tag:blogger.com,1999:blog-54412322876307559432024-03-12T22:34:56.977-04:00I Was Thinking..Thoughts, opinions, and ideas about Mathematics and other things from a retired teacher.Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.comBlogger70125tag:blogger.com,1999:blog-5441232287630755943.post-23542489949783712242017-07-25T11:27:00.001-04:002017-07-25T11:27:42.747-04:00When was America Great, Mr. Trump?When asked to relate when America was greatest (<a href="http://www.cnn.com/2016/03/26/politics/donald-trump-when-america-was-great/index.html" target="_blank">this quote is from CNN</a>):<br />
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<span style="font-family: inherit;">"If you look back, it really was, there was a period of time when we were developing at the turn of the century which was a pretty wild time for this country and pretty wild in terms of building that machine, that machine was really based on entrepreneurship," he told the Times.</span><span style="font-family: inherit;">Trump also pointed to the "late '40s and '50s," a time when, he said, "we were not pushed around, we were respected by everybody, we had just won a war, we were pretty much doing what we had to do."</span></blockquote>
As a clarification, it should be pointed out that at the turn of the century there was no federal income tax in the United States. Something must have happened during that stretch that led to the creation of the income tax in 1913. Not long thereafter came the great depression. Without a doubt the first third of the 20th century was filled with fiscal and financial turmoil. Not too great.<br />
<br />
The other time Trump termed "great" had a maximum tax rate at 90% or higher.<br />
Does "Make America Great Again" ask for a return to pre-depression policies, or to a 90% maximum tax rate?<br />
Can someone please explain?<br />
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Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-80585041660980050022017-04-13T12:23:00.000-04:002017-04-13T12:23:07.809-04:00Bottle deposit scamHannaford has evidently partnered with a bottle/can deposit redemption outfit called <a href="http://www.clynk.com/" target="_blank">Clynk</a>.<br />
<br />
Our Hannaford removed its machines for bottle & can returns and replaced them with a choice of using one machine (now broken) or their new service called Clynk. Clynk requires that you place empties in one of their bags, create an account, drop off the bag, and wait for your money to appear in an online account. Sounds simple, but for two items:<br />
<br />
<ol>
<li>Clynk gives you 10 bags when you start, but after that you have to BUY the bags. That is an added consumer cost.</li>
<li>Instead of getting funds immediately at the store, they are held (our store says 2 days). During that time you do not get the benefit of your money, either Hannaford or Clynk does. This may be small potatoes to you, but after accounting for all the customers, it could and would be massive.</li>
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<div>
This is sold as an easier way. Here is a quote directly from the Clynk web site:</div>
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<span style="background-color: white; color: #555555; font-family: "Open Sans", Verdana, Helvetica, Arial, sans-serif; font-size: 14px;">CLYNK is the only bottle redemption system that allows you to create a personal account, accumulate a balance and use your money at your discretion. CLYNK’s unique database lets you track your personal environmental impact every time you drop a bag at CLYNK. Start saving time, money and the environment.</span></blockquote>
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Wow! I have always wanted to have my own "personal account" so I could "accumulate a balance". It is too bad I can't use the money at my discretion, since when I return empties I do not get the cash in hand, but have to wait.<br />
<br />
<span style="font-family: inherit;">If Hannaford truly wants to do something for their customers, they will toss this Clynk business immediately. The <a href="http://www.dec.ny.gov/chemical/8833.html" target="_blank">DEC web page</a> states "<span style="background-color: white;">A consumer can return empty containers to any store or vendor that sells the same type of container.</span> " So Hannaford should get back to its won business.</span><br />
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<br />Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-49228300537670407792017-03-28T17:47:00.002-04:002017-03-28T17:47:22.042-04:00"Math rules" is a complete sentence!This is a reminder for my math blog at <a href="http://notesfromamathteacher.blogspot.com/">http://notesfromamathteacher.blogspot.com/</a>.<br />
<br />
<a href="http://notesfromamathteacher.blogspot.com/2017/03/is-kindergarten-too-old.html" target="_blank">March 28, 2017</a> I posted a GeoGebra link which I then tweaked and converted to an animated gif in my 14-year-old image editor (Macromedia Fireworks).<br />
<br />
Images such as this can and should be used to advertise the "A" in "STEAM" education. In cae that is new to you, STEAM is<br />
<br />
<br />
<ul>
<li><span style="color: red;"><b>S</b></span>cience</li>
