Transference

This is a topic many of us reflect on in regard to our students.  There are times we think they have it and should be able to apply the procedural knowledge to a similar task and they can't.  One place I see this is in adding numerical fractions where students have to make sure they have a common denominator to do this.  They learn it but when they apply the same principal to adding algebraic fractions, they have no idea what to do.

So what is going on?.  According to an article I read there are several reasons students have difficulty with transference.

1. Initial learning is necessary for transfer but it is not known about the type of learning needed to promote transference.

2. Abstract representations of knowledge helps improve transference.

3.  Transference is considered to be an active, dynamic process.

4. All new learning is based on transfer of previous learning.

It now appears that transference is based on the degree to which people understand the material rather than memorizing procedures and facts. In addition, it takes a lot of time to really learn the material, more than we usually schedule in class for any one topic.  It has been found that students need to know the when, where, and why to use the new knowledge.  In other words, build a connection.

According to another study done to see if students could transfer the mathematical processes to other subjects, it was found students had a difficult time because it required them to translate a problem stated in words into a math problem.  A second study claims the reason students are unable to transfer their learning from high school over to work because they do not have enough authentic based and project based learning.

This leads me to believe there are two types of transference:
1.  Transference of a skill from a simple problem to a more complex one such as requiring a common denominator for regular fractions and algebraic fractions.

2.  Transference of the skill from the classroom to other subjects or even to work.

In addition, there are two types of transference which might explain why I see the two versions.  The first type is near transfer where students are able to apply their knowledge to problems and situations similar to what they learned.  The other is far transfer where students apply their knowledge to a situation quite different from the context they learned it in.

It is noted that schools offer more opportunities for near transfer.  Students find far transfer more difficult because they have to seriously consider the situation in order to remember the rules and concepts needed to solve the new problem. 

Furthermore, there are three factors involved in transference, the person's representation of the problem, their background experiences, and their understanding of the problem.  Representation refers to how the person mentally solves the problem which is related to their knowledge of the content of the problem while experiences is often based on prior knowledge and understanding is linked to representation.

This gives us some understanding on why many of our students are unable to transfer math as easily as we think they should.  I know my eyes have been opened.  Let me know what you think!

Deep Learning.

  I ran across the term deeper learning in the computer field but wondered if it had applications to mathematics. In education, deeper learning refers to using presenting the material in innovative ways so students learn the material and apply what they have learned. 

It is also a way to encourage students to take control of their own learning because they are expected to combine communication, collaboration, with in-depth academic knowledge.

Many of the currently recommended practices such as project based learning,  assessments accumulated over time, etc help with deep learning but are not necessarily being practiced.  My school changed to a modified block schedule one year but it was done without putting a proper foundation in place so it failed miserably.

The collaboration requires students to work in small groups so everyone has a chance to participate.  Small groups also allows for peer teaching because students will ask each other for help when the teacher is unable to get over to answer questions.  Students are often able to explain a topic to each other in ways that are more easily understood. In addition, digital devices can help promote deeper learning because it allows a more personalized education.

The research is showing that students who graduate after using deeper learning often score higher on tests, graduate on time and are more likely to enroll in college than those that don't.

Some of the ways to promote deeper learning in the classroom include:
1. Master academic content in addition to providing real world applications so students see cross curricular applications.

2. Require students to think critically while solving more complex tasks. Integrate researching, brainstorming, and design thinking.

3.  Have students collaborate because it helps students gain a skill that is important in today's society.

4. Require effective communication because students need to communicate effectively and persuasively.

5. Students need to learn how to learn by learning to set and make goals, track progress, reflect on their own strengths and weaknesses.

6.  Students need to develop a life long growth mindset so they are willing to take the initiative and develop persistence.

These are excellent strategies, many of which are not that difficult to implement but often when we want to start using new methods in the classroom, we run across opposition because policy is often behind what research shows we need to do.

I know I have to do some of these things but I need to do it in a way my administration is comfortable with.  Let me know what you think about deeper learning.  

Note-taking in Mathamatics

  I grew up in an age where you wrote down everything the professor wrote on the board in the hopes you got it all, it made sense later, and you could decipher those same notes when you read them over.

I never learned to take notes properly but with today's technology, I can open an app, record, and go back later to take notes at my own pace.

The other day, I was working with my seniors and realized they do not know who to take notes as they read the textbook.  So  we took a class period to go over looking for information you want and how to go through examples.  I showed them my thought process involved in working through examples.  Most of them sat there and tried to write everything down but one girl watched and listened and heard what I was saying. 

