Rainy Day Activity: Make A Möbius Strip

Rainy Day Activities are back! This month, we explore the mathematical mystery of the Möbius Strip, which is which is a surface with only one side and only one boundary. By twisting a strip of paper 180 degrees, a circle with an interior and exterior becomes a continuous loop. Click here for a post from last fall showing how Project EXCITE students used Möbius strips to explore the relationship between art and science.

by Loretta Rice

Caution:  This activity starts small, but can lead to a colorful pile of fun!

The exploration of the Mobius Strip often comes up after discussing topology in math class.

Vocabulary: to·pol·o·gy/təˈpäləjē/

Noun: The study of geometric properties and spatial relations unaffected by the continuous change of shape or size of figures.

The investigation into the Möbius Strip will lead into a lot of questions that need to be answered. “What is going on here?” “How do these connect?” “What will happen if I change this?”

Understanding the relationship between objects and the way the objects are made is what the Möbius Strip is all about.

Materials Needed:

  • plain paper strips
  • scissors
  • tape
  • pencil
  • magic marker
  • flat surface to work on

Instructions:

1.  Start with a long rectangle. The exact width and length is not that important.

2.  Mark each corner in order. (ABCD)

3.  Give the rectangle a half twist.

4. Using a piece of tape, join the ends so that A is matched with D and B is matched with C.

InvestigationWhy does the Möbius Strip have only one side and one edge?

1.  Start midway between the edges of a Möbius Strip and draw a line down its center.  Continue the line until you return to your starting point. Did you ever cross an edge?

2.  Next, hold the edge of a Mobius Strip against the tip of a felt-tipped highlighter pen. Color the edge of the Möbius Strip by holding the highlighter still and just rotating the Möbius Strip around. Were you able to color the entire edge?

3. Now, with scissors cut the Mobius Strip along the center line that you drew. Then draw a center line around the resulting band, and cut along it. Did you predict what would happen?

Further Investigation:

Now, think about what would happen if you cut down the center of your Möbius strip. An ordinary paper ring cut in half would give you two separate rings, right?

If you cut down the center of a Möbius strip, what happens?

For yet another awesome result, try cutting the strip one-third of the distance from the edge. Have your camera ready to document this surprise!

Modifications for Younger or Older Students:

After some practice you can experiment with different flexible materials to create Möbius jewelry or art work for hanging or framing.

Additional Resources and Links:

Math is good for you!: The history and theory behind the Möbius strip.

Videos showing the different experiments that can be done with the Möbius strip:

http://www.youtube.com/watch?v=BVsIAa2XNKc&feature=related

http://www.youtube.com/watch?v=IRVOwuHU-M0

http://www.youtube.com/watch?v=6dEnz4tSKNk&feature=related

Have you ever made a Möbius strip before?

Since Fall 2007, Loretta Rice has taught math and science courses for Project EXCITE and Gifted LearningLinks. Some of her past courses include: “It’s a Puzzlement,” “Brain Teasers,” and the upcoming “The Geometry of Architecture” in Summer 2012. Register here!

Spy Kids: Secret codes for gifted kids

Source: PBS/NOVA

As a kid,  there’s nothing more thrilling than speaking or writing in a language your parents can’t understand. Add the challenge of creating and deciphering your very own secret code, and you’ve got an irresistible and mind-bending activity for gifted students.

Deciphering codes requires looking for patterns everywhere, and it’s somewhat mind-boggling how important this process of searching for and defining patterns and relationships is to everything we do.  It forms the basis of language, mathematics, science, and even art and music.

The starting point for many children are basic substitution codes such as alphanumeric codes (1=A, 2=B, etc.) and Morse Code.  Studying and “playing” with these codes can help younger children develop their language, reading, and spelling skills as well as their problem-solving strategies.

Studying codes is also an excellent example of an activity that can fulfill the need that many gifted students have for tasks that increase in complexity the deeper they dig!  Codes are at the heart of the concept of algebraic functions in mathematics; the development of scientific explanations and predictions based on patterns of observations in the natural world; rhythm and pitch in music, geometric transformations and the organization of space in art, computer programming, and genetic sequencing.

Maybe your gifted child has already begun “speaking in code.” Where do you start in helping your child cultivate their own code books (even if they don’t tell you what it means)?

