Happy Memorial Day

 

Warm-up

 If it takes 3 oranges to make one cup of juice, how many quarts of juice will you get from 305 oranges?

Warm-up

 If you get 305 oranges from a tree in one year, how many oranges will the tree produce after 20 years?

How Effective Is Computer Based Instruction?

 Last semester, I put my students on a computer based program as a way of helping scaffold their foundational knowledge.  I'd say about a third of the students worked hard, attempting to learn.  Another third did some but did it since it was assigned but they didn't make much progress, and the last third spent more time trying to play games when I was not near them so they never actually did any of the assignment.   I know that we've had to rely on computers so much more since COVID hit, that I wondered if computer based instruction is as effective as we think. 

I came across a paper which looked at the effectiveness of using computer based remediation in two and four year colleges.  The premise is that students do most of their learning via the computer and the instructor is a facilitator rather than the primary instructor. The study was done to determine if technology centered instruction works well for students who are underprepared for college, academically.

In general, students are tested when they enter college and if their skills are not good enough, they are enrolled in developmental classes designed to help them bridge their gaps and get the skills they need to pass the college level classes.  Research indicates that only half the students who enrolled in this type of class actually finished it, and those who do not finish the classes are less likely to complete the degree requirements.

The use of technology driven instruction can be both good and bad. A positive effect is that the instruction is consistent across all classes allowing for a more even levels of preparation for higher classes. Another advantage, is that time is freed up so faculty can spend more time providing individualized rather than whole class time instruction. This means they receive more assistance when they need it, rather than having to wait for office hours or lab time. In addition, there is a clear time line for assignments, quizzes, and gives the instructors a chance to mentor student mastery of the material.

On the other hand, not all students do well learning on computer based programs and have the self motivation to work their way through the material without getting bored. These programs are designed with the expectation that students have the ability to regulate their time so they work through the program at the appropriate pace to finish the material.  It also assumes students will not be distracted by the possibility of surfing the web, checking mail, etc.

Furthermore, it also assumes students will actually work the problems rather than guessing their way through the answers until they hit the right one which raises questions of potential mastery and engagement in learning.

The conclusion to the study indicates that taking a computer based class did not improve the number of students who passed their remedial math class.  In fact, the number of students who went on to pass their first college based math class actually decreased and they were less likely to complete fewer credits over a period of six terms. Furthermore, they were less likely to get an associate degree or finish any credential within a 6 year period.

One suggestion for why students didn't do well in their first college level math class is simply due to the class having an actual instructor rather than letting student progress at their own rate on the computer.  Although these results are for college level, I think it is important to keep these results in mind since many high schools rely on computer based classes as a method for students to recover credit for classes they've failed.  Later next week, I'm hoping to look at programs that adapt as students work their way through the program.  Let me know what you think, I'd love to hear?

How Far Behind Are Students In General And What To Do About It?

 

I ran across information the other night in which someone shared the research on how far behind our students have fallen during the pandemic.  The article went beyond just reporting the results, it also suggested several research based ideas to help students catch up within one to two years.  It was wonderful seeing some real data and real solutions to this question which plagues every school district.

First off, the research focused mostly on elementary and middle school students rather than high school students.  We know that when children get behind in elementary school, they struggle for the rest of their time in school and they are more likely to drop out.  In addition, the data came from across the nation so it was not focused on just one state. The research compared the student achievement growth in the prepandemic period from 2017 to 2019 against the growth in the pandemic years of 2019 to 2021. The data came from the information collected by NWEA, the people who offer MAPS testing which is used in multiple districts across the nation.

Fortunately, it is known that the pandemic officially hit the United States in early 2020 and just about every school ended up going remote with methods ranging from the use of packets, television broadcasting, zoom, or what ever method the district could manage. Unfortunately, the results were quite discouraging.  In low poverty schools, students lost the equivalent of about 13 weeks of of unperson instruction while students in high poverty areas lost the equivalent of around 22 weeks or a little over a half a year.  

If you look at the results along racial lines, African American and Latino students lost four to five weeks more than caucasian students.  They also discovered the districts that returned to in person instruction sooner only lost between seven to ten weeks of learning. In other words, the longer schools stayed closed the wider the gap.

