Modular Origami

 

Everyone knows that in origami, you take a square piece of paper, make certain folds in it and at the end, you have a box, a crane, a swan, or other item.  Modular origami, also known as unit origami, is when you take two or more pieces of paper that you fold into specific shapes and put together to create a larger 3-dimensional mathematical based form that is often symmetrical.

These smaller units are building blocks of the larger, final form.  The mathematics involved is with assembling the final form and discovering the patterns needed to create it.  Often the patterns use different base units but they use similar ways of combing the units into larger creations.

The shapes that have the most symmetry, are referred to as Platonic solids.  These are named after the Ancient Greek philosopher Plato.  A Platonic solid is a 3 dimensional shape made from two dimensional shapes aka regular polygons so every side and every is identical.  There are only five Platonic solids and they are the tetrahedron made up of four triangles, a cube made up of 6 squares, the octahedron made up of eight triangles, the dodecahedron made up of 12 pentagons, and the icosahedron made up of 20 triangles.

The basic building block is the sonoboe unit also know as the son oboe modules. It resembles a parallelogram with two flaps folded behind. They are easy to learn and can be used to create a variety of three dimensional shapes. The more units you combine the more complex the final product and you just might end up with more questions about what can be done. 

Historically, there is evidence of modular origami dating as far back as 1734 with instructions in the book Ranma Zushiki by Hayoto Ohoka in Japan. In this book, there is an illustration which shows several items but one is a modular cube. It is shown from two different directions and is referred to as a "magic treasure chest". The same model of modular cube appears in a book published in 1965 by Isoa Honda called World of Origami.  This modular cube uses a traditional paper fold called Menko to make the 6 parts needed for the cube. Modular origami was rediscovered in the 1960's first by Robert Neale who worked for NASA in the United States and later by Mitsonabu Sonobe of Japan.

If you need a place to start, head to  this site in Australia because it has instructions for the basic sonoboe unit and directions to make a cube, dodecahedron, and a faux dodecahedron.  These are a wonderful place to start while it gives students a chance to have fun learning more about 3 dimensional shapes.  If you want to explore other shapes, just do a simple web search and you'll find them. Let me know what you think, I'd love to hear.  Have a great day.

Math Neurons In The Brain

 

They finally did it.  They located the neurons in the brain that do the math.  That is so exciting. Research shows that certain neurons fire when a person is performing  certain mathematical operations.  Specifically, some neurons fired when the problems involved addition and others fired when subtraction was being done. 

The researchers used data obtained by the physicians at the Department of Epileptology in Bonn.  They specialize in performing surgical procedures on the brains of people with epilepsy.  They discovered that epileptic seizures originate from the same area when they implanted probes in the brain.  These probes were used to determine the place where the spasms originated.  At the same time, they were able to measure the activity of individual neurons.

The electrodes were planted in the temporal lobe of the brain so they could record the activity of the neurons. The researchers had these people do simple arithmetic problems.  The results indicated that certain neurons fired during addition and others fired during subtraction. To determine if the brain was reacting to the plus and minus signs, researchers replaced the signs with the words and the brains reacted the same way. This indicates the neurons actually encode instructions for the mathematical operation. In addition, they found the activity displayed by the brain showed quite accurately which operation it was performing as people did the problems. 

The researchers set up the program by instructing a self learning computer as to whether the activity indicated the brain was doing addition or subtraction to train it.  Then they ran new data through the program to see how well it learned and the program was able to accurately determine the correct operation being performed.  Furthermore, they discovered that cells in the parahippocampal cortex also fired during addition and subtraction.  It appears the cells used dynamic coding in regard to the carrying out the operations.

There has been limited research done on which parts of the brain is actually used when solving mathematical problems.  This will help scientists when they conduct research on the brain as they look for more data.  Up to now, they believed the brain consisted of billions of neurons capable of carrying out different functions but this is the first time that neurons have been identified as "math" neurons. 

This opens so many doors for future research.  It is cool they've identified how the brain handles addition and subtraction.  Let me know what you think, I'd love to hear.  Have a great day.

