Using Comic Strips In Math

 

Not long ago, I wrote about presenting word problems as comics to make it easier for students to understand and do.  Since then, I've been thinking about other ways one could use comics in the math classroom.  One reason I've been thinking about using comics is that they can be created by students or used in class as part of the instructional segment.

If students create comics, they have to learn to be concise when they are explaining anything.  This means they have to think about what they will explain, how to explain it, select vocabulary, and create the comic. It provides another way to "show their thinking" and their understanding of the process and concept.

Research shows that when the brain is able to process words and pictures together, it helps with better recall and transfer of learning.  What better way of putting words and pictures together than creating comic strips.   In addition, using comic strips require students use creativity when they design the comic strips, critical thinking as they determine what works or doesn't work, communication to share information with others, and collaboration if they work with others to make the comic strips.  

One can make comic strips either by hand or digitally depending on what resources you have available.  As far as topic go, students can explain how to do order of operations, multiplying or dividing fractions, multiplying binomials, or any other mathematical topic. They can also find real world examples they can share in comic form with others, share vocabulary, and so much more.

On the hand, comic strips can be integrated into the classroom as a teaching tool.  As far as things go, using comic strips as part of the instructional component can help students develop critical thinking skills because the comics require them to analyze and interpret multiple elements on one page from the layout, to the language, to visual and textual elements.

Furthermore, comic strips are unique in that they combine pictures and words to communicate a story or idea concisely.  In the process of presenting the material, comic strips involve a certain degree of creativity  to convey the material accurately and students see it as they read the material.  

There is a decent selection of material available. There is "The Cartoon Guide To Algebra" or other cartoon guides, "The Solution Squad" by Jim McClain, "The Manga Guide To Calculus" other math manga guides, Beast Academy books for ages 8 to 13 and a few others.  A simple check on the internet or at your favorite online store can show you others. 

The nice thing about having students create the comic strips is that they learn the math better because they are having to explain it to someone else.  It develop their understanding and vocabulary while doing something fun and it lets those who love to draw a chance to shine.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

Lake Ontario, New York received 95 inches of snow in a 24 hour period.  What was the average hourly snowfall?

Warm-up

 In 1997, Lake Ontario, New York received 40 inches of snow over a 12 hour period.  What was the average rate of snowfall per hour?

Solution To The Queen's Gambit.

I learned to play chess as a teenager only because my younger brother wanted to learn so he could join the chess club. I never got very good at playing it because no one bothered telling me that I needed to think several moves ahead so I constantly lost. 

There is a chess problem called the queen's gambit which says if you put 8 queens on a normal chessboard of eight squares by eight squares, how many different ways can these queens be arranged so they cannot attack each other.  Remember that a queen can move any number of squares in any direction as long as it's in a straight line.  It can move diagonally, horizontally, or vertically.   answer to this is 92 but what is it is extended to the N-queen version. 

The original n-queen problem was proposed back in 1848 in a German magazine when it appeared as the 8 queen problem. Someone found the answer within the next couple of years. Then in 1869, the question was again asked but using larger numbers and this question remained unanswered until recently. A post doctoral student worked out that there are approximately (0.143n)^n ways the queens can be arranged so that none of them can attack each other on an n by n chessboard, assuming n by n is a huge number.

The (0.143n)^n is not an exact answer but an equation.  The (0.143) represents the average level of uncertainty in the variables possible outcomes. This is multiplied by the value of n and raised to the power of the value of n. This works for larger boards such as n = 1,000,000.  If you multiply 1,000,000 by 0.143, it comes out to 143,000 and that is raised to a million which gives an answer with five million digits. 

This person was able to determine the equation because he understood how queens could be arranged on these huge chessboards. He determined whether they'd be distributed in the center or along the edges and applied the appropriate the appropriate well known mathematical techniques and algorithms to find this equation. Basically, it means that if you tell this person the size of the board and how the queens are arranged, can find the number of solutions for this situation.  In other words, its a type of optimization problem.

