#MTBoSBlaugust Day 12: Mathematical Mindsets Norms Video

Somehow in my stream of consciousness Google clicking this week, I ended up on Cathy Yenca's (@mathycathy) blog post from 2013 on First Day Favorites. I loved the professional and engaging look of her video and it dawned on me that I could use this to convey a message I wanted kids to hear from me clearly and consistently.

Thinking back to my readings about math class norms from Mathematical Mindsets, I decided to create a video to summarize what I want my students to walk away from the Post It Activity knowing I believe about them. It's brief and will ensure that all my classes are hearing a consistent message from me. Once we wrap up our class discussion, I'm planning on playing this for them and having them reflect as an exit ticket.

I have the file saved if anyone wants me to send it so it can be tweaked! Like Cathy, I used the free trial of VideoScribe to create this! It's super user-friendly!

#MTBoSBlaugust Day 11: AP Calculus AB Learning Targets

Here's my first draft of my AP Calculus AB Learning Targets for this year. My goal is to add these to my Student Unit Organizers so students have a road map as we go through the unit. In addition, these will help me design my assessments and my study guides- the next 2 tasks on my list. (How do I already feel behind when there are still 3 weeks left until I see kids?!?!?!)

These are a compilation of my own 2015-2016 course calendar, information from my APSI last month, the AP Curriculum Framework, and reading through other teacher's course calendars for student-friendly wording (This one in particular was helpful). They align to Calculus of a Single Variable, 8th Ed. by Larson, Hostetler, and Edwards. 

I've made a few big changes from other teacher's that I've read, the book, and some things I didn't like last year. The biggest ones are:

• Weaving transcendentals and trig throughout the units, rather than separating them like the textbooks and many other teachers do. I am still trying to wrap my brain around late transcendentals- I feel like the more practice kids can get with these intricate functions the better. If someone wants to convince me otherwise, please please please do. I am interested in hearing the other side. 
• I switched the order I taught integration. Last year I started with the "area problem" as a motivation and then switched back to general antiderivatives, but starting with antiderivatives seems more of a natural flow and allows us to revisit u-substitution multiple times, rather than just once. It's my major change for this year, but I'm excited about it. This also would logically put differential equations in the middle of the integration unit, which I think will be a nice break for the kids. 



https://www.lookhuman.com/design/
73407-girls-just-want-to-have-differential-equations/
6733-heathered_blue_nl-md

• (Side note: I just ordered the shirt at right...while I was writing these learning targets.... I need to start buying back to school clothes, but this was so much more fun.)
• I moved volumes of known cross sections to after disks and washers, but I'm not sure if I'm going to keep it that way. Disk and washer were simpler cases for my kiddos last year since they are always the formula for a circular cross section; I saw so much more struggle in using the formulas for other shapes.  I'm considering testing it out this year to see how it changes the understanding of the kids. Nothing set in stone here yet. 
• L'Hopital's isn't an "after exam" topic anymore- it's right in there, during my derivatives unit! Excited to get to spiral in some limits there to review. Also excited that it's only the simplest case, not all the other wacky indeterminant forms we'd normally get into after the exam. 

Let me know if you see anything I missed or have any sequencing tips that made a huge difference in your students' understanding! 

_______________________________________________________

AP Calculus AB Learning Targets

2016-2017

Chapter 0: Pre-Calculus Skills

• I can identify minimums, maximums, intercepts, intervals of positive/negative, and key values from a graphing calculator
• I can sketch all parent functions by hand and identify their domain and range
• I can write the equation of a line given a point and a slope
• I can rewrite an absolute value function into a piecewise function
• I can rewrite expressions using factoring
• I can rewrite expressions using rational operations
• I can rewrite expressions using long division
• I can rewrite expressions using completing the square
• I can solve equations and inequalities graphically and algebraically
• I can find the inverse of a function algebraically
• I can compose and decompose functions
• I can use properties of logarithms to rewrite expressions and solve equations
• I can identify exact trig values of important angles in the unit circle and use them to sole trigonometric equations

Chapter 1: Limits & Their Properties

• I can evaluate the limit using a table  
• I can use a graph to evaluate the limit
• I can evaluate a limit using properties  
• I can evaluate a limit by using direct substitution  
• I can evaluate a limit algebraically  
• I can write a simpler function to evaluate a limit  
• I can evaluate a limit using two special trigonometric limits  
• I can evaluate a limit using the squeeze theorem**
• I can evaluate a one-sided limit
• I can determine if a function is continuous (satisfy 3 conditions)
• I can discuss the continuity of a function on a closed interval  
• I can identify the type of discontinuity by name  
• I can use Intermediate Value Theorem (IVT) to analyze function behavior in an interval
• I can write the equation for a vertical asymptote  
• I can evaluate limits with function values approaching ±∞
• I can evaluate limits as x approaches ±∞
• I can find the horizontal asymptotes of a function

