Shape Sorter Activity CCSS.Math.Content.6.SP.B.5.b

By: Nick Spencer, Sam Marcoe, Grayson Windle, Elizabeth Englehart

With the Shape Sorter activity, which can be located at this website, we enable our students to work with several different concepts such as venn diagrams and geometric patterns.  The students work with a venn diagram in which they can set up two “rules” for each side of the diagram, and then are tasked with organizing the shapes accordingly to the rules.  If a shape meets both standards set by the rules, the student can place the shape in the center of the venn diagram, and if the shape doesn’t meet the criteria for either rule the student can leave it outside of the venn diagram.

In the figure above, my group of teacher candidates created our own venn diagram with these two rules:

  1. The figure has at least one line of symmetry (left side)
  2. The figure has rotational symmetry (right side)

In the beginning of this activity, we have several different shapes and begin organizing them in the matching part of the venn diagram based on their geometric characteristics.  The figures on the left side of the diagram have at least one line of symmetry while the figures on the right have rotational symmetry.  The figures in the center have both characteristics set by our rules, while the figures on the outside have neither lines of symmetry nor rotational symmetry.

This online educational activity allows the students to further develop conceptual understanding, and practice procedural fluency for geometric shapes and laws, as well as develop understanding for venn diagrams.  By allowing students to combine geometry with venn diagrams, students can open their perspective that venn diagrams can be utilized in various and unique situations.

 

Modeling a Digital and Global Age Learning Environment

Modeling a Digital and Global Age Learning Environment

 

Over the past decade our world has become increasingly more dependent on technology. Some school districts have made an effort to bring more technology into every classroom, but it is time for teachers to include technological tools in each of their lessons to assist each of their students’ needs. There are tools that can be used in the classroom such as clickers, smartboards, and document cameras. There are also so many web based resources that teachers can use in the classroom and that students can use at home, given they have internet access. Teachers need to take the time to be trained in technological tools that can improve their students’ learning environment and many will quickly find that these tools assist many of their teaching strategies.

Class Website

             A class website is a great communication tool for teachers, their students, and the students’ parents. My website includes a unit calendar that includes daily homework and quiz/test dates. Parents and students can check the website each evening to make sure all homework is being completed and to be made aware of future assignments. My website also includes links to online textbooks, helpful math websites and websites that provide fun games for extending each of my students’ mathematical skills. Many schools have web address set up for each teacher or more personalized websites can be created on KompoZer which is available for free download.

Math is Fun

Math is Fun is a resource website that makes learning mathematics fun and easy. The site covers k-12 curriculum and provides definitions and procedures for solving all types of math problems. This is a great review website as well as a resource for double checking equations and methods for solving problems you are not 100% comfortable with. This online website can help students complete their homework if they forget any needed equations and it can also help parents help their children. It is no secret that many parents have difficulty helping with their child’s math homework past the elementary years; this website will help parents quickly review some concepts and procedures and further their ability to help with homework.

Khan Academy

Khan Academy is another math resource website. This website is excellent for students who learn by example. There are 2-3 examples for each topic that begin with video instruction showing step by step procedures to solve a problem. After video examples there are practice problems that have to be correct before moving forward but also hints to help. My students love Khan Academy and many times when I know I will have a substitute I have them show the Khan Academy video as opposed to teach a lesson to my students. On my website I update the Khan Academy link to go directly to our current topic page.  This resource along with the textbook website, which also has instructional videos, is usually enough instruction for my advanced students to keep on top of work when they miss class. This allows me to not have to take time away from other students to teach students who were absent. For some of my struggling learners it is expected of them to review lessons on Khan Academy each evening because hearing the information twice greatly improves their success.

