Warm-up

 

Come up with an explanation to accompany the graph.  What would the x and y graphs represent and what would the steepness of the line tell you about the graph?

Reading And Interpreting Graphs

 

Over time, I've discovered that I do not spend enough time teaching students to read and interpret real life graphs.  They have trouble, reading what the X or Y axis means which makes it hard to interpret what is going on.  Graphs have to have a context so they can be interpreted. 

Line graphs should come with certain pieces of information.  It should have labels telling what the X and Y axis represents and the units such as per thousand or miles.   The horizontal axis usually represents a time element such as weeks, months, years while the vertical axis represent the quantity measurement such as 15,000 fish, or miles covered, or the amount a stock sold for. In addition, the graph's title should tell you exactly what the graph is about.

Next, check the scale on both the X and Y axis.  Is the scales by ones, 10s, 100s, 15000s for each axis or is one by 10's and the other by ones? It is important to know before one begins to interpret the data on the graph.  Then when you are ready to interpret the graph, know the question you need to answer.  When is the number of salmon being processed is at it's lowest or highest.  What is happening to the population in 1924?  

Finally to read the graph, one finds the point where the x axis value intersects the y axis. This is the answer to the question and students need to be able to explain the answer to others.  This requires several sentences such as "the processing plant was only able to do 15 fish per hour in May" or "After 15 hours of driving an average speed of 70 miles per hour they traveled 1050 miles.

On the other hand, most bar graphs compare two or more items.  These still have the X and Y axis but the x axis is often made up of items such as appliances or types of fish, and the y axis tells you what it is such as kilowatt hours used or number of fish caught.  These graphs should have a title and something to identify what is being compared such as weekend vs weeknight use or caught by subsistence or sportsmen. and they color that represents each. Again, you have to know the question in order to interpret the information on the graph and answers need to be given in complete sentences. 

TV411 has a really nice set of activities that require students to interpret data contained in a graph. The first set of questions go with a bar graph on the amount of electricity used by several appliances such as lights, heater, or refrigerator.  The two bars represent weekday use and weekend use and the site asks 10 different questions which require the student to really read the questions and the graph to find the answer.  At the end of each question, the student will submit the answer and they receive immediate feedback.  If the student gets the question correct, the program will tell them so and explain the answer in more detail while if they miss the question, the immediate feedback explains the answer in terms of the graph.  

The second activity asks eight questions on reading and interpreting a pie chart while the third activity provides two pie charts and a table for students to determine if there is enough information available to answer certain questions such as how many seniors voted for a certain candidate or if to tell if more senior citizens voted for a certain person.  I really liked the questions in all three activities because they require higher level thought to answer.  Let me know what you think, I'd love to hear.  Have a great day.

Using Current Events In Math

 

I've been reading up on using current events in your math class. It is a great way to show students connections between the math class and real world events so they can see real examples rather than situations that seem to have been designed to align perfectly with the material taught in the section.

When students use their math skills to understand and analyze current events, they find the math means more and it's relevant.  In addition, research indicates that students who study news and current events in school, tend to do better on standardized tests while improving in a variety of subjects and have higher reading levels and a better vocabulary.

A simple search on the internet will lead you to so many different sites with math news that can be utilized in class.  Or check your newsfeed for articles with graphs and data such as the latest data on Covid, or international countries that are open, or oil spills. One can also read the local newspaper for sales, discounts, tax increases, and so much more.  It's just a matter of looking around. 

Most news articles are set to be read by someone who is at a sixth grade level which means most high school students should be able to read the articles.  One way to use the article is to have students create a verbal summary of it via Flip grid or any other video program.  Another way is to create a page with a place for students to create a written summery and several math related questions on the rest of the page.  Students can do either individually or in small groups.

Another way of using the news is to teach students identify The Who, What, Where, When, and Why of the story before answering the following questions.  Can they identify where the math is in the article? How is the math used to help people understand the news? Finally, what questions or thoughts do students have about the math or story?  