Here are some  resources to help you keep up with your child’s secret code enthusiasm:

Learn about the fascinating history behind famous secret codes:

http://www.euclidlibrary.org/kids/tickleyourbrain/11-12-04/Secret_Messages.aspx

http://www.pbs.org/wgbh/nova/military/cryptography.html

A do-it-at home activity for making your very own secret code:

http://unplugyourkids.com/2011/01/10/secret-codes-cardan-grille

How to write in super-secret invisible ink:

http://unplugyourkids.com/2011/01/23/invisible-ink-messages

Real World Secret Codes:

http://www.pbs.org/wgbh/nova/physics/kryptos.html

Decoding Ancient Languages: Hieroglyphs:

http://www.pbs.org/wgbh/nova/ancient/cracking-maya-code.html

Patterns and Fibonacci Numbers in Nature:

http://www.world-mysteries.com/sci_17.htm

Decoding DNA:

http://www.pbs.org/wgbh/nova/body/cracking-the-code-of-life.html

For secret agents looking to create and break a variety of challenging codes, visit http://www.nsa.gov/kids/home.shtml .

Hungry for more? Check out our new summer Math Studio course, “Codes and Spies”,  which integrates math problems and concepts with fun, critical thinking activities like solving puzzles, finding patterns in music, and building a Rube-Goldberg machine. “Codes and Spies” is for students completing Kindergarten through Grade 3, and will be offered afternoons in Chicago and Skokie, IL on July 9-13. Find more information here.

Has your child caught on to the “spy” phenomenon? What is their favorite secret code?

Geometric Scavenger Hunt

by Stacy Levine

How are three-dimensional shapes a part of our everyday lives?  Taken from the Saturday Enrichment Program (SEP) course “Rational Numbers in Geometry”, this activity will give inquisitive second and third graders an opportunity to explore spheres, cylinders, pyramids, and more within your home (or other location, such as your neighborhood on a non-rainy day!).  Additionally, the scavenger hunt will help children learn to identify and categorize three-dimensional shapes.

Materials Needed:

  • paper
  • pencil
  • variety of three-dimensional household objects (e.g. cereal box, paper towel roll, etc.)
  • timer (optional)

Instructions:

  1. Decide on which three-dimensional shapes will be a part of your scavenger hunt and discuss the characteristics of each (see “Additional Resources and Links” for more information).
  2. List each of the shapes from step #1 on paper, leaving enough space next to each for students to give examples.
  3. Allow time for students to search for various shapes and record objects on paper.  If desired, a timer can be used.
  4. Reconvene afterward to check objects.

Modifications for Younger or Older Students:  Younger students could take part in this same activity, with some slight modifications.  First, if students are unable to write down the examples on paper, they can draw pictures.  Additionally, instead of searching for three-dimensional shapes, two-dimensional shapes can be substituted.  For preschool aged kids, assigned shapes can be basic:  squares, circles, etc.  First or second graders can look for such shapes as parallelograms, pentagons, etc.  For older children, try giving them a digital camera to use for snapping photos of the three-dimensional items.  Another variation for older students is to write out clues ahead of time that lead them to various three-dimensional objects around the house (e.g. I am a sphere covered with pentagons and hexagons.  I have a popular sport logo on my middle.  I am usually found being kicked into goals.  Answer: soccer ball).

Additional Resources and Links:
http://www.learner.org/interactives/geometry/

Have you ever held a scavenger hunt in your home?

Stacy Levin has been an instructor in the CTD Saturday Enrichment Program (SEP) since 2007. Her current course, “Final Answer” examines the multiple methods available to solve a math problem. She has a prechool-aged son.


New Research: Hate the Test, Not the Subject

A study released by the University of Chicago this week suggests that a stress hormone triggered during math testing can affect how high-ability students perform. For some, the hormone causes math anxiety and poor performance. For others, the hormone leads to excitement at the thought of a challenge, heightened awareness, and high scores.

“‘We found that cortisol, a hormone released in response to stress, can either be tied to a student’s poor performance on a math test or contribute to success, depending on the frame of mind of the student going into the test,’ said Sian Beilock, associate professor in psychology at UChicago and one of the nation’s leading experts on poor performance by otherwise talented people,

“Students with lower working memory exert relatively less mental effort to begin with, researchers found, so taking a stressful test didn’t drastically compromise their performance. Among people with large working memories, those who were typically the most talented, rising cortisol either led to a performance boost or a performance flop — depending on whether they were already anxious about math,”

This may not be news to some parents of gifted children: your child studies and knows the material, yet bombs the test.  Each gifted student handles testing in his or her own way, just as there are many types of giftedness. Gifted children, like all children, are emotionally complex, and can be prone to “overexcitabilites” such as anxiety. Too often, when a child is bright in one area, adults expect them to be bright in all areas. This can lead to a pressure cooker of expectations on the child.

