7th Grade: Exploring Polyhedrons

One lesson I learned well as a kindergarten teacher: allow lots of free exploration of materials before attempting to teach with them.

As a middle school teacher, I think this is even more imperative.

As an introduction to polyhedrons, I asked my students to play with wooden polyhedrons at their tables without any rules other than playing safe and fairly.

Out of an 80 minutes session, I gave them around 30 minutes just to see what they would do.

All the students were very involved talking and playing with the blocks. I walked around a little, but mostly reserved myself time to simply observe.

I noticed how happy and well adjusted my students seemed. They enjoyed working together at building towers or ramps or little villages. It was a relaxed environment.

When I called attention to the class, I simply asked them to choose 8 blocks to keep at the table and return the rest. It seemed that after 30 minutes, they were ready to move on. The clean-up was quick and painless.

It was interesting how they chose their 8 blocks. All the groups chose eight distinct blocks, allowing me to make a simple comment on the ways we begin to classify polyhedrons as prisms or pyramids. They immediately looked to see how this classification played out with their particular blocks.

We then went over basic vocabulary and I asked them to take careful notes as I wrote on the Smartboard: poly hedron = many faces. Faces=polygons or "sides" of the block, edges being where two faces come together and vertices or like the corners. I asked them to pass their finger purposefully over the faces, then the tips of their fingers over the edges and finally counting all the vertices. Then I asked them to choose one block and draw it as carefully as they could. I walked around helping certain students focus on the parallel nature of most of the lines. I asked them to notice that when we draw polyhedrons, it is the edges that best define the figure. I find that interesting and is a good benchmark for my students to think about polyhedra in more general terms (my ultimate goal).

My objective with this unit is to clearly define "volume" as a measure of space within a polyhedron: sort of line a stack of cards (base x height). I also want my students to understand what surface area is and the complexities involved with measuring it (particularly, that there are no easy formulas for most polyhedra). 

I have found that surface area confounds many people. So we ended the day wrapping up our polyhedra with visually interesting wrapping paper. I taught my students how to trace a net using the blocks. They cut out their nets and folded them up into polyhedra. Finally, they taped these polyhedra and we made a class poster. 

It was interesting for me to note that about 1/4 of the class really struggled with the concept of "nets". The continuously second-guessed themselves and ended up tracing inaccurate nets that then would not fold correctly. I suspect that these same students will struggle with the work around surface area in general and now I have a clearer picture of who they might be.

What I am still considering, though, is how to address this with them.

Some things I've learned over the years...

I've learned that it's taking me a long time to become the teacher I wanted to be.

I've learned that I can keep going long after I can't.

I've learned that we are responsible for what we do, no matter how we feel.

I've learned that either I control my attitude or it controls me.

I've learned that heroes are the people who do what has to be done when it needs to be done, regardless of the consequences. I want to be a hero like this

I've learned that money is a lousy way of keeping score. (I've always sort of suspected this, 2 b honest)

I've learned that sometimes when I'm angry I have the right to be angry, but that doesn't give me the right to be cruel.

I've learned that it isn't always enough to be forgiven by others. Sometimes I have to learn to forgive myself (this has been a hard, hard thing to learn).

I've learned that our background and circumstances may have influenced who we are, but we are responsible for who we become.

I've learned that two people can look at the same thing and see something totally different.

I've learned that my life can be changed in a matter of hours by people who don't even know me.

I've learned that credentials on the wall do not make any one a decent human being.

Learning Objectives and Assessment in Math

Ideally the assessment process informs the teacher and the learner about learner progress and at the same time, contributes to the learning process. In theory, good assessment:

• measures meaningful learning outcomes
• does so in a fair, reliable, accurate way
• is easy to administer, score, and interpret
• informs the teacher about student performance and how they are interpreting course experiences
• results in meaningful feedback to the learner
• is itself a learning experience

Measuring student learning is always a challenge no matter what the delivery format. Your choices are limited by time, resources and creativity. When thinking about student assessment in a course, the following questions may help decide how many and what types of assessments you will include in your course.

