# Tag Archives: math

## Congratulations, JSHS Student Researchers!

Last month, students from across the state of Iowa attended the Iowa Regional Junior Science and Humanities Symposium (JSHS), hosted by the Belin-Blank Center at the Marriott Hotel in Coralville, Iowa.

JSHS is a collaborative effort with the research arm of the Department of Defense and is designed to challenge, engage, and publically recognize high school students conducting scientific research in science, technology, engineering, or mathematics (STEM).  JSHS aims to prepare and support students to contribute as future scientists and engineers – conducting STEM research on behalf of, or directly for, the Department of Defense, the Federal research laboratories, or for the greater good in advancing the nation’s scientific and technological progress.

Students completed an original research project and submitted a research paper to the regional competition. The authors of the top 18 papers were invited to compete for scholarships and recognition by presenting their results before a panel of judges and an audience of their peers.  Students also toured various labs and facilities at the University of Iowa to hear about cutting edge research, potential career paths, and student opportunities.

After an intensive day of presentations, the judges had the difficult task of selecting five finalists based on their research papers and presentations:

1st place: Megan Ertl (Beckman Catholic High School) – “Quantification of Muscle Accelerations to Interpret Individual Fatigue as an Industrial Application

2nd place: Cheryl Blackmer (Ballard  High School) – “Development of a LAMP Assay for the Detection of Powassan Virus”

3rd place: Pranav Chhaliyil (Maharishi School of the Age of Enlightenment) –  “Metagenomics Analysis of Bedtime Oral Cleaning by the Novel GIFT Method, Shows a Reduction in Dental-Damaging Bacteria”

4th place: Aaron Wills (Central Lee High School) – “Engineered Environmental Containment: “Using Lemna minor L. to Reduce Nitrate Levels in Aquatic Environments”

5th place: Brianna Cole (Valley High School) – “Cumulative Effects of Recurrent Amygdala Kindled Seizures on Respiratory Function”

Additional presenters, who were winners by virtue of having their papers accepted, included Allison Brasch (Waterloo West High School), Mason Burlage (Beckman Catholic High School), Ava Depping (Madrid High School), Serenity Haynes (Central Lee High School), Sean Kluesner (Beckman Catholic High School), Pearl Krieger Coble (Winfield-Mt. Union High School), Kayla Livesay (Van Buren High School), Kathryn McCarthy (Sioux City East High School), Evylin Merydith (Keokuk High School), Tyler Montgomery (Kennedy High School), Elizabeth Smith (Waterloo West High School), Laura Stowater (Algona High School), Shelby Westhoff (Beckman Catholic High School).

The top five finalists will attend an expense-paid trip to the JSHS National Symposium next month in Hunt Valley, MD to present their research and compete for additional prizes.

To see all the fun we had, including tours of the IIHR – Hydroscience & Engineering, Iowa Flood Center, and Additive Manufacturing-Integrated Product Realization Laboratory (AMPRL) in the University of Iowa Department of Mechanical & Industrial Engineering, check out our full photo album! Congratulations to all, and good luck at Nationals!

## Curious About Research?

Do you know academically talented teenagers who show curiosity or promise in doing research, or are you one yourself? Then you need to know about the Perry Research Scholars Institute (PRSI), where students can experience lots of different types of research happening at a top public research university!

Students in grades 8–10 (academic year 2017–2018) may apply for the Perry Research Scholars Institute (PRSI), a two-week residential summer academic program at the University of Iowa’s Belin-Blank Center.

At PRSI, students will participate in seminars with university faculty, tour their research facilities, and study their publications. While students will spend some of their time learning advanced lab techniques, they will not be conducting original research in this program. Rather, they will be granted an exclusive, behind-the-scenes look at research while it’s happening, in fields such as anthropology, business, education, engineering, medicine, psychology, sustainability, and more. This “backstage pass” approach will help students develop an understanding of research that extends well beyond bench science.

