Rate of Change Graphs

In grade 12 advanced functions (pre-calculus) we look at rates of change of functions. Our students have been working with rates of change since grade 9, but we now look at instantaneous rates of change, or slopes of tangents to a graph, along with rates of change of slopes of tangents. This can all be rather confusing to students so I decided to make a game to help my students build up their skills and vocabulary.

I started by placing all the desks in pairs, facing each other. Each pair had 2 small whiteboards and markers. I randomly assigned the pairs and had them decide who would be person A and person B.

Here was their challenge -

They jumped right in and were asking each other good questions. I paused them at one point and we talked about the fact that, as they couldn't ask about the actual graph, the person drawing should be able to come up with the shape of the graph but there could be horizontal and vertical translations. 

They soon discovered that the orange piles had more challenging graphs and actually chose those ones! Once they had each gone through a number of graphs, I suggested they could make their own. And, boy, did they! Here are some samples:

Along the way there were great conversations and many misconceptions came to light.

My colleague suggested having students create velocity-time and acceleration-time graphs based on the green and orange graphs (assuming they were position-time graphs). That produced even more misconceptions.

It was great to see them engaged and challenging each other in a really positive way.

Here are the graphs that I used - the first 6 were the green graphs and the last 6 were the orange graphs.

Log Graphs

I was recently inspired by this tweet from Alex Overwijk:

I created my own set of logarithmic graphs - ten with base 10 (white paper) and ten with base 2, 3, 4 or 5 (green paper).

My students, in random groups on big whiteboards, had to determine the equation of the graph they chose. All groups started with base 10 graphs (many stayed with those for the whole activity) and checked their answers on Desmos (and with me). I had printed two sets so they had plenty of new graphs to choose from when they correctly identified their graph. I didn't give them a lot of hints as I circulated as I really wanted them to find strategies to help determine the equations. They were using what they could see about the graph (vertical asymptotes, for example) to help map points from the parent function to their graph. I was really impressed with their efforts. I did decide to write the base on the graphs that were not base 10 as I thought it was too much of a step up at that point.

Here is some of what I saw:

Here is the file. Thanks for the idea, Al!

My plan is to revisit these graphs when we solve logarithmic equations which will allow them to use algebraic skills to help find the equations without having to know/guess which point was mapped onto each new point.

Cofunction Angle Identities

Every year many of my grade 12 Advanced Functions students struggle with cofunction angle identities. Let me back up. They actually struggle with the entire second unit of trig. This includes proving identities and solving equations, but begins with cofunction angle identities. This is often where the confusion begins and it snowballs through the unit. I asked myself what I could do better, or differently, this time around so here is my attempt to help my students better understand the difference between related acute angles and cofunction angles. 

I started by creating this investigation.

They worked in random groups on large whiteboards and definitely needed some coaching from me along the way. I feel that they finished the class with a strong understanding of the relationship between sine and cosine of complimentary angles.

We consolidated together and then recapped the following day with this:

I think that changing the acute angle to the vertical to α (not q) helped some see the difference between related acute angles and cofunction angles. It also helped me refer to what they had seen in the investigation - "Remember how alpha was the angle between the terminal arm and ...".

I wanted to build on what they had learned as we continued to move forward with compound angle identities so I decided to make a warm-up. I created a "game" that we have played for the last two days and we will play round 3 tomorrow. Each student got a small whiteboard and marker. On day 1, they had to write T for transformation, R for related acute angle or C for cofunction on their whiteboard. Each expression was on its own slide and students put up their whiteboards with each answer, which we discussed briefly. Here is the "game" for day 1:

They did well for the most part. Well enough that both they and I wanted to do another game the following day, so on to part 2. This time I asked for three things: T/R/C (same as day 1), quadrant and equivalent expression. Once again they each wrote their answers on a small whiteboard and held it up when they were ready. Here is the set of questions - students did not get to see the answers which are shown below:

Again, it went well.

This morning I shared the first two "games" with a colleague and she suggested making another one. We settled on giving an expression and a choice of potentially equivalent expressions, of which more than one could be correct. We made three of them and my colleague did them with one of her classes today and her feedback was that they really made the students think. She also pointed out how perfect these were as we head into proving identities.

My hope is that this daily practice will solidify their originally tenuous understanding of new concepts which will in turn give them more opportunities for success when they are working with identities and equations.

If anyone would like any of the files, please let me know.