He looked at the spaces which had a greater probability of having a queen inhabit them.  He then broke it down sections and then determined a formula to produce a valid number of configurations and used a lower bound or lowest possible number of configurations. After he had that number, he went on to find the upper value or largest number of configurations and discovered the two bounds are quite close.  Thus the answer lies somewhere between the upper and lower bounds. 

This took about 5 years to complete and he got interested in it because he could apply breakthroughs in the field of combinatorics to this problem. The key to really finding this number was to understand that spaces were not equally weighted but some were more likely to have a queen.  This lead to a breakthrough in finding the solution. 

This was cool finding a real life problem that used upper and lower bounds to find the answer.  It was also great finding out that this problem is more of a combinatorics problem and that the answer is an equation.  I hope you find it as interesting as I did and it is one that could be shared with your students.  Let me know what you think, I'd love to hear.  Have a great day.

Ways To Make Worksheets More Engaging.

 

How many times have you googled interactive activities and gotten pages of nothing but worksheets.  Those worksheets filled with problems.  Every worksheet is expected to be completed.  Do the same search for worksheets + engaging or engaging worksheets and you'll get about the same answers.  Furthermore, if you pass out a worksheet, students moan and many of them are not actively engaged the way we want.  So what if with a little bit of work,  you can turn boring humdrum into something more exciting.

One way to do this is to pair up your students.  Each pair of students will get half of a worksheet.  One student has the even problems with the answers to the odd problems and the other student has the odd problems with the answers to the even problems. The student with the odd problems works the first problem while the other student provides coaching as needed until the first student arrives at the correct answer.  Then the other student does the second problem while the first student provides coaching.  They go back and forth until they have finished the assignment.

If you don't mind a slightly active classroom, you can cut the problems up into individual slips so each slip has one problem but before you start cutting, make sure the worksheets are not double sided.  Take each problem, crunch it up into a paper wad and put in a container.  Pass out two or three paper balls to each student but before they start the game, lay down the rules and expectations such as "Do not try to peg the person with it", or "throw it in a general area".  Once the rules and expectations are given, let students toss the balls, pick up ones that land near them and toss those for say 20 seconds.  At the end of the time, each student will pick up a certain number of balls to take back to their group. Then they open the balls and work the problems on whiteboards.  Once the problem is completed, they write the problem and answer on a paper to turn in.

One way to turn the worksheets into a more interesting collaborative activity is to divide the students into groups of 3 to 4 people. One person is designated as the "announcer".  The announcer reads the first question to the rest of the group who work the problem out on whiteboards, tablets, or paper but they keep their work secret from the others.  When students are done, the announcer asks to see the answers and students discuss any answers that differ from each other. Then the second student becomes the announcer and reads the second question.  Students repeat the process and once they've shown their answers and discussed the discrepancies, the next student takes charge and the process repeats.

For a variation on basketball, divide your students into teams of four.  Print off enough worksheets so each group has one worksheet worth of problems.  Cut the problems up and make sure each group has one set of problems.  One student in the group will select a problem and the whole group works on it together to get the answer.  Once the answer is found, a student checks with the teacher to see if it is correct.  If it is correct, the student gets to crumple up the problem and dunk it in a basketball hoop or in a trash can, or similar container. If it isn't then the student has to take it back to the group to rework it.  Once it's correct the student goes back to his group and another problem is selected.  Students continue until all the problems are done and first group who gets them all correct, wins.

These are just a few ideas to turn making worksheets more engaging and interactive.  Let me know what you think, I'd love to hear.  Have a great day.

Math For Presidents Day.

 

Today is Presidents day.  The day we remember those who were past presidents.  It's actually came about from combining Washington and Lincoln's birthday celebrations into one federal holiday.  While researching Valentines day math, I came across a very interesting math activity on Yummy Math.

Did you know that between 1789 and 2001, the presidential position only received raises five times?  I certainly didn't but this activity at Yummy Math looks at that.  The owners there created an interesting activity to go with the data.  Students are asked to graph the data and they are introduced to step graphs via this data. In addition, they are introduced to the consumer price index and how it is used.  They use the formula given to calculate what that salary is in today's dollars.  In addition, students are given questions to make conclusions based on the data.  This is an interesting topic.

