Faster or slower?

This semester, I’m driving between East Lansing and Grand Rapids once each week to teach a Math Methods course for our graduate student interns who are student teachers in the Grand Rapids area. A one-way trip is about 68 miles, so it takes about an hour.

Everyday-Driving

Last week, the drive to Grand Rapids took about 2 hours, because of an accident. I ended up being a half an hour late (thank goodness for my co-instructor!) to class.

Before I hit traffic, I was hoping to arrive at 8:00 AM (although the class didn’t begin until 8:30). At 7:30, I was about 37 miles from my exit. I wondered: How fast would I have to drive to arrive to the building by 8:00 AM? I thought, if I could drive ten miles over the speed limit, I could arrive before 8! But would it be worth it? There are a lot of reasons not to speed, mostly my reasons are: (a) fear of a ticket or (b) fear of an accident. A ticket is painful financially plus it takes time to be pulled over – so by driving faster, I might end up arriving later. An accident is worse, because it could hurt me physically, or it could hurt others. I could still get a ticket. And an accident would slow me down even more! What are other reasons I shouldn’t speed? And – (a question I ask myself all the time) what is “speeding”, really? That is, when is am I going to fast? Is it going fast enough to get a ticket? Is it going too fast to react well? (in that case, the top speed would change if I’m a little tired or distracted) As I thought about these things, I also realized that going faster for such a small distance didn’t really benefit me that much – was it worth it to go 20 miles over the speed limit to save seven minutes?

Everyday-Driving-Table-03

Traffic slowed to a crawl at 7:59 AM, I was still 13 miles from my exit, and I needed to be to class no later than 8:30. Once I take my exit, it usually takes me about 5-10 minutes to get to the building, get out of my car, and walk to the classroom. So, I would assume that, in order to get to the classroom on time, I should be at my exit no later than 8:20.

Everyday-Driving-Table-01

I thought through my options, and realized that as long as I could get up to 45 mph, I’d be fine. I had no worries at that point.  But about 20 minutes later (8:20), I still was 11 miles away. As long as I could get up to at least 60 mph, I’d probably be fine. But if I could go 70 (the speed limit) then that would be perfect. I noticed that – at that point – driving even 30 miles over the speed limit wouldn’t save much time, but my risk of an accident or a ticket would drastically increase. So speeding probably wouldn’t be a good decision, even though I wanted to feel like I was making progress.

Everyday-Driving-Table-02

Unfortunately, traffic didn’t start moving until about 8:40, and by that time I was already late.

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Ely, Nevada and its surrounding area

I discovered Ely, Nevada’s homepage recently and particularly enjoyed it: Ely, Nevada Home Page., but especially for the sentence (near the bottom of the page): “Centrally located in the intermountain west, Ely and the surrounding area contains a population of over 6 million people within a 500 mile radius.”

I loved the humor and creativity evident in that sentence. And then I thought, “What questions could I ask to make sense of this statement? How could I share this statement with a math class?” I considered the questions:

  • In order to engage in answering any of the questions below, what tools can I use?
    • Is there an online tool that, given a map, would create the circle for me? Is there a tool that would add up the populations for me?
    • Creating the circle (if using the map, figuring out scales, etc. is important):
      • Get a map of US (either a folded map or print out a map from Google Maps) and a compass, measure the size of “500 miles” and use the compass to create a circle
      • Get a digital map of US and use GeoGebra to draw a circle (depending on how precise I want)
    • Finding populations:
      • How precise should I be?
      • Should I choose the largest cities, and look up the metro populations? What is my “cut-off point” for city size?
      • How far would I be under-estimating, if I don’t count all of the smaller towns and cities?
  • What does “surrounding area” really mean?
    • If I draw my intuitive sense of what “surrounding area” means on a map, how big of an area does it include? Do my classmates have similar or different ideas about the size of “surrounding area”?
    • How big of a radius should I use to include the whole US? Where would I center it? How much of Canada, Mexico, or other countries would be included?
  • How big is a region with a 500-mile radius?
    • What cities / metro areas are included in the 500-mile radius circle centered at Ely?
    • Could they have chosen a smaller radius and still included an impressive population?
    • Are there particular cities they wanted to include that inspired their choice of 500 miles?
  • Where I am now (in East Lansing, MI):
    • If I created a circular region of radius 500 miles centered on East Lansing
      • How many people would be contained in that region?
      • What cities would be contained in that region?
    • If I wanted to include a population of 6 million people, how large should the radius be, if I centered the circle on East Lansing?

I will try to engage in this task when I have more time…

In order to engage in answering any of the questions below, what tools can I use?

