Best Way To Pay Off Student Loans

Now a days it is so difficult to graduate from school without owing money.  I managed to get my Master's degree without owing money but it took me twice the amount of time because I took one class a term, charged it to my credit card and paid it off before the next term.  I know so many others who borrowed money to finance their degrees and ended up owing in the $60,000 to $70,000 range.  In addition, many students who head to college from high school, end up having to borrow money to do it.  

Recently, I cam across an article that looked at the most efficient way to pay off student loans since most people either try to pay it off as soon as possible if they have the money, or they take the payment plan sending in so much every month and are able to get the remainder forgiven after 20 to 25 years of payment.

Several mathematicians decided to explore which of the above possibilities is best or is one should use a combination of them so they created a mathematical model to see which works best.  The model was designed to look at specific borrowers and their circumstances. For some borrowers designing a hybrid payment plan is best but for others it is best to stay with one of the standard choices.

Researchers concluded there are three possibilities to pay the student loans off quickly.  If you only had to borrow a small amount of money, the best thing is to pay it off as soon as possible because you'll only pay a little bit of interest. If you owe a large amount, enroll in an income based payment scheme that allows you to pay monthly payments based on the amount of money you make each year.  The hybrid version suggests you pay as much of the loan as possible over the first few years before switching to an income based payment schedule. 

The model these folks created is based on the idea that people have to pay tax on the amount that is forgiven in the income based scheme, and the compounded interest that is common for various student loans. This model finds the point known as the critical horizon when it is best to switch from regular payments to the income based payments.  The critical horizon is the point when the costs of compounding interest matches the benefits of loan forgiveness. 

Currently, this model asks students to estimate their future income level, tax rates, and possible living expenses but the creators are hoping to refine it so that lifestyle changes such as marriage, having children, or buying a house can affect the motivation of the borrower. Most people in order to choose a future income level have to look at career stats to get an average amount but it doesn't allow for people who might do extremely well in their chosen field. The current model doesn't take into account the possibility of someone changing fields within a couple of years.

This model was created because there has been little research done on this particular question and it is important to think about so people are able to pay their loans off in the most efficient manner with the least extra cost.  Let me know what you think, I'd love to hear.  Have a great day.

Ways To Increase Rigor

 I keep hearing we need to increase rigor in the classroom but that can be difficult if your school requires that you follow the textbook rather than being able to teach with more freedom.  By definition, rigor in math classrooms is referring to what helps students use both creative and effective methods to solve problems.  It allows students to experience frustration when something doesn't work out but keeps them moving forward. It also helps students gain a deeper learning.

We know that direct instruction with the "I do, we do, you do" is not as effective as allowing students to work with manipulatives since it provides students with an access point for the concept being taught. In addition, using manipulatives helps eliminate barriers for all students. 

Rather than always assuming that the mistakes students make represent their lack of understanding, use the mistakes to help every student learn.  Making mistakes allow students to grow new synapses.  One way to do this is to provide problems with mistakes, so students have to go over the problem to identify what was done incorrectly.  Students then show how the problem should have been done.  Unfortunately, as teachers we want to help students do the problem rather than letting them struggle which means they don't learn was effectively .

It is important to show students that there may be more than one way to solve a problem.  For instance, for a problem such as 3(x+3) = 18 is usually taught by having students multiply the x and 3 that are inside the parenthesis by the 3 outside. For some students it is easier to divide both sides of the equation by the 3 first. In addition, it is important to expose students to problems that are not as easy to solve as the textbook problems or might have fractions or decimals in the answers. 

Furthermore, do not tell students how to solve specific problems.  Instead, let them discuss different ways to solve the problem and once its solved, ask them to find another way to solve it.  This helps them think outside the box and encourages more creative ways to solve problems.  

Another thing is that students need to see the bigger picture and make connections between mathematical contexts rather than teaching students to memorize methods to solve problems because it makes it harder for students to transfer from one problem to another. For instance, if students learn to calculate the area for a square, encourage them to think on the formula for the area of a triangle with the same base and height measurements.

In addition, rather than moving on to the next lesson as soon as students begin to show mastery, change the context of a problem, rewrite the question, for extend their thinking so they are able to transfer what they have learned and are able to apply the concept in real life.  For instance, if students learn to calculate the amount of something based on a percent, provide a real life problem such as calculating the actual amount of fruit juice in the fruit drink. 