On the other hand, this information leads to a wonderful activity about money.  " A dollar bill is .0043 inches thick, weighs .03 ounces and 6.14 inches long." With this one sentence, students have enough information to calculate how high a stack of one million dollar bills is in inches, feet, yards, and miles.  Students can also calculate how many ounces and pounds one million dollars weighs.  Finally, you can ask students how many inches, feet, yards, and miles one million dollar bills form when laid end to end.  This gives a nice perspective of what the height, weight, and distance of a million dollars since most of us will never see it.

If you have students who struggle with graphing, this site has two graphing activities which have students creating pictures of George Washington and Abraham Lincoln.  Both graphs are done in the first quadrant so there are no negative numbers which is why it is also a great activity for beginning or struggling students. In addition, there is a multiplication and a division worksheet available for younger students who need practice three digit times one digit numbers.

This report about an activity has students analyze data on what states the presidents come from. Students make a prediction about which state had the most men who became president, creates a tally sheet, and then they make a bar graph.  Unfortunately, the original webpage is no longer active so if you want the students to do this, you'll have to provide the info or have them research it. It is a nice activity for students to interpret data and turn it into visual representations.

Most of the "activities" I found that sounded really interesting have been removed and are no longer available.  The others tend to be worksheets that are just worksheets with a president's day banner on it.  These are the ones that were a bit different.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If one pound of sliced strawberries equals 2.7 cups, how many pounds do you need for a strawberry pie calling for 4 cups of sliced strawberries?

Warm-up

 

If you have 4.5 pounds of strawberries and each pound has 18 berries, how many berries do you have?

Motivating Students In Math

Math seems to be one of those subjects that students sometimes need a bit extra motivation to get the work done.  It can be one of several reasons they don't want to finish an assignment but as the teacher, we need to help them get motivated.  

To begin with, motivation can be either extrinsic or intrinsic. Extrinsic refers to rewards given outside of the student's control.  Examples of extrinsic motivation include rewards such as pizza parties, movie nights, homework passes, or something similar.   

On the other hand, intrinsic motivation comes from within.  Examples of intrinsic motivation are the student wants to get A's only, needs to be the top of the class, or have a need to impress others.   For our students one of the reasons we use to motivate students intrinsically is to remind them they need the math class to graduate but it doesn't make much of an impression if they don't see a reason to graduate.  Consequently, we need different reasons and ways to motivate our students.

I'm always reading articles, looking to make myself a better teacher but on this topic, I ran into a list of suggestions that would never work with my students since it has the teacher showing students their missing pieces of knowledge, finding patterns, or setting up situations to encourage them to increase their knowledge.  I've gone with the suggestions I could implement with my own students.Not all the suggestions I'm presenting here will work in every case.  Some might be adjusted to fit your situation better and will depend on the students you have. 

Not all the suggestions I'm presenting here will work in every case.  Some might be adjusted to fit your situation better and will depend on the students you have.  One of the big ones is to establish relationships with your students so they know you are there when they need you and you are willing to answer questions.  Building a relationship with students is done both inside and outside of the classroom by greeting them everyday by name, asking about their day, weekend, etc, and even going to their games to cheer them on.  I do not like sports but I go so on Monday's I can make comments about that one play or the basket, or the steal, or the pin in wrestling.  

Think about cutting back or spreading out the lecture so you aren't requiring students to sit through it for 20 or 30 minutes.  It is much better to divide the material up while providing practice time between the lecture parts.  It is well known that today's students have a much shorter attention span than they had years ago.  Instead of lecturing, use task cards, question stacks, scavenger hunts, guided practice, pairs work, anything that is more hands on.  

Think about using math centers or stations.  That is something we typically associate with elementary school but in middle school or high school they can be used effectively. Each center might have different problems or a different set of problems they can work on, something using manipulative or drawings, or each center focuses on a specific skill.  An example of this might be center one has students identify the parts of a coordinate plane.  The second station might review how to plot points with practice and the third station might have students practice graphing lines using tables. All skills are associated with graphing but each focuses on a different part.

