Unconventional Applications Of Percentages In Everyday Life.

After looking at how weather forecasts, I decided to look a bit further into other uses since percentages are a fundamental concept in mathematics. They are commonly used in various everyday situations, from calculating discounts at stores to understanding probabilities in weather forecasts. However, there are several intriguing and less conventional ways in which percentages play a pivotal role in our lives, often in unexpected and fascinating ways.

One way is only found in this digital age where percentages are omnipresent in social media. Platforms like Instagram, Facebook, and X formallyTwitter utilize engagement percentages to measure user interaction. These metrics include the engagement rate, which calculates the percentage of people who interact with a post by liking, commenting, or sharing it, providing valuable insights into content performance.

Another way. is in gaming and gambling, where percentages are critical for calculating odds and probabilities. From determining the probability of winning a game of chance to strategizing in complex video games, understanding percentages assists gamers in making informed decisions and assessing risks. In certain games, it is good to know the percentages or odds of landing certain numbers, or rolls of dice so you know how likely it is to happen.

Furthermore, percentages play a significant role in health and medicine beyond simple dosage calculations. Medical professionals use percentages to communicate risks, success rates of treatments, survival rates, and prevalence rates of diseases. For instance, survival rates for certain illnesses are often represented in percentages, offering insights into prognoses and treatment outcomes. Percentages can also be used to determine which vaccination, flu shot is the better choice in a specific situation.

In financial markets and investments, percentages are integral to understanding returns on investments, analyzing market fluctuations, and determining interest rates. Traders and investors use percentages to calculate gains or losses, assess portfolio performance, and make informed decisions in the volatile world of financial markets.

Then Percentages are extensively used in analyzing population demographics and survey results. Polls and surveys often report results in percentages, providing a snapshot of public opinion on various topics, elections, or societal trends, aiding policymakers and businesses in decision-making. This becomes especially important in election years with major elections such as the presidential one coming up. In addition, you will find percentages used to describe how many turned out for a particular election.

In the realm of sports, percentages are employed for statistical analysis and player evaluation. Advanced metrics like shooting percentages in basketball, conversion rates in soccer, or batting averages in baseball offer insights into player performance and team strategies.

While percentages have conventional applications in daily life, their utilization in different unconventional areas showcases their versatility and significance in modern life. From social media metrics to medical statistics, gaming probabilities to financial markets, and beyond, percentages serve as a universal language, providing valuable insights and aiding decision-making across diverse fields. Understanding these unconventional uses underscores the pervasive nature of percentages in our multifaceted world, demonstrating their relevance beyond traditional mathematical contexts. Let me know what you think, I'd love to hear. Have a great day.

Understanding Percentages In Weather Forecasts

We've all read various weather forecasts. Most include different percentages. Sometimes you might see a prediction of 80 % chance of rain, or 50 % chance of snow. When I was younger, someone said the percentages mean that that percent of the area will experience rain or snow but is that true?

Mathematics and meteorology intertwine seamlessly in the realm of weather forecasting, where percentages play a crucial role in conveying the likelihood of specific weather conditions. Understanding these percentages in forecasts is key to making informed decisions and preparing for what the elements may bring.

In order to make forecasts, meteorologists use a blend of historical weather data, mathematical models, and cutting-edge technology to predict future weather patterns. However, weather forecasting isn't an exact science. Instead, it relies on probabilities and statistical methods to estimate the likelihood of various weather events occurring.

When you read about the chance of precipitation, the percentage indicates the likelihood of measurable precipitation occurring at any given location within a specific forecast area during a defined time frame. For instance, a 40% chance of rain means that there's a 40% probability that rain will fall at your location within the forecast period. That also means there is a 60 % chance it won't.

On the other hand, percentages are also used to forecast various weather phenomena, such as the likelihood of fog, thunderstorms, snow, or hail. These percentages offer insights into the probability of these events occurring within the forecasted area.

One also needs to learn to interpret and use the percentages mentioned in a weather forecast. It's important to interpret percentages in the context of the forecast timeframe and geographical area. A 60% chance of rain for an entire day implies higher confidence in precipitation compared to a 60% chance within just an hour.

