Right Sized Chunking For Teaching Math.

Teaching mathematics involves breaking down complex concepts into manageable chunks to improve learning and comprehension. Determining the best-sized chunk for teaching math is a balancing act between students' cognitive abilities, attention span, and the complexity of the mathematical concepts being taught. This process, known as chunking, aims to present information in digestible portions, ensuring students grasp fundamental concepts effectively.

Chunking involves breaking down large or intricate information into smaller, more manageable sections. In mathematics, it means organizing concepts, problems, or lessons into smaller units that align with students' cognitive capacity to process and retain information effectively. There are several factors that influence the size of the chunks taught to students.

First, the complexity of mathematical concepts plays a significant role in determining chunk size. More complex topics might require smaller chunks to ensure thorough understanding. Next, one needs to consider the age, grade level, and prior knowledge of students. Younger students or those new to a concept may require smaller chunks, while older students might handle more extensive chunks. 

Chunk size should consider students' attention span and engagement levels. Breaks between chunks can help maintain focus and prevent cognitive overload. In addition, chunking should allow for a progressive learning journey, with smaller chunks building upon each other to form a comprehensive understanding of the topic.

To find the proper sized chunk, begin with smaller, bite-sized chunks to introduce foundational concepts. Gradually increase the chunk size as students demonstrate understanding and confidence. It is important to strike a balance between depth and breadth of content is essential. Ensure that each chunk provides sufficient depth for understanding while covering the breadth of the topic gradually. Furthermore, one should include visual aids, diagrams, and relatable examples as they can assist in chunking information effectively. They break down complex ideas into more understandable visuals, aiding comprehension.

Incorporate practice exercises or problems aligned with each chunk. This allows students to apply their learning, reinforcing understanding before moving on to the next chunk. In addition, regular assessments after each chunk help gauge students' understanding. Provide timely feedback to address misconceptions or gaps in comprehension. The optimal chunk size for teaching math may vary based on individual student needs, the nature of the topic, and the teaching environment. Teachers should remain flexible, adapting chunk sizes as needed to accommodate students' progress and understanding.

In the realm of teaching mathematics, finding the best-sized chunk for instruction is a nuanced process that requires a balance between depth of content and students' cognitive capabilities. By breaking down complex concepts into manageable chunks, educators can ensure effective learning, understanding, and retention of mathematical principles. The key lies in adaptability, using varied teaching strategies, and gauging students' progress to determine the most suitable chunk size that optimizes learning and fosters a solid foundation in mathematics. Let me know what you think, I'd love to hear. Have a great day.

Quick Uses For Social Media Like Ideas In Class

 

Although there are ways to use social media in the classroom, I prefer to use activities that are similar to social media but are not the actual one.  Mostly it is because I'm afraid that students will end up having too much fun on TikTok, Instagram, or other program.  So today, I'll be sharing ideas that can easily be utilized in the classroom but still have all the fun of social media.

Back in 2012, Dan Mayer started a site called 101 questions on line.  There is a picture to check out or a video for to watch before they are asked to type in the first question they have in mind.  Although the site was hacked back in 2020, the information is still there but you can't sign in.  This is similar to twitter in that students can only use 140 characters.  This could easily be adjusted to have student ask a math based question.

Matt Miller, author of Tech Like A Pirate has some wonderful ways to recreate the social media like experience without having students on social media.  For instance, he has templates that allow you to have students do videos just like they would on Tiktok but using Google slides.  This means you can download the templet and assign it via google classroom.  He even tells you how to set it up for every student.  This is where you could assign students to create a short video on a math topic, explain how they attacked a problem, discussed where the mistake is in a problem or so much more.  

If you prefer using Instagram or Snap chat, he has templates for those too and he explains how to set each one up.  In addition, he gives a few ideas of how to use them so you don't have to be an expert in this type of technology.  I like these because I don't have to go into a great amount of detail since my students are experts when it comes to social media and I don't have to set up accounts on each one.