The final suggestion is to give students a choice.  For instance instead of assigning 24 practice problems, let them choose 18 out of 24 problems so they feel as if they have a chance to do the ones they know how to do.  For a certain topic, they might have the choice of a task, a online quiz, or creating a flip grid video explaining how to do it.

These are some easily implemented suggestions and they can be customized for your students. Furthermore, these can be adjusted according to if your school is under COVID measures or have loosened up a bit. Let me know what you think, I'd love to hear.  Have a great day.

Nerdle

 

Nerdle is a new web based math game that has captured the attention of the public.  It is similar to wordles but uses equations.  Instead of finding words, you look for the 8 character equation within the 6 guess limit.  The game is challenging but engaging and maybe it is something we can use for engaging our students.  

There are only a few rules for the game.  There are eight "letters" or spaces to use.  The "letter" represents a number from 0 to 9 or one of the operations, or an equal sign.  The "word is a mathematically correct equation with the equal sign. The number to the right of the equal sign is a number. 

It also assumes that you are following all order of operations so you need to multiply before adding.  The final rule is that order of the answer has to match exactly.  So if 15 + 20 = 35 is what the program is looking for, it won't accept 20 + 15 = 35 because it doesn't see it as correct. In addition, it uses color coding to let you know if your guess is correct.  If the "letter" is correct, it will turn green, if it is correct but in the wrong location, it will turn purple, and black if it is not there at all.  Every guess has to be mathematically accurate such as 25 = 3 * 7 + 4.

Nerdle opened on January 20th of this year.  It was created by a data scientist who came up with the idea during a conversation with his 14 year old daughter who wanted to know why wasn't there a wordles type game for math people. So he and his daughter worked out the rules and name. His son worked out all the possible equations that would work within the situation.  With a bit of help from colleagues and friends, they launched the site.   Since its release it has become wildly popular and used by teachers in their classrooms.  There is also instant nerdle but you only have one chance to get it correct.  

Each day, the site releases one new puzzle out of 17,723 possible combinations which is enough to last 48 years.   For those who get frustrated easily and don't have the patience to work through the whole process, there are sites that work it out and share the solution. They will tell you what one it is such as puzzle 26.  This is something students will discover quickly is let them work it out on line since many of them use "cheat" sites to help them through certain games.

If you have students who need a bit more of a challenge, let your students try the current puzzle to see how they do.  Go check it out, give it a try, let me know what you think, I'd love to hear.  Have a great day.

Valentines Day Math

 

Today is Valentines day and I like offering activities that use Valentine's Day as their focus.  It makes a nice change and allows students to have a bit of fun. I tend to go to Yummy Math for ideas because they always seem to have something.  

For instance, they have one that deals with those sweetheart candies.  Those heart shaped ones that come in pastel colors complete with messages like "I love you."  According to the activity, NECCO, the company that makes them, went bankrupt in 2018.  Within two years, another company bought the machinery, shipped it to Ohio, and began producing the candies again.  The activity has students answering questions about this candy before the company went bankrupt.  Some questions require students to figure out how much is made per month, or a per heart cost. They also ask students to draw conclusions.

Another activity has students reading and interpreting data off of two different infographics.  One infographic looks at where flowers are grown while the other looks at the cost of roses in several different International countries.  In addition, there are questions associated with each infographic that ask student to interpret data.  There are also questions asking students about how many hours must they work at a certain rate to buy a dozen roses in a different county.

Then there is the activity to help students work on estimation by asking students to estimate the number of food cans used in building a 3 dimensional heart.  Students are asked to look at a picture, estimate the number of cans used in the construction before watching a time lapse video showing how it was built. Students are asked to make more estimates after watching the video and they have to explain how they came up with their estimate.

For those who love to cook, there is an activity on creating a heart shaped raspberry flavored cake. They see how a square cake and a round cake are put together to form the heart.  Students are then asked to use information on cake batter to figure out how much cake batter is needed for various sized cakes. The amounts include fractions so they get some practice with real situations.

For chocolate lovers, one of the activities shows a heart shaped, bat shaped, and a turtle shaped chocolates on a grid and students are asked to determine which one is the largest in size.  Students are also asked to explain how they determined the greatest and least volume pieces.  It is a good exercise for students to explain their thinking.