In addition, percentages aid in risk assessment. Higher percentages indicate a greater likelihood of a specific weather event, allowing individuals to prepare accordingly by carrying an umbrella, dressing appropriately, or planning activities indoors. This is especially important for predicted hurricanes, tornadoes, or flooding.

Furthermore, businesses, agriculture, transportation, and emergency services rely on weather forecasts to make informed decisions. Farmers decide when to plant or harvest crops, schools being in session, and emergency responders prepare for severe weather events based on these forecasts. Airlines rely on weather forecasts especially to schedule or cancel flights for safety.

The weather models meteorologists use, use complex mathematical algorithms to simulate atmospheric conditions. These models analyze vast amounts of data, including temperature, humidity, wind speed, and pressure, to predict future weather patterns. The probabilistic forecasting involves statistical methods, such as ensemble forecasting, which generates multiple forecasts to account for uncertainties in weather predictions. These ensembles produce a range of outcomes, allowing meteorologists to assign probabilities to different weather scenarios.

In conclusion: mathematics forms the backbone of weather forecasting, and percentages serve as a tool to communicate the likelihood of various weather events. While weather predictions aren't foolproof, understanding percentages in forecasts empowers individuals and organizations to make informed decisions, adapt plans, and stay prepared for the ever-changing atmospheric conditions. So, the next time you check the weather forecast, remember that behind those percentages lie complex mathematical models working tirelessly to decode nature's unpredictable behavior.

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

Black Friday Activities

I realize most students are off today but that doesn't mean you can't plan something for early next week, especially since Monday is Cyber Monday.  The nice thing about looking at Black Friday activities is one can find them so students get practice reading, gathering data, and creating infographics or finding infographics so they can learn to read them.  They can learn to interpret graphs, practice finding discounts, etc. Black Friday sales are so math oriented.

If you do a quick search for Black Friday infographics, you'll find quite a few.  There is one about this history of Black Friday, including when the term was used to describe some Stockmarket crashes and its current usage. There is even one comparing Black Friday with Cyber Monday.

This lesson plan has everything you need from standards and lesson objectives to the actual worksheets.  This activity has students calculating the amount they will actually pay for Samsung or I  phones, a Sony camera, and a Dell computer.  The activity lists the price and the discounts, not how much they actually pay so students have to do the calculations for themselves.

Yummy Math has an updated activity for Black Friday that is similar. This one has students looking at a 48 inch TV, a 13.3 inch MacBook Air, and a lego building kit. Students are asked how much they save, the percent of discount, and at the end, they are asked to find some Black Friday advertisements on-line or out of the local newspaper.  Students are then asked to chose the deal they think is best, post it, and explain why it's the best deal.

Although this next activity is geared for grades 4 and 5, it could be extended to middle school or high school.   The first part of the activity is to divide students into small groups.  Each group will set up a pop-up store with school supplies, artwork, snacks, etc.  Once the stores are set up, the teacher conducts a short discussion on Black Friday before giving students 5 minutes to discount every single item in the store.  Once all the stores are ready, each student is given a certain amount of money to spend.  As they "buy" things, that money is given to the owner. The idea is to spend all the money if possible.  At the end, students discuss if they managed to spend all the money, what made it hard or easy, etc.  Since these are older students, they can talk about the amount of discounts, which items sold better, why, etc.

These are most of the activities I could find that didn't cost money or weren't for sale.  I did see one activity that looked interesting in a packet that looked interesting.  It has students create add flyers for Black Friday sales. Perhaps choosing a local store or an on-line one.  Most every town has some sort of store, even if it is associated with a gas station.

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

Happy Thanksgiving

 

Using Infographics In Math

 

When I trained to become a teacher, infographics did not exist or at least the professors in my program had not heard of them.  In fact, most were not particularly familiar with computers or other technology so we were trained in the old fashioned way.  Lecture to the students, give them lots of notes, finish with an assignment of 30 problems or so and let them work most of the problems at home.  Since then, infographics have hit the scene and it is important to include them as part of the regular curriculum so students learn to read and interpret data.

Infographics by definition are visual charts or diagrams containing images, pie charts, and minimal text which is used to present an easy to understand a topic.  It is a tool used to educate and inform. Infographics that contain graphs allow students to find patterns and trends.  In math, infographics are a great way to present various concepts and themes.  Infographics help bridge gaps of understanding, improve memory retention, and helps improve engagement.  

