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These [[InterfaceOptions]] for customising [[TiddlyWiki]] are saved in your browser

Your username for signing your edits. Write it as a [[WikiWord]] (eg [[JoeBloggs]])

<<option txtUserName>>
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<<option chkAutoSave>> [[AutoSave]]
<<option chkRegExpSearch>> [[RegExpSearch]]
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This concept map is intended to closely follow the one presented by James J. Vagliardo in his paper found at http://cmc.ihmc.us/papers/cmc2004-270.pdf
<html>
<iframe src="http://bubbl.us/view/962af/1f0a5a/61e0Urgy3MO4o/" width="600" height="400"></iframe>
</html>
You can also visit the bubbl.us site to view the [[concept map|https://bubbl.us/?h=962af/1f0a5a/61e0Urgy3MO4o]].
Examining the concept map makes it clear that there is a great deal going on. That said, true understanding of every component of logarithms in not necessary in Algebra II. While logarithms were conceptually complete by 1700, we have gone from using logarithms as the supercomputer of the age to using them to solve for an unknown exponent. While I do want my students to have at least superficial knowledge of the rules of logarithms and why logarithms were historically important, I want them to have a deep understanding of a different subset of the concept map. Namely:

# How to use logarithms to solve for an unknown exponent. This must include story problems and connections to other content areas.
# Be able to describe the graph of a logarithm and know key points on the graph.
## Asymptotic to the //y// axis as //x// approach 0 from the right. Nonpositive //x// values are outside the domain.
##The //x// intercept is at 1. Unless there is a transformation the graph always goes through (1,0) no matter the base.
##Be able to roughly sketch the graph without a table.
##Know that a logarithmic function grows much more slowly than a linear function.
##Connect the graph to a logarithmic scale and realize why we use logarithmic scales.
#Know that a logarithm is an inverse of the exponential function and why this fact is mathematically important.
##This again connects to logarithmic scales.
##The graph of exponential function and logarithm are reflections over //y//=//x//.

!![[Next|Technology]]
I am primarily a high school mathematics teacher at [[Bellevue Community Schools|www.bellevue-schools.com]]. The schedule changes significantly every year. However, I’ve been mostly teaching Geometry and Algebra II the last few years. Bellevue Community Schools are located in the [[Village of Bellevue|http://www.bellevuemi.net/]] and is surrounded by rural housing and farms. Like many other rural schools the district, and most families attending, are [[not affluent|http://www.greatschools.org/cgi-bin/mi/other/500#students]]. Many students have access to computers, a smartphone or a iPod like device; however, this is far from universal. High speed internet is simply not attainable from any carrier or not affordable in other cases. The school was blessed with a technology bond a few years back. [[Interactive Whiteboards|http://www.polyvision.com/solutions/interactive-whiteboards/eno-family/eno-classic]] are installed in every classroom. There are two computer labs shared from 6th to 12th grade. A classroom set of netbooks is also available. There is also a set of graphing calculators that is mostly used with ~PreCalculus students. There are classroom sets of [[Scientific Calculators|http://www.amazon.com/Sharp-EL-W535B-WriteView-Scientific-Calculator/dp/B000RZDSWK/ref=sr_1_10?s=office-products&ie=UTF8&qid=1342124364&sr=1-10&keywords=sharp+scientific+calculator]] for every math and science classroom.

[[Next|Content]]
[[ConceptMap]]
[[Introduction]]
[img[CalcVsToes_Web.png]] My name is Eric Fitton and I am working on getting my Masters Degree from [[MSU|www.msu.edu]] in [[Education Technology|http://edutech.msu.edu/programs/masters/]]. If you want to see my teaching applets, view my resume or otherwise know more about me, please visit my main page at http://efitton.net. This particular subdomain is for a project where I look at students using GeoGebra in my classroom and being able to do that by flipping my classroom.
I am fortunate that I will have the same special education co-teacher with me for the third year in a row. The first year she did the same assignments as the students. We work together extremely well. She gives me feedback when her notes from the previous year show another approach. In addition to talking to me when an approach, strategy or thought process makes sense to her; she will also give mini-notes on how she makes sense of the material or how she took notes to the students in the class. I have no doubt that she will be watching video lectures at home and using them in her resource room hour. In short, I am blessed that I will get feedback from a teacher that I trust and have been working with for multiple years. I know that she will let me know what parts of flipping the class worked and which parts did not. This will not only include the “lectures” but will also include GeoGebra applets, story problem group work, concept maps and student reflections.

