Mathematics

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 * Mathematics**

Using the following format, please add your PBL content.
 * 1) **Title:** In **BOLD**, please put the name of your project.
 * 2) **Name:** Please add your name.
 * 3) **Subject/Course:** Please provide the subject(s) covered.
 * 4) **Driving/Essential Question:** Please provide the project's essential question.
 * 5) **Objectives:** Please **bullet** 3-5 objectives of the project.
 * 6) **Planning Documents:** Using the "file" button in the tool bar above, after completing each of the items (1-5 AND 1-3), attach your planning documents to the wiki.
 * 7) **Example of Student Work:** Optional but desired. If you have a sample of student work, please add the file.


 * Reflective Questions:** For the project, please answer the following three reflective questions based on what you learned when completing the projects with your students.
 * 1) What went well with project?
 * 2) What did not go well?
 * 3) What will I change next time?


 * PLEASE LEAVE THE ABOVE FORMAT GUIDELINES AND ADD YOUR TWO PROJECTS BELOW USING THE FORMAT OUTLINED ABOVE.**

The Nature of Functions
 * Maria I Suarez **
 * Precalculus **

 **Essential Question:** Can you find a mathematical model to represent different things in nature and/or surroundings?

 **Objectives:** Open this file to see project details and samples Open this file to see the PBL Project form
 * ** To be able to find the mathematical expression that is represented in a graph. **
 * ** To understand the basic concepts of functions. **
 * **To present mathematical functions in different media**

1. What went well with the project? Right after we discuss the 8 basic functions and their properties we went around town trying to see shapes and figures that could be represented with one of the functions we discussed. They were very involved in the discussion and we work in a couple of pictures together. I assigned the project just before the Mount Nebo experience so they could have more natural things to use for their projections. They really enjoyed this project! 2. What did not go well? The students would have liked to have more time to work on their final product. They also talked about making different kind of final products (i-movies, slide shows, etc.) 3. What to change for next time? I will give more time for the students to work in their final product and leave the final product open to create different media.
 * Reflective Questions: **


 * FTC Robotic Competition **
 * Maria I. Suarez **
 * Logic and Problem Solving/ Applied Mathematics/ Technology **

**Essential Question:** How to build a robot to complete specific tasks using autonomous and driver controlled programs?  **Objectives:**
 * ** To build a robot that will have specific functions and abilities **
 * ** To program a robot to work in an autonomous and a driver controller mode **
 * ** To use the engineering process to complete the construction of the robot in an efficient manner. **



Engineering Journal Technical Notes (Ratios and more) 1. What went well with the project? The students were very enthusiastic about the project because of the competitive nature of the project. They got well organized and distributed the work load in an efficient way, They selected their project manager who was in charge in keeping up with the different phases of the project. 2. What did not go well? We had technical difficulties with the Robot C software and the blue tooth communication technology 3. What to change for next time? Make sure the communication issue is resolved before we start the testing phase. Check that the software license do not expire in the middle of the season.
 * Reflective Questions: **

The Projectiles Project


 * Maria I Suarez/Mark Hespenheide **
 * Precalculus/Physics **

**Essential Question:** How can we measure and model projectile motion?

 **Objectives:**
 * ** To acquire and analyze data from experimental situations **
 * ** To graph: position, velocity, acceleration, vectors. **
 * ** To use all data, horizontal and vertical motions to produce mathematical models. **




 * Reflective Questions: **

1. What went well with the project? The students were very involved in the experimental part of the project, analyzing the data, using Photoshop and interpreting the data into mathematical models.

2. What did not go well? The students would have liked to have more time to work on their final product. We had some problems finding computers with Photoshop installed to be able to manipulate the data.

3. What to change for next time? Make sure I have enough computers available with Photoshop.

**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 19px; text-decoration: none; vertical-align: baseline;">Stock Market Optimization ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Phillip Gablehouse ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Algebra 2, Economics, Business Math, Statistics ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Essential Question: Under what conditions do we achieve the best possible return? ** <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;"> Objectives:
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Understanding of stock market mechanics.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Develop research skills to make intelligent predictions.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Graphical representations of growth and/or decay over a period of time.



**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Reflective Questions **
 * 1) **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What went well? **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;"> The students were enthusiastic about the project. Once things got moving, there was lots of interest from the students
 * 2) <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What did not go well? **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;"> Time. The project is very slow to develop. To experiment with different investment strategies beyond simple day trading requires a greater time commitment. A commitment that takes away too much class time.
 * 3) <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What to change for next time? **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;"> Due to time constraints, it may be best to concentrate on various day trading strategies so that useful data can be gathered quickly and the project moves along, otherwise the project could take up to a semester with only a few achieved objectives, making the project extremely inefficient.

**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 19px; text-decoration: none; vertical-align: baseline;">Paper Airplane Design ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Phillip Gablehouse ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Geometry, Physics, IPS ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Essential Question: What properties make something fly? ** <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;"> Objectives:
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Use simple geometric designs to design and build paper airplanes to complete pre-determined goals.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Understanding of airfoil design and how varying geometric properties affect results.