So what can schools do about it.  The following suggestions have been researched and shown they produce demonstrable results.  These are the best solutions we have at the moment.  It will require a bit extra spending and time but if any of them are implemented, students can make up their loses in one to three years.

One of the best recommendations is for students to implement is through high-dosage tutoring where a tutor meets three times a week with one to four students for the whole year.  This can result in up to weeks of instruction.  That is almost equal to the amount of time students have fallen behind in high poverty levels. 

Another suggestion is to place students in summer school where they work on missing skills.  This type of extra work can help students recover up to 5 weeks of instruction.  If that is not an option, add one extra period to the instructional day so students receive one extra period of instruction in math every single day for the full school year. This can result in a gain of 10 weeks of instructional gain.  The final suggestion is to lengthen the school year  just like most do to make up for unscheduled school days for the next two to three years.

Unfortunately, all of these require spending funds that districts may have earmarked for other things.  The last district I worked for had already earmarked all their COVID funds to help run the district for the next few years and none of it was earmarked for direct student instruction other than buying new books.  It seems to me that we need to spend the monies on these suggestions, especially the first one so we can help students catchup rather than enduing up with a lost generation of learners.  

The tutoring idea could be done by adding in an extra period, using peers, aids, janitors, kitchen staff, and parent volunteers to work with the students.  Yes, they would have to be trained but that can be done. The other, adding in one extra period a day of math so they get a double dose is doable, especially if they turn a six period day into seven periods.

We have the research, we have solutions, lets see if we can talk our districts into implementing them so our students get the best chance to get the future they deserve.  Let me know what you think, I'd love to hear.  Have a great day.

Why Use Multiple Ways To Solve Problems

 

One thing that is emphasized is to show students as many ways as possible to do something, but unfortunately, many teachers and textbooks tend to show only one way to do something rather than exposing students to multiple ways.  For instance, FOIL seems to be the main way students are taught to multiply binomials yet there are so many ways to do it.

I know when I was in school, teachers only taught one way to do any problem including only the FOIL method for multiplying binomials.  It's only since I began teaching that I've learned multiple other methods or rather how to use methods they already knew from earlier situations that could be applied to a situation.  Research indicates when we teach students to solve math problems in only one way, the method taught may not be the "best" way of doing other problems.  We need to focus on having students learn multiple methods so they can choose the method that works best for the situation and for them.

Teaching students multiple strategies to solve problems helps make them more flexible mathematical thinkers. This is because they have multiple methods to choose from to solve any problem and can select the one that works best for the situation rather than trying to figure out how to apply the one method to every single problem. 

In addition, when we teach students multiple strategies to solve problems, we are providing scaffolding so they find a place that has what they need. In addition to finding the strategy they can use, they are exposed to methods that may be beyond them at that point but which they may be ready for later on.  Furthermore, these strategies can be used as they go from one skill to another.  For instance, an elementary teacher taught me how to multiply using the lattice methods.  I saw it and realized I could use the same method with my high school students when they multiply binomials, or trinomials. Many of my students had this teacher in 5th or 6th grade so they knew the method and could easily use it. 

Furthermore, when students feel they can't do math, or don't like math, or don't have that math gene, they often are more willing to try doing problems when they've been taught multiple strategies. Teaching them multiple strategies allows all students to have an entry point  and a way of attacking problems at their level rather than feeling lost because they don't understand or can't do the method taught by the teacher. 

One thing that comes up again and again in research is to teach students how to visually represent the problem no matter what age or level of mathematics students are in. Another thing that show up is one thing that I have not always done.  Research indicates that in addition to teaching at least two ways to do every problem, the teacher should take time to compare and contrast the two methods because it helps build analytical thinking.  I've never done the compare and contrast part and need to start doing that.  It has been found that students benefit when they are expected to use multiple methods to find solutions. 

Once students are used to using multiple methods, teachers can then take time to start looking at why one method might be better for solving certain problems for its easy and efficiency.  In addition, it is suggest that teachers show methods that look like they would work but don't and then take time to lead a discussion on why it happened this way.

Finally, it is important for the teacher to create a culture of problem solving by having discussions where students show the different ways they used to find the solution.  This reinforces the idea that there is not one way to solve a problem.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

If it takes four pounds of fresh grapes to make one pound of raisins, how many pounds of raisins will you get from 2548 pounds of freshly picked grapes?

