A great little card problem 1-5-2-4-3 Expect the Unexpected!

So many people have blogged about the awesomeness of Twitter Math Camp 2015. One workshop that I went to was given by Alex Overwijk on Vertical Non-Permanent Surfaces. He started off by giving us a problem that is best explained by watching the video below, from http://www.peterliljedahl.com/.

Essentially, if you take low cards (let's call the Ace a 1) and put them in the order 1-5-2-4-3, and then flip the first card down (Ace), and bury the next card (put it at the bottom), flip the next card (2), put the next card at the bottom, and continue until one card is left that you put down, you end up with the cards face up in the order 1-2-3-4-5. The question Alex asked is, can you do this with one suit (Ace through King) so that when you flip and bury, you get them in correct order 1-2-3-4-5-6-7-8-9-10-J-Q-K?

When given this problem at Twitter Math Camp, I immediately thought I knew how to do it...my group mates and I wrote it out on the board (thus, the vertical non-permanent surface), and then tried to prove it...with an epic fail. Unfortunately, there were so many great things to talk about that I did not have time to figure it out during the limited time Alex gave us.

Although I don't have the book with me at home to reference it now,  I bought the book Crossing the River with Dogs to help a bit with the problem solving course I am starting this year, as I have no text book and am just using 25 years of experience and a book I wrote 17 years ago to guide the course. I had forgotten about the card problem from TMC and was flipping through the book before school started--and am pretty sure I saw it and got an idea of a solution. I closed it immediately so as not to see the entire answer, but just to get the idea.

So, I thought this would be a perfect problem to give to my new Honors Problem Solving Seminar class on the first day. I actually like to give students problems without fully knowing the answer, so that I can't lead them in a direction and can learn with them. Like me, these students immediately used their intuition, only to find out they were WRONG!! They are a very bright bunch, but with only 10 minutes left in class to start, just like I had at TMC, they did not figure it out. I let the students take home 13 cards of mine each so they could play.

I got a message from a student asking me if, after an hour and a half during his free periods he did not figure it out, was it OK? I said it was.

A teacher came up to me and told me that he almost got said student in trouble for playing cards and not working in the library...till the student told the teacher he was working on a math problem for the problem solving class.

SOMEWHAT OF A SPOILER ALERT...

Before I went over it in the next class, I asked my colleague Liz next door (who also went to TMC with me) to help me with the solution. What I saw in the book was a circle with the cards, where the ace was at the top, and then there was a space, and then there was a 2, and then there was a space, and then a 4, etc...so the space represented the card that would be skipped. We thought we could do it right away. It still took us about 10 minutes, as we were almost right several times...I tell you, when we did get it, we screamed like giddy little girls and did the nerdiest high-fives ever!! We even took a picture of the answer, but I don't want to show it if you want to try it.

So, back in class the next day, we had A LOT to go over in our hour and a half block class. We started class with http://corbettmaths.com/5-a-day/, we watched the Grit video (a must watch) and then took a quiz on How Gritty Are You and talked about grit when solving problems. Then we created our interactive notebooks for the course...maybe I will blog about that at a later date. Before we knew it, there was 10 minutes left. So I opened up the card question again. 

A student went up to the board to explain his solution. Basically, he said that you take the amount of cards and divide them in half. So there were 13 cards, divide in half, and round up, and there are 7 that will be flipped (1-7) and the remaining would be buried. He then divided the remaining 6 by 2 and said that half of those would be buried, so he separated 8-9-10 from J-Q-K and then figured it out from there. Some students got this method, some did not. 

I showed them the circle method that I used to solve it...they hated that!?

I knew a another student had gotten it, so I asked her if she did it the same way.  She said she figured it out in two minutes. I was impressed. I said how did you do it? She said she worked backwards (genius!) like this:

1. Take k

2. Take q and put it on the top

3. Take k from the back to the front 

4. Take j and put it on the top 

5. Take q from the back to the front

Then just repeat the same steps:))))))

When you do this, you get, face up, the cards in the correct order with the Ace on top, ready to put down with the next card to bury. What!!!! Cray. So easy.

I loved that:

1. They did not get the answer at the end of class and had to think about it on their own for two days.

2. They had no way to Google the answer.

3. They organically came up with unique ways to solve, and that it clicked differently for different students.

4. They hated my way. 

5. A student the answer in a very simplistic way (an "elegant" solution) that was different from the way the first student got it.

I love this class, and I am so happy to be teaching it this year...I expect to have a lot of unexpectedness!

~Lisa

Three MUST USE online tools for Formative Assessment

Some buzz words I've been hearing lately are "formative" vs. "summative" assessment. It's taken me some time to wrap my head around them, but I think, for me at least, I've finally gotten the idea. The website from the Eberly Center of Teaching Excellence and Educational Innovation really sums it up well. In a nutshell, here's what it says:

Formative assessment monitors student learning and is low stakes...so it has little to no point value. It is a way to provide feedback to teachers to improve how they teach and to students to improve how they learn.

