Pythagoras vs. Einstein

Source: https://www.facebook.com/1698785500391556/photos/a.1698786577058115.1073741828.1698785500391556/1909519182651519/?type=3&theater

A Math Genius Blooms Late and Conquers His Field

I enjoyed reading this feature: https://www.wired.com/story/a-math-genius-blooms-late-and-conquers-his-field/

Some quotes from the opening paragraphs:

A bad score on an elementary school test convinced him that he was not very good at math. As a teenager he dreamed of becoming a poet. He didn’t major in math, and when he finally applied to graduate school, he was rejected by every university save one.

Nine years later, at the age of 34, Huh is at the pinnacle of the math world. He is best known for his proof, with the mathematicians Eric Katz and Karim Adiprasito, of a long-standing problem called the Rota conjecture.

Even more remarkable than the proof itself is the manner in which Huh and his collaborators achieved it—by finding a way to reinterpret ideas from one area of mathematics in another where they didn’t seem to belong. This past spring [the Institute for Advanced Study] offered Huh a long-term fellowship, a position that has been extended to only three young mathematicians before. Two of them (Vladimir Voevodsky and Ngô Bảo Châu) went on to win the Fields Medal, the highest honor in mathematics.

That Huh would achieve this status after starting mathematics so late is almost as improbable as if he had picked up a tennis racket at 18 and won Wimbledon at 20. It’s the kind of out-of-nowhere journey that simply doesn’t happen in mathematics today, where it usually takes years of specialized training even to be in a position to make new discoveries. Yet it would be a mistake to see Huh’s breakthroughs as having come in spite of his unorthodox beginning. In many ways they’re a product of his unique history—a direct result of his chance encounter, in his last year of college, with a legendary mathematician who somehow recognized a gift in Huh that Huh had never perceived himself.

Fun with pancakes

I’m not sure who had the idea to cook pancakes like these, but I’m sure glad he/she did.

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Source: https://www.facebook.com/CTYJohnsHopkins/photos/a.323810509981.46389.175118999981/10150793851844982/?type=3&theater

What Do You Do With 11-to-13-Year-Olds?

I greatly enjoyed this very thoughtful post about the unique joys and challenges of teaching middle school mathematics: https://mathwithbaddrawings.com/2017/07/05/what-do-you-do-with-11-to-13-year-olds/

Happy Phi Day!

In the United States, today is abbreviated 1/6/18, matching the first four significant digits of the irrational number \phi = (1 + \sqrt{5})/2, otherwise known as the golden ratio. For a good introduction to \phi, see http://www.foxnews.com/story/2007/06/18/happy-phi-day-perfect-time-for-some-phinancial-fun.html.

For today, I’ll give a fun fact that I learned last year; for its national flag, the country of Togo chose a rectangle whose dimensions matches the golden ratio.

 

Dividing fractions

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Source: https://www.facebook.com/MathWithBadDrawings/photos/a.822582787758549.1073741828.663847933632036/1767946789888806/?type=3&theater

Engaging students: Probability and odds

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Trent Pope. His topic, from Pre-Algebra: probability and odds.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now?

This website contains problems that would be great for odds. On the worksheet it has you solving problems about the chances of getting different gumballs from a gumball machine and chances of winning gift cards in a drawing. These worksheets would be great because there are real life applications with these examples. On the worksheet students are to solve what color gumballs they could draw from the machine. This will give them a visual representation of their odds. In order to find their odds they must know all the required information such as the number of total gumballs and the number of each color. Then the instructor can ask the students any question about what they can draw. The other problem is that there are gift cards, coupons, and free admission to a theme park that a student draws from a hat. This would be another great example of how students can find the odds of what they can draw.

http://www.algebra-class.com/odds-and-probability.html

 

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How could you as a teacher create an activity or project that involves your topic?

This project idea comes from the game show Deal or No Deal. The purpose of the project would be for students to see what the odds are of winning more money than the amount offered from the Banker. For instance, the banker will offer you $100,000 to leave the show without seeing what is in your briefcase. The contestant would then look to see how many briefcases are left that could contain an amount greater than $100,000. If there are five chances out of the twenty remaining briefcases, the student would have a 5/20 chance, or 25% chance, to win more money. So, the contestant might want to say no deal because there is a higher chance of winning more money should he/she stay in the game. Students could go multiple rounds of this and see if their chances increase as the game goes on. This would engage students and they would look forward to winning the game show.

http://www.teachforever.com/2008/02/lesson-idea-probability-using-deal-or.html

 

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How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

This topic has appeared in many examples through pop culture. One is in the movie 21. Also, I have found a YouTube video demonstrating probability and odds. The video gives an example of how a game show changes your odds of winning a brand new car. There are three doors and the host asks you which door you think has the new car. When you do this you have a 33% chance of selecting the right door. After you have made a selection, the host goes and selects one of the remaining doors to open. Remember that the host knows which door the car is behind. He opens the door to show you that it does not contain the car. Then, the host asks you if you would like to change your door or keep it. Because of variable change you are more likely to pick the car if you change your decision. This increases your chances to 66% of choosing the right door. I thought this was a great way to engage students about probability and odds because it is all about your chance of selecting the correct door. You have one chance to pick the right door, but three doors to pick from. This is all about odds. It increases after the host opens a door because you have a second chance to select the correct door. This would apply to all game shows, and people would be able to make personal connections.

