Warm-up.

Companies sold $6.22 billion in online Black Friday sales in 2018.  That was an increase of 23.6 percent over 2017.  How much did companies do in online sales in 2017?

Paolo Ruffini and Synthetic Division.

I love synthetic division.  I love it so much but unfortunately, it doesn't work on anything other than binomial factors. I love it's beauty and simplicity but I've often wondered about it's history.  Who provided the first consolidation of that knowledge?

From what I can tell, the creation of synthetic division is attributed to Paolo Ruffini in 1804.  He is an Italian Mathematician who lived from 1765 to 1822.  The work he did was a forerunner of the Algebraic Theory of Groups and he tried to show there is no solution to a quintic equation without radicals.

In addition to to being a mathematician, he was also a physician and philosopher.  Unfortunately after Napoleon took over, Ruffini lost his teaching position because he refused to take an oath of allegiance but that did not stop him.  He worked as a doctor while continuing his mathematical research but once Napoleon was defeated, Ruffini returned to his university position.

What we refer to as synthetic division in high school is also known as Ruffini's rule which allows people to divide a polynomial by a linear factor to find the zero's of the equation.  As you know, if there is no remainder, you have found a zero but if there is a remainder, you have found the y value of the equation at a certain x value .  It is also much more efficient than using algebraic long division to solve the same problem.

Around 1800 or so, the Italian Scientific Society of Forty opened a competition asking people to provide a method that could be used to find any roots of a polynomial.  In the end, the society received five entries but Ruffini's was declared the winner in 1804 and his paper was published as part of the award.  He refined and republished the paper in 1809 and 1813.

We have students practice it on paper by following certain steps but emathlab  has a really great practice section to help students learn the process.  If you get anything wrong, it will correct the work on the problem so you can see where you went wrong.

In addition, this article looks at ways to extend the use of synthetic division with modifications for x^2 - a and x^2 - bx + c problems.  I had never seen these before because I'd been told you cannot use synthetic division for anything other than linear factors.  I had trouble with the x^2 - a one but not the last one as it made perfect sense.  This site has a wonderful pdf with problems, explanations, a bit of history on synthetic division.

I've always wondered who came up with synthetic division because when I next teach it, I can include a bit of the history so that students will know more about it.  Let me know what you think, I'd love to hear. Have a great day.

Happy Thanksgiving

The Costs of Thanksgiving Dinner.

I absolutely love finding information that students can turn into graphs.  Information based on real life situations, not something a textbook author dreamed up to perfectly fit the section's topic.

Did you know that a full Thanksgiving dinner cost $5.68 in 1947 but in 2018, it ran $48.90.  If you adjust past prices to account for inflation, the prices have ranged between about $41.00 and $57.00.  If students graphed these, they'd have a better idea of the prices went up and down through time.

The figures can be found here and are based on a meal consisting of a 16 pound turkey with 14 ounces of stuffing, three pounds of sweet potatoes, 12 dinner rolls, a pound of frozen peas, 12 ounces of cranberries, a half pound each of carrots and celery along with everything needed to make two pumpkin pies topped with whipped cream and a gallon of milk.  Enough to feed 10 people.  This site gives a better breakdown of the cost of the turkey, and a combined cost of sweet potatoes, stuffing, and cranberries so students can break the meals down even further.  They can use the information to calculate what percent of the total cost the turkey is or the percent the sweet potatoes, stuffing, and cranberries make up.

If you are only interested in the price of turkey, The Chicago Tribune has multiple ads beginning in 1915 showing turkey sold for $0.28 per pound while as in 2015, it went for $0.48 per pound.  The newspaper also includes the price adjusted for the actual cost in 2015 dollars to give a better idea of how the cost relates.

On the other hand the Business Insider shows the breakdown for the Thanksgiving meal as it cost in 1911 so you know how the $6.81 breaks down but if you include inflation, the cost in 2013 dollars is way, way more.  It is possible to compare prices to 2019 using this article from Moneywise.

The World Economic Forum has a great article on the economics of Thanksgiving.  It looks at everything from the cost of dinner to the average cost of a turkey, to a map showing which states pay the most, the least, and in-between.  It also looks at the number of people traveling, the age grouping of people who end up shopping on Thanksgiving, and the average cost per person including travel.  It is filled with six different graphs so students get a lot of experience reading real graphs.

This site addresses how long a person has to work to afford a full Thanksgiving meal for a large gathering.  It explains where it got the data and how they used it so it is quite educational.