When students are asked to create infographics, they must synthesize information and data, how they see the topic, and understand how they visual the material.  In addition, you as the teacher can create infographics to help students learn the material, or use as a reference.  Let's look at how the teacher might go about creating an infographic for students.

The first thing is to select the correct topic.  What ever the topic, it should correspond with curriculum objectives and is appropriate for the grade level being taught. Math infographics can be used to change concepts into perfect visual representation, thus clarifying and making the topic understandable.  

Creating the infographic design requires that one begin with selecting key elements, conceptualizing them, before finding appropriate and captivating images, diagrams, and the right icons.  Any explanations will be concise and support the visuals rather than explaining them. Each step, concept, or equation can be explained using simple language students understand.  For instance, a math infographic can break each step down into digestible steps to guide students into solving complex problems.  If possible include an interactive element such as fill-in the blank, label diagrams, or even mini-quizzes that help students with self reflection.  

On the other hand, students can use infographics to show how well they grasp a particular topic on concept. Students need to select a topic that doesn't need to be on math per se but might focus on using and presenting data.  One topic might be the use of slang in middle school, the most popular candies according to various age groups, how many platinum records their favorite music group has sold. Before having students begin their research, they should be exposed to several different types of infographics so they know the differences. In fact, this can lead to a discussion on each infographic, what information they are best for, etc.

One they decide on the topic, they need to conduct research.  They need to research their topic and what is the best way to share the information with others.  Specifically, they should think about which information, facts, or data are important to the topic and which aren't.  Then they need to consider what colors they think will work best, which graphics do the best job of conveying the information to the viewer, and in what order should the information appear.  They should also be reminded to include citations showing where they got their information from. 

Then it is time to put them together.  This can be done easily using one of several free sites available online.  Once the infographics are completed, they can be shared with the rest of the class.  Let me know what you think, I'd like to hear.  Have a great day.

Real Thanksgiving Based Math

 

As a middle school/ high school math teacher, I hate looking for Thanksgiving themed math activities only to find they are regular worksheets with turkey or pumpkin pie disguised problems.  You know those ones that start "John has 3 pumpkin pies and his sis Mary has 4.......".  I want ones that deal with planning how large a turkey to buy, how long to cook it, how many cans of pumpkin do you need for 3 pies.  Problems that are much more practical in real life.  I'd love problems dealing with Macy's Thanksgiving day parade, the bowl games.  Real things!

Let's start with the Thanksgiving meal.  This site has a really wonderful activity that does not rely on numbers that are unrealistic.  This activity begins by having students create the menu for Thanksgiving dinner.  This is great because what is eaten varies from place to place, person to person.  My sister always had roasted goose while my brothers go for turkey and I don't eat meat so I won't have one.  In addition, it also has students make a shopping list so they know exactly what they need to buy for it.  There is a page for calculating total cost of supplies and another for determining the cost of preparing the meal using electricity or natural gas.  It even looks into the cost of appliances used to store foods in. At the end, the Natural Energy Education Development Project or NEED even provides their numbers for students to analyze. This is a multi day plan and can't be completed in just an hour.  It takes all week.

On the other hand, the EconEdLink people have a lovely activity for looking at the inflation and cost of a Thanksgiving Dinner.  This has just been updated for the current year.  The activity has students looking at real data to learn about price index, changes in the price index, and takes them through the process of seeing how it is done while referring to real references.  Although this is an economics lesson, it does use math and provides students with some real life math usage.

Want to give students an idea of how much it costs to put on Macy's Thanksgiving day parade?  Look at this article from Yahoo Finance because it is eye opening. Image spending between $30,000 to $100,000 for a float, or sponsoring a  brand new balloon for $190,000? This article isn't an actual activity but it gives the costs so you could have students do some problems based on these numbers.  Although the article is from 2018, the costs should be at least the same or even higher. 

On the other hand, The Street has a bit more detailed information on the actual parade route, certain bits of information including how much gas is used to fill a balloon and the actual cost to fill the balloon which is more than the sponsor pays.  There is also a bit of history included but it is quite interesting.  This article is from this year.

Do you know I couldn't find any activities on football games played on Thanksgiving. I found a history of football games but no activity.  I found a list of three games due to be played but no activities. So I went to Yummy Math who has a full page of activities dealing with Thanksgiving.  These range from how much it costs to do a meal, to consumer spending and Black Friday, to Macy's day parade, to football, and a cool construction out of soda cans.

