Driving - Miles Per Hour

This video has gone viral recently and reminded me of a funny incident of doing some quick math in my head to win a bet.  You might have seen this.  If not, take a peek!

This is probably how I'd react to simple questions about Harry Potter or Star Wars (no, I've never seen either - unleash your insults in the comments), so I empathize.  But this emphasizes thinking of such a common label (MPH) in a different way.  Oh yeah, 80 miles per hour means 80 miles traveled in one hour (or 60 minutes)!

So one night I was driving a bunch of high school kids back home from an away football game.  We were a long distance from home, and I told Jeff, who was riding shotgun, that I'd tell him the exact time we'd get off the interstate.  He wanted to bet something on it, so I told him a Coke.  I told him we'd get off the interstate at 12:18am.  Sure enough, when we reached our exit, the time was 12:18am!  I never made Jeff pay up, but he and I still talk about it and laugh 5 years later as our first real interaction!  So how did I do it?

It has to do with this miles per hour thing, along with the fact that the mile markers on the interstate are a mile apart.  Here was my thought process: We were driving 70mph.  That's 70 miles per 1 hour, aka 70 miles/60 minutes.  Reducing the fraction, that's 7 miles/6 minutes.  That's a ratio.  You could also write the ratio 6 minutes/7 miles.

I knew this route very well.  Here's the diagram.  Upon approaching the Mile 80 marker, I subtracted and saw we had 47 miles to I-75 and 17 miles to exit, so 64 miles left.  I knew that I was going to take either 7/6 or 6/7 of 64.  Since I'm going over 60 miles/hour (a mile per minute), I know it's going to take less time than 64 minutes.  Multiplying by 6/7 makes sense since multiplying by a fraction less than 1 makes it smaller (6/7 < 1).

64 divided by 7 = 9 1/7.  Multiply by 6, and I get 54 6/7 (fractions are better than decimals*, just sayin'), or almost 55 minutes to get there from that point!

So when approaching the Mile 80 marker, it was 11:23pm and I made the bet!

Of course, I was driving and could ensure we stayed around 70 mph.  If we got behind someone slow, I could speed up later. :)  Try it on a long trip sometime!  Maybe you could win some favors or a Coke!  But make sure you do the math right and start figuring it out a couple miles ahead of time to be ready to make the wager.  Try it by yourself first.

Notes:

Mathematically, you could set up a proportion.  6/7 = x/64 and solve for x.

If going 60mph, it's simple.  64 miles in 64 minutes

If going 75mph, 60min/75miles = 4/5.  Multiply distance by 4/5.

If going 80mph, 60min/80miles = 3/4.  Multiply distance by 3/4.

Don't feel the need to speed up to 120mph just to multiply by 1/2!

And wear your seatbelt, be responsible, and all that. :)

What you learn in class that applies:

1. Ratios

2. Being mindful of labels

3. Multiplying fractions

4. Reducing fractions

5. Comparing fractions (6/7 < 1 and 7/6 > 1)

6. Solving proportions

I wonder if anyone else has tried something like this besides me.

*Fractions are exact.  Decimals are not always exact. 1/7 is .142857 repeating.

Excel - Basketball League Scoresheet

     Every spring, we run a basketball league for 19-and-under-year-old kids as one of our programs.  One of the fun things we do is keep track of the kids' points and keep track of points per game throughout the year.  We also track fouls during the game.  5 fouls and you're out of the game.  To handle this, I've created an Excel spreadsheet that I print out for us to log these stats during the game.  I love Excel.  There, I said it.  I also set up a master list of all the team rosters that is similar to the scoresheet.  Here is a snapshot of each:

Before, each week I had been copying the entire team on the Rosters sheet, switching tabs, and pasting the team into the Scoresheet. (Ex: Select all of Team 2 on the right, copy to clipboard, paste over top of Team 5 on the left for next week's game)

On the Scoresheet, I wanted to set it up to where I could just type in the team number in cell A25, for example, and it would pull the entire team roster from the Rosters sheet for me!  In order to do this, I'm going to have to set up a bunch of formulas to map to the correct row on the Rosters sheet.

On the Scoresheet, say I put "1" in A25.  Then I want B26 to grab "Mike" from Rosters - B2.

If I put "2" in A25, I want B26 to grab "Chris" in Rosters - B13.

If I put "3" in A25, I want B26 to grab "Nate" in Rosters - B24 (not shown).

If I put "4" in A25, I want B26 to grab "Drew" in Rosters - B35 (not shown).

And so on...  I need a formula that grabs a different name based on the number in A25.

1 --maps to--> 2        (as in B2)

2 --maps to--> 13      (as in B13)

3 --maps to--> 24      (as in B24)

4 --maps to--> 35      (as in B35)

5 --maps to--> 46      (as in B46)

6 --maps to--> 57      (as in B57)

7 --maps to--> 68      (as in B68)

The difference between each number on the right is 11.

Ever remember learning y - y0 = m (x - x0)?  It's the point-slope formula you learn in algebra.  We're going to need it!  The rate of change, or slope (m), is 11.
Note that the number on the right is dependent on the number on the left. y-values are dependent and x-values are independent.  So the values on the left are x-values, and the numbers on the right are our y-values.

Plug in any number and it's mapped number from that list. We'll choose the first and easiest: 1 mapping to 2

y-y0 = m(x-x0)

y - 2 = 11(x - 1)

y - 2 = 11x - 11
y - 2 + 2 = 11x - 11 + 2

y = 11x - 9

Sure enough, that's the exact formula I need to transfer player 1 to his spot automatically!