Furthermore, one should set up learning with multiple steps to increase the rigor of a lesson.  Begin with the simplest type of problem before moving students on to a more complex problem of the same concept before having students work problems that challenge the most "proficient" student.  This helps build rigor. 

Finally, expose student to multiple sources so they synthesize the material.  Rather than having students rely on your teaching for everything, let them check out videos on the same material, use activities to explore and practice so they have to synthesize everything together.  

So these are some ways to increase rigor in your classroom.  Let me know what you think, I'd love to hear.  Have a great day. 

Warm-up

 

If a cow produces enough milk each day to make 2 gallons of milk each day, how many gallons of ice cream would a heard of 30 cows produce in a year?

Warm-up

 

If people consume 1.6 billion gallons of ice cream in a year and each person eats 23 pounds, how many people ate ice cream.

Percentages and Eating?

The other day, I was introduced to a hard seltzer drink.  It was mango flavored with 5% alcohol by volume. .  I thought 5% sounded a bit much and wasn't sure I wanted any.  I sat down to calculate how much alcohol was in the actual drink.  The can was 12 ounces so I multiplied 12 by .05 to get .6 or 6/10 of an ounce.  So just over half an ounce of liquor in the can. Not that much in reality.

So what about other liquids such as juices that often have 10% fruit juice.  This means if you have a 12 ounce bottle of orange juice with only 5%  real fruit juice, it means there is only 6/10 or .6 ounce of orange juice.  The rest of it is flavorings, water, and sugar.  

Another area we run across percentages everyday is in regard to the labeling on foods.  Every food has a nutritional label which helps you decide how much you are consuming.  For instance, if you have one serving of cheese and the label says it provides 15% of your daily calcium requirement, that means you are having 300 mg of calcium out of the required 2000 mg needed each day.  

Labels also include how many calories of the recommended intake of 2000 calories per day you are eating.  For instance, you will see when you look at the amount of fat or sodium on the label, you will see that it tells you what percent of your daily intake the fat or sodium.  For instance, if you look at a 300 calorie food with 60 grams of fat, the label will show you that the fat is 20% the calories and it is recommended you consume 50 to 80 grams of fat each day or no more than 30% so this would not be a good choice to eat.

Three examples from the labels on three different products.  There are so many more examples but most of us do not pay attention to those labels.  I had a friend who would just pick any juice off the shelves because she'd never been taught to read labels.  I showed her that the bottle of juice had only 5% juice.  She started looking at every bottle of juice on the shelf until she found one stating it was 100% juice.  After that she always read labels.

If you look at the recommendations, you'll see it is recommended one consume between 50 to 60 percent of your calories as carbohydrates, 12 to 20 percent should be taken in as protein, and no more than 30 percent should be in fat.  Again, percentages rather than actual numbers. 

This topic is a great way to combine real life applications of percentages with science or home economics so students see a relationship.  Let me know what you think, I'd love to hear.  Have a great day.

Productive Struggle

 

I think productive struggle is one of the hardest things to let students experience.  Most of the elementary teachers I know are not really into math so they instruct, following the method given by the textbook.  I have observed that many students would rather just have the teacher help them through the problem, rather than trying to work it out for themselves and many of us teachers were taught to be "helpful". 

Productive struggle is a necessary part of learning mathematics because it allows them to work through any problem, making it less likely they give up.  Productive struggle helps students look at making new connections while trying to find different ways of solving the problem.  It allows them to try new types of problems they've never seen before.

In real life, most mathematical problems do not have neat answers nor can they easily be solved.  These problems are messy.  They have answers with decimals or fractions.  The problems often have fractions in them.  Students have to persevere in the face of these types of problem.

It cannot be assumed that all students who arrive in high school are able to work their way productively through problems. We have to help our students learn to productively struggle through problems.  There are things teachers can do to help them.  First of all, call on students who may not have the right answer.  Many times, those who get the correct answer are rewarded, rather than celebrating the attempt.  If a student gives the wrong answer, ask them questions designed to help them question their thinking.  

Instead of praising students for being smart, praise them for sticking with it and trying to solve the problem.  When you praise the effort, students are more likely to face challenging  problems. To reinforce this, display problems that show creative solutions rather than the papers with the highest scores because it showcases productive struggle. 

