Warm-up

If the length of the jellybean is 2cm, estimate the radius of the bowl.  Justify your answer mathematically.

Warm-up

                         Justify your answer mathematically

Would You Rather?

 I would like to thank Mrs Bruner for this site.  Its called "Would You Rather?". This site has some wonderful pictures with questions for students to ponder.

The one she shared on Twitter showed a picture of bananas displayed in the produce section and a second picture showing the bananas being weighted.

The question asks whether you would pay so much per banana or so much per pound.  The author then asks people to justify their selection using mathematics.

The site is authored by John Stevens.  He has provided eleven pages of these "Would you rather?" situations covering everything from food, to clothing, to board games and other things.  Each one presents a situation with option A or option B.  These are designed so there is not a "right" answer but students are required to support their choice through mathematics.

This justification requires students to communicate their thoughts, stimulates mathematical thinking and discussion.  Furthermore, when they can communicate their thinking, they are improving their understanding of mathematics while applying it to real world situations.

Although I'm more interested in the ones designed for older students, John has created two pages worth for Kindergarten to Second so the little ones can begin learning to share their thinking mathematically.  In addition, he's grouped the material for K-2, 3-5, 6-8, or 9-12 so if you have a classroom filled with ELL students, you can say choose one from the 3-5 list but if you teach advanced 6th graders, you could pull one from the high school list.

Furthermore, John includes a sheet students can use as they answer the "Would you rather?" activity.  It has a column for option A, a second column for option B and a column between the two reminding them to break it down.  There is a spot for a conclusion under each column and two more at the bottom of the page where students fill in the prompt "I would rather" followed by "because."

For students who are not as able to communicate via writing, they can always communicate their thoughts via a medium such as Flipgrid or other verbal method.  There are other ways a student can express their answer regarding which they'd rather.

Check it out, add it to your day so students get a nice change of pace and work on learning to express themselves and their thinking.

Let me know what you think.  I'd love to hear.

Too Much Testing?

I'm beginning to wonder if we are testing our students too much.  Where is the fine line between getting the information needed and doing too much.

Today, our school looked at data to determine the school's priority of what we want to focus on first.  We looked at attendance, reading scores, math scores, behavior, and so much more.  Each bit of data providing a bit more.

Due to a lack of time, I never got a chance to look at the reading scores but I did get a chance to check the math results for grades 3 to 12th.  We looked at the once a year state test which showed only 4 students out of over 250 students were on grade level.  We also looked at the MAP results for this year which showed only 3 students at almost grade level. 

I'm told that we should also have our high school students take the AIMS math test because it will give us additional information while acting as a universal screener.

Ok so lets look at this.  They take the state test once a year and every spring all students classified as ELL must take another test to determine their English language proficiency.  The MAP test is taken three times a year so we can check their rate of growth but the data provided does not give me quite enough information.  For instance, it might say they need to work on solving equations but it doesn't tell me what part of solving equations are they struggling with.

Although the AIMS test is for students up to 8th grade, most of my students according to the MAPS are well below grade level.  They appear to be performing at a 4th to 6th grade level.  From what one of the elementary teachers said, the AIMS has two components in the math test.  Students can be checked for calculation and for application.  They say it will provide me with better information on what they struggle with but this would be another test they'd end up taking three times a year.

Each of the tests take two to three days to administer which means I'm loosing student contact time.  I am unable to work with students who desperately need the class time.  So if students are testing for about 30 days a year. I admit it is only for one test per day but still 30 days a year means they loose class time for 1/6th of the school days.

Furthermore, if the students do not take the testing seriously, the results may not be a true representation of their abilities especially when they rush through it and take less than 15 minutes to complete a test that should take much longer. 

In addition, most of the teachers at our school are not given enough time to sit there and analyze all the test results to see if we can find enough information to know what we need to scaffold our students in.  Its not enough to have the data from these tests, time needs to be allocated to examine results to plan a path of instruction. 

Data is worthless without the ability to use it to plan the next steps. I'd love to hear from others out there who face the same problems. 

Building Content Literacy Using Word Problems

 Unfortunately, defining literacy in math can be rather difficult because of the complexity of the subject.  Often we expect students to know how to solve equations, transfer knowledge from one type of problem to another but we forget to include the use of language.

Math has three types of words.  The words which have only mathematical means, those which carry both mathematical and regular meanings such as product, and words that are have only general means such as hello.  Many of our students who are below grade level are behind,  ELL, or special needs and struggle with understanding what they read.

Literacy includes both vocabulary and word problems.  If a student is struggling with language in general and the mathematical language in specific, they will find it extremely hard to solve word problems.  So in today's column, I'm sharing ways students can work on improving their literacy within mathematics.