Summative assessment evaluates student learning and is high stakes, so it has a high point value. An example would be a midterm exam or a unit test.

Now that I am teaching in a block schedule, I need to be able to see how my students are doing before moving on to the next concept. I am used to (for 25 years!) teaching, students doing homework and reflecting, and then, very generally, reviewing with a warm-up and adding to it or moving on. But now, I may need to teach two major concepts in a day, or I will not nearly get through enough concepts.

So what I am going to try is to do some online formative assessment activities that also "change things up" for kids during block. What I mean is, I am going to break the block up into several mini lessons using these online tools: Desmos, Kahoot!, Formative.

The first one is an activity builder from Desmos. 

Activity Builders just came out in August, and I was lucky enough to be in a Desmos workshop at Twitter Math Camp, where we played with this incredible DIY tool. It is absolutely AWESOME!!! It was so cool to work with Michael Fenton, and I got a little starstruck shaking the hand of founder and CEO Eli Luberoff.

Here is what one activity I made looks like from the teacher screen...students will answer one question at a time.

There are three screens you can put into the Activity Builder: (See directions here: https://teacher.desmos.com/activitybuilder)

• A Graph Screen: if, for example, you ask students to graph a line that contains two particular points, you can look at the teacher screen and see that all of the graphs are in alignment (YES! They get it!) or if there are a few students struggling.
• A Question Screen: you can ask a question, import a graph if you want students to base it off of one, and there is a place to check whether or not students can see any other student's responses...only 3 student answers will show up at a time, which is great because then students won't just copy an answer--if all were there, they might not even do the problem. But a selection of 3 is nice!
• A Text Screen: This is just to tell them what the activity is or to say it ended, or you could even give them instructions or a formula on it. 

Then you can mix and match as necessary...by the way, saying mix and match, if you haven't tried Desmos Polygraph, you ARE missing out. And now, you can make your own! My friend made one at TMC for Calculus, and it was SO COOL! And you MUST try it with Kittens!

The second one is Kahoot!

I had heard of Kahoot! before, but I didn't see it in action till Twitter Math Camp, when @jreulbach showed it as a My Favorite. It was really fun and competitive, and is based on getting high scores for being both correct and fast...most of my students enjoy competition, and for the ones that do not, it does not have to be played everyday. Also, if I'm not mistaken, only the first few student names are shown, so if students don't do well, I don't think it shows their score. It's a nice, quick change of pace that I know I will use to transition during the 90 minute block.

From jlavely64's Kahoot on blog.getkahoot.com

Here's one I made that I am using as a Precalculus slope warm-up. Rather than review on paper, students do it while playing an exciting game...aka "learning without even trying." I love that you can go to youtube and pick a song and video to play while kids are putting in the code, and then you get to add ridiculous pictures that make me laugh, like the one below--lots of room for pictures (so that also means graphs) on this one.

The third one is Formative.

And finally, I just learned about formative from Melanie at her blog When Life Hands You Lemmas (love that name). Please look at her blog, as she explains it really well, but you can assign a quick Do Now or Exit Ticket or Show Your Work, and you can see what ALL of the kids are writing/typing/drawing at once (i.e., LIVE RESULTS) on YOUR computer! 

Or you can import your own document, and it will look like this, and they touch the box and can type in it...

You can also import videos! AND you give them the right answers so they can check. AND you can write feedback for each student OR each question! It's pretty unbelievable. 

I can see using it these tools every single day in one of four different ways: 

• a warm-up that reviews material from the day before or to see what students know BEFORE you teach see This activity I made as a preview to sections 1.1 and 1.3 in Pre-Calculus or my Pre-Calculus preview of section 1.4
• a formative assessment to see what the students understand from what I just taught Algebra 2 Honors--do you understand Quadratics?
• a blended learning tool for differentiated learning...in other words, students go through a step by step activity that leads them to discover a new topic at their own pace. Like this algebra 2 honors preview lesson on 1.1: symmetry and circles
• Flipping the classroom and having students complete one of these ahead of time.

I hope you like these as much as I do...they are all very user friendly, which means you can make them quickly...and use them from year to year and class to class. They are all SO EASY!

One more thing...if you haven't seen Sarah's blog (if you haven't, do you live under a rock?? JKJK). Rather than blog once a day for 180 days, she is going to tweet a picture out each day, and she invited others to do it. She will use #teach180 (see her pic, below) and she asked if anyone else wants to join...why not?? I learn pretty much everything now from MTBoS...would love to learn from you as well. Let's do it!