<li><span style="color: red;"><b>T</b></span>echnology</li>
<li><span style="color: red;"><b>E</b></span>nergy</li>
<li><span style="background-color: white;"><span style="color: red;"><b>A</b></span></span>rt</li>
<li><b><span style="color: red;">M</span></b>athematics</li>
</ul>
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<br />Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-18674767420448810542017-03-21T09:58:00.001-04:002017-03-21T09:58:50.551-04:00Fake news?<div style="text-align: justify;">
<span style="font-family: inherit;">So we have a president (no capital "p" here) who recently had a "rally" in Louisville. Imagine, he goes to the state that elected Mitch McConnell, and afterwords tweets "<span style="background-color: white; color: #14171a; letter-spacing: 0.26px; white-space: pre-wrap;">Thank you Louisville, Kentucky. Together, we will MAKE AMERICA SAFE AND GREAT AGAIN!"</span></span></div>
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<span style="background-color: white; color: #14171a; letter-spacing: 0.26px; white-space: pre-wrap;"><span style="font-family: inherit;"><br /></span></span></div>
<div style="text-align: justify;">
<span style="background-color: white; color: #14171a; letter-spacing: 0.26px; white-space: pre-wrap;"><span style="font-family: inherit;">So now he is going to make America safe "again". It must have been safer when Kentuckians went into coal mines so that we "up top" could burn that coal.</span></span></div>
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<span style="background-color: white; color: #14171a; letter-spacing: 0.26px; white-space: pre-wrap;"><span style="font-family: inherit;"><br /></span></span></div>
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<span style="background-color: white; color: #14171a; letter-spacing: 0.26px; white-space: pre-wrap;"><span style="font-family: inherit;">Please, @realdonaldtrump, tell me how sending people into coal mines so that we can burn coal is going to make us safer? Is the high-sulfur coal of Kentucky a hidden key to health and safety? (By the way, one cannot be safer if one is made less healthy: anyone who has breathed NYC air over the last 70 years should know that).</span></span></div>
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<span style="background-color: white; color: #14171a; letter-spacing: 0.26px; white-space: pre-wrap;"><span style="font-family: inherit;"><br /></span></span></div>
<div style="text-align: justify;">
<span style="color: #14171a;"><span style="background-color: white; letter-spacing: 0.26px; white-space: pre-wrap;">I just cannot get over the thought that Trump wants to increase the defense budget for a country that he is trying to kill.</span></span></div>
Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-61175620284676754122017-02-24T09:32:00.001-05:002017-02-24T09:32:40.518-05:00Time for an attitude adjustment in the USA<div style="text-align: justify;">
The following is part of the platform stated by the US Conservative party (see it <a href="http://home.conservativepartyusa.org/national-platform/" target="_blank">here</a>)</div>
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<strong style="background: transparent; border: 0px; margin: 0px; padding: 0px; vertical-align: baseline;">* Repeal and Replace The Affordable Care Act (ACA)</strong></div>
<div style="text-align: justify;">
Support candidates who support legislation that repeals the Affordable Care Act of 2010, aka “ObamaCare”. Support federal lawsuits promulgated by State Attorneys General to halt implementation of said Bill. All new health care solutions should:</div>
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1. Provide meaningful Tort Reform that limits liability for pain and suffering. This cuts malpractice premiums and reduces the cost of needless “Defensive” tests.</div>
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2. Allow posting of local prescription drug prices on the Internet so that people can shop for the lowest price.</div>
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3. Allow individuals and small businesses to purchase health/dental insurance across state and regional lines.</div>
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4. Encourage the medical industry to digitize medical practice paperwork to reduce related waste. However, no digital process should ever be mandated that excessively burdens medical practice to the detriment of the doctor patient relationship.</div>
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5. Eliminate the Anti-Trust exemptions for Health Insurance companies which currently allow them to collude in setting prices and coverage.</div>
</blockquote>
<div style="text-align: justify;">
I pick on the Conservative party simply because the current administration seems to be beholden to "conservatism". </div>
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<br /></div>
<div style="text-align: justify;">