1.  Review vocabulary words frequently used in word problems.  Discuss their possible meanings along with common usage.  Identify the words as addition, subtraction, multiplication, division, power or root.  Take time to help students differentiate between fewer than meaning subtraction and fewer than meaning an inequality.

Divide the class up into small groups in multiple of four.  If you have 24 students, you might divide the class into four groups of six or eight groups of three.  You do not want them groups to be too large.  Give each group an envelope of word problems with a variety of words .  Have students go through the problems and highlight the important words which indicate the operation and other information needed to solve the question.

Once the groups have gone through their packet of words, it is time for them to solve the problems using the highlighted words. When all the problems are done, they can check with the teacher to see if they got the correct answers while explaining how they solved it.

2.  Teach students steps to learn to better solve word problems.  First, have students practice reading the word problems through at least two times. Teach them to break the word problem down into smaller parts so the problem becomes more manageable.  Take time to let students act out the situation or create something visual such as drawing a picture.  Just some way to let them visualize the problem.   In addition, help students learn to distinguish between fewer than meaning subtraction and fewer than meaning an inequality.

3.  For older students, I've had students learn to use the KFCW or Know - Find - Consider - Work.  My students refer to it as the Kentucky Fried Chicken Wings.   After reading the problem, they write down what they know.  Then they have to identify what they are asked to find.  The third step or consider is where they think about the math they have to do to solve the problem.  Do they have to do something like change ounces to pounds before they do anything else?  The last step is to actually do the work.  This helps because it makes them think about the mathematical operations needed.

4.  Other times, I had them identify the who, what, where, when, why, and how in the word problem.  I literally have them identify who is in the problem, what they are doing, where they are doing it, when did it happen, why did it happen and how did they do it.  Once everything is identified, they then work on solving the problem.  Since this is something they've done in English, it was easy for them to apply it in Math.

All of these activities help improve literacy in Mathematics, vocabulary and word problems.  If students understand all the words and comprehend what was written, they will do so much better.  Let me know what you think, I'd love to hear.

A New Pictorial Mathematical Language

  Over the past thirty years or so, there has been a movement to design proofs without words because sometimes a single picture conveys so much more than two or three paragraphs trying to explain the same thing.

When teaching mathematics, we are told to provide both written and visual information on the same topic so that students have a better chance of understanding the concept.

Since 1975, The Mathematical Association of America has had a column devoted to showing proofs of things without words.  So they use pictures and diagrams to show the proof since sometimes the picture is much clearer.  This article has a lovely multi-part explanation of the idea behind it in easy to understand pieces.

In 2017, three men from Harvard developed a 3-dimensional pictorial language called quon which can be used from math to physics and more.  The language is designed to transmit complex amounts of information in simple pictorial designs.  Although it was created to convey quantum information but in the meantime, it lead to the discovery of results in other areas of mathematics.

This new language uses images to convey the same information in the usual algebraic equations plus a bit more.  It visualizes the concept via a picture which allows people to "see" the concept rather than relying on written equations.  It also allows mathematicians and physicists to share the same frame of view when looking at the same equation rather than interpreting it differently.

Originally, the language operated in two dimensions but recently has expanded into a third dimension so  it can be swiveled, deformed or viewed in different ways, which leads to the creation of equations.  They discovered this language works well in expressing Pauli matrices, an intricate part of quantum information protocols. 

These three men have continued their work to expand the use of the quon language in other sectors of mathematics.  This language shows the promise of allowing mathematicians to experience new insights into all sorts of mathematics from Algebra to Fourier analysis, statistical physics to string theory. 

Furthermore, it is believed that this particular language may be the basis of a new field of study.  In addition, other people are beginning to use quon to prove some extremely complex equations using simple pictorial representations.

Imagine a way to express complexity in just a few three dimensional pictures.  So cool.  Let me know what you think, I'd love to hear.  Have a great day.

I See Math!

I finally sat down to read "Hacking Mathematics - 10 problems that need solving" by Denis Sheeran.  The first idea is one I love but I have limitations due to where I live but he provides a solution and I found a place on the internet to help.

He recommends changing  the usual bell ringer out every so often for an "I See Math" activity.  It is composed of three slides.

The first slide is the title, the second has the intriguing picture and the third contains some sort of question to get them going.  The picture he used had multiple sized cups of iced coffee and hot coffee complete with prices.  The students had to decide a way to answer his question of which size is the best. 

He said students discovered they could not use a unit price to determine the best buy because some students wondered about the ratio of ice to coffee in the iced version.  Did that decrease the actual amount of the liquid in the cup vs the hot coffee which might be filled with only coffee.

I love the idea but out here, we do not have that many choices for finding these types of opportunities.  We have one real place for picking up coffee in the village and that is at the school cafeteria which has a large pot of coffee available by eight in the morning.  In addition, the stores do not always have the per unit price available for an item but you are often lucky to find two items to compare.