Give problems that cannot be solved by following memorized processes. Using problems such as these require them to make sense of the problem itself, before deciding how to solve it and what math is actually needed.  It is important to provide students with feed back that is informative.  This means providing context to help them head towards the answer. 

It is important to refrain from giving easier problems to struggling students because it gives them the message that they are unable to do challenging problems which can reinforce their perception they can't do math.  It reinforces the closed mindset.  Provide students with time to ask questions and play with ideas to help with retention.  Allowing students more time can improve retention. Take time to encourage students to adopt a growth mindset as a way of reminding students that everyone is capable of doing math.  

By allowing students to productively struggle when solving non-standard problems, it helps them develop creative thinking and apply concepts to non-standard situations while experiencing rigorous problem solving.  This all leads to deeper learning of the material. 

On Friday, we'll talk about ways to develop rigor since many textbook problems do not contribute that much to rigorous problem solving. Let me know what you think, I'd love to hear.  Have a great day.  

Death in the family

I'll be back Wednesday.  My mother passed and I have to take care of things.

Warm-up

What do you notice?  What do you wonder?  What math do you recognize?  What is it used for?

Warm-up

What do you notice?  What do you wonder?  What mathy thing could you ask about this?

The Geometry Of Knives

 

Over the past few days, I've been binge watching of the show Forged in Fire.  I love knives, I love looking at knives, I have a nice collection of knives and I love the show.  If you haven't seen it, there are four contestants who over three rounds try to become the champion.  At the end of each round, the weapons are scrutinized and tested before eliminating one person.  By the end of the show, only one blade has survived.  

Through the multiple episodes I've watched, I've heard judges use the terms obtuse and acute along with referring to "The geometry of the knife."  This intriguing phrase captured my attention because I'd never thought of geometry being associated with knives.  This term refers to the grind used on the blade or how did they make the sharp edge.  

The edge itself is made up of two parts. The first part is the edge bevel which takes the thickness down from the top or spine down to almost the edge.  The second part is the micro bevel which finishes off the process.  The angle of the micro blade determines the use of the knife.  If it has a edge between 35 and 40 degrees, it is often found on an axe while an edge of 25 to 30 degrees is associated with a utility knife.  Pocket knives usually have a micro edge of about 15 degrees while cooking knives are between 15 and 20 degrees. Finally a razor blade is often around 10 degrees.

The grind is the type used to create the edge of the blade and there are eight a knife smith can choose from.  The first, the hollow grind, is a very popular one and has been for centuries. This grind is a concave one where the two sides of the blade curve in to a point, symmetrically. Unfortunately, it tends to dull quickly and isn't the strongest edge but it is the edge you see on cutting knives.

Next is the asymmetrical grind which uses two different lengths of sides and can combine two different grinds. One of the most popular asymmetrical grinds uses either convex grinds or flat grinds.  Using two different grinds produces a more durable edge with a decent sharpness to it.  This grind is often found on tactical knives.

The third is a flat grind which comes in three different types.  The flat grind is the easiest and simplest one possible. One is the full flat grind in which both sides are cut with the same angle to create a V shaped. It is most often found on Chef's knives because it provides an extremely sharp edge, good for pushing the knife into something.  Unfortunately, a full flat grind may provide a sharp edge but it is not very durable and a true full flat grind is not often found today.

The second type of flat grind is the high flat grind and is more popular because it leaves a small part of the blade with the same thickness and is located near the handle on the bottom of the blade. This is a good knife to use for survival because it is easy to sharpen anywhere.  The last type of the flat grind is the Scandi flat grind which is similar to the high flat grind but the part where the thickness is the same is longer.  

Then there is the convex grind where the bevelled part pushes outward some rather than inward like the hollow grind. Although the edge stays sharp longer and is more durable, it is much harder to sharpen due to the slight outward bulge in the blade. This grind is associated mostly with axes.

The compound grind begins with one grind and then adds a second V bevel to make a cutting edge.  This is the most common grind done today on knives because the edge cuts better and does not chip as easily. The chisel grind has only one side of the edge sharpened to between 25 and 30 degrees that starts in the middle while the other side remains flat.  Aside from being found on chisels, this edge is often found on Japanese cooking knives.

The final grind is the semi-convex grind which is also known as the asymmetrical convex grind which puts the convex edge and V edge together. This is not a particularly popular grind but it is seen once in a while. I found it interesting to learn all this.  Let me know what you think, I'd love to hear. Have a good day.