Now if you want to take this a step farther, ask students if it is possible to make prediction based on the math contained in the story? Are there any other positions or points of view that can be supported by the math or data in the story? Can the students connect the news to anything else they've learned?

Furthermore, when you are teaching a new topic like piecewise functions, or box and whisker plots, look for authentic data.  Authentic data might include having students measure the length of their feet and their height before creating a scatter plot of the data and creating a line of best fit for the data.  An activity like this gets students fully involved.

Try having students find ads on certain things in the newspaper so they can compare and contrast certain items such as tablets, pens, clothing, etc. Or they can compare the raw data from the ads to the graphical representation to determine is the information in the ad is accurate or slanted.  Do a quick search for information on population growth or decline for your town, city, or state to see how it has changed over the past few years and then have students make a prediction on what will happen to the population in 10 or 20 years.  

The same type of predictions could be applied to world records at the Olympics especially since this year the Olympics are being held over the summer, or weather patterns.  Weather patterns are good because much of the west has been experiencing a significant drought for the past 20 years.  Students could make predictions on the water level in Lake Mead and other lakes whose levels have dropped significantly.  They could predict when the water levels drop to being totally unusable.

Anytime there is an election, especially one for the president, have students look at the maps showing which states went republican and which went democratic so they can see how the states voted.  They should then compare these results with a map showing population densities and the distribution of electoral votes so students can see that having the most number of states does not mean you will win the presidency.  It depends on which states you get.

Lots of real math in the newspapers.  Let me know what you think, I'd love to hear.  Have a great day.

First Day Activity

 

I have an activity I love doing the first day of class before I pass out the textbooks, composition book, expectations, and the syllabus. It is one that doesn't require a ton of preparation, and only really needs a pen or pencil and paper.  

It is an activity that doesn't require students to remember actual mathematics after a long summer, nor does have them take notes.  It does help them ease back into school and it is fun.  Furthermore, the activity only takes 15 to 20 minutes to complete.

First thing that is needed is a regular sheet of paper that is divided into 16 squares.  Each box has a letter or set of letters so they are distributed as Box 1 is A & B, Box 2 is C & D, Box 3 is E & F, Box 4 is G & H, Box 5 is I & J, Box 6 is K & L, Box 7 is M, Box 8 is N, Box 9 is O & P, Box 10 Q & R, Box 11 is S, Box 12 is T, Box 13 is U & V, Box 14 is W & X, Box 15 is Y and Box 16 is Z.

Every student will get a copy of the paper.  They are given 10 minutes to try to think of at least one word for every letter of the alphabet. but take time to let them know if you will or will not accept variables as an answer.  In other words, if they can't come up with a word beginning with Z, can they use the variable Z?  I do not allow that.  They have to use regular words.

At the end of the time, have students stand up and leave their answers on their desk.  Give students 5 minutes to walk around the room, checking out everyone else's solutions for suggestions  but they cannot rush back to their papers to fill out words for the letters they haven't done yet.

At the end of 5 minutes, students go back to their desks and they are given another 5 minutes to fill out missing letters.  Once they've done this, you have two choices.  You can collect the papers and continue with your normal routine or you can go around the classroom writing all the answers for each letters.  What you'll find is that students will struggle with finding math words for certain letters such as Z, J, or W.  

I usually list all the words the whole class has come up for each letter.  If no one found a word for a certain letter such as J, I do not give it to them, I make them look it up online.  When they look up a word online, they also have to explain what it is if it's an unusual word.  They might find zero for the letter Z and that doesn't need an explanation but if they use jump for the letter J, they might have to explain that one.

I have done this so many times and each time the students do it, they have been involved and most students liked it a lot.  Let me know what you think, I'd love to hear.  Have a great day.

Warm up

If one coconut yields 425 ml of coconut milk, how many will you need to make 3.5 gallons of coconut milk?

Warm-up

If one coconut produces .75 cup of coconut water, how many coconuts will it take to produce 32 ounces?

Power Grids

 

It's that time of year again.  Between the hot temperatures and the out of control fires, the power grids are under a lot of stress.  Consequently, there is a lot of math involved in the production and distribution of power.  Power companies have to predict when there are decreased demands on the system to keep them running.  