• What is it you want your students to learn? 
• What do they already know? Is a pre-test needed to measure prior knowledge?
• Which assessment methods match your teaching style?
• What assessment method will best test what your students learned?
• Will you test memorization or performance?
• Will these assessments be low or high stakes? (what portion of final grade)
• Should you use adaptive testing? (will the test adapt to user responses)
• How many assessments are sufficient? How many papers should you assign? How many quizzes and exams will be enough?
• Will the number of students affect the type of assessment you choose?
• How quickly will students receive feedback?
• How much time will you spend correcting or commenting on assessments?
• How much grading time will you have?

Types of Math Assessments

Types of Assessment

• Pre-Testing

You might find it helpful to find out whether your students meet the basic knowledge and skill levels required to learn your materials. Use a pre-test to find out. Pre-tests are often paired with remedial materials.

Some teachers offer self-assessment pre-tests prior to the beginning day of class and offer students ways to catch up before the first day. Others provide time during the first week for students to do such things. Alternatively, you could pre-test prior to each module, week or topic.

• Objective Assessments

Objective assessments (usually multiple choice, true false, short answer) have correct answers. These are good for testing recall of facts and can be automated. Objective tests assume that there are true answers and assume that all students should learn the same things.

• Subjective Assessments

In subjective assessments the teacher's judgment determines the grade. These include essay tests. Essay tests take longer to answer and they take longer to grade than objective questions and therefore only include a small number of questions, focusing on complex concepts. Essay tests are best evaluated using some sort of pre-determined rubric of performance characteristics.

• Self Assessment

Self assessment types of assignments are provided for quick student feedback. Self assessments:

• help the learner check if they have mastered a topic
• provide opportunity to measure learning progress
• are usually voluntary and may allow multiple attempts
• inform the learner, but not the teacher
• can occur whenever a performance activity is linked with feedback about that performance.

Self assessment examples:

• practice quizzes
• games, simulations, and other interactive exercises
• practice written assignments
• peer reviews
• true-false questions

• Practice Exams

Practice exams and problem set homework are popular with students in courses which use exams for grading. Students who complete a practice exam usually encounter fewer problems on the official exam. Technical problems have been worked out, and the student knows what to expect in terms of types of questions.

It's important to let the student know that practice exam questions will be similar to what they will find on their exams. However, the specifics will differ based on course content. Students are very likely to complete a practice exam which parallels the real exam even though it does not count toward their grade.

• Group Projects

In real life many projects are team efforts. There is a great deal of learning value in discussion and collaboration. 

Smaller groups are more manageable. Teams of two are easier to coordinate than larger teams, although some courses do groups of 5 or 6. It is important to carefully assign the groups based on when they like to work and how they prefer to collaborate. Define clear roles, and include peer review of group participation as part of the grade. You can ask students to keep a log of their process and procedures. Provide a "panic button" for students whose team members are not participating correctly, so you can help them either decide to work alone or connect with another group.

• Peer Review and Audience

In the classroom, time constraints often prevent students from being able to review each others projects in sufficient detail. It is easy to post projects online where everyone can see them. The work is thus a public performance, a potential source of pride or embarrassment. It is helpful for other students to see the scope of work produced by others. They may be motivated on the next assignment by seeing other outstanding projects.

Peer review can be an effective learning technique. Taking on the role of judge is a different mode of understanding the goals of an assignment.

• Participation

Class participation can be an alternative method of assessing the student. A good way to encourage class participation is to make it part of the overall course grade. Class participation may include answering reflective questions in a course module, taking part in weekly class discussions, providing peer review critiques of fellow students' assignments, or locating and contributing online resources to a class-created knowledge base. As with essays, this assessment might best be implemented with a rubric of pre-determined characteristics. This allows the student to know what is being assessed.

