Coin Tosses Are Not 50/50

I just came across an article that says the results of a coin toss are not 50 percent heads, 50 percent tails as previously thought but instead does contain a slight bias.  Can you imagine that?  Apparently, someone ran an experiment in which they had 48 people toss the coins from 46 different currencies and the results they got were quite interesting.  This particular paper has not been peer reviewed but it is quite interesting none the less.

The man who lead the study believed that the reason for the bias has to do with the way the coin moves in the air when flipped.  He proposed that the side that is up when flipped is more likely to spin in the air with that side up so it has a better chance of showing when landing.

They concluded that if you look at the side that is up before you flip the coin, that is the side you should call.  According to their results, this strategy resulted in the correct call 50.8% of the time based on over 350,000 flips.  

For the study they used a specific method for flipping the coins.  Participants were to flip the coin with their thumbs and catch it in their hands.  If it landed on the table, it was not counted because the table added the possibility of a bounce, or a spin.  At one point they had people flipping coins for several days of 12 hour sessions to get enough flips for their study.

This study was inspired by a mathematician who was also a magician who said there was a slight wobble and an off axis tilt when someone flipped a coin with their thumb.  This observation was made back in 2007 and the researchers who ran this study proposed that the coins would land on the side facing upwards about 51% of the time.  

It was also found that the individual flippers results varied a bit because some managed to flip with almost no tilt or wobble while others had enough of a tilt or wobble to end up with the 50.8% result.  Further more, the coins nationality seemed to make no difference in the results.

Knowing this could help the person choosing the result to have a slight advantage in calling for who goes first in a sports game, who wins a mayoral election in the Philippines, or any other event that relies on a coin toss. If you were to gamble on the results of a coin toss, using this information could help you win money.  The solution to countering the bias is to not let anyone see which side of the coin is up before flipping it.

I found this interesting.  Let me know what you think, I'd love to hear.  Have a great day.

Math Activities For Halloween.

 

It is that time of the year again.  Halloween is swiftly approaching and it is fun to take time in class to do a few activities to celebrate it.  Give the students a breather from their regular topics so their brains have a chance to process material while they get something different.  When I suggest Halloween activities, I'm not talking about simple word problems with halloween things in them.  I'm talking about real activities that look at the amount of candy sold and more.

Start with some activities that look at probability such as asking students about the probability of Halloween landing on a Friday or Saturday night.  Add in to that the probability of the holiday landing on the night of a full moon based on the moon taking 29.5 days to orbit. To extend this activity, ask if the event of landing on a Friday or Saturday is independent mathematically.

Another activity is making the students the person who figures out what candies go into a 100 piece mixed bag of Halloween candy.  They should list the types of candy they'd love to see in a bag, survey their class mates to see which candies are most popular with them, and then decide on which ones in what quantity will end up in the bag.  They can then calculate the ratios for each candy so that every one gets the type of candy they want.  Ask them how the ratios change if the bag only has 20 pieces in it.  You can extend this exercise by asking how much should the 100 - piece bag be priced based on whole sale prices and enough to make a profit when they sell each bag.

Don't forget pumpkins.  For students who are more advanced, ask students to figure out the equation that models the shape of their pumpkin or they can find the circumference of that pumpkin.  Most pumpkins are elliptical in shape.  It is easy to find pumpkins at the store that you can get for the classroom and the pumpkins can include a couple small decorative ones.

If you have tangrams in the classroom, down load halloween themed ones such as bats, pumpkins, or whatever you can find and let the students use those. This site has some that are free and can be downloaded. This is not the only site so do a search and find more.  If you don't have any tangrams, download a set from the internet.

To have students practice coordinate graphing, find a few halloween themed graphs and let them have fun. There are several sites on the internet that offer free graphs so you won't have trouble finding enough for students to receive different ones.  Some range from quite easy to much more complex and can be found for one quadrant or all four.  

In regard to those math puzzles that have pictures to represent quantities such as three ghosts added together equals 30, one ghost and two pumpkins equal 40, and three bats equal 15. So you know each ghost represents 10, the pumpkin is worth 15, and one bat equals 5.  You can find some redone on the internet or you could have students create their own versions either by hand or digitally.  Pixabay.com is a good site for free copyright free photos in vector style. 

Back to probability.  Prepare a large bowl of M & M's for each group.  Have students sort through the candies, dividing them up into colors, and then creating a tally for each color. Next have students add up the tally's for each color and make predictions for how many candies of each color are in a fun sized package. Finally, they discuss or write up a summary on were their predictions right or wrong and why they were off if they were wrong.  At the end, they can eat the candies.  

These are just a few suggestions for having some fun mathematically based activities on Halloween.  Let me know what you think, I'd love to hear.  Have a great day.  

Student Misconceptions About Fractions And Decimals.