Warm up

 If this is a 45 degree angle, what would the slope be if expressed in delta Y/delta X?

Warm-up

 

If a blue slope has a gradient of 25%, express the slope as deltaY/deltaX.

What Makes The Perfect Free Throw?

 

If you wonder why the sudden interest in math and basketball, it is the season at my school.  Students are participating in regionals and state so there is a lot of interest in it.  I see students practicing their free throw shots all the time because that one shot can make the team pull ahead just enough to win the game.  Of course, there are folks out there who explored the idea that there is a perfect form to get a free throw and used mathematics to find it. 

One of the biggest things is that the fate of the attempt for a free throw is already set by the time the ball leaves a players hands. About 20 years ago, a couple of mathematicians created a computer  program which was able to imitate all the different trajectories of basketballs shot. They spoke with all the coaches and assistant coaches at their college to see what question they wanted answered.  The number one question was "What is the best free-throw?"

If you stop to think about it, basketball is made up of a bunch of trajectories because every time a ball is thrown, it follows the path of a parabola and the path is a trajectory.  The trajectory can suddenly change when the ball hits the rim or backboard and the program took this into account.

The mathematicians figured out a way to change trajectories into statistical probabilities.  They even included a trajectory that went through every physical obstacle except for one so they could see what happened.  Due to the question, they studied the free throw shot first since that was what everyone wanted to know. They studied this question for five years and made some conclusions.

First off, they discovered that players with the same consistency usually made the shot with a 75 to 90% accuracy.  Those who reached the 90% accuracy rate had the best trajectories. Since the fate was decided by the time the ball left the players hands, the mathematicians looked at the perfect "launch" conditions.  The shot is effected by factors such as the distance above the floor, the backspin rate, the launch speed, and the launch angle.

They discovered that a 3 hertz rate for the back spin was ideal because the ball takes one second to read the basket, and a 3 hertz rate allows for 3 spins between the player and the basket. If the ball is released at a point 7 feet above the ground, the ball needs to be launched at a 52 degree angle because it can be off one angle either way and still make the basket.  The launch speed is the hardest one to control.  If the ball is too slow it won't make it and if it's too fast, it will go past.  It is important for the shooter to know how their body moves so they can be consistent with their launch speed. They also discovered that the higher a player is from the ground, the better chance they have of making the free throw. 

The last and most surprising condition was the aim point of the free throw.  The best results come from the player aiming for the back rim because it is more forgiving than the front of the rim.  So over all, players need to aim for the back of the rim, launch the ball at around a 52 degree angle as high above the ground as possible, and launch it with a smooth motion. 

So if you have a student who needs a bit of help improving their free throw, tell them all about this and the research that helped them get there.  Let me know what you think, I'd love to hear.  Have a great day.

Geometry In Making Goals In Soccer.

There are those who love watching soccer when it's in season. There are those who grow up playing soccer from when they can kick the ball. Most of the young players are taught that the more of the goal they can see when they make the shot, the better chance they have making the actual goal.  

As children grow, they learn where the best place to shoot the ball is.  They learn early that if they overrun the ball in the box, they will hit the side netting instead of the middle of the goal but by the age of 10 or so, they become better at judging angles.

After a bit of formal analysis, there are three situations that are more likely to be used to attain goals.  For instance, if a 55 degree is chosen, the kicker has a good chance of making a goal but if they are situated to the left or right of the goal forming a 17 degree angle, there is less opportunity for making a goal. 

A person drew a line from the kicker to the goal and calculated the probability of making a goal from that position and he assumed the pitch was the same along the path, in other wards, flat. He determined The changes of scoring a goal at a 55 degree angle results in a 30 percent possibility of making a goal whereas someone shooting from a 17 degree angle has only a 6 percent chance of making a goal. So based on this, the wider the angle, the better chance of making a goal. 