So now you have a nice choice of activities to do this week.  Let me know what you think, I'd love to hear.  Have a great day.

Using Sports To Help Teach Math.

There are always several students in your class who live and breathe sports.  They can tell you which player has the best stats, which teams are heading for the finals, or which number is the most awesome.  In most cases, the two sports one hears about most often are basketball or football.  

Fortunately for most math teachers, there are some cool ways one can incorporate sports into the class.  It may happen on a day where students are being released early, or things happened so your planned lesson has to be tossed out the window.  One can also plan certain activities to accompany certain topics taught in the actual math class. 

Using videos showing some field and track events, or setting up your own, one can have students carefully find the time of various runners for 100m, 200m, or 400m events using stopwatches. The resulting data can be used to compare how each runner finished.  For instance, figure out how much faster the runner in first place was compared to the one in second place.  Then have students calculate a runners velocity by dividing the distance they ran by the time it took them. (V=d/t).  If students calculate the runners velocity for each event, they can see if the runner uses the same velocity for each event or if it changes.

Another possibility is to have students choose a sport such as basketball, cricket, or baseball.  Students can go online to find statistics for the chosen sport. Students can look for things like the average height of basketball players on the teams who won the most championships, or build a fantasy football team by selecting players and explaining why they chose each one based on statistics.  There are so many possibilities with finding ways to use statistics.

Speaking of sports, one can find activities that look at sports stadiums.  One can explore how much electricity is used to run it for an event, how much food and drink must be purchased to supply those who attend games at the stadium, what does it cost to hire ticket takers, the refs for a game, and other costs of having games.  One can also figure out how much tickets should cost and the total one might get if every seat in the stadium was sold, half the seats, etc.  Take this a step further to determine how much the stadium must bring in via ticket sales to cover the cost of a game.  Many of these numbers can be found on the internet.

In fact, when having students look for stats associated with their favorite sports, it doesn't hurt to go over the basic stats or the more advance stats.  The Teaching Channel has some really nice idea for these. I don't know about sports so I'd have to use this article to help me.  In fact, just a short trip to the internet will net you with some really cool ideas to use in your classroom.  Let me know what you think, I'd love to hear.  Have a great day.

The relation between compound inequalities and absolute value inequalities

 I finally ended up using a textbook that I like.  It is older but it groups many topics together so they flow well from one to another.  One such flow is the book teaches how to solve compound inequalities first and in the very next section, students are asked to use what they learned with compound inequalities to solve absolute value inequalities.  

I think this made it so much easier for my students because they could "see" why these inequalities had to be solved as two problems rather than one due to the definition of an absolute value. I was able to relate the AND's and OR's to the appropriate absolute value inequality even to the point of showing how can be solved the same exact way as you do for a compound AND statement.

It was nice to have the book actually teach absolute value inequalities by having students rewrite them into compound inequality equations after reducing any into the absolute value being greater than or less than to a value. When using the idea of rewriting the absolute value inequality into a compound inequality, most of my Algebra I students were nodding and it was like a light bulb going off in their heads.

The biggest issue they had was to remember that the absolute value inequality with a less than sign was solved using the AND while the greater than required the OR situation to solve.  In addition, I had to remind them about the special cases of no solution or all possible solutions because they got in a rhythm solving the inequalities.  It's the same as when students first solve inequalities and just whiz through one that equals a negative number.

Once I finish this section, I should take time to have students convert compound inequality problems into absolute value inequalities so students better understand it is a two way relationship.  The previous section did not take time to show how to do that so they only do the conversion one way.  I think I'll find a worksheet of mixed compound and absolute value inequality problems so they can practice going from one form to the other.  I don't know if I'll actually have them solve the problem but I want to have them comfortable going from one to the other.

The longer I teach, the more I realize how important it is for students to make connections between one thing to the next.  I've expanded how I teach things so I am introducing ideas before we actually get to them or add depth to a topic as I teach it.  This is one topic, I need to show to students.  Let me know what you think, I'd love to hear.  have a good day.

Why Do Our Brains "See" Or Perceive Smaller Numbers Better?