Fortunately for me, Alice Keeler has a blog entry explaining how to do this and a link to a file with at least 36 - 'I See Math' activities.  The activities cover so many different topics that you could do one a week for the whole school year and not run out.

I love that there is the file so I can use them and get an idea of how to put them together so  I can develop my own.  I always take pictures in the summer on my trips which I might be able to use to create some of these.  I only need one interesting photo attached to a question which will trigger discussion.

One of the best things I see about this particular exercise is aside from showing real world connections, it generates student discussion and questions which is needed.  I plan to try this on Tuesday morning with all my students.  Thank you Denis and Alice for this wonderful idea.

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

Warm-up

Warm-up

Mathematical Patterns in Real Life

  We know the beauty of mathematics is in the patterns. I'm not talking about finding the missing number in a list of numbers.  I'm not talking about solving dry equation after dry equation.  I'm talking about the mathematical patterns that produce the beautiful pyramids or the number of petals in various flowers.

Its the thirds used by Egyptians when they drew human figures.  A ratio that is not far off of the real human form.  For instance figures are drawn in units called heads.  The current ideal for the average figure is seven and a half heads high which is a perfect thirds, 2.5, 2.5, 2.5.  Something cool.

If we look at most things around us, we can see patterns in shapes such as roofs with a series of isosceles triangles which are used as  roof supports. Overall it has a specific pitch or slope. If you look around, you can find squares, rectangles, trapezoids, triangles, parallel lines, line segments, intersecting lines, perpendicular lines, etc.

Look at nautilus and snail shells to see the golden ratio or Pythagorean spiral depending on what article you read.  They spiral around, each segment getting larger and larger until the beautiful shape is created.

If you check butterflies or books, they have bilateral symmetry while maps provide real examples of vertical angles, corresponding angles, alternate interior or exterior angles, or same side angles.  Many cities have parts with parallel line cut by a transverse. 

Look at rates.  You'll find them in pulse rates such as 87 beats per minute, blood pressure, gestation periods for animals, corn when its on sale for 6 for a $1.00 or 4 different packs of veggies for $5.00.  There are also found in water or electric rates because you pay so much per unit or at the Olympics when new records are set.

Even the instructions for knitting, crocheting, tatting, fabric flowers, etc use repeating patterns to create the finished products such as scarves, hats, gloves or doily's.  Its the repeating patterns that make things interesting.  For instance, if you are making a pair of socks, you might cast on 64 stitches to start.  The pattern might be knit 2, purl 2 and repeat round and round until you've completed several inches worth. 

They are there.  Show them to the students, let them see the world is filled with mathematical patterns that can be used to explain the world.  Let me know what you think, I'd love to hear.  Have a great day.

A Few More

Most children have a person they look up to, someone they want to be like.  Sometimes its a sports celebrity.  Sometimes,  its a model or actress but how many times do you hear someone say "I want to be just like Katherine Johnson when I grow up." When I was young, I didn't know of any female mathematicians.  My high school Algebra teacher inspired me to go into Math.

I think its time we introduce students to some of these wonderful women who contributed to the field of mathematics.  The movie Hidden Figures introduced us to some fantastic ones but unless we see a movie or find something on the internet, most are unknown compared to Descartes, Galileo, or others.

1. Sophie Germain who lived during the French Revolution. When the revolution began, she shut herself in her father's study to read and learn.  It was upon learning about Archimedes that she developed an interest in mathematics. She taught herself Greek and Latin so she could study some of the mathematics in their original languages. 

Although she could not study at the local university, she managed to obtain study notes so she could learn.  Eventually one of the professors discovered she was submitting papers on a false name, so he became her mentor.  By the time she died, she'd become the first woman published by the French Academy of Sciences, proved Fermat's Last Theorem, and her work on the theory of elastisity.

2. Sophia Kovalevskaya born in to a Russia where women were not allowed to attend University so she married a paleontologist and they moved to Germany.  She was privately tutored until she received her Doctorate in Mathematics.  She was known for her papers on partial differential equations, Abelian equations, and Saturn's rings.

After her husband's death, she was appointed as lecturer at the University of Sweden, before becoming the first woman to be granted a full professorship in the region.   She won prizes from both the French Academy of Sciences and the Swedish Academy of Sciences before she died in 1891 at the age of 41.

3. Emmy Noether was lauded by Albert Einstein as the most brilliant and creative mathematicians produced since the higher education of women started.  She grew up in Germany where there were rules against women matriculating to higher at Universities.  Finally she received her PhD, when she wrote on a a topic in Abstract Algebra but she was unable to secure a university position for many years until she was granted an "unofficial associate professor" at the University of Gottingen but she lost it in 1933 because she was Jewish.