Math lends itself to post-tests and there will be formative and summative assessments through the unit and a summative assessment at the end of the unit. I believe that student reflections will be a useful formative assessment and should help me get a look at students understanding. Moreover, this will help guide what I do in the unit. I am not planning on a pre-test this unit as everything in this unit is absolutely new to students. It would be uninformative for me and could possibly hurt student confidence heading into the unit. That said, I should be able to compare how students did on the post-test this year compared to the previous two years. In addition to the actual numerical results, my co-teacher’s thoughts about how much better or worse students understood the material overall will be important.

Where the rubber really hits the road for me is how are students able to perform the following year or in a different subject. When students enter ~PreCalculus will they still be able to use a logarithm or will they have forgotten how? If there is a physics problem requiring a logarithm will they be able to work with the physics teacher or will fantasia or amnesia show themselves. The ~PreCalculus teacher gives a pre-test with logarithm problems at the start of the year. I will be interested in the following September if he sees a difference in how students perform.

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GeoGebra is wonderful dynamic software that connects Algebra, Geometry and Calculus. It can be found at http://www.geogebra.org/cms/. Instructional [[screencasts|http://math247.pbworks.com/w/page/20517545/Learn%20and%20Use%20GeoGebra]] are available.
Logarithms are incredibly powerful idea and tool in mathematics. They are relatively old, used before exponents. They are a central tool in computer science and physics. However, while students might be able to use logarithms to solve drill problems, deeper applications and understanding frequently eludes students. Looking back at my own mathematical learning, true understanding of logarithms came slower to me than almost every other facet of math^^1^^.

While my students can apply logarithms for their Final Exam, ~PreCalculus students simply did not remember to use a logarithm to find //x// on their pretest four months later. My hope going forward is to increase students’ perspective of logarithms, help students interpret the context of logarithms and make students able to explain logarithms by the end of the unit. With this background I believe there will be a reduction in amnesia; students should be able to apply logarithms four months or four years later.
I believe this unit lends itself to being “flipped.” It is inordinately difficult to activate prior knowledge with logarithms. While students can explore logarithms during activities and make connections during discussions; inquiry based learning where students //discover// logarithms is unrealistic. The background of notation and basic calculations simply needs to be supplied first. Using video, screencasting software, youtube, my website, and ~DVDs I plan on supplying lectures and notes as the homework for students. This will allow time for group activities and class discussion helping students put logarithms in historical and mathematical perspective.[>img[log.jpg]]
Freeing class time will allow for different types of assessments where I will be able to talk to small groups or individuals. This will help with regards to both formative and summative assessments. I will also be seeking feedback from my co-teacher during and after this unit. This will certainly give me perspective if the unit was more successful flipped. The rubber ultimately will meet the road the following September when ~PreCalculus students take their pretest.

^^1^^ A discussion of eigenvectors can wait for another day.

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!!Using Technology for Creativity and Critical Thinking
A great deal depends on how you use a tool. Many people cringe when then think of worksheets. A worksheet often is a great deal of practice problems. But a worksheet can also help students find patterns. Consider the following worksheet about the [[Triangle Inequality Theorem|https://docs.google.com/document/d/1zZlIEPwDhjhKmwbET4_KwVM_V2W4sBjQrnkMcGg-wjU/edit]]. It should help students find patterns and reason out ''why'' the Triangle Inequality Theorem works. This is how I intend to use GeoGebra in my flipped classroom.

!!Professional Development and Leadership
GeoGebra has a user forum and a website called [[GeoGebraTube|http://www.geogebratube.org/]] for sharing applets, worksheets and other materials. I have always shared my applets at efitton.net. However, I plan on becoming more involved with the ~GeoGebra community through ~GeoGebraTube and their forum. Similarly I will be looking to participate in an online community dealing with flipped classrooms. One of the advantages to meeting face to face in the hybrid program is I now have met two different math teachers who have already flipped their classroom and I certainly consider them part of my ~PLN.

While ~GeoGebra is limited with how many teachers can realistically adopt it at my school, it is ultimately a math tool, the same is not true of the flipped classroom. Social Studies and English teachers have assigned reading to students well before anyone coined the phrase, "flipping a class." That said, if I have success with flipping lessons and units I am sure other teachers will adopt and adapt my approach at my school.

!!Social and Ethical Uses of Technology
While important (we use [[Common Sense Media|http://commonsensemedia.org]] in our //Creativity and Computing// class), these are addressed in other areas of my districts curriculum rather than in this proposal.