 

 

Engaging students: Multiplying fractions

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Saundra Francis. Her topic, from Pre-Algebra: multiplying fractions.

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How could you as a teacher create an activity or project that involves your topic?

 

This multiplying fractions project found on Teachers Pay Teachers by Mix and Match gets students interested in the topic through creating a dog house. For the project students have to use the size of their chosen dog to discover the dimensions of the doghouse. The students will then scale down their doghouse by multiplying fractions to create a model doghouse. Once the students have discovered the dimensions of the model they can build the model doghouse. There are worksheets provided on the website that will guide the students through this process, the also have word problems related to the doghouse for extra multiplying fractions practice. This project would engage students because they will be able to create their own doghouse and they will be given an opportunity to build it. It also will help students understand how to multiply fractions through working out how it relates to scaling items.

 

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How does this topic extend what your students should have learned in previous courses?

 

In previous math course students should have learned how to multiply, reduce fractions to the simplest form, and how to covert mixed numbers to improper fractions. Using these concepts students will be able to multiply fractions based on previous understanding of fractions and multiplication. A YouTube video titled Review of Fraction Concepts created by mathtutordvd (https://www.youtube.com/watch?v=7Wrde6iFVcA) reminds students what a fraction is and what it represents. It also reviews term such as numerator and denominator, which are important terms for students to know when they learn how to multiply fractions. This will engage students’ prior knowledge by giving them a refresher and will prepare them for learning how to multiply fractions. It also might help students that were previously having a hard time understanding the concept of fractions once they watch the video. It also discusses the important of fractions, which will help students realize how it can apply to their daily lives.

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How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

The Multiplying Fractions Song by NUMBERROCK (https://www.youtube.com/watch?v=CcDGRLosAf0) is an excellent video to engage students and help them understand how to multiply fractions.. This video goes through two examples of multiplying fractions while rapping. The examples used about finding treasure and digging for dinosaur bones will catch students’ attention. The video not only gives student procedural knowledge, the steps to multiply fractions, but explains why we are able to multiply fractions through the images. They sing “multiply the numerator, then multiply the denominator” which students can repeat when they are working on problems later in the lesson. In the video, models are displayed that show students how to multiply using a model, which is part of the TEKS. The diagrams also show students why multiplication of fractions works and gives them a better understanding of the concept. The rap song and cartoon visuals draws students attention and help them remember the topic being learned.

References:

https://www.teacherspayteachers.com/Product/Multiplying-Fractions-Project-2351217

https://www.youtube.com/watch?v=7Wrde6iFVcA

https://www.youtube.com/watch?v=CcDGRLosAf0

Merry Christmas

Engaging students: Permutations

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Sarah McCall. Her topic, from probability: permutations.

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What interesting (i.e., uncontrived) word problems using this topic can your students do now?

In high school math, word problems are essentially unavoidable. They can be a pain, but they do help students to be able to see applications of what they are learning as well as good problem solving skills. So, if we must make use of word problems, we might as well make them as engaging/fun as possible. Some examples of ones that I found and would use in my classroom:

  1. Permutation Peter went to the grocery store yesterday and met a super cute girl. He was able to get her phone number (written on the back of his receipt), but today when he went to call her he couldn’t find it anywhere! He knows that it consisted of 7 digits between 0 and 9. Help Permutation Peter by figuring out how many combinations of phone numbers there are.
  2. Every McDonald’s Big Mac consists of 10 layers: 2 patties, 3 buns, lettuce, cheese, onions, special sauce, and pickles. How many different ways are there to arrange a Big Mac?

 

 

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How has this topic appeared in pop culture?

Many students are easily confused when they first learn the difference between permutations and combinations, because for most permutations is an unfamiliar concept. One way to show students that they have actually seen permutations before in everyday life is with a Rubik’s cube. To use this in class, I would have students pass around a Rubik’s cube, while I explained that each of the possible arrangements of the Rubik’s cube is a permutation. I would also present to them (and explain) the equation that allows you to find the total number of possibilities (linked below) which yields approximately 43 quintillion permutations. This means it would be virtually impossible for someone to solve it just by randomly turning the faces. Who says you won’t use math in the real world!

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How can technology be used to effectively engage students with this topic?
In a day and age where a majority of our population is absorbed in technology, I believe that one of the most effective ways to reach high school students is to encourage the constructive use of technology in the classroom instead of fighting it. Khan academy is one of the best resources out there for confusing mathematics topics, because it engages students in a format that is familiar to them (YouTube); not to mention it may be effective for students’ learning to hear a different voice explaining topics other than their normal teacher. In my classroom, I would have my students use their phones, laptops, or tablets to work through khan academy’s permutation videos, examples, and practice problems (link listed below).

References

https://www.quora.com/How-are-permutations-applied-in-real-life

https://prezi.com/q3aaem0k2xie/permutations-in-the-real-world

https://ruwix.com/the-rubiks-cube/mathematics-of-the-rubiks-cube-permutation-group

https://www.khanacademy.org/math/precalculus/prob-comb/combinatorics-precalc/v/permutation-formula