The Mathematics Of A Taffy Pulling Machines

Most of us grew up with some sort of taffy.  I'm used to the salt water taffy that is cooked and the pulled and pulled and pulled until it's just right.  Sometimes you can buy it a tourist places where its made using a machine.  One mathematician got the idea to apply math to making better taffy pulling machines.

A taffy pulling machine takes a piece of candy, stretches it, folds it again and again to incorporate air so eventually it develops its characteristic chewy texture.  As the taffy is stretched, its length increases exponentially by the same ratio.  It is the stretch ratio, that is what captured the mathematicians interest.

When a human stretches the taffy, they stretch it and then fold it in half.  They stretch it again and fold it in half again and continue until it reaches the proper consistency. Mathematically,  it uses a multiplication factor of two.  On the other hand, when mechanical pullers are used, the multiplication factor turns out to be an exotic irrational factor.

It was discovered that pulling taffy by machine could be modeled using topological dynamics which studies the long term changes over time in a mathematical space.  This is the same mathematics used in glass blowing and in drug preparation.  It also describes how viscous materials need to be mixed together.

As part of his research, Jean-Luc Thiffeault read patent application on taffy pullers.  He was looking at how a two dimensional material is stretched exponentially in one direction while shrinking in another.   The first taffy pulling machine was patented back in 1893 but never was made due to not being developed enough.  The first recognizable patent was filed in 1901 and granted in 1906.  This machine is mathematically the same as the 3 rod machines used today but no one is sure if it was ever made.

In 1908, two men patented a simplified 3 rod machine that formed the basis of today's machines.  In addition, businessman Herbert L. Hildreth of Maine built a taffy empire in the early 1900's. One of his employees patented a four rod machine in 1901 but it wasn't granted until 1903. Hildreth was not thrilled about this and ended up purchasing the patent for $75,000.

Due to all these machines being patented at about the same time, people began filing lawsuits and the Taffy patent wars went all the way up to the Supreme Court.  The court declared that the two rods would move the candy somewhat but they could not properly stretch the material.  To properly pull the material, the machines needed three rods.

Currently, there are two taffy pulling machines in use today.  One has three rods, the other has four rods but they produce the exact same stretch factor of 1 + sqrt 2 or the silver ratio.  Using this information and information from various patents, Thiffeault built a 6 rod machine based on a four rod which stretches the taffy about twice as much as current machines do.

I found this quite interesting.  I hope you enjoyed it too.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

Each gum drop has a volume of 1.5cm^3.  How much space do you need for 32 gum drops.

Warm-up

The average weight of a gum drop is 3.3 grams.  If there are 29 grams per ounce, how many gum drops are in a pound?

Ways to Help Students Learn How To Solve Multi-Step Equations.

 I've spent the last few days working on finding ways to help my students learn to how to solve multi-step equations.  I have developed several methods but I'm going crazy trying to find the online place I can do one of these activities.  After a bit of research it is the visual ranking tool by Intel but I don't know if it can still around

1. I create worksheets with no more than 6 problems on each side.  For each problem, I fill in the steps using the operation step by step so they fill in the information to show all the work.  For the first few sheets, I do this for all the problems.  The next set of sheets, I only do it for the problems on the front and have the exact type of problem on the back but the students have to do them on their own.  I finally provide worksheets with just problems to solve showing all their work.

2.  Another activity I like to use once they have a better idea of the steps is to write out the steps used to solve equations.  I cut the steps into strips and place them into envelopes with the equation written on it.  The idea is that they arrange the strips in the correct order, tape the pieces together and write it down on the accompanying paper.  This is especially good for problems with grouping symbols and combining terms before they can start isolating the variable.  Since it requires more steps to arrange.

3.  I like creating a lot of terms on halves of 3 by 5 cards.  I write a bunch of positive and negative numbers such as -12, -7, -3, 3, 7, 12 and the same for terms such as x, 2x, 3x, 7x, -x, -2x, -3x, -7x.  I make sure there are matching pairs.  I create problems using the terms so students place the terms together for a problem such as 2x + 6 = 5x - 4.  They then use the opposite terms to make zeros and isolate the variable.  Eventually, they will get x = an answer.  Once the problem is solved, they can copy it down on their worksheet.

4.  Once students are good at solving problems, they can go on keynote or power point to create an animated presentation showing the steps by having things move around.  For something that requires less work, one can have students use flip grid or other video program to explain how they answered a program.

5.  Take the deck of terms from # 3 and have students deal them out like cards.  They take their hands and create as many equations as they can.  They take each equation and solve it until they have all the equations solved.  This step lets them practice their skills without doing a load of practice problems.

6.  Use the strips from activity #2.  Instead of having the strips for each problem in an envelope, add several problems together so students sort the strips into individual problems arranged in the correct order so each problem is solved.  This takes their solving skills a step further because they have to identify the steps for each problem.

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

Odd Units of Measurement.

 Although it took a while for certain measurements to become standard and even some of the standard ones such as a stone can be a bit unusual.  A stone is 14 pounds and is the unit they used to weigh people in Australia but there are some other, really unusual units of measurement.  Most of the units I'm sharing are so unusual, I've never heard of them but they are still interesting.

1.  The Beard-Second is a unit of distance defined as the amount an average physicist's beard grows in one second.  This is about 5 nanometers and this particular unit got its inspiration from the light year.

2.  The Smoot is also a unit of distance.  It is defined as five feet and seven inches the same height as one Oliver R Smoot who in 1958 was a pledge at MIT.  His fraternity pledge was he be used as the base measure of the Harvard Bridge.  It was determined the Harvard Bridge is 364.4 Smoots tall give or take an ear.

3.  After investigating around 250 megaliths in Scotland and England, a scientist decided they were built using a standardized measurement. It was concluded the megalithic yard is equal to 0.9074 yards or 0.8297 meters.

4.  There is the Bloit, another distance measurement.  This distance found in the Zork games is used in the Great Underground Empire.  A Bloit is stated as the distance the King's favorite pet can run in one hour, or about 2/3 of a mile.