GeoGebra

When my students enter the geometry unit we spend a lot of time on laptops or in the computer lab. I have found that geometry is one of the toughest units for young middle school students to comprehend because it involves so much more than they are used to dealing with at one time. GeoGebra is a great resource for all students, but especially those who are just beginning their journey through geometry. My students use GeoGebra, which is an online resource that can help students create shapes and find their measurements. Students have difficulty with relationships between shapes and remembering differences between area and perimeter. GeoGebra helps students, especially visual learners, actually see what area means vs perimeter. I have GeoGebra linked on my class website so students can have quick access to it at home.

With an ever changing world technology allows teachers keep their lessons and students up to date with the latest mathematical tools. Providing students with the knowledge of how many resources are online and available for help with homework will make students more interested in completing their homework because going online and searching through websites is fun for young adolescents. It is important for teachers to be able to make the transition from paper and pencil, to online tools, and then back to paper and pencil. Resources such as GeoGebra are wonderful for helping students gain conceptual understanding of abstract concepts, but the procedures that are needed to solve problems without the help of technology cannot be forgotten. Students will be taught what a good digital citizen is and the importance of this will be reminded to students whenever we go online as a class. I will model each resource for my students before I expect, or encourage them to use it without my supervision. Trusting my students to make good choices usually makes them want to keep that trust and I rarely come across problems with online resources.

 

Resources

KompoZer

http://kompozer.net/

Math is Fun

https://www.mathsisfun.com/

Khan Academy https://www.khanacademy.org/

GeoGebra

https://www.geogebra.org/

Digital Citizenship

http://www.digitalcitizenship.net/Nine_Elements.html

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

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

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

Trigonometry and the Unit Circle: G-C.A Learning Progression

Triangles, and circles, and pi, oh my!  These, and many other terms are thrown around in class but what do they all really mean and when are they used? Geometry class, namely the trigonometric section of geometry. That is, taking students’ knowledge of right triangles and the associated properties, and applying that knowledge to find trigonometric ratios sine, cosine and tangent. To help students understand the connection between right triangles and trig, we introduce the unit circle using GeoGebra so students are able to visually watch the angles change and see the relation between angle measures in degrees and radians. This is just the tip of the ice burg that is geometry and all it has to offer to one’s mathematical arsenal. The attached learning progression and lesson plan take you through an alternative high school lesson which introduces students to trigonometry and the unit circle.

The image below is a screenshot of the GeoGebra application used to help visualize the correlation between degree measure, radians, and right triangles.

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Here you can find the learning progression and associated lesson plan with referenced worksheets. The interactive GeoGebra page was created separately in the free GeoGebra download available at geogebra.org

Learning progression final

 

7.G.B.6 Model this!

Studio 360 adds walls of modular shelving and storage to Slovenian apartment

Can you identify the basic geometric shapes in this living room? Given edge length measurements, can you find the area of the block panels, the approximate volume of the couch, or the surface area of the coffee table? To do this, you need spatial visualization skills and modeling skills. These are very important skills that students will need to develop in order to interpret and describe physical environments for solving tasks in everyday life.

The main idea of this lesson progression, intended for a high school Geometry class, is for students to develop spatial visualization and modeling skills to solve problems with real-world, 3-dimensional figures. It consists of three separate lessons, a formative assessment in between the lessons, and a summative assessment at the end of the sequence. The lessons gradually teach students the necessary foundational skills, from visualizing 2D and 3D figures as a composition of basic rectangles and triangles, to calculating the area, surface area, and volume of the figures. The assessments will provide the necessary feedback for both the students and teacher to determine what students know and need to review in order to reach the CCSS-Math standards aligned to this lesson progression.

F-BF 3-Act Math: Pyramid of Pennies

Use a 3-Acts task to engage your students in building functions.

Common Core State Standards:

3.MD.7: Measurement & Data, Geometric measurement: understand concepts of area and relate area to multiplication and to addition.

F-BF.1: Functions, Building Functions: Build a function that models a relationship between two quantities.

Learning Target: I will be able to build a function to solve for the volume of a square pyramid.

