What sort of questions might your students have about this train schedule? How long is this train at the station during a typical week? If this train is at the station in Montréal for a similar amount of time, then how long is each trip from Halifax to Montréal? What is the average speed of this train during its trip? What do you think the schedule looks like that is posted at the station in Montréal? How would this trip via train compare to a trip via car or plane (time, cost, etc.)?
I drive past this building every day on my way to work. It is Young Tower at 6080 Young Street in Halifax. I think it is pretty interesting... I used this picture as a problem solving warm up activity for a group of grade 10 math teachers recently. I gave each group of teachers a large piece of chart paper and asked them to divide the paper in half with a line. I asked teachers to brainstorm what they notice about this picture and record it on one half of their chart paper. I asked them to look at the picture using a number of lenses. What would an architect notice about this image? What would a person who worked at this building notice about this picture? What would a mathematician notice about this picture? After about 5 minutes of brainstorming, I asked each group to tell me one thing they noticed and I recorded it at the front of the room. Groups noticed things like the number and size of windows on the building ("about half the lateral surface is glass"), the shape of the building ("almost a cube"), the picture must have been taken on a weekend because there are very few cars in the parking lot, and the weather was really nice that day. Next I asked them to brainstorm what they wonder about this picture and record in on the other half of their chart paper. If this picture was the start of a math problem, what could that math problem be? What things that they noticed sparked their curiosity? After another 5 minutes, I asked each group once again to tell me one thing that they wondered. After looking at all the questions that the groups posed, we selected one and asked everyone to estimate an answer to that question. I also asked them what information would they need to make a more accurate estimate. Once they had an initial estimate, I gave them some additional information about the building and let them revise their estimate. We had several really interesting questions posed by groups. Some questions concerned the shape of the building, like "How close to a perfect cube is this building?" Other questions focused on finance such as, "How much revenue is generated by leasing all of the office space in this building?" One of my favourite 'wonderings' was, "How much wrapping paper would it take to wrap this building up like a Christmas present?"
This "I Notice/I Wonder" problem solving strategy is one that I saw shared by Max Ray-Riek from the Math Forum. He has a blog where he talks about Noticing and Wondering in High School. This strategy starts off with brainstorming to let students get familiar and engaged with a problem situation before jumping into a specific question to solve. By having students come up with questions, you'll often get more engagement and interest. It also allows you to respond to interesting suggestions from students that you might not have considered. It allows everyone in the class meaningful participation in the conversation because everyone has something that they can notice. This strategy might also create additional opportunities for differentiation by using several different questions that students suggested.
I really like some of the questions found on the Openmiddle.com website. It is a great resource for questions that really get students thinking. They are often formatted so that there is a very low threshold for entry to the problem but they allow for enrichment and extensions. I created the problem below for a professional development session for Math 10 teachers. Students in the Math 10 course are near the following outcome in the yearly plan: AN03 Students will be expected to demonstrate an understanding of powers with integral and rational exponents.
Fill in the boxes with whole numbers 1 through 6, using each number at most once, so that the value of the expression is as large (or as small) as possible.
In order to verify that these were correct values, I wrote a short program in Python (see below) to check all of the 720 possible (6!) values. It was my personal Hour of Code activity. I`m just learning to use Python so my code could definitely be more efficient. If you can write some better code, let me know and I`ll post it here and give you credit.
largest = 0
smallest = 10000
for i in ValuesList:
for j in ValuesList:
for k in ValuesList:
for l in ValuesList:
for m in ValuesList:
for n in ValuesList:
if i!=j and i!=k and i!=l and i!=m and i!=n and j!=k and j!=l and j!=m and j!=n and k!=l and k!=m and k!=n and l!=m and l!=n and m!=n:
if z > largest:
largest = z
if z < smallest:
smallest = z
Update: I submitted this problem to the OpenMiddle.com website and it has been posted there.