Friday, February 17, 2017

Weighted Averages

Accounting, Report, Credit Card, Payment  As part of my animation class, my students are learning about weighted averages.  Many of the students in the animation class struggle with math.  They don't all know their multiplication tables, they find doing regular problems but as part of the class, they have to understand weighted averages.

I started class by showing students how grades are calculated if the teacher has chosen to weighted averages.  I do and I know many high school teachers who do it.

Several students are athletes at school.  They know they must have a 2.0 or higher in order to travel but they had no idea how their grade point average was calculated.  They found it informative when i showed them.  I also explained how finding a college grade point average differed from the high school average.

These three examples are ones they are familiar with.  I took them outside of their area of knowledge to look at average balances or average sales per day in a month.  I brought up the idea that one of the students owned a business showing others how to set up an MMA fight in their town.  A few eyes brightened because two students are into MMA.

I talked about one of them creating a business where he teaches others to set up and run MMA fights in their location.  I made up some numbers like he earned $700 per day for 15 days, $1500 each day for 2 days and $800 per day for 13 days.  I lead them through calculating the daily average and pointed out the monthly income from this venture.

A few students looked stunned at the idea that creating a product they sold to others so others could do something on their own. This idea of creating a product to sell to others is beyond something they have ever thought of.  It is something that can be done from a remote village so they do not have to leave.

The village is pretty much accessible only by plane if they want to get to Anchorage.  I showed them the same math could be used to determine the average number of passengers carried per month or per year for the local airline.  Then I changed it to their bank account to find the daily average balance and daily average credit card balance using the weighted average. 

It was fantastic.  This was one of the first topics I had no trouble finding lots of examples that use weighted averages.  I loved it.

Now for a quick question to all my readers.  I am thinking of creating a series of lessons helping people create math videos using green screening techniques.  How many might be interested in this type of information?.  Please let me know.  Thank you.

Thursday, February 16, 2017


Airline, Airport, Gateway, Gate  I am in transit right now.  I'm on the way in to a conference.  I know most of my students have no idea of the cost involved in travel.  This would be a great topic for one day.

Have the students plan a trip, figure out the cost for them to go to their dream destination, of the hotel, rental car, admissions to things, food, etc.

They could do this for one person and for their family.  The final cost could be quite shocking.

The trip from my village to the hub is $500 round trip.  From the hub to Anchorage is another $300 to $400 round trip, so about $800 to $900.  The hotel in the winter isn't too bad but if you have to calculate the trip using summer rates, it means a cheap hotel is over $100 a night.  I'm not going to talk about the upscale ones.

Everything is so much more expensive in the summer so costs about double.  The food and admissions stay the same but it can be rather expensive normally.  

A week trip for one person is probably in the $3000 range for airfare, hotel, car, food, etc.  If the family is composed of 6 to 9 people, that makes the cost jump. 

For me, I can fly 2 people in August from Fairbanks to Los Angles for under $1400 round trip which is just a bit more than here to Anchorage.

If students choose to look at the costs at different times of the year to see how much seasonal changes affect the costs.  It would be easy to create an excel spreadsheet with the costs so they could create graphs to compare the different costs.

Yeah, a great real life problem.  I've got to go catch my flight.  Have a good day.

Wednesday, February 15, 2017

Famous Ratios and Ratios Used With No Thought.

Calculator, Three Dimensional, SymbolYesterday, I let my mind wander.  It began with division before heading off to scales as in scale models, eventually settling on ratios.  There are several famous ratios we use in real life.  Mathematical things we don't even think about being ratios.

The most famous example is Pi.  Pi is defined as the ratio of the circumference to the radius of a circle.  How many times have you had students try to measure the circumference of a circle using a string that was later lined up against a ruler.  The ruler is also used to determine the radius.

This activity has lead to great  discussions on why their results are no where close to the actual number.  We've discussed stretch, lack of precise measurement, and all sorts of other issues.

The golden ratio is another famous one.  Basically it is the whole length/long part = long part/short part or approx. 1.618....  It appears in art, architecture,  geometry, and other areas.  This is even mentioned in Numb3rs and the Da Vinci Code.  It is said the golden ratio was used by the Egyptians while it was used by Da Vinci when he painted the Last Supper.  In addition, people who create labels for soda and such use it so the ratios look right on the bottles.

In addition, dentists use the golden ratio when fixing teeth.  Both Notre Dame and the Parthenon were built using the golden ratio.  There are more places you see it, in certain instruments or even with insects.  The golden ratio has a tremendous influence without our being aware of it. 