The math involved includes differential equations, linear algebra, and graph theory which are used in grid simulations designed for operators to both monitor the systems and calculate the requirements of the system.

Power grids are actually complex systems that cover huge geographical areas and are composed of multiple devices.  In addition, the power system is made up of multiple generating sources connected by a network for transmission.  If any part of this stutters, the results are a rolling blackout or a blackout. The system has a weakness when power input changes from one minute to another when electricity is generated by wind power while it's pretty stable when generated from a coal-fired power plant.  So the mathematical models must include these because many systems combine wind, coal-fired, nuclear, and solar systems.

Unfortunately, writing mathematical models must incorporate a lot of different equations that govern every aspect of power, it's transportation, type of current, and so many other factors. The system is constantly solving all these different equations in real time so operators know exactly to correct any instabilities of the power flow. 

Another aspect of this situation is that communication and battery backups have to be factored into the situation because if the power goes out, how long will your cell phone work if the towers go down, or if they manage to stay up, how long can you be without power before your batteries die?  In addition, most power companies rely on cell phones now to communicate with their workers.

So some of the equations deal with the idea of setting repairs so that they are prioritized based on how long certain backup batteries at places such as hospitals or cell phone towers. In addition, they have to factor in where certain breakers lie in the subsystems and which ones have backup batteries.  

Thus the modeling programs used must take into account of so many factors and elements to provide everyone with the best information possible to keep the system working through all of the extra demands.  Math is responsible for the modeling used by power companies to be as efficient as possible. Let me know what you think, I'd love to hear.  Have a great day.

The Math Found In Cities.

 

Think about the term, urban geometry.  The geometry of urban areas.  Picture if you will, standing almost at the center of Madison Square Park in New York City.  The Math museum is behind you while the Flatiron building is in front of you.  The Flatiron building got its name because it's shape reminded people of a clothes iron but the block it inhabits is actually shaped like a right triangle.  The souvenirs sold are in shaped like an isosceles triangle because people prefer symmetry.

The cross street near the end of the building is 23rd and it intersects at almost a 23 degree angle which happens to be the tilt of the planet.  In addition, twice a year, near the summer solstice, the sun shines down the numbered streets. 

Furthermore, the "east-west" streets of Midtown Manhattan actually run northwest - southeast and when they run into the Hudson and East Rivers, they intersect at almost a 90 degree angle.  But in Chicago, the street grids are almost oriented to the North.  On the other hand, in older cities like London, there are no right angled grids.

All of these observations have lead to the new field of Quantitative Urbanism which is so new, it doesn't have a journal yet.  This field is trying to take observations like the above and translate them into mathematical formulas.  In other words, use math to explain cities.  

Studying cities dates back to the time of the Greek, Herodotus. By the time, we arrived in the 20th century,  scientists were studying things such as zoning theory, public health, sanitation, transit and traffic engineering, all apart of urban development.  This continued and in 2003, the new field emerged when a group of people met to discuss how one would mathematically "model" human society.  Out of the meeting came a paper on factors which might effect the size of a city. 

One observation is that as cities grow, many things increase at an exponential rate of between one and two which is referred to as "superlinear scaling".  For instance, private employment has an exponent of 1.34 while new patients is 1.27, serious crime is at 1.26 and gross national product ends up between 1.13 to 1.26.  As cities grow, a person's productivity increases by 15% because there are more people to collaborate with and innovation also increases.  It makes sense because as cities grow, the opportunity for interaction increases and other things increase. 

On the other hand, certain things end up at less than one because they are usually lagging as cities increase.  For instance, gas stations tend to run at 0.77, total surface area of roads is at 0.88, and total length of wiring in the electrical grid is at 0.87.  This makes a lot of sense.  I have a cousin who live in an area of Virginia which is within easy commuting of Washington D.C, so people began moving to the area. They needed places to live, streets to drive on, and the road system was way behind with tons of people driving on two lane roads which hadn't been redone to handle the increased traffic.