Due to the discrimination, she moved to America to teach and conduct research at Bryn Mawr College and the Advanced Institute for Advanced Studies.  Over time, she developed the mathematical foundations for Albert Einsteins general theory of relativity and advances in the field of algebra.

There are other examples I can share and will in the future.  I chose to omit Hypatia and Ada Lovelace because they are fairly well known but these three are not as well known.  If we want to convince women to going into mathematics, we need to show them some role models.

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

The Marquise Du Châtelet

  If you are like me, you read the name in the title and went "Who?"  This lady was listed as a woman who should be known in science but its another aspect of her that interested me.

The Marquise Du Châtelet born
Gabrielle Emilie le Tonnelier de Breteuil in Paris on December 17, 1706.  This is a time when wealthier women were expected to marry, run the house and bear heirs.  Its amazing that Gabrielle did this and so much more in her time.

She grew up in a traditional household of her time.  When she was 18, she married the Marquis Du Chatelet, a man with a title but little money.  Within a few years she'd produced two sons and a daughter while running the household and pursued traditional entertainments like the opera, etc.  But it was while she was pregnant with her second son that she picked up a mathematics book and began her road to studying and becoming fluent in the subject. 

Eventually, she studied Descartes Analytic Geometry with two of the leading mathematicians of the time.  All through the 1730's and 40's she continued to raise her children, run the house hold, read, study, and publish a few works.  She was even published by the Royal Academy of Sciences which for her time was extraordinary.   

She became pregnant again late in 1748 or early 1749, she worked hard to finish translating Isaac Newton's Principa into French because pregnancies late in life tended to result in death. She did not just translate Newton's work, she either corrected or completed many of his ideas so they are what we know today.

This work was published ten years after her death in 1759 and in time for Haley's comet.  Imagine, a woman in 18th century France who educated herself enough in mathematics that she could read, translate and understand all the concepts and mathematics involved in Newton's work.  That awed me so much.  I know how hard it is to translate from one language to another and get it so it means the same thing in both languages. Even in today's world, her translation is the only full version of Newton's work.

In many ways, she was typical of her time including having lovers outside of marriage.  One of her lovers - Voltaire - she was with for 15 years while the other fathered her last child.  Still, through it all, she wrote many papers, translated, and continued running her house for her husband. When she died in 1749, she was only 43 years old but she left her mark in a male focused world.

Thank you for letting me share this wonderful lady with everyone.  Let me know what you think, I"d love to hear.  Have a great day.

 

Bringing Math into the Classroom.

The other day I read about a third grade teacher who organized her classroom to resemble a veterinarian clinic with several math stations.  Each station provided a different aspect of the math involved in the field of veterinary clinic.

I realized there is no reason, high or middle school teachers should not invite representatives from the bank, a store, the sewer department, the police, or any business to come in and speak to students on how math is used in their jobs.

We have a man who is in charge of the water and sewer department who is always willing to come in to explain the math involved in his job.  It's amazing on how much they have to monitor in the sewer department as they move waste and process it.  Out here, most of the wastes are moved using suction through pipes on the ground.  I don't think we have any buried pipes because the ground freezes.  They also have to keep something running through the pipes to keep them from freezing.  

He is always willing to talk about all of that.  He brings manuals to show the information they use to keep the system running.  He is also great about discussing how he has to keep passing certain math tests to move up the management ladder.  The more classes he passes, the more responsibility he gets.  Its cool.

We don't have any banks out here but I could have the manager of either of the stores come in to discuss the math they use.  They have to order supplies in and if they want the bigger items like washers, dryers, refrigerators, and freezers, they have to figure out shipping via the barge.  They can explain how they calculate the price mark-ups so they make a profit.

There is a local engine repair place who charge a per hour fee plus supplies.  Someone could explain how they arrive at an hourly cost for their labor and how they price the parts.  The business also offers to help people purchase boats and four wheelers (ATV's) and the representative could explain how that works mathematically.  

The only other business of any consequence is the health clinic manned only by health aides.  It would be possible for one to visit and explain how they use mathematics in their jobs.  I know its used when they calculate how much medicine to administer medicines among other things.

In most places, there are so many more businesses to call to see if anyone would be willing to visit your classroom to discuss the math they use.  Most places have access to Insurance companies, larger stores, gas stations, movie theaters, doctors, lawyers, temporary agencies, etc.  

This would add a touch of reality to the mathematics we teach in school.  It can help them see how mathematics is found in the world around them and exists everywhere.  Let me know what you think.  I'd love to hear.  Have a great day.

Warm-up

Warm-up

Mortgage Math.

  In many small schools, consumer math is no longer being taught.  The school only offers math classes designed for college bound students.  Unfortunately, many of the consumer math classes do not take time to look at mortgages and how a simple eighth of a percentage can change the over amount paid.