PBS Math Club

 

I am always on the lookout for videos, interactive sites, and real life applications of math but it is often hard to find all three together.  PBS has a site that has so much to offer all grades.  This general site has six different resource area ranging from K-8 mathematics, to various high school strands.  I like that they have the strands for high school broken down because that makes it easier to find exactly what one is looking for. 

I clicked on High School Functions to see what they had.  On the left side, there is a list of the main topics being covered under the umbrella of functions. If one of those topics is clicked, it shows subtopics associated with the general topic.  For instance, if you click on building functions, you'll find two sub topics.  One is on modeling functions between two quantities and the other is on building new functions from existing functions.   

Next you decide what grade level you want to look at and select that.  I chose 9 - 12 because that is the usual group of students I work with.  I have my choice of 5 videos that run about 26 minutes long, two interactive activities where one deals with valentine cards and the other is on building virtual rockets. In addition there are 12 webpages linking to the appropriate videos in Khan Academy, and for the teacher there are multiple links to professional trainings and to RTI videos.

I clicked on the interactive valentine cards which allowed students to explore the relationships between two variables.  It could either be assigned directly or used with google classroom.  It provided both teacher support materials and support materials for the students. The only issue I had was that when I clicked on the launch button, I ended up with something exploring the relationship between arm distance and height on a scatter plot.

Since this site uses Google classroom, you can use a google e-mail address to log in.  When you first log in, you can choose the subject, type of lesson, and how you want it sorted.  Under type of lesson, they list video, interactive, interactive lesson, lesson plan, media gallery, audio, image, document, webpage, or collection.  Unfortunately, they don't have all of these available for each topic but with a simple search you can find some interesting lessons.

Under interactive lessons, I found things like square rimmed tricycle - radius and circumference, or cooking with Bill Nye for independent and dependent variables, or shower versus bath, ratio and rate.  I looked at the Bill Nye lesson and it took you to a website called cooking with Bill Nye.  The first thing it asked you to watch a 40 second video in which Bill Nye talks about cooking pasta in two different pots, one with a lid, one without and the student is asked to make a prediction.

After the prediction is saved, the student moves on to the second page where we watch a 27 second video where Bill Nye explains why it heats faster with the lid on.  The student is asked if they changed their prediction from step one.  Once done, the student moves on to the third page where the student watches a 46 second video and then fills in the missing data in the chart.  Then the student decides whether time or temperature is the dependent or independent variables and explain their thinking. The final page talks about how much money can be saved by using a lid.  

The lesson plans are set up as actual lessons which come with the grade level, links to movies, links to resources such as charts, etc, and the answer key. It tells you how to do the lesson, which item to show or do when and includes assessment information along with the standards covered.  This lesson is all set to go and can be assigned via google classroom or it could be run by the teacher as a class lesson.

There is so much material here that can be used to create a student centered learning experience, have something special available when a sub is there or used in a flipped classroom.  If you have time, check it out.  I'd love to hear what you think about this.  Have a great day.

Rounding In Context.

 

By the time students arrive in high school, they have been exposed to the normal rules associated with rounding.  You know the one where they look at one place past the one they want to round and use that to determine if they round up or down.  That works well in a theoretical situation but it doesn't always work in real life.  Not everything that is rounded follow the rules of if it is five or more, round up and four and lower round down.

Bankers have their own way of rounding that doesn't follow these rules for taxes.  If the calculated value ends in five, and the value in the cents place is even, it is rounded down. For instance if you have a 8.5 percent tax rate then you would have $1.085 after adding in the tax. So it ends in 5 but there is an 8 in the cents place but the rule says if the number in the cents place is even, it rounds down so it will show as $1.08.  How ever if you spent $15.00 with a tax rate of 8.5 percent, you end up with a tax of $1.275.  This ends in 5 but there is an odd number in the cents place so this is rounded up to the next digit so tax is $1.28 for a total of $16.28.  Now if you have a weird rate like 6.8% and the tax ends in any other number but 5, the normal rules of rounding apply.  So you spend $15.28 and tax is $1.039. Since 9 is not a 5, this means the regular rules of rounding are applied and this rounds to $1.04 for a total of $16.32. This type of rounding also applies to transaction fees and other such items.