  • Is there an online tool that, given a map, would create the circle for me?
  • Is there a tool that would add up the populations for me?
  • Finding populations:
    • How precise should I be?
    • Should I choose the largest cities, and look up the metro populations? What is my “cut-off point” for city size?
    • How far would I be under-estimating, if I don’t count all of the smaller towns and cities?

How big is a region with a 500-mile radius?

Radius for Ely

  • What cities / metro areas are included in the 500-mile radius circle centered at Ely? The map shows some large cities are: Salt Lake, Ogden, Provo, Las Vegas, Reno, Phoenix, Los Angeles, San Jose, San Francisco, Fresno, Sacramento, San Diego, Tijuana (Mexico), Boise, Idaho Falls
    • I’m going to estimate the population. I notice that all of Nevada (2.839 million) and Utah (2.943 million) are contained. So that’s already about 6 million people! Most of California is contained, except for a tiny piece in the northwest that includes Crescent and Eureka, and California has population 38.8 million – let’s be conservative and knock off about 8.8 million for that tiny corner and we still have 36 million contained. The Phoenix metro area adds about 3 million. Tijuana adds about 1.483 million. Boise (0.616) and Idaho Falls (0.130) add a little less than a million. I could keep adding on, but a low estimate is still over 40 million.
    • I found that the same site I used to create the circle also will estimate population within the radius, and it says “The estimated population in the defined area is 49,997,320.” 
  • Could they have chosen a smaller radius and still included an impressive population?
    • I’m going to guesstimate and say 250-300 mile radius would probably still include 6 million people.
    • Let me test my conjecture. (Note that I needed to reset the site before changing the radius, or it told me the same population as above.)  I entered 250 miles and was told “The estimated population in the defined area is 4,331,681.”  300 miles was closer with 5,438,487, and 315 was slightly more than 6 million with 6,396,013.
    • Radius-315-for Ely
  • Are there particular cities they wanted to include that inspired their choice of 500 miles?
    • I’m guessing it is cool to say that San Fran and other California cities are “in the surrounding area.”

Where I am now (in East Lansing, MI):

If I created a circular region of radius 500 miles centered on East LansingRadius-500-for East Lansing 

  • How many people would be contained in that region? Well, it includes all of Michigan (9.91), Wisconsin (5.758), Illinois (12.88), Indiana (6.597), Ohio (11.59), West Virginia (1.85), Washington, D.C. (0.658), and Kentucky (4.413). So, as a low estimate, a 500-mile region would include a population of at least 53.656 million.
  • I was much, much too low! With the population finder, I found  the estimated population in the defined area is 101,594,400
  • If I wanted to include a population of 6 million people, how large should the radius be, if I centered the circle on East Lansing? I conjecture 90 miles. Metro areas for Detroit (3.734), Lansing (0.464), Grand Rapids-Muskegon-Holland (1.321), Kalamazoo (0.326), and Flint (0.425) would be just over 6 million.

I’m really under-estimating though, because I choose to balance time over precision (time of looking up populations), but the radius around East Lansing of 90 miles would definitely include 6 million people. Using the population tool, I found that 75 miles radius would include 6,795,008 people.

Radius-75-for East Lansing

Gambling with Woot

MysteryBox-01

A few weeks ago, Woot.com advertised a “Mystery Box of Electronics.” One box cost $50. Woot said: “Not knowing is half the fun!” Each box would include:

  • Random consumer electronics items
  • You get 3 items per order in a brown box (nothing special about the box!)
  • Condition will be refurbished on most items. Some could be new

So, I decided, why not? And I gambled on Woot – I bought three mystery boxes of electronics! (Because not knowing really is kind of fun…)


I received my last box today! Here is what I received:

2015-09-17 13.34.58

Box 1:

I received three items:

  • A very pink “beat mixr” headset. Other listings on Amazon.com show prices for new/used (for some reason some new are cheaper than the used!), starting as low as: $110.
  • A small Acesori PowerStick. Other listings on Amazon.com show prices for new/used, starting as low as: $4.
  • An Acesori Glass Vault screen protector for an iPhone 5. Other listings on Amazon.com show prices for new/used, starting as low as:  $7.

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Box 2:

In the box, I found:

  • ifocus “Deluxe edition” educational system for kids (a set of CDs: one is an educational game and the other is a fitness program). The company that sells these says they are $199.95, but it doesn’t look like I could resell them
  • An Acesori Glass Vault screen protector for an iPhone 5. Other listings on Amazon.com show prices for new/used, starting as low as: $7.
  • Acesori Bluetooth Noise-canceling Neckband Headset with Built-in Microphone. Other listings on Amazon.com show prices for new/used, starting as low as: $30.
  • LG Electronics Gruve Bluetooth Stereo Headset. Other listings on Amazon.com show prices for new/used, starting as low as: $30.