When I was growing up, my mother always told me to pay extra every month so the mortgage would be paid off sooner.  This does work because you end up making at least one additional payment per year.

Furthermore, most students don't realize that the larger the down payment, the lower the amount of money they have to borrow which is important to planning ahead.

Lets look at the down part of buying a house.  Most down payments are either 3%, 10%, and 20% of the agreed upon price.  It is good to point out that with a 20% down payment, the purchaser does not have to purchase PMI or private mortgage insurance to protect the lender in case you cannot make payments.  PMI costs between 1/2% and 1 % per year added to the mortgage payment.  Once the loan balance amount reaches 78 percent of the original amount, the PMI is removed.

When securing the mortgage, there are several different types such as a conventional loan for a flat percent over a 15, 20, or 30 year term.  Rates change frequently so its important when choosing a rate, to choose one of the going rates.  In addition, to conventional there are ARM or adjustable rate mortgages which are often guaranteed to remain fixed for a set period of time such as 5 or 10 years before the interest rate is recalculated every year.  This type of mortgages is often appealing in a time when mortgage rates are rising frequently.  Another type is the interest only where the borrower pays for the interest only and often requires the borrower to pay the whole borrowed amount after 2 to 5 years. The idea is the borrower will secure a proper loan before the balloon payment is due.

The last element to look at are interest rates.  Every lender offers a slightly different rate based on your credit scores but many online real estate places such as Realtor.com and Zillow.com offer current interest from several sources so students get an idea of the different rates.

So here are three elements students can explore using a spreadsheet while using math formulas for interest, payment, and down payments to decide which deal is the best.  Use this information to provide a spreadsheet based project where they:

1.  Find a house they would like.
2.  Determine the down payment for 3, 10, and 20 percents.  How much is each.
3.  Find the amount being borrowed.
4.  Calculate the monthly payment for each balance based on mathematical formulas using a low, medium, and high interest rate.
5.  Use the figures for taxes, etc from the websites to be included in the monthly payment.
6.  Calculate any PMI payments at 1 percent of the loan per year.
7.  Once the spread sheet is set up, let them play with paying off the loan by adding $50, $100, or $200 per month to see how it changes the payoff time. 

This is real world application of Math which prepares students for the real world in which they may consider buying a home.  The more they know, they better job they will do planning for buying the house.

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

Gambling and Probability

Although most of our students will not grow up to be professional gamblers, the study of probability involved in gambling is worth looking at since it is one way to peak student interest.

I know a guy who used to live in Los Angeles before moving up to Alaska.  At that point, he'd pop over to Las Vegas to earn extra money gambling.  He said the reason he never became a professional was he needed to spend time with his kids. 

Its interesting what things a gambler has to keep in mind while gambling.  I'm not talking about the old folks who hit the casinos once a month, or those who hit a gambling establishment occasionally.  I'm talking about those people who make a regular income with the game of chance.

There are three things both the casinos and the gambler must consider about the game.  First, they are dealing with definite possibilities.  Second, the expected value or the amount of money one can expect to get from the game.  Finally, the volitility index or standard deviation when the game is played.

Lets look at these in more detail.  As stated there are definite probabilities within the game.  A gambler knows the probabilities depend on the number of outcomes or sample space.  When you roll a six sided die you have a one in six chance of landing on a specific number but when you are talking poker which uses multiple decks, the probabilities are quite small.  In poker, trying to draw a four in five card is only 0.00024 while drawing a flush is even smaller.  Knowing these odds, helps guide a professional's betting choices.

A second factor in gambling is the expected return per game.  In other words, if the game were based on flipping a coin where you get $1.00 every time the coin comes up heads or  you lose $1.00 every time you get a tail, you would expect to end up with nothing because the probability is 50/50.  Mathematically, it is EV = (.5(1.00) + .5(-1.00)) = 0. Because the odds are equal, this situation is considered fair because no one has the advantage. 

If on the other hand the dealer gave you $1.50 every time a head came up and you lost $1.00 every time a tail came up the odds would change to EV = (.5(1.50) + .5(-1.00)) = .25 or you'd be 25 cents richer per game on average.  This means that for every 100 games played, you'd expect to be $25 ahead.

Usually, gambling casinos have negative EV's so they have the advantage.  They want to have enough money to pay all their bills.  Even thought the EV is negative overall, professional gamblers still come to play because the actual amount they win is often different than the theoretical or EV. 

The volatility index or the standard deviation is what gamblers are concerned with.  This tells them whether they can win or loose a bit more than what is normally expected. They use the deviation for a specific number of rounds to help them decide when to continue or when to stop playing.

Furthermore, its not just mathematical odds, its also their ability to read the body language of the other players to determine if they should stay in the game or fold.  Its not a straight calculation because there is a human element.

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

Golden, Silver, Bronze, and Aluminum Ratios.