There are several other exercises so teachers have a chance to select the one that works best for their students.  We know something might be rated as 8th grade but due to the pandemic, students may not be quite there.  In addition, the activities have students practicing different skills.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 If it takes 3.5 pounds of grapes to make one quart of juice and each pound of grapes cost $2.99, how much does it cost to buy enough grapes for one quart of juice?  5 quarts of juice?

Warm-up

 

If it takes 3.5 pounds of grapes to make one quart of juice, how many pounds of grapes will be needed to make 5 quarts of juice?

100% Fake Snow

In the past, most places hosting the Winter Olympics, snow was not an issue because it was a normal thing in the area or fairly close but China chose a location where snow is not a regular part of winter. If the area gets snow, they don't get more than about 7 inches which is not enough for all the events. Since they needed snow,  China had to make 185 metric liters of snow because they didn't have any. 

Although it is labeled fake, the headlines are referring to the fact they are using artificial snow rather than relying on the natural stuff.  There are cases in the past where artificial snow was used to supplement the supply but this is the first time, a country has had to make all the snow required for the events. The Chinese government hired a single company to provide snow for all four venues. 

Normally, when people create artificial snow, they use water under high pressure, compressed air, and a special nozzle to blow extremely small droplets of water that freeze in the cold air.  Unfortunately, pure water doesn't really freeze until the temperature gets low, usually around -40 degrees F. However, if the water is in tiny droplets, it can freeze at 32 degrees F but humans usually use something like sodium iodide to start the process of creating snow while nature relies upon a tiny ice crystal.

When humans use tiny droplets of water, the snow is actually a small spherical ball of ice that looks like snow to the naked eye but isn't.  This is why artificial snow feels so hard and icy. Most recreational skiers want a nice fresh powder but the competitive skiers want snow that will help them go faster, turn sharper, etc. 

Another issue with making snow for China, is that the outdoor temperatures have to be down at freezing and the area around Beijing has not gotten that cold in the last 30 years. On the other hand, the venues in the higher elevations will not have this problem. The company hired used a bunch of snow cannons, fan driven snow generators, and cooling towers to produce the snow. 

Due to the demands of needing to produce around 1.2 million cubic meters of snow over an area of 800,000 square meters, the demand for water is extremely high. It has been estimated the company is using around 49 million gallons of water which is the same amount needed to fill 3,600 average sized swimming pools or provide drinking water to around 100 million people for the day.  The cost is estimated to be over $60 million dollars just for the machines that has to be calculated as part of the expense of hosting the olympics. 

The use of artificial snow changes the way competitors react and sometimes it is more dangerous because the snow is too icy and too fast.  This is the way China met the challenge of no snow so it could host the demands of the Winter Olympics.  Let me know what you think, I'd love to hear. 

Comparing Methods Of Solving Problems

 

When we teach students the steps needed to solve certain types of problems, students end up memorizing the steps rather than taking time to understand the mathematical principles behind each step.  It's also hard to "tell" students those principles because it often goes in one ear and out the other but researchers at Vanderbilt University has found a way to help students with this. 

They discovered if you ask students to compare different ways of solving the same problem, it encourages higher level thinking while demonstrating a deeper understanding of the concepts.  It has been found that comparison is a more natural way of learning for humans. People use comparison so much in real life such as when they shop, they look to see which deal is a better one, or they compare routes from point A to point B.  

These researchers even took time to determine which problems are the best to use comparison with.  As mentioned earlier, one type of problem is to compare the different ways to solve problems. Another type of problem to compare are those problems which are confessable and finally, comparing correct with incorrect strategies. They discovered when students compared these three types of problems, they acquired greater conceptual knowledge and were more flexible when solving problems.

Furthermore, the researchers looked at the best time during the learning process to use this comparison technique. One thing they discovered was that students could compare methods if they understood at least one method.  After some practice, students showed they were able to compare the similarities and differences between two or more unknown methods but this needed more support from teachers.

