Geometry Curriculum Aligned with CCSS-Math

7.G.B.4,6 and 8.G.C.9: Volume and Surface Area

cospers Assessment of CCSS-Math, CCSS-Math Learning Progressions, Common Core State Standards Mathematics, Geometry Curriculum Aligned with CCSS-Math

This learning progression is designed for a high school Applied Geometry course in which the students have access to computers. The Common Core State Standards that align with this learning progression are: 7.G.B.4, 7.G.B.6, and 8.G.C.9. The reason that we combined 7th and 8th grade standards is because the students need a chance to review and solidify their understanding of the basics of area and volume but also be challenged by having to find real-world applications. The students will also be assessed on the following standards of mathematical practice: MP4; model with mathematics, MP6; attend to precision, and MP7; look for and make use of structure.

pizza

The unit will begin with a discovery based lesson about pi. The students will be required to partner up and measure the circumference and diameter of two different objects. Each partner will be assigned different circular objects. There will be a master list at the front of the room that the students can record their measurements on and then write what the circumference/diameter of their objects was. The purpose of this lesson is to show the students that, no matter what the size of the circle, if they take the circumference/diameter of their object they will get approximately 3.14, or pi. By realizing this pattern, the students are looking for and making use of structure (MP7). The students will also be required to graph the data, with the circumference of the objects being represented on the y-axis and the diameter on the x-axis. This will create a relatively linear line. The students will then be asked to identify the slope of this line, which will be 3.14. Finally, the students will be required to answer some questions in order to elicit further evidence of their understanding. Some of the questions that will be asked are:

1. What is the average ratio of circumference to diameter?

2. What observations did you make?

3.  Can you derive the formula for circumference of a circle by knowing Pi is the ratio of diameter to circumference?

After the students understand the origin of pi, they will be required to use the concept of pi to find the area of circles. They will also learn how to find the surface area and area of rectangles and triangles and their respective 3-D prisms. Finally, the students will learn what the difference is between surface area and area. They will be asked what real-world applications that are related to area and surface area. In using the equations to correctly calculate volume and surface area, the students are attending to precision (MP6). The students will be formally assessed via a worksheet. Problems on this worksheet include

1. Find the surface area of the origami figure you made in class

2.  circle      radius:                     Equation:                      Area:

3. What do you notice about the shapes of a cylinder when calculating surface area?

 

Once the students understand how to find the area and surface area of these basic objects, they will move on to finding both surface area and volume of more advanced objects such as pyramids, cones, spheres, triangular prisms, cubes, and cylinders. The students will be provided with a list of the necessary equations to know. The purpose of this lesson is for the students to be able to apply the given formulas and to identify how volume and surface area apply to the real-world. In order to allow the students to practice the equations for finding the volume and surface area of these objects, they will use Google Earth. The students will download the Volume of Solids kmz file from realworldmath.org. They will then follow the directions to virtually travel the world and find the volume and surface area of different buildings in different countries. The program itself will give the necessary dimensions for the equations. By having the students find real-world applications of volume and surface area, they are modeling with mathematics (MP4). The students will record their work and answers on a worksheet that they will be given. One of the questions on the worksheet looks like this:

Leaning Tower of Pisa

pisa     Surface Area:

 

Volume:

 

In order to assess the students ability to connect this lesson to the real world, they will also be asked on their worksheet “In what career would you use volume and surface area? Why do you think that?”. This is an important question since the main purpose of this lesson is to get the students to realize the real-world applications of volume and area.

 

Learning Progression: LearningProgression

Discovering Pi lesson plan: discovering pi

Area and Surface Area lesson plan: LP Area

Google Earth Volume and Surface Area lesson plan: GeoMath lesson plan

CCSS Cluster: HSG Congruence Learning Progression

nelsokar Common Core State Standards Mathematics, Geometry Curriculum Aligned with CCSS-Math, HS Geometry, Uncategorized

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This learning progression would take place in a 10th/11th grade Geometry class and is aligned to multiple Common Core State Standards regarding “Congruence” – specifically HSG.CO.B.6, HSG.CO.B.7, and HSG.CO.B.8. This geometry courses uses the Holt Rinehart Winston Geometry textbook aided with an online resource, TeachersPayTeachers – All Things Algebra. This specific learning progression aligns directly with the TeachersPayTeachers worksheets. The guided notes and assignment are taken directly from this online resource.

