tag:blogger.com,1999:blog-27503558848165562272024-03-13T06:03:59.809-05:00Long tails of \int_e^r estCalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.comBlogger52125tag:blogger.com,1999:blog-2750355884816556227.post-35055758844221677412015-08-20T14:28:00.000-05:002015-08-20T14:33:33.885-05:00Counting on Day 1I usually like to take the first day of a course to talk about some interesting and simple math that students might not get elsewhere. In the past, I've talked about why we write numbers the way we do, interesting facts about large numbers, and a basic overview of math subdivisions (topology, analysis, algebra, statistics). This year I was inspired by a <a href="http://mr-stadel.blogspot.com/2015/08/counting-dots.html" target="_blank">blog post</a> from Andrew Stadel about counting dots.<br />
<br />
The link to my slideshow is <a href="https://docs.google.com/presentation/d/1I9yjSk-gRezOry17YJ5y29tvF1u9f7-tbmFX51VtP0o/edit?usp=sharing" target="_blank">here</a>. Feel free to use as is or save a copy and modify to your own needs.<br />
<br />
I begin with having them do a math meditation that I learned from <a href="https://twitter.com/gelada" target="_blank">Dr. Edmund Harriss</a> at Twitter Math Camp 2014. I ask the students to do a visualization exercise (close your eyes if you need to, or stare of into space if that helps). Begin with a blank canvas -- it could be any color -- then we are going to add a certain number of dots to the scene. I then call out the numbers 1 through 5 (maybe a number every 10 - 20 seconds). I pause at five and tell them that this is about where we run into the limits of being able to put dots anywhere. They might think about starting to organize the dots as we progress. Then I continue on in the same manner until about 15 to 20.<br />
<br />
<table align="center" cellpadding="0" cellspacing="0" class="tr-caption-container" style="margin-left: auto; margin-right: auto; text-align: center;"><tbody>
<tr><td style="text-align: center;"><a href="http://www.printable-flash-cards.com/wp-content/uploads/2008/12/numbers-dot-preview.gif" imageanchor="1" style="margin-left: auto; margin-right: auto;"><img border="0" src="http://www.printable-flash-cards.com/wp-content/uploads/2008/12/numbers-dot-preview.gif" /></a></td></tr>
<tr><td class="tr-caption" style="text-align: center;">http://www.printable-flash-cards.com/wp-content/uploads/2008/12/numbers-dot-preview.gif</td></tr>
</tbody></table>
<br />
<br />
When I stop, I ask students to describe to me the picture in their head. It might be a 4 by 5 rectangle. It might be two 2 by 5 rectangles. It might be two groups of 3 by 3 squares with 2 dots to the side. One person from Calculus said, "Three dice with the six side up and one with the two." (D&D player anyone?)<br />
<br />
I then proceed with the sideshow and tell them I want them to try to figure out how many dots are on each of the next slides, but they will only appear for 0.6 seconds. We do a few of these and they love the challenge of trying to get it quickly. I'll explain what they are doing is called "subitizing" and how many animals (including humans) have the ability to do it with small numbers, but only up to a point. Imagine if 97 dots flashed up there. Could you count them in half a second?<br />
<br />
I also like to stop after a couple of them and ask them how they figure it out. I don't know much about the subitizing literature, but my guess is that with small numbers of dots you can make shapes with them easily (three dots make a triangle or line, 4 into some kind of quadrilateral or rows, etc.) then count after the image disappears if necessary. Making a decagon is quite a bit harder, though. Even with seven dots, some students will say they saw a row of 3 and a group of 4 (like we chunk phone numbers).<br />
<br />
I then show the <a href="https://youtu.be/Pxb5lSPLy9c" target="_blank">Vsauce video</a> which talks generally about numbers (including logarithmic number lines as well as subitizing). We discuss any questions, comments, or other thoughts they may have after viewing and then go back to trying out our counting techniques.<br />
<br />
<div class="separator" style="clear: both; text-align: center;">
<iframe allowfullscreen="" class="YOUTUBE-iframe-video" data-thumbnail-src="https://i.ytimg.com/vi/Pxb5lSPLy9c/0.jpg" frameborder="0" height="266" src="https://www.youtube.com/embed/Pxb5lSPLy9c?feature=player_embedded" width="320"></iframe></div>
<br />
<br />
In the next few slides, the dots are organized into easier to think about shapes. Even if they don't get the number correct after it flashes on the screen, we talk about how they might find the number of dots without counting each and every one. The students sub-organize the shapes into smaller shapes (eg this section was a 3 by 3 square, then one more to the side) and it's interesting to hear the different ways they group them. The switch to blue squares is not too different.<br />
<br />
Once we do a few slides of blue-square counting, I ask them what percentage of the whole shape is made from blue squares. In the slides I made, I have it animate to move the squares around and see the same amount in a different way. By moving a blue square to a white space in an organized way, it's easier to determine the percentage.<br />
<br />
To finish, I tell the students I wanted to show them this stuff to highlight some interesting psychology behind our number sense, but also to motivate the need for math as a subject. <br />
<br />
By organizing and moving things around in a meaningful way, it is easier to interpret what is quantitatively in front of us. Especially with my algebra students, I tell them that our notation and algebraic manipulation will cause some growing pains, but once we get some techniques under our belt (like we did with chunking shapes into smaller pieces or moving blue squares to fill white holes), it will make the process easier.CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com0tag:blogger.com,1999:blog-2750355884816556227.post-76378426614217679362015-03-10T09:45:00.002-05:002015-03-10T09:45:36.699-05:00Trusting in An AnswerOne thing I've been noticing this year as I find "challenge" problems online or give sets to my own students is that the first thing I acknowledge is the trust.<br />
<br />
Often, we ask students to "Prove that..." or "Solve" in problems. All well and good in a classroom setting, but in a more "real-world" situation, the first thing that needs to be assessed is whether that problem CAN be solved. I think my own problem-solving strategies are sometimes directed by knowing that there is an answer before I even begin the work. <br />
<br />
I trust that my teacher or problem-giver is giving me the opportunity to solve the problem when it is posed. That is, if I am not allowed a calculator, I trust that it CAN be solved without a calculator and that my answer will most likely be a whole number or at least a simple fraction. If I am in a 6th grade classroom then my problem will be "6th grade appropriate" (ie should be solved without calculus or other advanced methods). <br />
<br />
Of course, this is not the case in higher mathematics where the main question can often be, "Is this solvable?" Certainly less so in every day life when we are faced with innumerable unsolvable problems. I'm not suggesting we give students unsolvable problems, but it might be interesting to mention this to students that we are in an artificial, safe environment in this classroom where these problems have already been solved and I (the teacher) am giving them to you knowing you <b>should</b> be able to complete them.CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com3tag:blogger.com,1999:blog-2750355884816556227.post-65641654729623625592014-10-24T18:56:00.000-05:002014-10-24T18:56:17.948-05:00My Favorite Unit Circle QuestionIn Precal, I love the development of the unit circle. It brings together so many ideas and concepts they've learned from algebra 1 and geometry and makes sometimes unexpected connections.<br />
<br />
Two years ago I developed this question that I think really helps students see how the circle comes together. It originally stemmed from students memorizing THE Unit Circle with it's 16 (or 17 if you count 0 and 2pi differently) special angles and coordinate pairs (although they often mixed up the coordinates since the "patterns" weren't quite what they thought). Later they would begin to believe that the circle only held those 16 angles and other angles just popped out of the calculator.<br />
<br />
So, the question goes like this:<br />
<ol>
<li>Draw a right triangle in Geogebra. Use it to measure the lengths of all three sides and the angles.</li>
<ol style="list-style-type: lower-alpha;">
<li>Using the ideas from similar triangles, shrink or expand your triangle so that it will fit within the unit circle. [This means to divide each side by the hypotenuse, but we'll have gone over the idea of similar triangles and ratios before a quiz.]</li>
<li>This triangle can be flipped and flopped to fit into the unit circle in 8 different ways. Draw a picture with all 8 triangle orientations. [It looks like a double-winged butterfly with all right angles against the x-axis and hypotenuses radiating (pun intended) from the origin.] </li>
<li>Find all associated 8 angles and their coordinate pairs based on your triangle.</li>
</ol>
</ol>
<div>
For example:</div>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-zXlnOkHS1mI/VErlgxdwOEI/AAAAAAAADQ8/N-g9mfsh4EY/s1600/triangle1.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://3.bp.blogspot.com/-zXlnOkHS1mI/VErlgxdwOEI/AAAAAAAADQ8/N-g9mfsh4EY/s1600/triangle1.png" /></a></div>
<div>
<br /></div>
This would correspond to the angles:<br />
<ol>
<li>40º (0.766, 0.643)</li>
<li>50º (0.643, 0.766)</li>
<li>130º (-0.643, 0.766) [Got this from 180º - 50º]</li>
<li>140º (-0.766, 0.643)</li>
<li>220º (-0.766, -0.643) [180º + 40º]</li>
<li>230º (-0.643, -0.766)</li>
<li>310º (0.643, -0.766) [360º - 50º]</li>
<li>320º (0.766, -0.643)</li>
</ol>
<div>
I think this question really helps students see the utility of the unit circle. And how we could use it to figure out any of the 6 trig functions for any angle if we were precise enough with a compass, protractor, and a ruler.</div>
<br />
<br />CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com1tag:blogger.com,1999:blog-2750355884816556227.post-27437649173265803652013-08-21T15:39:00.000-05:002013-08-21T15:39:26.348-05:00My SBG LetterThis may or may not be of any interest to you, but this is what I hand out to my students along with their syllabus to discuss my version of Standards Based Grading.<br />
<br />
<h1 align="center" style="text-align: center;">
Standards Based Assessment<o:p></o:p></h1>
<h2>
Philosophy<o:p></o:p></h2>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="mso-tab-count: 1;"> </span>This class will be graded on a
system known as “standards based assessment.”<span style="mso-spacerun: yes;">
</span>Grades will be assigned for content knowledge based on understanding of
the main standards needed in the course.<span style="mso-spacerun: yes;">
</span>Thus, the grade for the class will accurately reflect the student’s
understanding of the material from the class.<o:p></o:p></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="mso-tab-count: 1;"> </span>This system has many positives for
the student, teacher, and the student’s family.<span style="mso-spacerun: yes;">
</span>Grades will be posted based on the concepts.<span style="mso-spacerun: yes;"> </span>Seeing a 74% on the Chapter 4 test in the
gradebook is not informative to any of the participating parties.<span style="mso-spacerun: yes;"> </span>The student may have a wonderful grasp of
three quarters of the material while missing only a single key concept or he