Using Desmos Activity Builder in Unconventional Ways

I love how Desmos Activity Builder has given students the opportunity to discover many concepts in mathematics at their own pace. A well designed activity will get them to predict, test and validate their ideas, helping misconceptions come to the surface along the way. That light bulb moment when you hear your students exclaim "Oh, I get it!" is amazing. The activities on teacher.desmos.com are all fantastic, however I thought I would share a few less conventional ways of using Activity Builder.

#1.

I was helping create a test recently and wanted to include some "student" work for my students to analyze. To accomplish this I create an activity with a graph screen and then a sketch screen. Here was f(x):

And here is "Martha's" graph of the reciprocal of f(x):

Using the sketch feature to create work for students to discuss is quick and easy. It really helped me see what relationships they understood. 

#2.

If students are creating their own graphs you can collect them into one activity to allow you to discuss or show them off more efficiently.

If you add a graph screen to a new activity you can paste the URl into the first line of the graph screen and that entire graph page will be loaded.

Paste a link like this: https://www.desmos.com/calculator/sr04cmo3vk as shown below.

You can then preview the activity to see each graph in turn.

#3.

Although you can make them part of a larger activity, both Card Sorts and Marbleslides can be stand-alone activities. These are options under the Labs tab. (You may have to turn this option on - I'm not sure if this is still required.)

You could create a card sort as a warm-up or exit ticket. Assuming all students have access to technology, they can complete one in a very short amount of time and you get really quick feedback (see green/red below).

Marbleslide challenges can be used at all levels of graphing and are delightful! Sean Sweeney has posted 36 Marbleslide challenges here. I will stop on that note so that you can go try them out yourself. This is the one that I am currently working on, from Set 14:

From the #MTBoS...

Annie Forest shares ways she uses Desmos with primary students here.

Thinking Classroom in MHF4U

My advanced functions classes (grade 12 - similar to pre-calculus) are doing really well so far. I am mixing up VNPS (vertical non-permanent surfaces - i.e. whiteboards) with some direct instruction and a lot of explorations with Desmos. Here is the link to the Desmos activity I created to introduce increasing/decreasing intervals. Using the pause button was great to help focus everyone's attention as common misconceptions came to light.

Yesterday, we started investigating the remainder theorem. In their random groups, they divided f(x) = x² + 5x + 6 by x + 2, then by x + 3, then by x - 1. They also looked at f(-2), f(-3) and f(1). Then I asked what they noticed. Some really didn't notice much so I asked them to divide f(x) by (x+4) and find f(-4). They saw that the remainder from the division was the same as the value they calculated and that the value they were calculating was zero only if the divisor was a factor of f(x). (Note to future self: this was too scaffolded; fix for next year.)

So today we started with this: Find a factor of x³ + 5x² - 22x - 56. They were in new random groups of 3 and clearly did not make the connection to what they had started yesterday. Here is some of what I saw - lots to talk about!

We discussed our objective here - to factor this polynomial, which would allow us to sketch it. I asked something like "If something is a factor, what do we know?" The light bulb went on and they ran with that, finding at first one factor, then the remaining two using a variety of methods. Seeing that some were trying all integer values of x in f(x), I created a new question for them: Factor x³ + 6x² - 8x - 7. Those who had found all the factors to the previous question by systematically trying all integers starting at 0 soon got tired and asked if there was a better way (that was the point of the question). I suggested they look at the constant terms in their factors and the original polynomial. They were remarkably quick at putting those pieces together.

So, soon groups knew that only needed to try ±1 and ±7. They made more mistakes along the way (see below!), but there was progress. They found that they could not determine the other factors using the factor theorem, but had to divide by the first factor they had found. They understood the process and had ownership of it, having tried many paths that didn't take them where they wanted to go before figure out what would work consistently.

We didn't get through very many examples, but I firmly believe that it's better to work through one or two examples in depth, allowing students to find the pitfalls along the way and find their way out of them, than to spoon feed students a multitude of examples.

It's been ridiculously hot here the past few days so I feel somewhat incoherent and am not sure what point I was trying to make with this post anymore... I guess the takeaway I see is that using VNPS and VRG to let students explore and make mistakes is really powerful. I am really trying to get my students to do the thinking, not fall back on memorizing an algorithm, despite the fact that many think that is the best way to learn (ack!). I am trying to convince them that understanding the mathematics will take them so much further and be far more beneficial to them in the long run. That making attempts, some of which will fail, will prepare them for other times when they will struggle and not know where to start when solving a problem.