Lots of different ways of enjoying the same information and information one does not usually see associated with Thanksgiving.  Have fun coming up with creative ways to use it in class.  Let me know what you think, I'd love to know.  Have a great day.

Macy's Thanksgiving Day Parade.

The Macy's Thanksgiving Day parade is the parade many people think of immediately because it has strong associations with Thanksgiving Day since the 1920's.

It its also a great event to look at for costs and how costs have changed over the years.  One event that caused an increase of costs was 9/11 because it required all parades have additional police protection.

Lets look at the cost of putting the parade on in 2016.

The parade is 2.5 miles long and takes 3 hours to complete.  The total cost of the parade in 2016 ran between 10.4 and 12. 3 million with an additional 2 millions for costumes, and the property taxes associated with the parade ran another $139,000.

The breakdown of the main amount is as follows:
1. Logistics and coordinations - $1.5 to $3.4 million.  This is the cost of workers, parade supplies, and helium.  Even though the parade is a once a year event, it employs 26 full time workers, and 10 to 15 part time workers.  These workers take a budgeted $1.3 million for salaries.

2. The balloon floats require 50 to 90 people for each one to wrangle them down the parade route.  The balloons also need between 300,000 and 700,000 cubic feet of helium.  The minimum cost is $510,000 to fill the smaller balloons.

3.  It costs $90,000 to sponsor a returning balloon or $190,000 for a new balloon.  If one wants to build a new float, it can cost a lot because the price covers the cost of building a new float over four to nine months and can cost for basic expenses between $30,000 and $100,000.

4.  There is also the cost of hiring police officers, drug sniffing dogs, and rooftop snipers that has to be added in.  No one is sure how much it runs but it is at least $200,000 in overtime costs and that does not include the regular costs.  It is thought the full cost for the police presence is several million.

It is hard to get exact figures because Macy's is extremely tight lipped about how much they spend.  It is hard to even find costs from any of the earlier parades other than 2016 or so.

There are some infographics which contain this information such as the one here or here.  This site has a graphic showing the number of balloons by decade while this site has graphics that compare warmest with coolest days and one on the number of new floats between 2001 and 2016.  This graph shows the breakdown of viewers by percent for the 18 to 64 age groups.

Keep your eyes open here on Friday because I'll be providing links and information on Black Friday Sales.  I hope the information I gave is enough for students to create their own infographics or even graphs.  Let me know what you think, I'd love to hear.  Have a wonderful day.

Warm-up

If farmers are paid $0.82 per pound, how much would one receive for selling 687 pounds of macadamia nuts?

Warm-up

If a macadamia nut weighs 0.12 ounces, how many make up one pound.

What Does Show Your Work Really Mean?

Having students show their work is a struggle especially when one is told by the student "But I know how to do it" to explain the answer only.  Since I heard a college professor comment that showing work is part of finding the solution, I've come to the conclusion that having students show their work is important.

I've started telling students that by showing their work, they are actually communicating to both myself and others how they got from the question to the answer.  Showing their work is showing their journey and path.

I remember reading journals in college filled with papers written by various mathematicians.  None of those papers showed the problem with just an answer.  Every paper showed the process the mathematician used to get from the problem to the conclusion.  The work is how they communicated their thoughts, assumptions, etc to others.

Showing your work does not have to be showing every step in written form, it could just as easily include pictures showing how the students get from the problem to the solution.  I use pictures and drawings to "explain" problems to students so I don't see anything wrong with a student providing their work using drawings.

Furthermore,  I've discovered many students who showed only answers, came up with the answers in one of two ways.  First, they did the work in their heads but they were unable to explain how they got from the question to the answer.  They were unable to remember the steps they took and sometimes could not repeat the process on future problems.  They couldn't even provide any sort of illustrations to show this.

The other possibility is they did the work on an app or calculator that provided only the answer so they wrote down the answer but had absolutely no idea how the app got the answer.  This is apparent when they take a quiz or test.  Yes, I know there are apps which show the steps but most of my students tend to use Siri or other app which gives them only an answer.

Unfortunately, it is often easy to do math mentally in Elementary school but when the problems become more complex and requires multiple parts, it becomes more difficult to complete mentally.  For instance, factoring a quadratic with a leading coefficient or performing polynomial long division often requires numerous steps and is so much harder to do in your head.  The other thing, is when students continue doing the work mentally, they sometimes miss certain steps so they are unable to get the correct answer due to that gap.