 

Have you ever wondered why it is easier to guess the correct number of items in a bottle if there are fewer items?  Or why you might remember smaller numbers you've seen on posters than the larger numbers?  Well back in the 1870's, Willam Stanley Jevons wondered about the number 4.  He noted that if he tossed a handful of black beans into a box and fleetingly glanced at them, he was more likely to remember the number of beans if there were 4 or less.  If there were 5 or more beans, his guesses were more frequently wrong.

The article he published in 1871, lead to a long debate on why there is a limit on the number of items in a set that we can accurately remember. Recently, a new study was published which takes us a step closer to understanding why 4 seems to be the limit.

Apparently, the brain uses two mechanisms in order to judge the number of objects it sees.  The first mechanism is used to estimate quantities while the other improves the accuracy of those estimates but only for small numbers. This study connects debated ideas with the neural underpinnings.  This is a huge step because there is little out there in cognition where scientists have been able to connect with biological foundations. 

This study does not finish the debate or fully answer the question but it does begin to untangle the biological underpinnings for how the brain is able to judge numbers. The understanding of how the brain judges quantities could help solve bigger questions about memory, attention, and mathematics.  It turns out that the brains ability to judge the number of items in a set has nothing to do with counting. It's a number sense people are born with as demonstrated by infants and other animals such as fish, monkeys, bees.  

This innate number sense is often associated with survival for animals because they have to judge how much food, how much competition, where the most flowers are.  Since more than just humans have this ability, it is thought that innate number senses has been around for a very long time. 

In 2002, a paper was published that was able to link numbers to specific neurons. The authors studied monkeys and found that numbers are linked to neurons in the prefrontal cortex where higher level processing takes place.  These neurons lit up on a brain scan when the brains preferred number was seen. So if the brains favorite number is 3, the neurons fire more when the brain sees three objects.  In others, the number might be 5. It was also discovered that the neurons fire for the numbers next to the favorite number but do so less often.

Ten years later, in 2012, these same researchers discovered that these neurons respond when they are estimating a set of sounds or visual items that correspond to their favorite number. Unfortunately, no one had been able to find these neurons in the human brain due to brain imaging tools which did not have the resolution to study individual neurons and most scientists are unwilling to put individual electrodes deep into the brain.

Later, the same group in conjunction with a group in Germany were able to use people who already had the electrodes planted in their brains and discovered humans appear to have neurons that fire with the preferred numbers and in a later study, they analyzed additional firings to discover that for items above the number 4, the neurons fired less precisely than they did for numbers below 4. 

These findings align with the idea that the brain can only hold a limited number of items in their working memory which is 4. It may be the mechanisms are connected but there will have to be more research to investigate both the firing of the neurons and the number of items humans can hold in their working memory.  This is just the beginning of learning more about this area and will have to be explored in more detail.

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

What Do Digital Devices Do To The Brain?

 

I keep hearing that people who use digital devices regularly undergo certain changes.  I have a student who is upset because his parents have designated certain days to play games instead of letting him do it as much as he wants.  I notice the students who use their devices frequently all seem to have difficulty spending time getting assignments done so today, we are looking at that question.  "What do digital devices do to the brain. 

First of all, it appears that technology can change the structure of the brain.  When you have your phone set to allow notifications to happen, all those pings, alerts, and rings, can be a huge distraction. This makes it harder for people to concentrate consequently they do not perform as well in school, at work, or even in daily life.  It is possible that frequent multitasking on the device can lead to a diminished gray matter in the anterior cingulate cortex.

In addition, using digital devices can wear out the pleasure center of the brain.  When we read text messages, watch youtube videos, check out tiktok, or play video games our pleasure centers in the brain become hyperstimulated by the production of dopamine. Over stimulating this part of the brain can make other enjoyable activities less enjoyable.

Furthermore, the use of digital devices often leads to a decrease in physical activity.  These devices capture our attention so we are less likely to get up and move around. Unfortunately, we need to have physical activity to strengthen muscles and improve our cardiovascular system. There are several studies which link increased digital/technology device to a more sedimentary lifestyle and increased obesity.

The increased of digital devices can also lead to remembering less.  We don't use our memories as much because we can look information up on google.  This leads to a reliance on using technology instead of our memories because we know it will be there if we need to find it again.  Consequently, we do not develop our memory as well as in the past.