On the other hand, paint is something you do not want to round down on since you want to have enough paint to finish the project.  It has been suggested by Bob Villa that you round up to the closest quart so you buy just a bit extra but not too much.  If you calculate all the area you want to paint in a room and come up with 4.3 gallons of paint, the normal rules say you round down to 4 gallons but 4 gallons may not be enough since .3 is a third of a gallon.  Instead, you might round it up to 4.5 so you have enough paint but not too much.  If how ever you end up needing 4.03 gallons, you might round it up to 4 gallons and 1 quart.  When dealing with items such as paint, primer, water seal, etc that are expressed as one gallon covering so much area, you might think in gallons with quarts at .25, .5, and .75.

Another area for rounding is when you are buy carpeting. The easiest way to figure out the amount of carpeting needed for a room.  If you want to measure it yourself, it is recommended that all measurements be rounded up to the next half or whole foot.  So if one dimension is 4 feet 3 inches, round it from 4.25 to 4.5 and if the other dimension is 8 feet 7 inches, round it up to 9 feet.  This way people make sure they have enough carpeting for the job.  This is a always round up situation.  It is also suggested that one add in five percent to cover seams.  The total number of square feet are divided by 9 because there are 9 square feet in one square year.  The answer is then rounded up to a whole or half a yard.

These are examples where the standard rules of rounding do not apply since in many situations one has to round up so there is enough material to finish a project.  These are just three situations but there are more where one rounds up.  Let me know what you think, I'd love to hear.  Have a great day.

Warm-up

 

What do you notice? What do you wonder? What mathy question can you ask about the picture?

Warm-up

 

How many triangles can you find in this picture?

Math TV

 

The other day, I came across a website I've never seen before.  It's called Math TV and it is filled with videos created by student instructors.  While the videos at this site show how every example is solved in the accompanying XYZ textbooks, these examples are similar to those in other textbooks. 

In addition, most examples are worked by two or more people, including one done in the Spanish Language.  The nice thing about having multiple explanations is that students can watch a different video to get a different perspective on the problem.  It allows students to feel as if they have some control over their learning.

This site has videos for enrichment, basic skills, elementary algebra, intermediate algebra, college algebra, trigonometry, pre-calculus, calculus, and applied calculus.  Unfortunately, if you click on the course, there are topics listed but the topics are not clickable so the easiest way to find a topic is to use the search option.

 When you put a topic into the search bar, it brings up all the videos that match that description from all the classes. Then when you choose the video, you'll see a list of tutors who have made a video showing how to work the specific problem.  There are even videos in Spanish for students who need it in that language.  The nice thing about having multiple videos is that if one doesn't work, there is another one a student can watch.

I looked up interval notation and linear inequalities. This topic is found in the intermediate algebra class and there are eight videos ranging from an introduction to two word problems.  The examples go from solving a two step inequality, to one with terms on both sides to problems with parenthesis. So there are examples from fairly easy to quite complex. 

I am not sue I would use these in class for instruction. Where I see using them in class is as extra support by providing the url to the video that is similar to the assigned problem. Students need to learn to watch videos showing how to work similar problems so they can use the steps to solve the current problem. With multiple explanations available, it makes it easier for students to find one that works for them.

This way learning is being transferred from teacher to learner. Let me know what you think, I’d love to hear.  Have a great day.

Digital Classroom Tools

It is always nice having certain tools in one place rather than trying to find them in different locations or via different apps but I found a website that has many of the ones any teacher wants for day to day use.  I found one site that has so many different tools all in one place but you might not need to use all of them.

The site, Super Teacher Tools, has so many things one can use in the classroom. They offer a seating chart tool, a group making tool, and a random name generator through the instant classroom section.  It was easy setting up the account to use this and the first thing I had to do was set up the class group.  

Once the group was set up, I could then use the seating chart tool, the group making tool, and the random name generator because they rely on the group list.  The seating chart publishes the names in the order entered in the list but you have the opportunity to rearrange all the names into the configuration you want.  The group maker is nice because you can either set up groups based on the number of people you want in a group or the number of total groups you want to use. Finally, the random name generator takes the list of students and put them in random order showing just one name at a time.  The programs are quite basic, doing exactly what you want without any bells or whiles.  I don't know if I'd use the seating chart but I would use the other two.

To help things run smoother, there is a classroom timer which offers either a timer that operates like a stopwatch or you can set a countdown timer for days, hours, minutes, and seconds.  You can choose the exact time you want.  the clock is one of those that flips as it counts up or counts down, basic but it does the job you need. These clocks are easy to use and very intuitive.