When introducing comparison to students, teachers need to provide clear and visible examples to students while showing the solutions at the same time. In addition, students did better at figuring out the similarities and differences between mentors when teachers used well defined vocabulary, terms, gestures and visual clues.  Furthermore, teachers should provide questions to prompt clear explanation of key points and summarization of the major points brought up during the comparison. 

These same researchers are getting ready to explore how using the comparison method impacts student attitude towards mathematics, or for teachers to use it to decide if students need additional support. Fortunately Vanderbilt University has a site with the materials in a downloadable form to show you how to implement this method.  They have activities for linear equations, functions and linear equations, solving systems of linear equations, polynomials and factoring, and solving quadratics.  

I downloaded the first topic on linear equations.  The 27 page module has several activities from "Which is correct" to "Why does it work?" to "How does it work?" to "Which is better?" for multiple scenarios.  Each scenario shows two students doing the same problem and students are asked specific questions they discuss during a think, pair, share activity with a big idea at the end.  It provides the worksheets students need to do the activities.

Furthermore, there is a teachers guide which provides suggestions on when each activity should be done during the lesson, at the beginning, the middle, or the end. There are also questions provided to help guide the discussion students are asked to do.  I like they've provided the materials needed to try this method in class.  Let me know what you think after you've checked it out because I'd love to know.  Have a great day.

Storyboard That for Comic Strips

In January, I wrote a piece on turning word problems into comic strips.  I mentioned a couple of websites but since then I found one that works so much better and is easier for me because it has so many more objects to use.  Today, I'll be doing a step by step visual instruction on doing the comic strip so you can see how it's done.

Storyboard that is the website I discovered and like so much.  It is the program I used to create the comic strips you've seen the past couple weekends for warm-ups.

When you open the program, you are shown to this screen.  It automatically comes with three separate frames but you can add in more frames if you need them.  There is a button that says add/delete. 

I chose this problem because it has a comparison and is fairly easy but the process is the same no matter what grade level you are doing.  

"Aaron’s candy container is 20 centimetres tall, 10 centimetres long and 10 centimetres wide. Bruce’s container is 25 centimetres tall, 9 centimetres long and 9 centimetres wide. Find the volume of each container. Based on volume, whose container can hold more candy?"

I have to sit down and figure out what I need to do this.  I think I'll do it in 6 frames so that the information is given in smaller chunks but I could do it in three.

1.  Aaron alone with his box.

2. Bruce alone with his box.

3. Aaron with box measurements

4. Bruce with box measurements

5. Aaron and Bruce bragging they can have more candy

6.  Question.

Now that I have my plan, I'll set my storyboard for 6 frames and select a background.

After I click and drag the background into the frame, I go through the characters tab looking for a tab.  I don't have to use the names in the original problem, I can change the names to reflect the ethnicity makeup of my class.  I chose teens to reflect the age of my students.

Next step is to find boxes for the characters so go to the search bar and type in box or boxes to see which has the most for the best choice.  

Then I put in all the speech boxes and filled them out. I put all the information in and changed all the text from 10 to 14 so it was big enough to read.

So now you have a full strip.  I usually just take a screen shot of the strip itself and print it off to use in class.  I included the headings and such so you can see what it does.  If you want, you can log in using Microsoft, Google, or a variety of other programs and if you do, you are allowed to make two comic strips a month.  If you don't sign in, there does not seem to be a limit if you do the screen shot method.

This is a step by step guide using the Storyboard That program online.  Let me know what you think, I'd love to hear. Have a great day.

Warm-up

 

Warm-up

 

Ways To Help Students Understand Math Better

 

I know we all have that point in teacher where we know we've taught the material, thought they understood it and then nothing. It's like you have to completely reteach everything and it seems like no matter how you choose to teach the material, it ends up the same.  There are some things you can do to increase the likelihood of having students remember and understand the material better.

We want them to understand what is taught, to apply the skills they learn, and remember the material in the future when referred to.  It does no good for students to only remember it for the space of a few days.