During off-hours, students can expect plenty of fun getting to know other bright teenagers who are also interested in research! They will even experience an authentic taste of life on a university campus, complete with two weeks of living with a roommate in the residence halls. Evening activities include special seminars, off-campus field trips, and cultural and recreational activities. Social events are scheduled, and students will be granted access to the University of Iowa libraries, computer facilities and study areas.

Don’t miss this unique chance to see how research works, up close and personal; experience college life for two weeks; and meet new friends with similar abilities and interests! Applications are open through March 16 at www.belinblank.org/students. The program will run from July 8–July 20, 2018.

Looking for more research programs for high school students? Check out the Junior Science and Humanities Symposium (JSHS) and the Secondary Student Training Program (SSTP). PRSI is great preparation for programs like these!

## Hints for the Yellow Brick Road

Samuel J. Ferguson, a mathematics PhD student and friend of the Belin-Blank Center (check out his other posts here), made a video about how to solve a simplified version of a problem posed by the Yorkshire Post in 1976.  Now he’s offering a few hints as to how to solve this tricky problem.

It is a long journey, through a country that is sometimes pleasant and sometimes dark and terrible. . . . The road to the City of Emeralds is paved with yellow brick, so you cannot miss it.

—L. Frank Baum, The Wonderful Wizard of Oz

‘Pooh’, said the wizard, ‘is where we are now. The railway starts here and runs in a straight line via 39 intermediate and equally spaced stations to the terminus at Oz.

‘Unfortunately, the railwaymen are on strike, so you’ll have to go by bus. The bus goes along the Yellow Brick Road, which runs in a straight line from here to the outlying village of Bah, where it turns through a right-angle and goes in a straight line back to the first railway station after Pooh. From there it goes in a straight line to the next outlying village, where it turns through a right-angle and proceeds in a similar zig-zag fashion all the way to Oz, alternately calling at railway stations and outlying villages. Each of the 80 straight stretches of road is a different whole number of miles long. Rail distances are also whole numbers of miles.

‘The fare is one ozzle per mile, but you needn’t be alarmed, as all distances are as short as they can be.’

I was alarmed, and it turned out that I had good reason to be. My money was running short, for the Wizardry of Oz had been suffering from hyper- inflation lately.

Unfortunately the Wizard had vanished before I could ask him the vital question, how long is the yellow brick road?

—1976 Yorkshire Post Christmas problem of the Yellow Brick Road, reproduced courtesy of the Yorkshire Post

What if there’s only one intermediate rail station between Pooh and Oz, instead of 39? If you go to YouTube and search for “How long is the Yellow Brick Road?” you’ll see my video answering this. It has been viewed in more than a dozen states and in such countries as England, India, Germany, and Finland! Here is a drawing of my solution. How did I find it?

The thing to notice, when there’s an intermediate station, is that the rail distance c between Pooh and the station is the same as the distance c between the station and Oz (I found that c = 25 is the smallest c that works). The rail distance from Pooh to Oz is now 2 × c. We let

$a^2 + b^2 = c^2$,

with a the distance from Pooh to Bah, and b that from Bah to the station (this relation between a, b, and c comes from the Pythagorean theorem). Moreover, with f the distance from the station to the next village, and g the distance from there to Oz, we have (since c is also the distance from the station to Oz)

$f^2 + g^2 = c^2$.

(We skipped a couple of letters, in case you like to use d for “distance” or e for “Euler’s number,” roughly two and a half.) Now we can write out our problem precisely.

Problem 1 (Yellow Brick Road with 1 station). What is the smallest positive whole number c (rail length) such that

$a^2 + b^2 = c^2$

and

$f^2 + g^2 = c^2$

for some positive whole numbers a, b, f, and g, all different? Having found c, what is the sum a + b + f + g (length of the Yellow Brick Road)?