In reality, when analyzing the real life data of  soccer players, it was determined these shooters were more likely to make a goal when in a circular area in front of the goal.  The best position resulted in a 20 percent chance of making a goal and chances decreased the further away from the the center, facing the goal.  On the other hand, if the same data was applied using the angles method, instead of a circle, the area squashes and indicates that one can still shoot from a bit wider area and still make a goal. 

Another analysis looked at the math of soccer on the field and it backed up the analysis above.  In addition, it looked at questions such as the area the defense needed to cover. It was found that the closer the kicker is to the goal, the wall of defenders needs to be larger but the idea distance is 10 yards from the kicker. 

In addition, this analysis looked at chances of making a goal using a curved or straight kick from various locations on the field. Basically, the closer the ball was to the goal, the better chance of making the goal. In addition, the kick should be done about 5 yards to the right of the midline when using a curved kick that arcs to the right.  If it arcs to the left one should kick from 5 yards to the left of the midline.  For a straight kick, the closer to the goal, the bigger the angle and better chance of making a goal.This is the article should you like to read it yourself.  Let me know if what you think, I'd love to hear.  Have a great day.

Mathematics Of When To Shoot Basketballs.

 

Basketball can be a fast paced game with the players running up and down the court.  At any time, they have a split second to make the decision to try for a basket or wait for a better opportunity,  Sometimes they try for it and make it, other times they try and miss, or they blow past what looks like a perfect opportunity to the spectator.  It doesn't matter if it is a neighborhood, high school, or professional game, the decision is there.

Back in 2011, graduate student Brian Skinner created a model to help shooters to determine the best time to shoot based on mathematics.  In 2007, he heard a talk on traffic flow where every driver attempt to minimize their individual travel time rather than trying to reduce the average travel time for all drivers.  

This talk reminded him of the Patrick Ewing theory applied to basketball in which it was noticed by analysts that when Patrick Ewing and other high scoring players were absent, the team was more likely to win. In fact, it was noticed that the diagrams showing the movement of the players resembled traffic flow. Brian Skinner suggested that every equation and variable used in traffic flow theory could as easily be applied to basketball.  He took the routes in the traffic models and transformed them into different places.

Then he took his equations and made them follow the ball from the inbound pass to the hoop.  He factored in the probability a specific shot would make it into the basket, the quality of future shots the team is likely to make, and the number of seconds left for a player to make the shot or forfeit the ball to the opposing team. He found that the more time left for the play, the choosier the player should be in selecting which shots to make. 

In the situation where you have two teams and both have the same chance of making a certain shot but the first team passes twice as fast as the second team, one assumes that the first team should shoot twice as often as the second team to win.  According to Brian's model this is not correct.  He calculated that the first team should shoot every 13 seconds to the second team taking a shot every 20 seconds because those extra three shots allow the first team to be more selective about the shots they take.  This gives them a better chance of winning. 

In addition, he has probabilities and scenarios for all sorts of situations. Unfortunately, the shooter trying to take a bit more time to determine when to shoot is very difficult to do because it  requires split second responses and not everyone can do it. On the other hand, players can look at their stats after the game is over to compare their their actual decisions to theoretical ones to see how well they choose the best time to shoot.  

There you have it.  Applying traffic flow equations to basketball plays to get a better idea of when to shoot the ball.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 The biggest pig is the world is the Hungarian Mangalista.  The average one Hungarian Mangalista weighs 900 pounds.  If a regular pig weighs 500 lbs, what percent more does the Mangalista weigh?

Warm-up

 

The worlds smallest pig is only 10 inches tall and weighs about 18 pounds.  If the average pig is 26 inches tall and weighs about 500 pounds, what percent smaller is the smallest pig than the average pig in both height and weight?

Computer Model Of A Cell

 

All of life is made up of cells.  Some cells are plant cells while others are animal but up to this point in time, no one has made a complete model of a cell.  Recently, a team at the University of Illinois at Urbana-Champaign created the most complete three dimensional computer simulation. I chose it for today's column because it is digitally based. 