WP_20150922_17_20_56_Pro

Box 3:

  • BlueAnt Pump – Wireless HD Sportbuds – Black. Other listings on Amazon.com show prices for new/used, starting as low as: $26.
  • Turtle Beach Ear Force Z11 Amplified Gaming Headset. Other listings on Amazon.com show prices for new/used, starting as low as: $24.
  • TruGuard Tempered Glass Screen Protector for iPhone 5/iPhone 5S & iPhone 5C. Other listings on Amazon.com show prices for new/used, starting as low as: $6.

So the big mathematics question is: Did I get my money’s worth?

In the Woot discussion (http://electronics.woot.com/forums/viewpost.aspx?postid=6481553&pageindex=60), many customers say, “No!” Others say, “Yes!” I’ll give several comments here, and let you decide!

  • The value is there for at least what everyone paid for when they ordered it. The problem is people discount the value simply because it’s something they don’t want or can’t use, or they deem to be useless. That’s on them, not Woot.
  • This sale rather was marked as getting mystery items that were either new or refurbished worth at least $200. So for $50 bucks, if thats the case, how could we go wrong. We should all consider that that this $200 value should not by any means be gauged on our perception of value, but rather fair market value, which is really the point of taking up a mystery box. It may be worth it to you, but it may not. If its not, you can go through the legwork of selling it off and making back your money.
  • (received my box 3 above) I am not satisfied because I don’t feel this box is worth the 50 bucks I paid. But still I will get use out of some of it. I understand that with these boxes you run the risk of getting items you don’t want or won’t use but you expect the value to still be there.
  • As to “value”, that is in the eye of the beholder. The great marketing dilemma. Do we want to talk about MSRP, street price at release, current street price, or the best deal available? Naturally, very few items ever sell at MSRP, but it is still used everywhere to convince everyone how much money they are saving at the current street price.
  • Overall – the value of the boxes I received is total crap – at least for me.
    IMHO no quality items. Protectors for outdated devices. MSRP that are years old and cannot really count against today’s value. Overall inflated MSRP assumptions of products that cost cents to make (like cases and protectors)… no value there.

So…. what do you think? Was it a good decision or bad?

Paying a tithe (10% of earnings)

In Sunday School, I teach kids who are turning 7 this year. There is a lesson manual and a schedule to follow, but I like bringing in authentic tasks (especially when it is math-related!) whenever possible.

Today the lesson was “I can pay tithing.” We talked about the meaning of tithe (Merriam-Webster says: “to pay or give a tenth part of especially for the support of the church”).

I brought a sack of pennies (about one-year’s accumulation!) and gave each of the kids a pile. I asked them, “Find out how many pennies you have and then tell me how many pennies you’ll give for tithing.”

Multiple Answers

One child said, “I’ll give them all.” I responded, “Sure. You don’t have to give all of your pennies, but you can. Do you still want to?”  “Uh-huh.” “Why do you want to give all of them instead of just 10%?” “Well. I don’t need them. I could keep some.” After church was finished, he came running back to tell me “I kept some of my pennies but I put some on the ground so that someone can find it for a lucky penny!”

Another child said, “I’ll give 4 because I’ve got 43 pennies.” I responded, “Okay. How did you figure that out?” “It’s easy! Every time I count 10 pennies, I take one.”

Another child said, “Can I keep the extra pennies?” I responded, “Sure!” “Okay, I’ll give 17 because I have 39 pennies.” “Okay, how did you decide to give 17?” “Because it looks like half.” “Ok. You can decide how much you want to give. The church asks for 10% but it’s okay to give more. Do you still want to give 17?” “Uh-huh!”

I gave them each a tithing envelope, they filled out their own slip (mostly), put the pennies in the envelope, and brought them to the bishop.

Assumptions

Sometimes in textbooks a word problem is given and the question asks: “Which should you choose?” or “What’s the best choice for Rohit?” Those problems make me cringe because it is making an assumption that “optimization” means the same thing for everyone, especially when some context is given. In this example, the kids all have different reasons for wanting to keep or give away the pennies – some told me they have $20 or $30 at home to spend, so maybe they don’t value pennies much. Others were already making plans for how they would spend their pennies. Some kids are just too darn nice.

But, if I pushed them to only pay 10%, then it feels like I’m making an assumption that you should always give the minimum. At the same time, by not pushing them to pay only 10%, I feel like I’m telling them they should always pay more than the minimum. So, even for me, I’m not sure what the “right” response is in this real-world math problem.