Furthermore, he explains how to use certain websites to create fake Facebook pages, creating things with hashtags, and even a site where students can create fake text message interchanges.  So by just going to his page, checking out all the possibilities, and a bit of imagination, you can have students "use" social media in a safe environment.  Remember, students can make short videos explaining how to do problems, how to problem solve, and so much more. 

Check out Class Tools if you want to create fake Facebook entries. This could be used to create entries for mathematicians through history.  There is also a SMS generator that could be used to show students a discussion between two mathematicians, or two students working to solve a problem. or have students read the interchange to see if they can pick up where the error was made.  In addition, I found a tweet like tool that allows you to create fake tweets. Check it out for more things they have.

Hope this gives you a starting point for using fake social media in your classroom.  Let me know what you think, I'd love to hear.  Have a great day.

How To Safely Use Social Media In The Math Classroom.

In today's digital age, social media has moved past its original purpose of networking and entertainment. It has emerged as a powerful tool in education, offering innovative ways to engage students and facilitate learning, even in subjects traditionally viewed as less interactive, such as mathematics. When used thoughtfully and safely, social media platforms can transform the math classroom into an interactive, collaborative, and stimulating environment.

There are multiple benefits to using social media in the classroom. Social media platforms provide a familiar and engaging medium that can capture students' interest in mathematical concepts, provide interactive content, trigger discussions, and host multimedia resources that can make math more relatable and accessible. In addition, platforms like X formally Twitter, Facebook Groups, or specialized educational networks offer avenues for collaborative problem-solving and peer-to-peer learning. Students can share ideas, ask questions, and work on math challenges together, fostering a sense of community.

Furthermore, social media can illustrate real-world applications of mathematics. Teachers can share examples of mathematical concepts in everyday scenarios, connecting abstract math theories to practical, relevant situations. Platforms like YouTube, Pinterest, or math-focused blogs offer a vast array of instructional videos, infographics, and tutorials. These resources cater to different learning styles and allow students to explore math concepts at their own pace.

However, one has to follow certain guidelines to ensure student safety. First, opt for dedicated educational platforms like Google Classroom, or platforms designed specifically for math education. These platforms offer controlled environments with features tailored for learning. Next, set clear guidelines and expectations for the use of social media in the classroom. Educate students about responsible online behavior, emphasizing respect, etiquette, and privacy.

It is important to monitor and moderate content. Always maintain active supervision of discussions and content shared on social media platforms. Moderate interactions to ensure they remain focused on academic discussions and are free from inappropriate content. Safeguard student privacy by limiting the sharing of personal information and ensuring compliance with privacy laws and school policies. Obtain necessary permissions and consent from parents before using social media platforms.

Take time to promote positive interactions. Encourage constructive discussions, peer support, and positive feedback among students. Foster a supportive online community where students feel comfortable sharing ideas and asking questions without fear of judgment.Incorporate lessons on digital literacy, critical thinking, and online safety into the curriculum. Teach students how to discern credible sources, evaluate information, and navigate social media responsibly.Social media integration in the math classroom holds tremendous potential to transform the learning experience, making mathematics more engaging, relatable, and collaborative.

By leveraging the benefits of social media while prioritizing safety and responsible usage, educators can create a dynamic learning environment where students actively participate, explore mathematical concepts, and develop essential digital literacy skills. When utilized thoughtfully and in accordance with established guidelines, social media becomes a valuable ally in enhancing mathematical understanding and fostering a lifelong appreciation for the subject. Let me know what you think, I'd love to hear.

Overcoming Math Anxiety

 

There was a paper recently published in which the authors concluded that math anxiety can cause students to disengage, thus creating a significant barrier to learning. Although mathematics is often considered a universal language, many students find it a source of anxiety and stress.  They have a fear of numbers, equations, and/or problem solving which can show itself as math anxiety. This can lead to a lack of confidence that hinders academic performance.  There are things we can do as parents, or educators to help students get rid of their anxiety and begin to like math.

First off, math anxiety is more than just a dislike of a subject.  It is an emotional response where people feel tension, fear, or apprehension when they have to complete a mathematical problem or task.  The anxiety can be triggered by a past negative experience, the pressure to do well, or even the belief that the person lacks that innate mathematical ability that makes you good at math.