So, what to do with your overexcitable learner?  Help your child with stress relief before a test.  Reinforce what they are good at, support them when they are feeling overwhelmed, and if they struggle with testing, help them to practice without a clock. More stress relief tips can be found here.

For even more information, check out the book by Sian Beilock: “Choke: What the Secrets of the Brain Reveal About Getting it Right When You Have To”.

Does your gifted learner sometimes come down with a case of math anxiety? Do you have stress relief tips that have worked for your child? We would love to hear them.

Museum of Modern….Math?

by Lindsey Wallem

That’s right, The Museum of Mathematics (nicknamed MoMath) and is set to open in Midtown Manhattan in 2012. Read the full story in the New York Times here.

With fun hands-on math experiments like riding a bike on square wheels while the surface underneath them moves in a circle, the museum will be seriously cool for all visitors. The concept for the museum was initiated by Glen Whitney who, as a former gifted kid, knows first hand what stimulates talented youth. Like the students in our Summer Program, Mr. Whitney began his “career” taking math classes in the summer months with other exceptional students. He majored in math at Harvard, and eventually became a hedge fund manager, where his talents could be used for profit.  But his career choice was missing that spark he felt surrounding his passion, so he left the hedge fund and began fundraising for the Museum of Mathematics, where other kids like him could feed or possibly even discover a love of math the way he did.Will his museum raise math test scores across the country? “I’m not holding my breath,” Mr. Whitney said.  But it can make math fun for the kids who visit, and a change in perspective could be huge for one child.  If there is one constant theme in the stories we tell here at Talent Talk, it is that gifted kids are capable of amazing things.

Will you and your child plan a trip to the math museum? What kinds of beyond-the-worksheet “math activities” can you mimic at home?

“Ask Paula”, July 2011

by Paula Olszewski-Kubilius, Ph.D

This month: What to do with a 4-year-old who shows impressive mathematical ability.

Q: My 4-year-old is able to solve mazes that are complicated for his age without taking a single wrong turn. He says he can see the path. I have an older child in the gifted program. Please tell me if this shows any sort of cognitive ability in my 4-year-old. His math skills are pretty high (can add 3 digit 2 tier carry overs, sub with borrowing). He can speak clearly and with higher level examples but reads at an emergent level. -Suparna Basu

A: From what you have described, your 4 year old does appear to be able to reason mathematically well above what is expected for his age. His ability with mazes may be an indicated of exceptional spatial skills. Some common characteristics of math-advanced children include advanced computational and problem solving skills, interest in mathematical symbols and representations, an interest in patterns  and relationships in nature, the capability of doing problems in “his head” without concrete aids of manipulatives, pleasure in solving difficult problems, and enjoyment of mathematical games and puzzles. It is also common for young children to be very advanced in some areas and average in other (e.g. his reading).

I recommend that you stimulate your son with more mathematically and spatially oriented games and puzzles and possibly have him take enrichment classes through the Center for Talent Development or other programs in math or science .The advantage of these programs beyond the advanced curriculum is the opportunity to explore and enjoy mathematics with other students who are similarly interested in and excite by math. Research shows that adolescents who have advanced abilities in math and spatial reasoning are often interested in the physical sciences and pursue careers as physicists and engineers. So, books about physical aspects of the world or enrichment classes in physical science might be a good match for your son. Your son might enjoy tinkering with objects at home to discover how they work or building and constructing things such as robots or small airplanes.
Soon, your child will be entering kindergarten, and it will be important for you to bring his advanced mathematical reasoning abilities to the attention of his teachers. He may need some more formal assessment of where he is at in terms of his learning of mathematics and placement above grade level. To help guide school officials, I would recommend that you keep track of your son’s work at home–e.g. problems he has solved, puzzles he has worked on, things he has created–so as to be able to share them with his teacher and school administrators and give them insight into his advanced abilities.

In the meantime, enrich, enrich, enrich with puzzles, games, computer programs, and other materials at home.

Paula Olszewski-Kubilius, Ph.D,  is director of Northwestern University’s Center for Talent Development (CTD) and a professor in the School of Education and Social Policy.