The Pyramid of Pennies problem can be found on the Dan Meyer’s 3-Act Math task website. This problem asks the students to determine how many pennies it takes to build the pyramid. It includes a 26 second video on the process of building the pyramid.

Pyramid of Pennies: http://mrmeyer.com/threeacts/pyramidofpennies/

Act 1:

The students will watch the video first to get a better idea of how many layers there are in the pyramid. The students are then prompted with the following questions:

 

pennypyramid1. How many pennies are there?

2. Guess as close as you can.

3. Give an answer you know is too high.

4. Give an answer you know is too low.

 

Before proceeding to act 2, I will ask my students what information is needed to find the amount of pennies used in the pyramid.

Act 2:

After the students had made an educated guess about how many pennies the pyramid is made of they will move on to learn some more information about the structure.

5. What information will you need to know to solve the problem?

They are shown 3 images about the base layer of the pyramid. In the first image, the base of the pyramid dimensions is 40 by 40 pennies. In the second picture, a single stack of pennies has 13 pennies in it.  In the third image, the picture shows the dimensions of a single penny. There is also a link that takes the students to a Wikipedia page that tells them how to find the area of a square pyramid.

Act 3:

The students learn the answer to the Pyramid of Pennies problem as well as get to see the newspaper clipping to the construction of the pyramid.

The students will obtain the learning target by seeing the construction of the pyramid and using all of the given information. The students will first need to set up an equation to find out how many pennies make up the base of the pyramid and how the next levels of the pyramid change as the structure grows. The students can create their own formula to determine the amount of pennies by using variables and the given information. In Act 2 the students will be able to adjust their original guess to an answer that comes from a generated formula. The students will be assess with the follow up questions at the end of the Pyramid of Pennies activity. The questions are harder

6. I have $1,000,000.00 in pennies, how big of a pyramid can I make?

7. Each stack has 13 pennies which is a strange number to choose. Why do you think Marcelo Bezos chose it? [Hint: not out of an abundance of superstition.]

8. Bezos says he can tell you the number of pennies in a pyramid with this equation:

where s is the number of pennies in a stack and b is the number of pennies on one side of the square base of the pyramid. Does this work? If so, prove it.

9. The Wheat and the Chessboard problem.

10. Give groups of students a dollar in pennies. See how fast they can assemble seven stacks of thirteen pennies. Then ask them to use that as a sample to determine how long it would take them to build the entire pyramid.

11. If Bezos now wanted to add more levels to the pyramid, what’s the best way for him to do that?

12. How heavy is the pyramid?

This activity is great for students because it is a really simple idea. One way that the students can be engaged is to have them measure the dimensions of the penny on their own and record their findings. The students will work in groups to complete this activity in the computer lab where they will be able to use technology view the video and images.

G-MG.1: Feeding Minds

The 3-Act Math Task, Meatballs by Dan Meyer, can be found at http://www.101qs.com/2352-meatballs.  This activity is ideal for those spaghetti lovers in your classroom. It involves learning how to calculate how many meatballs can be placed in a pot of spaghetti sauce with out causing the sauce to overflow. This can easily relate to the majority of your students, whether they cook the spaghetti and meatballs themselves or they just eat it. This activity allows students to be constantly engaged through a subject of interest, various discussion starter questions, and mini videos.

Act 1 of this task begins with a video of boiling spaghetti sauce that ends with a bowl of meatballs about to be entered into the sauce pot. This is followed with the following discussion starter questions:

1. How many meatballs will it take to overflow?

2. What is a number of meatballs you know is too high?

3. What is a number of meatballs you know is too low?

This allows students to be fully aware of their central focus for the day and begin to brainstorm of the answer possibilities.