I do believe it is time to get rid of the phrase "Show your work" because it is not as accurate as it might be to rephrase it into something like "Document your mathematical thinking" or "Write down notes explaining the process", or "Communicate your full solution".  Perhaps we can explain that letting us see how they got from the question to the answer is the same as writing the answer to a short answer question in English.  It is time to get past having students see "Showing your work" as a negative activity and instead see it as a positive form of communication.

I would love to hear what you think about this.  Let me know please.  Have a great day.

8 Ways To Transfer Knowledge.

 One thing students have difficulty doing is transferring their knowledge from the textbook to other parts of their lives even with real world examples.  Often, students are unable to see the direct link between the textbook and real life but there are ways to help students learn to transfer their knowledge so it becomes habit.

One issue is that memory relies on context which makes it harder for the brain to transfer material learned in a classroom to a work environment.  So one has to learn how to take prior knowledge and give the information a new context so it applies to the new situation.

1.  When teaching new material have students suggest ways it applies to either possible future jobs or life before tying it to their long term or short term goals.  Teachers can suggest connections but if the students do not provide their own connection, they find it more difficult to transfer knowledge because they see no relevance.  In addition, when students find relevance to the material, they are able to apply it to other situations.

2. Incorporate activities where students are able to reflect on the new material to see if they understand the concept in depth.  Then they need to explain it to themselves because this offers them the opportunity to discover if they have misconceptions, correct them which leads to a deeper understanding.  Furthermore, when students work on explaining it to themselves while using their own words, it is easier for them to connect it to previous knowledge while transferring it to a new situation.

3. Have students use a variety of media to learn the material.  This means using text, videos, audio, and other methods so students are able to retain more material and score higher when tested on the material.  Visuals can make it easier for students to learn or at least they perceive it that way.

4. Help students change the way they study so its more random rather than being at the same place, with the same music, at the same time everyday.  Although changing the place, time, and way of studying is more difficult initially, it is easier to retain the information later on.  This change also called "desirable difficulties" actually leads to a deeper learning which is what students want.

5. It is important for students to learn to identify any gaps they have in their foundation because these gaps make it more difficult to transfer knowledge. Furthermore, it allows students to strengthen their knowledge based once these gaps have been identified.  One way to identify gaps is to take practice tests because tests also identify topics students have mastered so they can focus only on the gaps.

6.  Teach students to set both learning goals and success criteria because it tells students what they are going to get out of their learning and how they know when they learned it.  Goals need to be realistic and specific so the goals can be met.  A realistic goal might be learning all the facts above  7 x 7 over the next 6 weeks because you know you struggle with those and you've set a reasonable time period to do this.

7.  Help students learn to generalize the knowledge from the classroom to a different situation.  An example of this might be taking what they learned about inequalities in the classroom and find as many situations in real life that are described by inequalities such as filling a gas tank, the height restriction of a amusement park ride, or buying patterns on sale.

8. Every day, have students find situations to apply what they learned.  These situations should be outside of the classroom such as buying enough pizza's for a party and thinking of this as using an inequality or look at starting a business in town and calculating the startup costs.  Finding real life applications of the material provides that direct connection between the classroom and application to work.

These are some ways for students to learn to transfer what they've learned in the classroom to other situations such as a work environment.  Furthermore, it prepares them so future employers are more likely to feel as if the student is ready to step into the company.  Let me know what you think, I'd love to hear.  Have a great day.

Math, Infographics, and Graphs

 Most standards are already out of date by the time they are printed.  In fact, we have new ways of communicating information that are not taken into account by standards.  I love reading infographics because they condense information into easy to understand graphical representation.

The great thing is that infographics can be used to teach math while showing real world statistics.  Furthermore, infographics provide students with immediate understanding of certain topics.

One way to use infographics are to have students start with a graph.  It might be a graph on last 50 years and what happens with poverty rates, football statistics for the NFL, divorce rates, or the number of people traveling over Thanksgiving.  Once they have the graph, they might look into what causes the rate to go down or up.  This requires a bit of research possibly  but once the student has determined the reasons for a fall or increase, they are ready to create the infographic that will provide a story to accompany the graph.

Another way is to find two different graphs that are marked in the same unit and graph them on the same coordinate plane.  Once this is done, the student can compare and contrast the two graphs before looking for positive correlation, negative correlation, or there might be no correlation.  It is important to remind students that correlation does not necessarily imply causation.  This site puts two totally different graphs together to show correlation but also takes time to discuss how correlation can imply causation.

