Student Demonstrations of the Pythagorean Theorem

Students were asked to demonstrate the Pythagorean Theorem, using 5 geometric figures that each team was given.  (The problem is presented in an earlier post from today.  Scroll back!)

Here is a poster that one team submitted.  It clearly shows how to take the 5 geometric figures and place them to create a visual image of a-squared plus b-squared.  Then they take the same 5 figures and create a picture of c-squared:

An Eighth Grade team submitted this video as a way to demonstrate the theorem.  This video is the essence of mathematical elegance!  Tell us what you think:

Student Solutions to the Palindrome Problem

As we wrap up I Love Math Season for 2017, we want to share some excellent student work.  For each problem, student teams submit solutions in whatever form they found most communicative. They are encouraged to be thoughtful and creative, finding the best ways to explain their mathematical thinking.  They are encouraged to use whatever form helps communicate the mathematical essence of the problem.

Students worked hard on the Palindrome problem (see the January 26 post).  The problem could not be more simply stated:  How many of the numbers between 10,000 and 100,000 are palindromes divisible by 36?

Here were two very creative submissions from the Sixth Grade:

You can clearly see the 22 palindromes identified on the wooden sculpture.  And the sculpture veritably screams the count.  This team was thoroughly successful explaining how to derive the palindromes that meet the requirements.

Another Sixth Grade team came up with a machine to generate "5-digit palindromes, divisible by 36":

This team recognized that to be divisible by 36, the rightmost wheel must be an even number, so they only gave the spinner even options.  If it is to be a palindrome, then the first digit must be the same as the last.  Therefore the first spinner also contains only even numbers.

If the palindrome is to be divisible by 36, then it must also be divisible by 4 and 9.  The team was figuring out how those demands would dictate the second, third and fourth digits.  While all nine digits are needed on the inner-spinners, limits begin to arise once the outer digits are set.  Using this machine, all 22 palindromes can be generated, working from the outside in each time.   (The final two digits need to be divisible by 4; and the middle number must be set so that the sum of all five digits is divisible by 9.  When these criteria are met, the palindrome is divisible by 36!)

Who would think divisibility rules could come in so handy?!?

The Final Problem -- Pythagorean Theorem

The fourth and final problem that students were challenged to solve during I Love Math season was this classic:

Demonstrating the

Pythagorean Theorem

You know what a right triangle is.  Right?

It has one right angle. 

(Could it have more than one?  That is worth thinking about for a minute.)

A right triangle always has one side that is longest.

The longest side is always across from the right angle.

It has a name:  hypotenuse.

The other two sides will always be shorter than the hypotenuse.

They always come together to “create” the right angle.

A long time ago, a mathematician named Pythagoras figured out that there is always a relationship between the lengths of the three sides of a right triangle.  We could measure the lengths of the two short sides and call those lengths a and b.  We could measure the length of the hypotenuse and call it c.

Here is the relationship:

a² + b² = c²

Pythagoras liked to picture things geometrically.  So the way he really thought about this formula was like this:  If I take the right triangle, and if I draw a square on each of the three sides, then the areas of the two smaller squares will always add up to equal the area of the big square.

Demonstrate the Pythagorean Theorem using the square and the triangles you were given.

If it helps to see another set of geometric figures, here is another set.  These are slightly different than the ones above, but they work just as well to demonstrate the theorem.  Enjoy!

Distinguished Mathematicians Coming Soon

I Love Math Day is next week!  February 14 is set aside to celebrate our love for mathematics, all day long.  Three guests will visit the school and discuss how they use mathematics in their careers, hobbies and daily life activities.  We are so grateful that they want to spend this day with Mustard Seed School students.  They are giving us the generous gift of time and wisdom.

Emily Magaletta is a registered nurse working in NY.  She will be sharing how numbers and mathematics help health professionals care for their patients.  She received her BSN from the College of Misericordia, and her MSN from New York University.  She has also earned her Nurse Practitioner license.  She has worked at NYU since 2001, primarily caring for patients before or after heart surgery.  Often her patients are unconscious, so various monitors and the numbers they produce are essential to assessing the patients.  Emily is the mother of two Mustard Seed students:  Dominic in second grade and Alexander in the pre-school.