I'll end this with part of an e-mail I received from a student who graduated last year - I can't tell you how much it meant to me: "I have to say, I think your calculus class has been the most useful so far. The problem solving skills  I learned in that class have taught me to set up equations and approach problems from a different point of view (in multiple classes... especially chemistry)! The self learning technique also helped because that's pretty much all I do now before I go to a lecture. "

Be You, Not Whomever

In the spirit of Carl's keynote at TMC17, I thought I would pass along my best advice to all those who are taking on new challenges and have half a dozen #1TMCThing: don't let who you are get lost as you try to implement others' great ideas.

It took me a long time to figure this out - I can't be you so I need to make your idea work for me. And sometimes that means it just won't. After TMC15 I so wanted to jump on the High 5 bandwagon (giving every student a high 5 on the way into class), but I just couldn't. Merely thinking about it made me cringe, despite all the great things everyone was saying about it. I have also wanted to be more like <insert teacher's name here> and it took me some time to realize that I can't teach like them because I'm not them. So be sure to sift through all the great ideas you have collected and find the ones that you can actually put into action. Make them yours, adapt them as needed, and make them great. If you have found an activity that you think you can implement well with your students, work through it and tweak it so that it represents what your students need. Cultivate your own style while stretching yourself to be better, always. Be you, not whomever.

I really hope this doesn't sound preachy. Not sure whether to hit Publish, but #justpushsend and all...

Desmos Class at Exeter 2017

This is a thoroughly unhelpful blog post except for those of you who were in my Desmos class at Exeter. It's just a whole bunch of links...

Desmos Bank
blog.desmos.com

Sunday:

• Exeter Day 1 Activity
• Marbleslides: Lines
• learn.desmos.com

Monday:

• Calculus Day 1 and answers on one page
• Match my Parabola
• Scavenger Hunt (calculator)
• Polygraph: Lines
• Card Sort

Tuesday:

• The Desmos Guide to Building Great (Digital) Math Activities
• Activity Builder Scavenger Hunt

Wednesday:

• Sean Sweeney's Marbleslide Challenges
• Labels on Graphs

Thursday:

• GIFsmos
• Daily Desmos
• Parabolic Art

Linear Matching

A while back, Pam Wilson shared an old linear matching activity. It had students match up a graph, two points, a slope and two forms of a linear equation to form a set. I really liked it, but it used old calculator screen captures for the graphs. I cleaned it up and ran it with my students. It went well, but I learned that it works better if each type of card (graph, slope, equation, etc.) is printed on one colour of paper so that students have a complete set when they have one card of each colour.

Here is the .docx file and here is the .pdf file. I'd love to hear if you use it and how we could make it better!

Curve Sketching in a Thinking Classroom

I ended my first full unit teaching in a Thinking Classroom (you can read about the beginning of this journey for me here) in my calculus classes just before the March break. I surveyed my students and had mixed feelings about the results. Many suggested 20 minutes of notes at the beginning of class, yet many of these same students also said that they never looked at the notes I post on Google classroom each day. I e-mailed Alex Overwijk to ask for advice, saying that I felt like I was doing something wrong, or at least not entirely right. His response went along the lines of "they don't like being uncomfortable, they don't like having to struggle and you aren't doing anything wrong". I think that what led to me feeling as I did had a lot to do with the very skill-based nature of the unit. They were taking derivatives and taking more derivatives and then they took some more derivatives. This led to groups taking turns doing questions. The questions were not challenging enough to require them to work together as a group to solve them. I'm not sure how to change that for the let's-learn-how-to-take-derivatives unit, but I will ponder that some more before we get there again next year.

This past week we have been working on the elements of curve sketching. Using the first and second derivatives to help determine intervals of increase and decrease, local maximum and minimum points, intervals of concavity, points of inflection, etc. I have continued to use visible random groups and they have continued to work on the VNPS (whiteboard/chalkboard) for almost the entire class each day. I have tried to be more intentional about what I do work through with them - mostly at the beginning of the class. The questions have been more interesting and more challenging for them. I am really pleased with their efforts. I am finding that they are putting all the pieces together more easily and that I am also thinking more deeply about the material. And it's fun! At least twice this week the bell rang at the end of my afternoon class without anyone in the room being aware that it was the end of class. They didn't want to stop. It's incredible how much fantastic work they are producing and how well they can explain it all to me. I am not quite sure how much gushing is appropriate, but my students are awesome. I snapped this picture of some of them this morning and it makes me happy and proud to look at it.