In the financial realm, there are many ratios used and the following are just two.
1. Price to earnings ratio which is used to determine if the price of a stock is reasonable.
2. Profit margins = Net income/sales.
I hadn't heard of these but if you are in business, you are likely to be quite familiar with them.

Even in maps, there are certain standard ratios found.  The USGS uses 11 different scales on their maps depending what the map is of.  If its of Puerto Rico, it will have a 1:20,000 ratio while the map of the United States is 1:1,000,000.  

Anytime you look at  road map, an atlas, or anything else that has a map, you are going to see a ratio which is often referred to as a map scale.  It might be 1 cm represents 20,000 cm or 1 inch represents 100 km.  It depends on how the scale is set up.

In addition, look at building plans whether for furniture or for houses.  They are all done to scale with a ratio such as 1:20, 1:50, or 1:100 in S.I. units or 1/4" or 1/8" for US units.  the 1/4" inch means 1/4" on the plan represents one foot when its built. 

All these ways and we don't give it a second thought we are using ratios.  We don't think about it at all.

Tuesday, February 14, 2017

Mistakes in Division

Time, Split, By, Divide, Multiplication  Over the past few years, I've noticed a trend in my incoming students.  The majority of the students seem to make the same two mistakes when dividing.  I do not know where it comes from.  I have no idea where or when the misconception developed.

It is frustrating because I am not sure how or when to reteach the topic so students start doing it correctly.

First is when students divide, they do not place a zero in as a place holder.  An example would be dividing 5020/10.  So 10 goes into 50 five times.  The student writes 5 above and brings down the 2 but does not put a zero above it.  They bring down the 0 for 20 and put 2 above it because 10 x 2 = 20. 

Instead of 502, they come up with 52.  These students do not recognize they need a zero in there as a place holder.

I've been thinking of having students do their division either on graph paper so they can place one digit per column or use lined paper sideways.  If there is supposed to be a number in each column for the answer, it might provide an automatic reminder to put the zero in for a place holder.

The other situation is thinking the remainder is the number you place next to the decimal.  An example would be 13/5.  The student knows 5 goes into 13, twice.  They write 10 and subtract so 13 - 10 is 3.  They put 5.3 rather than 5.6 because 3/5 = .6.  It is something that occurs with great regularity. 

This one is a bit more challenging.   I am not sure how to have them think about converting the remainder into a decimal.  The only idea I have is to have them create a picture showing the remainder as a fraction of the original. Once they have a fraction, they can convert it from the illustration into a decimal. 

If anyone has any suggestions on ways to help students overcome these misconceptions, I would love to hear.  I realize I could just let them do the math on the calculator and not worry about these misconceptions but the first issue could translate into dividing rational expressions.  They might not put a zero when needed if dividing x^2 + x -3/x+1. In addition, they might not use the remainder properly.

Monday, February 13, 2017

Positive and Negative Numbers

Athletic Field, Ground, Lane, Lines  Last night, I thought about how we use positive and negative numbers in real life, yet we teach it using a horizontal number line.

We teach  children to count in a positive or negative  direction based on the numerical sign.  Even now when students stumble, I place a dot on the board and point in a direction, to prompt them.

I don't think we spend much time giving students context for applying these two concepts to real life.

We have mountains we can see from the porch of our school.  Its nice to connect positive numbers to climbing it since we are increasing elevation, and negative numbers to going down because elevation is decreasing.  The same could be applied to airplanes as the rise to their cruising altitude or descending to land on the ground. 

The stock market gives regular reports with positive and negative numbers as the Dow Jones shoots up or drops.  Daily temperatures rise or drop depending on the time of day. There are times when both the high and the low are negative numbers so the rising would be the positive part and dropping would be the negative.

Population growth can use positive numbers indicating it is growing or negative because the population looked at is declining.  Most people discuss weight as gaining or loosing it both number which fall into today's topic.  In addition credit card balances are shown with positive (items purchased) or negative (amount paid). 

The situation that hits most of us immediately is our bank account with deposits causing our accounts to increase, or withdrawals causing our accounts to decrease.  If we spend enough, we could owe the bank because we know have a negative balance.

Even general elevations such as Mount Whitney in California is over 14,000 feet above sea level while Death Valley is over 200 feet below sea level.  Then we have high and low tides, especially the Bay of Fundy which has tremendous differences.

Everyday, we hear words that indicate the use of positive and negative numbers such as "I'm short this month" meaning a negative amount compared to the amount needed.  I traveled 80 miles further than I anticipated today which is + 80 miles.

I think I'm going to use this topic as a welcome back to school activity where I have students brainstorm all the uses we have for positive and negative numbers.  There are so many situations my students could relate to.  Yeah.

Let me know what you think.