One thing we can do is create a supportive environment, where we help students understand that math ability can be developed with some dedication and effort.  Take time to emphasize the importance of a growth mindset where mistakes are seen as opportunities to learn rather than failures.  Celebrate small victories and efforts.  Always encourage students when they make progress, solve problems, or show improvement since positive feedback boosts confidence and motivates students to engage more actively with math. 

In addition, recognize that each student has a unique learning style and pace. Offer personalized approaches to learning math, incorporating various teaching methods like visual aids, real-world applications, or interactive activities to cater to diverse learning preferences.

The next thing is to help build student confidence through practice. Gradually increase the difficulty of problems, allowing students to build confidence by mastering simpler concepts before advancing to more complex ones. Small, manageable steps can ease the fear of overwhelming tasks. Encourage regular practice to reinforce understanding. Provide access to ample resources, such as worksheets, online tutorials, or math games, allowing students to practice independently at their own pace.

Provide relatable examples. Connect mathematical concepts to real-life scenarios to illustrate their relevance. This approach helps students understand the practical applications of math and makes it more relatable and less abstract. 

Take time to foster an environment where students feel comfortable asking questions and seeking help without fear of judgment. Encourage peer-to-peer learning, group discussions, or tutoring sessions to facilitate collaboration and support. In addition, teach effective study strategies, including organization, time management, and breaking down complex problems into smaller, manageable parts. Equipping students with these skills can alleviate anxiety by providing a structured approach to tackling math challenges.  

Overcoming math anxiety is a gradual process that requires patience, understanding, and consistent effort from educators, parents, and students themselves. By creating a supportive environment, promoting a growth mindset, providing ample practice opportunities, and offering guidance, individuals can gradually build confidence and develop a positive attitude towards mathematics. Empowering students to view math as an accessible and intriguing subject can unlock their potential and pave the way for academic success. Let me know what you think, I'd love to hear. Have a great day.

Essential Mathematics For Pilots

Flying an aircraft involves more than just skillful control of the plane; it requires a solid understanding and application of mathematics. Pilots rely on different mathematical concepts and calculations to ensure both a safe and efficient flight. From navigation and flight planning to understanding aerodynamics, mathematics plays a crucial role in the world of aviation.

During navigation and flight planning, math is used in several different aspects. First, pilots use basic arithmetic to calculate time, speed, and distance. These calculations help determine travel time between destinations, fuel requirements, and optimal cruising speeds.Second, pilots need to know how to dead reckon. Dead reckoning involves estimating an aircraft's position based on its previous position, speed, and course. Pilots use trigonometry and basic geometry to calculate headings, wind corrections, and groundspeed. Third, during navigation, pilots need to understand concepts like coordinates, vectors, and bearings is crucial when using navigation systems like GPS (Global Positioning System) or VOR (VHF Omnidirectional Range) to determine position and course.

As far as the weather, pilots analyze weather charts that use various symbols and measurements. Interpreting meteorological data involves understanding percentages (for precipitation chances), wind speed, temperature, pressure, and visibility calculations. In addition, As far as wind and drift calculations, pilots use trigonometry, to calculate wind direction and speed to adjust for drift, ensuring the aircraft remains on the intended course.

Pilots also need to understand the principles of lift and weight and that involves mathematical concepts related to forces, pressure differentials, and aerodynamic coefficients. In addition, pilots use mathematics to analyze aircraft performance data, including takeoff and landing distances, climb rates, and fuel consumption rates. Furthermore, pilots apply mathematical risk assessment techniques to make critical decisions during emergencies, considering factors like probabilities, decision matrices, and safety margins.

Finally, pilots need to calculate fuel consumption rates and managing fuel reserves is vital. Pilots use mathematical calculations to ensure they have enough fuel for the flight duration, including contingency reserves. So pilots use a lot of math in their daily job. Let me know what you think, I'd love to hear. Have a great weekend.

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.

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.