could barely have a working knowledge of the entire chapter’s material.<span style="mso-spacerun: yes;"> </span>On the other hand, an 80% on Trig Graphs is
much more informative to all the stakeholders to show where the student may
need to improve his understanding.<o:p></o:p></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="mso-tab-count: 1;"> </span>The main goal of the class is to
teach students the mathematical material for the course. <span style="mso-spacerun: yes;"> </span>Thus, a student’s grade should reflect his
knowledge and understanding of that material without being clouded by
completion or participation grades, extra credit for material not related to
concepts from class, or behavioral issues.<span style="mso-spacerun: yes;">
</span>Any behavioral issues, nonparticipation, or incomplete homework will
certainly be addressed, but will not be directly reflected in the grade for the
class.<o:p></o:p></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="mso-tab-count: 1;"> </span>Since mathematics often requires a
good understanding of a topic before being able to work well with the following
topic and because I wish for all students to understand all of the topics from
class, I will give students the opportunity to retake assessments over concepts
covered in class.<span style="mso-spacerun: yes;"> </span>No graded assignment
will be dismissed since every assignment is given for a purpose.<span style="mso-spacerun: yes;"> </span>Retaking an assessment can help students more
accurately show their understanding of the material and helps students to learn
the material before getting too far ahead in the curriculum.<span style="mso-spacerun: yes;"> </span>If a student has a bad day or does not fully
understand the topic on the day of the quiz, he will have the opportunity to
show his improved understanding at a later time.<span style="mso-spacerun: yes;"> </span>I care less about when the material is fully
grasped than the idea that it actually is understood at some point.<span style="mso-spacerun: yes;"> </span>That being said, grades and understanding of
concepts can be time sensitive, so students will need to complete all retakes
in a timely manner.<o:p></o:p></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<span style="mso-tab-count: 1;"> </span>The new grade will completely
replace the original grade.<span style="mso-spacerun: yes;"> </span>Thus,
students should never give up on the class, no matter how low his grade
gets.<span style="mso-spacerun: yes;"> </span>As long as he is keeping up with
the material in class and working on reassessing the material for which his
understanding has improved, an F in the class can quickly become an A as zeroes
are replaced with better scores.<span style="mso-spacerun: yes;"> </span>The
grade is constantly in flux and should be viewed as such.<span style="mso-spacerun: yes;"> </span>Until the final grading period, parents
should concentrate on the individual grades to see what their child should work
to improve rather than the overall grade for the course.<o:p></o:p></div>
<div class="MsoNormal" style="margin-bottom: .0001pt; margin-bottom: 0in;">
<br /></div>
<h2>
Policies<o:p></o:p></h2>
<div class="MsoListParagraphCxSpFirst" style="mso-list: l0 level1 lfo1; text-indent: -.25in;">
</div>
<ul>
<li><span style="text-indent: -0.25in;">Students may retake only ONE section per day.</span></li>
<li><span style="text-indent: -0.25in;">Students must retake the entire section (even if
it is more than one question)</span></li>
<li><span style="text-indent: -0.25in;">Students may retake each section two times (in
addition to the original)</span></li>
<li><span style="text-indent: -0.25in;">To have the retake ready for the following day, students
must sign up before 8pm (according to Google’s clock).</span></li>
<li><span style="text-indent: -0.25in;">Students should be able to show proof of work
done to improve understanding since the original assessment to merit a retake</span></li>
<li><span style="text-indent: -0.25in;">Students may take it in during encore or before
or after school.</span></li>
<li><span style="text-indent: -0.25in;">The new grade replaces the old grade entirely
(whether it’s better or worse) to reflect the current understanding.</span></li>
<li><span style="text-indent: -0.25in;">Retake questions will assess the same concept,
but may be drastically different in format or more difficult than the original.</span><span style="text-indent: -0.25in;"> </span><span style="text-indent: -0.25in;">Students should be prepared to demonstrate
mastery of the topic, not just hope for an easier question.</span></li>
<li><span style="text-indent: -0.25in;">Students must take all the retakes for this semester
before the date to be announced.</span></li>
</ul>
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<h2>
Procedure<o:p></o:p></h2>
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The procedure for signing up for retakes will be done
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CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com1tag:blogger.com,1999:blog-2750355884816556227.post-30841416983208804762013-08-01T17:16:00.000-05:002013-08-01T17:16:16.639-05:00Our Little Baby All Grows UpI lived for 11 years in Music City, Nashville, Tennessee. I found myself putting music in a bunch of holes I found in my life (after breaking up a 5 year relationship, after leaving school, etc.). Being there, when I was, I found a number of outlets to consume local, indie music. In the early to mid 2000s, enjoying indie music was even cool, so there was the weird thing of being part of a popular movement whose whole point was to enjoy things that were not mainstream.<br />
<br />
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<br />
I feel like this is where we are with our online community of math teachers, affectionately known as the <a href="http://mathtwitterblogosphere.weebly.com/" target="_blank">mathtwitterblogosphere</a> (or MTBos). We're the rising indie band who is a couple steps away from having Sony call to offer us a contract. It's a little bit of a scary place.<br />
<br />
Many of the things that define this community are directly related to the smallish size and "grass-roots" popularity that we enjoy. Will we "sell out" if our little group grows too big? <br />
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<br />
<br />
On the one hand, we want our ideas and the group itself to be open to anyone and everyone who feels moved to join. After all, what we do is for the kids.<br />
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On the other hand, it feels like we will lose a lot of the things I love about the group if it grows too big. <br />
<br />
I was trying to tell my wife about my confliction with my vision of the future of this group. Sarah asked if I wanted TMC to just be a little get together of my friends. It hurt when she said it, because it sounds bad and exclusive, but I'll admit that I selfishly kind of do want that. I have found some people that I would consider friends for life through this community and I am continuing to find more here and there, so I guess I shouldn't be worried about that part. Whatever happens to the community at large, I can hope that people I really want/need to interact with will still be there for/with me.<br />
<br />
<a href="http://samjshah.com/2013/08/01/tmc13-the-state-of-things-for-me/" target="_blank">Sam</a> posted about the close-knit community we have. How can we maintain that while growing much larger? Will we split into groups (#geomchat, #algchat, #statschat, Northeast, West, Central, early adopters, newcomers, etc.)? What happens when we're bigger than physical space will allow us to get together during the summer? Would corporate sponsors at meetings help or hurt?<br />
<br />
I guess what I'm saying is that change is hard.CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com4tag:blogger.com,1999:blog-2750355884816556227.post-86094794995983850492013-05-16T13:41:00.001-05:002013-05-16T13:41:27.315-05:00Phrases that Annoy MeThere are a couple phrases that students use that really annoy me in lessons.<br />
<br />
"So, is it just...?"<br />
<br />
and<br />
<br />
"So, do you always...?"<br />
<br />
The main reasons they annoy me is because I can tell students are just looking for an algorithm rather than learning the background reasoning. They want a formula they can plug into. They don't want to think about it.<br />
<br />
I mean, I get it if they're looking for a generalization and if they truly understand what is happening and are putting the pieces together to make a formula, then I'm all for it. <br />
<br />
The worst thing about the phrases, though, is that they shut out learning. I will try to turn the questions back around and ask them, "You tell me. Is it always...?" Then they get frustrated that I didn't answer their "simple question" and solidify their, "Well, I guess I'm just going to fail this quiz" mentality. If I answer, even if I follow it up with, "Yes, but here's WHY it works (or doesn't)," they shut off their brains after I say, "Yes." CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com6tag:blogger.com,1999:blog-2750355884816556227.post-68702498861845981192013-05-15T19:38:00.003-05:002013-05-15T19:38:59.285-05:00Perks of SBGThis deserves a longer post, but <a href="http://shawncornally.com/wordpress/?p=3763" target="_blank">this post</a> made me think I should post now.<br />
<br />
There are at least 2 times where Standards Based Grading saves my butt with no real work.<br />
<br />
1. Talking to parents. So much responsibility goes to the students for this type of grading. They have no reason to fail. Help is always available There are opportunities out the Wazoo for them to succeed. So parents who try to point a finger have to just turn back around and look at their own kid.<br />
<br />
2. Final exams. As a teacher I get so sick of students talking about how in their chem class they can turn in a blank exam and still get a B. They take the last 2 weeks of school to just be bodies in seats and not work at anything. With SBG, the kids have to prove their knowledge on this final assessment and their grade can jump up or die fast with their performance on it. So, even my A+ students who might be able to get a decent grade with half the exam blank in a traditional system know that they have to still put in the effort to show me that they understand the material and deserve the grade they are shooting for.CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com3tag:blogger.com,1999:blog-2750355884816556227.post-21506902140538103802013-04-29T20:05:00.001-05:002013-04-29T20:05:50.402-05:00Precal ProblemsI'm having fun writing some problems for my precal exam and review, so I thought I'd share some of what I've got so far. Feel free to use. Where necessary, numbers were researched and are accurate:<br />
<br />
1. (Law of Sines/Cosines) You Quantum Leap into the body of an artist who works on building furniture from reclaimed wood. Ziggy and Al agree that you have a 98% change of moving on if you create a triangular table top from the three pieces of wood that remain from your wife's crib. The pieces measure 3 feet, 2.5 feet, and 4 feet. Without cutting any of the pieces, at what angles should you attach the ends of the pieces to one another to create this amazing table? (Find all three angles and pray that this one will be the leap home!)<br />
<br />
2. (Right Triangle Trig) You are marching your army towards "The Wall" away from the White Walkers towards the 7 Kingdoms. You know this wall is 700 feet tall, but need to figure out how far away you are to know how much food you need to take from Craster's stores. One of your scouts runs ahead 4 miles and from there sights the top of the wall at 1º from his position. How far is your camp from the Wall? (Answer in miles. 5280 feet = 1 mile).<br />
<br />