Cute Book Store

 I am in Dubrovnik, Croatia.  I found two of these in the Old Town area and it turns out they are bookstores.  Cute.  Enjoy.

Sales Tax And Scientific Method.

 

I was all set to write about one topic and I decided to discuss a different topic since it is one that only works in places where taxes are collected at checkout.  I am currently in Dubrovnik Croatia where I landed last night.  I've been out and done a bit of shopping since I tend to like to eat most of my food in my rented place.  

In the United States, tax is added on at the counter and it varies from jurisdiction to jurisdiction.  In some places in Alaska, it can be as low as 1/2 percent while in other places such as California, it can be over 10 percent.  Even within some states, the rates vary according to the county you are making your purchase in.  I know folks who carry a variety of sales tax cards so they are set for where ever they are.

Now, in all the places I've visited, the tax is included in the price of the item.  If the item says 5 Euros, that price has already had the taxes added in.  This is known as a Value Added Tax and it is usually a flat rate so you don't have to keep track of a variety of tax rates.  I know that in some countries like Canada or Iceland, if you keep the receipts and you've spent over a certain amount, you can get the tax refunded at the airport when you leave.  

The only thing I know of in the United States that has a value added tax is gasoline and diesel rather than adding in tax once you've paid for the gas. Of course, those taxes are road, federal, and some states that are added in but again, those are not the same everywhere so there is a large variation across the land while in most European countries, the price is consistent through the whole country. 

In addition, jurisdictions often have different rates of tax being charged depending on the items you are buying such as food.  In some places, they do not charge you for food that you buy but if you eat out in a restaurant, they do collect it.  In others, they have a different rate for regular things versus cars.  Sales tax can be quite confusing for many due to the different rules.

The thing I've wondered about is this - If you are operating using a value added tax, does it work out the same as if you add it later.  In other words if the country adds an 11 percent tax to the price so you pay the 11 percent on each item, does it work out the same as if you added up the cost of everything first before adding the tax?  This question might be a good one to pose if you'd like to teach students about formulating a hypothesis in math and then asking students for ways to test that question.

This is a nice class lesson to see which way the government might collect more funding.  Discuss the hypothesis with students, let them make a prediction, have them help create ways to test the hypothesis, do the experiment, and collect the answers. This activity shows how one can use scientific method in Math.  let me know what you think, I'd love to hear.  

Math and Bus Routes

Have you ever travelled by bus around town?  I have but usually when I'm off traveling.   Of course, bus schedules and the actual times buses arrive are not always the same. Some folks did a mathematical study to see why buses often arrive in groups of two or three rather than being spaced out as planned. The official name for it is "Bus Bunching". 

This is a problem that transit companies have been trying to solve for decades. Overtime, researchers have been using mathematical models to work on understanding why it happens and how to prevent it. Unfortunately bunching causes riders to wait longer and more variable. In addition, bunching makes the system itself more unreliable and off schedule.  This makes people change to other forms of transportation.

It was discovered that the bunching occurs because bus routes are inherently unstable. At first, when buses are on time, everything works well. The bus makes it's stops, people get on and off, and the world is great but once the bus gets behind, it is almost impossible for it to catch up. In fact, once the bus is behind schedule, it gets even more behind as time passes until the second one catches up to the first one. The same thing happens when a bus starts out on time but gains time so it is early and early until it catches up with the bus ahead of it. One slows till it meets the one behind and one speeds to catch up with the one ahead.

The time between buses is referred to as bus headway so when it begins to run late, more people arrive at the stop and the time needed to load and unload passengers increase making the bus run later. On the other hand, when the bus is early, there are fewer passengers waiting so the time needed for the stop decreases and the bus travels the route quicker.  Either way, one bus ends up catching up or slowing down to another.

Proposals to solve this issue include skipping stops if no one is there or needs to get off or limit the number of people who get on the bus as ways to help make up time for late buses. A different strategy is to build additional time into the schedule referred to as slack to help allow for the variable travel times. For buses that run early, companies instruct buses to wait at stops to use up some of the extra top. Unfortunately, slack doesn't always help buses running late. Skipping or adding slack tends to reduce the speed at which people move along the routes.