This group created a computational model of a very complex system.  They've used the simulation to make some very interesting discoveries about both the psychology  and reproduction of the cell.  In addition, it provides people with an opportunity to generate more ideas for experimentation. This simulation looks at the metabolic functions of the whole cell rather than the biochemical reactions for an artificial system.  They are getting results as if it is a real cell.  

In the past, others have tried to model cells but have not been successful because most cells are quite complex.  This simulation differs in that it uses a much simpler cell with fewer genes so it is much easier to figure out how the system works.    Researchers can monitor the simulation for waste, use of nutrients, gene products and other biochemical processes in a three dimensional environment.  This model allows scientists to understand how the simplest cells work and what the minimal requirements are for life.

The results of this simulation lead the way to creating more complex models so eventually they will be able to create simulations of specific cells such as the common intestinal E.Coli. 

Researchers started with a version developed by J. Craig Venter Institute back in 2016 and was based on the simple bacterium cell  Mycoplasmas mycoides.  They stripped any parts out of the cell that didn't have anything to do with survival so it only had 493 genes which is about half of what the original had.  The 493 is about one eighth of the number of genes in the E. coli.

Scientists are not sure what 94 of the 493 genes do but they are aware that when the cell dies, these 94 genes are not present. It is suggested that these genes help with life but we just haven't figure out what they do.  It is hoped that this simulation will help them figure out what these 94 genes do. 

The group who created the simulation combined all the data available for cells and how they function before using flash frozen thin sliced images of the minimal cell developed by the J. Craig people to position its organic machinery precisely. They then sprinkled all the necessary proteins throughout the cell using a protein analysis before relying on a detailed chemical composition of analysis of the cell membrane from the Dresden University to correctly place molecules around outside of the cell. They also used a map of biochemical interactions to determine the appropriate interactions.

They observed the digital cell as it grew and divided, all the while monitoring thousands of biochemical interactions so they could see how the cell acts over time as it grows.  In addition to seeing the results from the original cell, they noticed some new, previously unobserved such as how the cell distributes energy and how fast messenger RNA degrades.

One of the most surprising results had to do with the speed of growth and division of the cells.  Normally, the cell needs the enzyme - transaldolase - but there does not appear any is present so either the cell is able to do it without the enzyme or it exists in a different form. Although this simulation does have some problems, it is one of the best available right now and will lead to better models as more is learned.  Let me know what you think, I'd love to hear.  Have a great day.

Math Protects You Against Zombies

 

It's amazing how zombies took over the imagination of people in the movies, television, books, and even in music.  People love watching the fight between humans and zombies.  The fascination with zombies has reached the point where math has been used to explain how to survive a zombie apocalypse. 

We've seen articles covering this topic but from a survivalist point of view rather than a mathematical view.  Numberphile on Youtube created a video explaining the math of zombies using the diffusion equation.  

We know that zombies tend to move around randomly while diffusing through a population.  In addition, there are three possible outcomes to the scenario.  First is the zombie kills the person, the person kills the zombie, or the zombie infects the human, making another zombie.  So the person giving the explanation ends up using partial differential equations to determine the best course of action to survive the encounter. 

One conclusion based on the relationship between distance and time is that one should run away rather than staying to fight. In fact, the object is to kill zombies faster than zombies are being made. This is because zombies invade at a rate proportional to their speed and distance from us.  Earlier predictions said the Rocky Mountains is a great place to retreat to but any real rural area away from cities would be good since it would take a while for zombies to get through the population before heading out.

Back in 2017, a study conducted by students at the University of Leicester and published in the Journal of Physics Special Topics determined that within 100 days of "ground zero", the number of zombies would radically outnumber the uninfected. The study based it's math on the premise of one zombie converting one person each day. The conclusion stated with a 90% certainty, there would be only 273 survivors left after 100 days.  They used the SIR model which is a model used to calculate the spread of disease within a population.  The students included the idea that over time, the survivors are less likely to become infected due to learning to fight better, In a follow up study when they included human reproductive rates with better rates of killing zombies, it was determined that over time, humans would win out over zombies. 