3. (Right Triangle Trig) It's 1895 and you are emigrating to the United States! To prove your worth to your new country so they will let you in, you decide to do some quick calculations while on the boat ride to Ellis Island. From some distance away, you initially sight the top of the Statue of Liberty at an angle of elevation of 3º. Based on the ship's speed you calculate that you travel 1460 feet closer right when the angle of elevation becomes 4º. To the nearest foot, how tall is the Statue of Liberty? Your future citizenship may depend on it!<br />
<br />
4. (Angular Motion) The Curiosity Mars rover has a weird pattern on its wheels that includes the morse code for ``JPL'' (Jet Propulsion Lab, the arm of NASA driving the rover). This helps the scientists figure out how far the rover has traveled by seeing the pattern in the tracks left behind while driving. The wheels have a 50 cm diameter. How many copies of JPL are left in the ground when the rover has traveled 5km? If it takes the rover 3 days to travel this distance, what is its average speed in kilometers per hour?<br />
<br />
5. (Angular Motion) A "45" was a vinyl record with a single on each side. It was called that because it would spin at 45 revolutions per minute. The standard size is a 7 inch diameter. <br />
<span class="Apple-tab-span" style="white-space: pre;"> </span>(a.) A fat ant wants to slim down his exoskeleton by running along the edge of the outer edge of the record while it plays the entirety of The Monkees' ``I'm a Believer'' (2 minutes 47 seconds long). How far will it have run by the end of the song?<br />
<span class="Apple-tab-span" style="white-space: pre;"> </span>(b.) The ant's friend wants to keep her company, but doesn't really need as rigorous of a workout. So, she's going to run around the hole which only has a 1.5 inch diameter. By the end of the same song, how far will this ant have run?<br />
<br />
6. (Angular Motion) Your car came with 17 inch diameter wheels. <br />
<span class="Apple-tab-span" style="white-space: pre;"> </span>(a.) How many times will a wheel rotate in one mile? (1 mile = 5280 feet. 1 foot = 12 inches.)<br />
<span class="Apple-tab-span" style="white-space: pre;"> </span>(b.) You want to pimp your ride up to 22 inch rims. How far will you have actually gone in the new rims when they rotate the same amount as the answer from part (a)? (Answer in miles.)<br />
<span class="Apple-tab-span" style="white-space: pre;"> </span>(c.) If you don't recalibrate the sensors in your car, your spedometer may say you are going 60 mph on the interstate, but how fast are you actually going in the new wheels? (Does your answer make sense? Should it be faster or slower than the car says? This has happened to me!)<br />
<br />
7. (Right Triangle Trig) After the Mars Curiosity rover landed, it needed to assess where it was in relation to one of its goals, Aeolis Mons (aka Mount Sharp). NASA knows that the mountain is supposed to be 18,000 feet tall (about the same size as Mount McKinley, the tallest mountain in North America). Initial pictures show that the angle to the top of the mountain from Curiosity's current position was 18.71º. What is the horizontal distance from the rover to the mountain (round to the tens place)?<br />
<br />
8. (Law of Sines/Cosines) You wake up in a daze in the middle of a forest. You've been kidnapped! Fortunately, your captors left your school bag with you and (oddly) a map. From the map you can tell that if you walk at a bearing of 53º, you'll find your way to civilization. In your bag you also have a ruler and calculator. You find two straight sticks and measure them to be 8 inches and 5 inches. If you put two ends of the sticks together, you should be able to form a 53º angle and find your way. How far apart (in inches to three decimal places) should the other ends of the sticks be to create a 53º?CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com2tag:blogger.com,1999:blog-2750355884816556227.post-41832569033495207372012-12-10T18:40:00.000-06:002012-12-10T18:40:20.084-06:00Hobbit RatiosNothing amazing with this little project, but it's something to get started and you guys can run with it.
<br />
<br />
In geometry we're talking about ratios of similar shapes. Comparing a 2x2 square with a 6x6 square, we see that the side lengths triple, but the area increases 9-fold. I had them discover this by doing a number of different shapes and finding their areas then comparing. Similarly, for volume, the shape whose sides have ratio N has a volume that is N^3 times as much.<br />
<br />
Anyhow, with The Hobbit movie coming out this week, I thought I'd tap into that excitement a bit and have them do a little map reading. I found this map online: <a href="http://mearth.lords-of-blah.nl/">http://mearth.lords-of-blah.nl/</a> It's huge, but printable. There are other online maps, but not with a scale and with different resolution screens or whatever, I thought I'd give them all a print out so we can be a little consistent.<br />
<br />
Then I typed up these <a href="https://docs.google.com/document/d/1-ZzMJGgyG0aJ2yr0Cczc5_SdZTjUhkAkn1lR1o-7aZg/edit" target="_blank">instructions</a> on what to do with the map. I couldn't think of how to incorporate volume into the instructions at the time, but now I think I might include making a scale model to the same proportions of the map. Maybe how wide and tall a mini-Isengard should be made to fit on the map or something.<br />
<br />
Anyways, feel free to steal and/or enhance this lesson for yourself.CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com1tag:blogger.com,1999:blog-2750355884816556227.post-65959276714530664052012-11-15T21:22:00.003-06:002012-11-16T12:39:55.667-06:00A Day in the Life... Nov 15, 2012<div class="separator" style="clear: both; text-align: center;">
<a href="http://2.bp.blogspot.com/-5mN9xUoL1F0/UKV_gc3m0II/AAAAAAAADBo/QHGlEzx6u6Q/s1600/a-day-in-the-life-2.png" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="112" src="http://2.bp.blogspot.com/-5mN9xUoL1F0/UKV_gc3m0II/AAAAAAAADBo/QHGlEzx6u6Q/s320/a-day-in-the-life-2.png" width="320" /></a></div>
<a href="http://samjshah.com/2012/10/30/a-day-in-the-life-math-teacher-2012-edition/" style="text-decoration:underline;">Sam</a> suggested that many may not know what the life of a math educator is like. Others may know the generalities, but even other math teachers can find comfort in knowing the details of another person's day. It can be nice to know that you don't have it as bad or that we're all in the same boat. Anyhow, here's how my Thursday, November 15, 2012 went.<br />
<br />
Last night, I went to bed at 10pm. I go to bed every night at 10pm. I keep a very strict bedtime on school nights. When the clock ticks to 10, I drop what I'm doing and head to bed.<br />
<br />
I got up at 3:45 this morning. I usually get up around 4, but I've been feeling a little under the weather lately, so maybe my internal clock is being weird. I make a quick trip to the bathroom (normally I'd leave that detail out, but you'll see how it comes back later). Breakfast is important to me, so I had some cereal and juice as well as a chewable vitamin and head to the living room.<br />
<br />
I click on the TV for the early AM news where the banter almost makes me want to turn it back off, but I guess they don't care much this early in the morning. I just want to see if traffic is ok (St Louis has lots of bridges closing and other work being done all the time) and what kind of weather we'll have today. They just want to tell me about all the Christmas sales going on (it's the Ides of November for crying out loud!) and what in my house could be poisoning my children (I have none).<br />
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I open my computer and check my RSS feed for any actual news that is important. I have about a <a href="http://www.qwantz.com/index.php" style="text-decoration:underline;">dozen</a> <a href="http://www.sinfest.net/" style="text-decoration:underline;">webcomics</a> <a href="http://wondermark.com/" style="text-decoration:underline;">that</a> <a href="http://www.smbc-comics.com/" style="text-decoration:underline;">update</a> <a href="http://www.asofterworld.com/" style="text-decoration:underline;">often</a>, any <a href="http://www.futilitycloset.com/" style="text-decoration:underline;">math</a> <a href="http://fuckyeahmath.tumblr.com/" style="text-decoration:underline;">blogs</a> that may be <a href="http://what-if.xkcd.com/" style="text-decoration:underline;">interesting</a>, a quick scan of <a href="http://www.ohgizmo.com/" style="text-decoration:underline;">tech</a>/<a href="http://www.wired.com/" style="text-decoration:underline;">science</a> blogs, and a visit to youtube to see if anything interesting has come up that I can <a href="http://www.youtube.com/watch?v=wS7CZIJVxFY&feature=share&list=PL0902DFCE3065270D" style="text-decoration:underline;">use in class</a> or just for <a href="http://www.youtube.com/user/dailygrace" style="text-decoration:underline;">some</a> <a href="http://www.youtube.com/user/LivePrudeGirls" style="text-decoration:underline;">morning</a> <a href="http://www.youtube.com/user/MyHarto" style="text-decoration:underline;">laughs</a>. Overall, this takes about an hour to catch up.<br />
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So, now it's 4:45. I pull out a blanket and take a morning nap on the couch. My actual alarm (first of the day) wakes me up at 5:45 from a dream I was having where my students were all failing and my inbox was full from parents being upset with me--basically full of anxiety.<br />
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But, I'm awake now, so time to make sure that doesn't happen in reality. Jump in the shower and shave, then to sneak back into the bedroom (so as not to wake my wife) and turn pull the dimmer switch up enough just enough to keep myself from making a horrible fashion choice for the day. I think I made a bad choice with the stripes-on-stripes and the gray + silver, but whatever -- I teach at an all-boys school and I don't think anybody will notice.<br />
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<tr><td class="tr-caption" style="text-align: center;">Selfie Thursday</td></tr>
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As I do every day, I grab my bag, feed the fish, and touch my wife on the shoulder to wish her a good day, she mumbles back the same wish for me, a quick "I love you" and I'm out the door around 6:25am.</div>
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It's a chilly day (my computer said 26, but the car said 34), but I have Wisconsin blood in me, so it doesn't bother me. I had my phone ready because I knew I'd be doing this today, so I snapped this without looking while driving over a bridge:</div>
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I'm the first one in the teacher's parking lot at 6:45, but I like to get there early to set up for class and be ready for any students that come by before school. Whenever they ask, I tell them I get there around 7:00 and can help them with any questions they have then. With a test in both courses tomorrow, it was a real possibility some would show up this morning.<br />
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Swing by the faculty lounge to check my mailbox on my way to class. There are two tickets I requested for our fall play "Count Dracula" that my wife and I will attend this weekend. For me, it's about 50% actually wanting to see the production and 50% supporting my students.<br />
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<tr><td class="tr-caption" style="text-align: center;">Fancy tickets may be a sign of good things to come?</td></tr>
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I teach at a Catholic school, so I pass by this interesting piece every morning. It's made by one of the brothers who lives on campus. I guess he's somewhat famous since I'd heard of <a href="http://www.melsmart.com/main.php?g2_view=html:HtmlPage&page=ABOUT%20THE%20ARTIST" style="text-decoration:underline;">Brother Mel</a> before I got here.<br />