I found two videos on note-taking on you tube.  I went straight to the videos because these students can watch for themselves or you can show in class to give students a better idea of how to do it.

1. Math notetaking from the textbook.  The creator of this video uses colored sticky notes to designate vocabulary, key concept, direct quote, equations and she writes on these notes, sticks them to the pages as she reads the book. This is the first step in note taking.

2.  Next is the lecture notes from class.  She uses highlighters which are the same colors as her sticky notes.  She goes through her lecture notes highlighting certain concepts, then adds the appropriate sticky note to the page.  She shows how to combine the two for a set of well done final notes.

Cornell note taking is often recommended to students because it divides the paper into three sections.  The right hand side is for key points or topic headings while the left side is for the notes and the bottom part can be used to clarify and identify the main points or summarize the notes.  In math, we find that harder to visualize but it is possible to do it.  If you check this site , there are great examples of both good and bad note taking using this system in math.

This pdf shows how to set the page up with notes on how to use the three column note taking method which is very similar to Cornell.  To follow up, this 20 minute video shows how to use the Cornell format in Math done by a teacher who has her students use this method.  In addition, she includes information on how to use the notes later. 

As for the actual notes, it is recommended students:
1. Ask questions to clarify the material as needed.

2. Identify important elements of the lecture - could be done by reading the material before coming to class.

3.  Review notes after class.

4. Consider using a tablet or computer to take notes if you feel this would allow you to take notes better. (my students often snap pictures of notes for later.)

5. Record the lecture for use later.

6. Skip words, not numbers.

7. Use color for emphasis.

8. Use a form of shorthand.

9.  Use a three column paper for notes.

These suggestions are strongly recommended as a way of helping students take better notes in math.

Check these out or have your students check these out if they need help learning to take better notes in math.  Let me know what you think.

Warm-up

Warm-up

Black Friday and Football

Today is quite well known due to the extreme sales offered by some stores while others are excited due to the extra number of football games being offered.  The question becomes, how do you integrate mathematical activities with this theme to get the attention of all students.

Yummy Math has some lovely activities for the whole thanksgiving weekend.

1.  Macy's Thanksgiving Day Parade.  This activity has students look at the route of this famous parade to determine how far people must march, beginning and ending times, volume of a couple balloons, and a few other things.  Although this is geared for upper elementary, it could easily be adjusted for upper grades.

2. Black Friday Sales. This activity makes students look at various "sale items" to determine the discount and the actual amount saved on popular items like televisions and game boxes.

3. Consumer spending - this activity has students examine graphs on our spending to see if they can spot historical patterns to get a better idea of what might keep spiking.  Almost the same thing that retailers use to determine their best sales.

4. Home Team Advantage for the NFL - helps students learn to read and interpret infographics.  This infographic shows the wins and losses for games played at home and away.  Students are asked to prove whether the team will win more at home or away by analyzing the data.  Perfect for the sports enthusiast.

Yummy Math offers 6 more activities dealing with topics associated with this weekend.  Topics from cooking Turkey and mashed potatoes to football to food banks and shelters.  All very appropriate and very relevant. 

Check it out and have fun.  Monday, I'll be talking about ways to take notes.  Have a good weekend.

Happy Thanksgiving

Perfect Time of Year.

We are heading into a six week period that is perfect for creating and using infographics.  Infographics are a great way to present information to the world in an easily understood way.

Students need to find and present information in a variety of ways.  The first thought is using graphs, or posters but infographics are the newest way to share information.

The reason I say this is the perfect time of year for creating infographics is that there is a ton of information being collected by companies which students can use.

For instance, one student could easily research the most popular varieties of candy bought on Halloween while another student looks at the most popular costumes bought by adults for themselves or for children.

At thanksgiving time, students can research information on black Friday to see what are the most popular categories of items sold.  They can check into the top brands sold in each category or even what items are best purchased during black Friday sales.

In addition, they can check out the number of people who travel the day before thanksgiving, the day of and the day after by air, train, or car.  There is also the types of meat such as turkey, duck, or geese as the main meat.  What food items are purchased the day before thanksgiving and the day of thanksgiving?

Furthermore, they could check out the projected sales numbers between black Friday and Christmas day. They could check various categories such as electronics, cars, etc over the past 10 to 15 years to look for trends that could be covered in an infographic.

This is the perfect project for just before winter holidays where you need something to keep the students interested but you can't start anything new.  Let me know what you think.