The formula part says 11 times the number in A15, then the "-10+A16" part...A16 is "1", so "-10+1" is -9...so 11x-9, just like our formula above! (I made that last part more complicated for flexibility with the formula copying and pasting etc.  If I copy this to the next line down, I want it to be 11x-9+1 or 11x-8.)

This is the same process I used throughout the whole spreadsheet to automatically pull the correct names from the Rosters based on that one itty-bitty number in that one cell!  If I change the number in that cell, all the names change on the Scoresheet for me!  Not only did I use this method here, but also in another spreadsheet where I can enter the scores for the games and all the standings and team records update for me!

This took some work, but because of it, I am literally doing 10 times less work every week on this.  It is a breeze now.  I didn't have to do this, but this knowledge and skill saved me time here and will in the future, too... on this and other useful sheets!

What you learn in class that applies:
1. Rate of change or slope
2. Mapping x-values to y-values
3. Point-slope formula
4. Dependent (y) and independent (x) variables
5. Balancing an equation (adding 2 to both sides)
6. Distributive property (multiplying 11 into the x and 1)

Making Mass Lemonade

      There are a ton of instances where we have to make mass quantities of lemonade or iced tea.  We run 3 booths at our town's Heritage Festival to help kids raise money for camp, and we serve both at our booths.  To dispense the lemonade or iced tea, we have many 5-gallon coolers.  We are using the one below this weekend to serve drinks to our kids working our Pancake Days event at the local mall.

     Inside the canister of mix, there is a plastic scoop with a chart of how many scoops make different quantities of lemonade.  

The problem: There is no way we are scooping out 16 scoops of mix for two gallons of lemonade.  We might need 5 gallons!  During a festival, we don't have time to scoop out 40 scoops of mix, especially when workers have to leave a booth to go to get the water, leaving the booth under-staffed.

How can we make this easier?!?!


Well, we notice from the label that a 1/2 scoop is a serving, as if anyone uses that amount.  And there are about 136 servings in the container.  I determine that 136 servings/container * 1/2 scoops/serving = 68 scoops/container.  Since 8 scoops makes a gallon, 40 scoops would make 5 gallons.


Well, half the container would be 34 scoops, less than it suggests for what we want.  We decide: let's try just half the can for five gallons of lemonade with junior high kids at our After School Break program.  Junior high kids can be the true judge of lemonade quality!  Sugar!  The can suggests it will be watered down a bit, but usually they tell you to use a bit more mix than necessary to keep you buying more.  When we tried out the half-can theory, the junior high kids approved!


So our rule now: just eyeball half a can of mix for a 5-gallon cooler.  Need 2 gallons for our smaller high school After School Break?  2/5 of 1/2 can. 2/5 * 1/2 = 2/10 = 1/5 = 20%.  Just eyeball about 1/5 or 20% of the can (or take the time to scoop out 16 scoops of mix).  If you want to go 1/4 of the can (1/2 of 1/2), just fill the cooler to 2.5 gallons (1/2 of 5 gallons) with water.


This is simple math, but the benefit is huge.


When junior high kids need more lemonade at their festival booth, they don't need to pull us away from what we're doing anymore to fill their cooler.  We can just tell them, "half a can of mix, fill 'er up with water."  Our high school kids or leaders can eyeball a certain amount of mix for a varied amount of lemonade.  We don't have to scoop anymore (thank goodness).  We save so much time now.


What I love is that the real-life problem starts with no particular correct answer. The goal is to just make the problem less of a problem.  Another thing: with all those numbers flying around, using labels is important for sanity's sake.


What you learn in class that applies:
1. Multiplying fractions
2. "Of" means to multiply
3. Factor-Label method (servings/container * scoops/serving = scoops/container)
4. Converting fractions to percents and vice versa
5. Making & testing a hypothesis (part of the scientific process)
And of course, many things behind the scenes such as problem solving and spatial reasoning.

The Common Question

"When will I use this?"  This is the question that would frustrate me most as a math teacher.  Of course you will use this in real life. "But I'm not going to be a math teacher," they would say. Insert the word engineer, architect, programmer, accountant, or other math-centered field here.


My struggle was communicating how different topics would benefit kids when they got older and into the "real world."  Personally, with time, managing our finances, completing projects around the house, a factory job, and now a ministry job, there are a number of times I wish I could go back and say, "Here!" "Here!" "Here!"


I think there are at least two reasons kids ask this question. (1) They really don't see why a particular concept is important in the real world and/or (2) It's an easy go-to excuse to try to legitimize dislike for the content/subject.  After living life a bit, and with a math background, I feel like I can address - to some extent - the first one.


Here is something that Patriots receiver Chad Ochocinco posted on Facebook.

To be clear, this is not actually an equation.  This was actually the second time I had seen this same status.  My response is at the bottom.  But this exemplifies how a lot of kids feel in math class.

Future posts on this blog will have more practical examples.  But I want to throw out there right off the bat that regardless of whether or not you feel you can use the content in real life, you are learning the following skills (which you will definitely use in real life) while learning math:

- Learning logic

- Making logical arguments (which can in fact make you sound smarter/intelligent)

- Using rules to come up with more complex rules

- Applying known rules to an unknown situation (problem solving)

- Thinking critically

- Thinking abstractly

I would almost argue that these are more important than the content since you use them so often.

Also, there may be many adults that feel that they don't use a lot of mathematical content in their life.  I don't believe math is the answer to life's most important questions, but there are many times when if you were to use it or knew where to use it, it would make life much easier and heck, maybe even save money!


I look forward to this!