Here is the progression I used for the week (apologies - I got lazy and didn't include all the answers). They did not all get through every question yesterday and today, but I believe that they all have a solid grasp of the material. I continue to post filled-in notes at the end of each day should they wish to review the work or try any of the questions on their own.

Full Unit in a Thinking Classroom

I was going to blog about the last few days in my calculus class, including "Leibniz Day", but I just can't. I found out at the end of the day today that a student I taught all of last year passed away over the weekend. Sometimes teaching is really hard.

Here is the docx version of what I did and here is the PDF version.

Quotient Rule in a Thinking Classroom

I learned a lesson yesterday when my students far exceeded my expectations. One group had completed the entire sequence I came up with to "discover" and apply the quotient rule within 30 minutes of the start of class. This is when I am really glad that I have taught the course many times before and know the material inside out. I let that group start on their homework, something that never really happened during class time even before I switched to a the thinking classroom model, while I found another question worthy of their time. I could have just thrown an uglier question up on the board for them, but I wanted one that would make them think and, hopefully, challenge them. I found a good one and they got back up and worked at it for most of the remainder of the class.

The challenge for me is to set up my sequence of questions in a very intentional way, making sure the progression is neither too little nor too much at a time. But I also have to make sure that I have planned enough challenges to keep them going and keep them thinking. Getting that right will take a little more practice.

After that class I fixed things up for my afternoon class, adding the new question into the sequence along with another harder question that would make any algebraic misconceptions come to light. It turned out to be a little too tricky for most groups and they all got stuck. I adjusted by giving them the answer so they knew what they were trying to get, but many groups eventually abandoned that question and moved on. After that class I rearranged my sequence again, putting that question last.

As I learn to adjust and plan better hopefully I will get better at finding that sweet spot of just-right difficulty progression and quantity.

Product Rule in a Thinking Classroom

Today was day 3 of running my classroom as a thinking classroom - visible random groups (of 3) working on vertical non-permanent surfaces. We started with a quick demonstration to show that the derivative of a product is not the product of the derivatives. Students then worked through a product rule "discovery" activity that I have been using for years. You can find it here. I wish I could give proper credit for it, but I do not remember who shared it with me. 

Here is the setup:

Students worked out expressions for length, width and area and for their rates of change before completing the following table.

Their job was to work with the numbers they came up with in the table to figure out a pattern that worked for each row - that would be the product rule.

It is always interesting to see who comes up with it quickly and who takes a little longer (often because they are trying really complicated things!).

Once I felt like the majority of students had found the pattern, I sent them off to their VNPS to work through today's sequence of questions. 

The last of these asked them to come up with the product rule for three terms. I loved what some of the groups did. They extended the introductory activity to 3-D, added height and worked through the numbers again! (sorry that the picture quality is terrible)

Here's a group that came up with a conjecture for the product rule with three terms and tested it out. I would like to say they did this instead of asking me if they were right, but they did jump right to it when I said they should check it for themselves.

It was really exciting to see such fantastic work, at such a high level from all my students. I love how they trust that they will be able to tackle all the questions I give them and believe in themselves enough to try.

P.S. All of my planning is on one getting-bigger-by-the-day Word document. I'll post the whole thing at the end of the unit.

Why Should You Try a Thinking Classroom?

Overheard in calculus class this morning (day 2 of full thinking classroom):

• "I don't think this is going to work"
• "Why?"
• "Maybe we can try..."
• "Shouldn't that happen because..."
• "What does the graph look like?"
• "What if we..."
• "Can you verbally explain what you did?"
• "Can you tell us what we did wrong?" (one group to another group)
• "We were real mathematicians today"

Thinking Classroom - Day 1

After last week's workshop with Peter Liljedahl I decided to go full-on thinking classroom in both my calculus classes. I told them that they wouldn't be taking notes today and that they would be working in groups at the boards around the room. We talked a little about what the derivative function is and how we find it, along with the issues that would arise if they tried to find the derivative of y = x^729 from first principles. Next they each chose a card to determine their group. Off they went to their whiteboards/chalkboards and they started on the first question. Even though many had already been told the power rule, I made them "convince me" (and themselves) by finding each derivative from first principles. Here is the order of the questions they did:

They noticed patterns in parts (a) and (b) and were able to explain why the derivatives of (c) and (d) were the same as (a). Part (e) went better than expected and generally confirmed their conjectures. The results from parts (f) and (g) were confusing for many and I found that it was helpful to rewrite the question and get them to write the answer in the same form in order for them to see the pattern still held. They got stuck trying to do part (h) from first principles so needed to find the derivative another way.