Or you could have students investigate how the interval and scale change the way we look at the graph.  For instance, a graph showing unemployment might change the time scale to every month instead of every six months to see how it changes the overall graph.  Another way to look at that same graph is to change the y axis so its every five instead of every one running only from 55 to 60 percent.

What about taking the cost of attending colleges from colleges the students are interested in and creating their own graphs comparing tuition, fees, etc to see how much they would owe after 4 years. The next step would be to present this same information in an infographic.  I know some colleges do not charge Native American students tuition while others do.

Another possible thing would be to have students look at the historical trend of something such as the Alaskan Permanent Fund Dividend and comparing it with the price of oil to see if there is a relationship.  Take this a step further by predicting what might happen to the check over the next few years based on what they conclude has happened so far.

Let students choose a topic, research it before creating a graph on that topic before creating an infographic to accompany it.  It might be something such as the cost of cars over the past years, or the  drop in value from new to 10 years old for a variety of models to see which has a history of the best resale.

There are so many possible topics.  One place to find wonderful graphs to use in some of these exercises is at the New York Times complete with ideas, graphs, and lesson plans.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

If the radius of the front wheel of the bicycle is .75 meters, what is the circumference of the wheel?

Warm-up

The radius of the front wheel is .75 meters.  What is the area of the front wheel?

Increasing Positive Emotion in Math.

 If you read the blog the other day, you'll know that our brains remember more if they have an emotional attachment to the material.  Unfortunately, too many students have negative emotional connections associated with math which creates several problems.

First, many people believe they are not good at math and will never get good at it so they have a mental barrier to learning.


Secondly, if a student does not like math, they are more likely to have lower achievement scores and as soon as they can, they quit taking math or are more likely to take lower level math classes.  This means they are closing themselves to certain career opportunities, especial those of higher earning potential.  It also limits how they will interact with society and it's possibilities.

Right now, if a student struggles with math, one of two things happen.  They are encouraged to work harder or they are moved into a lower level of mathematics.  Unfortunately, neither of these solutions may not help improve student attitude.

So the question becomes, what can we do as teachers to make class more emotionally engaging so students have getter emotional associations with math?  There are ways to do this, some work better with younger students, some with older students but all can help students associate happier emotion with math.

1.  Create authentic inquiry by setting up a situation where students can ask the questions.  It might be drawing a rectangular prism on the board and have students create questions the can be answered from looking at the drawing.  This makes the students operate more as real life mathematicians who guess, question, and narrow their solutions based on repeated results.

2. Always show more than one way to do the math but don't be afraid of asking students to see if they can figure out a way to solve the problem on their own.  This gives them a chance to find counter examples on their own, or set conjectures.  In addition, it motivates students and allows them to develop mathematical creativity.

3. Instead of having students sit in one place the whole period, create opportunities for them to move around.  It could be as simple as asking a multiple choice question, assigning one answer per corner and having students go to the corner they think represents the correct answer.  Or organize a scavenger hunt with at least ten half sheets folded in half.  On the front, write a problem, and on the second page, write an answer but it is not the answer to the problem on the front.  It is the answer to another problem.  These are hung around the room after assigning a letter to each problem. Students have a sheet with boxes.  In the corner of each box is a smaller box where they write the letter of the problem.   The idea is that students start at one paper where they write down the problem, solve it, and then look for the answer.  Once they find the answer they look at the new problem, write it down and it's letter in the box, solve it and repeat it.   I've been known to use QR codes with a problem and answer contained in the code so it uses student devices but they still write the problems down on paper.

4.  When you create an assignment, quiz, or test, set them up so students can choose the problems they want to do.  For instance, you might want to write a quiz with 15 problems and have students answer any 10 problems.  This increases student motivation and makes them feel as if they have some control.

5. Place both examples and non examples on the board before asking students to figure out what makes the topic.  For instance, when discussing spheres, one might post pictures of basketballs, baseballs, or an orange.  For non examples, you could post pictures of bananas, footballs, or boxes.  Let the students use this information to create a list of what characteristics a sphere has.

6. Establish situations where students can collaborate together in a learning environment.  For instance, you can set up Jigsaw activities for sharing vocabulary, create a slide show or book using google slides to have small groups of people share their material with others.

When math becomes more engaging and students feel as if they are taking over more of their learning, they develop positive emotions so they learn the math.  Let me know what you think, I'd love to hear.  Have a great day.  I'll be back Monday with a new topic but I will continue to post warm-ups.

Reading and Vocabulary.