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<a href="http://1.bp.blogspot.com/-F1dH5LC9KkA/UKV-xE-1KrI/AAAAAAAAC-A/e6GZ9h6H_IA/s1600/IMG_0271.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://1.bp.blogspot.com/-F1dH5LC9KkA/UKV-xE-1KrI/AAAAAAAAC-A/e6GZ9h6H_IA/s320/IMG_0271.JPG" width="240" /></a></div>
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Here are some pictures of my classroom:</div>
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The box under the TV is supposed to let me wirelessly project from my computer, but something's messed up and it won't stay on for more than 10 minutes at a time, so I don't use it. The posters on the left are "Parent Functions" and the unit circle is on the right next to my math clock. Desks are in rows, but they won't stay that way throughout the day.<br />
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<a href="http://1.bp.blogspot.com/-GmpLlp-5Y0o/UKV-1KMGY8I/AAAAAAAAC-Q/H3JXyAebDjg/s1600/IMG_0273.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://1.bp.blogspot.com/-GmpLlp-5Y0o/UKV-1KMGY8I/AAAAAAAAC-Q/H3JXyAebDjg/s320/IMG_0273.JPG" width="320" /></a></div>
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My desk faces the wall, because I never use it. These posters are of our daily prayers.</div>
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<a href="http://4.bp.blogspot.com/-0HZNsdY8bnY/UKV-5lhki3I/AAAAAAAAC-k/Y7CpAmAsffo/s1600/IMG_0275.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://4.bp.blogspot.com/-0HZNsdY8bnY/UKV-5lhki3I/AAAAAAAAC-k/Y7CpAmAsffo/s320/IMG_0275.JPG" width="320" /></a></div>
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From the back of the classroom.</div>
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<a href="http://4.bp.blogspot.com/-ugYCLp33NTA/UKV-9Y0A7MI/AAAAAAAAC-0/rZEFCPjTcCo/s1600/IMG_0277.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://4.bp.blogspot.com/-ugYCLp33NTA/UKV-9Y0A7MI/AAAAAAAAC-0/rZEFCPjTcCo/s320/IMG_0277.JPG" width="240" /></a></div>
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The desk I do use. The power outlet on the other side of the room doesn't work, so this is the only place my computer can go and be powered. Whatever, I don't use it during class except to put in attendance.</div>
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<a href="http://3.bp.blogspot.com/-EepqpF2u3CY/UKV-_FT71qI/AAAAAAAAC-8/PbJ6E4X7z9U/s1600/IMG_0278.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="240" src="http://3.bp.blogspot.com/-EepqpF2u3CY/UKV-_FT71qI/AAAAAAAAC-8/PbJ6E4X7z9U/s320/IMG_0278.JPG" width="320" /></a></div>
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A closeup of my desk. Mostly just a flat surface to hold all the papers and books I let float around. Kids will use that rolling chair sometimes, but as long as they don't fight over it, I don't care.</div>
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<a href="http://3.bp.blogspot.com/-TR_1ae4Qp00/UKV_BZLAM6I/AAAAAAAAC_M/7UHrOEdn41g/s1600/IMG_0280.JPG" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" height="320" src="http://3.bp.blogspot.com/-TR_1ae4Qp00/UKV_BZLAM6I/AAAAAAAAC_M/7UHrOEdn41g/s320/IMG_0280.JPG" width="240" /></a></div>
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I have to check the schedule for the day, because I can never remember. We're on a block schedule, so only 4 periods for the day. Today is a "Regular Day," but every Wednesday is a Late Start (for the kids, we come in at standard time for meetings) and every other week has a mass when we'll follow the Liturgy schedule. We also rotate which period starts the day and today is a "B Day" (meaning B block corresponds to first period).</div>
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One of the pieces of paper on my desk reminded me that I have to recommend math courses for my current students. This is to be turned in by Friday, but I should take care of it now while I'm thinking of it. It's already been sitting on my desk for about a week.</div>
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I don't have control over the temperature in my room and the guy next door (who has control over his room and mine) doesn't get to school until 7:30. Fortunately, my key works in his door, so I run over there to set the thermostat at 68, since it's pretty cold in my room already.</div>
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Maybe it overcompensates, though, because as soon as class starts, my geometry kids want to open windows (it's still in the 30s outside). Then they fight over how many to open and how much.</div>
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I closed the two wide open ones and left the cracked one to settle the argument.</div>
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The intercom buzzes and we begin school with a prayer from an assistant principal and some morning announcements. Most of the announcements have been sent to me in an email that I am to read to the students in a bit, but he highlights the important ones and adds anything that was not on the email.</div>
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October was a month the Church focused on right to life things and at one point we had students participate in standing at the Planned Parenthood building to talk with expectant mothers about their options. The teacher in charge of that came on the announcements as well to discuss their successes and remind the students to continue collecting their change to help fund the purchase of an ultrasound machine to allow willing women to see their child before making their choice. Some kids roll their eyes and this, but most politely listen.</div>
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Anyhow, I then read our other announcements which takes about 10 minutes to get through everything. Only about 5 of my 25 students are listening to them, but you never know when something comes up that they need to hear, so I go through them all.</div>
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I am beginning today with a geometry class. They have a test tomorrow over vectors, so today is a review day. I put up problems on the main whiteboard and they get in groups with their whiteboards and collaborate to review the problems. I'll bounce around from group to group to help with any issues they can't resolve.</div>
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<tr><td class="tr-caption" style="text-align: center;">A group that decided to use different colors and also split it up so that one person did #1, another #2, etc. </td></tr>
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<tr><td class="tr-caption" style="text-align: center;">A student using his phone for a calculator and another group that split up the problems. I tried to make sure they discussed them so they all knew how to do each one, but who knows.</td></tr>
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They did fairly well with the problems given them and seem prepared for the test tomorrow. They finished a bit early, so I let them play hangman or tic-tac-toe on their boards for the rest of the period.<br />
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On "Regular Days," we have a period of time called "Encore" after the first period. It's 30 minutes of free time for students. They can grab a snack at the cafeteria, attend a short eucharist service, make up tests, meet with teachers for tutoring, play outside, etc. If they have a D or an F in a class, though, they are assigned a room to go to so they can work on that class. I have 13 students assigned to me currently and we worked on some practice problems for their test in Trigonometry tomorrow. <br />
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Their grading term was done last Friday, so that's when the new checklist came out. We are asked to inform parents about their low grades and offer ideas about what we can do to help them climb out of the hole. SBG is a lifesaver here, so I just reiterate that point to these parents. <br />
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I always dread emailing parents. I am not a parent myself and in general am not a fan of telling adults what to do, so I always fear that they'll come back aggressively and I won't know how to respond. While that happened sometimes back in TN, this doesn't seem to be the case here. The families here are paying good money for their kids to attend this school and most are not rich, so they are sacrificing for their child's education. I'll post more about this in a separate post, but they are very supportive of the teachers and trust us to help the kids.<br />
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Nevertheless, I took until Tuesday to email the parents about their child's grades and felt guilty every day for not doing it earlier (even losing sleep over it some nights--maybe that was the cause of my morning dream?). So, some of those students actually took the initiative to come in during this free time for extra help and it made me happy. So, in the 5 minute passing period, I emailed their parents back to share the good news that we were making things happen. Most wrote back quickly, too, to say they were happy and to let them know if they could help in any way. It feels so great to have helpful parents.<br />
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So, on to the second period of the day, Trigonometry. They, too, have a test tomorrow over Trig Identities. They have severely struggled with it. Not only in the usual way of not having an algorithm to follow and needing all of their algebra skills to come together at once, but apparently their algebra 2 teacher was pretty horrible. Most students had barely heard of factoring at all, so I have needed to teach (rather than just review) that alongside the added complications of sines and cosines. They get frustrated with it, too, but we're working through it.<br />
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We worked through the three problems that I had given them for homework and I posted another 7 more for them to work through in class. For whatever reason, these juniors and seniors are not as enamored with the whiteboards as the sophomores in geometry, so they work on their own paper when they get in groups.<br />
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D block (which is third of the day today), is my planning and also the lunch period, yay! So, I should have an extra long time off to grade my stack of papers, grab some lunch, and generally take it easy. SHOULD.</div>
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Our school is on a sprawling 40 acres of land, so the actual school is all on one floor which is kind of nice except that the halls really bottleneck. So, I took an alternate route to the faculty lounge and ended up behind the assistant principal who needed the course recommendations, so I handed those off on my way.</div>
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At this private school, when a teacher doesn't come in because he is sick or takes a personal day, we have to cover for each other when possible. They inform us of this with a pink piece of paper in our mailbox. I guess I got there too early this morning to find the paper in my box earlier, so when I get to the lounge for my break, I see the paper and have to dash back off to another room to cover for a sick teacher.</div>
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The plans on his desk say to let the students discuss their answers from the previous day's question for about 15 minutes, then they should read in their books and answer the questions at the end of the chapter.</div>