In act two students are asked what information would be useful to know here? As students begin to answer this question, the teacher will show the students images of the things that would be useful to know to answer the leading question (How many meatballs will it take to overflow?). The images consist of the height remaining in the pot, the diameter of some sample meatballs, the diameter of the pot, and the number of meatballs. During this act the teacher will also either teach or review volume formulas of cylinders and spheres so that they can proceed to the next act. Through this act students begin to think of ways in which they can approach this problem and what information is needed to begin to solve for the solution.

_1303_emptyheight_1304_meatballdiameter_1305_t_1306_t

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

During act three students will use the information collected during act two to calculate the space left in the pot and the volume of the meatballs and predict a number of meatballs that will fill the remaining space or cause the pot to overflow. Once students have made their calculations/predictions, the teacher will show a video of meatballs being placed in the pot one by one until the pot slightly over flows, providing students with the answer to check how close their predictions were.

This task is aligned to CCSS.Math.Content.HSG-MG.A.1: Use geometric shapes, their measures, and their properties to describe objects.  This task allows students to compare one shape (cylinder/room remaining in pot) to another (sphere/meatballs), as they are trying to determine how many meatballs fit in the remaining room in the pot. In order to do so students must be taught the volume formulas of these geometric shapes and be able to determine how one shape would fit into the other.

Meatballs is also aligned to CCSS.MATH.PRACTICE.MP4: Model with mathematics. Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. For this task students used their knowledge of cylinders and spheres to determine how many meatballs they can put in their spaghetti sauce before it overflows. This can be applied by all students in their every day lives and shows students how math is a part of our daily lives and doesn’t only arise in school settings.

G-MG.1-G-MG.3: Record Breaking Hot Coffee!

Screen Shot 2015-01-14 at 12.45.07 PM

Did you know that over 50% of Americans over 18 years of age drink coffee every day?! And that among these 150 million coffee drinkers, the average coffee consumption in the United States in 2010 was 3.2 cups of coffee per day! (Data gathered from the 2010 National Coffee Association Drinking Trends Study.) Since coffee consumption is such a huge consumer market in the United States and using an interactive video to represent a real-world scenario, the Hot Coffee math problem by Dan Meyer is a great math problem to do with your students!

The Hot Coffee math problem follows the Gourmet Gift Basket team as they try to Guinness World Record for the biggest cup of coffee! This compelling math problem can be found on Dan Meyer’s 3-Act Math task website using the following link: http://mrmeyer.com/threeacts/hotcoffee/. This problem is also a great way to address several high school geometry Common Core State Standards!

The High School Geometry: Geometric Measurement and Dimension: Common Core State Standards that are addressed in this problem are:

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

G-MG.3: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Act One

In the first act students will watch a one-minute short video of the Gourmet Gift Basket team building, transporting, setting up, and beginning to fill up their coffee cup! Then, students will be given the questions shown below:

  1. The Gourmet Gift Baskets team wants to break the record for the biggest coffee cup. Will that cup be enough to break it? How many gallons of coffee do you think will fit inside?
  2. Guess as close as you can. Write your guess down.
  3. Write down a guess you know is too high.
  4. Write down a guess you know is too low.
  5. How long do you think it’ll take them to fill up the cup?
  6. How many regular-size cups of coffee would fit inside that super-size cup of coffee?

After making their individual guesses for questions 2 through 4, by the end of the problem students should be able to answer questions 1, 5, and 6.

During this lesson, it is up to your digression whether you would like your students to work individually, in pairs, or in teams. My recommendation, however, and how I would group up my students would be in pairs or groups of three. So, students can challenge and work together to answer the problem’s questions.

Act Two

In the second act, students will ask themselves and answer the question: What information will help solve this problem?

First, I would have my students brainstorm in their small groups the type of information they think they need to complete the problem. While students are coming up with ideas, I will be going around and listening in to conversations and helping students who need assistance. After it seems that all students have finished their list, I will hold a whole class discussion. Where I will ensure that students understand the facts of the problem (shown below) and understand what procedures need to be taken in order to solve the Hot Coffee’s questions.