I have recently started doing things differently in my Geometry classes because I'm trying to make students more self sufficient and make the class a bit more student centered. I need to teach a bit less while making the students more independent.

One of the first things I did was create a paper filled with a table of 16 squares.  In each square I wrote one of the vocabulary words for the chapter and nothing else.  I divided the students into small groups of two to three people depending on the size of the class.

Once I passed out the vocabulary sheet, I assigned three words to each group so no other group had the word.  I gave them time to work on defining the terms using their own words. One girl asked if they could draw pictures to help define the word.  I said go ahead because research shows it helps to have pictures to go with words.

After about 10 minutes, I had students get up to new groups so everyone in the group had defined different words.  They shared their vocabulary words with the others so by the end of the time, every person in the group had shared their words and had a chance to fill in all the other vocabulary words.  This is a variation on the Jigsaw activity.

What I liked was the variety of methods used to complete the assignment.  Some groups split the words up so one person defined a word before sharing it with the rest of their group so everyone had their three words defined before sharing with the other groups.  Others looked the words up together, discussed what the definition should be before agreeing on one.

The important thing with this activity is that the words are ones they've used before in their two column proofs so these words are not new but they've not had time to really look at the words in detail.  By waiting this long before doing a serious vocabulary activity, students had a chance to build prior knowledge.

The other thing I've started doing is again dividing students up into small groups and having them read the new material.  After they complete the reading, they have to decide what the three most important ideas or concepts contained in the material.  At the end, I asked each group to give me the one concept they thought was the most important and they had to explain their thinking.

I do this activity on one day and give them 24 hours before taking it to the next step because it gives their brain a chance to process the information.  One day later, I go back and ask them to prepare a more detailed explanation of that concept to share with other students.  This presentation can be done via google slides, a short video, or some sort of animation but they have to explain the concept and why it is important.

I've found that most groups came up with different choices so there is very little overlap.  If I do have two groups choosing the same topic, I often ask one group to think about why a certain other topic is important.  Most of the time, they don't mind the change of topic because it was one on their list of the top three.

Both of these activities encourage dialogue, communication, and explanations so they learn to discuss and communicate math.  I would love to hear what you all think.  Let me know, have a great day and I'll be back with something new on Friday.

Emotions and Teaching Math

We all have those students, the ones who arrive in your math class, convinced they can't do math or don't try because they see no use for it.  They have a negative emotional connection with math which can slow down their learning.  There are ways to help students gain better emotional connections.

Remember, emotions help us think because they pull up memories we have associated emotions with.  The question is then, how do we take advantage of this in our classes.

One thing we are told is to make math fun by incorporating games and prizes but these are considered a quick fix.  What works better is to help students develop a sense of accomplishment by making math interesting, satisfying, and personally fulfilling.  There are some steps each teacher can take to change student dislike into appreciation and wonder.

First, rather than letting students see math as some sort of torture to be tolerated until the math requirement is met, we should explain why these math concepts matter.  Mathematics helps students think both theoretically and logically.  Furthermore, math helps the brain think abstractly which is a skill that can be used in so many different ways.  There is research that indicates students who study complex math topics tend to do better in life.

Second, give students problems where they see how math is used in real life.  If they see connections, they are more likely to participate.  Practical applications range from keep track of their finances, to how fast oil spreads on tissue paper which gives students a real life application of direct variation and understanding of how fast oil spills spread.  Find stories in the news where a need to understand math is important such as in a story on price fixing, or climbing interest rates.  If you have students read an article on climbing. interest rates before setting up a spread sheet to show how much an increase of one-eighth of a precedent can effect the overall amount of money paid on a lone.

Third, take time to discuss mathematicians who helped change the world or make changes to the world such as Pythagorus, Rene Descartes, or people like Alan Turing who helped break Axis codes and allowed the Allies to win the war.  For the sports minded, introduce them to Nate Silver who uses statistical analysis to predict winners in elections and Major League Baseball.

Finally, look at ways to decrease the stress students feel when learning new material or taking tests.  When introducing new material, always try to relate it to previously learned material such as when solving one and two step inequalities, we can related the processes back to solving one and two step equations.  Relating it back to previously learned material also helps students learn the same material better or even get it when they didn't get it the first time.

This is an introduction to the topic.  I will be exploring it in more detail as I have a few students who see absolutely no reason to learn math and do their best to disrupt instruction so others have difficulty learning.  I want ideas to counter this "If I don't want to learn math, why should you?"  Knowing how emotion impacts learning is my first step to countering this.