5. It is said they had a Pyramid inch back in Ancient Egypt which equaled one - twenty fifth (1/25) of a "sacred" cubit or about 1.00106 inches.

6.  Another unit of distance called the Sheppey or how close you can be for sheep to remain picturesque or about 7/8 mile.

7.  A Slug is an English unit of mass defined as mass that accelerates by 1 ft/sec^2 when a force of one pound force is exerted on it.

8.  The Barn - Megaparsec is a measure of volume or about 2/3 teaspoon.  The unit combines a barn which is an extremely small measurement used in nuclear physics and the megaparsec used to measure distance between galaxies.

9.  A Shake comes from "two shakes of a lamb's tail" or about 10 nanoseconds.

10. There is a jiffy coming from the computer field.  One jiffy is the length of one tick of the system timer interrupt or about 0.01 second.

11. Hobo Power is a unit coined to describe how bad something smells.  Hobo power ranges from 0 to 100 with 100 meaning the smell is so bad, you'd suffocate.

12. The Dol is used to describe the amount of pain.  It came about in the 1940's and 1950's and is based on "the justifiable difference in pain."

13. The Big Mac Index first coined by economists to describe purchasing power described in terms of the cost of a Big Mac.

14. Finally, the Nibble again comes from the computing field.  It is equal to 4 bits or half an 8 bit byte which fits one hexadecimal digit.

There are more but I thought these were cut and fun.  Let me know what you think, I'd love to hear.  Have a great day.

How Games Can Help Struggling Students.

 It is well known that giving students tons of problems to complete as a way of teaching them a skill is not always successful.  There are other ways to have students practice skills without doing lots of problems.  One very popular way is through the use of games.

First of all, math games offer a structured situation with specific rules, a final goal, and obstacles.  These things mean a student has to choose valid strategies, perform problem solving on the fly, and make decisions with immediate feedback.

Most students enjoy playing games even if they are math based.  In addition, games often reduce math anxiety and stress.  When students play they are less self conscious and are more likely to try. Furthermore, they don't mind making mistakes because they don't see it in the same context as doing worksheets and they don't undergo the same stress as when they complete worksheets.

When students play math games, they converse with each other, communicating important information while helping each other.  It also provides an opportunity to observe students while they play providing a great chance to assess informally.  Furthermore, game playing also provides automatic review with immediate feedback to students so they learn the material better or gain an understanding they didn't have before.

Another thing about games and struggling students is that games support different learning styles while providing an inbuilt differentiation while increasing student engagement and motivation.  There are some studies out there indicating games provide an effective way for students to transfer skills from short term to long term memory so they develop a deeper understanding.

The thing is that many students who have struggled with math for many years often will object to any game proposed or they will just throw in an answer and hope for the best so the game has to be appropriate for their ability and interest.  I have some students who hate any form of Jeopardy because they don't want to try but prefer Kahoot because they can guess and have a chance of being right.

On the other hand, the use of either an online game or app designed to focus on a specific skill such as solving one or two step equations, develop knowledge of the coordinate plane, can really help students. It's a matter of finding a game, your students are willing to play.  Let me know what you think, I'd love to hear.  Have a great day.

7 Math Oddities

Today, I'm taking a look at math oddities such as the Fibonacci sequence resembles a snail shell the way it spirals around because most students see only the standard math we teach but we don't always look at math oddities as much.

1. Automorphic numbers are numbers whose square number has the same ending digits as the original number.  For instance 25^2 give 625 or 76^2 is 5776.  It turns out that 25 and 76 are the only two automorphic numbers found in the first 100 numbers.  One other interesting thing is that 625^2  gives 390,625 so you have a number squared and then squared again.

2. Strobogrammatic numbers are numbers that can be rotated 180 degrees and still appear the same.  The only three strobogrammatic numbers are 0, 1 and 8.  They look the same up right or upside down.  The digits 6  when rotated 180 degrees becomes 9 so 69 when rotated turns into 96 but 101 can be rotated 180 degrees and still remain the same.

3. Paladromic numbers are numbers that read the same both front ways and backwards.  These numbers include all the single digits 0 to 9, 11, 22, 33, 44, 55, 66,  77, 88, 99, 101, 111, 121, 131, 141, 151, 161, 171, 181, 191, 202, 212, 222, 232, etc.  There are also the ones like 1, 11, 111, 1111, 11111, etc.

4.  Jeff Tucker created this formula - 1/2 [mod([y/17]2^(-17[x]-mod([y],17)),2)] which produces the same formula in graphed form.  Its called a self referential formula.

5.  If you multiply 12345679 by 8 you get 98765432 but if you multiply 12345679 by 9 your result is 111111111.

6. Perfect numbers are numbers whose factors add up to the original number or largest factor.  For instance, the factors of 6 are 1,2,3,6 and 6 = 1 + 2 + 3 = 6 or 28 = 1 + 2 + 4 + 7 + 14.

7. Figurate numbers are numbers that look like the shape they are named after.  There are three types.
Perfect squares are numbers that can be made into squares like 4, 9, 16.  The second is triangular numbers such as 3, 6, 10, 15, 27 which when built into shapes look like triangles.  Then there are Center Hexagonal numbers are numbers that when arranged around a center make a hexagonal shape such 1, 7, 19 and 37.

There you are, seven mathematical oddities which are fun and entertaining.  Let me know what you think, I'd love to hear.

Pros And Cons Of Calculators

 People have strong opinions on the question of using calculators in the classroom. I used to be one of those who hated them except for the upper level maths but since I started teaching a class with students who have failed through their middle school career, I've adjusted my opinion.

There are both pros and cons to allowing calculators into the classroom.  I'm going to take time to look at both sides off the question today.


PROS

1.  As long as students are writing down the steps and using the calculators only to make sure their arithmetic is correct, they can focus on the process and concepts rather than getting stuck on the arithmetic.

2. Just about every mobile device have calculators built in as an app or they can be installed.  Few people have a dedicated calculator.  With this technology, its important for students to become used to using them.