Differentiating Textbooks

As you know, I work with ELL students who have problems reading the textbook fully.  They have not been taught the skills needed to read a textbook so I have to take time to do it.  I recently came across a book titled “Differentiating Textbooks” by Char Forsten, Jim Grant, and Betty Hollis.

Until I’d run across this book, I’d never heard of differentiating textbooks.  The only method I know to use when reading a textbook, especially a math textbook is what I learned back in high school.  We just read it.  In college, I learned to go over examples but not to do much more. 

This book has so much information from grouping students to creating smaller books to techniques and graphic organizers. 

I like a couple suggestions made by the authors in chapter two on selecting and adapting textbooks.  I do not have time to create new textbooks but I like the suggestion of substituting headings and subheadings in the form of questions.  This simple move requires students to find the answer to the questions.

They also suggest that students circle and box their math problems because this helps reduce on careless errors.  They suggest circling one type of problem while boxing another type of problem so as to distinguish between the two.  Students work all the circle ones first and the other ones second. 

The rest of the book is divided into pre-reading, reading, and post reading strategies with examples.  One of the pre-reading strategies is the Clear Up Math Visuals one.  It is suggested that students had the word problem and then together decide on the visuals one should use to represent the problem.  This is one I need to use with my students.  Its perfect and has them taking more ownership of their work.

A during reading strategy which resonates with me is called Power Thinking.  Students use powers to indicate main idea down to details all on the same idea.  So Power 1 is the big thought such as sports.  Power 2 might be Wrestling while Power 3 names some of the wrestlers they see on television.  Another Power 2 could be Basket Ball while the Power 3 could be the teams.

In math it might look like Power 1 is exponents.  Power 2 might be positive exponents, Power 3 could be examples. So over all it could appear like this:

Power 1  Exponents.

            Power 2 - Positive

                        Power 3  - Makes the result bigger.

            Power 2 - Rational

                        Power 3 - Seen as Fractions

                        Power 3 - Represents roots.

            Power 2 - Negative

                        Power 3 - represents fractions

                        Power 3 - numbers get smaller.

This is a nice way to summarize the material.

For the after reading strategy, you might try three facts and a fib where people create groups of four facts but only three are true.  The other people have to determine which is wrong.  This one could easily be used in math.  For instance:

A pentagon is made up of 3 triangles.

A hexagon is made up of 5 triangles.

A decagon is made up of 8 triangles.

A octagon is made up of 6 triangles.

A student has to decide which one is wrong.

Although not all of the suggestions can easily be used in math, there are enough suggestions that I can use in the classroom to make this worth it. I would also say that this book is not so much about differentiating the textbooks as giving students additional reading strategies they can apply.

Perfection

  There is beauty and perfection in certain numbers in math.  Look at the 3-4-5 triangle which is one of our standard examples in math to explain the Pythagorean Theorem but its also considered the standard in industry.

Its one of our perfect triplets of all time.  I love it and so do carpenters or concrete forms.  In order to create the perfect 90 degree angle, carpenters and people who make concrete forms know to use the 3-4-5 triangle to create the perfect right angle.

This particular triangle has recorded uses back in Ancient Egypt.  They are not sure if the Pythagorean Theorem was known back then but they do know surveyors used the concept for building.

The way to apply this is to take a corner.  You measure 3 feet from the corner and mark it. Then you measure 4 feet from the corner, in the other direction and make a mark.  If you get a measurement of 5 feet between the two marks, you have a 90 degree angle.  If its less, the angle is less than 90 degrees and if its more, the angle is over 90.

I know how important this concept is in building because I helped build a small cabin when I was in college.  They laid the flooring and then build each side so when hefted into place, it was supposed to fit together perfectly.  When we went to put it together we discovered no one used this particular mathematical concept when building.

The floor was not square and each side listed slightly so they were not square.  After a lot of swearing, shoving, and pushing we got it to fit but I will not guarantee how long it stood since the right angles were put in under pressure.  I don't think our supervisors had ever built anything and I didn't know you used the 3-4-5 triangle under this circumstance.

It could also be used to make sure tile or carpet is perfectly square so that it fits in the corner.  So this lovely triangle can be used in any situation where you want a perfect 90 degree angle.  Let me now what you think.

I am off working a local wrestling tournament. Be back Monday!

Calculating volume

  The other day in class, during warm-ups, we discussed the words much vs many.  Since I work with ELL students, they are always mixing the two words up.

I explained that much is usually used with quantities that cannot be counted while many indicated countable quantities.