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Fortunately, it's a good group of kids and they were more or less on task. I'm never sure of other teachers' norms, though, so it's awkward to be in someone else's space like this. The kids discuss the answers for the questions they are working on. Should I let that happen? Are they being graded on this stuff? Is it just busywork? Are they learning by talking about the answers or are they just copying? This is too much stress to just be watching these other kids!</div>
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They know me by reputation, though, so some asked whether it is true that I taught at Vanderbilt. They're impressed when I said that I did. Then they asked what my ACT score was and that impressed them more. I feel awkward talking about myself in a classroom, especially when they are meant to be focusing on work and even more so when they are not my own students. I know too many students who try to derail learning by asking personal questions, so I try to answer quickly and point them back to their work.</div>
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Halfway through the period, another teacher came in to replace me. I now have 15 minutes to grab lunch from the line and eat it myself. Then my own lunch duty time comes. Since it's supposed to be my free period, I have one of the four lunches to watch over to make sure no food fights happen, nobody uses a water cup to steal soda from the fountain, no more than 8 kids sit at a table, etc. None of this ever happens (we do have good guys at the school), so it's mostly baby sitting, but I have to be their Jiminy Cricket standing in the corner anyways.</div>
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After lunch, we hand out buckets of soapy water with rags to one lucky table and they get to stay after to wipe down the tables. I like that they have some ownership of the space and help clean the tables, but I feel weird choosing a table for the chore.</div>
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<tr><td class="tr-caption" style="text-align: center;">The center table in the back got it today</td></tr>
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The internet went down at school. Teachers were freaking out all over the place because they couldn't check attendance (or twitter) and even the printers are wireless, so we couldn't print. Oh well, my lessons today did not use the internet at all, so I selfishly ignored the conversation.</div>
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<tr><td class="tr-caption" style="text-align: center;">The offending server?</td></tr>
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The last class of the day is Trigonometry again and it went as smoothly as the previous class. They kids really are good at working with each other and actually making sure they understand rather than just copying in my classes -- especially on review days. The job is so much easier with cooperative students.<br />
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Thus, school ends at 2:40 with a prayer over the intercom and any afternoon announcements. </div>
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My duties don't end, though. I was chosen to be the sponsor of the chess club this year. A retired teacher is the one who actually runs the show, but they need a current faculty member to be the sponsor.<br />
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<tr><td class="tr-caption" style="text-align: center;">The guy who actually runs the club wrote this three volume tome of lessons for the kids to practice with.</td></tr>
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I guess the chess hall of fame is here in St Louis (not far from where I live) and this guy has a major role there. All to say, the school takes chess seriously (won nationals two years ago and are consistently top 10). We have about 16 members on our team and they have practice every day after school for 2 hours and tournaments on weekends every now and again. The one last weekend lasted 12 hours (6am to 6pm) on Saturday.<br />
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Today, we split the freshmen into two groups. The other guy took half and I took the other half to go over the lesson of how to finish the game when you have two bishops (and your king) and he only has his king. Then we did some lessons about how to make "double attacks" (in the format of how you might see puzzles in the paper where you set up the pieces in a certain way and then play it out from there).<br />
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After that, the kids get to play on their own for the second hour. I just supervise mostly. In the picture below, kids are playing a chess variant called "bughouse" where you have teams of two and after you capture a piece, you can give it to your neighbor to use on his board. It's fast paced (5 minute time control) and fun for them, so they like playing with it.<br />
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Meanwhile, the other guy plays 4 games at once against some kids. If they can beat him and have correct notation for it, he'll give them a 2 dollar bill. They think it's fun to have the challenge. Only the guy in the foreground came close, but he moved the wrong pawn at the end and lost.</div>
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So, now it's 5:00 and time for me to finally head home. Here's a picture of the cathedral I pass on my way home every day (took while at a stoplight).</div>
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I got home at 5:30 and the rush is on. All at once, I feel the need to strip off the tie and formal clothes for more comfortable PJs, grab a snack, and go to the bathroom. Again, I'd leave off this detail except that I have included all of my restroom stops for the day here already. Once at 6am, once at 5:30pm; there's no time in the middle of the day for such trivialities.</div>
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I check over all the internets I missed during the day (which takes longer than usual because it was off for half of the day at school) while turning on the afternoon news and eating a snack. This is my time to relax before my wife gets home from her work around 6:30. I really should do all my schoolwork when she's not here so I can enjoy my time with her, but I've already been schooling for 12 hours and need the break.</div>
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I write the two tests for tomorrow. I don't write them ahead of time because I like to get a feel from the review day about how much they can handle and how difficult to make the questions. My early arrival tomorrow will be used in making copies of the tests for classes. I use <a href="http://macromates.com/" style="text-decoration:underline;">TextMate</a> on my Mac to write in LaTeX so the math looks nice. I use <a href="http://geogebra.org/cms/" style="text-decoration:underline;">geogebra</a> to make the figures and <a href="http://evernote.com/skitch/" style="text-decoration:underline;">skitch</a> to take the screenshots. Put the files in my Google Drive folder so that I can download and print/copy them when I get to school tomorrow. I guess I should use a flash drive tonight, too, just in case the internet is down when I get back tomorrow. It took me about an hour to come up with the problems, make the figures, and put it all together.</div>
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I have a feeling the tests will be short and they'll finish early, but I decide to err on the side of too easy this time. Thanksgiving/Fall break is right around the corner and I know they're looking forward to that. I know they also need a bit of a grade push about now to keep them motivated. Anyways, I'll test the important things and there's no need to make it unnecessarily difficult when I can tell how well they understand the material with easier problems.<br />
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Wife gets home and 6:45 and immediately gets to work making dinner. I hate that she doesn't have her own down time, but she says she enjoys it. I guess she has her down time in the morning before going to work when I'm not there.<br />
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I got an email from at student at 7:30 tonight saying that he had signed up to retake a section of a quiz a while ago and wanted to actually take it tomorrow. He mostly was writing to apologize on waiting so long to take it because I originally stressed that I wanted them to take it the day after they sign up. He's a nice kid and I told him the email wasn't necessary, but I appreciated the thought.<br />
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I checked my Google Drive at 8:30 after dinner. The policy in my classes is that they have until 8pm the night before to sign up for retakes. So, if there were any results from the google form, I could write those retakes quickly for tomorrow. Tonight there were none and there have not been any all week. Sometimes there will be as many as seven in an evening, but they have fallen off recently. Maybe after break students will feel the fire more. Plus, we have a test tomorrow, so many like to wait to only have one graded assessment per day.<br />
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There's grading to be done (especially since I didn't have my planning period today), but I can use the internet being down as an excuse for not putting the grades in. Hopefully I'll have my plan tomorrow and can enter them then. Some students mentioned today that their weekend might depend on their grades going up, so I should try to help them get ungrounded. I'll see what I can do.<br />
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Just for reference, I don't feel like today was all that busy or easy of a day. This is fairly typical of my weekdays. It is now 9:30pm. So, I think I'll take this last half hour of being awake to help my wife make cookies (we have company coming through town tomorrow on their way to Thanksgiving elsewhere) and think about what to do with the time after they finish early tomorrow. Maybe tomorrow's youtube mining will turn up something interesting.</div>
CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com3tag:blogger.com,1999:blog-2750355884816556227.post-11678730057277883222012-10-10T11:27:00.002-05:002012-10-10T11:27:38.330-05:00A Quick Mnemonic for Fractional ExponentsI forgot that I invented this mnemonic for fractional exponents, so I thought I'd share in a quick post between periods.<br />
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My students often have trouble remembering whether the cube root of x^4 is x^(4/3) or x^(3/4). So I use the analogy that fractional exponents are like trees.<br />
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<a href="http://www.denverwater.org/docs/assets/E1D5B7B0-E97B-67AD-CE384F57221C57E3/TreeGraphic_web.jpg" imageanchor="1" style="margin-left: 1em; margin-right: 1em;"><img border="0" src="http://www.denverwater.org/docs/assets/E1D5B7B0-E97B-67AD-CE384F57221C57E3/TreeGraphic_web.jpg" /></a></div>
When the sun shines, the POWER for the tree comes from the TOP (photosynthesis) while the ROOTS are on the BOTTOM.<br /><br />CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com0tag:blogger.com,1999:blog-2750355884816556227.post-48379396378335421762012-08-04T21:39:00.002-05:002012-08-04T21:39:55.337-05:00New JobI got a new job this year at a Catholic, all-boys school on the block system. So, in addition to teaching new subjects (geometry, statistics/probability, and algebra I), I'll have to adjust to all those things. If any of you have tips on any piece of it, feel free to share here. I'll try to post when more specific problems present themselves.CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com1tag:blogger.com,1999:blog-2750355884816556227.post-18789485987363625902012-07-23T16:57:00.000-05:002012-07-23T16:57:36.389-05:00Matrix Multiplication<p>Another quick tip. This time how to multiply matrices.</p>