In another study done by PhD students at the University of Sheffield explored the question of what happens if people choose to stay and fight the zombies.  These students also used the SIR model to answer the question.  They determined that more people would come back as zombies if they stayed and fought the zombies.  These students also ran the numbers for the scenario of sending in the military to fight but the result came back about the same. In their testing different scenarios, hiding from the zombies came back as a decent choice as long as they were not found.  If they were found, they would end up infected.  The premier solution would be to domesticate the zombies.  

In the first situation, the person looked at the math of spread using partial differential equations while in the other two studies, students relied on the same mathematical modeling used to determine the spread of disease in humans and in nature and what happens when a vaccination is used which is equivalent to the domestication of zombies in the zombie apocalypse.  So mathematics applied in two different ways based on whether it is a disease or something that is diffused.  Let me know what you think, I'd love to hear.  Have a great day.

Happy Pi Day

It is March 14th, the official pi day for those of us who love to celebrate it.  One year, the community college I worked at arranged a Pi day celebration. We had a scavenger hunt of facts, a pi throw, jokes, created art work and we even played music based on the digits of pi.  It was fun.

There are facts such as:

1.  The symbol used to represent pi has been in use over 250 years.  

2.  Since pi is an irrational number, we can never calculate an accurate area of a circle or volume of a sphere. 

3. Certain people loved pi so much, they created a language based on pi and call it pi-lish. In this the number of letters in a word is matched to the digit in the value of pi. In fact, one person wrote a whole novel called Not A Wake in this language. 

4. Before it was decided to use pi for the name of this value, it was called "the quantity which when the diameter is multiplied by it, yields the circumference". A Welsh Mathematician came up with using pi as the symbol in 1706.

5. There is a circular parade that happens every year on Pi day at the Exploratorium in San Francisco. This might have stopped due to Covid but before then it was a regular occurrence.

6.  To stress test a computer, they have it calculate pi to see how well it works.  It acts as a digital cardiogram for the computer.  

7.  Early Egyptians believed that the great pyramids of Giza were built on the tenants of pi. They believed that the vertical height and the perimeter of the base had a direct relationship with each other. 

8.  In the movie Contact, scientists are able to find hidden messages in the digits of pi from the creators of the human race.  In the series, Star Trek, there is the episode "Wolf in the Fold" where Spock defeats the evil computer by having it calculate pi to the last digit.

9.  If we calculated the circumference of the earth to 9 places, the error would be no more than one quarter of an inch over 25,000 miles which is an extremely small error.

10. Pi day began back in 1988 at the Exploratorium by physist Larry Shaw.

Just a few facts to use as a way of celebrating Pi day.  If you want to know more about pi, do a quick search and you'll find all sorts of cool facts. Let me know what you think, I'd love to hear.  Happy Pi Day to all.

Warm-up

 Today, if it took the boat 462 days to float 8000 miles, what was it's average speed in miles per hour.

Warm-up

Students At Rye Junior High built a 5.5 foot long boat they released into the Atlantic Ocean.  The boat landed in Smola, Norway, 462 days later after covering 8000 miles of ocean.  How fast did it travel per day?

Ways To Help Change Student Opinion Towards Math.

Since the pandemic started, too many students have to been in attending school virtually. This has made it harder for them to learn and teachers have not been able to help students develop a positive attitude towards math.  In addition, since students have gaps in their foundational knowledge, they display avoidance behavior to get out of doing the work.

One of the easiest ways to help students change their attitude so it's more positive is simple.  If the teachers attitude is positive towards math, it can spill over to students.  In this case, the teacher is providing the positive example for many students since parents have been providing a more negative attitude towards math their whole life. 

Instead of telling students they will need to learn math for their future, show them how they might use the math in their lives.  If you have students who talk about building their own house, show how the math you are teaching applies.  If your student wants to be a mechanic, show them how math makes their job easier.  If they want to be say an engineer, show them how the math they are taking now, will help them in college.