Modern technology has helped with these issues.  Transit companies are now able to provide real time feedback as the buses are observed operating in real time.  Buses can be directed to skip stops, slow down, speed up, or stop a few minutes to get them back onto schedule.  Researchers have created programs containing algorithms which monitor individual buses and can provide information to dispatchers who relay the appropriate information.  

It is hoped that one day, these algorithms will get rid of the bunching issue, help buses run on time so everything happens as projected.  Let me know what you think, I'd love to hear.  Have a great day.

How Are Bus Routes Set?

I don't know how often you travel by bus.  I generally travel by bus when I'm traveling internationally because its easier than trying to rent and drive a vehicle.  I often wondered how cities determined where the bus routes went and their schedule.  When I lived in Dallas, I took the bus but I hated it because I had to take a local bus to the transit center and get a bus to downtown.  Going to the transit center wasn't bad but coming back, the local bus already left before I got there and I had to wait quite a while for the local.  I hated it.

So lets look at how cities decide to set up their bus routes and tomorrow we'll look at issues bus routes have that are being helped by mathematics.  Transit companies spend quite a lot of time figuring out where bus routes known as fixed routes are placed.  The idea is for these fixed routes to help move people from where they live to where they work. We will start with some terms.

We have radial routes which are the backbone of the transit system.  They run around the central business area and the urban core.  These routes form the backbone of the transit system.  They are characterized by frequent stops, slow bus speeds, and shorter passenger trips.  The other type is the cross-town routes link major activity centers with direct routing or running through other with high intensity corridors that are not in the business center.  They are designed to have the best transfer connections.

When getting ready to set the different routes, they have to take into account the number of riders, the distribution of riders throughout the day so they can decide the number of buses and the times they should run.  For instance, there will be a surge when people want to head off to or come home from school or work. There are likely to be fewer travelers late at night.  

In addition, they have to determine how many bus stops to have because the spacing of bus stops does effect the number of riders.  The spacing of stops can also effect traffic since in many cities the buses stop on the street and end up blocking a lane thus slowing down traffic.  Furthermore, they have to take in to account the disabled, those who have strollers, bikes, and wheelchairs.  

The transit authority also has to do planning for short range (3 to 5 years) and long range (20 to 30) years so as to keep up with the demands of a growing area. Short range planning looks primarily at the current routes and predicts any changes to schedules over the next three to five years.  Long range planning look at what the city area will look like in the future.  They try to predict the population, employment, population density, and the where traffic congestion will occur as they plan. They also have to account for funding they think they will get to pay for it all and also helps provide political support.

So these are all the facts used to set bus routes in places.  Next time, we'll look at the math behind trying to get buses to run so they don't bunch along the way.  Let me know what you think, I'd love to hear.  Have a great day.

 

Warm-up

 

If you have 56,278 cocoa beans and you need 400 for a pound of chocolate, how many pounds of chocolate can you make from this number of beans. 

Warm-up

 

If each cocoa pod has on average 40 beans.  How many pods do you need for 56,275 beans?

Mark-ups At The Grocery Store

 

Today, I'm looking at how grocery stores determine the amount of markup they use since most of our students end up going there to shop for something. They are familiar with them so taking time to talk about markups and connecting them to a personal experience can help students relate to the topic.  

Even when discussing the amount of markups used by grocery stores, it isn't a clear cut percent because it depends on the type of product. Some items are marked up only a bit while others have a huge markup.  In fact, there are so many different factors, my mind is going wow.

In general, the people who determine the markup in grocery stores base it on the category such as competitive, destination, innovative, etc, market intel, demand, gross margins, and sales target.  There are staff members who negotiate prices for every product such as milk, vegetables, etc, are trying to get prices that are best for the store but allow them a better margin but some like milk are subject to both state and federal pricing regulations and it's price for store brand is set so there is a single digit or negative margin so they match other retailer pricing.  If the milk is more of a specialty item that is locally produced with cream and undergoing low heat pasteurization, it might have a much higher margin of 20 something.  The idea is that one looks at the amount of sales and margins across the whole category so the gross margin where the store wants.  So they have to look at what margin the whole category brings in with all the individual product margins taken into account.