3. Calculators allow students to be accurate in their calculations so their answers are correct but they still have to make sure they've entered the information correctly.

4. Calculators can be used to promote higher order thinking so students can problem solve more complex problems.

5.  Calculators can help students recognize and extend numeric, algebraic, and geometric patterns in math.

6.  The use of calculators can help students develop a positive view of mathematics.

7.  The use of calculators can be used to prepare high school students for their use in college.

CONS

1.  Using the calculator to do all the math without writing out the steps, eliminates the possibility of knowing the steps so they don't understand what is going on mathematically.

2.  Students develop the idea that the answer displayed by the calculator is always correct and they do not double check to make sure they entered it correctly.

3. Students do not develop their math skills nor do they keep their math skills.  Continual use of the calculator can make students too reliant on the calculator rather than keeping their skills up.

4. On the more advanced calculators, students can store notes and examples on the calculator which they can use on tests.  If they are not allowed additional notes, this could be construed as cheating.

5.  The cost of a physical calculator can range from $10 for a cheap one to over $100 each for the advanced graphing calculators.

6.  Students may think they do not need to use paper and pencil methods if they use a calculator but the physical representations can help them plan how they will solve problems.

7.  They end up with everything in decimal format rather than in fractions when the original problems are in fraction form.

Since most of the students I have in my 9th grade Algebra I class have been unsuccessful in previous math classes, the use of calculators are helping them learn to do the more complex math problems.  I require them to show all their work in terms of writing down the steps but the calculator helps them make sure the simple arithmetic work is correct.  I also allow my pre-calculus students to use calculators because they will be heading for college and need to know how to use these calculators for sequences, etc.

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

Warm-up

One room in this Bed and Breakfast goes for $125 per night including breakfast.  The local and state taxes add 15%per night and there is an energy tax of 2%per night.  How much does the room actually cost per night?

Warm-up

You own 6 rooms standing over the water.  Each place is rented for $535 per night plus 12% per night in taxes.  If you rent all 6 rooms every night for 22 nights, how much tax will you need to collect?

Sales Graphs

I was reading up on the history of Kleenex tissues.  The product started out as a tissue to clean cold creme off women's faces but their husbands kept using them to blow the nose.  This lead the company to advertise it as a disposable handkerchief.

I wanted to know about their yearly sales.  I found one graph which compared the number of boxes used each month between 2011 and 2019.  They divided it up as none, one box, two, three, four, five, six or seven, and eight and more.  According to the graph, most people use two boxes in a 30 day period.  The same place also offers a graph comparing the sales of boxes of tissues by company.

The government published a graph and a table showing the amount of paper products generated, recycled, composted, put in the landfill, and combusted with energy recovery for the years 1960 to 2017.  The way this graph is done, it is easy to see how the amount of paper products ending up in the landfill has decreased while the amount recycled has increased over the same period of time.

The government also has a table and graph focused on nondurable products which cover both disposable paper and plastic products for the years 1960 to 2015.  Although this includes paper products, the graphs are slightly different which offers a great opportunity to implement a compare and contrast activity using both graphs.

This page also breaks down the nondurable graphs into paper and paper products as noted in the other graph, newspapers and mechanical papers, plastic plates and cups, trash bags, disposable diapers, clothing and footwear, towels, sheets, and pillowcases, etc.  This means nondurable goods is divided down into seven subgroups, each with its own graph.  This again provides an opportunity to compare and contrast various subdivisions.  It opens up the possibility to have students calculate what percent of the whole section each subdivision is and the information can be published in a circle graph or pi chart.

On the other hand, to get a historical perspective, this site has the data for the amount of newsprint, newspaper consumption from 1919 to 1939 and possibly later.  It can be done monthly, semi annually, annually, so the ups and down are easily found.

This site has information on the production of paper products from Europe over the 20 years from 1991 to 2011.  One graph looks at the over all paper and pulp industry while another looks at average size of paper and pulp mills.  Another focuses on domestic consumption while a fourth looks at where pulp was imported from.  There are several graphs focusing on the paper mills themselves in Europe and exporting of products from Europe to the rest of the world.

A final site looks at the international cellulose or pulp market with quite a few graphs in three dimensions and stacked.  Some of the graphs look at the softwood and hardwood Kraft market pulp, changes in the paper and pulp market, paper and pulp output, breakdown of various pulps, and per capita paper and board consumption.

So many different ways to look at paper, paper products, and associated products.  All these sites provide real life data, not data that look as if it had been constructed for a textbook.  In many ways, it is good to let students read graphs based on real life situations because these are not always as nice as those found in a textbook.

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

Practical Applications of Mathematics In Real Life.

 There are at least five areas in our lives which demand a knowledge of math or people end up in trouble.  We go over many of the formulas in class but many of the students I teach do not relate to them because they do not have experience with these things.

The first area is financial management which covers a wide variety of things.  In the old days, it would have included managing and reconciling a checkbook but few people use those anymore.

What has become more important is a knowledge of how credit cards work since those are used so much more now a days. It's important to know how they calculate interest, how the minimum payment does not do much for paying off the debt, and how a slight variation in credit rate can cause you to pay more.  There are also costs associated with renting vs buying a house or an apartment, cars, or even furniture.

Add into that the different types of insurance - car, health, life, house, renters, and more.  In addition, there is retirement to think about as in Social Security, investing in 401K's, IRA's, and Roth IRA's. Or they can think about the stock market, investing, bonds, which are something many people think about.  If nothing else, there is the importance of students learning more about balancing budgets which apply both in their personal life and in a business.

A second area deals with home improvement that uses quite a bit of math.  If you decide to redo the kitchen, you have to know how to measure for counters, cabinets, calculate the amount of flooring, back splash, sinks, refrigerators, stoves, ovens, paint, and odds and ends for a new looking kitchen.  If there is a room, math is needed to calculate the flooring, wall coverings, and ceilings.  Not all flooring is sold in the same units so you have to know how to convert so you can compare carpet vs linoleum, vs wood.