Find activities and projects that excite them.  Let their excitement help them find ways to solve the assigned problem. As they finish these activities or projects, they are building their confidence and ability to do math.  The project might be designing their dream house and figuring out how much it costs to finish, or planning their dream trip which will include an itinerary and budget. 

This is important: teach your students how to learn.  Most of the time, students go through their whole educational life without learning.  They don't know how to take notes, study, or apply what they've learned. If they don't have these skills, we have to take time to teach students to do them.  Once they've learned these skills, they need to practice them. 

In addition, it is necessary to set things up so they can be successful in math.  I don't mean "give" them the good grades but set up a framework that allows students to become successful.  For instance, on Friday's I provide time for students to correct previous work and finish work they didn't get done.  This way, they have the opportunity to finish everything and get better grades than they are used to.  Furthermore, it helps students who take a bit longer to do the work, a chance to keep up and succeed.

Take time to show students that there is seldom only "one" way to do a math problem.  Encourage them to explore other methods so that they find one that works for them.  When I teach binomial multiplication, I teach 5 ways and one of the methods uses a pictorial representation.  One of the methods relies on the standard vertical multiplication form and another turns it into two distributive problems.  My students usually find one that works for them.

Encourage collaboration between students.  If one grouping does not work, move students around until you find groups that work well.  In addition, encourage students to "teach" each other since peer tutoring is a powerful tool.  Another plus with this method is that the student doing the "tutoring" gains pride when they are able to explain it to another and it helps students improve their communication skills.

Finally, if you have students who are competitive, divide them into teams so they can "complete" against each other.  This works with students who are intrinsically motivated and want to "win".  No matter what suggestions we read, we have to remember that nothing changes instantly and it will take time.  Let me know what you think, I'd love to hear.  Have a great day.

Literature and Math

 

Too often we just teach math in our classes because there is the idea that each subject is separate and it supports the student idea that one cannot integrate say Language Arts into Math. This is something that would be nice to do so students see more interrelationships among the subjects.  

Literature is one way you can show the relevance of math to daily life and if math is part of the story line, it shows students that math is not dry and boring.  

Many of the books that could be used are picture books one would normally use in elementary school such as Sir Cumference series or A Place For  Zero, or Spaghetti and Meatballs For All.  These books are very elementary but middle school and high school students can read these books and discuss the math covered by the book.  In addition, students can try writing picture books about math topics for elementary students. This one is best done after having students read several picture books so they can see how it is done.

Although most of the above books are designed for elementary school, it can be done in middle school and high school.  For instance, there are several books out there in the Science Fiction and Fantasy realm that use math.  One is Mathematics by Elizabeth Moon.  In her fantasy world, the wizards cast spells using derivatives and those spells are countered by antiderivatives.  In addition, all the chapters are numbered mathematically such as chapter 2 is labeled as sqrt 4.  Another is Achilles heel in which athletes compete and the competition includes a section on mathematics.

It is also possible to take books such as The Crazy Horse Electric Game and have students perform calculations to make things more understandable.  For instance, the character had to make a trip so ask the students to figure out how much it costs and the time schedule involved.  The main character is also in an accident so students could calculate the cost of medical treatment and how much they might have to pay depending on the amount the insurance company is willing to pay.  Since it involves baseball, students can complete player stats, the distance x rate formula for traveling by boat to the hospital, and more.  

This can be done with any book.  If there is travel involved, a character going shopping for something, building something, anything, just take that, expand upon the mathematics involved to make it more relatable.  One can also take a title of a movie such as Good Burger and have students calculate how much it costs to make repairs, determine profit of the burgers based on different materials used, or letting them calculate how long it takes to earn enough money to pay for the repairs.  As far as earning money, students can look at different hourly rates to see how that effects the length of time needed to earn money to pay for the repairs. 

Need some help getting started?  Check out this site which has some recommendations for articles such as buildings and architecture, or children's picture books, or sample chapters from certain books, all with a literature to math tie.  Although many of the suggestions are for the very young, some of the ideas can be taken, adjusted and applied to older students with a bit of thought.  