• Other types of assessment

Alternative methods of assessment are limited only by your imagination. Consider assigning reflective journals, one minute papers, contributions to digital archives, or portfolios.

Philosophy on Assessment in Math Education

What is the most important part of teaching? 

Many people may say that it is the teaching. But in reality, isn't it the learning? Along those line, I feel that assessing what has been learned is most crucial. But assessing student understanding is the most complex part of the teaching cycle.

Teachers must be fair and respectful when assessing students. I believe all students need to be assessed by the same standards. All students should be assessed by all methods, not different methods due to level or activity. In assessing a student, I will use both formal and informal methods of assessment. I also want to consider whether my assessments (which my students always consider to be "tests") are formative or summative. Do they inform and influence what/how I teach or do they inform me on what my students know? Finally, assessment does not always need to be considered "formal". Informal observations are just as valid, and in many instances, even more valid, then a higher stakes "test".

That being said, the use of tests and quizzes are indispensable to the assessment of mathematics education. But in too many classes, tests are the only form of assessment in math. I believe that in addition to tests, students should be able to apply the knowledge they possess at a higher level in "real world" projects. These projects allow the teacher not only to assess the mathematical component, but also the logical thought process as well as the ability to work in a group.

In addition to the periodic checks that tests and projects can provide, portfolios allow students together with teachers to assess their knowledge and progress over time. The problem with portfolios is finding the time to analyze them efficiently and usefully. I actually believe portfolios are good for the student and his/her family, but not as useful for the teacher who has too many students to begin with. With both the student and the teacher making decisions about which items of work will become artifacts, both can see how the student's knowledge has changed. Portfolios allow students to demonstrate change over time, rather than just a static item from one day.

Overall, I feel that each student needs to be assessed in several different methods. This allows the teacher to provide a fair and unbiased view of how well the student has learned the material.

As a teacher, I will use several types of assessment to make sure that all students are informed of their progress. Assessment is the teachers most difficult task, yet the most crucial.

Things I ponder: Enactivism

Occurring somewhere between the surety of the known and the uncertainty of the unknown, the act of listening is similar to the project of education. It is, after all, when we are not certain that we are compelled to listen. Our listening is always and already in the transformative space of learning.


Brent Davis from Teaching Mathematics

8th Grade POW: Chapter 10: Unravelling

Cynthia lay in her hammock, exhausted yet unable to rest. Nothing on this trip seemed real now. Not even 24 hours earlier she was on the Rim with the Adults. There had been hope of finding a way off this confusing island. Then the rocks came hurling through the sky. She saw Jesse hit in the head and fall, then, in short order, Margie and Jeanie and several Adults around them were hit. Then Miguel. Confusion ensued and Cynthia felt hands grab her over the edge of the Rim and into a series of covered trenches that offered protection from the rocks raining down on them. The Verdania Adults were shouting and running through the trenches. It felt like an old World War II movie. 

Dula ran up to her, “The other in your group, the Captain, is down that way with two of our people. They will evacuate you. You must leave, now!” Dula pushed Cynthia down the trench, “Go!”

Cynthia ran up to the Captain, who was strangely distant and silent.  The trenches led down the side of the Rim and back into the Verdania Caldera. They moved quickly. The walked briskly through the night, along moon lit paths through the forests until finally arriving to a small village.  They left Cynthia and the Captain in a small cabin with several hammocks hanging from posts. The two adults nodded as they left, closing the door behind them.

“What has happened? Oh my god, what about the kids? What are we going to do, Captain?”

The Captain paced along the floor, impatiently. “The kids are fine, at least ours. Dula told me that attacks on the Rim happen only occasionally, but to date, nothing has ever reached the Caldera floor. The children don’t even know about this. It’s all some sort of elaborate scheme to protect them from the Deesors.”

“Who are the Deesors?” asked Cynthia.

“Don’t know. There wasn’t time to find out. I think we are ok here, though. Feel bad about the others on the Rim though.” She looked down for moment. Cynthia wondered if she would cry.