Another thing to look at is the wholesale price they pay.  Wholesalers are the people who sell the products to the stores.  Sometimes the wholesalers are a part of the grocery chain but more often they are a third party who sells these products to make a profit. These people markup their products from a low of 2 to 3 percent for mass market products to 30 percent for specialty items but most range in the 5 to 15 percent range.  Next we'll look at the product producers who sale to the wholesalers.  These people sell their products to wholesalers based on what they believe retailers want.

Farmers who sell fruits and vegetables sell their products by the case and price them according to what the market will bear so they cover their labor and overhead and what the wholesaler is willing to pay. They hope to end up with a price that covers their operational costs while hoping to make a profit. For multi-ingredient products, pricing gets trickier but depending on the product can have a high markup.  Again it all boils down to setting a price that wholesalers are willing to pay.

So it boils down to the farmers who sell their products either to a wholesaler who sell it to the stores or sell to a manufacturer who turns it into another product to sell to another wholesaler who sells it to the grocery store. Basically, everyone from the farmer to the grocery store have to include a markup to cover operating expenses.  In addition, the markup is often determined by whether its a name brand, off brand, is a specialty item or a mass product, demand, and overall categories. 

Furthermore, once the products hit the stores, they do things to add more markup such as anything that is chopped, cubed, or riced fruits and vegetables can have up to a 40 percent markup but if you chop up, cube, marinate, or just cut up meat, fish, or chicken can have even higher markups of up to 60 percent.  If you look at the baked goods in the bakery, the markup could be as much as 300 percent.  These markups are on top of what the grocery store paid the wholesaler.   

So there are markups all along the way.  Let me know what you think, I'd love to hear.  Have a great day.

Calculating Markup For Real!

 

The other day, I realized that we teach students how to calculate markup using a standard formula but how do businesses determine the percent markup or why an adequate markup is important.  Most every markup problem I've seen in math books tend to be ones you are given everything you need to do it without any explanations.

For businesses to sell products so they can. make a profit, they have to figure out the best amount so they can sell enough products.  So a markup is the difference between the unit cost of the item and its sales price.  If you choose a price that is too low, you won't make any profit and end up in bankruptcy. 

In addition, one has to look at the type of business, margins, and your clientele to determine the markup you want. For a professional service, one has to look at past experience to help set the price and one needs to connect the brand to its financial availability.  Then one has to factor in the unique selling proposition which is determined on the accessibility.  For instance, in the cyber security training market, the only real limit is capacity but this changes from market to market. 

Finally, is the pricing strategy which could be one of several.  It might be pricing penetration where the business penetrates the market by offering services at the lowest price they are comfortable offering to get clients, testimonials, and recommendations before raising the price. One also has to keep in mind the number of competitors and their prices because if one hopes to attract clients, it is important to price slightly lower than the competition and not higher than the majority of them. 

In fact, for most businesses, it is important to find a niche where one does not have tons of competition, find a price that is slightly lower or consistent with them, and if you have to make the product, consider the cost of manufacturing it.   In other words, keep in mind the break even point for the product so you don't go below that and what price the market is willing to tolerate.  

As you can see, the markup is not a simple, add a specific percent to the unit cost of the product.  It has to take into account so many different factors in order to determine what it should be.  Although it may seem as though customers want the cheapest item they can find, the reality is they want to feel as though they are getting their moneys worth so if the price is set a bit higher, the customer needs to feel it is worth the extra cost.  

I admit, I never realized that the markup we see in the classroom does not always reflect markup in real life.  Now there are some industries where the markups are fairly standard such as in catering, gas stations, etc and often do not run more than 50 percent.  I've looked at markup in business where it varies more.  Let me know what you think, I'd love to hear.  Have a great day. 

Video Games And Math

 

Video games are a part of our lives. It doesn't matter where you look, they are there.  Many are point and shoot while others are look for three of a kind.  For students who want to make game design a career, math is extremely important.  Today, I'll be discussing the math behind Super Mario.

Math is used to determine how characters move, items fly across the sky, even juggling things. If math wasn't used, the game would not exist.  Now, there are some programs such as Maya which is a math based program that calculates vertices and normals as the artist uses a tool to create three dimensional graphics without calculating the math themselves.  Other math is done when the program is run by the game engines.  On to the specifics of Super Mario. When Mario jumps up and down, he is not going straight up and down but actually jumping in a parabolic shape so it shows movement.  