In this platform I see 5 items with which I wholly agree, even though I will never support them if they involve tearing down The Affordable Care Act. Each and every one of them, if presented properly and calmly and clearly, would probably receive bipartisan support. Each and every one of them, if presented under the guise of repealing "Obamacare" or as a we-alone-can-solve-this problem solution, would get ungraciously slammed into oblivion.</div>
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<br /></div>
<div style="text-align: justify;">
A message to all political parties: if you wish to guide society to change its ways, your first step must be properly, clearly, and calmly articulating the problem (without placing blame) and your solution (without claiming credit) and try to get society to agree with you by persuasion and not by command. Anything else is doomed. </div>
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<div style="text-align: justify;">
Our largest problem here in the US of A is that far too many are in the business of claiming credit and placing blame. Neither of those businesses help anybody, and the more they remain in control, the quicker our country will implode and explode. If you wish to kill the USA, keep doing what you are doing.</div>
Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-65265095795992219212017-02-09T11:11:00.000-05:002017-02-09T11:11:32.252-05:00Boycott Kentucky!<div style="text-align: justify;">
Mitch McConnell, Senator from Kentucky, has done enough damage, and the positions of Speaker of the House and Senate majority leader have done enough damage.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
No person in Congress elected to serve a state or a congressional district should have the power to control what Congress can or cannot do. That applies no matter which political party claims the position. </div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
There is no requirement that the Speaker of the House even be in congress. Given that, I have never understood why the Speaker is second in the line of Succession.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
Back to McConnell: he has got to go. The rules may never change, but he has got to leave.</div>
<div style="text-align: justify;">
<br /></div>
<div style="text-align: justify;">
I propose that the people of the United States adopt a "no go" policy towards the state of Kentucky, until such time as Mitch McConnell has left government. By that I mean:</div>
<div style="text-align: justify;">
<br /></div>
<br />
<ul>
<li style="text-align: justify;">Do not visit the state. (No Mammoth Cave visit, ni Kentucky Derby stay, no Louisville Slugger trip)</li>
<li style="text-align: justify;">Do not drive through the state unless you can do so without stopping. Most cars should be able to go the distance across the state on an interstate without needing gas, provided they are topped off before entering the state.</li>
<li style="text-align: justify;">Do not fly into, out of, or through Cincinnati's main airport. (Apologies to the city, but they put their airport in Hebron, Kentucky. The people of Hebron must have been irate when they called this airport CVG, in honor of what looks to be a gerrymandered city about a 15 mile drive away). Watch your Delta connections!</li>
</ul>
<div style="text-align: justify;">
This is not an original idea of mine (see <a href="http://petitions.moveon.org/sign/boycott-kentucky-135414312" target="_blank">here</a>), but it is time to bring it on. </div>
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Unless and until McConnell is gone from Washington (and only Kentucky can make that happen), shun the state itself. </div>
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<br />Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-43445962009441985442017-02-01T13:23:00.000-05:002017-02-01T13:23:43.465-05:00Political parties should not be deified<div style="text-align: justify;">
As we survive the thirteenth day of the Trump realm, I have to comment on something that bothers me. That something is currently known by the moniker "party loyalty". (Even some of those who voiced opposition to Trump are voting in favor of his antics simply because it is a "Republican" thing to do.</div>
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<br /></div>
<div style="text-align: justify;">
What bothers me is that far too many senators and congressmen give the impression that they see the country of the United States only as a necessary evil, required as an arena for what they really value, their political party. Currently it is Republicans who display it most, but Democrats have had their times as well. It has been covered in the media for ages.</div>
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<br /></div>
<div style="text-align: justify;">
The mere fact that politicians' initial response to a legislative proposal is to love it or hate depending upon which party it came from or who is backing it. Mitch McConnell even verbalized his goal of making President Obama a "one term president", basically vowing to block anything he proposes. Ain't that a good ol' boy way of showing your love for your country?</div>
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<div style="text-align: justify;">