The difficulty is that a, b, f , and g are all different. Otherwise, we could use c = 5 (because $3^2 + 4^2 = 5^2$), taking a = f = 3 and b = g = 4, getting 3 + 4 + 3 + 4 = 14 miles for the length of the Yellow Brick Road. Since we actually need $a\neq b, a\neq f, a\neq g, b\neq f, b\neq g$, and $f\neq g$, it’s hard to see in advance how to solve this. The number c = 5 doesn’t work because, if we took a = 3 and b = 4, then the only positive whole numbers less than 5 left to use for f and g (which must be different) would be 1 and 2, and $1^2 + 2^2$ is 5, not 52.

Since we can’t use c = 5, we have to use bigger numbers. Will 6 work for c? It turns out that $6^2$, like $1^2$ through $4^2$, can’t be written as a sum of two squares at all (let alone two ways). We’ve gone from an uncommon property (being a sum of squares) to a property that’s genuinely rare (being a sum of squares two different ways). It’s hardly clear that there are any numbers with the latter property, but a calculator-aided search showed me that there are some. The smallest number which is a sum of two positive squares in two different ways is 50. Indeed, 50 = 2 × 25, so certainly $5^2 + 5^2 = 50$, and since 50 is one more than the square 49 (the square 7 × 7), we have $7^2 + 1^2 = 50$ also. Nevertheless, we can’t get a suitable c from this, because there’s no whole number c with $c^2 = 50$; c = 7 is too small, c = 8 is too big, and there are no whole numbers in between.

There are so few squares (1, 4, 9, 16, 25, 36, 49, 64, 81, 100, . . . ) that perhaps the smart thing to do is to test them individually for the property we want (“sum-of-two-squares-two-different-waysness”). How do we test them? Grab a calculator and a square, such as $10^2$. If this number, 100, is a square then it’s of the form $a^2 + b^2$  for some positive a and b less than 10. So if, at some point, you subtract a square from 100 and obtain a square (like when you write 25 9 = 16), then you can write 100 as a sum of two squares (like when 9 + 16 = 25). Let’s do this for 100.

• 100 81 = 19 (not on our list of squares);
• 100 64 = 36 (success!), so 64 + 36 = 100, that is, $8^2 + 6^2 = 10^2$;
• 100 49 = 51 (not a square);
• 100 36 = 64 (another winner!), so $6^2 + 8^2 = 10^2$;
• 100 25 = 75 (no luck);
• 100 16 = 84 (nope);
• 100 9 = 91 (nada);
• 100 4 = 96 (still nothing);
• 100 1 = 99 (no dice).

If you follow this procedure and never get a square from such subtractions, it must be because what you started with can’t be written as a sum of two squares at all. Let’s see this for 36. We successively subtract off each of the positive squares smaller than 36, looking for a square answer.

• 36 25 = 11;
• 36 16 = 20;
• 36 9 = 27;
• 36 4 = 32;
• 36 1 = 35.

In no case did our procedure give us a way to write 36 as a sum of two squares. The only remaining possibility is that it must be impossible to write $6^2$ as a sum of two squares, and our calculations prove this fact.

Returning to our problem, we ask if c = 10 (and the square $10^2 = 100$) gives the answer. While $8^2+6^2 = 10^2$ and $6^2+8^2 = 10^2$, these aren’t sufficiently different to finish the problem. While $8\neq 6$ and $6\neq 8$, it’s not true that all four of the bases (8, 6, 6, and 8) are different.