Another game, the Kerbal Space Program, relies on math so much.  The game has to simulate Newtonian physics using math so the rocket can get off of the ground and out into space.  One has to calculate the amount of thrust needed for the rocket to get off its launch pad, through the atmosphere, into space.  Without math, this game would be quite boring and flat.  

When designing the game, math may not play as much of a roll as it does when the game is actually created but when the game is finished, everything has to mirror real life and math does that.  Using math such as dot product, cross products, scaling vectors, unit vectors, and vectors in general, reflection, matrices, scalar manipulation, trig functions such as sin, cos, and tangent, delta time, range and domain, makes the velocity of particles correct, the spread of a shotgun blast, and the proper bounce of a ball as it bounce along the road.   In general, the more complex the requirements of the world, the more math required to make it real.

Furthermore, math brings entertainment to the game by blending everything into the whole world so it is fully interactive rather than semi interactive. It brings the excitement of bullets flying around your character as you work your way through the situation.  Math in the game simulates fluid water, animates everything, runs algorithms, sets up the game engine architecture, all the movement such as walking, shooting, and jumping, analyzing character interaction, runs timers, replicates the physics, and so much more. 

So if a student wants to know where math is used in real life, you can point to video games and you can let them know that without math, their favorite games wouldn't be as much fun or as real.  Let me know what you think, I'd love to hear.  Have a great day.

Happy Mothers Day

 

Warm-up

 If Mother's Day became an official U.S. holiday in 1914, how long have we been celebrating it?

Four Things We Should Keep In Mind..

 

Today, I'm looking at four things we should have in every lesson since many of us have been struggling through switching from in person to distance learning on a regular bases.  In Alaska, most of us ended up sending home packets that never got done or even returned so many of the students are way behind.  Unfortunately, our fist instinct is to want to start at where they are, not where they should be.  

The first thing to keep in mind, is to teach on grade level. Instead of going backwards, start where they should be and integrate the missing skills into the lessons as sidebars, or using techniques to help scaffold students abilities while allowing them to manage math at their grade level.

There are several easy ways to do this.  For instance, you might include similar problems to remind students of the process such as finding common denominators for regular fractions and then repeating the process for algebraic fractions.  In addition, it is easier to break larger topics into smaller chunks to give students a chance to really learn the material.  Choose topics that allow you to focus on additional skills so they can practice skills associated with the topic to strengthen their foundation. 

Rather than giving a test to cover the whole topic or chapter, look instead at using lots of mini-assessments.  This could be a small two to five question quiz instead of a warm-up.  It might be a thumbs up or down, a show with fingers of one to five where one is that you still have no idea what to do up to a five where the person feels as if they can teach it.

When teaching, don't rely on only one way to do things.  Use as many different representations as possible or show multiple ways to do it. For instance, many books and teachers rely on teaching the FOIL method for multiplying binomials.  I have four to five methods I teaching including a visual one that is a drawing they fill in as they do the multiplication.  This visual method works well for multiplying two digit by two digit numbers and for binomials. It can also be used to help factor a trinomial.

If you are teaching fractions, don't always use pizzas or circles.  Show the same material using divided rectangles, beans, etc so they see more ways fractions appear.  The multiple ways of doing things and multiple representations allow students to find the way that works best for them and allows them to "see" things so much better. 

Finally, see if you can collaborate with other teachers at your school or district so you can learn more, share best practices, become more aligned with others, and the lessons themselves become more interesting and engaging for the students.

In summary, teach at grade level while incorporating unfinished learning into the lessons. Consider using tasks to help students fill in their gaps.  Allow time for mini-assessments and student reflection. Use multiple methods when teaching material.  Finally, collaborate with others.  Let me know what you think, I'd love to hear.  Have a great weekend.

A New Solution To An Age Old Problem.

 

Recently a mathematician working at the University of Oxford, proposed a new solution to a problem that has been around for centuries.  The problem involves unit fractions or fractions with 1 in the numerator such as 1/7 and was first considered back in Egyptian Times.  The reason they worked with unit fractions is because it was the only type of fraction found in their number system. 