Our politicians will not change their ways unless forced to. Perhaps we, the people, might make that happen by making a conscious effort to identify and vote out every congressman or politician who does not cross party lines at least once this session and speak out in national media in favor of at least one bill proposed by the other party. It is a small step, but that is how every journey starts.</div>
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<br /></div>
<div style="text-align: justify;">
Another small step would be the minimizing of the effect of congressional gerrymandering. A small step would be a requirement that all congressional districts be convex in shape.</div>
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<br /></div>
<div style="text-align: justify;">
Convexity is very simple. A district would be convex if, for any pair of points in the district, all points on the line between them are also in that district. Exclusion would be made only when the line connecting two points in a district passed through a neighboring state. An example of what would not be allowed is shown here. The red segment starts and ends withing a New York congressional district, but passes through another district. That would be a no-no. (Unless the other district was in a different state.)</div>
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<br />Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-87485431869977862712017-01-15T11:12:00.001-05:002017-01-15T11:12:40.840-05:00A dare for Donald TrumpDonald Trump has claimed that America was "great" back in the early 20th century and also in the 1940's and 1950's (see <a href="http://www.cnn.com/2016/03/26/politics/donald-trump-when-america-was-great/" target="_blank">here</a>)(and <a href="https://qz.com/648386/donald-trump-says-america-hasnt-been-great-in-116-years/" target="_blank">here</a>)<br />
<br />
I will not go into why he thinks those years were the greatest for America, but I can tell you one thing that did not exist at all back then: Twitter.<br />
<br />
Twitter began in July 2006. According to Donald Trump, America at that time was not great at all. The best I can sense is that Trump believes that America has gotten worse during the time of Twitter.<br />
<br />
So, Donald Trump, I dare you: live up to your goal of "make America great again" and stop using Twitter.Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-77509305640268719372017-01-11T12:12:00.002-05:002017-01-11T12:15:46.619-05:00Please, please...I just heard Donald Trump chastise news outlets for supplying "fake news" while allegedly speaking at an alleged news conference.<br />
<br />
Is there any chance that "Trump wins" will be discovered to be fake news? One can only hope.Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-62456361078926257832016-12-18T10:46:00.002-05:002016-12-18T10:46:27.058-05:00Electoral College: Perform Your Task WiselyMany people seem confused by the process of selecting a President.<br />
<br />
At no point has this country been other than a republic of states, and the US Constitution specifies the means for those states to elect a President. That process is the Electoral College.<br />
<br />
No matter what is on your ballot when you vote, you are not voting for a President. You are voting for an elector, a member of the Electoral College. ALL your voting takes place within your state. There is no vote for a national office.<br />
<br />
Tomorrow the Electoral College meets to vote for the next President.<br />
<br />
I pray and hope that it exercises this task with utmost care and diligence, and does not merely make its selection because of a misconstrued sense that the "public" voted for one person or another. The public did not vote for anything more than the electors.<br />
<br />
Electoral College: this is your opportunity to confirm the wisdom of our Founding Fathers.Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-6410281246200161192016-11-29T22:06:00.002-05:002016-11-29T22:06:56.586-05:00No doubtEvery time I think the Donald is an idiot, he opens his mouth and removes all doubt.Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-85858880429815612052016-11-19T12:28:00.000-05:002016-11-19T12:28:09.512-05:00A must see<iframe title="New York Times Video - Embed Player" width="480" height="321" frameborder="0" scrolling="no" allowfullscreen="true" marginheight="0" marginwidth="0" id="nyt_video_player" src="https://static01.nyt.com/video/players/offsite/index.html?videoId=100000004777637"></iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-26396061769497917872016-11-16T10:46:00.002-05:002016-11-16T10:46:18.860-05:00Is this news?<div style="text-align: justify;">
The New York Times of November 16 has an article entitled "the Two Americas of 2016", splitting the country in two by counties, based on whether or not they went for Clinton or Trump in the recent elections. (Article <a href="http://nyti.ms/2eZDRgW" target="_blank">here</a>). Why counties? Why not Congressional districts, why not voting districts, why not by elevation? (Who knows! Is height above sea level positively correlated with election results?)</div>
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The Times let me down with this article. It said nothing new. Urban areas tended to go democrat and rural areas republican. Who didn't know this BEFORE the election?</div>