We've all heard of the golden ratio during our days of playing with math.  We've learned of its applications, its origin, everything, and we teach it to students.  Basically it is the ratio of the shorter side of a rectangle of length one to the longer side whose length is (1 + sqrt5)/2 or 1.618

We've seen it in ancient architecture such as the Parthenon, or the Pyramids of Giza. It is said that Leonardo Di Vinci's "Vitruvian Man" illustrates the golden ratio.  In addition, it is said he applied the golden ratio to his "Last Supper" Painting.  Even the Fibonacci sequence and pentagons are tied to the golden ratio.

If you read yesterday's column, I mentioned something about the silver ratio and has nothing to do with money.  It turns out there are at least three more ratios out there that are not seen as frequently.

The first is the silver ratio which is when you cut off two squares and still have a rectangle left with the same ratio as the first.  Or you get one 1 x 1 square, a 1 x sqrt 2 square and a rectangle of 1/ (1 + sqrt2).  because the whole rectangle is 1 x 1+sqrt2.  In other words, its 1:1.4.

In addition, the silver ratio is revered in Japan as the most beautiful ratio through history. Their name for it is "Yamato-hi", meaning Japanese ratio. They've used it in their architecture, statues of Buddha, and in the art of  flower arranging.

Another one, I'd not heard of is the Bronze ratio or mean.  It is the ratio of a side of 1 to the other side of (3 + sqrt13)/2 from a rectangle.  The idea of this one is to cut the rectangle into three parts instead of two.  This would be the third in the Metallic Mean family.

The Metallic Means which include the golden and silver ratios then bronze, copper, aluminum, etc, all based on the quadratic formula of x^2-px-q = 0 so that you get a solution of p + sqrt(p^2 + 4q)all divided by 2.  If p = 1 and q = 1 you get the golden ratio or if you use p =2 q = 1 you get the silver ratio.

So by changing the values of p and q, you get a different ratio.  So far though, only the silver and golden ratios are found in real life.  The other ratios seem to be theoretical only.  One interesting fact.  someone completed a survey of historical taffy pulling machines provided an interesting ratio based on the lengths the machines could pull the taffy.  The data showed some of the machines pull length are based on the silver or gold ratio.

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

Infinite Series.

  I don't know if you remember shows like "Sesame Street" designed to teach children their numbers and letters while "Where in the World is Carmen San Diego?" taught geography in a fun, entertaining way but there were not that many math based problems at that time except for Mathnet which ran from 1987 to 1992. 

In Mathnet, detectives used math and logic to solve crimes. The crimes range from figuring out who stole a signed baseball to determining the innocence of someone charged with robbery. 

The other night, I came across a new math based series from PBS called the "Infinite Series".  In it, mathematician, Kelsey Houston-Edward, provides situations showing how math is used all around us.  The episodes are not very long but can be seen on YouTube.  So far there have been two seasons made and they cover some interesting topics such as:

1.  Why individual honeycombs are built in a hexagonal shape rather than squares.
2.  Did you ever wonder why computers are bad at algebra?
3.  Picture a mathematician explaining mathematical probability to a gambler.
4. Why social networks allow us to model and analyze?
5. How about an explanation of Arrows Paradox?
6.  How to build an infinately long bridge using the harmonic series?
7. How does set theory handle infinity?
8. What are numbers made of?
9. We all know what the golden ratio is but have you heard of the silver ratio?
10. Is the analog clock a circle or a torus?

So far there have been two seasons of this series made.  The first season has 32 episodes while the second season has 19 before it was cancelled. I listed a taste of 10 out of the 51 episodes already made.   The series is produced by PBS digital studios and are easily accessible by everyone.

The nice thing about each video is that they range from 7 to 20 minutes long which are a perfect length for use in the class.  The video begins with someone explaining the problem before going on to cover it in detail.  The person explaining it might ask you to think of it in a slightly different way so as to make it easier to see.

I think I'll be spending time after school exploring these in more detail so I can figure out where to put them in my classes.  Check them out and let me know what you think, I'd love to know.  Have a great day.

Examples Aging Out

The world is changing so fast with all the innovations in the digital world that companies have had to adjust their way of doing things to keep up.  Three industries in particular have changed enough that we can no longer use them as examples for linear equations.

The first used to be rental cars.  Students could check with a variety of rental agencies to find the going rates for renting one of their cars.  Usually, they'd be quoted a daily rate plus so much per mile such as $20.00 per day and $0.03 per mile.

These quotes could be compared to find which one provided the best deal under various circumstances such as which would be best to rent for a couple days or for a week.

In addition, students could check the price of flying to a location to compare it to renting a car to make the same trip.  Throw in the cost of a hotel room and let the students determine which mode of transport was more economical.  This type of exercise brought up questions by the students of how much is a person's time worth? 