Attention “Ask Paula” fans: Paula will be on vacation next month, so we will be trying a new feature called “Ask The Community”. We will ask for your questions on Facebook as usual, but then go to you, our readers, to help with the answers. Look for more details in mid-August!

“Ask Paula”, June 2011

Welcome to this month’s “Ask Paula” feature, in which you ask our resident gifted expert (and CTD Director) Paula Olszewski-Kubilius, Ph.D, your  burning questions regarding gifted education.

Q: My son just tested at a 98th percentile for critical thinking and numeration on GEP testing for school (he is going into first grade), but only 85th for computations. He is in the 98th percent for all other areas. They have decided to leave math off his GEP. Can you help me understand what numeration and critical thinking are in regards to math? -Shannon Trostle Rowe

A: Numeration typically refers to one’s understanding of numbers, what they are, what they mean or represent (e.g. quantity, order, magnitude), relationships between numbers, and operations with numbers (e.g. adding, subtracting). Computation refers to actually being able to add or subtract or perform some operation with numbers correctly. Critical thinking has many definitions but in mathematics it is likely to mean being able to solve problems using mathematics, most likely word problems that involve applying mathematical concepts. All of these are aspects of mathematical reasoning ability.

In general, schools or programs give more weight to assessments of mathematical concepts (i.e. numeration) and problem solving than computation in terms of  identifying children who are mathematically talented or need an above grade level curriculum or placement in mathematics. If you are looking to understand mathematical ability or giftedness, the best resource is a book by Susan Assouline and Ann Lupkowski-Shoplik, Developing Math Talent, published by Prufrock Press.

Watch our Facebook page for your chance to ask Paula a question about gifted education next month.

Read archived “Ask Paula” answers here.

Beyond “Angry Birds”: The Art of Problem Solving Through Computer Games

Our second Rainy Day Activities installment focuses on computer games as a fun way to engage in rigorous mathematics activities. CTD instructor Jason Major uses technology to engage students in the CTD Summer Program and in teaching of gifted students within the Chicago Public Schools, where he has taught for four years. In the post below, Major recommends online programs as a way for students to gain practice on their own as well as a vehicle for fostering friendly competition.

“Sometimes in my classroom, we will get the laptop cart and play a game with all 32 class members!” says Major.

In addition to reinforcing mathematics principles, online learning can provide challenges beyond classroom lessons and give students agency over their learning process.

“Students learning in an accelerated environment have to take responsibility for their own learning a lot,” says Major. “They need to be self-motivated and dedicated. It takes discipline.”

And within classrooms? “Rather than merely progressing to the next level, courses oriented toward problem solving go deeper into subjects,” says Major. “I would like to see schools give kids harder material rather than passing them along.”

Ready, set, start problem solving!

Introduction:

Killing time at home on the computer?  How about instead of your 57th game of Angry Birds today, you learn some new math and try some interesting problems?

There are many online math games and programs, but the Art of Problem Solving (AOPS) website  has completely free offerings like “For the Win” and “Alcumus,” which are unique.

The target audience for Art of Problem Solving site is gifted and talented math students–students just like those served at CTD.  While students through high school (and even beyond) have plenty to keep them interested and challenged, students as young as fifth grade could be ready to attempt some of the problems found in these fabulous applications.

Materials Needed:

computer

web access

pencil and paper

calculator (or try it without one!)

Instructions:

Special note: If students are younger than 13, they will need their parents’ written permission to obtain a login for the web site. Since the web site is not a “closed” environment, as with any online activity, parents and/or guardians should be aware of students’ activity.

There are two free applications on the site.  Alcumus is an online learning tool that tracks students’ progress in Algebra, Number Theory, and Counting and Probability.  As students get problems correct, they are given “experience points” and can move up to higher “levels.”  Students can also choose “focus topics” that supplement what they are learning in their every day math class (and at a much higher level in most cases).  If students don’t know how to do a problem or get the problem incorrect, they are able to read the solution before trying the next problem.  Students are also able to “master” topics and move on to the next one.  Simply put, it’s the best supplemental activity for gifted math students that Major has found.

The other free activity offered by AOPS is For the Win, which is modeled after MATHCOUNTS’ “Countdown” round.  Students challenge each other online to see who can solve problems the fastest.  They can play against anywhere from one to dozens of other students at the same time.