 

One issue I've seen in most of my classes has to do with remainders and how they relate to their placement in decimals.  So many of my high school students can begin the process correctly but then they mess up.  For instance, they divide 27 by 4.  They start it correctly and know that 4 goes 6 times into 27 so they write down the 6, multiply 6 times 4 and write 24 below the 27 before subtracting.  They get 3.  This is where their misconception arises.  They then put the 3 after the decimal and get 6.3 for the answer even though the answer should be 6.75.  I don't know if they are just trying to finish the problem as quickly as possible or if they honestly don't know how to take the problem all the way to its normal conclusion.

So I figured I would address several misconceptions when it came to changing fractions into decimals and fractions - decimals in general, that we, as teachers, are likely to run into.  One issue is that students often see fractions and decimals as two different types of numbers, hence they do not see 1/2 as equal to .5.  They believe that fractions and decimals cannot be equal representations of the same number.

Along the same lines, they don't see that both fractions and decimals are designed to express parts of a unit quantity.  Sometimes they see the decimal point in a decimal as separating two different numbers and this may be the issue my students are having.  They may see the 4 as one number so in their minds the remainder of 3 goes on the other side of the decimal thus they have the two different numbers listed.  

Another issue is when students don't see the relationship between decimals clearly.  When they see a fraction like 4/5, they know the denominator tells them how many equal units the whole is divided into.  In a decimal, the information on the denominator is hidden and they have to rely on their knowledge of place value in order to do things correctly.  If you divide the numbers correctly, you end up with 0.4 as the decimal equivalent but students don't always recognize 0.4 as four-tenths or 4/10.  

In other cases, they might see 2.6 as two and 1/6th reading the decimal as the denominator and this interpretation leads to students association a fraction with the wrong equivalent decimal. In addition, students might see decimals with the idea that the more places you have in the decimal, the larger that number is.  For instance, students look at .621 and interpret it as 621 parts so it has to be bigger than .7 which only has 7 parts. The equivalent of this in fractions is when students try to compare denominators as 4 being smaller than 8 so 1/8 is larger than 1/4.  

Furthermore, since they don't understand that the denominator in fractions refers to how many equal parts something is divided into, they don't "see" that when you cut something into 4 parts, the size is larger than if the same sized cake was divided into 8 pieces so each piece is half the size.  In regard to decimals, they don't see that place value helps tell you the size of pieces.  For instance, 0.7 means that you have 10 pieces with 7 of the pieces colored red where 0.621 says you have 621 pieces out of 1000 colored in so each of the 621 pieces is smaller.

In my day, we just learned it without understanding the sizes of the parts involved.  I believe many of the misconceptions involved with both fractions and decimals create issues for when students convert from fractions to decimals and vice versa.  These misconceptions often drive how students arrive at an answer as they convert from fractions to decimals so they more than not get the incorrect answer.  

Let me know what you think, I'd love to hear.  Have a great day.

Calculus Lessons Are Understood Better When They Are Interactive.

 When I took calculus, the professor wrote all the notes on the board and we wrote down the notes.  There was never an idea any lesson being interactive.  Apparently, there was some research conducted to see whether students learned more in an active learning course where they solved problems in class versus sitting through a traditional lecture class. 

Furthermore, the researchers discovered that students understood complex calculus problems better and received higher grades. In addition, these results seemed to hold true across so many different racial, ethnic groups, gender, socioeconomic levels, first time and transfer students.

The study lasted over three semesters from fall 2018 to spring 2019 and involved over 800 students at a public university. Students were placed randomly in either a traditional lecture based class or in an active engagement focused  learning course.  

The active engagement based course had students working through exercises designed to build calculus knowledge during the class period. The traditional course has students sitting through a lecture class where they take notes, and work on developing understanding outside of class. In the active based class, students worked together, explaining ideas to each other because this class is having students understand the why behind calculus rather than just trying to memorize it.

Students try out their ideas, learn from their mistakes, and develop an understanding just like mathematicians. They make and test educated guesses, make sense of what they discover, and explain their reasoning to others.  In this set up, faculty play an important part in the process by asking probing questions, demonstrate mathematical strategies, monitor student progress, change the pacing, and provide activities to foster student learning.

In the past, calculus has been a barrier to people wanting to go into any STEM careers.  Only about 40 percent of the students are able to go into their choice career and it has been found that calculus is that one class that derails many students. In fact, it has been found that females are leaving the field at 1.5 times that of men, while Hispanic and Black students are likely to fail calculus at a much higher rate than white students.

This study is just the first one in the general subject. The next study needs to address barriers, lack of time, and why aren't professors allowed to bring active learning into their classrooms. Although this is a small study per say, it is a good first step.  Let me know what you think, I'd love to hear.  Have a great day.