First, create a good class opener, bell ringer, or warm-up to start the class with. The first five minutes of class set the tone for the rest of the lesson. It is suggested teachers post the agenda telling students what material will be covered that day. In addition, there should be information on the learning objective or essential question for students.  It is best to form the learning objective as an I can statement rather than saying "Students will learn......." This allows the teacher to have students reflect on how well they met the learning objective by asking them at the end of the time. There should also be a short active warm-up to review or assess prior knowledge or show a short video to expose students to upcoming materials.

Second, when introducing new material to students it is best to provide multiple representations because the more representations offered, the better the chance students have of understanding it. One should show the concept with manipulative, pictures, breaking down the problem, or a symbolic representation.  Depending on the concept, it might involve showing the equation, on a number line, a picture, or words.  This is so you find a way to communicate the material to the students.

Next, think about solving the problems in multiple ways to students see there is more than one way to solve a problem and it encourages students to come up with their own creative way to solve the same problem.  The more ways students are shown to solve a problem, the deeper the conceptual understanding of the material students develop. In addition, once students solve the problem using one method, encourage them to solve it again using a different method. 

Furthermore, it is important to take time to show students how the concept is used in the real world.  When we can show how the equation or concept is applied in the real world, it increases student understanding of the math. If we can't find a real world application, we should look for how its applied within math or another subject or show how it was developed over time by looking at the history of it. 

Then one should take time to have students communicate their reasoning behind solving the problem.  They should communicate using both written and verbal forms and teachers can use this to help assess how well students understand the concept.  It is easy to incorporate 10 minutes during class for students to explain to each other their reasoning. 

Finally, set aside the last seven to ten minutes of the class to have students do a self assessment of how well they understood the concept using a rating scale of 1 to 5 with one meaning I'm still lost and 5 meaning I've got it and can teach others the concept.  This is the perfect opportunity to give students a preview of the object for the next day's class.  The teacher can take time to discuss where the lesson is going the next time.  The final thing to do during this time is to preview the homework assignment to eliminate any misunderstanding.

If you make sure you do all of these things during each lesson, you will increase the chances of your students understanding and retaining the concept.  Let me know what you think, I'd love to hear.  Have a great day.

The Math Of The Mollusk Shell

 

I love finding articles that help support the point of view that math explains the world around us.  I came across on where scientists looked at an extinct mollusk to figure out mathematically how it got its shell shape. The cephalopods include octopus, squid, cuttlefish or nautilus. Historically, the number of extinct species outnumber the current populations with the highest number during the Paleozoic and Mesozoic ages.

Quite a few of the 10,000 species of the extinct cephalopods had tightly coiled shells.  One of those, the Nipponites mirabilis had a shell that twisted so it looked like something M. C. Escher might draw.  The convoluted shell twisted itself so it seemed to have to beginning or ending.

At first, it looks like nothing more than a tangled mass but if you examine it more carefully, you begin to notice there is a pattern to all that twisting.  Several scientists created a mathematical mannequin for this and other shells that did not display the standard pattern.

The mathematical mannequin shows how mechanical forces twist the shell so that it is uneven. In addition, the mannequin also shows how some snails develop their spiraling form.  This mannequin is able to show how three different shells were formed, specifically the spiral shell, the helical shell, and the meandering swerves of the Nipponites mirabilis. In fact, this came out of a desire to understand the physics behind the formation of seashells. 

Mullosks actually create their own shells using the outer mantle which is a fleshy organ.  The mantle actually secrets calcium carbonate in layers that harden into the shell. The scientists designed the mannequin to capture the interactions between the mantle and the outer shell. Although the ancient ammonite the mannequin is based upon died out around 68 million years ago, they had bilateral symmetry similar to their squid cousins. 

The mannequin was to help answer the question of how could a bilateral secretion cause an uneven finished product. One possibility is if the actual body grows faster than the shell, then it can become too big for the shell and cause enough physical stresses to twist the body and cause a twisted shell.  If the conditions are right, it twists unevenly to make the weird shell of the Nippoinits mirabilis. 

This is a cool way to use mathematical modeling. The scientists used currently knowledge and mathematics to create a mathematical model (mannequin) so they could figure out why an ancient mullosk ended up looking like a tangled mess. Let me know what you think, I'd love to hear.  Have a great day.