~Lisa

Save teaching time at the beginning of the year with "Are you ready for Pre-Calculus?" worksheet (and Algebra 2 Honors)

This year, we are going to block scheduling. There has been some talk of losing precious teaching time, and so the thought our department has been discussing is, how can we make that time up? Is there a way to do this without teaching a mile wide and an inch deep? Especially since we already felt like we could not get through enough material with the previous schedule.

I have always been a big proponent of teaching "Chapter P" at the beginning of the year, which means reviewing all of the prerequisites with the students. I have felt like I can get everyone on the same page, basically, and I will know that students really DID learn this (if they claim that they didn't in the previous year...though we all know they did!) This takes up a good 2 to 3 weeks of school, but my theory has been that if you teach/review the prerequisites well, you save time later when teaching concepts that are based on them.

However, at the end of the school year, our new department chair threw out the idea that we sort of skip over the prerequisites. I was appalled. WHAT?? There is no way I could do that. Reviewing at the beginning gives students confidence to start slowly with things they understand. They begin to trust you early. And they feel like they can DO THIS. How could I get rid of prereqs??

My colleague suggested the idea of a diagnostic test in my honors class at the beginning of the year, so that I could see what the students know and could skip anything they are all comfortable with. Though I love that idea and started off making that this year for my students, I began to think about the fact that we have so many different schools that feed into our 9th grade--not only several different high schools, but from several countries as well...so I worried that they would be all over the place with what they knew and did not know as a group, and I wasn't sure it would save me as much time as I needed.

Plus, I liked my knew way of teaching since after spring break. When I got my new classroom then with whiteboard wall paper, I really delved into the idea of students collaborating and teaching each other. I noticed that kids quickly became comfortable discussing what they knew with each other rather than working at their desks in a more isolated fashion. So I also wanted to use the idea of collaboration in getting all of the prerequisites out of the way in a few days. And I remembered a former colleague used to use an  "Are you ready for Calculus" worksheet...so looming fear started to turn itself into something I could possibly use in the classroom...

I don't teach Calculus, so I had to make up worksheets for my other classes. I will teach Algebra 2 Honors (out of Algebra and Trigonometry 9e by Ron Larson) for the first year this year, and Pre-Calculus (out of Pre-Calculus with Limits 3e by Ron Larson) for the second year. Before I go any further, let me first talk about http://www.calcchat.com/, which is THE ABSOLUTE BEST THING THAT HELPED MY STUDENTS THIS PAST YEAR WHEN WORKING ON HOMEWORK. @calcchat gives free solutions to odd homework answers to Larson textbooks, and my students (mostly) diligently used them, and I freed up SO MUCH class time going over homework. Yes, occasionally I caught a student copying answers, but it really only hurt them...my D student still remained a D student. Did I mention they also give live help from 4pm - 12am????

But I digress. So, in keeping with making students take ownership of their learning WHILE collaborating, I came up with two documents; one for Algebra 2H and one for Pre-Calculus. On the first day of classes, I will go over some class rules and assign the packet for homework with a time limit...I will likely tell them to spend 30 minutes or so just working on what they know. Then on the first block (90 minutes), I will divide them up into random groups (ala @AlexOverwijk from Twitter Math Camp) and have them work on the rest of the worksheet. I am hoping to use @pegcagle's idea of having red, yellow, and green cups for the groups to let me know if they are good, need help (they are frustrated, but "good" frustrated), or if they are freaking out in need of some major help.

After this class, I will decide if there is something I really need to spend time on, but rather than me teach everything, they will work together. There are videos (most from the Larson site on instructional videos ) and students can use our math lab for help. I think I will set up a google classroom (have not used this yet) link so that they can talk to each other while they are working on it outside of the classroom...or perhaps a twitter hashtag...any ideas on that one?

So here are the two worksheets.




Hopefully it will go well, and if not, I will adjust as necessary.
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Also, here is something fun that I just got...don't be jealous, you can get them, too. They are yoga pants that my colleague told me about from www.amazon.com. There is another kind, too...but you have to google "math leggings" rather than math yoga pants. They have been so much fun to wear at yoga...you should see all the jealous, I mean, weird looks I get from other yogis, haha!! I see a great Halloween costume in the making...or maybe my math club will design their own next year! Hmmmm....

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Don't know what to make for dinner? Go to Pinterest, which is where I always get my ideas. I just made this Chicken Quinoa Burrito Bowl from Gimmesomeoven.com tonight, and it was seriously delicious. I am not the best cook, and this came out really good, which means it's no fail. If I can make it, anyone can. My husband cleaned everything up (he usually does when I cook) so that I could finish this blog!

Gah! School is almost upon us. Where did this lovely summer go?
Until next time,
~Lisa