This is just a short introduction to the topic.  It is quite fascinating.  Let me know what you think, I'd love to hear.  Have a great day.

Religious + Atheists = Math Advances

 

Imagine if you will, a mathematician, Pavel,  in Moscow who trained as an Orthodox theologian and another man who lived in St. Petersburg.  This other, a man hated the church, a self proclaimed atheist, and constantly wrote letters to the Editor about various social matters thus gaining the nickname of "Andrei the Furious" Markov. 

The disagreement between the two arose when Pavel thought he'd come up with a mathematical proof of free will which supported the position of the church.  Andrei felt this mystical nonsense was wrapped up as mathematics. 

In 1902, Pavel applied the law of large numbers to the debate of free will versus predestination. He decided that voluntary acts are like independent events  in probability with no casual links between any of them.  The law of large numbers applies only to those independent events.  Since crime statistics conform to the law of large numbers, those voluntary acts must also be subject to the law of large numbers. 

Andrei Markov found a basic error with Pavel's thinking because Pavel assumed that the law of large numbers required the principle of independence and Andrei set out to prove that the law of large numbers applied to dependent events as long as they meet certain criteria.  

Consequently he created the Markov chain to show that random behavior could be produced mechanically but had the same features as those that Pavel used for free will.  Markov applied this first to digits and then the English language when he applied the idea to Puskin's poem Eugene Onegin. He broke it down into consonants and vowels before analyzing the placement of each.

He discovered that the letter following a consonant has a 67% chance of being a vowel and only 33% chance of another consonant. On the other hand, the letter following a vowel has almost a 13% chance of being another vowel and an 87% chance of being a consonant.  He applied his ideas to other pieces of literature to find that the author could be determined by the results and probabilities.

He presented his discovery to the Imperial Academy of Sciences in St. Petersburg in January of 1913. This move extended probability in a new direction.  It took chains of linked events and determined the next step based on the current state of the system.  

Today, Markov chains are used to identify which legislative maps have been brutally gerrymandered, how Google determines which websites are the most important, or it can teach the computer how to create human like text.  It is used to help identify genes in DNA, creating algorithms for voice recognition software, and so much more. All that is needed is to know are the probabilities for the next step in the process based on the previous step. 

Thus arising out of a disagreement between two mathematicians, we have something that extended probability to a something with so many applications.  Let me know what you think, I'd love to hear, have a great day. 

Warm-up

$126.73 million dollars of root beer is sold each year.  How much root beer is sold every week?  Every day?  Every hour?  Every minute?

Warm-up

 Selinsgrove, PA used 33,000 bananas, 2,500 gallons of ice cream, and 150 gallons of chocolate syrup to make a banana split that is 4.5 miles long.  How many bananas, ice cream and chocolate syrup is that per foot? 

The Math Of Crumpled Paper.

As a teacher, I see lots of crumpled paper ending up in the trash can.  Occasionally, I've used crumpled paper in an activity to help students practice their math but I didn't realize that someone had explored the mathematics behind crumpled paper. 

Some mathematicians at Harvard discovered that the lines in crumpled paper follows certain predictable patterns.  When you crumple a piece of paper, it creates stresses in the paper.  To relieve the stress, the paper breaks into a series of flat facets and each face is separated by ridges that are raised.

If you open the crumpled piece of paper, facets and ridges appear disordered but the mathematicians discovered that the total length of the paper increases logarithmically each time your unfold and recrumple the paper again and again.  The mathematicians knew that this process could be repeated multipole times with the same results but they didn't know how it happened.

They used sheets of Mylar which is a type of shiny, polyester film used by NASA in their spacesuits.  They studied how the flat areas in the Mylar broke up into smaller and smaller pieces as they crumpled, uncrumpled, and crumpled the sheets again and again. Unfortunately, the networks of ridges are irregular making it difficult to to define the flat area or facets in the sheet.  So one of the mathematicians hand traced every facet out by hand using Adobe Illustrator and Photoshop.

