Pre-Lesson Reflection: Reviewing Slope

I have a really good relationship with my students this year. They feel like they can be honest with me, and I LOVE that. On a couple of occasions this year, they have told me (in droves) that the one thing they really don't understand from previous years is the concept of slope. We're about to hit slope in two or so weeks, so I've been thinking hard about how to help clear up their issues with slope.

Here are my thoughts:

Use a life-sized coordinate plane in gym. I made two a few years ago out of drop cloth and duct tape. Call origin point A.

Make up a scenario. Let's say we're going to make Banana Bread. In fact, let's say we're making this recipe.

Ask the kids to find how many bananas we need. 3 bananas. 3 bananas to make 1 loaf.  Let's label the x-axis bananas and the y-axis # of loaves. So, we get the ordered pair (3,1). Give a student a picture of 3 bananas and a picture of 1 loaf. They go stand on (3,1). Call this point B.

Same scenario. You happen to have 6 bananas sitting on the counter. How many loaves of bread can we make? 2. Give student a picture of 6 bananas and 2 loaves. They go stand on (6, 2). Call this point C.

Same scenario. You happen to have 9 bananas sitting on the counter. How many loaves of bread can we make? 3. Give a student a picture of 9 bananas and 3 loaves. They go stand on (9, 3). Call this point D.

Extending the line into quadrant III is more of a stretch. Let's says you needed to buy three bananas at the store, but you forgot them. You are three bananas in the hole from what you need. Since we are three bananas in the hole, how many recipes are we in the hole? Give a picture of -3 bananas and -1 loaves (maybe make them red, so students know they are negative). Student goes and stands on (-3, -1). Call this point E.

Same scenario. You meant to buy 6 bananas but forgot to purchase them. How many loaves are we in hole? 2. Give student a picture of -6 bananas and -2 loaves. They go stand on (-6, -2). Call this point F.

Same scenario. You meant to buy 9 bananas but forgot to purchase them. How many loaves are we in hole? 3. Give student a picture of -9 bananas and -3 loaves. they go stand on (-9, -3). Call this point G.

Okay. Now, have rest of class "tour the line". I would start with a quadrant 1 tour. Student stands on the origin. Then, he/she walks to point B. How many bananas did s/he increase by? How many loaves? Record on a sheet. Walk to point C. How many bananas did s/he increase by from Point B? How many loaves from Point B? Record on sheet. Have student continue this tour for all points, eventually going back and picking up Quadrant III.

After they have recorded all data, ask them to make a comparison. What is happening from each point to the one to its right? We are increasing by 3 bananas and 1 loaf.

Since slope is change in y/ change in x. Let's test this theory. Have student start at (-9, -3). Our y is loaves, so let's rise/increase by 1 loaf. Our x is bananas, so let's run/increase by 3 bananas. Have them continue to step off 1/3 until they get through the line.

You could repeat this scenario with quiz grades and # problems missed for negative correlation.

So... what do you think? Will this work or be a disaster? Anybody have better scenarios that make sense in Quadrant III for positive correlation and Quadrant IV for negative correlation?

Comparing Distributions - Paper Planes

I have a GREAT group of AP Stats kids this year. They are super intelligent, they want to learn (well... they want to talk A LOT, too), and they seem to really love statistics. As I have gotten better at teaching the course, my kids really seem to like stats. Other AP Stats teacher have said "Do your kids dislike the class?" Theirs do. My first year... mine did too. I was trying to figure the content out. I didn't have time to make it fun. Things have sure changed. Anyone that knows me personally knows that AP Stats is my baby. I LOVE the course. I LOVE the content. I LOVE the application.

I also love coming up with new activities. The paper airplane activity is an oldy but goody. I basically revamped it and created teams. When kids entered the class, they had 10 minutes to google/research/YouTube/whatever and build the best airplane they could build. I have never seen kids as absolutely engrossed in anything as they were this. After 10 minutes, I put the kids in groups of 2-4 and gave them a clipboard and data collection sheet. Here is the handout:

DOWNLOAD HERE

Then, we went out to the football and each conducted 10 flights and recorded distance traveled. We used the football field since I have so many kids in stats and I only had access to four long tape measures. Is the data perfect? No, but it works.

One student came up to me and said "Ms. Boles, this is REALLY fun." Which, I counted as a big win in teenager speak. The bigger win, though, is when we returned to the classroom to draw modified boxplots and compare distributions. 