Dr. Kurt Schaefer is Chairman of the Economics Department at Calvin College and has directed the Calvin Center for Social Research.  In addition to his degrees in mathematics and economics, he has a Master of Divinity from Calvin Seminary.  He has served as the Secretary/Treasurer of the Association of Christian Economists and managing editor of Faith and Economics.  He co-authored the book The Uses and Misuses of Data and Models, concerning the appropriate use of mathematics, data and models in research and analysis.  He has interests in social justice issues such as fair pay, food, and poverty:  What are the best ways to get food to countries that don't have enough?  What are the best ways to help people who are poor? 

Dr. Susan Wieler is also an economist!  She was a panelist on our first I Love Math day in 2007.  We are grateful for her return.  Dr. Wieler did her undergraduate work in Philosophy at Colgate University, received her Masters in Educational Philosophy from Rutgers, and her Ph.D. in Economics from NYU.  She studied labor and poverty issues for the New York Federal Reserve Bank:  What are the causes of poverty in NY City?  What are the determinants of wages for foreign-born workers in upstate NY?  As a Senior Policy Associate for the Citizens' Committee for Children of New York, she worked for the well-being of NY children and managed a task force for the prevention of homelessness.  Dr. Wieler and has written a variety of articles regarding, inter alia, housing, tax and income issues.

Frequently Asked Questions

Today's post is designed to answer questions that frequently come up as students work on problems.  Problem solving is a huge part of the preparation for I Love Math Day, so students and parents often want to know what happens and what the rules are:

How many problems are there?
Four.  One each week for a month.

How long do I have on each problem?
About a week.

Can I just tell you my answer?
No.  Each team should turn in a submission for each problem.  A submission can be almost anything.  A poster.  A slide presentation.  An essay.  A puzzle.  A book.  A piece of artwork.  One team wrote a play to explain one of last year's problems.  The play explained the problem and how it could be solved.  One team created a bound, 12-page booklet.  Some submissions are puzzles.

The key is to think of a format that most supports the mathematical issues at play, and display the insights that lead to a solution.  The answer is rarely as important or interesting as the thought process behind it.

Can I ask my parents for help?
Of course!  We hope that students will spend some time at home puzzling over these problems.  A really good problem might come up at the dinner table.  That is success!  It is great for children to see their parents and siblings taking an interest in a math challenge.  No need to solve the problem.  No need to come up with profound insights.  It is enough to enjoy a few minutes thinking, asking questions, drawing pictures, or trying one or two ideas out.

Can I do research online?
Would a mathematician do that?  Yes, of course.  So you can also.  After all, you are a mathematician!  More than being permitted, it is encouraged.  Anything that a mathematician would do is what you should do to make progress on the problem.

So is anything NOT permitted?
Well, yes.   A mathematician would never, never turn in someone else's work as their own.  Submissions that students present must be their own.  All of the thinking and all of the work should clearly be the student's.  Each child should be able to explain in their own words how the problem works, how they arrived at the solution, and the mathematical ideas that underpin their solution.  

Palindromes! Palindromes! Problem 3

This problem could not be more simply stated.  Yet it requires layers and layers of thought.  It can involve lots of calculation, computation and heavy lifting;  some students are using their laptops to generate lots of numbers and narrow down the possibilities;  some are making progress by persistent, careful, creative thinking.  A good strategy can spare you a lot of heavy lifting.  The best problems often work this way.

PalindromemordnilaP

Consider all the palindromes between 10,000 and 100,000. 

(You remember what a palindrome is.  It is a series of letters or numbers that read exactly the same forward and backward.  So 10,000 is not a palindrome.  10,001 is.)

Of all the palindromes between 10,000 and 100,000, how many are divisible by 36?

The Boxing Tournament! Problem 2

Student have enjoyed working on The Boxing Tournament, problem number 2 as we prepare for I Love Math Day.  In hopes that this challenge can get discussed at home, here is the problem.  If you have any questions, any student in grades 6 through 8 can help.

The Boxing Tournament

An elimination boxing tournament was organized.  There were 114 participants, so the first round had 57 matches.  (In this tournament, no match results in a tie.  Each match determines a winner and a loser.)

The 57 winners were paired up for the second round, resulting in 28 matches, with one boxer left over.  That lucky boxer was allowed to proceed to the next round automatically.

This procedure was followed until one boxer won the tournament. 

How many total matches were required in order to determine the winner?

Suppose there were 199 boxers participating in the tournament.  How many total matches would be required?

Can you generalize?  If there are  n  boxers total, how many matches will be required to determine the winner?