But then the Captain looked up, dry eyed, and started pacing around the cabin again. She stopped at a table in the back corner of the room. There was a small bookcase on the wall behind the desk. The Captain rummaged through the books with idle curiosity. They were ancient, dusty old books, some of them handwritten in fancy calligraphy. Most of them were in Spanish or French, though a few were written in English as well. She picked up one book and saw the title Our History. 

“What a silly title,” she thought to herself. But as she breezed through the first couple pages, she realized that perhaps she could finally come to understand the origins of the people on this crazy island. She read:

Jurakan, Captain of the Ferdinando, made his life's fortune on seven perilous voyages around the Caribbean, each of which took him to a different island where he acquired a famous treasure from his dangerous opponents. From the following notes taken from Jurakan's log, you can find where Jurakan went, what prize he acquired, and whom he outsmarted on each of his seven voyages?

1. Jurakan was on Jutía Island on an earlier voyage than the one when he bested Macana; his meeting with Macana was on an earlier voyage than the one on which he acquired the Inrirí Diamond.
2. The Jurakan's attack against Viejo Tomás wasn't the one in Baya or the one in Goya.
3. Queen Nasa wasn't the foe from whom Jurakan got the Mimé Emerald.
4. Neither Chief Fotu nor the Pirate Guanín was the victim when Jurakán gained possession of the Lamp of  Ditas.
5. Jurakan outfoxed General Buré on the voyage immediately after the one that took him to Amabala and immediately before the one that brought him the Mimé Emerald as a prize.
6. The Caribe Ruby wasn't acquired in either Natia or Tabatá.
7. Immediately after tricking Viejo Tomás on one voyage, Jurakan got the Sapphire of Sezu on his next adventure.
8. Jurakan acquired the Caribe Ruby on a voyage later than the one on which he had to defeat Zerena; the voyage on which he beat her was later than the trip to Lukiyó Island.
9. Jurakan's adventure in Baya immediately followed his journey to the land ruled by Nasa.
10. The encounter with Macana wasn't in fabled Tabatá.
11. On his 7th and last voyage, Jurakan stole the Princess Parlina, who became his wife. On his 1st voyage, Jurakan sailed to the Goya Island.
12. King Fotu wasn't the foe who possessed the Black Pearls of the Puño.
13. The Sapphire of Sezu wasn't the prize Jurakan outwitted the Grand Buré to get.

Am I a teacher or a leader?

The teacher drives students; the leader coaches them. 

The teacher depends upon authority; the leader on good will. 

The teacher inspires fear; the leader inspires enthusiasm. 

The teacher says ‘I’; the leader says ‘we.’ 

The teacher assigns the task, the leader sets the pace.  

The teacher fixes the blame for the learning breakdown; the leader fixes the breakdown. 

The teacher knows how it is done; the leader shows how. 

The teacher makes work a drudgery; the leader makes it a game. 

The teacher says, ‘Go’; the leader says, ‘Let’s go.'




This all comes from some unknown source about bosses and leaders, but as I read it, I saw all sorts of connections with teachers. The title to the post: Am I teacher or a leader, presents a hard question to answer. I know what I want to be, I know that I manage to be the leader at times, but I also know that at other times I am falling back on known teacher/student paradigms.


How about you?

On Shoulders of Giants: Historical Math Cubes

One of my favorite topics in math is actually HISTORY.

All too often we treat MATHEMATICS like some monolithic and pure subject, devoid of the quirks of human nature.

But as Spock (Star Trek) so clearly stated, "We are essentially irrational beings." And mathematics is a part of it all.

Each year I have my 7th graders do a mini research project I call "On the Shoulders of Giants" (a quote attributed to Isaac Newton). They are asked to research an important historical figure in mathematics and present this information on a 5 inch cube. I do not tell them how to construct this cube so that they have to muddle through the measurements and design. When all the cubes are completed we stack them up in class. It is a fun way to present information.