Information students need to solve this problem: 

The dimensions of the cup: 7 feet by 7 feet

The rate the cup is filling up:2.1 gallons/minute

The conversion from cubic feet to gallon: 1 cubic foot = 7.48052 gallons

The conversion from gallons to cups: 1 gallon = 16 cups

Information on the old record: Which can be reviewed as a class during whole class discussion and handed out in paper form to each student or small group as a resource

Information on the Guinness World Record guidelines: Which can also be reviewed as a class during the whole class discussion or handed out in paper form to each student or small group to serve as a resource

Act Three

In act three students solve for the size of cup needed to break the old largest cup of coffee record, solve for the amount of coffee gallons that would fit in the cup, how long it would take the Gourmet Gift Basket team to fill up the cup, and how many regular-size cups of coffee fit inside of that super-size cup of coffee.

Students will meet the first standard when they are able to solve for the size of cup needed to break the Guinness World Record for the largest cup of coffee. Students will show evidence of meeting the second standard when they solve for the volume and area of the coffee cup to figure out how many gallons it will take to fill the inside of the cup. Students will meet standard three by using ratios to solve how many regular-size cups fit in their recording breaking coffee cup.

This is a great lesson because this 3-Act Math problem is exciting to solve! It really helps students stay engaged and motivated to solving the several answers pertaining to the cup of coffee, but probably the most interesting and conclusive question being: Did Gourmet Gift Baskets actually beat the old Guinness World Record for the largest cup of cup?! On the Dan Meyer’s webpage there are also great sequel ideas that align to other Math Common Core State Standards that you can do!

G.SRT – Map My Walk

CCSS.Math.Content.HSG.SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.*

G.SRT.C.8 Map My Walk

It is no secret that society is using more and more on technology for everyday things. As teacher we need to try to incorporate this kind of technology that our students are using on a daily basis into our curriculum.

apps

One way to incorporate technology into a lesson is by having students use what they can’t live without, their cellphones. In both Google Play and the iTunes store the Walk with Map My Walk app can be downloaded for free. This app tracks where the person is walking, what distance they have traveled as well as the duration of the walk. One standard that can be met using this app is G.SRT.C.8, which is a modeling standard that uses trigonometry and the Pythagorean theorem to solve right triangles. Using this app two students can start at a marked point and then walk away from each other at a right angle (maybe walk on the sidewalk of an intersection), while the students walk away from each other the app will track the distance traveled by each student. Once the students have walked the desired distance they can come together and draw their triangle using the data from the app. Now with the triangle drawn the students can now use the Pythagorean theorem to solve for the hypotenuse, which is the distance between the two students when they stopped walking. Once the students have solved for the hypotenuse then they can now use that information to solve for the angles using trigonometric ratios. To challenge the students a bit more they can be assigned to make their own triangle using only one student.

app

This activity will get students out of the classroom and doing something they probably did not think could be done with their phones. If this standard was not taught using technology it would be taught just by story telling and having students draw out the triangle according to the story. This activity is not only getting students to learn math in a more practical way but it is also getting them to move around. This is also something the parents and the community can get involved in. Parents can use the app to assess their child when they are out on family walks or when they are at the store. Once the student is comfortable with the app they can involve their friends and community members in an activity.

The students will be exited to get an opportunity to go outside and move around rather than just sit at a desk and work from a textbook. This technology is very helpful because students always have their phones on them. Instead of trying to fight them to put it away why not incorporate this into the curriculum where they will be using their phone in a positive and productive way. When students are able to see the how math is really applied to the real world they tend to listen better and be more involved. If the students can see the correlation between the lesson and the community they will be more enthusiastic. When students start to get into higher-level math they tend to lose interest because they feel like that math does not apply to everyday life but by having students use things like their phone in lessons they will see that math is still apart of it.

Here is the worksheet where the students will collect and write all their information. MapMyWalk

For more pictures and information about the app go to http://www.mapmywalk.com/app/