I am slowing down to three topics a week plus warm-ups on the weekend because of the number of sports events happening at school.  I have been helping out and will continue to do so but it is not leaving me as much time for me.  So the next time I add new material, will be on Wednesday.

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

Warm-up

Write a compound inequality to describe the situation.

Warm-up

Write a compound inequality to fit the situation.

Emotion and Memory

Did you know that emotions help your memory remember information?  Emotion plays an important part in how we remember things.

In general, emotions highlight certain parts of our experiences to make them more memorable.  It works in so many ways with our memories.  Some may be new to you, some may not.

First is attention which helps our minds determine what is most important in our lives.  Emotions such as novelty or surprise, helps us focus on conversations so we are aware of a few items which allows to use our limited attention span to its optimum.

Second is consolidation of memories.  The truth is that most of the information we are exposed to, do not make it into our long term memory.  When we have an emotional connection to something, we are more likely to remember it. For instance, most people of a certain age can tell you about 9/11 because of the emotions it brought up in everyone.  This is because certain stress hormones such as cortisol and epinephrine help us consolidate our memories.

Thirdly, it appears that those memories associated with painful events stay with us longer than memories associated with physical pain.  Contrary to the rhyme about "sticks and stones may break my bones but words will never hurt me" we do remember the words that hurt us.

Next is priming or triggering behavior through the use of unconscious suggestions.  For instance, if people unscrambled sentences that contained suggestions of self control, they would immediately make better choices when it came to eating snacks.

Then we have mood memory which is where we recall memories which have the same emotional feeling as the way we feel at that moment.  So if we are sad, we are more likely to remember things that are sad.

Blanking out is something most of us have experienced at one time or another.  It has nothing to do with drinking but being so stressed out we forget things.  This usually happens during tests.  It turns out that if we are bored or too stressed, our performance is likely to suffer.  When we are bored, our minds are unfocused but if we are too stressed, our minds are too focused and we miss information.  The best place to be is when we are in moderate stimulation.

Finally, we have a duration memory because we do not remember everything involved with the memory.  Instead we remember the best or worst moment and the last moment.  We do not remember everything of the event.

This is why it is important to connect emotion with any learning, otherwise it won't stick.  In the future sometime, I'll cover emotions and learning math.  Let me know what you think, I'd love to hear.  Have a great day.

Chunking In A Digital Age

 With the way children's brains have been changing due to the use of digital devices, I wondered if chunking was still recommended or if it's been replaced by something else.

Chunking is one of the ways to organize information.  In reading, it is putting together letters and sounds to create meaning while in math it is organizing the information into smaller, more understandable pieces.

According to research, the human brain can only handle seven pieces of information in short-term memory at any one time.  Chunking the information helps the brain avoid traffic jams in short term memory.  Chunking also helps the brain remember more information because they have combined pieces together.

Chunking is also a way of identifying the most important information out of a section or chapter.  This is done so students know what they need to focus on and learn.  Furthermore, when chunked properly, it is well organized and logical and increases student information on what is going on.  It helps them see the bigger picture while remembering the information more effectively.

Furthermore, there is another theory, the cognitive loading theory, that states the more the brain has to remember in a shorter period of time, the more difficult it is to learn because the brain is overloaded.  This supports the use of chunking because chunking does not overload the brain. The chunking also allows them to build on previous chunks thus providing the repetition the brain needs to retain information.

It is known the brain seeks patterns to make sense of material because the brain stores information as patterns.  Furthermore, the information must make sense to the brain otherwise it will not be stored.  This includes new information that is not yet familiar so when we introduce new concepts, our brain does not always store it so we cannot expect our students to remember it after one example.  So it becomes necessary for teachers to help students identify the new patterns while associating them with older patterns in order to formulate new ones. Patterns have been described as the roadways for memories to travel.

In addition to looking for patterns, the brain searches for personal meaning so it is important to help students find a way to relate to the new patterns because that helps students learn the material better and easier.  If the brain cannot find personal meaning, it will lose it and we see it as students who don't "get it."  This often manifests as the "I don't care" attitude which students see as better than a "I don't understand" confession.

Since most of our students are used to working through quite a lot of data, we need to know how to organize multimedia materials into usable chunks.  It is recommended one group related information into each chunk.  This includes buttons, images, graphics, etc as related information.