This year I had my university students do this project. Here are some samples:

They brought their cubes to class and without following a real script, each present the info to the group. It was interesting to hear how much they had found out and been impressed by their historical figure. The feedback I received was that the project was quite a bit harder than they had anticipated. In particular, it was challenging to create a sturdy 5 inch cube and then condense the information to the 6 faces in meaningful ways. One student found herself trying to layout the required information in a logical manner on the faces, which gave her a new way to think about what she had learned and wanted to present.

Overall, a successful project.

A song worthy of my middle schoolers attention

7th Grade: Working from Pythagorean Theorem and 30-60-90 Triangles.

I had great fun working with my students on the following problem:

Based on what we know about this triangle, find it's area and perimeter.

Winter Olympics Math with 8th Graders.

I had a terrible day with my two 8th grade classes on Wednesday. They came back from a 4 day weekend in a very distracted mood, while I came back all fired up to start work on factoring quadratic expressions. I finished the day probably as frustrated with them as they with me.

That evening I came home complaining about how the day went. I needed to vent and let go of the frustration. This morning I came to school determined not to repeat two difficult days. I decided to let go of factoring for this week.  Across my Twitter feed I saw some mention of NCTM Illuminations and an activity sheet for the Winter Olympics. While I haven't been following the Games much, I have been hearing my students talking about them. I looked over the activity sheet and saw that the problems were of high quality and not only taught something about the sport, but also led students through a mini math investigation along the way.

I thought, "What the heck."

I copied the sheets and gave them my students with explicit instructions to work on the problems, choose one they like, make a nice looking poster so that at the end we could make a school wide visual presentation.

It was remarkable to watch them work through the problems. Some struggled with the math, but most came to a deeper understanding, as evidenced not only by their questions but also by their product.

Here are some examples:

NYT Opinion on Negative Numbers

http://opinionator.blogs.nytimes.com/2010/02/14/the-enemy-of-my-enemy/

The Enemy of My Enemy

By STEVEN STROGATZ

It’s traditional to teach kids subtraction right after addition.  That makes sense — the same facts about numbers get used in both, though in reverse.  And the black art of “borrowing,” so crucial to successful subtraction, is only a little more baroque than that of “carrying,” its counterpart for addition.  If you can cope with calculating 23 + 9, you’ll be ready for 23 – 9 soon enough.


Subtraction forces us to expand our conception of what numbers are.  Negative numbers are a lot more abstract than positive numbers — you can’t see negative 4 cookies and certainly can’t eat them — but you can think about them, and you have to, in all aspects of daily life, from debts and overdrafts to contending with freezing temperatures and parking garages.At a deeper level, however, subtraction raises a much more disturbing issue, one that never arises with addition.  Subtraction can generate negative numbers.  If I try to take 6 cookies away from you but you only have 2, I can’t do it — except in my mind, where you now have negative 4 cookies, whatever that means.

Still, many of us haven’t quite made peace with negative numbers.  As my colleague Andy Ruina has pointed out, people have concocted all sorts of funny little mental strategies to sidestep the dreaded negative sign.  On mutual fund statements, losses (negative numbers) are printed in red or nestled in parentheses, without a negative sign to be found.   The history books tell us that Julius Caesar was born in 100 B.C., not –100.  The subterranean levels in a parking garage often have names like B1 and B2.  Temperatures are one of the few exceptions: folks do say, especially here in Ithaca, that it’s –5 degrees outside, though even then, many prefer to say 5 below zero.  There’s something about that negative sign that just looks so unpleasant, so … negative.


Perhaps the most unsettling thing is that a negative times a negative is a positive.  So let me try to explain the thinking behind it.How should we define something like –1 × 3, where we’re multiplying a negative number by a positive number?  Well, just as 1 × 3 means 1 + 1 + 1, the natural definition for –1 × 3 is (–1) + (–1) + (–1), which equals –3.  This should be obvious in terms of money: if you owe me $1 a week, after three weeks you’re $3 in the hole.