During the conversation, the topic of calculating the amount of fuel remaining in a tank came up.  This is something that has to be calculated when you buy or sell a house.  A nice real world application.

This skill is also great to know if you want to know how many gallons your fuel tank takes.  Up here in Alaska, most people heat with fuel oil so we buy several gallons at once and its stored in cylinders connected to the house.  In addition, there are water tanks which are cylindrical although I know someone who bought a rectangular shaped one for collected rain water.

So how does one calculate the volume of a tank sitting in your yard? If its cylindrical you measure the radius of the tank and the height(length)  Next you square the radius, multiply the result by the height or length. The final step is to multiply the result by 3.1415 to get the cubic volume but you still have to divide this figure by 231 to find the number of gallons.

To find the amount used, you could use a stick, dip it and based on the depth you could easily do an approximation of the volume left.  For instance if the diameter is 30 inches and the stick is covered up to 12 inches, then 12/30 or 2/5 is left so you can multiply 2/5 by the volume such as 200 gallons so you'd have 80 gallons left.

This could be extended to certain types of travel cups, cylindrical water troughs, columns to calculate the amount of cement needed, etc. 

On the other hand if you tank is rectangular, volume is simply length times width times height.  So to find the number of gallons by dividing by 231.  This would be great for calculating the amount of water for a pool, aquariums, holding tanks, etc.

So many real world applications and possibilities for students to work through.  Where are some places students might need to know this?  Let me know what you think.

Green Screen part 1

I have just started playing around with green screens and video making in Math.  Since I'm still learning, I chose a project aimed at kindergarten and first grades.  Its just counting cats but its a start.

This video counts from one to ten cats by the sea.  The rocks are cut out of a picture I took in front of the Waikiki Aquarium using preview. I placed them on a keynote slide with a green background (fern) and exported it as an image.  The numbers and cats are from a felt board app on my iPad.  I took individual screen shots of the numbers and   The background is a picture I took from my brother's deck looking out across Birch Bay to Mount Baker. 

I put it all together in iMovie so the first layer was the background shot with each photo on top and each was identified as green screen.  Add titles so the kids can read along and voice on top.  I exported it as 460 or so and voila, I have a movie.

I shared it with a few people already. The first grade teacher in the language immersion class would like me to make it again using seals and a local background so she can record a voice over in the local language while the regular first grade teacher wants a copy to share with his class.

You might want to know why I'm exploring this particular venue.  I want to know how to use green screens in a variety of ways so I can create better videos for my students and also to be able to help my students when they create films.

I have several ideas such as showing how to use transformations to prove congruent triangles or dilation to prove similar triangles.  I can put a coordinate plane in the back of something using green screen. 

I want to create a video on Dunkirk and how long it would take the fleet of boats here in the village to move 350,000 the same distance here.  I am working on one with the solar system, planets, distance, eccentricities, etc.

I'll probably post another video in a while when I get it done but I wanted to share this with you as it is the first time I've made a green screen movie.  Let me know what you think.

Wind Chill Charts

Over the weekend while being a line judge for the volleyball scrimmage, I wondered how certain charts would look if placed into a more linear form.  In other words, what if I took the information from a standard wind chill chart and created a line graph out of it?

I know that sounds crazy but a standard chart does not give the reader a good feel for what is happening but sometimes a line graph shows the information better.

So after finding a wind chill chart with the wind speed in knots and the temperature in Fahrenheit, I typed the data into an Excel spreadsheet.  I could have used mph for wind speed but knots is the usual around here. 

I enjoyed the results once they came up.  I did not bother putting all the axis and labels in just because I wanted to see how they looked.  What can one see in the pattern of the temperatures as the wind increases.

Isn't it a beauty?  Each color represents a different row of data for the temperatures.  On the other hand, I could have shown it by column and it no longer has a linear appearance.

Notice this time the lines are a bit more curved.  Which one is better?  The one that is based on temp being the primary focus with the wind speed secondary or vice versa?

This could lead to some great discussions on which factor is the dependent and independent or perhaps its harder to tell.  One could ask the students is this a true linear equation when showing the first graph?  They'd have to look to see if there is a consistent change?

I am looking at this as a fun exercise to get students thinking about graphing and ways of interpreting the data. Even exploring the idea of is this a valid way to represent the data?  What do you think?

Thoughts on Ways to Review The Material.

