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.

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.