From there it’s a short hop to see why a negative times a negative should be a positive.  Take a look at the following string of equations:

–1 × 3 = –3

–1 × 2 = –2

–1 × 1 = –1

–1 × 0 = 0

–1 × –1 = ?


Now look at the numbers on the far right and notice their orderly progression:

–3, –2, –1, 0, ?

At each step, we’re adding 1 to the number before it.  So wouldn’t you agree the next number should logically be 1?

That’s one argument for why (–1) × (–1) = 1.  The appeal of this definition is that it preserves the rules of ordinary arithmetic; what works for positive numbers also works for negative numbers.

But if you’re a hard-boiled pragmatist, you may be wondering if these abstractions have any parallels in the real world.  Admittedly, life sometimes seems to play by different rules.  In conventional morality, two wrongs don’t make a right.  Likewise, double negatives don’t always amount to positives; they can make negatives more intense, as in “I can’t get no satisfaction.”  (Actually, languages can be very tricky in this respect.  The eminent linguistic philosopher J. L. Austin of Oxford once gave a lecture in which he asserted that there are many languages in which a double negative makes a positive, but none in which a double positive makes a negative — to which the Columbia philosopher Sidney Morgenbesser, sitting in the audience, sarcastically replied, “Yeah, yeah.”)

Daily Thought

"You have to be unique, and different, and shine in your own way."


 Lady Gaga

Some Teaching Mantras:

Life isn’t fair, but it’s still good.


Nobody has every said that life is fair, but it seems that in teaching, many people lose their way lamenting this fact. We are the lucky ones. We work in meaningful ways. We get automatic and lovely feedback on our efforts. We sing songs, paint pictures and enjoy a good fable now and then. True: we may have too many papers to grade, too many students to teach, and a heater that isn't working properly or a window that won't open. But it is all good!


When in doubt, just take the next small step.


Yesterday I was talking to a veteran teacher who was "tired" and unwilling to try out the latest math program. In fact, she was paralyzed by self doubt. Not understanding it, she decided not to try it. We need to embrace our doubts. Name them. Give life to them in order to see them for what they really are. Then we need to work with them. Taking the next "small step" is a way to respect our doubts and still move forward beyond them.


Don’t take yourself so seriously. No one else does.


I need this mantra: daily. Education is not about "life or death". My students can fail an exam and the sky won't fall. The quadratic formula is not life changing, even for me, who loves it. Homework not completed does not mean the student is destined for failure in life. Lighten up. Play a game. Forgive an assignment. Go outside on a sunny day. Don't do this every day (you'd shortchange education by that), but do on occasions.


Over-prepare, then go with the flow.


This allows the above to happen. Nothing is worse to always be "catching up".


However good or bad a situation is, it will change.


This is a lesson for professional longevity. We never arrive. We are always journeying. Our teaching companions move on. Our best students graduate. Our most favorite teaching unit becomes obsolete. We need to see this as an intrinsic "good". No fight the current. No lamenting the situation. And above all, if all seems good, remember that it won't always be so. That makes the inevitable change palatable. You will be a far stronger teacher in adversity.


If we all threw our problems in a pile and saw everyone else’s, we’d grab ours back.


Remember that. Daily.


The best is yet to come.


If we always live in the past, we miss the beauty. We shrivel up and fade away. Refuse to do this. It is good now and will be better later. Make it better.


No matter how you feel, get up, dress up and show up.


No more "manic Mondays". Each day is a gift. Dress your best! Even if you aren't sure, answer "Great!" when asked how your weekend went. Be excited outwardly and the excitement will seep inward. Complaining only leads to more complaining.




Life isn’t tied with a bow, but it’s still a gift.


Bottom line : others should be so lucky to be teaching such wonderful kids in such wonderfully inventive times. It isn't always easy, but it is always interesting and meaningful. It is all a gift. Remember that.