  We all know most students prefer to wait till the last moment to cram and hope they remember it long enough to pass the test.  I've heard stories of students who stayed up all night and then were so tired they filled an entire blue book with a single word yet were convinced when they walked out, they had written the best essay ever. Until they got it back with a big red F scrawled across the front of the book.

A few years ago, I took a summer course which promoted the idea of introducing the material a few days before it was taught and continue exposing children to the material after.  The idea is to expose students to the material everyday over a 21 day period so they retained it better.

Recently, a new idea has emerged where rather than concentrating on one skill, one should study several related skills together so as to mix the skills up.  This is shown to improve retention and understanding.  Now, a new study shows if a person breaks up two study sessions with sleep, it can create long lasting and effective learning.

The study selected 40 adults to either work on learning certain Swahili words.  The group who studied at night, slept, and studied the next morning did much better in three ways.  They had better retention, relearned words faster, and retained the information better than those who studied during in the morning and again at night.

This seems to support something I heard years ago.  I was told that if you read the material each evening a little while before going to bed, you are more likely to retain it and this new study seems to give credence to that idea.

Remember, sleep is important in memory formation so perhaps our brain is working on memorizing the material while we sleep.   There are three stages to learning material.  First is the acquisition where we are introduced to new material while the second stage is consolidation or the time when the memory becomes stable.  Finally is recall where the brain is able to access material stored in the brain. 

It appears that sleep helps the brain consolidate memories by creating stronger neural connections that form our memories.  Consolidation is the only step which occurs during sleep.  The other two only happen when the person is awake.

This may be why sleeping between two study sessions improves learning.  I'd love to hear what you think.  Thanks to Josh Fisher for arousing my curiosity to find out more.

Conic Sections Through Folding

Last week I reported on creating hyperbolas using a paper folding technique.  If you check the picture to the left, you can see the results of the activity. 

I started by passing out the written instructions so they could get started.  Once they began asking questions about the instructions, I showed them a video out of you tube showing how to do the exercise.  After that, they took off and had a blast. 

I collected one of the papers to share the results.  Guess what?  It is possible to do the same type of thing so you produce an ellipse, a circle, and a parabola using a line or circle and lots of folding.  The parabola uses a straight line rather than a circle to begin with but otherwise it uses the same type of folding technique to create the final product.

For the circle and ellipse, the shapes are created inside the original circle but you see the shape based on the folds.  The only technology needed is either a compass or a circular object students can trace and a marker.  I used an empty disposable coffee cup to provide the circle.

If you are interested you can check out the instructions for all four at this site.  If you want to include a video which provides clarification on how to fold, check out You Tube.  This video explains creating a ellipse from wax paper and lots of folding, while this one discusses using the same technique to create a parabola.

Although having already made one shape, its much easier to follow the written directions for other shapes.  A couple of my students chose to start again when they realized they were not folding the paper properly and they were extremely happy with the results.

I loved the activity because it only required I grab wax paper from home and it really worked.  I am going to try the other shapes in a day or two just to see how well they work.  If you want, go check out the directions and play with this at home.  I think you'll find it is lots of fun.

Warm-up

Warm-up

Math in the Newspapers

  Most people have access to newspapers.  I do although it is only published once a week and arrives on Wednesdays.  Its not big and only covers the region but it has some math in it.

When we read newspapers, there is quite a bit of math used.  Most people read the newspaper without realizing that.  Even I skim over things without paying too much attention but if you slow down and look, you'll find the math.

1.  The stock market - most newspapers include the current.

2.  Sports results with averages, etc.

3.  Recipes and cooking.

4.  Ads with sales information.

5. Weather forcasts.

6. Statistics, graphs, etc.

This article suggests ways to integrate articles into the class room so students see the amount of math found in the newspaper.  It has several examples to see how to use actual articles in the classroom.

The Newspaper Association of America Foundation has a great 48 page Teachers edition on mathematical connections in the newspaper for middle school students. The activities start off with two different scavenger hunts which have students looking for certain things in the newspaper and making note of the page.

Then it has students work through various things such as calculating the commission earned if you sell a house, salaries for jobs, pet ads, and all sorts of other activities based on things found in the news paper.

The Oklahoma Newspaper Foundation published a 22 page list of activities in each subject using a newspaper.  The suggestions are broken down into elementary and secondary activities.  Suggestions for secondary math begin on page 17 and include things like:
1. Figure out the percentage of space for type, photos etc.
 2. Look for examples of things which could be used to explain congruence.
 There are 28 different suggestions for secondary activities and I like that some of the activities deal with the many facets of newspapers.