Finally, the use of digital devices/technology can decrease the amount of sleep we get.  Many people take their device to bed so they can check their email, social media, or other media just one last time before going to sleep.  This often leads us to surfing and losing track of time so by the time we get to sleep, we only have five or six hours instead of eight or nine hours. In addition, people lose sleep because they may be thinking about something they read or saw.  The other thing is that the blue light emitted can disrupt our regular sleep cycle.

This is a good thing to know.  It can explain why some of your students are up to all hours.  Let me know what you think, I'd love to hear.  Have a great day.

Teaching Compound Inequalities.

 I am getting ready to help my Algebra I students learn compound inequalities.  We've been through learning to solve one step, two step, multistep inequalities.  They had to learn when the direction of the inequality changes and I think they've got it all but now we hit a topic that can be quite difficult to learn. 

I think the hardest thing about compound inequalities is that they have two different solutions.  Up to now, they have been working on solving inequalities that result in one answer such as x < -5.

It has been suggested that one start with or type inequalities because the student solves two different equations and gets an answer for each problem.  This is similar to what they have been doing except it is for two problems, not one.

In addition, it is good to have students fill out compound inequality graphic organizers so they have a quick reference for or versus and. The first part of or should be filled out as students are learning how to work with the or so they become familiar with it.  

After they are comfortable with doing or, introduce the and so they understand that although there are two equations, the answers overlap more so than any with the or.  Add what they've learned to the graphic organizer for and so they have complete notes.  

No matter which is being taught - or, or and, there needs to be some sort of visual representation such as the use of a number line.  Rather than just shading in the number line, one should use two different colors so that students connect the visual representation with the equation.  In fact, if you want to solve one half in one color and the other half in a second color, you can then draw the answers on the lines with matching colors.  This provides a visual connection with the eye.

This can be re-enforced by having students watch videos by Khan Academy, Math Mashup or any other reputable site.  Khan academy usually walks students through the process but their language might be of a higher level than where your students are at.  However, Khan Academy usually has quizzes and such to help assess their understanding.

Furthermore, one can explore Desmos, or Geogebra as both have activities that can be used for this topic.  Geogrebra has several graphing activities designed to specifically to practice graphing of inequalities.  So have fun teaching compound inequalities.  Let me know what you think, I'd love to hear.  Have a great day.  

What Are Math Snacks?

Have you every had a math snack?  No it isn't food.  Math snacks is a site that is filled with all sorts of learning games.  These learning games are designed to focus on specific skills.  Technically, math snacks are educational animations, games, and interactive tools that help students.

Math snacks is designed for students in grades 4 to 8 but could be used for older students who need a bit more reinforcement. The page called teaching with provides information on how to use these snacks as part of regular instruction.  

There is a short video show how to use the snack, and provides a bunch of resources to help use it.  There is a teacher guide, a learners guide, answer key, common core standards it meets, the ability to download it, and a transcript.  Not all activities have the full list of resources but most have at least a teachers guide so the teacher has help.

Each snack lists the name, whether its an animation, game, or interactive tools, along with the title. I checked out Curse reverse. This is an animation focused on helping middle schoolers learn to build algebraic expressions.  It can be done in either English or Spanish.  The game it took me to, was well done as far as artwork went, it uses the up, down, left, or right arrows to move the person through the maze.  I got stuck shortly into it because I couldn't quite figure out how to get the person over something but I need a bit more time to figure it out.

I took time to look at the teachers guide for Curse Reverse which listed everything you should do before teaching this snack. It even includes discussion questions the teacher asks students after letting them play for 10 minutes.  There is also a section designed to support the students in doing the activity and finally, if you need to enrich this activity, there is a section for that.  It is a very complete guide and has everything you need to do a good job teaching this.

Math Snacks was developed by people at New Mexico State University so it has been created by several educators.  They even took time to do some research on how well it works, and a list of publications associated with the snacks.

This is something my middle school students could use and enjoy.  I hope to include a few of these activities for my 6 to 8th graders because I think they might enjoy them.  Let me know what you think, I'd love to hear.  Enjoy your day. 

Math Bits For Math

The other night I played a game that said it was created by Math Bits and one of the universities, so I looked the site.  It is chock full of information for all levels of math.  Some of the material is free while others requires a subscription.  Either way, I like the simple interface and easy access to each section.  