Using Dice In Algebra

 

Most schools have sets of dice in every math classroom.  Usually, dice are used to teach probability because they are very visual and many standardized test questions ask questions about dice.  However, there are other ways to use dice to actually help teach algebra and we'll look at ways to use them in Algebra. 

Dice can be used to help create algebraic expressions or equations.  If you use two different colored dice, one represents positive numbers while the other represents negative numbers.  Think about rolling two dice and you have the coefficients for your equation.  Role multiple dice to find the coefficients for a bunch of terms that students can then combine and simplify.

In addition, dice can be used to generate the terms of two different equations that make up a system of equations.  Students then solve the system to find the values for x, y, and even z if you have three equations you are solving.  Rolling the dice allows students to create unique equations to solve and it involves students more in the whole process.

Another use of the dice is to find a number that can be factored as the first step in factoring.  The dice can also be used to find the coefficients for a binomial or trinomial equation or expression to be factored. Again, when you roll the dice, you have equations that are being made immediately and every student has a different equation.  

Furthermore, dice can be used to find numbers that students use to create inequalities.  The numbers could be whole, fractions, or even in decimal form.  One can even set up multiplication problems to be compared such as 2 x 5          3 x3  and students are asked to compare them.  So many different ways to create inequality problems with dice.

Finally, create games that have students create the equations, variables, inequalities, etc and then have them solve the equations.  Those who solve the problems win points.  When you use dice and have students use them to create problems, they become more involved.  Let me know what you think, I'd love to hear.  Have a great day.

Using Playing Cards In Algebra.

 

I love finding new ways to help engage my students.  Usually, when I look up how to use playing cards in math, I end up finding suggestions designed for students who are much younger than the ones I teach. Today, I actually found something that addressed using playing cards in an Algebra class and I am so happy.

It is possible to use playing cards in Algebra to help students learn to calculate certain probabilities such as an ace, a red card, or a 3 of clubs.  Every student is expected to answer questions on probability on most standardized tests and most of us do not have time to squeeze in proper instruction so a game here and there can help with that. Card decks can also be used to illustrate specific probability problems so students can "see" what the words are asking.

In addition, playing cards can be used to help students explore combinatorics and permutations.  This topic may not be covered in lower level algebra but it can be in Algebra II or higher level mathematics. Specifically, cards can be used to find the number of ways to arrange or select cards from a deck which relate to fundamental principles.  Although many students are able to follow a verbal explanation, having a way to illustrate the same material is so much better for others and for many, it puts everything together. 

You can also use cards to represent variables, show expressions or equations.  Cards are a different way to represent algebraic concepts so students can connect basic ideas and concepts so students can transfer from one way to another.  In addition, this can make algebra fun and engage students.

One can also create word problems using the cards of a playing deck and the problems can be translated into more real world situations.  This makes it more interesting and fun.  Finally, playing cards can be used to help with random variables and expected values.  It is a great way to introduce the topic by assigning values to cards and then calculate the expected value of drawing certain cards.

So a few ways to use playing cards in Algebra.  Let me know what you think, I'd love to hear.  Have a great day.

Mathematics At Bedtime May Help Improve Mathematical Memory.

 Two researchers in the United Kingdom created a study looking at how sleep intersects with mathematical memory.  They looked at how sleep impacted the learning of multiplication factors.  More specifically, they examined whether learning complex multiplication problems before going to sleep helped them recall the material.  

Their study involved adults between the age of 18 and 40.  In this study, they learned complex problems either before they went to sleep or after they woke up in the morning. Some of the problems were on new material or previously learned material.  The sessions were both timed and untimed. In addition the tests were given 10 hours after subjects studied the math.

Results indicated that those who worked with math prior to going to sleep tended to better than those who did it in the morning. The results were the same even when you looked at the differing learning abilities.  One theory indicates that the material is temporarily stored and reactivated in the brain when people sleep. Thus material learned at night has a better chance to consolidate than that learned during the day.

The results have some interesting applications for elementary children who are learning their times tables or other mathematical memorization skills.  If these results translate to children, then if they have a math lesson before bed, they could possibly better learn the material than at school.

The authors also suggested that the sleep helped adults better recall possible due to the lack of external stimuli. During the day, people are surrounded by more noise and stimuli than at home so the brain takes advantage of it. They also thought the brain is able to lock in the information at night due to having no other competition.  When the learning takes place in the morning, the brain is exposed to conversations, media, reading, digital devices. or other classes.

It is possible that the competition for memory during the day may be the cause in the differences between learning material at night and day but more study will be needed to determine that. Further more, there needs to be some study done to see if having students practice mathematical skills that require memorization are better learned at night.  Let me know what you think, I'd love to hear.  Have a great day.

Building On Prior Knowledge.

 It is well known that students do better when they are able to connect what they already know with what they are currently learning. I've found over the years, that the way to connect previous knowledge with new knowledge may be unusual but if they can "see" things, they are more likely to learn it.