This blog has some great quickie lessons from the Washington Times which are based on the normal things found in a newspaper.  Check it out.

Have fun checking these suggestions out and maybe you could use them in the classroom.  Let me know what you think.

How Are Hyperbolas Used In Real Life?

  When we teach conic sections, we include hyperbolas but have you ever wondered how they are used outside of the classroom?  I have but this is the first time I've really looked at the topic.  The ellipse activity went well so I want to an activity I can have students use for hyperbolas.

The picture above is found at the Dulles Airport.  The building is a hyperbola if viewed from one location but a parabola if viewed from another so we can see hyperbolas in architecture.  The shadow from a cylindrical lampshade or flashlight forms a hyperbola if you look carefully at the wall.

The open orbit of a comet as it goes around the sun forms a hyperbolic shape along with the area of interference between two circular waves.  Did you know the property of hyperbolas is used in radar tracking stations where they locate an object by sending out sound waves from two sources?  The circular sound waves intersect in a hyperbolic shape.

If you roll a steel ball by a strong magnet, the ball's path is changed from a straight path to a hyperbolic path.  This is easy to explore, just borrow a few things from the science department and you have an experiment.

I found this hyperbolic paper folding activity on line.  The activity uses wax paper and something circular .  The students form the hyperbola from the folds they create.  In addition, this You Tube video shows how to do the folding.  I think I'm going to try this activity as an introduction on hyperbolas with my students on Friday.

Illuminations from NCTM has a nice applet to explore the graph and conic sections of hyperbolas by playing around with various factors such as height, slant, m and b so students can explore to see what happens as each one is changed.

Another site is https://www.explorelearning.com/index.cfm?method=cResource.dspView&ResourceID=140Explore Learning with a gizmos app students can use to play around with the a, b, h and k to see how each term effects the hyperbola.  You can get 5 minutes use without signing up for the service.

I think my students are going to have fun later in the week with the paper folding activity.  Let me know what you think.

Planital Orbit

Yesterday my students finished up the assignment on ellipses where they created and graphed the equations for each of the planets in the solar system. 

I realize that I did not include gravitational pull and other factors but I created a simplified version so students could see the use of ellipses in real life.

I had them begin with the information on The Universe Today for both the closest and farthest distances from the sun.  The farthest distance was designated as the a value while the closest distance was designated as the b value for the basic equation.

Equations

After creating the basic equation, they calculated the value of c using c^2 = a^2 - b^2 because they needed it to figure out eccentricity or e = c/a.  After getting this done for each planet, they used Desmos to graph the orbits.

 

 

Graphs

 The hardest thing was getting students to recognize that what Desmos produced was not necessarily correct because the window had to be adjusted so the major axis appeared correctly.  It was a learning experience and this activity gave students a better understanding on how the resultant value of eccentricity. 

What Does Mark-up Cover

  Yesterday, the last thing I looked at was mark-ups but have your students ever asked why we even discuss markup?  Where does all that money go and are there businesses with bigger markups than others?

Mark-up is used to pay for the rent, benefits, employees, supplies and profit.  I've never really discussed it in detail with my students because I don't usually take the time.

In most retail businesses the standard markup is considered to be 50% which is just doubling the price.  The logic is if you buy a can of soda for $1 and sell it for $2, there is a 50% markup because $1 is half of $2.  This is a great way to show the logic behind the 50% number but mathematically its really a 100% markup because you start with the base number and calculate it that way. 

Add into this topic the idea that a business in rural areas are probably going to have a higher markup than an urban area due to higher transportation costs, fewer customers and other costs.  I live in a place where they have to barge in the fuel before the river freezes in the fall.  By the time the store sells it, it is about $6.25 per gallon where if I were in Anchorage, it would only be around $3.50 per gallon.  Over double the cost.  A half gallon of ice cream is sold for $12 to 15 dollars due to paying air freight to get here.

Most everything sold in the world has a markup but so do services such as temporary agencies.  They charge more than the rate the worker is paid.  The markups are between 20 and 60% or higher but cover the benefits of the worker and money for the agency so it can pay all its bills and make a profit.  The markup depends on the number of workers needed and if the worker has very specialized skills. Someone with specialized skills might have a 200 to 300 percent markup.

The Small Business Administration has a great seven page worksheet you could use in the classroom to show how one goes about identifying fixed costs, variable costs and other factors to determine the amount of markup one should use.  The first two pages are the only ones needed and the students could look up costs or call around to get prices or you could provide the numbers.