!!Universal Design for Learning
GeoGebra again plays a staring role. Not only will we be representing a great deal of the content visually, they can be quickly and easily modified. This allows students to dig for patterns while watching the image change. In addition to set applets, graphs can also be created on the fly by either me, individual students or groups. The purpose of flipping the class is to increase student engagement. By having student reflect in writing and in concept maps they will be expressing how they are thinking on multiple platforms that are usually ignored in a math classroom.

!!Information Literacy and Technology Skills
This is not part of this proposal at this time.

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There is a great deal I love in this unit. Exponential Growth and Decay is spiraled back in. We find and use Euler’s number //e// and we really connect to the sciences. That said, the motivating question for the unit is to give students: 5^^//x//^^=2069 and ask them to solve it. Someone will eventually guess and check their way there; however, we do not have a workable approach other than logs. Given the plethora of exponential equations available from science and economics, being able to solve these problems is motivating and the goal is straightforward.

The pedagogy this unit will be wide ranging. There will be direct instruction in the form of videos for students to watch at home. Students will drill to make sure they can switch between logarithmic form and exponential form. Students will practice finding //x// no matter if it is the base, the exponent, or the answer. However, students will be asked to use higher order thinking skills as well. We will “discover” //e// by using the interest formula with an ever decreasing amount of time between compounding. There will be story problems that will be solved in groups. Students will make concept maps linking logarithms to different areas of math and science. And students will reflect on their understanding of logarithms and revise their understanding as we go through the unit.

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!!Logarithms

http://cmc.ihmc.us/papers/cmc2004-270.pdf
A highly informative article that shows a number of ways logarithms can be thought of historically. This article is where I found some of the best concept maps.

http://www.math.umt.edu/tmme/vol5no2and3/TMME_vol5nos2and3_a14_pp.337_344.pdf
How logarithms were historically used. Would fit well with the facet of perspective with an eye towards history.

http://www.thocp.net/reference/sciences/mathematics/logarithm_hist.htm
This gives a nice history of logarithms. An table with relevant dates is laid out.

http://www.rapidlearningcenter.com/mathematics/Preview/RL404/CS/ALG_CS15_LogarithmicFunctions.pdf
A logarithm "cheat sheet"

http://summit.sfu.ca/system/files/iritems1/7821/b39308625.pdf
A substantial piece of writing and research. I actually used this article more several years ago. My Dream IT project focuses much more heavily on graphing than the historical aspect.

!!Flipping the Classroom

http://usergeneratededucation.wordpress.com/2012/05/29/udl-and-the-flipped-classroom-the-full-picture/
A look at flipping a classroom paying attention to UDL and conceptual understanding.

http://flippedclassroom.org/
A social network of educators using screencasting and flipping their classrooms.

!!~GeoGebra

http://www.geogebra.org/cms/
Home of all things GeoGebra. This is the website with the actual software. On this page you will also find an introduction, a user forum and classroom materials.

http://math247.pbworks.com/w/page/20517545/Learn%20and%20Use%20GeoGebra
Screencasts, worksheets and scripts to learn GeoGebra. This is how I first learned how to use the software.
Logarithms with a flipped classroom and GeoGebra.
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While I make sure logarithms get a large share of time in Algebra II, there is still a limited amount of time to learn the material. Traditionally in my classroom for this unit I go over homework questions (usually story problems), followed by a lecture, followed by guided practice and students then have a book assignment or worksheet which is a combination of drill and story problem. I have made time for Newton’s Law of Cooling as a demonstration and always have had students “discover” //e//. That said, most days follow the same routine which misses students make deep connections or working out the story problems.

This unit is shaping up different in the coming year. Using screencasts and video with a variety of distribution mediums, the “lecture” is going to happen at home. The practice part of the assignments are not a large time commitment. While I still will have guided practice as a regular part of the day; how the end of the day looks can and will vary. Having groups work together on story problems rather than watching me will get students more involved. Having students concept map will get students thinking about connections. Introducing mini science labs will help students uncover what a logarithm is and how it works. Similarly, worksheets designed to get at understanding can be used and start a discussion. However, this would all be much more limited without being able to use technology to flip the classroom, making the lecture the homework.