Exemplary 7th Grade POW Essay

This week’s POW was called “Join The Bacteria Team.” In the problem, Justin was in Mr. Bott’s math class, on Friday afternoon right before spring break, and he notices that the tables are dirty (there were 2 bacteria on the table originally). To solve this problem, I had to figure out how many bacteria would be on the table after spring break was over, if the number of bacteria doubled every hour. I had to make a table to show the growth of the number of bacteria as well as a graph. This problem was about using scientific notation and finding patterns.

My first step in this problem was to make the table. To do this, I selected a time in the afternoon for the growth to start (2 pm on Friday), and end (9 am on Monday). Then, I carefully filled the table out, starting at 2 and doubling each number, (for example 2+2=4, 4+4=8) making sure to use scientific notation when the numbers got larger than a couple thousand. After filling out all of the table, and rechecking my work multiple times, I started making the graph. To make the graph, I split up the data into sections, meaning that I graphed for everyday separately. This allowed me to see a pattern that the data goes up at an even pace very quickly (see sample of graph and table below). I used a calculator to ensure that all my calculations were correct.

The total number of bacteria after spring break was about 1.13 times 1071. I know my solution is correct because all the calculations were check and I compared my results with other students after I had solved the problem. There are other solutions to this problem because some people might have chosen to start at 1 pm or later and chosen to end at 8 pm or earlier. Plus, since this problem will never have an exact answer, there will be many variations.

This problem was very interesting to me because I had never really thought that bacteria grew that quickly till I solved the problem. This problem challenged me because at first I did not realize that the graph would be so difficult to make, especially when there are so many huge numbers. This POW reminded me of counting beans in 1st grade, because there were so many beans and the number just kept getting larger and larger. I managed my time fairly well, however I could have done more work over the week rather than the weekend. Next time, I would like to try and find a rule for the POW which will make it easier to complete the table.

This problem is to figure out the lengths of 3 different routes bisecting the circular Caldera rim and valley. The pertinent information is that the Kid’s village is in the exact center of the circular valley which has a 100km radius. The northern-most point on the rim is Sentry Point 1. Hope Lake is 30km south of there. Sentry Point 2 is due west of Hope Lake. The Adult’s village is 15km south of Sentry Point 2. We need to figure out the direct overland distance (“as the crow flies”) from Sentry Point 1 to the Adult’s village. We also need to figure out the distance on the usual adult route from Sentry Point 1 to the Adult village, going south to Hope Lake, then West to Sentry Point 2 and south to the Adult village. Finally, we also need to calculate the distance if you went on the rim from Sentry Point 1 to Sentry Point 2 and then South to the village. The main math topic imbedded in this problem is geometry.

My first step in solving this problem was to draw out the situation on a diagram/map. Heres what it looks like:

I decided to figure out the easiest thing first which was the usual adult route, since I knew the distance from Sentry Point1 to Hope Lake is 30km and the distance from Sentry Point 2 to Adult’s village is 15km. I needed to figure out the distance from the Hope Lake to Sentry Point 2. I realized that this length represented one side of a right triangle formed by Hope Lake, Kids Village and Sentry Point 2. I knew the distance from Hope Lake to Kids village was 70km. I also knew the distance from Kid’s village to Sentry Point 2 was 100km, because it is the radius of the circle. I used the Pythagorean theorem as follows:

4900 + b2 = 1000
b2 = 5100
b=71

So I added the 3 lengths and got the total usual adult path was 116km.



Next, I decided to figure out “as the crow flies” distance. I realized that this distance was the hypotenuse of yet another triangle formed by Sentry Point 1, Adult’s village and a point 15km due south of Hope Lake. I knew the lengths of two of the sides were 45km and 71km. I used Pythagorean theorem (see above) to figure out that the hypotenuse was 84km.