The person hand traced the flat areas for 24 (4 inch by 4 inch) sheets of Mylar.  Each sheet could take between hours and days to finish. On average, the sheet had 880 facets after a few rounds of crumpling but one had over 3,800 facets. Many of the sheets when completed looked like abstract art hung in a museum or like the colored world maps you find in many classrooms.

As the paper is crumpled again and again, the larger regions break down into smaller regions much like a pebble on the beach breaks down. Physists have a theory called fragmentation theory that explains why rocks break down into smaller pieces and the mathematicians discovered their data on crumpled papers fit that theory.  This allowed them to predict the logarithmic scaling they'd observed in sheets of Mylar.

During their research, the took one sheet of Mylar and crumpled it up 70 times.  They noticed that after a few times of scrunching it, they were unable to discern new ridges but upon closer observation, they noted that the sheet never stops forming the ridges but the ridges increase at a logarithmic rate.  

I can just hear a couple of my students giving me the "So!"! in response to this.  This discovery has applications in other fields such as the macroscopic folding in the Earth's crust, the microscopic crimping of graphene membranes found in high end batteries and superconductors, and the future design of small thin smartwatches. 

I love reading about things like this since it reinforces the idea that math explains the world.  Let me know what you think, I'd love to hear.  Have a great day.

Beauty And The Golden Ratio

I admit, I tend to watch all sorts of competitions from decorating, to fashion, to makeup. I find them quite fascinating and I'm always learning something.  The other day, I was watching a makeup competition.  One where people are giving a challenge and if they mess up, they are out.  One of the contestants mentioned doing something with the golden ratio face.  Of course, my ears perked up and off I went to explore more on the topic.

As you know, artists and architects of the Renaissance period used the golden ratio to plan their masterpieces.  A scientists discovered our eyes perceive beauty based on the golden ratio applied to the face.

A perfect face is going to have a length - width ratio that equals 1.6 so the face is one and a half times longer than it's width.  If one divides the length of the face into three part, each part should be equal.  The three parts are from the top of the forehead to the spot between the eyebrows, the spot between the eye brows to the bottom of the nose, and the bottom of the nose to the bottom of the chin.

On a perfect face, the length of the ear is equal to the length of the nose, and the longest part of the eye is equal to the space between the eyes.  So if the face matches all of these criteria, the eye "sees" the person as beautiful.  However, most faces that are seen as beautiful are not necessarily symmetrical so symmetrical is not a factor.

It is interesting that the model Bella Hadid is the person who comes closest to having the perfect face based on the Golden ratio.  She meets it with a 94 percent success rate.  The measurements were carried out by a plastic surgeon in London by  using computerized mapping techniques.  

Now there are several activities available so students can use the golden ratio to see how close a person is to perfection.  One is Is She Beautiful - The Golden Ratio which has students compare two measurements of Shania Twain to see if the ratio is close to either 36% or 46% based on the appropriate measurements.  This just looks at two ratios.

Another one is Math Behind the Beauty which explains the Golden Ratio in a bit more detail and has students trying to fit a perfect mask to photos provided by the site.  The site also allows students to do their own face, or the face of a star.  The masks are based on the work done at this site so one can read up on the research before assigning the activity.

For those who don't believe in the Golden Ratio face, this article lays out the argument against this whole idea. It appears to have some valid arguments and this could be used in a compare and contrast activity between the two ideas.  Let me know what you think, I'd love to hear.  Have a great day.

 

The Fractal Shape Of Cauliflower

Cauliflower is quite popular right now.  You can find it riced and served instead of rice, used as the dough of pizza, and in so many other ways but mathematically, it is a fractal and someone finally figured out how to explain it's fractal shape.  

First of all, if you look at the cauliflower carefully, you'll notice that many florets look like miniature versions of themselves while looking so much alike.  This is called self-similarity which is one of the properties of fractals.  Furthermore, cauliflowers show a high level of self-similarity, often seven or more copies of the same "bud".

All califlower but most especially romanesco cauliflower show spiraling pyramidal buds as shown in the picture. Bud after bud, spiraling endlessly around. The spirals follow the fibonacci series of 0,1, 1,2,3,5,8....... and in the cauliflower there are 5 spirals clockwise and 8 counterclockwise or vise versa.