Now for the times you need dice, they have a simulated dice app where you can choose the number of dice you want to use.  For instance, you can roll 5 dice to get numbers for students to use when they are practicing order of operations and want to add specific operations to find a certain number.  Or you can use two dice to provides numbers for binomial multiplication or even three or four dice to determine the trinomial you want students to factor.

There is also a customizable spinner where one can use numbers, names, and what ever else you want.  I see possible use of adding binomial factors to the spinner so the spinner can be used to find two factors for students to multiply together. Or put trinomials on the spinner for students to factor.  A spinner can be used for things other than stats.

Finally, there are four games one can use for reviews.  There is a jeopardy type game, a rocket review game designed to review basic facts, who wants to be a millionaire, or speed match.  All four allow you to  create your own games that target specific skills.  Although most of these give you the option to search for a game, I found very few and the ones I found were a few years old.  

This site has several things teachers use on a regular basis but they are all basic without any bells and whistles.  This may have something you like so check it out. Let me know what you think, I'd love to hear.  have a great day.

Ted-Ed Site

 

We all love to watch TED talks because they can be so inspiring and provide us with information we want but did you know that TED talks has a site that is for educators, parents, and students?  I didn't until the other day.  They have a whole set of topics which all follow the same plan but are of so many different topics.  

I didn't know about the site until I followed a link for something and there it was.  It was easy to sign up for as an educator.  Once in, I looked for math lessons and came up with such a variety.  I like that they allow you to look for specific strands such as Algebra, data and probability, geometry, measurement or numbers and operations.

Some of the titles for Algebra have some interesting titles such as "How folding paper can get you to the moon" or "Gridiron physics: scalers and vectors", or "Accuracy vs Precision".  The last one I could have used in my trades math class to help students understand the difference.  Under data and probability, you'll find activities like "The monster duel riddle", or "How to spot a misleading graph?", or "Why do airlines sell too many tickets?".  

If you check geometry, you'll find topics like "The art of folding origami", or "Can you solve the honeybee riddle?", or "football physics".  Then in measurement you'll see "Which voting system is best?", or "Why don't perpetual machines ever work?", or "How to visualize one part per million".  Finally under numbers and operations there are topics such as "The race to make quantum computers work" or "Where do math symbols come from?" or "How does your smartphone know your location?".  Note that many of these videos show up in more than one group so could be used in different ways.

All lessons are set up in the same way.  The lessons begin with a short 4 to 5 minute videos. The videos can be watched together in the room or can be watched individually depending on your situation.  The next step referred to as the Think stage involves a set of about 8 multiple choice questions based on the video just watched.  This is followed by a dig deeper section where students receive additional information on the topic.  The finally part is the discuss with one open ended questions and a guided question for the class to do.  In addition each one has the option to customize as needed.

Furthermore, TED-ed offers a variety of collections revolving around specific topics including calculating distance , exploring other dimensions, the math of online dating, fractals, and so many other topics.  Again, these are set up with the watch, think, dig deeper, and discuss in the same way the individual lessons are set up. 

TED-ed also offers teachers the opportunity to create their own lessons beginning with a video from Youtube or other place.  Once the video is located, you are asked to create an introduction to go with the video before moving on to creating multiple choice questions or open ended questions to go with the video.  Dig deeper asks for additional information on the topic and a finally discussion question for students to answer.  It allows teachers to add a final concluding comment to wrap up the whole lesson.

The information on having students get an account and assigning lessons is found under the help page.  You have to do a search to find the information but once it is found the information is quite clear.  Let me know what you think, I'd love to hear. Have a great day.

Warm-up

If 500 beans make one pound and can be sold for $18.00 per pound, how much will you make if you have 150,000 beans?

Warm-up

On average, a cocoa tree produces 30 pods a year and each pod contains 40 beans, how many pounds does one tree produce if there are 500 beans per pound?

Emergency Lesson Plans

Most places I've taught at usually request each teacher have a binder with emergency lesson plans available.  Most people I know, end up having a stack of worksheets in a binder with a list of students and a short set of instructions.  I admit, I've done that but then I realized I don't have to do that for those short one or two day absences.

Instead, I try to do something a bit different.  One year, I set up a two day lesson with everything the sub needed.  I kept one season of the show "Numb3rs" in the folder so students could watch a specific episode while filling in the guided notes sheet I created to go with it.  