Because this group is so competitive, I asked some, in their comparative paragraphs, to argue why their airplane was the best of the group. For one member - this was usually easy - their plane had the largest max. Others, however, had to argue that theirs was the best because it had a smaller spread and was therefore more reliable. Another argument was that one student's minimum flight distance was quite higher than others. Other students had to argue why their plane wasn't the least desirable of the group. Regardless of the argument, the kids really understood the concept, had GREAT comparative paragraphs, and had a great time in math class **faint**.

Target's $1 Aisle is a Stats Playground

I live in a very rural area. We have a Wal-Mart, a really sketch K-Mart, a Food City, other smaller "cheap" grocery stores", and LOTS of fast food. Developers are just finishing up a shopping center with a Marshall's and Hobby Lobby, though. No Target in sight. Probably ever.

The closest Target is about 2 hours away. For some background - I also grew up in a similar area - no Target. When I lived in DC, I was first introduced to Target's wonderfulness (to be clear - DC Target is a little less wonderful than average Target wonderfulness).

Reminiscing aside, I traveled to said nearest Target yesterday. And - let's be clear - the most awesome part of the whole store is obviously the $1 section as you come in the door. I found about 5,280 pieces of wonderful $1 items that I thought - at the time - were absolutely essential to purchase. After some deep breathing and reasoning with myself a little, I finally purchased these...

"What is that?" you ask. I'm so glad you did. It's a dozen bags of small, plastic multicolored woodland animals.

I opened one bag. Here is the assortment of colors/creatures I received:

There are 6 creatures (all very cute) in three different colors.

There are little foxes and deer.

Also joining the party - squirrels and rabbits.

I hear you. This is a math blog. Tell me about the math. Okay - if you insist, but isn't the fox cuteeee? 

Here are my thoughts: this little bag is a probability play ground.

Kids usually have so much trouble processing the general probability rules, and these little guys can work like manipulatives for said topics.

Some examples:

Put all woodland creatures in bag. If you mix the bag up well and select one animal, what is the probability that you pick a red animal or a fox? (Addition Rule for Nondisjoint Events)

Put all woodland creatures in bag. Manipulate assortments so bags only have species of one color. For instance - green foxes, red deer, purple squirrels, and red rabbits (I forget other two animals right now). Question: If you mix the bag up well and select one animal, what is the probability that you select a purple animal or a fox? (Addition Rule for Disjoint Events)

Put all woodland creatures in a bag. If you mix the bag up well, what is the probability you select a fox, return the fox to the bag, and select a rabbit? (Multiplication Rule for Independent Events)

Put all the woodland creates in a bag. If you mix the bag up well, what is the probability you select a fox, keep it, and then draw a rabbit? (Multiplication Rule for Dependent Events)

Are color and species independent? (Independence Rule)

Given that a green fox has been drawn and not replaced, what is the probability you draw a red squirrel? (Conditional Probability)

SO MANY possibilities! SO MUCH fun. 

The main part, though - is that, with these manipulatives, kids can actually predict/see/count the answers, which really, really helps the formulas and calculations make sense.

Now, quit reading this, and go to Target.

Starting the Year Off Right: Growth Mindset

I always heard teachers talk about doing days of team building, mindset, goal setting, and other "soft" skills at the beginning of the year. And, yes, I would half-heartedly do something like this because I felt like I had to. But it was half-hearted, and I was concerned I was losing valuable math hours.

Things change. I had a pretty yucky year last year. To be fair, it was a tough crew, and every teacher that taught them had the same feelings. It was like we were playing Tug of War... the whole class was at one end and I was at the other dragging them through the content. They didn't like group work. They didn't like projects. They didn't like lectures. They didn't like pretty much anything. Why? This group (for the most part) struggled. If something was hard; they disengaged. They quit. Enter behavior issues. This wasn't a new development. It had been happening for years. As a class, they had lots of skill gaps, so when a new topic was presented that hit on one of those weaknesses, a lot of students had very negative emotional responses. It was a class that basically believed they couldn't, and so they wouldn't.

At the beginning of this school year, I vowed I would do something to help my kids have a better mindset about mathematics and learning in general. Enter two days of growth mindset work. I know. You're talking about the down-to-business math teacher here giving two whole days for "soft" work. And, to be frank, I'd do it again in a heartbeat.

Here's how it went down:

DAY 1

Marshmallow Challenge. Just assign it like any other group project. Be explicit. Give kids the rules and materials. Set a timer on the board and let them go. Most groups will fail (their towers will fall). LET THEM. By the way, this ties in nicely with geometry curriculum since it gives you a chance to eventually discuss the strength of a triangular structure.