Extending our list of squares ($11^2 = 121$, $12^2 = 144$, and $13^2 = 169$) and following our procedure (you’re invited to do this with a calculator), we find that neither 11 nor 12 can be written as a sum of two squares, while $169 - 144 = 25$ shows that $12^2 + 5^2 = 13^2$. Similarly, $5^2 + 12^2 = 13^2$, which is a neat fact (the Egyptians knew it) but not different enough to solve our problem. Continuing on in this way, extending the list and calculating (the author didn’t go beyond 25), we achieve no success for would-be c’s, except for the bases 15, 20, and 25. In fact, $15^2 - 12^2 = 81$ and $20^2 - 16^2 = 144$, so

• $9^2 + 12^2 = 15^2$ and
• $12^2 + 16^2 = 20^2$; similarly,
• $15^2 + 20^2 = 25^2$,

as you can check with your calculator. This news isn’t much of a surprise, however, since we can get these more quickly by taking $3^2 + 4^2 = 5^2$ and multiplying all of the bases by 3, 4, and 5, respectively. The bases 15 and 20 don’t solve our problem but, remarkably, right when we start our procedure for 25 we get

• $25^2 - 24^2 = 49$,

showing that $7^2 + 24^2 = 25^2$. We see that taking $a = 15$, $b = 20$, $f = 7$, and $g = 24$ solves our problem. Our procedure shows that the number $c = 25$ is the smallest whose square can be written as a sum of two squares in two (completely) different ways. Thus, the length of the Yellow Brick Road, the sum of the lengths of the four straight stretches, is $15+20+7+24 = 66$ miles, as you can check. Problem solved!

What about the original problem, where there are 39 stations? What is the smallest c such that $c^2$ is a sum of two positive squares in 40 different ways? We could start by looking at numbers like 5 and 13, whose squares can at least be written as a sum of two positive squares in one way, namely

• $3^2 + 4^2 = 5^2$ and
• $5^2 + 12^2 = 13^2$.

Using the “multiply the bases” trick, we can find c’s which can be written as sums of squares in more than one way. For example, mutiplying all of the bases in the first equation by 13 gives (since $3\times 13 = 39$, $4\times 13 = 52$, and $5\times 13 = 65$) the equation

• $39^2 + 52^2 = 65^2$,

while multiplying all the bases in the second equation by $5$ gives

• $25^2 + 60^2 = 65^2$.

On the other hand, $65^2 = (5\times 13)\times (5\times 13) = 5^2\times 13^2$, so, by both equations, $65^2 = (3^2 + 4^2)(5^2 + 12^2).$
Then, by using the incredible pair of algebraic identities (from Wikipedia’s entry on the “Brahmagupta–Fibonacci identity”)

• $(a^2 + b^2)(c^2 + d^2) = (ac + bd)^2 + (ad - bc)^2$ and
• $(b^2 + a^2)(d^2 + c^2) = (bd-ac)^2 + (bc + ad)^2$, with $a = 3$, $b = 4$, $c = 5$, and $d = 12$,

we get

• $65^2 = 63^2 + 16^2$ and
• $65^2 = 33^2 + 56$.

It turns out that $c = 65$ solves the problem of the yellow brick road when there are 3 stations (8 stretches of road)! In that case, the length of the road is $39 + 52 + 25 + 60 + 63 + 16 + 33 + 56 = 344$ miles. With the “multiply the bases” trick and the above two algebraic identities, you can really find a c, in fact the smallest such c, such that $c^2$ is a sum of positive squares in forty completely different ways—solving the original problem. As a final hint, if you get stuck, try applying what we’ve just done to the equations $15^2 + 8^2 = 17^2$ and $21^2 + 20^2 = 29^2$.

## Furthering STEM Education

Recently, we sat down with the Belin-Blank Center’s STEM Initiative Team to talk about their vision for the future of STEM (Science, Technology, Engineering, and Mathematics) at the Center, in Iowa, and beyond. The team is made up of Kate Degner, Administrator for IOAPA and SSTP; Leslie Flynn, Clinical Assistant Professor, Science Education, and Administrator for STEM Initiatives, Belin-Blank Center; and Lori Ihrig, Administrator for Summer Program Faculty and Commuter Programs.

Can you talk a little bit about your background in STEM?

