of a cylinder and then submit the task.
Khan Academy. (2011, April 12). Cylinder Volume & Surface Area. [Video]. https://www.youtube.com/watch?v=gL3HxBQyeg0&disable_polymer=true
You can read more about volume and surface area of a cylinder here: https://courses.lumenlearning.com/prealgebra/chapter/finding-the-volume-and-surface-area-of-a-cylinder/
Task: Find an object in your house that is in the shape of a cylinder. Find the volume and surface area of the cylinder. Post a picture of your object, your answers and a screenshot of your work. Check the work of two of your peers and comment on any changes, if necessary.
Content Standard: G-GMD.3: Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
Math Practice Standard: CCSS.MATH.PRACTICE.MP2 Reason abstractly and quantitatively.
Lesson objective: Students will practice finding the surface area and volume of cylinders.
This module challenges students to find the volume and surface area of non-rectangular prisms and cylinders. These problems are challenging because students can no longer count the cubes to come up with the answer. Rather, they need to generalize the process of finding the volume and surface area by using the base of the shape. They will have a chance to find a real life object and apply their new learnings.
The pdf below is your opportunity to build a solid structure that will be tested with weights. Please read very carefully and complete all components of the project. Before you test your structure, your peers will have the opportunity to give feedback and make design suggestions in order to improve your model. For the peer review, you will be given a picture of another student's complete structure. You will give feedback on the following: ways to improve the structure, check the height requirements, and suggestions on the included research questions. Once peer reviews have been complete, students will have an opportunity to make adjustments. Once adjustments are made, we will take the structures to the weight room for the final test.
This module applies the learning of volume and surface area into a project. This project is adapted from www.teachersnetwork.org. Students will have a week to complete each task and will work in groups to decide what structure is the most effective at holding weight. Ideas to choose from include all solids studied in prior modules, or a combination of solids. A video below shows a final product being tested so you have an idea what to expect on the evaluation day.
Gathright, B. (2010, October 27). Geometry Project- First Attempt. [Video]. https://www.youtube.com/watch?v=JUWop3pC_wE
Education at Illinois. (2016, April 19). 8.1 Representation, Communication and Design [Video]. YouTube. https://youtu.be/zJ3UVat1Llk
[Collection of 3D figures]. [Image]. (2005). https://www.onlinemathlearning.com/solid-geometry.html
Cope, William & Kalantzis, Mary. (2009). “Multiliteracies”: New Literacies, New Learning. Pedagogies. 4. 164-195. 10.1080/15544800903076044.
Core Connections Geometry. (2013). www.ebooks.cpm.org
Gathright, B. (2010, October 27). Geometry Project- First Attempt. [Video]. https://www.youtube.com/watch?v=JUWop3pC_wE
Geometry Solids. [Image]. (n.d). https://slideplayer.com/slide/7913816/
GreeneMiddleSchool. (2012, October 28). Volume vs. Surface Area. [Video]. https://www.youtube.com/watch?v=PsJd-BOBNGY
Khan Academy. (2011, April 12). Cylinder Volume & Surface Area. [Video]. https://www.youtube.com/watch?v=gL3HxBQyeg0&disable_polymer=true
NUMBEROCK. (2015, July 28). Volume Song. [Video]. https://www.youtube.com/watch?v=LZxXUb9iAZc&vl=en
Padlet. (2017, February 02). Introduction to Padlet. [Video]. https://www.youtube.com/watch?v=U3P5QySmLeU