At this point we stated the power rule as a group and turned to proving it. In the past, I have gone through the proof with my classes and many students' eyes have glazed over as they completely tuned me out. This time I gave them the expansion of x^n - a^n, we talked about how many terms there would be in part of it and let them try the proof. At least one group in each class finished the proof on their own! And all groups made good headway with it which helped them stay engaged when I showed them the full thing. I think they thought it was kind of cool!

I gave them two more questions after the proof:

The first was no problem and the second was done incorrectly by almost 100% of groups. We stopped there for today and I asked them to write down a summary of what they had learned. I didn't do anything else to close the lesson as I felt like it wasn't needed.

Here is the sequence for tomorrow:

Overall I thought today went well. I have done enough of this type of work with students that I was very comfortable and my students were great. There were a few times when I took a marker (there was only one marker/piece of chalk for each group) and handed it to a particular student, but in general they took turns doing the questions. Those not writing the solutions were watching what was going on, looking for errors. There were some good discussions going on today, but I anticipate more tomorrow due to the nature of the questions. There were some groups that would call me over to check their work, but they got a lot of "What do you think?" and "Convince me" and "Are you sure?" so I suspect that will diminish as we continue. I had to ask a few students to put their phones away, but it was not really an issue. They all did math and were all thinking and even those who came in knowing the power rule learned something new.

Here Goes!

As I wrote in my last post, I am jumping into the full VNPS-VRG-thinking-classroom in my calculus classes. I have two classes - one in the morning, the other in the afternoon - so I will be able to tweak my plan in between and hopefully really make progress with both my planning and implementation. We have finished our first unit on limits and introducing the derivative function. On Wednesday we will start with the derivative rules (power, product, quotient, chain) so that's what I'm trying to plan out. Sheri Walker and I thought through this progression together yesterday so I've tried to tie things together and include some of what I expect to see. The intro will be to the entire class, the sequence are the questions I will give each group of 3 at their whiteboards/blackboard. Groups should progress at different rates so my job is to circulate, observe and help keep them in flow. The last question (part i) may be for the speed demons or for everyone - we shall see! I am also trying to keep in mind what comes next.

You may notice that I have a number of unanswered questions. If you can help me think through those, I would be most appreciative.

My plan is to edit my document after I have done both classes and then post the new file in case it might be useful to others.

A Thinking Classroom

I was fortunate enough to be able to attend a half-day workshop on Thursday with Peter Liljedahl. I first heard about Peter's work a few years ago after he had spoken at a conference in Ottawa. There was much buzz from those who attended about VNPS and VRG, most notably from Alex Overwijk. Al was happy (!) to share all he had learned about vertical non-permanent surfaces and visible random groups (he may or may not have stopped strangers on the street to tell them about it). He has become Peter's #1 fan, even entitling the Ignite session he did last year "Things Peter Says". Although I had heard the vast majority of what Peter shared with us, there were a couple of important puzzle pieces that got filled in which make me believe that I can create what he refers to as a thinking classroom. This post is not intended to explain it all to you - for that you should visit Peter's website here or read this post from Alex.

I have used VNPS in my classroom for a few years now, but not every day and not instead of teaching/facilitating lessons. I use them for review stations and for tasks. I do visible random groups whenever I do group work, so that's not new for me (best group size = 3). I also spiral some of my courses with lots of activities, so I think that in many respects I already have a thinking classroom. My students tell me that even when I teach the same lesson as other teachers, I do it differently - I make them do the math, I don't just give it to them. But one can always do better... What convinced me was hearing about an actual curricular example of how to structure VNPS. Peter only spoke about students learning how to factor for a couple of minutes, but with enough detail to make it all click for me. I will attempt to share how I see it all working, but apologize in advance if my thoughts have still not gelled.

VNPS is a means to a goal, not the goal itself. It is, according to Peter's research, the most effective vehicle to creating a thinking classroom. One where all students are engaged in meaningful mathematics - doing the math, not watching someone else do it. They are learning to because autonomous, to look for the next question and persevere when they get stuck. The teacher's role in this is hard to nail down - you need to adjust to what you see constantly. It's structured chaos at its best. And I don't think it can be successful without really good planning (and I would highly recommend the book "5 Practices for Orchestrating Productive Mathematics Discussions" to help). Thursday's experience helped me see how all the pieces fit.