The other industry which used to use a flat rate plus so much per minute is the cell phone industry.  Its changed from a flat rate per month plus so much per call to a flat rate per month with unlimited calling and so much per text sent to totally unlimited. 

These examples were great because you'd run across articles in the news about people who were not paying attention to the contents of their contract and would receive ridiculously high bills for texting way past the limit.  Those articles were great because I could find a basic rate with a per text cost and let the students determine how many texts the person sent. 

I found one article which resulted in a calculated number of texts somewhere over 200,000.  That day, the kids really got a shock when they found the number because that is way more than they usually do in a month.  Overtime, most cellular plans evolved into unlimited so you can download anything, use your google maps, send texts, and just talk on the phone.

The final industry is the personal transportation industry which used to be only populated by taxi's. Taxi's also worked on the base rate plus so much per tenth of a mile, or mile.  It was like $3.00 to arrive and $0.06 per tenth of a mile.  Another wonderful real world application of a linear equation but due to Lyft and Uber, this is slowly going away. 

For these two companies, you get a flat rate from pick-up to destination, the same as regular shuttle companies such as Prime Time and Super Shuttle.  These newer companies charge a flat rate per person or per group.  I use shuttles or public transit more often than taxi's. 

Its hard using any of these examples now because most of my students do not relate to them, so its time to find new examples which operate on the same principal so we have updated examples.

Let me know what you think, I'd love to hear.

Warm-up

Warm-up

Math and Computer Science.

 I've been trying to get some sort of math based coding class started at school.  The idea is for students to learn coding to write mathematical routines. I thought of using python as the base language for students to program these routines.

Thanks to someone on Twitter for this website.  Its called CS and Math started by Mike Larson and Ashley Tewes.  This site provides free lessons for both primary and secondary classes.

The primary lessons set has seven lessons all done in scratch designed to explore different topics.  One lesson is on writing a program to convert Fahrenheit to Celcius while another is designed to test the probability of a coin toss.  Then there is the one which compares discounts, or another to locate places on a coordinate grid.  Check out the program which is a game that subtracts integers, or check the one designed to calculate ratios in recipes.  The final one allows students to create polygons.

Each of the lessons come with a well thought out lesson plan showing the instructor the steps needed to create the app.  I love the instructions because they are so clear.  In addition, there is a link to a video to use just in case you get stuck and how to do it in Python.

The secondary lessons have eleven routines, three from the primary group.  Of the last eight, they cover the Pythagorean Theorem, Solving systems of equations, Hypnotic squares, Translations, Graphing slope intercept form, solids of revolution, and solids of extrusion.

Again, the secondary routines come complete with the instructions needed to write the routines so it comes out properly.  The directions are interspersed with questions so students change certain parts of the program to see what happens.

 I like the work has been done for me so I don't have to take a lot of time to prepare.  In addition, each lesson has a goal and a standard listed so you know approximately which one it meets. 

I see these programs being used as part of the teaching such as when teaching students about graphing linear equations, it would be easy to slip the lesson on graphing in slope intercept form.  Or when introducing the coordinate grid, it would be easy to include the primary graphing exercise to help students learn more about the topic.

Its great to have a way for students to get some real world experience on a topic.  It definitely answers the question of "When will I ever use this?"  Let me know what you think, I'd love to hear.

Frequent Testing

I've read that one way for students to develop greater recall of material is to test or quiz them on the material learned on a regular basis.

One way to do this is with the traditional once a week quiz where the students know they are being tested.  The quiz doesn't have to be for points but some students freak out if they take a quiz that has no points attached. 

So I make it worth a few points. This way if they do not do well, it won't hurt them much at all.

Another way to sneak in a "quiz" is by using a math based Jeopardy.  My kids love it. They have to work together in partners to get the answer which they write on a whiteboard.  I check the answer on the whiteboard and announce how many got the right answer before I unveil the answer.  This builds suspense and they get excited.

The really enjoy the game.  It is a good way to test their knowledge in a fun and unthreatening way.  Often, I give them a small halloween sized piece of candy as their reward.  Another game they love is Kahoot because it allows them to compete and the winner is always changing.  You should hear them whine about not getting in faster or darn, it took them too long to find the answer. 

I think Kahoot is their favorite game of all time.  Unfortunately, the bandwidth out here is so narrow that the minute 45 students are testing, no one else can use the internet and that restricts my use of it.  With Jeopardy, I can download the games but as far as I can tell, I cannot do it with Kahoot. 

Several of my students enjoyed the mathematical bingo we played the other day.  I need to make a couple for next week so students can play it.  Its not hard to do and using a rolling random name selector makes it more fun.  In addition, if you make them fill in their own Bingo cared from all the possible choices. This gives them a better buy in.