I learned early on if you ask students to write a paragraph on the show, they would talk about the leaves falling off the tree, or the guns being fired but not a summery so I make sure I leave a guided notes sheet for every student.  

The second day, I have the sheet that is associated with the lesson so students get to apply the concept associated with the episode they watched. Fortunately, they are still available on several different sites such as this one or Cornell University.  These two sites are active as of when this column was published.

Other times, I've left the URL for one of the original Mythbuster shows.  Again, I leave a two day lesson plan for this. The first day, students spent time watching the video and filling out a guided notes sheet so they focus on the important part.  This site has a nice generic worksheet where students identify the myth being investigated, background, controls, etc.  Although, it is focused more on the scientific process, it can be used with any episode.

For the second day, I have students research the general topic to see what math they can find that is associated with the topic or I might have them investigate a topic in more detail such as the history of duck tape.  The topic depends on the episode I have students watch.  By the end of the second day, I expect a nice three to five paragraph write up on the topic.

Other years, I've been known to leave a lesson on stats but not the standard lessons.  I might ask students to analyze graphs that are misleading such as one on "How to spot a misleading graph".  This lesson has students watch a short (5 minute) video on the topic at the beginning.  The next step is for students to answer 8 multiple choice questions about the video. Students are given a chance to practice what they learned and ends with a discussion.  

Sometimes, I have empty bingo cards available where I have students fill in the spaces with numbers between 1 and 100.  I have a can of problems whose answers range from 1 to 100.  The problems are made up of several numbers with a variety of operations such as 1 x 3 + 5^2 - 1= 27.  I give the problem and students have to do the math to find the answer.  This is great practice for students.  I've done this one in class before and I have students working together to find the answer.

Finally, I've left some "art" type activities such as coordinate grid pictures where students work their way through a list of coordinates until they end up with a picture.  Once they've created the picture, they need to color it in.  Other times, I've left some sort of impossible heart, impossible square, pi based city scape, or other fun activity.  These art activities appeal to the students who love to draw.

So if you want to do something a bit different, try some of these.  Let me know what you think, I'd love to hear.  Have a great day.

Helping Students Learn Mathematical Discourse.

 

Although most teachers know that having students discuss their thoughts, their choices for attacking the problem, and how they completed the problem, it is not unusual for students to arrive in high school missing the ability to express themselves. Often times, they are more concerned with getting the "right" answer rather than explaining their process.  Unfortunately, we have to help students learn the processes involved in mathematical discourse.

Fortunately, there are things teachers can do to ease students into mathematical discourse as long as we don't expect them to immediately "get it."

One of the first things a teacher can do is to spend time teaching students variatious communication strategies.  Make it a part of the lesson and block out time for students to practice each communication skill be it restating what another student said or participating in a think-pair-share.

At the same time, it is important to teach students listening skills so they "hear" what another student is saying as they explain how they did something or what is being said during a brain storming session.  Listening is an necessary skill so they can restate or explain how another student solved a problem or are able to give feedback. 

In addition, teachers need to incorporate comprehension lessons to these mini-lessons so students understand what the vocabulary is asking them to do.  For instance, they should know the difference between simplify and rewriting, expressions and equations or be able to decode and decipher word problems.  

Furthermore, teachers need to identify places in the lessons where mathematical discourse and conversations can be placed so students practice what they are learning on a regular basis.  These could be whole groups, small groups, or pairs but the conversation should be focused on the content being taught that day.  

It has been suggested that the teacher also incorporate opportunities for students to express themselves in written form either through the use of math journals, sentence starters, or digitally.  Students need to express themselves both in oral and written form since many standardized tests ask students to explain their thoughts and they have to do it via writing. 

Teachers might also ask students to interpret data from an activity in some sort of written form since many times at work, workers are asked to report on something and this provides practice.  Finally, students might also be asked to brainstorm different ways they could use to solve a problem.

In addition, teachers need to ask students questions such as having them explain how they solved a problem, explain their own thinking or why they chose a particular method to solve it.  They might be asked to restate something another student said, or comment on an idea.  They might share something they wonder about, noticed, or concept.  The questions require students to share thoughts rather than just the answer.  The questions take everything to a higher level.

Unfortunately, without giving students a chance to develop mathematical conversations, they lack an important tool in their communications.  Let me know what you think, I'd love to hear.  Have a great day.