If all else fails, get together with the English or Language Arts teacher to select a book that he or she can have students read in their class and they will do some calculations on topics related to the book in your class.  If you are reading "The Diary Of Ann Frank", students can calculate the size of the hiding area, the number of square feet allotted to each person, the cost of creating the disguise for the entrance to the room, etc. Let me know what you think, I'd love to hear.  Have a great day.

Student Written Word Problems.

 

There are reasons to have students write word problems.  Many of the problems that are found on the internet or in textbooks are not easy for my students to relate to.  For instance, anything involving a car and traveling from Los Angles to Seattle make no sense to my students because there are only one or two in town and you can't drive very far with one.  I think you can go eight miles to the new school and that is it.  Problems of that sort need to be rewritten using snow machines or all terrain vehicles.  Even problems with school buses do not work well unless there is one in town.

One reason to have students write their own word problems is that it gives the teacher a way to see if they understand the concept. In addition, it gives students a chance to create a real world application of the mathematical concept.  For instance, if students are learning about unit prices, they could write problems about shopping in the store and comparing two items to determine which is the better purchase.  Furthermore, writing word problems requires students to develop higher level critical thinking skills.  

When students write word problems, they write them with an idea of how they would solve the problem and if there is more than one method involved, they will think about which method might be better.  Writing word problems help students connect the mathematical concept with their background knowledge and with their own life's experience. 

One piece of research showed that although there was no indication of growth of students ability to write better and better word problems, there was a recordable change of attitude towards word problems. Furthermore, students who could solve an equation in standard form had difficulty with the same information when it was in a word problem. Once students began writing their own word problems, they started seeing a better connection between the equations and word problems. In addition, it also helped decrease student anxiety towards completing word problems. 

For the actual process, it is best to give students the answer or a few numbers to use in their word problem.    The answer might include variables such as x^2 + 4x + 4 or it might be 28 or you might give them numbers such as 3, 5, X.  You might even go so far as to tell them you want them to write a problem that uses multiplication and has an answer of 36.  

The majority of word problems usually involve a certain number of people who are doing something.  With this in mind, students choose the characters and what they are doing.  Usually, the characters are doing something like traveling, sharing things, buying something, deciding what happens if they add a foot to one set of wall or subtract a foot from the other set. 

So for an example, I might give my middle school students 45, 3, each and uses division.  So I have the parameters of the problem.  Since there is a 3, I might decide to have 3 friends, Joe, Mabel, and Frank.  Since I have to use division, it usually means they are sharing something so maybe there is a pile of 45 pieces of candy.  I have to use the word each, so if they are dividing the candy, I'd want to know how many pieces each person got.  

The final problem might read. Three friends,  Joe, Mabel, and Frank ,have a pile of 45 pieces of candy.  They want to share it equally among them.  How many pieces will each person get?

This is not something students can do one time and be experts.  It is going to take time for them to learn to write their own problems and it is not an easy process.  You should acknowledge that it will take time to learn to do this and that it can be difficult but with practice, it becomes easier. In addition, it should be taught, retaught, and taught again but over the span of weeks so students have an opportunity to get the understanding of the structure of word problems.

This is a good way to help students become less fearful of word problems and connect the words with the equations.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 In 1966, the country of Reunion received 71.8 inches of rain in a 24 hour period.  What is the hourly rate of rain?

Warm-up

 

Back in the 1950's, it snowed 62 inches in just 24 hours at Thompson Pass just outside of Valdez, Alaska.  What is the hourly average rate of snow.  

Pretesting Students Helps.

 

If you notice, just about every math program includes some sort of pre and post test for each chapter.  Unfortunately, not all of us use them because we are trying to cover a certain amount of material in the time we have with the students.  I know that in the past, I've skipped them because the district expected me to cover the material to match the suggested pacing guide.  I never kept on track.  I came across an article where someone took time to study the effect both pretesting and post testing have on student learning.  The results were surprising. 