<p>Begin with the matrix multiplication problem:</p>
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<p>Then move the first matrix down. [Note: Since matrix multiplication is not commutative, this is important. Although it should be noted that the same effect can be accomplished by moving the second matrix up. But under no circumstances should the reverse be tried.] The answer will go in the new space you have created in the bottom right corner. Immediately you can see (if the product is possible) the shape of the answer. In this example it is a 2x2 matrix.</p>
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<p>Pick a position in the answer matrix and follow across from the left and vertically from above to figure out which numbers you will use. Multiply pairs beginning with the outermost numbers (the blue 1 and 7 in the example) and sum with the product of the next pair in until you run out of pairs. The answer will go in the position where the arrows meet.</p>
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<p>Remember not to use numbers from your answer when computing other spaces. For example, the 58 was not used to find the 64 below.</p>
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<p>Continue with each position until the answer matrix is complete!</p>
<p>But what if the matrices in question are not able to be multiplied?</p>
<p>Consider the following case. Although it initially looks like our answer will be a 2x2 matrix, we see that the 3 does not have a pair, so these matrices cannot be multiplied in this order.</p>
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<a href="http://1.bp.blogspot.com/-vlBfuvM4LOw/UA3HS6MewuI/AAAAAAAAC78/Gl75KLZFSSU/s1600/Slide5.jpg" imageanchor="1" style="margin-right:1em; margin-bottom:1em"><img border="0" height="130" width="320" src="http://1.bp.blogspot.com/-vlBfuvM4LOw/UA3HS6MewuI/AAAAAAAAC78/Gl75KLZFSSU/s320/Slide5.jpg" /></a></div>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com3tag:blogger.com,1999:blog-2750355884816556227.post-85597756753892668292012-07-20T11:08:00.000-05:002012-07-20T11:20:33.826-05:00Factoring cubic binomials<p>To be finished later, but here is a quick idea that came up:</p>
<p>Factoring binomials of cubes (x^3 + y^3) or (x^3 - y^3)</p>
<p>My mnemonic is "SQuiggy CHases Many Purple SQuirrels." It stands for:</p>
<ul>
<li>SQuare (the first term)</li>
<li>CHange (the sign)</li>
<li>Multiply (the two terms)</li>
<li>Plus</li>
<li>SQuare (the second term)</li>
</ul>
<p>So, (x^3 + y^3) = (x+y)(x^2 - xy + y^2)
and
(x^3 - y^3) = (x-y)(x^2 + xy + y^2)</p>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com3tag:blogger.com,1999:blog-2750355884816556227.post-76825400391977419732012-03-05T08:07:00.000-06:002012-03-05T08:07:24.066-06:00Hunger Games Sequence<p>I have a 45 minute drive to school each way. So, I've been picking up audiobooks to keep me company on the drive. Many of you have suggested reading <u>The Hunger Games</u>, so that was my latest acquisition. The whole thing is about 11 hours, I think, so I should finish it in about a week.</p>
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<p>Anyways, getting through chapter one this morning, I listened to this bit:</p>
<blockquote><p>The
reaping system is unfair, with the poor getting the worst of
it. You become eligible for the reaping the day you turn
twelve. That year, your name is entered once. At thirteen,
twice. And so on and so on until you reach the age of eighteen,
the final year of eligibility, when your name goes into
the pool seven times. That’s true for every citizen in all
twelve districts in the entire country of Panem.</p>
<p>But here’s the catch. Say you are poor and starving as
we were. You can opt to add your name more times in
exchange for tesserae. Each tessera is worth a meager year’s
supply of grain and oil for one person. You may do this for
each of your family members as well. So, at the age of
twelve, I had my name entered four times. Once, because I
had to, and three times for tesserae for grain and oil for
myself, Prim, and my mother. In fact, every year I have
needed to do this. And the entries are cumulative. So now,
at the age of sixteen, my name will be in the reaping twenty
times. Gale, who is eighteen and has been either helping or
single-handedly feeding a family of five for seven years, will
have his name in forty-two times.</p>
<p style="text-align:right;font-style:italics;"><span style="text-decoration:underline;"> The Hunger Games</span> by Suzanne Collins.<br> Chapter 1 as found <a href="http://www.scholastic.com/thehungergames/media/hungergames-chapter1.pdf">here</a></p></blockquote>
<p>Being a math teacher about to begin a unit on sequences and series in my Algebra II class (as well as a deeper study in Precalculus), I thought this would be an interesting problem to use. They are also all reading this book and seem to like it.</p>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com1tag:blogger.com,1999:blog-2750355884816556227.post-20822800685244152472012-02-08T17:54:00.000-06:002012-02-08T17:54:10.142-06:00Teaching Resolution: Make More Mistakes<p>I have a problem with how I work in my classroom: I don't often make mistakes. I don't mean for that to sound conceited, but I generally like for people with authority to show few weaknesses and be both precise and accurate in their communication of concepts. It irks me when the principal (a former English teacher and currently moonlighting as a college professor for English education courses) writes an e-mail to the school that says something like, "We are starting this business that will be ran by..." or when our superintendent sends an e-mail to the entire district full of spelling and grammar mistakes.</p>
<p>I thought about it harder this past weekend, though and have come to a conclusion that this may not be the best strategy in the classroom. By limiting my own mistakes it sends a few false messages to students.</p>
<ul>
<li>It implies that it is bad to make mistakes.</li>
<li>It implies that the concept or problem is (or should be) easy.</li>
<li>It puts me on another plane than my students so that they think I am way above their level and they will never be able to attain that level of understanding of the subject.</li>
</ul>
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<p>So, I am resolving to make more mistakes in my classroom. Some will be intentional; some may not be. Just today in precalculus, for example, I was trying to number our examples as we went and I put up numbers 1, 2, 3, 6, and 7, then I asked them to work through those seven problems. Those students who were paying attention got confused and corrected me. Yay!</p>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com6tag:blogger.com,1999:blog-2750355884816556227.post-50948528544884536652012-02-08T17:31:00.000-06:002012-02-08T17:44:56.457-06:00Vocabulary or Jargon?<p>While we're entering a new era or spelling in our culture, the debate reigns whether to join the revolution or fight for the original ways. It's a debate on which I can argue both sides and about one in which I have a hard time deciding my own position. While "kids these days" are writing things like "lol ur rong" does seem somewhat uneducated, it also gets the point across and isn't the point of language to communicate ideas?</p>
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<p>While that example may fall more squarely into the English teachers' realm, I have an internal debate about it in my own classroom, too. How important is it really that I use the words "denominator" and "numerator," when it's so much quicker (and more easily understood) to say "bottom" and "top?" When a line with negative slope is changed to get more vertical, how wrong is it for me to say that the slope gets "more negative?" </p>
<p>I realize that there are some words that need to be defined and used correctly so as to avoid confusion. I just finished a section of Algebra 2 where we talked extensively about the difference between permutations and combinations. The words here are important to the understanding of the logic and the formulation of the mathematical expressions needed to solve these problems.
</p>
<p>On the flip side, though, in precalculus we are discussing vectors. What is the real difference between calling two vectors orthogonal versus just the familiar "perpendicular" word they already know? Later we introduce the term "normal" for this same concept. Do we really need three words to express the same concept? (Or are they not the same concept and I'm thinking about it incorrectly?)</p>
<p>I realize that even the words "denominator" and "numerator" are important when we involve more complex expressions. But even they get confusing when we discuss things like "multiply both the numerator and denominator of the entire fraction by least common denominator of the terms in both the denominator and the numerator" to simplify an expression like <a href="http://www.codecogs.com/eqnedit.php?latex=\inline \frac{\frac{1}{x}@plus;\frac{2}{x@plus;1}}{\frac{3}{x-1} @plus; \frac{5}{x^2 - 1}}" target="_blank"><img src="http://latex.codecogs.com/gif.latex?\inline \frac{\frac{1}{x}+\frac{2}{x+1}}{\frac{3}{x-1} + \frac{5}{x^2 - 1}}" title="\inline \frac{\frac{1}{x}+\frac{2}{x+1}}{\frac{3}{x-1} + \frac{5}{x^2 - 1}}" /></a></p>
<p>So, what say you? When teaching my students math, how important is it to indoctrinate them to the traditional language of mathematics and somewhat confusing vocabulary? Is it possible to communicate some concepts using more colloquial language?</p>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com4tag:blogger.com,1999:blog-2750355884816556227.post-62770550179934154272012-01-26T17:16:00.001-06:002012-01-26T17:16:29.911-06:00iTextbook Thought<p>I have been intrigued by Euclid's <u>Elements</u> ever since I saw this website: <a href="http://aleph0.clarku.edu/~djoyce/java/elements/elements.html">http://aleph0.clarku.edu/~djoyce/java/elements/elements.html</a>. I'd always thought it would be a cool way to explore geometry. Begin with the basics and then see what you could do with those things.</p>
<p>
So, my idea for an iTextbook is based around those ideas. You begin with some "simple" definitions and postulates as in Euclid's book. Maybe give some reasoning behind the postulates by asking questions like, "Why do we need 2 points to draw a line? Challenge: I am thinking of a line in this window. See if you can guess it with zero hints. (Student draws random line and it won't match what's in the computer.) Here. I'll give you one hint: This point is on the line (point appears, kid draws another line, it won't match). You missed because here are some example lines that it could have been (show a bunch of lines through the point). Here is a second hint: This point is also on the line (connect the dots = winner)."</p>
<p>
Expand by asking students to create some tools needed. For example, show a picture of a pile of items that you might find in a junk drawer. These include, but are not limited to, some string, some push-pins, a pen, tape, scissors, a straightedge of some sort, etc. Ask students "How could you create a perfect circle (see <a href="http://aleph0.clarku.edu/~djoyce/java/elements/bookI/defI15.html">definitions 15-16</a>) from these items?" Once they arrive at a correct answer, they "level up." A screen appears with something like, "You have acquired the circle tool!" and the glittery circle button appears in their tool-bar at the top which works like it would in any drawing program (click for the center point, then click again for a point on the circle to fix the size).
