I had a private lesson yesterday which was a lesson that started off in an unexpected manner but by the end we were both hooked.
Alix arrived into the lesson and, as I usually do with a new tutee, I asked her what she hoped to achieve from having private lessons. She said that she wanted to learn more advanced mathematics. Woah, back up a minute – you’re not struggling? You believe yourself to be good at mathematics and need to be stretched? Okay…wasn’t expecting that. Let me think… where do we start?
I don’t think that my first response was all that bad. I asked her if she wanted to read through Paul Lockhart’s book, “Measurement” together and tackle each problem that she thought was worth the effort. Needless to say she didn’t take to kindly to reading a book during a mathematics lesson, even if it was a maths book.
My second line of attack was the only other plan I had at the time. Here’s what happened…
“How about we look at a video together and you tell me what you think?” – A rhetorical question since I was already loading up the video on Dan Meyer’s blog (http://threeacts.mrmeyer.com/tacocart/).
We watched the video and I asked her what she thought – she shrugged and said nothing.
“You don’t have any questions about this? There’s nothing that you’re curious about?”
“No, I know the answer already so why would I need to ask any questions!” – Okay, didn’t expect that one. Let’s turn it up a notch.
“Great! If you have an answer then I can only assume that you must have a question? What is that question? Can you show me the answer?”
“The question is what is the quickest route to the taco cart.” – she runs her finger over the path she thinks is quickest.
“Okay, so how do you know that is definitely the quickest path to the taco cart?” Can you prove to me that this would be the quickest path because I’m not sure if I agree with you.”
“No because I don’t have any information about the problem. I need to know all of the lengths on the diagram…Oh, and also the speed that they walk on in sand and the speed that they walk on the path…I guess though that they would be going at different speeds if they’re not walking together.” – Wow, that was pretty quick thinking – she knows what she’s doing! Furthermore she picked up on a key assumption made in the task in that they both walk at exactly the same speed. I hadn’t thought about this much before Alix pointed it out to me – interesting.
“Great stuff Alix! You’re really starting to pick apart the problem. And you’re right about the fact that they would almost definitely be walking at different speeds when separated. Shall we see what information we have on offer?”
“Right, the information here suggests that we make an assumption. We can assume that they do infact walk at the same speed even when they’re walking separately. I guess maybe we could think about your point after we’ve tackled the problem with the given information – Is that okay by you?”
“Yeah, that’s fine. I want to firstly figure out who would get there first out of Dan and Ben and then I’ll show that my way is quicker. I need to start by finding the distance of Ben’s path. I’ll have to draw a diagram.” – She takes out a ruler and pencil and starts thinking about how she’s going to draw the diagram. I’m thinking…what’s so hard about sketching it out?
“What are you thinking about?”
“I’m just wondering how to draw this so that I can measure the length of Ben’s path. I don’t know what scale to use.” – Oh, she’s drawing a scale diagram. She hasn’t met Pythagoras’ theorem before?
“Okay, that sounds like a good idea. Can you think of a scale to use on this page that would be approptiate for the problem?” – I could have taken her to Geogebra there and then but I wanted her to work through a problem that she was obvioulsy having difficulty with. It’s a skill worth having to ascertain an appropriate scale for a scale diagram.
“Maybe 1 square of the graph paper could represent 5ft…No it would have to be 1 square to 10ft because otherwise it wouldn’t all fit on.” – She draws the diagram and initially does it inaccurately. I prompt her to see if the answer makes sense and she corrects the diagram. She gets an answer of 670 ft for the length of Ben’s path.
“Are you happy with that answer? Is it accurate enough for you?”
“Erm, I guess so. What do you think?” – that means NO, it isn’t accurate enough for me. I just don’t have any other tools that’ll do it better. Help!
“How about we try to do it more accurately on some computer software called Geogebra.”
I talked her through how to do each part on Geogebra. As you can imagine, Alix is impressed. There’s no turning back from it now. She’s hooked.
“Are you happy that we now have it accurate enough?”
“I do think it’s accurate enough but I was wondering if there’s a way to do it using maths, like Pythagoras or something?” – Yes Alix – back of the net! She wants to learn some new mathematics to make solving this problem more efficient. I’m hooked now.
“That’s a great question. How about I show you an applet I made on geogebra which will hopefully help you understand what the Pythagoras theorem is?” – Click the picture to link to the applet. Read the instructions below the applet.
Alix then went on to discover Pythagoras’ theorem and with guidance wrote this down as a formula. At this point it was the end of the lesson.
This is the first ever private lesson which neither I or the student wanted to end. She became hooked on the question and I was hooked on her learning. We’ve all seen the problem before, usually presented in the form of a lifeguard wanting to find the quickest path across beach and water to rescue someone. Dan Meyer’s well-edited visual media adds so much more to hooking students in than the problem printed on paper ever could. It doesn’t get any better as a Maths teacher when a student wants to learn more Maths to help her solve a problem. I can’t wait to take this problem all the way and much further. We’ve got some great lessons in store delving deeper into this problem and hopefully making some 3 Act lessons of our own.