There is an advantage to graphing using tables. It is universal and at times helps us uncover the material. A table can help us find slope, intercepts and connect to a story problem. Graphing point by point using a table is slow. When exploring a family of functions it is much faster to graph using a computer. A GeoGebra (similar to Geometers Sketchpad) applet can help students explore graphs of logarithmic functions and exponential functions. Changing variables with sliders helps clarify that an exponential function and a logarithmic function remain reflections. Connections can be made to the [[Graph-Translation Theorem|http://www.geogebra.org/en/upload/files/wadler/6_3_Translations.html]] and students can again use sliders to look at horizontal shifts, vertical shifts and dilations. The graph also helps with the discussion about domain and range. That you simply cannot take the log of a negative number is an essential piece of understanding and a graph can help students with that understanding. A great deal can be done using an automatic grapher and this unit lends itself to a program like GeoGebra.

Here is an example of an applet that I wrote for this unit showing how GeoGebra works. When I mention creating a worksheet, it is intended to be an exploration worksheet for students to look for patterns and make connections.
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There are actually two different ways I will need to make use of technology. The first component is the technology to “flip” the classroom. To create the videos I will be using a number of technologies. At times I may be using a flip camera, webcam, document camera, interactive whiteboard, a variety of microphones and an input tablet. Various software packages, including one to create screencast will be necessary. I also need to be able to distribute the videos. I will be uploading the videos to a personal website and to [[YouTube|http://www.youtube.com]]. However, not all students have access to the internet or a data plan, I plan on using jump drives for these students. I will also need to burn ~DVDs for students who do not have access to a computer. This does limit the type of work I can send home to either book assignments, worksheets or video lecture. It is possible a grant would allow for netbooks or tablets with java for more experiential homework and ease the workload on distributing the lessons.

As a result of the flip, I will be using more technology in my classroom. Some of this technology might be relatively simple. A thermometer with either hot or cold water can have students using logarithms to estimate a future temperature using Newton’s Law of Cooling. I will plan on using GeoGebra both on my interactive whiteboard and in the computer lab for students to manipulate graphs of logarithmic functions. Additionally, students will be writing about their understanding of logarithms throughout the unit. There is also an opportunity to create concept maps using a Web 2.0 tool.

That said, GeoGebra will be playing a leading role in my classroom and in this unit this coming year. I have created two applets already to go with this unit, am willing and able to make more and am sure I will be able to find more on [[GeoGebraTube|http://www.geogebratube.org/]]. I have spent approximately two days each year with Geometry and Algebra II students teaching them how to create and use GeoGebra. While pre-written applets are wonderful, having students start working from scratch in GeoGebra brings an entirely new twist to the classroom and the computer lab. There is certainly a place for GeoGebra to be used with an entire class, with worksheets / applets prepared ahead of time and for students to create on their own.

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[<img[construction_web.png]]In prior years I have focused almost entirely on two aspects of understanding in this unit. The largest focus was on applying logarithms. I wanted students to be able to apply logs to practice problems, story problems and to small projects. With the difficulty in activating prior knowledge I also tried to help students gain perspective understanding. We would connect examples with exponents, often reworking an entire example backwards using exponents. I would emphasize multiple times throughout the unit that logarithms are the only way to find an unknown exponent other than guess and check.

Clearly application matters. Logarithms are a building block of math and science. While I previously tried to incorporate perspective understanding, I believe I need to increase the amount of time spent on connections between the areas of mathematics and increase the quality of activities related to perspective understanding. I also think that class time will better be used having students manipulating models of logarithms; probably graphically, than how class time was used in previous years. Hopefully students will be able to interpret how logarithms work by adding this facet of understanding. Moreover by creating stories with thermometers and exploring Newton's Law of Cooling thoroughly, I expect to see greater understanding in the facet of interpretation. Where my instruction was most lacking; however, was dealing with the facet of explanation. Students never explained their thinking on logarithms. This is not only a vital assessment piece, it will also help students organize their thinking and help them make more connections. This can be gradually teased out to help students' with their self-knowledge. Are they aware of what they do and do not understand?

Technology can be an asset to my students understanding. While many of my “lectures” are interactive with students at the board, my lectures on logarithms are not. By moving the least interactive piece of instruction to the home we free up class time to not only work on drill assignments but engage in activities. We can solve story problems in groups and have group and class discussions. We can manipulate logarithms on graphing calculators and with GeoGebra. Students can explain their thinking by writing, and as we learn more, by revising. They can make sense on the material and demonstrate their connections between different aspects of logarithms and in math in general with concept maps. The writing can easily be a google document and there exist many Web 2.0 services to create concept maps.

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