</p><p>
Basically, you build geogebra tools by finding out how to construct each thing. You have the line-segment tool and circle tool, so <a href="http://aleph0.clarku.edu/~djoyce/java/elements/bookI/propI1.html">Challenge 1</a> is available to you: construct an equilateral triangle using the tools you have.
</p><p>
Once you show you can bisect a given line segment, you get a new "midpoint" tool. This unlocks new challenges in which you need to use the midpoint for other constructions.
</p><p>
After you construct an object, you get follow-up questions like: In constructing the midpoint, do the two circles need to be the same size? Is the bisecting segment always perpendicular to the original segment? Does the original segment bisect the segment you created? etc.
</p><p>
It'd be dynamic as with geogebra or sketchpad or anything like that, but you can only use the tools when you "level up" by figuring out how to get build them from more basic tools.
</p><p>
There's a "sandbox mode" where you can play around with tools you have or preview what will become available later. Challenges available are based on the tools you have acquired so far, so they are somewhat ordered, but not necessarily linear. They could be labeled with a difficulty level. Unlocking new tools unlocks new challenges. Hints can be available for the tougher constructions/proofs, like in most games, too.
</p><p>
Anyways, what do you think? The idea just struck me this afternoon, so I haven't thought through all the angles (pun intended), but it seemed like an interesting way to introduce parts of geometry.</p>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com1tag:blogger.com,1999:blog-2750355884816556227.post-91531067751895665282011-11-16T07:13:00.001-06:002011-11-16T07:17:05.735-06:00Differentiating the Holidays<p>I am shamelessly stealing this from a coworker. It is a horrible way to come back after a long hiatus of no blog posts. I apologize. I have blog posts in my head, but am not motivated enough to write. </p>
<p>I also hate the idea of FWD FWD FWD e-mails, but I did think this one was fun enough to share beyond our school:</p>
<p style="text-decoration: underline;">The Top Ten Ways to Differentiate Thanksgiving Dinner </p>
<p>10. Serve all of your guests on different-sized plates<br/>
9. Make each of your guests focus on just one or two foods instead of the whole buffet<br/>
8. Eat in shifts in different rooms<br/>
7. Let your visual guests just enjoy looking at the food while your kinesthetic guests get to eat it<br/>
6. Allow the turkey to have a say as to whether he should be oven-roasted or deep-fried<br/>
5. Serve dessert first, then the hot dogs<br/>
4. Pair everyone up with an eating buddy<br/>
3. Serve the simpler foods first like mashed potatoes and work your way into the more complex foods after your guests have showed mastery of eating<br/>
2. Allow your quick eaters to put their food in a blender for faster consumption<br/>
And the number one way to differentiate Thanksgiving Dinner…<br/>
1. Serve Thanksgiving, but call it a Fourth of July celebration</p>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com0tag:blogger.com,1999:blog-2750355884816556227.post-59247279408475904342011-08-25T19:15:00.000-05:002011-08-25T19:15:52.650-05:00"Real World" Problem Solving: I Give Up!I have a SMART board in my room. One of maybe 3 in our school. I also share my room for one period a day with a teacher who doesn't like to use it. So, the board is on a wheeled mount so I can move it out of the way for the 45 mins a day he is in my same classroom. This means I have to orient it at least twice a day and don't really get to ever lock it into place.<br />
<br />
We take attendance online in our gradebook program and it is the expectation that attendance be done in the first 10 minutes of class (to catch skipping students). Since doing this on my own daily leaves the first 5 minutes of class where the students goof around instead of working, I decided to use the SMART board to my advantage.<br />
<br />
This is what our attendance looks like (names cropped to protect the young):<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://2.bp.blogspot.com/-WYQMrYLBKOc/Tlbiz6dm6FI/AAAAAAAAC2s/CjFnlwo0qXc/s1600/Attendance.jpg" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="400" width="165" src="http://2.bp.blogspot.com/-WYQMrYLBKOc/Tlbiz6dm6FI/AAAAAAAAC2s/CjFnlwo0qXc/s400/Attendance.jpg" /></a></div><br />
<br />
I set everyone to the far right column (absent) and when students arrive, they are to hit the leftmost bubble next to their name to mark themselves present (the middle option is for tardies). Generally, they like being able to come up to the SMART board and having it work for them. Win-win.<br />
<br />
Here's the issue: Since it's never perfectly calibrated (especially with students going to the board and slightly moving the mount to one side or the other), hitting the board in the right spot is often difficult. When you hit the board, a time diamond-shaped cursor shows up and blinks where it thought you touched, but even moments after I orient/calibrate it, some sections of the board will be off by a centimeter or two.<br />
<br />
Many of my students have the <span style="font-style:italic;">hardest time</span> getting it to work. I understand that it's somewhat unpredictable on your first touch, but they will continue to hit the same spot (usually harder and harder) and continue to get more and more frustrated. <br />
<br />
Part of me wants to laugh and part wants to cry. Is it really that hard to compensate for the issue? You hit it on the button exactly the first time and you see the cursor a bit below your name so it doesn't work. Try hitting it a centimeter higher, right? They will try four or five times and then either try to get a friend to do it or give up and walk away mad and throw their hands up in the air saying, "I can't do it. You just do it for me."<br />
<br />
Then I try to teach them how to problem solve math problems and they react the same way. Is it surprising?<br />
<br />
I'm all for trying to tap into student intuition and their own internal motivation (a la <a href="http://shawncornally.com/wordpress/">Shawn Cornally</a>, but what can I do with students like this? Am I being overly dramatic in this observation of student behavior?<br />
<br />
(NB: Not all are like this. About half of them figure it out and there are no problems, but about half have the issue as described.)CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com2tag:blogger.com,1999:blog-2750355884816556227.post-38659249534843529422011-08-09T18:19:00.000-05:002011-08-09T18:19:35.326-05:00Pretests (2011-12)Presented without real comments, here are my pretests for Algebra 2 and Precalculus this year:<br />
<br />
<a title="View Alg2Pretest on Scribd" href="http://www.scribd.com/doc/61970303/Alg2Pretest" style="margin: 12px auto 6px auto; font-family: Helvetica,Arial,Sans-serif; font-style: normal; font-variant: normal; font-weight: normal; font-size: 14px; line-height: normal; font-size-adjust: none; font-stretch: normal; -x-system-font: none; display: block; text-decoration: underline;">Alg2Pretest</a><iframe class="scribd_iframe_embed" src="http://www.scribd.com/embeds/61970303/content?start_page=1&view_mode=list&access_key=key-29kg67gv49q55x0sude8" data-auto-height="true" data-aspect-ratio="0.772727272727273" scrolling="no" id="doc_29582" width="100%" height="600" frameborder="0"></iframe><script type="text/javascript">(function() { var scribd = document.createElement("script"); scribd.type = "text/javascript"; scribd.async = true; scribd.src = "http://www.scribd.com/javascripts/embed_code/inject.js"; var s = document.getElementsByTagName("script")[0]; s.parentNode.insertBefore(scribd, s); })();</script><br />
<br />
<a title="View Pre Cal Pretest on Scribd" href="http://www.scribd.com/doc/61970378/Pre-Cal-Pretest" style="margin: 12px auto 6px auto; font-family: Helvetica,Arial,Sans-serif; font-style: normal; font-variant: normal; font-weight: normal; font-size: 14px; line-height: normal; font-size-adjust: none; font-stretch: normal; -x-system-font: none; display: block; text-decoration: underline;">Pre Cal Pretest</a><iframe class="scribd_iframe_embed" src="http://www.scribd.com/embeds/61970378/content?start_page=1&view_mode=list&access_key=key-23qyzbetpbkx2cfabryn" data-auto-height="true" data-aspect-ratio="0.772727272727273" scrolling="no" id="doc_82856" width="100%" height="600" frameborder="0"></iframe><script type="text/javascript">(function() { var scribd = document.createElement("script"); scribd.type = "text/javascript"; scribd.async = true; scribd.src = "http://www.scribd.com/javascripts/embed_code/inject.js"; var s = document.getElementsByTagName("script")[0]; s.parentNode.insertBefore(scribd, s); })();</script>CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com0tag:blogger.com,1999:blog-2750355884816556227.post-41637087157675419592011-07-11T19:44:00.000-05:002011-07-11T19:44:10.449-05:00Transparency With Students<div class="separator" style="clear: both; text-align: center;"><iframe allowfullscreen='allowfullscreen' webkitallowfullscreen='webkitallowfullscreen' mozallowfullscreen='mozallowfullscreen' width='320' height='266' src='https://www.youtube.com/embed/B5cAwTEEGNE?feature=player_embedded' frameborder='0'></iframe></div><br />
One thing I'm wondering about for the coming school year is how much to talk with my students about the background of the logistics of teaching. Certainly many of you have said that it is important to regularly talk about the meaning of SBG to get students on board with the system. Providing outlines for units is also important for connecting the concepts to one another.<br />
<br />
I wonder, though, how much of it just becomes "execu-speak" as in the clip above. Of course I get excited about the whys and whats of not only the content, but the teaching methods, but how much do they care about it? How much SHOULD they care about it? <br />
<br />
How much detail should I go into when explaining why I don't take homework grades? Should I even bring up the issue that I don't think I should take off for late work, but the school put a policy in place to make me do it? Do they care WHY I'm just repeating their questions back to them and never actually giving them the "real answer" (as seen in <a href="http://www.wired.com/wiredscience/2011/06/when-a-class-makes-me-scared/">Rhett's awesome post</a>)? Is it worth mentioning the reason I have the classroom set up in the way I do, the reason I go out of order from the book, the reason I grade on a 10 point scale, the reason I spend hours each night after school thinking about all the little details to give them the best possible learning experience that I can?CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com4tag:blogger.com,1999:blog-2750355884816556227.post-87526853077826405272011-07-02T21:40:00.001-05:002011-07-03T08:37:31.881-05:00Real-world Math: LampshadesMy wife wants to make something like this:<br />
<br />
<div style="text-align:center;"><img src="http://www.chicaandjo.com/wp-content/uploads/2009/02/lampshade4.jpg" height=200><br />
</div><br />
The issue is that she wants the shadow part to be words, so getting the angles right is essential.<br />
<br />
So, how should she design it in photoshop so that we can print it out and glue it on? Enter math.<br />
<br />
She measured the top and found that it has a circumference of 26 inches. The bottom of the shade has a radius of 32 inches. The "slant height" is 7 inches. The "true" height is harder to measure because you have to keep it level and ours is already on the lamp, so it wasn't measured, but could be "if you REALLY need it".<br />
<br />
So, what to do? Well, here's what I knew:<br />
<br />
<ol><li>This shape is called a "frustrum." In particular, this is a <i>conical</i> frustrum.</li>
<li>You can create a cone by cutting a sector from a circle and gluing the ends together.</li>
<li>Since this frustrum shape is a large cone with a top cone cut off, it could be created by taking a sector from an annulus (a "ring" made by taking a large circle and cutting our a concentric, smaller circle).</li>
<li>Formulas I knew and may or may not need here: <br />
<ul><li>For a sector created from a circle of radius R and sweeping out angle T (in radians), the arc length of the sector (S) is S = R*T</li>
<li>Circumference = 2 * pi * radius</li>
<li>Pythagorean theorem (in particular, using the "true" height, slant height, and radius of a cone)</li>
</ul></li>
</ol><br />
<h3>So, here's what I tried first (aka The Hard Way):</h3><br />
(Background info: I've never taught geometry and am more of a functional analysis person, so that's why this stuff came to me first.)<br />
<br />
Consider the sector that would form the large cone (from which we cut the top off to make the lampshade): I knew the arc length is 32, but that's about it. The angle I called T and the radius of this large circle I called x.<br />
<br />
Things I learned about this cone by playing around with the formulas above: <br />
<ul><li>If we call the radius of this large cone R, then 32 = 2 *pi * R, so R = 16/pi.</li>
<li>32 = x * T</li>
<li>If we call the "true" height of the cone H, then H^2 + R^2 = x^2 (and could use R as above).</li>
</ul><br />
Consider the sector that would form the cone we cut off the other one: I know the arc length is 26 and that's it for that one. This angle is also T, but the smaller circle would have a radius of y (so that y < x above).