Spinning your wheels and productive struggle

A couple of years ago, my parents bought a cabin a few miles up Blacksmith Fork Canyon in Cache Valley, Utah. It’s about a 40-minute drive. My dad asked to drive me up there today to see the progress he’s made on the cabin – it’s supposed to snow tomorrow so this would be our last chance for the winter.
2014-12-19 16.30.32

We drove on clear roads for some time, but eventually got high enough that the roads were covered in snow and, eventually, ice. 

At the first transition from snow to ice, my dad lost control of his truck and we slid around and landed in some trees. The front tires were off the road, but the rest of the truck was at a 90-degree angle to the road.

My dad switched gears to reverse out and get back on our way. The wheels just spun. My dad is very calm in emergencies and, even though my first thought was, “We’re in the middle of nowhere. We are going to die.”, he calmly began telling me what he was doing as he spun his wheels.

He told me that the friction caused by spinning the wheels would melt the snow or ice down until the tires would hit the ground and we would be on our way. So, he would spin the wheels for a minute or two, and then let the truck cool down a bit and let the ice settle a bit, and then spin again.

2014-12-19 16.30.53After he repeated this process over and over for about 15 – 20 minutes, the wheels finally caught and we reversed back on to the road, made up to the cabin. We got stuck one more time, but luckily it was closer to the cabin and some of his neighbors were there with a truck and chains that they could use to pull him out. He calmly explained that if they weren’t there, he would go get his tractor from the cabin and use that to pull the truck out.

The reason I include this story on a math ed blog is because, even though the content is not strictly mathematical, we often talk about the importance of “productive struggle” for our students. The trigonometry book I taught from mentioned that students should be encouraged to regulate their work time and try to keep from “spinning their wheels uselessly.”

This experience taught me a couple of things: First, that spinning your wheels can actually be a productive thing to do: Even when working on mathematics, it can be helpful to dive in and struggle for a while and then take a break to go on a walk, take a shower, or do the dishes. Spinning your wheels and taking a break and then jumping back in can actually help you reach solid ground. Second, I have not had as much experience getting stranded as my dad. So, I can imagine I would spin my wheels for a few minutes and then just give up. But because my dad has had experiences where he was forced to find a way out, he has learned that he just needs to persevere and eventually it will work. So helping our students have experiences where they work hard for a while and reach a solution is important. Third, my dad asked for help and had back-up plans in his mind in case asking for help didn’t work. He suggested a solution to his neighbors, showing them how they could be useful. He also considered other resources available to him, and what he could use if his neighbors were not able to help. So giving our students opportunities to plan for multiple eventualities and to flexibly consider potential strategies is also important.

Choosing a Bus Pass

Mathematics Problem

Students can receive discounts when they ride the bus in three ways: (1) Show your student ID and one trip is 60 cents; (2) Use your student number to buy a 10-ride pass for $6.00; or (3) Buy a semester-long pass that lasts 5 months for $50. Which choice would you prefer?

Quick answer: Choices 1 and 2 are the same value. Choice 3 will be cheaper if you ride the bus at least 84 times over the five months, so if you plan to ride the bus about 17 times per month, or about 4 times per week, then it will be cheaper to buy a semester pass, so you should prefer to buy the semester pass.

Real-world problem

“Prefer” often indicates in mathematics problems that you should optimize the situation. For this problem, optimization probably means finding the cheapest way. But in the real-world, we have to think about other constraints that might not be named in the problem. Here is where you can ask your students (if they have thought about this in their own life) to share their thinking and interpretation of the problem.

I have thought about this situation each semester in the past four years. I choose not to buy the semester pass, because I will feel pressured to always ride the bus. But I want to choose to walk or ride my bike most of the time, except when weather prevents, so that I am incorporating exercise into my daily routine. So my choice really becomes between options 1 and 2.

They seem like the cost should be the same, but in practice option 1 is much more expensive for the following reasons: First, I don’t use cash a lot so there are times I don’t have cash at all. When I do, I generally have quarters or dollar bills. If I use quarters or dollar bills, I receive a change card from the driver. Somehow because of the coins I usually have, I end up with lots of nickel change cards. I can only use one change card at a time, so (for me) the nickel change cards only result in more nickel change cards. So I throw those away. Second, I sometimes misplace my student ID and so at those times I have to pay $1.25 as the full fare.

For both of those reasons, option 1 is more expensive. And yet – I end up using it much more often than option 2. Why? Because option 2 requires me to buy the 10-ride card online, one at a time, and wait for it to arrive in the mail. So it is not convenient, even though it is cheaper.

State Requirements for: Algebra I, Geometry, Algebra II

Click here to download table: Table created by Eryn M. Stehr, based on data from websites as listed below retrieved on or before 12/15/2012

As is shown in this table, at least 38 states currently include mathematics courses with a minimum requirement of algebra as a high school graduation requirement. Algebra is also being offered earlier in some states.

USMath