There are more.  In geometry, you could create a mathematically based charades or pictionary game for members of the class to play.  This type of game integrates physical movement with learning so some students will enjoy it more.

The whole point behind most of these ways of testing student knowledge is with a game framework.  Most people would rather play a game that tests their knowledge than take something labeled a quiz. Much of it has to do with the 'testing' connotation associated with the word quiz vs something fun.  By alternating what we do, I can sneak in frequent test a couple times a week.

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

Distance and Midpoint formulas

 Distance and midpoint can be difficult to teach to students, especially ones who do not live where there are distance signs or city blocks.  Out here everything is one hour by air, or about 20 minutes by boat, or maybe an hour by snow machine.

The town has two stop signs I think but no one really pays attention to them.  So when it comes time to teach the distance formula in Geometry, it becomes very difficult.  Even the city does not have any real city blocks so I can't even use that as an example.

Often times, I bring in a map of Anchorage, the closest city to me with real blocks, and copy a part of it onto paper with a grid over it.  I do this by making a copy of the map, then making a second copy of a grid from an overhead on top of it.  Voila, I have a wonderful grid.

I have the students use a marker to label the left bottom corner as 0,0 assuming map is only the first quadrant.  Then they label the lines as 1 to the end on both the X and Y axis so we can create ordered pairs to use in calculating the distance. 

The first time I have to find two intersection on the map before discussing how we'd find an accurate distance between the two points.  They often suggest we count blocks but I ask how do they count accurate blocks since they've suggested we go down and over or diagonally?.  This leads to a nice discussion on how could they do it.

Eventually, I lead the discussion to the Pythagorean Theorem and its application by having them create a triangle from those two points.  this leads to how it can be applied with points, on to the formula itself.  It helps them see how the formula works.

We do this for a few places in Anchorage in terms of going from Walmart midtown to Freddy's near the airport.  It really gives them something to relate to.  Once they've got it down, I move them to a regular grid with points to have them practice the distance formula.

As for midpoint, I get out a ruler for them to measure the distance between the two points on a general sheet of paper.  I ask them to locate the midpoint on the line and then ask they explain their method to me.  I get a lot of "Duh's" from them but it works well.  I extend this to a coordinate grid by having them find two points on the line, connect the points with a line and then they measure the line to find the midpoint.  They mark the midpoint on the line and find the coordinates.

I ask them how we might accomplish this without using a ruler, or pencil and just find a formula for the midpoint.  Eventually, they get to the formula which we add to their notes.  This can take one or two periods to complete but they have a better understanding of the concept rather than just knowing the formula.

Please let me know what you think.  I hope you have a great day, I'd love to hear from you. 

Project Math

 I was searching the internet for an activity on points, lines, and planes that was more interactive than just filling out standard worksheets when I stumbled across the Project Maths website.

They have a pdf for this topic which starts out with students connecting certain dots and then measuring the lengths in centimeters and millimeters.  In addition, there are some true and false questions to test their knowledge.  It comes with everything needed for the lesson.

I decided to check out the website itself to see what else it offers.  The site is out of Ireland and is from the Mata Maths Development Team.  It appears to be for upper levels rather than elementary which is nice for me.

In the teacher resource area, they a lesson plan library, new resources  for your whiteboard, resources for teaching algebra and tutorials on geometric constructions. Furthermore, the area contains resources for the junior and leaving completion certificates. I took time to explore all of these areas but I'll report on that after I discuss the other areas.

The student area has video tutorials for geometry and  videos on using hand held calculators.  In addition, there are sections for learning material for the junior and leaving certificate materials.  The two completion areas are divided up into the major topics needed to learn before completing the certificate. 

The topics contain Geogebra files that can be downloaded or used directly on a computer to show how it works and there is a student worksheet to be used to practice the material. For instance, if you are teaching or reviewing adding integers, the Geogebra file will show how it works on a number line.  Then the worksheet has a second example using the number line at the top before students are expected to do it themselves.

Many of the student activities require the students to make changes to the Geogebra files so they can observe what happens before writing their observations on the worksheet.  Almost like a science experiment. 

As far as the teachers area goes, the resources for the junior and completion certificates are broken down by topic so you can choose a specific topic to work on.  It even has proofs available teachers can use. 

The lesson study area has over 100 lessons listed by name, topic with a description of the lesson, strand, year, and level.  The lessons are sortable according to topic, year, or level.  I did a sort for Geometry lessons and found 43 different lessons. 

The lesson tells you who created it, ages, standards it meets, goals, unit plan overview, flow, what happened, and just about everything you need to teach the lesson with photos and everything.    These are actually lesson study proposals so they have some academic substance to them and are part of the professional development of the group.

I am impressed with the amount of usable material I can access to incorporate in my lessons.  Check it out, give it a look and let me know what you think.  Have a great day.

Happy Labor Day

Warm-up

Warm-up