Give it a thought because this is a way to get students to apply markups in a real situation.

Percents in Context.

  We all teach percentages in our classes all under the category of mark-up, mark-down, etc but how often do we take time to help students determine if the percentage is good or bad.

For instance, if you get a 90% in a class we consider that great but if you look at a 2% discount on something, it might not be as good as a 10% discount.  I think its important to take time to help students understand percentages in context.

For instance, did you know the most successful professional sports betters only win 53 to 55% of the time.  Although that is just a bit over half the time but considered the best rates possible.  Yet if you look at basketball, good rates are a bit different.  If you look at team rates, the best rates I've seen for field goals is just over 50% while individual rates may seem impressive but you need to look at how far from the basket, the shooter is standing.

If you look at Tyson Chandler, he has a 68% shooting rate which sounds pretty good but if you look at the distance of his shots, you'll find 96% of those shots were made within a seven foot radius of the basket.  He only made 2 of 14 shots beyond the 7 foot range.

In baseball, a good slugging percent is between 41 and 44 % and anything over 50% is while a batting average of 27.5% is not bad.  However a 20% on base is bad while around a 35% is considered pretty good. 

If you look at a different field such as sales you'll find the rates are different.  If a sales person uses cold calling as a way to set appointments, they will only manage a 1 to 3% rate which is horrible but if you make your living that way, you try for the 3%.  On the other hand, if a sales person uses a referral, the rate jumps to 40%

On the other hand certain jobs are paid via a commission which is based on a certain percent of the total amount sold.  In other words, the more you sell, the more you make.  This is usually the pay which sells people such as car sales people, some telemarketers and retail sales people.

Another area is mark-up of common items such as soda from a fountain.  Did you know the mark-up for that is usually in the 20 times range or several hundred percent?  Most things like tea have a 3 to 400 percent markup which means they make a killing on it.

A large cheese pizza often has a 600 to 800 % markup so its a good seller.  In addition, pasta is another item with a huge markup because the dried boxed pasta is a few cents per ounce and commercial sauce is only like 30 cents per ounce.

A wide range of percentages whose meaning changes based on the context of the situation.  I'd love some feedback on this idea. Tomorrow, I'm looking at what things do mark-up cover.

Warm-up

Warm-up

Kepler's Law and Ellipses

  The other day in class, I took time to look at Kepler's first law and ellipses.  I included a bit of history of how at one point, people believed all planets had a perfect circular orbit but that was disproved in the 17th century.

I found a short article at Khan Academy which gave a great description of this.  Besides providing a bit of history, it also includes some animations which help illustrate it.

What is so cool about teaching ellipses with Kepler's law is the sun is the origin of the ellipses.  A coordinate plane could be placed over the orbits so students calculate the formula for the ellipse.  This article has great information on eccentricities.  I teach it but this is a great topic for for showing its application so its not something taught in isolation.

Just think, Neptune and Pluto have interesting paths because at certain points, Pluto is closer to the sun than Neptune.  This has to do with differences of eccentricity of orbit and is a great way of showing students that not all orbits are the same. 

This site has the distances between each planet and the sun for its closest point and its furthest point.  Using this information, students can create an elliptical equation for the planet's orbit. Yes, I am aware there are a lot of factors involved in the orbit but I'm looking at students creating the equation from the data.

Once they've created the equations, they can use the information to calculate the eccentricities for each orbit and compare their answers to the actual answers.  This leads to a great line of questioning on why they might be different.

Yes I'm going to be doing this today in my advanced math class.  I'm interested in seeing how well it goes.  I'll report back on Monday and let you know how it goes.

Parallel Lines are Where?

  As you know by know, I'm always looking for places math is used in real life so as to show students it is practical.  Usually, I look for connections in Algebra I or II but today, I'm looking at parallel lines which appear in those two plus Geometry.

We teach parallel lines as lines that never cross. The lines have the same slope but different y intercepts.  Do we really take time to really discuss when students will see these in real life or why its important to know how to find them?

Look at the picture, they are railroad tracks which have to be parallel because the distance between wheels will never change.  If the tracks are not parallel, the trains will derail and it could cost the company millions of dollars.  What about all the lanes on the roads or highways?  Those lines have to be parallel so none of the cars will get close enough to run each other off the roads.

What about the rows of shelving in the supermarket.  Most are set up as parallel segments so as to allow enough room for carts to pass each other in each aisle.  Some of those shopping carts are getting rather wide especially the ones set up children.  The parallel shelving can be found in libraries, book stores, hardware stores and so many other places.