Another area math comes in handy is when looking at the cost of appliances and the amount of electricity they use.  The appliances which save the most energy may appear to cost more but within a short time have payed off the difference in savings.  Furthermore, its important students learn to read bills to see how much electricity they use every month.  They can take the information, create charts to see what time of the year they use more electricity.  Is it in the summer when temperatures are warmer or is it in the winter.  In Alaska, we usually use more electricity in the winter due to the cold temperatures.

If students look carefully they can see how math is used outside for landscaping.  There is the cost of the plants to place in the yard, supplements to keep the plants healthy, water used to keep them alive, and the cost of keeping the yard up versus hiring someone.  What about planting a garden?  This involves the cost of setting up raised beds, soil, plants, fertilizer, tools, etc to get it up and running.

What about looking at the cost of installing a pool or a spa and checking all the costs associated with their upkeep.  I have friends who had an above ground pool they could only use in the summer because it got a bit too cold in the winter.  She had to buy an alternative to chlorine to keep the water clear while balancing the pH, etc.  Maintaining a pool can be quite expensive especially if you fill the pool at the beginning of the season.  Many people forget about the cost of filling a regular swimming pool, heating the water, chemicals, etc.

These are just a few places math is used in everyday life and are examples students can connect to.  Let me know what you think, I'd like to hear.  Have a great day.

Journals and Quick Writing In Math

Getting many students to write can be quite difficult.  I spend time asking my students to justify their answers so they can work on improving their reading and writing skills.  One way to increase reading and writing is via the use of journals or learning logs.

Journals or learning logs help students clarify their thinking in much the same way as talking about it with others.  It also helps students improve their communications by giving them an opportunity to organize their thoughts and utilize mathematical notations within their communications.  Furthermore, the journals offer an opportunity to assess their understanding.

Journaling allows students to record their discoveries, reflect on new concepts and ideas, and provides a place to record any questions or concerns they have about their learning.  Other possibilities for journal questions include things like asking which part of the homework was challenging, or what two things would you like to know more about, or summarize what was learned today.

Another way to increase writing opportunities in class is to implement quick writes.  A quick writing is defined as a something short written in response to a specific prompt.  During a quick write, students are expected to write as much as they can, as fast as they can, as well as they can.  Quick writes usually last from one to five minutes.

Some possible quick write questions include things like "What do you know about this concept?" before you begin a new topic so you know what they already know. When you know what they already know, you can adjust your lesson to begin where they are at, rather than deciding where they are.   This technique can also be used to bridge old knowledge to new knowledge.  You might also ask if they've solved a problem like this before and if they have, how did they do it.  Quick writes can easily be used at the beginning of a lesson, the middle or end of a lesson depending.

Since quick writes are quick, it makes a students thinking more transparent, and visible.  It also helps a teacher identify misconceptions students have, and it helps students keep an eye on their learning.  It is also a place where they can express hat they liked or disliked about the lesson, or monitor their understanding of the concept taught.

One nice thing about quick writes is they don't have to be grammatically correct, rewritten, or graded on language usage.  The writing just flows from the mind, and is informal so it is less stressful for the student.  Furthermore, quick writes do not have to be based on a statement or question.  It could be based on a picture or a photo. Show a picture, then ask students to do a quick write on what they see, or create a word problem based on what they see.

It doesn't take much to begin using quick writes in class. Let me know what you think, I'd love to hear.  Have a great day.

Histograms

I've often wondered where one finds histograms in real life.  We find them in the textbooks but I've never really used them in real life so I needed to find something I could share with the students.

When it comes to creating histograms, this article is a great one for introducing histograms, types of histograms, example of taking data and turning it into a histogram, creating histograms using excel, and information on interpreting histograms.  There are five types of histograms students are likely to run across in real life.

The first is a normal distribution which is lower on both ends, rising to its maximum in the middle so both sides are roughly the same.  Then there is the binomial distribution has two equal peaks with a dip in the middle.  A histogram could be a right skewed distribution which has the peak on the x = 0 and it decreases from there while the left skewed distribution begins at its lowest point at x = 0, and increases to its peak on the right side.  A right skewed distribution is a positive distribution while the left skewed distribution is negatively distributed.  The final distribution is a random distribution which has a variety of peaks and valleys running through the whole graph.

Scholastic has a lesson designed to help students understand that a histogram is a type of bar graph showing frequency, connecting the definition of mean to histograms, and they learn how actuaries use the histograms.  The activity has students graph claims paid out from a number of Hurricanes and Tropical Storms.  Once the information is graphed students then interpret the information in order to answer certain questions.

Shoder has a nice lesson utilizing interactive activities to help students learn more about interpreting histograms, intervals, scales, and sample problems.  The lesson plan includes everything needed to complete the lesson along with full instructions.

I also found a lovely lesson using a histogram from the New York Times online edition as the basis of a "What do you notice?" and "What do you see?" activity.  The histogram looks at the meals sold by the Chipolte restaurants in terms of calories and dishes.

The distribution shows all the meals sold by the restaurant to the public.  The most of the dishes sold are in the 1000 calorie range with a full day of sodium included.  The dish is a meat burrito with cheese, salsa, guacamole, sour cream, along with rice and beans.  The article goes on to present two more histograms, one looking at the amount of sodium and the other grams of saturated fats.  At the end, it gives sample meals and the total calories.

The lesson begins by having students look at the histogram and write down their answers to "What do you notice?" and "What do you wonder?" but its done in pairs.  One student brainstorms to the first question while the other records their thoughts and then the students switch.  They repeat the process for the second question. They then share their thoughts before developing inquiring questions to investigate in deeper detail.

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

Debating In The Math Classroom

When I heard about incorporating debate into the math classroom, I thought "It can't be done."  Debating is something you do as a class or you do it when running for a political position but in the math classroom?

Debate is something that can be incorporated in the math classroom. It also meets MP3 which encourages students to construct viable arguments and critique the reasoning of others.