After the Marshmallow Challenge (don't clean up yet), show the following two videos:

Khan Academy: You Can Learn Anything

and

The Power of Belief

Afterward, we made observations about each group's tower. Usually there is at least one tower that stands per class. Why did it stand? How was their planning and building process different from the other groups' processes? Let the kids make the observations. I did have to prompt mine sometimes to get them to notice the important things. Here were the takeaways I wanted:

• Groups that stood largely had triangular structures.
• Groups that stood went for structural integrity rather than height.
• Groups that stood started with the marshmallow (the weight) and built the structure underneath to support it. Not vice-versa.

Students recorded these observations in notebook. We cleaned up the mess. End of Day 1.

DAY 2

As soon as students were seated, we review the takeaways from Day 1. This is done in the vein of "What did we learn from yesterday and how can we apply it today?". The same groups repeat the same marshmallow challenge. Here's what will happen - almost every group's tower will stand. I had 1 or 2 towers not stand over the course of the entire day, but that was 4 periods - pretty good. Even in the 2 or so that didn't stand, they were dramatically improved.

After the marshmallow challenge, I didn't declare a winner. You can if you want. We were all winners, though (and not in the give every kid a trophy way - I hate that). You see, we'd failed, we'd made observations about how to improve, and we'd used those observations to be more successful. The culture that this activity builds is awesome.

As a final activity, I talked a little bit about growth mindset (read: gave a motivational speech) and then we filled out a growth mindset handout.

Here it is:

DOWNLOAD HERE

Handout was lovingly borrowed from Dallas at nerdynerdynerdy.com.

Months later, kids (even kids I don't teach!!!) are still talking about growth mindset and ending sentences like "I don't know what I'm doing"... yet. (By the way, the power of "yet" was part of my motivation speech). It's a classroom culture changer and maybe the best two days I've spent this year.

 

Using Google Slides as Group Whiteboards - Angle Puzzles

If you've read my other posts, you've probably gathered that my geometry classes are all 1:1 now. I had a few kids with laptops last year, but now - virtually all of my geometry students have them.

This new opportunity pushes me to rethink the way I structure and deliver all content. In previous years, lots of practice time was somewhat traditional - student would do a problem, instructor would provide feedback. The problem here is the delay inherent between the pencil/paper feedback cycle. Don't get me wrong - I'm usually all over the classroom helping, prompting, asking questions, etc., but there are still delays with this traditional method (I mean there are 30 kids and 1 of me).

Enter our 1:1 devices. I automatically ask myself how I can restructure the lesson so it's more effective, students get better/more immediate feedback, and it's truly more personalized. To do this, I analyze the takeaways I want students to have, the structure and necessary supports needed for a particular lesson, and the way in which I need students to practice.

One such lesson is my annual day of "angle puzzles". Basically, these are complex drawings composed of sets of parallel lines and transversals that ask students to reason and apply their knowledge of angle pairs. HERE are the puzzles I use. Yes. This is a link to Teachers Pay Teachers. No. I'm not a fan, but I couldn't find them anywhere else when I was looking a couple years ago.

Having used these angles puzzles for a while, I knew the following:

1. Low to low middle kids struggle with angle puzzles and need lots of peer/instructor support. Groups are a good thing.
2. On these puzzles in particular, students get "stuck" and either disengage or wait on instructor help.
3. These kind of puzzles need lots and lots of instructor feedback.
4. When kids work in groups on these puzzles, one student tends to dominate the group.

To address these problems, I decided I needed the following features from a technology solution:

1. Collaborative work from multiple devices on same file
2. Instructor ability to see student work on own screen at all times
3. Instructor ability to provide continuous feedback to groups/teams
4. Groups/teams need opportunity to respond to feedback
5. Ability to include multiple puzzles to groups in one file

Having analyzed these needs, I determined that Google Slides could offer all these features. So, I took the snipping tool and created an image file for each angle puzzle. Then, I inserted these files as the backgrounds of individual slides. This is important. Since the image is the background, students can't move the puzzle around or delete it. Since we don't have Google Classroom, I took the slides link and shorted it with bit.ly before providing it to the groups.  Here are some samples of my students' work:

Takeways: My students (for the most part) loved this activity. Google Slides worked just how I'd hoped. The one issue I found though, was when the students inserted the text boxes. In image #2, the text boxes are large (standard size when you click to create the text box instead of drag to create them). If students do not resize the text boxes, they have trouble clicking on the correct text, since the text boxes begin to overlap. The group in image #1 have resized their text boxes appropriately and had few issues. Image #3 shows the back and forth comments/feedback between students and me.