There are some ways to help students find personal meaning.  One is to ask students to relate to the information in a personal way by writing it down on a KWL chart.  Another is to ask students to share a story with other students about something that happened to them that uses the concept you are teaching.  For instance, they might talk about when they ran out of gas because they didn't check the amount of gas in the tank before heading out.  This story might relate to a compound inequality about filling the tank with gas.

So with chunking and personal meaning, students can learn the material better and for longer periods of time.  let me know what you think, I'd love to hear.  Have a great day.

Mobile Devices + Children's Brains = ?

It seems like everyone in today's society owns some sort of mobile device.  Most seem to own a cell phone which they use for everything.  Unfortunately, it appears these devices are used by all age groups.

It appears that one quarter of the children between the ages of 2 and 5 have their own smartphone.  The percent increases to over half of the children aged 11 to 13 own a cell phone but over 70 percent of that age group know how to use them.

Most education is based on Piaget's work.  He believed that children through their experiences of the world around them construct new ideas based on what they observe.  Researchers have noticed a decrease in interactions between parents and children meaning these devices are interfering with the process of bonding.

Furthermore, children learn about language, emotions, how to have a conversation, learning to read facial expressions, but decreased interaction time means children are missing out on developmental milestones.

Although it has not been proven that cell phones emit radiation, they do emit radio waves which can damage brain development.  In addition, it does appear that the use of technology changes the parts the brain uses.  One doctor did brain scans of two groups of people.  The first group was made up of people who were very technology savvy while the other group didn't use it.  The results show the first group showed lots of activity in the dorsolateral prefrontal cortex area but the second group show that activity.  However, the second group began showing that same type of brain activity after learning to use the devices.

Initial research indicates that children who use mobile devices are not using the brain circuits that control the learning of reading, writing, and concentration.  Screen time also effects their ability to communicate, the development of their emotions, and impact their people skills.  In addition, there is evidence to indicate that too much screen time can decrease cognitive skills with as little as two hours a day of use or cause changes to the brain if the child is on the device for more than 7 hours a day.

There are indications that using a device can make your brain lazy because the brain no longer has to remember phone numbers or information because the device can remember it or help find answers via  various search engines.  It is much easier to look it up online than to remember it.

Research has also found that use of mobile devices can impact the amount of sleep children get each night.  Devices throw out a blue light that fools our bodies into thinking it is daylight by decreasing the amount of melatonin produced so it no longer follows our natural circadian rhythm.

There are studies out there that are currently underway to monitor children's brains over the next few years but many educators have serious concerns about how these devices effect the development of the brains.  This could lead to an overall change of educational theory and teaching methods.  Let me know what you think, I'd love to hear.

Small Paper Books.

 My school district is having a three day inservice with all the teachers meeting in one place.  The district flew everyone in from the various sites to enjoy some really awesome training.

One of the sessions I attended was on art in the classroom.  It had some things like tableaus I was not particularly interested in because I wasn't sure how to integrate them into the math classroom.,  There was one activity I could see integrating into my classroom.

It was on using a six page book made out of one piece of paper.  Basically, you fold one sheet of  paper into 8 equal parts.  Once this is done, you fold the paper in half using a hamburger fold and cut from the fold in to the next fold so you have a slit running ac ross two sections in the middle.  Push it together, wrap and you have a book.

 The instructor recommended we create a book on a topic we'll be teaching sometime before December.  The idea is that we make a book so we know how easy it is and then we'll have our students make some.

I decided to make one on compound inequalities for my Algebra I class but honestly, I'm not sure I can get them to make any because they are more "Math is to torture humans", than wanting to try.

I drew the front page using pen but colored in the words using a colored pencil.  I didn't know about the colored pencils until I got it done.

The next page, I devoted to real life examples of "and" compound inequalities for putting gas in the tank or bullets in a handgun.  Out here, students regularly use guns for hunting but since I didn't know how many bullets a normal rifle held, I settled for using a hand gun.

I wrote an explanation for the and so they could see the and is actually two equations combined into one.

The next set of pages focused on "or"'s using distance.  The fish are up to 12 miles south or more than 16 miles north.  South being the negative direction while north is positive.

I then wrote out the equations for the or and the final page showed the graphs for each one.  I think I need to redo the book and improve on my writing and the way its organized.

This is the best suggestion for art the presenter had in her arsenal.  I really enjoyed making the book but I had more fun making geometric lights.  I'll talk about that later.

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

Compound Inequalities are All Around Us.

  I am getting ready to teach compound inequalities to my Algebra I students and have been trying to think of some real world situations they might relate to.  It is fine and good to teach it using variables and such but without a context, it remains a mystery for most.