Now the question is "Why does this happen"?. Well it turns out the answer is based in science. It was discovered that the main spiral begins at the microscopic level extremely early in its development. The genes work together to form gene networks that are beyond simple intuition and mathematicians rely on differential equations to predict the the gene networks behavior.

People made a model to see how cauliflower grew into their unique shape after the first few leaves show up. The model was based off of two things. First is the spiral formation of the cauliflower and second is the gene network seen in plant Arabidopsis because this plant has it's own version of a cauliflower due to a one pair gene mutation.  

In addition, Arabidopsis is related to the cabbage family and cauliflower.  There is a lot of data available on this plant since it has been studied extensively which gave mathematicians some basic information. They discovered that there are four genes involved in the development of cauliflowers are S, A, L, and T.  In the Arabidopsis, the A is missing in the parts that become like a cauliflower and it is the gene that causes parts to become flower. It appears the lack of A is what causes the buds in the cauliflower making it look as it does.

The model used actually produced a visual result similar to the cauliflower. Due to the lack of the A gene, the cauliflower attempts to make a flower but instead they develop into stems, after stems, after stem with the same angles between each. This is cool that people can now explain how the cauliflower is a fractal.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

A farmer harvests 34,200 pounds of potatoes per acre.  How many pounds is he going to harvest if he has 15.5 acres of potatoes planted?

Warm-up

 

If a three ounces of French fries is equal to 13 fries, how many fries will you have in 6.25 pounds of fries?

Math + Minimum Payments

 

Most everyone has at least one credit card in today's society.  You need a credit card to rent a car, book a stay at a hotel, or even pay when overseas.  I've found I can use my card to buy currency from the place I'll be traveling to.  

Most savvy people know that one should always pay more than the minimum payment for the month but young people have not always learned that yet.  I've often wondered how credit cards determine the minimum amount listed on the monthly bill.

Many credit card companies use the interest plus one percent formula to calculate the minimum payment.  This means they take one percent of the current balance and add it to the amount of interest accrued for the month to get the minimum payment unless the balance is $25.00 or less in which case, the minimum will be $25.00.  The other way used by some credit card companies is to calculate the minimum payment as two percent of the current balance.  

If the card has an interest rate of 15.5% which was the average rate in 2018, it will take 226 months to pay off a balance of $5500 with no additional charges using the first calculation for the minimum payment.  Imagine taking almost 19 years to pay off this amount.  On the other hand, using the second method to calculate the minimum payment, it would take a person about 290 months or about 24 years to pay it down.  In addition, the first method means a person saves just over $2,300 of interest and about 5 years of payments.

If a person has a smaller balance of say $1,000, it will take a person about 57 months using either method because they work out about the same on smaller balances.  It is with the larger balances that the differences become more apparent.  Although most of the largest credit card companies use the interest plus one percent, many of the smaller ones including Discover use the two percent method.  Note that the lower the interest rate, the faster the balances are paid off but it can still take a long time to do it using only minimum balances.  Sometimes, any late fees are included in the minimum payment while other times they are added to the balance which again depends on the company.

When a person applies for a credit card, the method of calculation should be located in the pamphlet that arrives with the card, otherwise it should be found online in the person's card account.  When in doubt call the credit card company and ask a representative how the company calculates the minimum payment.

This website has some nice activities to show students some basics on balances and the minimum payment. The video showing how to calculate things only looks at a minimum payment of $25.00 rather than either of the above methods but it would be easy to extend the activity to have them calculate the minimum payment using both methods.  The second video discusses how much interest is saved when more than the minimum payment is made. The videos are in Edpuzzle so they are interactive and require the student to participate.  

In addition, there are additional worksheets so students get to practice calculating interest and balances for other situations. Furthermore, students are required to graph the total interest paid over 5 months and the remaining balance after 5 months for three different people. At the very end, are suggestions for teachers to simplify the work or extend it.  

Now, as a teacher you know how they calculate the minimum payment.  Let me know what you think, I'd love to hear.  Have a great day.