Similar to the above things, but with a couple added to relate the two cones I now know:
<ul><br />
<li>If we call the radius of this smaller cone r, then 26 = 2 * pi * r, so r = 13/pi.</li><br />
<br />
<li>26 = y * T</li><br />
<br />
<li>If we call the "true" height of this cone h, then h^2 + r^2 = y^2 (and can use r from above).</li><br />
<br />
<li>x = y + 7 (the 7 was measured as the slant height of the shade, see above)</li><br />
<br />
<li>H = h + m, where m is the measured "true" height of the shade.</li><br />
<br />
</ul><br />
Putting these pieces together, I ended up with these two equations:<br />
<ul><li>H^2 + (16/pi)^2 = (y+7)^2</li>
<li>(H-m)^2 + (13/pi)^2 = y^2</li>
</ul><br />
Two equations with two unknowns may be solvable, so I tried to actually measure m and got about 55/8 inches. I could solve these for y, then find the angle T = 26/y and have the sector for the small circle. Adjusting a bit to get x would give the large circle.<br />
<br />
This was messy and gross. In fact, I gave up from here and decided to find a more "elegant" solution.<br />
<br />
<h3>Second attempt (The Easy Way):</h3><br />
This time I tried to go back to basics of what I remembered about geometry (bear in mind I've not had anything really geometry related since 8th grade which was almost 20 years ago now).<br />
<br />
I know there is such a thing as "similar triangles," so maybe there's such a thing as "similar cones" (seeing as how they are just rotated triangles). So, I went with that. Variables are already defined as above, so here are the ratios I used:<br />
<br />
x/y = R/r = (2*pi*R)/(2*pi*r) = 32/26 = 16/13<br />
<br />
So, clearly y = (13/16)*x. We also know that y + 7 = x and using substitution, (13/16)x + 7 = x, so (3/16) x = 7, so x = 112/3. Now that I know x, I can get T = 32/x.<br />
<br />
Whoa! That's easy, but were my assumptions ok? Well, I'm not <i>really</i> doing a general proof here, so I'll just check it with these numbers: To the Bat Geogebra!<br />
<br />
<applet name="ggbApplet" code="geogebra.GeoGebraApplet" archive="geogebra.jar"
codebase="http://www.geogebra.org/webstart/3.2/unsigned/"
width="606" height="578"mayscript="true"><br />
<param name="ggbBase64" value="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"/><param name="image" value="http://www.geogebra.org/webstart/loading.gif" /><param name="boxborder" value="false" /><param name="centerimage" value="true" /><param name="java_arguments" value="-Xmx512m -Djnlp.packEnabled=true" /><param name="cache_archive" value="geogebra.jar, geogebra_main.jar, geogebra_gui.jar, geogebra_cas.jar, geogebra_export.jar, geogebra_properties.jar" /><param name="cache_version" value="3.2.47.0, 3.2.47.0, 3.2.47.0, 3.2.47.0, 3.2.47.0, 3.2.47.0" /><param name="framePossible" value="false" /><param name="showResetIcon" value="false" /><param name="showAnimationButton" value="true" /><param name="enableRightClick" value="false" /><param name="errorDialogsActive" value="true" /><param name="enableLabelDrags" value="false" /><param name="showMenuBar" value="true" /><param name="showToolBar" value="false" /><param name="showToolBarHelp" value="false" /><param name="showAlgebraInput" value="true" /><param name="allowRescaling" value="true" />This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com<br />
</applet><br />
<br />
I'm still getting to know this application, so I'll explain how I made this:<br />
First I put A at the origin and defined a to be 112/3 as discovered above. Create a circle with center A and radius a (the program called this c as you can see in the variables to the left). I needed another point on this circle, so I just used the one on the x-axis by creating line b (y = 0) and then intersecting the circle with the line; this created point B. <br />
<br />
So, create a circle centered at B and radius a; this is circle labeled as f. (I now realize this is somewhat redundant since I could just switch the roles of A and B without this extra step, but I went into it thinking A would be a point on the circle's edge and would create the center of the circle elsewhere.) Anyhow, I then figured out the angle (which I called T above, but is labeled e in this applet). 0.86 is in radians, though, and since my wife will do the measuring, I figured degrees would be better, so I converted it which is labeled as d. <br />
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Use GGB to find a point which is d degrees away from A which it labels as A'. Just to make sure, create the arc between A and A' and check the length: 32. Perfect!<br />
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Now to finish, I made a circle centered at A and radius 7 (since I knew that was the slant height of the shade). Where that intersects the x-axis is the point C which will be the beginning of the smaller arc. Make a circle centered at B and radius y (which you may remember is equal to x - 7 as above) and find where that intersects with the line between A' and B. Double check by finding the measure of arc CD (26 yay!) AND the angle CBD (49.11 degrees yay!).<br />
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So now we have it. Measurements can now be passed back to the wife who is the real Illustrator/Photoshop guru and she can bend the text she wants for the lampshade to fit.<br />
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Pshew!<br />
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I'm still not sure what she has in mind and maybe this project won't even get finished by her, but I'll post a picture if it ever gets there!CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com0tag:blogger.com,1999:blog-2750355884816556227.post-57181150722254739582011-06-27T19:27:00.000-05:002011-06-27T19:27:57.152-05:00Secondary Skills and OpportunitiesSometimes I wonder how things would be if we didn't need the secondary skills to go with certain things or how the world would be if everyone had the same opportunities. <br />
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Consider this: Something within Sam would make him the best violin player ever to walk the earth.<br />
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Unfortunately, Sam has grown up in the slums of Chad and never even hears a violin, much less gets the chance to practice or own one. OR perhaps Sam has all the skills and brains to be a great violinist, but he has stage fright or he loses a hand in an accident or some other issue that keeps him from actually showing this talent to the world.<br />
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Then I wonder how that plays out in my classroom. Bad test takers? Less-than-ideal backgrounds? Other handicaps physical or mental that might keep a kid from doing her best?CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com0tag:blogger.com,1999:blog-2750355884816556227.post-37023983820701374232011-06-25T10:45:00.000-05:002011-06-25T10:45:56.681-05:00Cheap GasHow far is it worth driving for "cheap gas?"<br />
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Let's say I have a gas station near me (or on my route or whatever), but that I know another gas station elsewhere is selling gas for ten cents less. Is it worth it for me to go out of the way to buy it cheaper?<br />
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When I fill up, I usually get about 12 gallons. So, the trip must be worth $1.20 for me to make it. Since gas is about $3.60 right now, I am saving about 1/3 of a gallon of gas to get to this other station.<br />
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My car gets about 27 miles to the gallon, so to be worth the 1/3 of a gallon, it must be less than 9 miles (round trip) for me to go out of the way to get it and come back.<br />
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This seems like a decent open-ended question for students to fill in with their own numbers for their cars.CalcDavehttp://www.blogger.com/profile/14039458440867020542noreply@blogger.com2