It was a great day!

Teaching Histograms - Histogram Buckets!

I feel like most statistics teacher have the problem of getting kids to understand histograms. Yes. They kids can tell you that histograms are for quantitative data and the bars touch, yadda yadda yadda. That's not the problem. The problem is getting kids to really conceptually understand histograms, rather than thinking they are some sort of glorified bar graph.

I set out to fix this little histogram problem this year, so I came up with the idea of histogram buckets. First, I came up with a data set and printed the numbers on magnetic printer sheets. Each number is a block.  I cut them out individually.

When the students came in the door, I gave them each a square. When prompted, the students brought their magnetic blocks and placed them in the appropriate pre-labeled bins.

From the bins, I then took the magnetic squares and placed them on the whiteboard behind them. I used electrical tape to make the axes.

Then, we discussed what would happen if we changed the bin widths. So, we did... from 5 to 10. I handed the magnetic blocks back and changed the widths on the bins.

Students, again, placed their blocks into the appropriate bin. The histogram looked like this:

This helped to easily discuss that the histogram was representing the exact same data, but that, when we changed the bin widths, the displays looked different. This activity helped us to really have great discussions on what appropriate bin sizing is and is not.

I also had an INB handout for kids to fill out while we completed this activity. Here's what it looks like:

 I'm linking my histogram blocks and the handout files below.

Downloads:

Histogram Buckets for Magnetic Sheets

INB Student Handout

AP Statistics - Free Response Journals

I promised @DruinOK that I would write this post at the beginning of August. I obviously lied. I hope it's better late than never.

In previous years, I have done a fairly poor job at regularly and meaningfully integrating Free Response Question practice into my AP Statistics units. I mean, yes, the kids would engage with one or two on their exams and quizzes. Then at the end of the year, I did basically an FRQ blitz. It killed me AND the kids. I knew this wasn't as meaningful as it should be. It was kind of like saying - Oh hey - I taught you all this content, but I didn't teach you how to relay it, so let's cram it all here at the end.

Last year (about halfway through the year), I knew I was "doing it again". So, I tried to invent something that would change it. The AP Calculus teacher at our school did Free Response journals. Basically, the kids just had a separate spiral-bound notebook in which they'd write all their free response answers. I really liked this idea, but I wanted it to be a little more for stats. After all, we all know that Stats is, for a better word, English picky, so I wanted the kids to be able to refer to the original FRQ, their answer, and the grading rubric.

Enter my FRQ INB journals. (See those strange black/white/green ones in middle? That's them! It's actually extra cute "Bah Bah Sheep Duct Tape" from Amazon. Be jealous.)

Anyway.... here's how they work. First they have a grading rubric in the front (not shown). Then, a TOC. After that, each page is basically one FRQ from a previous year. I use the FRAPPY format from @StatsMonkey, but then I also include the grading rubric.

To include both the Frappy and the grading rubic, I create a merged PDF. I usually use smallpdf.com to do this. I also always have a one-page blank PDF on hand to use. To create a blank PDF file, save a blank word file as a PDF.

Here are some basic rules (assuming Frappy is 1 page long):

• If rubric is 2 pages, then your merged PDF will be page 1 - blank, page 2 - Frappy, page 3 - rubric pg. 1, page 4- rubric pg. 2
• If rubric is 3 pages, then your merged PDF will be page 1 - rubric pg. 3, page 2- Frappy, page 3- rubric pg. 1, page 4 - rubric pg. 2
• If your rubric is four pages, then your merged PDF will be page 1 - rubric pg. 3, page 2- Frappy, page 3- rubric pg. 1, page 4 - rubric pg. 2. In addition, you'll make a separate merged file of the last rubric page. You'll want this additional file to have rubric pg. 4 on both pages 1 and 2.

Okay. Now you're going to open the merged file up in Adobe Reader and print it. You'll want to select the multiple pages per page option, landscape, and have it print double sided, flipping on the short side.

For instance, here is an example of a 2 page rubric:

When you have either 2 pages of Frappy and 3 rubric pages OR 1 Frappy page and 4 rubric pages, it gets a little harder. Basically, you print the last page off separately (see directions above for 4 pg. rubrics). This single page gets glued down before the actual folded page gets taped in above it. This sounds complicated, but once my kids did it once, they got the hang of it. They now assemble this book when I have a sub - by themselves.