PhET Interactive Simulations

 

It is always nice to find a real interactive activity for students to explore rather than relying on worksheets of any sort.  PhET is run by the University of Colorado in Boulder and is free.  It has 806 simulations for math, physics, chemistry, earth science, and biology currently.  What makes this site different is that the majority of stimulations are written in HTML5 rather than Flash so they can be used on the iPad or Mac.

When you click on Math, there are 44 total simulations but only 39 of them are done in HTML5. They cover topics like ratios, proportions, algebraic concepts, number lines, fractions, equivalents, quadratics, vectors, graphing, and more.  

Each simulation comes with a basic version to explore and play with and at least one more complex version to use what you learned with the introduction.  I did not find instructions for the student to read but it was quite easy to play with the simulation to figure out how to use it.  Fortunately, the site has extensive materials for the teacher.  

Furthermore, each simulation can be assigned via google classroom.  You also have the option of downloading the simulation or embedding it in something else.  At the bottom there is a description of the activity, the topics it covers, sample learning goals, and standards alignment.  Under the teachers section for the simulation, there are suggestions and teacher created lessons but you need to join the site to access them.  If you are working with students who are not proficient in English, there is an option to translate the activity into another language. 

In addition, they let you know about related simulations. For the curve fitting activity, there is a least square regression, and a calculus grapher.  The last two parts cover software requirements, which digital devices it can be used on, and who wrote it.  Lots of great information.  

There is a short about section which provides general information on the site if you look under teaching. In addition, there is a huge section of tips for using these simulations in the lecture, with clickers, designing and facilitating their use, and videos on facilitating the simulations.  There is also a professional development workshop available for this site.

Then there is a browse section which is actually a search engine so you can click on the type of simulation you'd like such as area model or function machine, choose the type of simulation such as remote, homework, lab, guided, or other choices, choose the subject, grade level, and/or language.  I tried the search engine by clicking on build a fraction, homework, and English.  The engine came back with 18 suggestions that I could choose from

The 18 were all listed under one title and if the title is clicked, there is information on who wrote it, when, language, and a downloadable document to go with everything but you do have to set up an account to do this.  It doesn't take long to sign up.

Overall, this site offers quite a lot of activities and support so it is something to think about using when you need an exploration or homework for class.  Let me know what you think, I'd love to hear.  Have a great day.

Get The Math!

 

I love it when I stumble across interesting websites.  I came across one titled "Get the math" which is a lovely site geared for middle and high school students by showing them real life applications of algebra and other levels of math.  

Each lesson has an introduction where someone shares basic information on the topic.  This is followed by a challenge which requires students to use material from the introduction to help solve the problem.  

The challenge is set up so it introduces the issue and provides enough help to work on it but not enough to fully give the answer. It even allows students to try again if they don't get it right. Once the challenge is done, students are given the opportunity to view a video on how others solved the challenge.  The section has one more challenge so students can practice what they've learned in the previous challenge.  

This site covers math in music, the fashion industry, video games, restaurants, basketball, and the use of math in special effects.  The math challenges are not the standard problems one associates with textbooks. Instead, they are the things that people face within that line of work.  For instance, in the math in music section, students learn about beats per minute and how to mix two different tracts into one so they are moving at the same speed and matched beautifully.

In restaurants, students learn how the prices they see on the menu are set by the chef.  Students are giving the job of calculating the price for guacamole next year based on previous costs.  This activity has students making predictions based on previous prices to come up with the future price.  In fashion, they get to change a design so it is able to be sold at a specific price point after learning how math is used by fashion designers.  For special effects, they learn about distance and light to get just the right exposure for the look they want.

The site lets you look at only the challenges, or the videos if you want but it also includes lesson plans for each situation.  The lesson plans include an overview, learning objectives, list of resources and materials needed for the lesson.  It also includes a list of things to be done before the lesson, the lesson itself with two activities and the culmination along with end of course questions that cover the same algebraic concepts as covered in this lesson.  There is also a long list of common core standards the lesson touches on.  