Debating does take preparation prior to the actual day and it can also take up most or all of a period to do.  It allows students to create points in support of  or against a mathematical proposition.  Debating gives students a chance to develop and synthesize ideas while creating arguments to support their side of the proposition.  The process means they have to utilize ideas they understand and can explain.

In addition, to prepare students have to collaborate, research, and combine ideas into coherent arguments. Debate can be used to review material in preparation for a test.  Imagine asking students to debate substitution versus elimination as being the best method to use to solve systems of linear equations.

Students do not have start with a full blown debate with the two minutes to give their points and 30 seconds for rebuttal.  It is possible to ease students into debating using "Which one does not belong?" to a variety of others.

There are several formats that can be used in the classroom.  Some are quite easy to use while others require a bit more work.  For instance, there is the four corner debate method.  The teacher makes four signs with strongly agree, agree, disagree, or strongly disagree written on them and placed in the four corners of the room.  Students are divided into groups before the teacher reads a statement.  The groups discuss the statement, prepare an argument, before going to stand in the corner with their level of agreement or disagreement.  Students then share their reasoning.

Tag-Team debate. The teacher creates two groups with no more than five students in each one.  The teacher provides a statement, the groups are given time to prepare and when the debate begins, each student is given one minute to present their part of the argument, then the other team gets their turn and then back to the first team gets to speak but the first speaker must tag the next speaker before they can go up to continue the argument.  The same applies to the other team and it goes back and forth until they have finished presenting their arguments.  If they cannot do it with the 5 people, they can start again with the first speaker.

Inner circle - outer circle debate.  The teacher divides the students into two equal groups.  One group sits in a circle facing outward, while the other group forms a circle around them facing inward towards them.  The teacher reads the statement to be debated.  The inner circle is given 10 to 15 minutes to discuss the topic while the outer circle listens and write downs arguments based on the inner circle discussion.  The outer circle cannot talk during this process.  Then the outer circle gets to discuss the topic, their arguments for 10 to 15 minutes , while the inner circle listens and takes notes.
At the end, both groups share their arguments and counter arguments like a debate.  Once this is done, they are done.

The fishbowl debate occurs when the teacher appoints several students to sit in a semi circle at the front of the room.  The teacher reads a statement or poses a question and the students at the front discuss it but students from the audience may pose questions to the students at the front but otherwise may not speak.  Students may switch out from the front with someone in the audience and the discussion continues.  This works best for prior knowledge.

The ball debate.  Divide the students into two groups, then rearrange the desks so there are equal numbers on each side of the room.  The teacher poses the question or the statement and then have the students sit on the desks based on their position to the question or statement.  The for group is on one side and the against on the other side.  After giving them a few minutes to think, the teacher throws a ball to one student on one side.  Only the student with the ball can speak.  When they are done, they throw the ball to the other side so a student there and speak.  They go back and forth till the teacher determines they are done.

Have fun incorporating debating in the classroom.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

You rented a yacht at $9714.30 per day for 7 days. At the end you had to pay an additional $23,800 for taxes, etc.  How much did you owe?

Warm-up

You rented a super fancy sports car at $1487 per day.  There were $380 of insurance, etc and the total you spent is $13,383.  How many days did you rent it for.

Strategies to Support Reading Part II

 I began this column yesterday because we've been told that we have to incorporate ways of teaching reading in all courses due to the low reading scores.  I've never really been taught reading strategies so I'm sharing some I've found.

8.  There is the use of a code students can use while reading the word problem or reading the text.  X is placed next to a key point of information.  ! means new information and ? indicates confusion about the information.  These are then used as a basis for discussion.

9.  Another strategy they teach readers is for them to either visualize or draw the picture they see in their mind.  The same can be said for word problems.  Students can either visualize or draw a picture of the information contained in a word problem.  They might draw an engine and 8 cars with the distance between the cars because that makes it easier to figure out the problem.

10.  After reading the word problem, or the text, students can ask themselves some questions about the material.  They can make connections using Text to Self by asking what math experience was similar to the one they read about, Text to Text by asking how it connects to previous work, and Text to world asking how is this used by people?

11. Students can also reflect after reading by asking themselves what strategy went well, ask what they can do differently next time, and what did they learn from other students.

12. Students are commonly taught to restate or summarize what they read in their own words but in math that takes a slightly different form.  It comes through by asking a student to explain what they did, and what they learned.  Then they need to generalize what they learned by thinking about how what they learned can be applied to other problems.

13. There are also strategies students can be taught to apply to vocabulary.  For instance, teach students to see that word chunks have meaning, and include the origins of words, and their histories.  Add clues to help students understand the definition within the context of the math situation or problem.  Always provide examples to help students.

14.  Use analogies to help students make connections and clarify relationships.  Have students write them, use them, practice them.  An example would be " Numerator is to top as denominator is to ________" or "Circle is to cylinder as _________ is to cube."

15.  For students who like to move around, try to find definitions that are active and allow them to move around.  Don't be afraid to look at creating songs, raps, cheers or finding them online because students often remember songs, raps, and cheers.

Another day, I'll share more strategies for writing.  Let me know what you think, I'd love to hear.  Have a great day.

Strategies To Support Reading Part One.

 Yesterday at our staff meeting, we got the results of the scores from last spring's state wide testing.  What surprised me is that twice as many students were in the proficient range in math as they were in language arts.  So the principle told us we need to support our students in developing better language skills by working on it in our subject.

My first thought came out as "I teach math, how to I support language arts in Math class.  I know about math texts being more concept dense than most textbooks.  I also know Math uses three levels of language but what about actual reading strategies?  How can we use the same strategies in our math classes that are used by reading teachers?  That was my big question so I found some suggestions.

1.  We can still identify characters and setting in word problems.  We can also activate the schema or the the topic of the action.  For instance if the word problem talks about two people going to a movie and buying tickets for the latest Lion King Movie, the characters are the two people, the setting is a movie theater and the schema is the change from buying two movie tickets.

