Have you ever been on a road trip with a math teacher? They're always calculating distances and estimating how much farther they can go on a tank of gas and noticing things...
This was just what happened during a trip with my wife. We were headed towards the New Brunswick boarder from Halifax and we saw a highway mileage sign. It said it that Amherst was 42 km away and New Brunswick was 47 km away.
"So how far is it from Amherst to New Brunswick?" I asked. 5 km? But is it exactly 5 km? If we assume that these distances are rounded to the nearest kilometer then it might be as far as 6 km (41.5 km to Amherst and 47.4 km to NB) or as short as 4 km (42.4 km to Amherst and 46.5 to NB)?
(46.5 - 42.4) <= distance <= (47.4 - 41.5)
A few minutes driving brought us to the next mileage sign and some additional information.
So the next mileage sign down the road seems to indicate that there are only 4 km between Amherst and the NB boarder. Using our previous logic however we could say that it is anywhere between 3 km and 5 km right? (39.5 - 36.4) <= distance <= (40.4 - 35.5). Now we are getting somewhere. We know that it is between 4 and 6 km and between 3 and 5 km.
Now we can narrow it down and say with some certainty that the distance between Amherst and NB was between 4 and 5 km. I'm feeling confident now. A few more kilometers down the road we pass the next mileage sign.
What the what! What is going on here? Is this a conspiracy of cartographers? What strange and inscrutable measurement system is at work here? Time for some Google Earth mapping. I used Google Street View to find the second road sign along the Trans-Canada Hwy 104 where the sign says there is 36 km to Amherst. A straight line from this spot on the road to the geographical centre of Amherst is only about 27 km so I think we can assume that the road signs are measured as the distance along the road.
So lets use Google Maps to measure the distance along the road. From the mileage sign to the city centre, Google Maps says that it is still only about 32 km away. Still about 4 km off what the sign says. I wonder how accurate these signs are? How long ago and using what technology were these signs created?
Do you know how road distances are measured in Nova Scotia? Have you encountered any inconsistent mileage signs? Stay tuned for updates. I've sent an email to the Nova Scotia Department of Transportation and Infrastructure Renewal to ask for some clarification in the method of measurement. I'll keep you posted of any responses.
Buzzfeed recently had a hilarious post profiling some of the infamous people from math problems who buy random items (often fruit) in ridiculous quantities. It got me thinking about the contexts we associate with some mathematical ideas.
I once walked into a math class and the teacher was posing the following problem:
"If I had 4745 roses and 26 vases, how many roses can I put in each vase?"
As you might have guessed, the class had recently been working on division with 2-digit divisors. I know the teacher's intention was to have the students practice dividing large numbers and I also know the teacher understood the importance of putting the math in context. I just think by asking the predictable "how many roses in each vase" question, they missed an opportunity for come creative and critical thinking.
When given this scenario about these roses, more than a few questions and thoughts came to mind;
Boy, that seems like a lot of roses.
Why would someone have that many roses?
Is it Mother's Day soon? I should send my mom some roses.
Why 26 vases? I can barely find one vase when I need it. Who has 26?
Roses usually come in dozens; how many dozens?
How would you store 4745 roses? How big (or how many) refrigerators would you need?
How much does a florist make on Mother's Day?
What is the profit margin?
How much do they have to pay to get all those roses? Is there a bulk discount?
What do they have to sell them for to make "x" profit? How many roses (and at what price) do they have to sell to break even?
I could go on for days like this. One question made me think of two or three others.
Scenarios like this on their own are disconnected from a student's experience (and interest). And we wonder why kids are disengaged from the math. I'm not saying there isn't some good math in these scenarios; there's lots of problem solving and reasoning. But this question needs a hook; the students have to want to solve it.
One way to "hook" the students is to embrace the ridiculousness of the scenario. Tap into the kids' silly side and involve them in creating the questions. You are not giving up control of the content - chose your scenarios carefully with the math in mind. Create some of your own questions to keep in your back pocket before you even begin the lesson.
The teacher in the above situation wanted the students to divide. Let the students tease out the different division questions (and let them think it is all coming from them even though you may have created the questions beforehand!). Pick a few questions from the student generated list to work on. Give students a choice of which questions they want to do. Done early? Pick another question or make up one of your own.
So back to the Buzzfeed post. I'll use one of their pictures to reiterate what I mean.
This question is pretty boring on it's own:
"Coach T buys fifteen 12 pound bags of cheese puffs for Coach P. How many ounces is that?" (Whatt?? Who cares?)
Now, add a picture to connect to the ridiculousness. Or better yet, start with the picture and think about all of the questions that come to mind.
As you might have guessed, the class had recently been working on division with 2-digit divisors. I know the teacher's intention was to have the students practice dividing large numbers and I also know the teacher understood the importance of putting the math in context. I just think by asking the predictable "how many in each vase" question, they missed an opportunity for some creative thinking and problem