 

Learning Progression – edTPA

HSG.GMD.A3 – Calculating the volume of grain silos

tamayop Geometry Curriculum Aligned with CCSS-Math, HS Geometry, Picture Problems, Technology Teaching Issues

shutterstock_Silo

Depending on where you live, these grain silos can be a pretty familiar sight.  Silos are used to store the excess grain harvested from the fields and are especially popular in more agricultural areas.

For this problem, students will be tasked with finding the amount of grain in a silo given the measurements and the percentage of the silo that is full.  This problem can be easy – simply finding the max volume of the structure given the heights and radii – or it can be made more difficult – finding the amount grain if the silo is only partially full or even comparing the volumes of multiple silos using the concept of similarity.  The picture provides a visual representation of what the silos actually look like, making it easier for students to see the silo as two cones and a cylinder and not just a single structure.

CCSS.Math.Content.HSG.GMD.A.3

Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.*

8.G.C.9 GeoMath: Finding Volume and Surface Area

cospers Geometry Curriculum Aligned with CCSS-Math, ML Modeling

“Today’s students are used to and expect continuous connection to information and people. Forcing them to put their personal technology away during class contradicts their lives and gives students one more reason to expect what they learn in school will have little relationship to reality.” – Robert Beichnergoogle earth

It’s that exact expectation from students, that what they are learning in math class has little to do with the real world, that is going to drive me to the brink of insanity. I know that my geometry students are in desperate need of an answer to their two daily questions “when will I ever use this?” and “why do I have to learn this?” Well, I finally took the advice of Robert Beichner and decided that, for this lesson, I won’t ask the students to put away their “personal technology”. Instead, I am encouraging them to use their computers to learn about the importance and relevance of finding surface area and volume of different geometric objects by applying it to the real world.

How? Well, it’s quite simple really. If we want students to understand real world connections, what better way than to look at the ‘real’ world? Okay, Google Earth isn’t technically the real world. However, it’s a pretty good visual representation. Teachers and student can now go to the website www.realworldmath.org to find a wide variety of ways in which you can use Google Earth to teach math, science, and history. One of the lessons involves “flying” around the world to different famous buildings to find the volume and surface area of each. Google Earth even provides a brief history lesson along the way. Students will be able to see the applications of different formulas for volume and surface area unfolding before their eyes. Google Earth and the link mentioned above are great resources to get the students involved in real world applications of math all while using their beloved technology. I’ve used Google Earth in the classroom before and it’s quite amazing how willing students are to stay on task because navigation the app is fun, visual, and mathematical relevance is obvious.  You never know, you may get one day’s break from the “when will I ever use this?” and “why do I have to learn this?” questions. Trust me, the opportunities are endless. Literally…Google Earth even includes outer space.

blockhead

googleearthpyramid

 

 

 

 

 

Lesson Plan:  GeoMath lesson plan

Worksheet: GeoMath- google earth worksheet

Pump Up the Volume! Volume of Cylinders (CCSS.MATH.CONTENT.8.G.C.9)

dollingerm Geometry Curriculum Aligned with CCSS-Math, Geometry Teaching Issues, Middle School Issues, ML Modeling

ContainersCylinders

 

This is a great activity for any middle school class that enjoys going beyond math problems from a book and into the realm of kinesthetic learning. With this activity, students will gain a deeper conceptual understanding of the volume of a cylinder as they determine which cylinder has a larger volume when two different cylinders are formed using an 8.5” by 11” piece of paper. Students will make a prediction, then confirm or reject their prediction using their knowledge of the dimensions of paper and how it relates to finding the volume of a sphere.

Students will also learn how this activity has real world application through the opening exploration activity involving soup and a cylindrical container that ends up being too small. A discussion will revolve around what could have been done to prevent the soup from spilling over the end of the container. Students will then consider the possibility of change the dimensions of the container to increase the volume so that the soup will fit.

Created by: James McInroy, Jehnna Keshishian, Megan Ellis, and Michelle Dollinger

 

Explanation of Engagement

Mathematical Practices

Lesson Plan

Worksheet

How Tall is That? 6.RP.A.3

sinclairs Geometry Curriculum Aligned with CCSS-Math, Geometry Teaching Issues, Middle School Issues, ML Modeling, ML Ratios & Proportional Relationships , , , ,

MATH VisualCan’t measure the height of that tree with your ruler? No problem! Let’s use ratio’s to find it’s height!

This lesson is a great modeling activity where students are able to apply ratios and proportions to real world situations. They will be using the concept illustrated above with objects they find around their school. The students will find their own height and shadow length, and use this ratio to find the heights of several tall objects. Some of these might be basketball hoops, portables, school buildings, soccer goals, railings, and even the height of their own teacher! Continue reading →