Monday, March 9, 2009

A Middle School Mental Math 'Practical' Problem

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You're on a superhighway in the middle of nowhere at 10 PM, your fuel tank shows about a quarter of a tank remaining, your trip odometer shows you've gone about 180 miles since the last fill-up and you just passed a fuel rest stop. The sign reads "Next Rest Area and Gas - 58 miles." Are you in trouble or will you make it?

(1) Unrealistic scenario? Rest areas rarely that far apart on the Interstate? Why didn't the driver realize that he was low on gas and just stop! Why not just get off the Interstate and look for a local station or call AAA or Onstar if needed or use a navigation system to locate another gas station? (Do all of us have access to these technologies?). Does the student have to know that one still has some fuel remaining even when the gauge points to empty? Should a discussion of these kinds of practical issues be an integral part of 'real-world' problem-solving?

(2) Is this type of question fairly common in the texts you're using?

(3) What prerequisite skills and conceptual understandings are needed for this problem? Should most 7th or 8th graders be able to do this?

(4) Why do you think I suggested a mental math approach? Should we eliminate the mental math/estimation aspect altogether or use it as a starting point for further discussion?

(5) How do you think your students would fare on this problem? How many students would recognize the 1:3 ratio between gas remaining to gas consumed? Is this part:part construct something stressed in texts and in our classrooms? Should it be? Is it worthwhile discussing several approaches to this problem?

(6) Do you have a favorite visual model for this kind of ratio problem? Your thoughts?

Anonymous said...

You're on a superhighway in the middle of nowhere at 10 PM, your fuel tank shows about a quarter of a tank remaining, your trip odometer shows you've gone about 180 miles since the last fill-up and there is a sign "Rest Area and Gas - Next gas 58 miles." If you pull over it will wake the damn dog who drives you nuts with his barking? Stop anyway?

Anonymous said...

Wow, I worded that badly. Meant to present the problem a choice - get gas now, or wait 58 miles, if that's safe to do.

Totally_clueless said...

I remember that facilities were that far apart on I-40 between Barstow and Needles, CA.

TC

Anonymous said...

How accurate is your gas gauge?

Dave Marain said...

tc-- I was thinking of Montana where one might drive for a day between stops!

Jonathan--
I feel as though my feeble attempt at a "real-world" problem is getting in the way of "real" math! The truth is that if my wife or one of my daughters were in the car with the driver there would be no way that the exit would have been missed if that driver wants to keep his/her sanity!

Anyway, is there any chance of getting back to discussing how many students would "see" the 1:3 ratio and, more importantly, how do we bring students to that kind of understanding instead of assuming only the 'few' will figure it out!

Hilary said...

This problem would be one my students enjoy. They would not only get to work through the math problem part of it, but would also inevitable challenge if this is a realistic situation.

I do agree that maybe it is a bit contrived, but I think that contrived situations sometimes do a better job at making a point and teaching a concept than true real world problems.

Dave Marain said...

Thanks, Jonathan. I do agree that the more contrived, the sillier the problem context, the more students may want to play around with it (while they're making fun of the problem!).

Dave Marain said...

Sorry, Hilary. I gave credit to Jonathan for your supportive comment! Thanks! Let me know if you do use this in the classroom.

Denise--
If you ask my wife, my gauge has been stuck on empty for years. Uh oh, I may have to delete this comment...

Kate said...

I think it's a nice problem, but I also think that you are giving away the store. I'd do something more like this:

You're on a superhighway in the middle of nowhere at 10 PM. The sign reads "Next Rest Area and Gas - 58 miles." Are you in trouble or will you make it?

Make them figure out what more information they need and be prepared to tell them once they ask. That's how you make it realistic.

Dave Marain said...

Excellent points, Kate! Leave the problem more open-ended. You're also getting into the issue of data sufficiency, i.e., what additional information is needed, if any, to solve a problem.

I see it both ways here. If my primary focus is on ratio concepts I may not want to take too much time to discuss how we can make the problem more realistic. However, the kind of questions you're suggesting promotes reasoning and engages their interest. As educators we need to keep them on track when the dialog goes too far astray. It can be a difficult balancing act, the art and practice of teaching. And, remember, many adults believe anyone can do this job!

Hilary said...

Kate, I really like your idea of having the students determine what information they need and then give it to them when they ask. It adds a whole new level of critical thinking to the problem. So often my students struggle with knowning what information they need to solve a problem and I think by using your approach thier skills would improve.

Dave Marain said...

Hilary,
That's the wonderful sharing part of a blog - the whole is greater than the sum of its parts! Kate is one of the best at keeping problems open-ended, forcing students to THINK about what information is needed. After all, isn't that our ultimate goal for our students: Enabling them to solve problems in unfamiliar contexts.

We all know students do not want us to "EXPLAIN!" They want us to show them the procedure or simply give them the formula so they can "plug in." Teaching Kate's way is far more challenging both for the educator and her students but the payoff is huge. Of course, you need to tailor this to the abilities and backgrounds of your students, but, most youngsters are capable of so much more...

sooophisticated said...

I think these types of questions force our students to think outside of the box, in search of the information they need to solve the problem. However, I find that students loose interest when we explain in an effort to move them along. Any suggestions on to how give those motivational hints, without giving away answers?

Dave Marain said...

sooophisticated--
You're asking the kind of question that motivated me to start this blog over two years ago. There is no simple answer because there are so many factors at play.

If my intent was to give an example of a standardized test question for practice or to focus on ratio concepts, then I would not make it open-ended, i.e., I would not leave out details for discussion purposes. "Here's the problem, boys and girls, now individually or in your group find a way to solve it." All those questions I posed to educators in this post were designed to provoke reflection and discussion about different instructional strategies or using this problem for a variety of purposes.

The issue of when and how to give hints is an ongoing struggle for all of us. It's all about the 'art' and 'practice' of our profession and one can only get better at this by taking risks, making mistakes and learning from them. I made many of these mistakes but over time I figured out that 'less is more' was my guiding principle. I could sense when the group was getting frustrated and I made a suggestion or decided "That's enough!" At that point I either invited a student to give the solution or I gave it or told them to finish it at home or ...

Kate's approach is appropriate if your goal is to help them develop their problem-solving prowess and we all need to devote time for this - it cannot be rushed. If we are driven, as high school teachers often are, by the desire to cover as much content as possible, then we will rarely take this time. BUT the time is worth it in the end when our students display the ability to tackle that nonroutine problem they've never seen before!

Again, thank you for your question. My response may not help but it might keep the conversation going...

Suzanne said...

Good problem, kids don't realize they do math everyday when they try to figure things like that out.

michael mcmullin said...

Interesting.