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Maybe the times should display a rainbow map where colors indicate percentage of population with Twitter accounts, or maybe Facebook accounts. Maybe show us a colored contour map colored so that height relates to available broadband speed and color to election results.</div>
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There are many ways in which creative maps could be used to educate the public about its country. This creation may be unique, but not newsworthy.</div>
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What would be newsworthy would be for the Times to show a map that encourages some form of unity between urban and rural areas. Perhaps displaying by creative use of color the average drive time to the nearest supermarket. (Maybe the urban people might realize they have a distinct advantage here. Perhaps the Trump vote's underlying cause was accessibility of grocery stores!)</div>
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However a newspaper packages its news, let it first be sure that it is news, and not just beating a dead horse.</div>
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PS: I would have written this as a comment on the Times web site, but the article had no such option.</div>
Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-78185126679017779032016-09-22T14:05:00.001-04:002016-09-22T14:05:13.484-04:00The dream continues<span style="background-color: white; font-family: Verdana; font-size: x-small; line-height: 19.2px;">Martin Luther King said</span><br />
<blockquote class="tr_bq">
<span style="background-color: white; font-family: Verdana; font-size: x-small; line-height: 19.2px;">I have a dream that my four little children will one day live in a nation where they will not be judged by the color of their skin but by the content of their character.</span></blockquote>
It is time for all people to begin to judge all people by the content of their characters.<br />
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A small step might be for people to take the oath that for the rest of the day they will talk only about individuals, and make no references to any groups to which those individuals might belong.<br />
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Talk about a baseball player's talents without referring to the team they play for.<br />
Talk about a politician without referring to his/her political party.<br />
Talk about a dinner recipe without referring to its ethnic or national origin.<br />
Talk about a person in the news without referring to race or origin.<br />
Talk about a neighborhood without clumping people into categories.<br />
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Be they good, bad, or indifferent in your eyes, talk about the individual.<br />
Not about the group.<br />
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Note that I use "individual" to mean more than just a person. The individual could be a gas station, a bank, a movie, a book.<br />
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Say what you want to say about a TV role without referring to any skin color, ethnicity, or religion.<br />
Wax poetic about your neighbors, just not as a group.<br />
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To help as all do this, media could take a proactive step and try it too!<br />
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If we can do this for one day, we can then try a second. Then a third and fourth, and so on.<br />
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New habits can take over for old habits, but it takes time and effort. The time we all have, the effort we must supply.Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-48919203335376508892016-09-19T11:05:00.003-04:002016-09-19T11:05:47.843-04:00Blog ChangeOver time most of my mathematical blog postings have found their way into my other blog, so from this day forward I will reserve this site for my non-mathematical tidbits.<br />
Stay tuned!Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-9734149783841664622015-10-27T22:10:00.000-04:002015-10-27T22:11:12.504-04:00GeoGebra PagesI am gathering a group of my GeoGebra demos for a presentation at AMTNYS next month.
<iframe height="631px" scrolling="yes" src="http://www.angelfire.com/ny3/golfgolf/GeoGebra/Geogebrapages.htm" style="border: 0px;" width="590px"> </iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-4476240932109885852015-10-19T11:09:00.001-04:002015-10-19T11:09:16.589-04:00A question for pre-calc studentsThis question is easily stated, but is it easily answered?<br />
Stay tuned!!<br />
<iframe height="660px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/1849953/width/590/height/660/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" style="border: 0px;" width="590px"> </iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-26755780421900973042015-10-05T17:33:00.000-04:002015-10-05T17:34:32.794-04:00GeoGebra or Desmos?Here is my 3-leafed rose plot in Desmos. Below you will find the same plot in Geogebra. The GeoGebra version, to me, can be presented to students earlier in their math program.