To summarize, those students who participated in both the pretest and post test did so much better than those who did neither. What surprised me is that those who participated in the pretest did the best on subsequent tests.  So participating in a pretest has more effect than those to did post tests after learning the material. 

It appears that when students take a pretest prior to the lesson or activity it results in less mind wandering and they did better when they took the test on the material. Unfortunately, most teachers use a pretest only as an assessment tool rather than an opportunity for students to learn. Since it can be used as a learning tool, it is best to use either low testing or no stakes testing.  In addition, it is a good way to introduce the topic. The pretest can be either digital in Google forms, Quizzes, or other site, or it can be paper and pen, or even posted as a question on the board for students to provide a written answer. 

To keep student stress levels down concerning pretests, stress these are nothing to worry about. Let them know they don't need to worry about failing it or even being able to answer all the questions because they will learn more about it in the upcoming lessons.  Although it is often felt that testing does not teach students, when students test, they are recalling information and trying to formulate an answer which influences learning.

In addition to teachers often using pretests as one of the assessments, students can also use it to see what they know and it allows their mind to process things so when the material is actually taught, they have that "aha" moment and they can see their knowledge is growing. In addition, when the material is presented, they will remember the problems they saw in the pretest.  In fact, one can use problems from the pretest as bell ringers and ask students to explain how they solved it or how did they know the answer was correct?  

As far as using pretests as assessment, the information from the pretests can also be used to show the teacher is meeting the needs of all the students. The data from both pretests and post tests can be used to show student growth to the administration, even for the students who struggle. In addition, this same data can be used to show parents their students have made progress in the class.  The data can also be used to plan differentiated instruction and assignments for students so their gaps are filled in and students who need additional challenges will receive what they need.

Furthermore, the data from pretesting allows for better pacing because the teacher can go through the parts of the course they already know and slow down as needed. The information can also help identify any misconceptions students have such as when adding two fractions with different denominations, they cannot just add the numerators and denominators.  Instead, they need to rewrite equations so they have the same denominator.  

So keep in mind that pretests can be used both as a learning tool and an assessment.  Let me know what you think, I'd love to hear.  Have a great weekend.

Students Interviewing Students

I just came across an article on students interviewing students using specific questions as a way of helping each other and learning to communicate better.  It states that when students ask each other conversational, academic, and reflective questions, they are able to deepen their understanding. 

The idea behind having students ask each other questions is to help students learn to use questioning language because most students do not know how to do it.  I have a rule of talking to three other students to get help before coming to me and usually the either say "Are you at number 4 yet?"  Or "Can you help me?  No?  Ok."  all in one breath without waiting for an answer.  

When we teach students to ask each other questions to elicit information, we are helping them become more independent.  Furthermore, it is important to have the questions written out at the beginning since most students don't know how to begin.  These questions are used in conjunction with a partner activity.  So for the first few times before doing the activity, list the questions on the board so they have guidelines. 

In the activity, two students work together, each is give a task or a problem.  They spend the first 5 to 7 minutes working on the problem without talking to each other.  During this part, they may not ask each other for help.  When time is up, they each get a list of questions and they write their partner's name on this interview sheet.  The first person begins to interview the second person using questions from the list. 

Some of the questions are as follows but the first question is always the first question.

1.  "When you read the question, how does it make you feel?"

2. "Do you understand the problem was you were asked to solve?"

3.  "How did you solve the rest of the problem?" "What steps did you take and why?"

4.  "Was there a point you got stuck?"

5. "Did you give up or did you figure out a way to move forward?"  "What did that step help you know or figure out?"

6. "How did you know what to do?"

At the end, the two students switch and the second student interviews the first, asking many of the same questions.  Once the interviews are over, the students can help each other to finish their problems.  Not all these questions will be asked every time but the choice will be based on what the answers are.  

These questions help guide students as they formulate verbal expression of their thinking.  It helps them put their thoughts together, learn to use vocabulary, and explain their thinking.  I think it is a cool idea.  Let me know what you think, I'd love to hear.  Have a great week.