In addition, you can look at parking lots because there are rows up rows of parallel lines and perpendicular lines.  Even buildings have parallel and perpendicular segments which make the walls of the building.  Windows have both parallel and perpendicular segments to create the whole effect. 

It seems like every where you look you see parallel lines be it in wood flooring, brick walls, stairs, ladders, and so many other places, even on cement sidewalks.

Just think what type of brainstorming you could have the students do when starting a unit on parallel and perpendicular lines.  You could even introduce the idea when state build certain intersections, they are required to make them meet at a 90 degree angle which means they are perpendicular.  Why would the government require that?  It requires students to think about the reasons behind such a requirement. 

So many fun things to discuss when you talk about this topic.  What do you think? I'd love to hear from you all on your opinion on this topic. 

Is This Possible?

  As most of you know, I live in the middle of bush Alaska. The reality is that most of my students will not go to college and probably never will.  A few are willing to head out for training but even those are few and far between.

A parent commented to me that the high school does not offer enough vocational math for the students who are not interested in attending college.

So what do I have to do to help these students so they don't get lost in the shuffle and do not have a chance to get the math they want.  I'm looking at integrating applied math into my standard math classes.  For instance, I can integrate some carpentry math during Pre-Algebra.  There is a part of the class where I have to review fractions. 

Fractions are an intricate part of carpentry.  In addition, I can include road grade and roof pitch when I'm teaching slope in several of my classes so why not add in a few roofs for students to find the pitch.  Then there is area and calculating the amount of paint, flooring, and ceiling tiles.

Of course cooking can also be incorporated while studying fractions because most recipes have fractions in them.  Add in the skill of enlarging or reducing and you've added in multiplication or division of fractions. 

Throw in pricing for items which allow you to take a discount when you buy more items.  You'll find this type of pricing at Fire Mountain Gems and Beads.  Let some of your artistic students create a design on grid paper and decide what they need to order to complete the piece of jewelry and the complete price.  Once they know the cost of materials, they can make an estimate of time and calculate a finished price for the jewelry.

Another place that uses this type of pricing is the same so have students who are into electronics, figure out what they'd like to order, discover the prices and calculate the cost.  In either case, students can calculate the rate of discount for each level to decide if buying the extra is worth the discount.

Back to the original question, is it possible to integrate things like this into the classes we are teaching for the students who are college bound while still meeting the needs of those who are going a different path?  Will it help those heading to college because they will see a real application of the math they are studying?

Let me know your thoughts.  I would love to hear from people.

Why Do Signs Work This Way?

Most of the students in my afternoon Algebra I class are having trouble understanding why a negative times a negative is a positive visually.  These students are very ELL and struggle with mathematics every day. 

I've been able to create illustrations for adding two negative numbers, adding one negative and one positive, multiplying a negative by a positive, and dividing a negative by a positive but I have not managed to create drawings for multiplying a negative times a negative or dividing a negative by a negative.

I can find all sorts of examples showing the usual but most of the information I find is with the if it works this way, it has to work that way but after a lot of searching I finally found an analogy explaining it.

" If you film a man running forwards (+) and then play the film forward (+) he is still running forward (+). If you play the film backward (−) he appears to be running backwards (−) so the result of multiplying a positive and a negative is negative. Same goes for if you film a man running backwards (−) and play it normally (+) he appears to be still running backwards (−). Now, if you film a man running backwards (−) and play it backwards (−) he appears to be running forward (+). The level to which you speed up the rewind doesn't matter (−3x or −4x) these results hold true.

backward×backward=forward

negative×negative=positive""

I got the above from the Stack Exchange.  It was really one of the only explanations I found that my students might be able to relate to.

Dr Math at the Math forum uses the idea of a mortgage payment to illustrate this particular operation.  If you pay $700 per month for your house payment each month you'll spend $8400 every year which is subtracted from the money you have in your bank account.  So a total of -$8400 or 12 times -$700 illustrating a positive times a negative is a negative.

But what if your employer decides to pay the 12 months for you instead so you are not paying the 12 months which is minus 12 months of -$700 or the payment so its -12 x -$700 or a positive $8400 because you have that much more in your pocket at the end of the year.

So for subtracting a negative from a negative could possibly be viewed as you are going forward, someone calls your name so you turn to face backwards.  You don't see anyone so turn to face forward and run forward so its a positive.

I'd love to hear your thoughts on this topic?  Do you have other ways to show it other than using the usual mathematical methods.