2.  Reading teachers also have students identify the main idea, implied main idea or inference, and supporting details.  In math, the main idea would be the skill or concept needed to solve it. Students need the details or facts in order to make an inference.  The details are often the steps in a process needed to solve the problem or if its a word problem, the details are the pieces of information needed to solve the problem.

3. Often times, word problems require students to infer things from the information given in order to actually solve the problem.  For instance, Matilda and her three friends went to Hanover's Pizza to eat.  They all purchased the special all in one meal for $4.50 each, how much did they spend?  Many students see the number three so they so $4.50 times 3 for $13.50 rather than inferring Matilda is one of the people eating so it will be 4 x $4.50 or $17.50.

4. To help students improve their reading of word problems, work on having them reread the problem multiple times.  The first time they read, it is for the introduction, sort of like reading a travel guide to learn about a new place.  The second time is looking for specific information such as what explicit information has the problem provided.  The third time is to determine what it is they have to find.  The fourth time would be to see if there is anything they have to do before they can actually answer the question.

5.  It also helps to teach students to think aloud or talk to themselves as they read.  There is nothing wrong with talking to ones self to extract important information.

6. When reading the textbook it is good to teach students to make predictions based on the clues in the titles and subtitles and diagrams found in the book.  Once they've made predictions, one should then read the material and at the end, they can discuss how accurate their predictions were.

7.  Another strategy is to just look over the pages in the section to "see" what is there.  Another strategy is to make a reading guide with 5 to 7 statements covering key concepts of the material.  Students decide if each statement is true or false or if they agree or disagree with each statement.  They can look at each others answers, talk about them, prior to reading the material.  When they read the material, they have to prove or disprove the statements based on what they find in the section.

Tomorrow, I'll continue this topic with more reading strategies that can be used in reading.  Let me know what you think, I'd love to hear.  Have a great day.

The Show Your Work Battle.

Write now, I am fighting the age old battle of getting lower performing students to show their work.  I have a few students who refuse to show their work because they know the answer.  All the work I've given them so far have included the answers so they can check their work as they go.  When you have the answers, its easy to figure out the solution.

I've had them work a few problems without the answers available to check and they run into problems. They don't buy the "Its easier to find a mistake if everything is written out" or "if the problems are more complex it is not as easy to do in your head so if you work on learning steps now, when you need them, you'll know them".  I also fear they want to wait till they get home to run the problems through one of those apps that shows them the steps and the answers.

So I have two choices here, let them just put the answers down without showing their work and count it totally wrong or continue fighting the battle.  Honestly, I'm not ready to go that way yet, because the group I work with have struggled with math and I'm trying to help them build a better foundation.  So I've come up with a few ideas that will keep them "showing their work" while learning the steps so they are prepared for more complex problems.  These activities are for two step equations and higher.

1.  Divide the students up into groups of four,  Each person in the group is assigned a number from one to four and each is given a problem.  The first person writes the problem on the board.  The second person performs the next step, the third person will write the next step and the fourth person writes the answer.

2x + 1 = 7
     -1.     -1
 2x = 6 
 2.      2
   x =3

When they've finished the problem, they can all write it down.  For the next problem, The second person begins by writing the problem, the third person does the first step, the fourth person completes the second step and the first person writes the answer.  They continue with the third person beginning the a problem, and cycles through.  This way each person has a chance to do every step of the process over four problems.  At the end they will have four problems completed and they can begin the process again for another four problems.

2.  Set up slides so the first person writes the equation, the slide is duplicated so the second person adds the second line, the slide is duplicated again with the two steps so the third person adds the third line, the slide is duplicated again and the fourth person writes the answer.  At the end the slides are connected with transitions to create a video and they can add a voice over explaining each step.

3. If a student thinks they know the answer, they can verbally explain their thinking on how they came up with the answer while writing down the answer.  I think if they can explain how they got their answer, that will work just as well as showing all the steps.


These are just three ideas I've come up with to get students to learn to show work because once we add in the distributive property, like terms on both sides , throwing in some absolute value and a few other things, it is not as easy to do it all in their heads. By learning to show it with the simpler problems, they know what they should be doing to solve the problems.

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

Mathematical Based Art.

 When one comments about mathematically based art, most people automatically think about tessellation's.  It's easy to have students cut a base pattern piece out to use repeatedly to create art but that is not the only thing we can do that is mathematically based.

This site has a bunch of different math based art projects that students will enjoy and still get a dose of math in them.  The first activity they discuss is the "Curves of Pursuit" which is based on the three bugs problem.  Basically you have three bugs traveling at the same speed and if you follow their paths, you end up with some wonderful curves.

This activity comes with a presentation to show students example of this art work before explaining how to construct these beautiful pieces of art work.  Furthermore, there are templets to get the basic shapes and examples of what the finished product might look like.

Another activity suggested is to have students create Celtic Knot work.  This is a bit more complex than the previous exercise.  It comes with a presentation giving information on both historical and modern examples.  The presentation also includes step by step directions to do basic Celtic Knot work.  They also have a handout showing how to create one step by step with illustrations.

They also have directions for creating impossible triangles, impossible rectangle, and an impossible three pronged object.  As with all their art, they include a great presentation showing various impossible objects and directions.  An impossible object is one that looks like it twists into itself.   Furthermore, there are activities designed to explore insect symmetry which has students use tracing paper to create the symmetry of the insects.

There is also an activity where students learn to create both mazes and labyrinths using square grid paper.  One on creating modern art in the style of Ellsworth Kelly.  Ellsworth Kelly is a modern American painter known for his hard edged paintings.  He was born in 1923 and died in 2015.  This uses fractions, decimals, and percentages to create the effect.

Then students could make cardidoids using a curve stitching technique which is the use of straight lines moving around a point to finally produce the shaped item.  This technique could be done with string or with pencils.  Students can also use shapes to create paper patchwork patterns similar to those found made in cloth.  This activity opens up an opportunity to develop rich mathematical conversation.

The last two possibilities for art activities are snowflakes made out of paper using reflective symmetry and Islamic Geometry using a four fold pattern.  Each and every activity has background information and handouts to make it easier to teach if you've never taught art before.  This site has enough activities to spread out over a year and is not restricted to only elementary school.