Their free section is referred to as Math Cache.  The section has math for middle schoolers all the way up to Calculus students and it even has a section for the T-I 84 hand held calculator.  I clicked on the Algebra I math cache to see what it had.  First off,  I noticed two games, one for pre-algebra and one for Algebra 1.

The games are actually short practice sessions that students do and if they do the problems correctly, they find their way to the next one because the answers give part of the URL for the next one.  If they do it right, they will work through 10 different ones.  These can be done individually or in a group and there is a worksheet one can download for students to use.

Then there is a section designated for students only.  It contains a link to the Math bits notebook filled with Algebra 1 topics.  If you click on a topic, you go to the lessons and practice associated with the overall topic and when you choose on the desired subtopic, you are sent to a page that looks like a notebook page filled with notes, examples, etc. There is also a section for online quiz practices, jeopardy, and bingo.  In addition, there are links to songs, a dictionary, study tips, and free graph paper.

In the teacher section, there is information on what a subscription for the materials available on site gives you and there are free samples.  One can look at the in depth listing to show exactly what one receives and what the cost of an individual subscription or school district subscriptions.  There is a section for Geometers sketchpad, estimating ages, Fibonacci faces, and information on using Algebra Tiles.  Finally, there is a section on Math in the movies.

If you want to access the math bits notebooks, there is a link on the main page so you can go there directly.   Furthermore, most everything is listed on the resource page so it is easy access for most of the resources.  

I think this is a good reference page.  It would work to support what students are learning or when you are out and you want a bit more material available for students to look at.  It can be found at mathbits.org and mathbits.com.  Let me know what you think, I'd love to hear.  Have a great day.

New Information On Prime Numbers.

 My apologies.  I was traveling with one computer and it died while out of the country so I didn't get anything done for Monday.  Now on to today's topic.  I remember having a math professor who did two things.  One was when he wrote his tests, he did it in a bar and wrote the problems on a napkin.  The second thing he continually worked on was finding the largest prime number he could.  I think he retired before he did that.

Now there is a new generation of mathematicians who are working on figuring out the distribution of prime numbers and understanding that distribution. Remember that Eratosthenes came up with the first real method of finding prime numbers between one and 100 known as Eratosthenes's sieve.  There are sieves to find all sorts of primes such as twin primes, etc.

Although we have the sieves, people do not understand the distribution of primes.  Since it is much harder to find primes in a sieve that contains much larger numbers, mathematicians have to estimate the number of primes they think are in the range. In addition, it is often harder to make predictions when looking at other sieves such as the ones for twin primes due to the size of the remainders.  

As you move to larger numbers, it has been found that the remainders fall into a statistically predictable pattern and eventually even out. This means say you take the remainders of 1 and 2 when divided by 3 and place them in one of two buckets. eventually, the two buckets will have the same number of primes.  Mathematicians need to know when the buckets even out and how soon that happens in order to know more about primes.

There were spurts in investigation in the 60's and 80's but nothing more happened until recently. A mathematician investigated the question about buckets evening out and how soon that happens, and calculated that the level of distribution was 0.6 for commonly used sieves. 

His grad students extended it to 0.617.  To do this, they used a technique of inclusion/exclusion which is similar to what students do when they work with the sieve of Eratosthenes.  They exclude 2 and all it's multiples which eliminates about half the numbers or 50%.  Then they exclude 3 and all its multiples which throws out another 1/3rd of the numbers. Ok, this means that due to the way things are counted, many numbers are double counted such as 6 and its multiples because 2 and 3 are factors of 6.  So you add the 1/2 + 1/3 and then subtract 1/6 to account for the twice counted numbers.  This way you do not have an over estimation.

So then you eliminate all numbers for 5 and its multiples but you have to subtract 1/10 and 1/15 to account for any numbers double counted due to 2 and 3.  Thus the process continues with the denominators getting bigger and bigger.  This creates and upper and lower boundary rather than an exact answer. 

This lead to someone to propose the idea that the buckets even out based on the generalized Riemann Hypothesis. This means we are looking for all the primes up to N and the remainders are equally divided up into the number of buckets equal to the square root of N.  So this opens the way for more possibilities in determining the number of primes but it will be a while.  

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