Chapter 2: Differentiation

• I can explain how the slope of secant lines can approximate the slope of a tangent line
• I can use the average rate of change (slope formula) to approximate the derivative of a function
• I can identify derivative as an instantaneous rate of change  
• I can find the equation of a tangent line at a specific point  
• I can find the equation of a normal line at a specific point
• I can find the general derivative using the limit process
• I can find the derivative at a point using the limit provess
• I can explain the relationship between the limit definition formulas (at a point and general) and the slope formula from previous courses
• I can use a graphing utility to find the slope at a specific point, sketch a possible graph of the derivative of a function  
• I can apply to alternative form of the derivative  
• I can find where a function is differentiable
• I can differentiate using the power rule  
• I can find where horizontal tangents occur
• I can find the derivative of sine and cosine
• I can understand how the derivative applies to Position/Velocity/Acceleration
• I can differentiate using product rule  
• I can differentiate using quotient rule
• I can find the derivative of tangent, cotangent, secant, and cosecant  
• I can differentiate using chain rule  
• I can differentiate using more than one rule
• I can understand function notation to find derivatives, including differentiating from a table
• I can find derivatives implicitly
• I can find the second derivative of an equation implicitly  
• I can find horizontal and vertical tangents of an implicitly defined function
• I can identify when L’Hopital’s rule applies to an indeterminant form (p.567)
• I can evaluate a limit of the form 00 or using L’Hopital’s rule (p.567)
• I can evaluate (f-1)'(a) (p.341)
• I can find derivatives of functions involving the natural logarithmic function
• I can find dy/dx using logarithmic differentiation(p.322) **
• I can find the derivative of the exponential function (p.350)
• I can find the derivative of a function involving a base other than e  (p.360)
• I can find the derivative of an inverse trig functions (arcsin and arctan must be memorized)

Chapter 3: Applications of Differentiation

• I can apply the extreme value theorem  
• I can find critical values of a function  
• I can find relative extrema of a function  
• I can find absolute extrema of a function using the closed interval test
• I can use the First Derivative Test to find relative extrema of a function
• I can relate the First Derivative Test to the second derivative  
• I can find points of inflection of a function  
• I can find intervals of concavity of a function
• I can sketch the graph of f ′ and f ″ given the graph of f(x)  
• I can sketch the graph of f(x) given the graph of f ′ and f ″
• I can use a tangent line to approximate function values  
• I can use differentials and the graph of f to approximate values**
• I can verify that a the criteria for Mean Value Theorem apply
• I can apply the Mean Value Theorem
• I can write an argument to justify my use of Mean Value Theorem
• I can apply Rolle’s Theorem
• I can apply the Second Derivative Test to find extrema
• I can solve applied minimum and maximum problems
• I can identify important quantities and equations in a related rate problem
• I can solve related rate problems involving distance
• I can solve related rate problems involving area and volume

Chapter 4A: Integration (General Antiderivative)

• I can find the general
• anti-derivative of an algebraic function  
• I can recognize why we need a constant of integration  
• I can define the indefinite integral and its parts
• I can find the general anti-derivative of a trigonometric function
• I can find the original function from the graph of the derivative  
• I can find a particular function given certain conditions  
• I can find the anti-derivative of a natural logarithmic function (p. 332)
• I can find the anti-derivative of a function involving a base other than e
• I can find the anti-deriative of an exponential function
• I can integrate functions whose antiderivatives involve inverse trig functions
• I can use the method of completing the square to integrate a function
• I can integrate functions using u-substitution
• I can integrate functions using long division

Chapter 6: Differential Equations

• I can sketch the slope field to represent a differential equation  
• I can sketch the solution curve to fit a given slope field  
• I can choose a differential equation to fit a given slope field
• I can use separation of variables to solve a simple differential equation  
• I can use exponential functions to model growth and decay in applied problems
• I can use exponential functions to model compounded continuously problems  
• I can identify a problem as exponential (y=Cekt) when it discusses the rate being proportional to the amount present
• I can find the general solution of a differential equation
• I can find the particular solution with conditions

Chapter 4B: Definite Integration and the Fundamental Theorem of Calculus

• I can estimate the area under a curve using a Riemann Sum  
• I can estimate the area under a curve using the Midpoint Rule
• I can approximate the area under a curve using the Trapezoidal Rule  
• I can compare Left, Right, Midpoint, and Trapezoidal approximations of the area under a curve
• I can evaluate problems involving summations both with and without calculators
• I can identify the limit definition of the definite integral
• I can represent the area of a region using a definite integral  
• Ican recognize that the value of a definite integral can be found using geometry
• I can use properties to help evaluate definite integrals
• I can use the Fundamental Theorem of Calculus to evaluate definite integrals
• I can accurately calculate a definite integral using u substitution and change of bounds
• I can evaluate a definite integral using a graphing calculator  
• I can find the average value of a function using the Mean Value Theorem for integrals  
• I can use the Second Fundamental Theorem of Calculus to find the derivative of a definite integral
• I can use the differential equation and a given point to find the function
• I can use an accumulation function to answer application questions

Chapter 7: Applications of Integration

• I can finding area between two curves
• I can use the disk method to find the volume of a solid of revolution
• I can use the washer method to find the volume of a solid of revolution
• I can use the shell method to find the volume of a solid of revolution**
• I can find the volume of solids whose cross-sections are known

#MTBoSBlaugust Day 10: Mathematical Mindsets

Just finished Jo Boaler's incredible Mathematical Mindsets. Can you tell it gave me a few ideas? 