One of the simplest ways to decide what they already know is to ask.  Ask them what they remember about this topic or what do they know about that topic.  It might be something like "What do you remember about combining like terms?" or "What is a like term?"  This questioning should be done before moving onto a new topic.

Another way to access prior knowledge is to assign a warm-up task. In reality, they are asked to finish the task and the task is based on what is being taught that day.  When a student is given a task before the lesson, they have a chance to explore the concept, try out various strategies to solve the task, and it prepares them for the new topic.

Then one could always use Think - Pair - Share where they discuss questions in small groups.  This discussion takes place before the whole group lesson.  This particular activity allows students to process the information, look at prompts designed to activate prior knowledge, while helping to check for understanding.

For any of these activities, teachers can set a time limit, ask students to write down their thoughts on white boards, or even model think a louds so students have an opportunity to observe what they should be doing. 

Other activities that might help activate prior knowledge include having regular or advanced graphic organizers because organizers help students understanding of the information.  In addition, these organizers provide a structure for the topic or lesson. Another possibility is to use an anticipatory guide.  An anticipatory guide if filled with statements that students agree or disagree with.  These guides are designed to have students think about what they already know, think about what they don't know, and possible see what has changed after receiving instruction.

One can also provide problems that are just a bit above what the student can already do.  Then have students discuss parts of the problem they recognize and are similar to other problems or concepts. The problem needs to be based upon what they know, and what they need to know to answer the problem.

Start class with a question students look at for up to three minutes.  The opening question can be a problem, a concept question, or even an opinion question.  The idea is the question will help students access prior knowledge so they are ready for the new material.  

These are just a few suggestions to integrate into the classroom to help students access prior knowledge.  This way, they are ready for the new material and have the best chance of learning it. Let me know what you think, I'd love to hear.  Have a great weekend. 

Tips For Teaching With Videos

I am getting ready to offer more support to many of my students so I'm looking at ways to have them watch videos actively.  I know from past experience that you can't just give students a video and expect them to come out having learned much. 

First, it is recommended that one set up a visible timer so students are aware of the passing time.  In addition, it helps everyone figure out pacing so they are able to complete their work.  

Secondly, it is not the best idea to show the video on full screen always.  Most people think it is best to view videos in full screen mode but when it is that large, it can make you forget the purpose of watching. In addition, you can always learn with text alone, or with images but you learn more with both together.  One suggestion is to post a key question or guiding questions as a reminder to students of what to look for.

Always make sure the start and stop points are specific.  Rather than showing a 25 minute video for a 5 minute segment, edit the video so that it is set for exactly what you want the students to see.  Take notes of time stamps within the video and trimming options for things like google slides.  Instead of relying on the voice over provided with the video, ask students to record their own voice over.  This can be done by watching the video on a google slide before utilizing available tools. This allows students to focus on the material rather than worrying about their appearance on screen.

Furthermore, all videos should be no longer than six minutes but specifically in the three to four minute range for the best attention span for students.  If the content is longer, stop and check for understanding regularly through the video.  When you check out a video, look at frame rate and resolution because many people think that with a lower resolution, the video is not as good.  It is known that the higher the resolution, the better people see the video so make sure it is going to keep student attention.

So now you have a few ideas to use when having students watch videos in class.  Let me know what you think, I'd love to hear.  Have a great day.  

Counting By Ten Shows Sophistication.

I recently came across an article that looked at how young children are able to count by 10.  Although I do not teach young children, this information is still interesting. When people take time to understand how young children learn to count, it impacts the type of materials used in the classroom. Furthermore, it impacts how those materials are designed to help students develop strategies.

A recent study examined how mostly first graders used a hundreds table to perform age-appropriate tasks for counting.  Upon examination, it was noted that the children who counted in a left to right, top to bottom manner, tended to outperform those who used a left to right and bottom to top method.

The study used two different digital hundreds table.  One was a top down table where one was at the upper left corner and 100 was at the bottom right corner. The other was a bottom up where one appeared at the bottom left corner and 100 was at the top right most corner. One group in the study used the first table, the second group used the second table, and a third group used the bottom to top table but with a small symbol showing that the higher one goes on the table, the higher the values of numbers.

At the end, it was discovered the children who used the top down hundreds table were able to use a more sophisticated strategy of counting by 10's and moving down vertically.  The children who used the bottom up table tended to count by ones and moving horizontally, row by row. 

Researchers speculate that students relate to the top to bottom table better because the movement is to read left to right, up to down.  It was a familiar movement as most of these children are also learning to read at the same time.  The movement was easy to transfer.  

In addition, students who can count by 10 are counting more efficiently than those who counts by one. Researchers are not stating that all children will gravitate towards the left to right top to bottom table but the results will provide teachers with an idea of how their students process numbers and addition.  Let me know what you think, I'd love to hear.  have a great day.