**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Reflective Questions **
 * 1) **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What went well? **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;"> (to be completed at a later time. Already discussed with Academic Dean)
 * 2) <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What did not go well? **
 * 3) <span style="font-family: arial,helvetica,sans-serif; font-size: 13px; line-height: 19px;">**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What to change for next time? **

**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 19px; text-decoration: none; vertical-align: baseline;">Game Design ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Phillip Gablehouse ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Logic and Problem Solving ** **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Essential Question: What does it take to make a game fair? Do optimization strategies always exist? ** <span style="background-color: transparent; color: #000000; display: block; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Objectives:
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Apply concepts from game theory, probability, combinatorics, etc, to develop a “game engine”.
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Analyze and reflect on whether that engine is fair. Is there anything that can be exploited by a player to improve their chances of winning?
 * <span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Be able to apply these concepts to any situation to help determine appropriate actions to arrive at a desired result.

**<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">Reflective Questions **
 * 1) **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What went well? **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;"> (to be completed at a later time)
 * 2) **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What did not go well? **
 * 3) **<span style="background-color: transparent; color: #000000; font-family: Arial; font-size: 13px; text-decoration: none; vertical-align: baseline;">What to change for next time? **

=**Accuracy of Data Collection**= James Taylor AP Statistics

Can the combined inaccuracies of many students abilities to time an object falling create an accurate measurement? What factors contribute to accuracy or inaccuracy in measuring with a stop watch the time it takes for an object to fall?

• Use statistics to find correlations between various attributes and the ability of a person to accurately time a falling golf ball. • Use statistics to determine if a reasonably accurate measurement can be found in the collection of a number of inaccurate measurements. • Use statistics to convey the story told by a meaningless onslaught of raw data.

=**eCyberMission – Energy Savings through a Shower Heat Exchanger**= James Taylor Math and Science
 * 1) The students did a great job of brainstorming on what attributes might or might not contribute to accurate or inaccurate timing. They were able to come up with hypothesis on why one attribute or another might have an impact. The students were reasonably diligent in doing their best to accurately time the falling golf ball and enter the data into the database.
 * 2) The results were disastrous. Some students put in hardly any work, others put in clearly a tremendous amount of work, but none even came close to the task of “conveying the story told by the data” or correctly analyzing the data to arrive at a reasonable conclusion.
 * 3) Two problems caused the disastrous results: lack of understanding of what the expectations were and lack of time to meet those expectations. Obviously there was a failure for me to sufficiently convey what was wanted – you can’t blame 100% failure on the students…. Second the project was done right before a large break. When the results were turned in, there was no time to correct the expectations and have the students turn in a re-draft, let alone have the students continue to work on their results until the results met the expectations. The timing caused this project to be a one-shot effort and that was not successful. Next time I would make sure I gave sufficient instructions so the students better understood the expectations for the project and had enough time to do multiple revisions of the analysis.

Can the heat in shower waste water be used or recycled to reduce the cost of the shower?

• Use Science, Technology, Engineering, and Math (STEM) to create a solution for recycling the heat in shower water after it has left the person taking the shower. • Test the solution and make refinements using scientific equipment and processes. • Present the results with analysis in a meaningful, significant, and convincing way.


 * 1) All around, the 8th grade students did a terrific job. They worked together to create, modify, and discard many ideas until they settled on one concept. Initially in the testing process they were hesitant and waited for adult instruction, but as the testing went on, they increasingly took over the process until at the end the adults were nearly superfluous and almost in the way. The students presented their work both in written form and in an oral presentation, winning first place for the State of Utah and making it to the top 4 teams in the NW Region.
 * 2) An earlier start would have given us more time to do a better analysis of the results and possibly conduct new tests.
 * 3) Start earlier.

=Geometry and Architecture=

**Kari Varga**

 * Geometry**


 * Essential Question:** How are aspects of Geometry used in architecture?


 * Objectives:**
 * to gain a better appreciation for a practical application of Geometry in the world around them
 * help them to become more familiar with and be able to identify different types of angles in a real world setting
 * help them to become more familiar with and be able to identify different types Geometric terms in a real world setting



1. **What went well with the project?** The students were very successful at being able to identify different angles and Geometric terms. They had a lot of fun with this project and were surprised at how applicable Geometry is in the real world. 2. **What did not go well?** Some of the students got a little bit frustrated about their lack of artistic skills when it came to drawing the structures. 3. **What to change for next time?** It would be nice to bring in someone from the art department to just give the students a few pointers about how to draw before they started.
 * Reflective Questions:**

=Sugar Cube Project=

**Kari Varga**

 * Geometry**


 * Essential Question:** How can you make a proportional model of a famous structure out of sugar cubes?


 * Objectives:**
 * figure out what dimensions are needed for the model to be proportional to the original structure
 * to be able to identify the scale factor that was used for the model
 * be able to identify congruent and similar aspects of the model and the original structure



1. **What went well with the project?** I put the students into groups of four for this project. They enjoyed working together and problem solving together. They were able to learn a lot about proportions, scale factor, congruency, and similarity in a low pressure situation. They enjoyed this project very much. 2. **What did not go well?** The students would have like to build something even larger (most were about 12" x 12") so they would have liked more sugar cubes to build with. (Each group had around 850 to work with.) Also, the glue I used was a type of frosting and sometimes didn't hold together very well. 3. **What to change for next time?** I would like to use more sugar cubes per group and find a better glue.
 * Reflective Questions:**