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.