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

Word Problems Increases Mathematical Communications

It is only recently I realized that word problems serve a purpose other than driving students crazy.  I admit some of the problems are very hard to relate to when you live in the middle of a village of under 900 people with no traffic lights, no buses, a few stop signs, and no roads that go anywhere other than the dump or the river.

So I figured out a reason to work word problems other than they are part of life.  Word problems present the perfect opportunity to help students increase their mathematical communications.  Often times, student struggle to understand the written problem which makes it more difficult to turn the language into symbols in a mathematical equation.

There are several factors contributing to this problem.  Students often see there is only one type of word problem rather than understanding there are multiple types.  They are also generally taught specific methods to solve word problems which does not work if the word problem is not written to match one of those methods, and many students are not good at their number fluency.  In addition, its been found that many students are unable to write the answer to a word problem using correct mathematical terminology.

The process of solving word problems involves a lot of communication.They need to have a good grasp of mathematical language so they can properly communicate their thoughts, the method they used to solve the problem, and their final solution.  This all has to utilize proper syntax and grammar within the language of mathematics.

One of the best ways to increase mathematical communication is through the use of writing, because writing requires formal use of language.  Unfortunately, most teachers assume since students take English, they know how to write but with the specialized language in math, teachers need to take time to teach students how to write their way through the process.

Solving word problems involve two steps, the problem representation and the search for solutions.  The problem representation can be further subdivided into three parts, comprehension, extraction, and  construction of the equation or equations.  This boils down to understand the problem, figure out what they have to find and what information is needed, and creating the equation needed to solve it.

It is important for the student to represent the information in a way that is meaningful to them thus making the problem more accessible to them.  It doesn't matter whether they use diagrams, notes, or drawings to represent the information, because this step helps them organize their thoughts.  This step requires the student have a good basis in mathematical language.  It is strongly recommended students read the problem several times before they reword the information to show their understanding. The better they comprehend mathematical language, the easier it is for them to create the problem.

The second part, the search for solutions, involves solving the actual equation.  Once the answer is found, it is important for the student to identify if the answer is reasonable and double check their calculations.  Once the answer is verified, the student needs to write out the correct solution while using proper syntax and language to communicate what they found.

So as a math teacher it is important we take time to discuss writing our answers correctly using mathematical language in addition to giving students guided practice in learning to do that properly.  This involves more than just going over the steps used to solve the equations, it also involves language used in writing word problems and in explaining our answers.

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

Warm-up

 

How are they the same mathematically?  How are they different mathematically.

Warm-up

How are these the same? How are they different?

Ratio in Cooking.

When we talk about fractions, we always seem to talk about cooking because most recipes have measurements in fractions such as 1/2 cup, 1/4 cup or 1 1/2 cups but not always.  It is said one way for people to become great cooks is to use ratios rather than standard recipes.

Many types of food are set up so that only the main ingredients are counted in the ratio set up.  Smaller things like salt, pepper, etc are not counted.  For cooking the ratios are preset proportions of ingredients so the results are always consistent.

Furthermore, since the basic ratios make up the recipe it is still possible to adjust them to individualize them by adding a bit of spice, flavoring, etc.  There is even a cookbook dedicated to teaching people about every major ratio used in cooking.

Some of the standard ratios used cover everything from bread, to vinaigrette, to crepes, to cookies. The ratios are general but they hold consistent as long as the units are the same.  If you have a 3 to 1 ratio, it will be the same using cups - 3 cups to 1 cup, or 3 cans to 1 can, or 3 tablespoons to 1 tablespoon.  It all depends on how much you want to make.

 1. Vinaigrette or salad dressing uses a ratio of 3:1 or three parts oil to one part water.  Then you add any flavorings you want such as herbs, mustard, or even spices to individualize it.

2. Brines are 20:1 or 20 parts water to one part salt.  This brine can be used for meat, poultry, or pork.  It's when they add herbs, etc that it makes it more personalized.

3. Basic soup stock is 3:1 or 3 parts water to one part bone.  Throw the bone in the water, add your onion, carrots, etc and cook.  One word of advice, weigh the bone so you can get the correct amount of water.  3 ounces of water to one ounce of bone.

4. Pie Crust is a 3:2:1 or three parts flour to two parts fat to one part water to make the perfect pie crust.  Always make sure the fat is super cold so it blends properly with the flour.

5. Bread is 5:3 or five parts flour to three parts liquid with a bit of yeast or baking powder to make it rise.  They don't count the salt used for flavoring or the bit of sugar to feed the yeast.  It also doesn't count the herbs, spices, raisins, or nuts used to add character to the recipe.

6. Pasta uses a 3:2 ratio or three parts flour to two parts eggs so this ratio requires both the flour and eggs be weighed to get the proper ratio.

7. Crepes require a 1:1:0.5 ratio or one part egg, one part liquid and half a part of flour.  The type of flour can change but the ratios remain the same.

8 Pancakes use a 2:2:1:0.5 ratio or two parts flour, two parts liquid, one part egg and half a part fat.  So the ratio are a bit different than the one used in crepes.

9. Biscuits are a 3:1:2 ratio or three parts flour, one part fat, and two parts liquid. This will give you those wonderful flakey biscuits. Again, they do not count the salt or flavorings to personalize these.

These are just a few basic ratios for making everyday things.  A quick search of the internet will pull up additional ratios used for cooking.  This is a great place for using real life ratios.  Let me know what you think, I'd love to hear.  Have a great day.

Why Incorporate Movement Into the Classroom?

Since the school year started, I've been trying to incorporate more movement into the classroom because research indicates that students who move around have better focus, increased cognitive processing, and better memory retention than those who sit still all day long.

Furthermore, movement also increases the amount of blood flowing to the brain, which improves mental clarity.  It has been noticed that due to the high usage of mobile devices, student attention span has decreased. In addition schools are cutting back on recess and P.E. time are being scaled back due to budget cuts.