Example of 5 page FRQ (if you can tell from the picture):

I make a point to do these FRQ days at the end of every chapter and/or unit (whenever it fits best). These days not only make me intentionally and purposefully teach FRQ strategies, it also gives kids a great cumulative FRQ review book to look through near test time. I pick 2-3 per day to work on. The kids basically just read and answer the FRQ. Then, we read the rubric as a class and discuss how it would be graded. The first day we did this, I had samples from AP Central printed out and students graded those according to the rubric before grading their own. Sometimes students will also switch notebooks and grade each others' responses. I just depends.

I KNOW my directions are probably confusing and merging/printing these PDFs at first might be complicated (in the end, it's not, though). I'd love to help you! Tweet me @KLBoles !

Organizing Interactive Notebooks (INBs)

My first year doing interactive notebooks in both my geometry and AP Statistics classes was last year. I really like the INB concept for multiple reasons, and after I found a happy medium between trying to the the INB Van Gogh and just giving taped-in worksheets, I began to love what INBs were doing for my classes. What I didn't love, however, was how I was storing them.

Last year, my storage idea just didn't work. I had a bookshelf at the front of room and a bookshelf at the back of my room. Because there were no divisions in the shelves, the books were thrown in them every which way. Being the organizational freak I am, I absolutely could not stand the way they looked or the amount of time some students had to look for their notebooks at the beginning of class.

This summer, I was browsing my favorite deal site - slickdeals.net (you should check it out), and I saw that ClosetMate stackable shoe organizers were on sale at WalMart for $21. $21 bought you a 3'x3' organizer. I wanted to have 6 rows, so I bought two and stacked them. I also went to Lowe's and had a sheet of plywood cut into 2 3'x3' pieces. I then painted the plywood to color code it by class before attaching it to the back of the organizers. My class INBs already have a piece of duct tape on the spine to color code the books themselves.

In the end- I LOVE it! It fixes ALL of my storage issues with INBs, and it looks really great - at least I think so.

P.S. If anyone is at the Dollar Tree and sees those little pails in yellow - I need a set desperately. The random green one in the yellow row is KILLING me.

Pictionary with Socrative

I meant to write this post earlier in the year (say - oh - August?), but that obviously didn't happen, so here we go...

This is my fifth year teaching geometry, and even though I love teaching the course, the basic naming and definitions portion at the front of it can often be boring. I mean, sometimes, it's really hard to spice up how to name a plane.

After we've discussed basic nomenclature, I always ask students to flip the process, meaning I give them a description, and they give me a drawing. This part is usually a little more fun. In the past, I've made index cards with the situation on the front and a possible solution on the back. One partner reads the text to the other partner, who does his/her best to draw the accompanying image on a whiteboard. I then run around like a chicken with her head cut off trying to look out for misconceptions and errors. Hey - it worked, but you never really knew when you had missed out on that one special moment... you know, the one where something was slightly incorrect, and you, the instructor, could have posed a question or a clarifying statement that could have really helped the kids build a solid understanding of the content.

This year, my students are 1:1 with Dell laptops. No. They are not touch screen. And yes - I'm bitter (but don't tell anyone). Think of what else my kids could do if they were touch capacitive. Anyway - that's not important right now. I'm off track. Here's what we did...

I used @sandramiller_tx 's graphics, since hers were digital and mine were hand-written. Go visit Sandra at https://tothemathlimit.wordpress.com. Anyway, I took these images and put them in Socrative. I used a standard multiple choice quiz format - with only choices of A or B.  Choice A was to be selected when student answers matched the exemplar or when they didn't but partners were able to discuss and remedy the mistake. Choice B was to be chosen when the partners disagreed or did not understand the proposed solution. The way the activity worked - one student had the laptop and read the written explanation. This student was also looking at the proposed image solution. The second student in the pair had a whiteboard and was drawing as the first partner was giving the directions. When partner 2 was finished, they both compared the whiteboard to the computer solution and discussed then appropriately chose either option A or option B. On my teacher dashboard, A answers showed up as green (I marked them the correct answers), and B answers showed up red. Even from a distance, I could immediately see when a student was struggling and who to assist. It's almost the same idea as the Red/Yellow/Green cups, but less troublesome.

Here are some screenshots from the activity:

If you'd like to try this activity, the share code for this Socrative quiz is  SOC-17217490.