Finally the site allows you to download the lesson plans, all handouts, and end of course questions for each section so you have the material on your computer.  In addition, if you or your students have issues with the internet, it is possible to download all the material including the videos and challenges so it can be done locally.  

I love this last part because not all schools have great internet and if this is something you want your students to do as a flipped activity, it can be easily done by sending the videos home on thumb drives.  I am impressed by the way the math is introduced to students.  It is done in a friendly way without all the mathematical equations we see in textbooks.  I think students might find it easier to understand the concept.  Let me know what you think, I'd love to hear.  Have a great day.

Happy July 4th

 

Warm-up

 

In 2018, Washington DC spent 6.5 million on fireworks.  In 2019, they spent 13 million.  What percent increase was that?

Thinking In 3D = Better Math Results

Every so often I check out Mathematical news to see what is happening in the field.  In the process, I found several studies which indicate the children who have better spatial reasoning, do better in mathematics later on.  For instance, a study done by the University of Chicago found that children who could see how shapes fit together did better with number lines and performing computations.

Although the world is three dimensional, we teach everything as if it were flat. When we teach numbers and letters to students we don't take time to talk about them in terms of how rounded or pointy they are, or how their heights compare, or even how close or far they are.  This helps children develop their spatial reasoning which has long term benefits.

Two ways to help young children develop their spatial skills is to let them play with puzzles and blocks.  The study discovered that when parents play with their children using blocks, they are more likely to use spatial vocabulary such as "above", "over", or "through". Another study indicates that parents who use words like "curvy", "edge", or "face", their children are more likely to use that terminology later on in school.  In fact, the more terms the parents use, the better their children do in mathematics.

This is important because spacial skills are used in the STEM fields such as engineering, science, technology, and mathematics. Furthermore, a different study found that students who score well in spatial reasoning on tests in high school were more likely to go into STEM subjects in college.  Students who do well, usually come from homes where parents used the terms when they were young and introduced them to spacial reasoning as children.

Another study, this one from Switzerland, found that a students success in Primary school can be determined by their spatial ability at the age of three.  They also found that students who had a lower spatial relationship ability at three, did improve but they never caught up to the ones who did well at three.    They also found out that at the age of three, there is little difference in the abilities of boys and girls but over the coming years, girls tend to develop slower than boys. 

It is suspected this happens because the toys designed for boys tend to encourage the use of spatial reasoning and appropriate use of language while toys for girls focus more on social skills.  It is also possible that there is a bit of gender bias in regard to the perceived roles of men vs women.  

Fortunately, there are things we can do as parents and as teachers to help improve spatial reasoning from childhood on.  First, it is important to use the appropriate vocabulary in everyday situations.  Rather than using words such as this or that, be specific with language such as "the ball on the second shelf on the left hand side." This serves two purposes.  First, it helps children with developing their spatial ability and second, the person has to visual position to describe it.

Next,  try taking up chess because it requires the players to visual multiple moves ahead of the current ones.  Each time a move is made, the board changes and by looking at things ahead, one visualizes all the changes that happen each move.  You are also aware of all the possible moves your opponent has after each move.

Go back to playing with Legos.  This is a perfect activity to work on spatial exploration when putting the modular pieces together in creative ways.  How do you use Legos to make a building, a spacecraft, or even a tree? Another way to improve spatial reasoning is to picture places like your apartment, your favorite coffee shop, or other building in your mind.  Picture everything you can in it's proper location in your head.  This is called creating a memory palace and can be used to remember telephone numbers and anything else.

Or you could play video games because you use spatial intelligence as your character moves through the game.  Have fun creating place in minecraft because it helps develop spatial reasoning as you put things together.  Want something a bit more fun, buy yourself a drone and fly it.  Again, you use spatial intelligence as you determine where the drone has to fly, changes one has to make to avoid things, and plan ahead to where it is going.

As a parent, do any of the above with your child.  As a teacher, spend a bit of time using these in class.  When it is fun, people are more likely to do it.  Go ahead and do this regardless of age.  Let me know what you think, I'd love to hear.  Have a great day.