Kate: I began my teaching career in 2003 in Lone Tree, Iowa. I was the only regular education 9-12 mathematics teacher in the building, meaning I taught every mathematics course offered from Consumer Mathematics to Pre-Calculus. During the summer of 2005, I was invited to be part of a 6-person writing team for the University of Chicago Mathematics Project 3rd edition Algebra textbook. Shortly after completing that project I began teaching upper-level mathematics courses (AP Statistics, Trigonometry, Pre-Calculus, and Discrete Mathematics) in Williamsburg. During that time I also went back to school and earned my M.A. in Mathematics. I’ve also had experience teaching concurrent credit classes, as well as night classes at a community college. During the last few years I also taught Calc I and II at the high school and college levels. Last year I graduated from the University of Iowa with my PhD in Curriculum and Supervision, with an emphasis on Mathematics Education and Educational Leadership.

Leslie: I have worked in STEM education for 25 years as a high school and college science and mathematics instructor, school administrator, professional development director, and professor in our STEM K-12 licensure programs. I became interested in STEM as a 4th grader engaged in specialized courses in STEM. I was fortunate to have programs where I could attend college courses and STEM competitions while still participating in school athletics and general education courses. My exceptional STEM female teachers opened my mind to the idea that girls can excel in STEM and they provided me with the skills and confidence to pursue college degrees in Chemistry.

Lori: I graduated with a B.S. in Science Education in 1999 from the University of Iowa and worked as a grades 7-12 science teacher for the Williamsburg Community School District. In Williamsburg, we participated in Iowa Excellence through a partnership with the Belin-Blank Center, and I began teaching an AP biology class. During this time period I also worked with the Center, teaching for the Junior Scholars Insitute (JSI) and WINGS, and earned my MS in Science Education from the University of Iowa. In 2007, I began working at ACT writing science curriculum and facilitating science teacher professional development for Quality Core, a project that was a partnership between ACT, the Gates Foundation, and the National Governors Association. In 2010, I began working on my doctorate in Curriculum and Instruction at Iowa State University with an emphasis in Science Education. My doctoral emphasis is on secondary science teacher education and the implementation of reforms-based science instruction by novice science teachers.

## Free Webinar on Math Acceleration

Dr. Susan Assouline, Associate Director, Belin-Blank Center

Register here for a free Webinar – You Already Know This: How to Use Your Teaching Skills & Current Resources with Math-Talented Students (Grades 3-7). On Thursday, April 14, from 3:30 to 4:15 PM (US CDT), Kate Degner, a doctoral candidate in math education, will demonstrate a technique for accelerating the math curriculum. I will give a brief explanation of the reports generated from our new online system for making informed decisions about math acceleration, IDEAL Solutions® for Math Acceleration.

As you know, mathematically talented students have varying academic profiles. This aspect is described in Developing Math Talent (Assouline & Lupkowski-Shoplik, 2011); you can read about one very talented student, Zach, here.

## Supporting STEM Innovation

Dr. Susan Assouline, Associate Director, Belin-Blank Center

Almost daily we hear about the weak performance of American students in math and science when compared to their international counterparts.

Many of the national reports that convey this message have issued a “Call to Action.”  In 2008, the  National Mathematics Advisory Panel released its final report about math education in the US and  recommended that districts ensure that all prepared students have access to algebra by Grade 8.  For general education students, this is great – but for mathematically talented students, the need for challenging math comes well before Grade 8.

The Belin-Blank Center is  responding to the “Call to Action” with a brand new website: IDEAL® Solutions for Math Acceleration.  This website is designed to assist parents and educators of mathematically talented students in understanding the degree to which their students would benefit from additional challenge.  After entering data about the student, parents and educators receive a report that provides individualized recommendations for the student.  This report also offers a detailed summary of the research related to acceleration and documents the information about the student for both parents and educators.

An IDEAL® Solutions for Math Acceleration report provides a starting point in the discussion about how to meet a mathematically talented student’s academic needs.  To learn more, visit www.idealsolutionsmath.com.

If you are an educator, contact us about becoming an IDEAL® Solutions for Math Acceleration School.