Finally, I needed to figure out the distance in the around-the-rim route. I found a formula online that was based on angles that was like this:

I needed to figure out the angle of the two lines between Kids village and Sentry Point 1 and Kids village and Sentry Point 2. I decided to use a protractor to measure the angle after measuring my lines carefully. Here’s what my diagram looked like:

With the protractor I measured the angle at 45 degrees. Plugging the numbers into the formula. I got:

200km (the diameter) * 3.14159 * 45/360 = 78.5km or 79

I then added the extra 15km south to the Adult’s Village from Sentry Point 2 to 79 and got a total of 94km. The resources I used to solve the problem were my Mac, keynote (for diagrams), the Internet (for formulas) and my Mom.



My solution is that the longest route is the usual adult route at 116km, the next longest is the rim route at 94km, and the shortest was the “as the crow flies” at 84km. I think my answers are correct. I do not think there are other answers. It makes sense that the shortest route would be the most direct (a line between 2 points).

This problem was hard because it took a while to discover the triangles that could lead to the answer. I also wasn’t sure using the protractor was the only way to get the arc length, but it was the only way I could figure out. Because I used a protractor, my answer might be less precise than if I had just used a formula. I think I managed my time poorly, but it was all I could do as I had a lot to do. 

7th Grade: Theodorus' Wheel and Art Project

Time to take Pythagorean Theorem and the idea of square roots to a new level.

I love the patterns discovered so clearly while working on Theodorus' Wheel in class.

After they make the wheel, I ask them to imagine them as part of an art project. They make hats, baby carriages, snail shells, Princess Lea's hairdo etc.

Here is what Theodorus' Wheel is:

The Wheel Of Theodorus, or Square Root Spiral, is a rather simple geometric construction that allows you to construct line segments with length of the square root of any integer. Observe the figure below.

The construction is fairly simple. Starting with a 1-1-root 2 right triangle, use the hypotenuse as one leg, and add a perpendicular segment of length 1 to create another right triangle. Pythathagoras says that the resulting triangle has a hypotenuse of root 3. And so it goes. And it creates this very nice spiral effect. The main point is that complex numbers provide a very convenient and very elegant way to express the geometric relationships in the figure.

Platonic Solids will be covered tonight in my university class

Platonic solids were named after Plato, who was one of the first philosophers to be struck by their beauty and rarity. But Plato did more than admire them: he made them the center of his theory of the universe. 

Plato believed that the world was composed entirely of four elements: fire, air, water, and earth. He was one of the originators of atomic theory, believing that each of the elements was made up of tiny fundamental particles. The shapes that he chose for the elements were the Platonic solids. 

In Plato's system, the tetrahedron was the shape of fire, perhaps because of its sharp edges. 

The octahedron was air. 

Water was made up of icosahedra, which are the most smooth and round of the Platonic solids. 

And the earth consisted of cubes, which are solid and sturdy. 

This analysis left one solid unmatched: the dodecahedron. Plato decided that the it was the symbol of the "quintessence," writing, "God used this solid for the whole universe, embroidering figures on it." 

Plato's description of the universe made a deep impression on his disciples, but it failed to satisfy his most illustrious student, Aristotle. 

Aristotle reasoned that if the elements came in the forms of the Platonic solids, then each of the Platonic solids should stack together, leaving no holes, since for example water is smooth and continuous, with no gaps. But, Aristotle pointed out, the only Platonic solids that can fill space without gaps are the cube and the tetrahedron, hence the other solids cannot possibly be the foundation for the elements. His argument struck his followers as so cogent that the atomic theory was discarded, to be ignored for centuries.

Aristotle's analysis contained a famous error: the tetrahedron does not fill space without gaps. 

Incredibly, Aristotle's mistake was not discovered for more than 17 centuries. Aristotle was so highly esteemed by his followers that they confined themselves to trying to calculate how many tetrahedra would fit around one corner in space, rather than considering the possibility that the great man was mistaken.

Funny how one guy could have SUCH in impact on history.