The structure looks like this:

• Start with a quick (~2 minute) lesson or prompt to activate prior learning or give students enough to build upon. Instructions should be oral as much as possible. If you give instructions in writing students have to decode them individually, whereas if they are oral instructions, students immediately start talking to each other.
• The questions they are working on must be sequenced in a logical way to develop the skills while keeping all students in "flow". These don't have to be incredible task questions - they can be everyday textbook questions. The importance of proper selection and sequencing is huge!
• There must be a lesson close that will level the class to the bottom. This could be a full-class debrief (going through a different example) or a gallery walk that is also thoughtfully sequenced.

I spend part of Friday with Sheri Walker working on sequencing questions for calculus (she was gracious enough to work on topics that she has already covered). Although not finalized, I think it helped us both think through how to make this work.

I still have lingering questions/concerns.

I worry - maybe that's too strong of a word - about the introverts in my classes. Especially the shy introverts. Because I am one, and I know how exhausting working in groups where you may not be entirely comfortable with the material or people can be. I am already purposeful about making my classroom a safe space for learning, which includes making mistakes, but I still worry... 

I wonder about the number of markers and who is using them. We worked through two problems on Thursday and each group only received one marker. There were no rules around who should do the writing, but Peter was going around the room taking the marker from some and handing it to others. I was very aware of how long I had the marker when I was in my first group and did my best to always put the marker down when I had finished with it. It is much easier to pick a marker up than to take it out of someone's hand. There were questions about all of this and suggestions of either using a timer so that each person had the marker for an equal-ish amount of time or that the person with the marker could only write others' ideas. This takes me back to the introverts issue - I would hate to be the one who had to write someone else's ideas if I didn't understand them. But I also know that I did not touch the marker in my second group, so it's not that hard to step back a little which is not what we want.

I don't know if I can make this work with my grade 10 applied class. MFM2P is generally made up of students where one half to two thirds have IEPs. Many require written instructions. Many cannot work in groups, only pairs. Many cannot work with certain other students in the class. Many (most?) hate math and are often very unwilling to do any work. There seem to be so many obstacles with that group, that I'm not certain this is the way to go. Spiralling with activities has really helped with engagement and success, so I think the VNPS may continue to be an every-so-often thing. If you can convince me otherwise, I would love to hear your thoughts!

Peter says we shouldn't give students notes. I agree, however, I will still continue to post notes on Google Classroom in calculus because there would be a mutiny if I didn't. There are only so many battles that I will take on! I like the idea of finishing the "lesson" at around the 50-60 minute mark, doing the lesson close then leaving ~15 minutes for students to write down, in their own words, what they have learned. They will then be able to reference my posted lesson with examples as they need.

There is so much more to all of this, but this is where I am for now. I am not promising to blog every day, but I will write about how it's all going. I would love to hear your thoughts in the comments. Thanks.

Skyscrapers

I spent a lot of time thinking about what activity I should do with my grade 10 applied students on the first day of semester 2. I wanted them to be engaged in mathematical thinking, preferably with something hands-on (but nothing that would cause complete chaos!) and I wanted them to work with someone else in the class. What I ended up choosing was Skyscrapers from BrainBashers - here is the link to their site. These are logic puzzles with only a few rules:

A completed board would look like this, where the numbers in the grid represent the height of each skyscraper:

I first learned about these puzzles from Alex Overwijk last year and I'm fairly certain that he heard about them from Peter Liljedahl (I spelled that correctly this first try!). Alex let us try them at our math PD day last year using linking cubes as the skyscrapers.

I set up the skyscrapers ahead of time for my class, using a different colour for each height so that it would be easy to see if they had more than one skyscraper of a particular height in the same row or column. It turned out that I found this feature more useful than them as I circulated and checked their work.

Each pair got their first puzzle and these:

Here is an example of one column. We reasoned through the fact that there is only one way to place the skyscrapers if you can see all 4.

Here are a couple of views of the completed puzzle:

I had printed out 6 different puzzles for them to work through, each on a different colour of paper so that I knew which one they were working. I also had the solutions printed on the same colour of paper to make checking their work faster. I checked the first two and then let them go. I should have printed some harder ones as some groups flew through these. I did have blank ones for them to make their own, but I really think this could have been a much richer experience if they had tried some of the harder ones, like these:

This is what I tweeted out:

What you may not realize is that very few grade 10 applied students ask for more "work", so it was awesome that they wanted more! Day 1 was definitely a success. I got to interact with all my students and see a little of how they think, whether they are able to follow directions easily, and how they work with others. It was a good day and a great start to the semester.