Most of the examples I've seen are ones my students will not be able to relate to such as using more gas if you drive too slow or two fast but if I change it to talk about filling the gas tank up it will make more sense.  If the tank takes a maximum of 5 gallons gas in their ATV, then they will add between 0 and 5 gallons or 0 <x< 5. The above example will make more sense to them because its a better context.

Another example is one involved video games.  On the level they have a certain number of tries to eliminate all the trees in front of them.  Say they are given 6 bombs to accomplish the feat so the compound inequality would be 0 <x< 6 because once they run out, they cannot do more.  This type of inequality appears in numerous video games.

You are planning a trip.  You are looking at going to either Mexico or Hawaii and the airfare is about the same but the hotel room cost is quite different.  The room in Mexico is $75 per night while the room in Hawaii is $152 per night so you have an or.  x > $75 or x> $152 and the x is based on how many nights you will spend there based on a minimum of one night. This is an example from my life.  I ended up going to Mexico due to the cheaper cost of hotel rooms for the Christmas Holidays.

Another example is when looking at renting cars.  Depending on the time, the company, and length, one rental company may be a much better deal.  For instance, Hertz may offer you a full sized car for $87.50 per day while Avis has the same for for $92.00 so your inequality becomes x > $87.50 or x > $92.00.

You always see examples that are like x > 32 or x < -15 so what would that be in a real world situation?  If you think of the minus sign as a direction, then you can look at it this way.  You are visiting your aunt in Fresno.  You want to go to a game store but you are between two.  One is 32 minutes heading towards the mountains, or 15 minutes towards the ocean.  These do not involve equals because anyone who has driven in traffic knows the time is usually the minimum time. Another scenario for x > 32 or x - 15 is that my friend and I went on a diet, she lost at least 15 pounds on it while I gained 32 pounds using the same diet.

One compound inequality I see all the time involves buying things on sale at fabric or grocery stores. When fabric stores offer patterns on sale for $1.99, they put a limit of 10 patterns on it or 0 <x < 10.  Often there are limits on butter at $1.69 per pound instead of $2.69 per pound.  You might be restricted to 5 pounds.

Even when traveling on the airline, you are allowed to have up to 50 pounds in your luggage so it has to be between 0 and 50 or they might charge you an additional fee.  Most airlines also allow you so many suitcases free before you are charged for extra bags.  Some airlines in Alaska do not charge by the piece, they charge by a total weight for your check through and carry on bags.  I've used an airline that had a maximum of 100 pounds before they charge you almost $1.00 per pound overage.  So you made sure the weight was no more than 100 pounds.

So as you can see, the world is filled with lots of compound inequalities.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Your device has 60,252 minutes of music, how many hours of music do you have?

Warm-up

You have 22 hours of music on your device. If there is an average of 19 songs per hour, how many songs do you have?

Amusement Parks and Rides.

There are so many of them spread across the United States.  Many you'll recognize like Knotts Berry Farms, Disney World, Disneyland, or Six Flags but there are  so many more.  I love visiting Disneyland when I can, even with the long lines yet each ride is an investment  for the company.

The price of a ride depends on things like its size, the cost of the materials used to build it, and demand for the ride.  An outdoor ride can run between $90,000 and $300,000 but it is possible a single ride could cost more than $800,000.

Now if you want to look at the price of building an amusement park from scratch, its cost depends upon the size of the finished park.  For instance, if you want to build a mega park like Disney or Universal, it is projected to run between $2,000,000,000 and $4,000,000,000 while a regional theme park will probably cost between $200 million and $500 million.  Although it is possible to build a water park for less than one million, most cost in the $10 million to $40 million to build.

On the other hand, only the mega parks are capable of producing over $1 billion in revenues but they have to spend to get those revenues.  For Disneyland, it is not unusual to spend between $30 million and $100 million for one attraction.  One reason they spend so much money is that they are competing with themselves and forms of entertainments that didn't previously exist.

If you want to provide some in-depth information on the business of amusement parks, check this site.  The article is in two parts.  The first part covers the amount of money parks make while the second part focuses on the amount the parks cost and what they earn.  Furthermore, this two part article is filled with lots of graphs, detailed information, and figures.  The figures can be used to create graphs to interpret the data.  It also explains why certain costs change.

I chose this topic because there is a lot of math involved from the land to the final product and it covers things you don't normally think about.  Let me know what you think, I'd love to hear.  Have a great evening.