<iframe height="600" src="https://www.desmos.com/calculator/xqfoak6xjg" width="100%"></iframe><br />
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<iframe height="700px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/1762703/width/600/height/700/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" style="border: 0px;" width="600px"> </iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com1tag:blogger.com,1999:blog-5441232287630755943.post-27452924795917666872015-10-01T16:03:00.002-04:002015-10-02T17:18:37.951-04:00Use truth values wisely!<div style="text-align: justify;">
This blog post is in reaction to S<a href="http://mathleticism.net/?p=1360" target="_blank">helby Aaberg's post</a> at <a href="http://mathleticism.net/" target="_blank">Matheleticism </a>regarding defining piecewise functions, Read that posting first! I submit a simple solution that eliminates the complex solution supplied.</div>
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All we must do is take advantage of the fact that the truth value "false" yields 0, whereas "true" yields 1, and dividing by zero is undefined. Thus multiplying by the truth value equates to multiplying by either zero or one, and dividing by the truth value goes right along with that. So dividing by the truth value is the same as dividing by either 1 or 0. </div>
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The same works at Desmos, as shown in the second graphic below.</div>
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f<iframe height="600" src="https://www.desmos.com/calculator/oklbp62weh" width="100%"></iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-42016453925409780542015-09-15T13:41:00.000-04:002015-09-15T13:41:07.158-04:00GeoGebra is Power!!!Step-by-step go from a square to a regular pentagon!<br />
<iframe height="610px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/1618069/width/590/height/610/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" style="border: 0px;" width="590px"> </iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-57744640955208653542015-09-03T12:19:00.001-04:002015-09-03T12:19:26.838-04:00Take it the other way!Here is a discovery I made while doodling around with GeoGebra. I searched around and could find nothing about it. Perhaps it's new? Who knows. It was new to me!!!<br />
<iframe height="580px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/1558431/width/580/height/580/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" style="border: 0px;" width="580px"> </iframe>
Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-37957073675019044712015-08-21T12:50:00.000-04:002015-08-21T12:51:04.037-04:00If only graph paper were elastic...One of the troubles trig students sometimes have deals with the amplitude and frequency of a trig graph. One approach I stressed was to recognize that the basic shape of the sine curve, for example, never changed. Just its position.<br />
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This is more evident when we look at the sine curves on an elastic graph paper.<br />
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In this example, drag the coefficient sliders to new values. You will see that the graph paper stretches or shrinks in each direction so that the graph stays put.<br />
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Neat, isn't it?</div>
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Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-63169844503205173032015-04-21T12:00:00.000-04:002015-04-21T12:00:56.271-04:00Students need time to exploreGreat mathematical discoveries can be made by students when they have the time to freely explore and "play around" in the world of mathematics. This is just a little example of how perceived complexities can have an underlying common theme. Have fun!<br />
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<iframe height="580px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/1054277/width/580/height/580/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/false/at/auto" style="border: 0px;" width="580px"> </iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-49295976495253717012015-04-01T14:50:00.002-04:002015-04-01T14:50:28.106-04:00Ellipse are simple!!Are you aware that an ellipse is actually a simple object? Take two circles, and have one point on each orbit as below, and just trace the midpoint between them.<br />
<iframe height="558px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/90684/width/590/height/558/border/888888/rc/false/ai/false/sdz/true/smb/false/stb/false/stbh/true/ld/false/sri/true/at/auto" style="border: 0px;" width="590px"> </iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0tag:blogger.com,1999:blog-5441232287630755943.post-10605828500683823832015-03-31T11:59:00.001-04:002015-03-31T12:03:24.579-04:00Inscribed angle is half of its central angle!!You can believe it! Just make sure you know that this is not a proof!! You can drag the points wherever you want before you start. Made in GeoGebra!!! Find my complete file <a href="http://tube.geogebra.org/student/m942113" target="_blank">here</a>.<br />
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<iframe height="500px" scrolling="no" src="https://tube.geogebra.org/material/iframe/id/942113/width/550/height/500/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/true/ld/false/sri/false/at/auto" style="border: 0px;" width="550px"> </iframe>Anonymoushttp://www.blogger.com/profile/12422213459156870716noreply@blogger.com0