Modifications for Younger or Older Students:

Older students can be part of “Math Jams” that take place periodically on AOPS that often are geared towards upcoming math contests like the American Math Competitions (AMC)  or the American Invitational Mathematics Examination (AIME).  AOPS is also working on modifications for younger students as well.

Additional Resources and Links:

Mathletics ($60 per year)

ALEKS system  ($20 per month)

Beach Balls & Books: Spending the summer studying hard and having fun—with long-term rewards!

After nine months of homework, books, and being cooped up inside for the odd Chicago-area “snowpocalypse,” the idea of spending the oh-so-few summer months in school may underwhelm. Nonetheless, Bria, a goal-oriented student in middle school is considering it. She spent at least a portion of her 2010 summer indoors studying Pre-Algebra because she wanted to test into Honors Algebra I in her local school. She was surprised to find that in the Apogee program, part of Center for Talent Development at Northwestern University’s Summer Program and where she chose to go, she could work hard and play hard too.

What isn’t always as well known are the long-term effects of participating in an accelerated summer program. Students are more likely to take a more rigorous course of study in school and attend more selective colleges (Barnett & Durden, 1993). Research also suggests that females in particular benefit in mathematics achievement. Grant and Olszewski-Kubilius (1996) found that females who studied math tended to accelerate more often and earned more honors in math during high school. And, they gained more confidence—something that Bria’s experience seems to demonstrate.

“I thought it was just going to be all academics, all the time, and not a lot of fun,” said Bria whose Mom signed her up for the Apogee three-week program for students in grades 4 through 6 and located in Evanston, Illinois. “It was really good because when we were in class, we worked really hard. By end of second week, the class was really intense. We had to study and go over things. But when we relaxed, we had a lot of fun.”

Bria remembers a long list of last summer’s activities: playing capture the flag, volleyball, Jeopardy, going to the beach and downtown Evanston, indoor camping and making s’mores in the residence hall kitchen, a team race with several challenges including the teammates spelling a difficult word using their bodies, foosball, Ping-Pong, weekly dances, ice skating, a Hawaiian luau, and a carnival.

Bria met a lot of people, and still keeps in touch with her classmates from Apogee, many from other countries, via e-mail and text.

All the summer fun didn’t distract Bria from achieving her goal of getting a firm grasp of pre-algebra and a sense of what was to come in Algebra. Now in grade 7, Bria is doing well and is even helping her classmates. “I find myself saying a lot of the time—even just today—that I already know what is new to everyone else in the class from taking Pre-Algebra last summer at CTD. People at my school call me ‘the math genius’ because I’m the person that they come to when they need help. I enjoy math a lot.”

Bria says she recommends the program especially to students who have a specific interest in a subject because it offers the opportunity to focus in that area all day. This summer, Bria says she is thinking about returning to CTD to study Geometry Honors.

What does this math whiz want to do? “My dream job changes every year or so,” she says. “At one point, I wanted to be a forensic anthropologist. Now I want to be and OB/GYN or a neurosurgeon.”

Want to know more about how to choose the best summer program for you or your child? Visit the National Association for Gifted Children’s (NAGC) web site for a the resource “How to Choose a Summer Program”.

Interested in attending the Summer Program at CTD? In addition to the Apogee program, CTD offers programs for all age groups from PreK through grade 12 (Leapfrog:  PreK through grade 3; Solstice for grade 4; Spark, a two-week program for students in grades 4 through 6; Spectrum for students in grades 7 and 8, and Equinox for students in grades 9 through 12). Check out the CTD web site for a complete list of courses, full course descriptions, and to apply.

Barnett, L. B. and Durden, W. G. (1993). Education patterns of academically talented youth. Gifted Child Quarterly, 37(4), 161-168.

Olszewski-Kubilius P., and Grant, B. (1996). Academically talented women and mathematics: The role of special programs and support from others in acceleration, achievement and aspiration. In K. D. Noble and R. F. Subotnik (Eds.) Remarkable Women: Perspectives on Female Talent Development (pp. 281-294). Cresskill, NJ: Hampton Press.

Making Math Cool

We recently saw this article, and were inspired by Vi Hart’s unique way of approaching a subject kids sometimes view as dull:

http://www.nytimes.com/2011/01/18/science/18prof.html?_r=1

Want to become a mathemusician? Or your own combination of two subjects you love? Check out Vi’s blog here to read more about her projects including “making math cool” with balloons, umbrellas,  laundry baskets, and even plastic swords. But most importantly, don’t be afraid to be creative!