It is agreed by experts that students need at least 60 minutes of movement per day and by incorporating activities needing movement, this helps students get that.  The movement gives students a chance to work off stress.  Many students find learning stressful, especially if they do not have a good foundation. It can also make the class more interesting and enjoyable. By the time, students reach high school, its harder to find activities which will get them moving around.

I've read about using the four corners instead of a written multiple choice exercise.  Let students work a problem, then they choose the appropriately lettered corner to show their answer. My experience with this is if it is used with students who do not perform as well, they are likely to choose the corner the person they think does the best chooses.  They are unwilling to stand apart but in a classroom with students who are more sure of themselves it works.

I've used a scavenger hunt with students that goes well.  I take a several sheets of 8 1/2 by 11 inch paper and cut each into two pieces.  I fold each piece in half in preparation for the work.  On the outside, I place a letter in the lower right hand corner.  Then I write either problems or vocabulary on the outside, one per paper.  On the inside I write the answers but not the answer with the problem, another answer.  For instance, I might right x - 2 = 5 on the outside but on the inside the answer x = 2 is written.  The idea is to start at one paper, do the problem and go look for the answer. Once the answer is found, the student does the problem on the front, finds an answer until they've worked ever single problem.

I did it with vocabulary in Geometry and it went well.  There was a lot of discussion about meanings and words because I defined them using my own words so they'd get a slightly different definition.  It took students a bit longer than I expected but they stayed engaged and busy the whole time.  I noticed that if they were not sure of the word's definition, they'd race to their books or notes to check.

There are gallery walks.  I do mine a bit different.  I have students complete the assignment, then I assign one problem per student or per two students.  They write it out on a blank sheet of paper showing all their work before hanging it on the wall.  Once all the problems are hanging, I have students take their answers around and compare what they got with what is hung on the wall.  If they have questions, they can ask for the "Author" to come and answer the questions.  When everyone has checked their answers, I collect the work.

I do not have time to incorporate fancy movement such as jumping jacks but I do have time to incorporate movement associated with learning.  Let me know what you think, I'd love to hear.  Have a great day.

Lagging Homework

Homework is a hot topic with people believing one should give it and the other group believes it is a waste of time.  I'm not sure how I feel about the topic but I have students demanding homework be assigned yesterday.

I've been reading the book Hacking Mathematics: 10 problems that need hacking by Denis Sheeran.  In it, he mentions lagging homework suggested by Henri Picciotto.

The usual way of assigning homework is to give a bunch of problems from the section being taught right now.  Instead of doing this, it is suggested you hold off a week, move on to the next section before assigning homework.  It gives teachers time to introduce and teach the material while giving students a chance to fully learn the material rather than both trying to hurry through the material.

Furthermore, Henri suggests teachers wait another week to give a quiz and an additional week for students to turn in quiz corrections. In addition, lagging homework gives students time to really process the new material, and had time to make connections.  Lagging homework is one version of spaced practice which is heavily recommended.

Another reason lagged homework or spaced practice is better is because people tend not to listen as well when the material is constantly repeated as it is when it is taught with homework in the same block of time.  In addition, the lagging homework is much better when assigned towards the end of a topic or after its concluded.

One person suggests that the lagged homework assignment should not take more than 20 minutes and should be done from memory.  The important idea is not that the student has to finish every problem but they do need to make an effort because by trying to recall the information, they are going to make stronger connections with it.  This work is not corrected by the student is expected to go back and check it, making notes about misunderstandings, etc.

The thing about lagged homework is that most of us automatically think about assigning another set of problems from the book.  Instead, we could assign problems from an online place for students to work.  I'm thinking of one of the places that provides automatic feedback if they miss a problem so they learn more about what they had trouble with.  If they do not understand the explanation, they could write the problem down and ask the teacher after school the following day.

The bottom line is to have students practice recalling information for this spaced practice.  Let me know what you think, I'd love to hear.

Student Written Word Problems

As we know students often have difficulty solving word problems because problems can have irrelevant information, use mathematical terminology, have higher vocabulary, and the syntax can be complex.

When students write word problems, they are able to base it on their frame of reference and interests.  For students who live in the bush of Alaska, trying to answer train problems for two trains, one leaving from New York City and one from Charleston, South Carolina make absolutely no sense.  Partly because there is only one railroad line in the whole state of Alaska that runs between Anchorage and Fairbanks.  There is a train running from Anchorage to Fairbanks during the day that stops at Denali National Park, and it returns at night, making the same stop.  The other possibility is the coal train heading to Healy to supply the electric plant.

Up here, students will write their problems using snow machines, four wheelers (ATVs) or boats because that is what they are used to.l. They do not have high rises so they'll use things like the main store, the airport, or possibly a hotel if there is one in town. They can use the problems in the book as a guide but adjust it to their particular circumstance.

Another advantage to having students write their own word problems is they see a connection between their lives and the real world.  In addition, they have to use their mathematical vocabulary outside of standard equations, use their creativity to plan the problem and they have to know how to solve it so they can provide a solution.

Furthermore, they have to translate the mathematical equation into words rather than words into equations.  Most word problems in textbooks do not ask students to create a problem, they ask them to solve it so they learn to create equations without necessarily understanding the concept.  The other thing is that most of the steps they've learned to use for solving word problems do not help them when writing the problems so it gives them an opportunity to see how people who work applied math problems go through.

If you stop to think about it, most of the worlds math questions tend to be based on observations which are expressed in words before it can be turned into equations.  By having students writing their own word problems, they are learning to express certain situations in written language which is part of learning to communicate mathematical ideas.  In addition, it helps students gain an understanding of the key steps needed to solve word problems.

According to several studies I read, having students write their own word problems, helps increase student ability to answer word problems.  In addition, it is a good way to incorporate reading and writing into the math curriculum and it helps students improve their ability to communicate.  Let me know what you think, I'd love to hear.  Have a great day.

Happy Labor Day

Warm-up

If an Arctic Tern flies 24,000 miles every year and lives for 25 years, how many miles will it have flown over its lifetime?