In order to express larger fractions they added two smaller ones together.  For instance, to represent 3/4, they'd add 1/2 + 1/4.  In the 1970's, two mathematicians suggested that if you had a sufficiently large proportion of whole numbers, there would be a subset of reciprocals that add up to one. Even if the numbers were chosen to make it difficult to find a subset, the subset still exists.

One paper, from about 20 years ago, used a color method to find the solution.  The idea is that the whole numbers are sorted into different colored buckets and at least one bucket is likely to have a subset that adds up to one.  

This mathematician used principles from harmonic analysis which is a branch closely related to calculus and applied them to the problem. He relied on an exponential sum which is used to determine the number of integer solutions to a problem.  It was used to determine the number of subsets that contained a sum of unit fractions that equaled one. He was able to confirm that that there did indeed exist the subset but not predict the density or how many subsets existed because the harmonic analysis uses the bucket sorting idea, not the density one.

When the methods from harmonic analysis were used, he wanted to avoid composite numbers with lots of factors because they tend to make large denominators larger. So he focused on proving that if there are lots of numbers with relatively small prime numbers, there will be a subset whose reciprocals add up to 1. In addition, he was able to show that there was always at least one bucket that had enough numbers for this to occur.

Unfortunately, this criteria cannot be applied to the density.  One cannot just choose a convent bucket. If they choose a bucket with nothing in it, this doesn't work.  As another mathematician was preparing to read this particular paper to others, he was able to figure out how to use this idea with numbers that had a large number of prime numbers. 

The problem with using the exponential sum is that you cannot get a precise number as it will always be an approximation or estimation. Rather than looking for fractions that added up to one, he looked for subsets whose reciprocals added up to smaller fractions and used those smaller fractions to find one.  Although there are still questions regarding this out there, this solution was one of elegance and beauty.  Let me know what you think, I'd love to hear.

How Is Trigonometry Used In Architecture?

 

I find it interesting that people can take a drawing and translate it into a real building that can either look normal or like something that came to us from the future. The people who create these visions and translate them from their heads to paper for the builders to turn it into reality. Did you know that architects use trigonometry in their work?

Trigonometry helps architects calculate the slope or pitch of the roof, ground surfaces, light angles, structural loads, and the heights and width of structures.  All of these things are used to create a mathematical representation that a contractor can follow when constructing the building.

Often time the slope of the roof is given in degrees and must be changed into actual measurements. This requires people to set the tangent x = to rise(y)/run(x) to change from degrees to actual lengths for the difference of y and the difference of x. Usually, the architect has the degrees and one of the values of x or y.  For instance you might what to know for a tangent of 38 degrees what the rise or y value is if the run or x value is 12 feet.  so Tan 38 degrees = y/12 and solve to find out the value of y. Ground surface is defined as the intended or finished grade of the surface of the ground at the site of the well or ground.  The grade is found using trig much in the same way as done for the slope of the roof.

As far as light angle goes, the trig is used to figure out how much area one light will illuminate.  For instance, if the architect is looking at using LED lighting, they have to determine if the lights are omnidirectional and will light everything evenly in a circle or if the lights are directional and light only in one direction.  The beam angle refers to the angle in which the luminous flux leaves the light fixture and depending on how far the light travels, it will create a cone of the corresponding diameter. Usually beam angles are listed in degrees and range from 10 to 120 degrees depending.

The beam angle only looks at where the light is at least 50 percent intensity.  Field angle refers to the part of the light ranging from 1 to 49 percent intensity, so the beam angle is inside the field angle and the only place the light is a full 100 percent is in the center of the cone of light.  It is recommended that architects use 120 degrees to provide basic lighting in a room and 90 degrees for corridors and hallways. 

As mentioned before, architects can use to calculate the structural loads of the building and other things.  Architects need to understand how all the forces work on a building using vectors which have magnitude and directions.  By applying trigonometry to the vectors, they can determine loads and forces. In addition, trig is used to figure out the load that the tresses are being exposed to.

So there are lots of ways trig is used by architects when they design buildings.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If there were 23.8 million flights in 2004 and 38.9 million flights in 2019, what is the percent increase of flights between 2004 and 2019?