For anyone who hasn't read it yet, Boaler does an incredible job synthesizing recent research on everything from the influence praise has on toddlers to the influence assessment and classroom environment have on students. She looks at achievement from many angles and offers tons of practical advice and strategies for all grade levels. I feel this insane desire to put one in the mailbox of every elementary teacher I've ever met who told me they "weren't math people." It's that kind of read. 

I wanted to go through some of my favorite ideas in the book and try to marry those to some ideas for my classroom, so bear with me here. Feeling a lot of inspiration mojo today. 

1. No one is "born" with a math-phobia

My brother is a computer scientist and my sister-in-law is a nurse. They both work in extremely math-centric careers and have a large amount of formal math training. Yet the insist that my 2 year old niece already needs me to set some free time aside to tutor her in high school because they won't be able to help and she'll probably struggle just like mom and dad did. It drives me insane. Math phobia is learned (maybe even- dare I say- taught) and I never want her to experience that. I see the unbounded love she has of experimentation right now....I never want her to lose that. 

This particular passage shot out at me the minute I read it...both when thinking about my own family and thinking about my kiddos:

"...researchers concluded that the difference between high- and low-achieving students was not that the low-achieving students knew less mathematics, but that they were interacting with mathematics differently. Instead of approaching numbers with flexibility and number sense, they seemed to cling to formal procedures they had learned, using them very precisely, not abandoning them even when it made sense to do so. The low achievers did not know less, they just didn't use numbers flexibly- probably because they had been set on the wrong pathway, from an early age, of trying to memorize methods and number facts instead of interacting with numbers flexibly. The researchers pointed out something else important- the mathematics the low achievers were using was harder mathematics."

The frustration so many students that have been labeled low face is that they have always trusted that the procedural way a teacher taught them was the only way. When that procedure became too complicated, too distant from their own intuition, they began to think of themselves as failures. And no wonder they've lost trust in themselves and their math teachers by the time they get to high school. I just imagine that kid in the back of the room who refuses to do what's asked of him because he's defeated before he starts....that's the kid who has been trying to keep up by doing harder math all along. Your heart just breaks for them. Of course they hate math. 

2. Homework isn't just a question of practice, it's a question of equity
My husband has always refused to assign homework in his science classes and as a math teacher I've never been able to go quite get there. I was different than many math teachers I've met in that I assigned only a small amount of homework and gave my students a talk on the first day that if they were struggling after ____ amount of time on homework (depending on the grade level), they should stop and come in the next day with questions. But this book really pushed my thinking on the topic.

What was particularly interesting to me was to see the amount of research backing up the neutral or negative effect of homework on student achievement, as well as the discriminatory effects of homework grading practices. In a sense, backing up what I have seen every day since I started teaching. I have taught those kids who have a job (or jobs) to help support their family, don't really eat when they aren't in school, and more or less raise their siblings when a parent is absent, working long hours, or deceased. I've always had a positive enough relationship with these kids to work one on one with them and decrease their amount of work or give them an extension on it. But I saw in their eyes that it stressed them to know that I was making an exception for them. Seeing these observations confirmed with so much research was huge for me. 

What I also loved was that Boaler gave an alternative if you are at a school that requires homework. Instead of giving a list of problems from the book, use that time to help students reflect and self assess. Boaler's homework reflection questions can be found here. 

Boaler also offers 6 strategies to purposefully make math class for equitable for all. Homework is only one part of a much bigger puzzle. I created this graphic so I can hang it right above my desk. I want these challenges nearby whenever I'm planning. 

Equity is how we better kids lives. It's the whole point of education. I'm excited that I'm becoming more cognizant of it through studies like this. 

3. The 5 C's of Mathematics Engagement

Love these. Some things to shoot for every day. 

4. Designing & Adapting Math Tasks
Boaler offers 6 questions to ask yourself to try to adapt a mathematics task to your classroom and I love them! Another little graphic for my desk so they're never far away! 

5. Heterogenous Grouping Helps All Students
So often when a student is grouped with those deemed "outside his or her ability level," you run into issues with the outside world. Parents worry their student will be left behind or not challenged enough and can be extremely vocal about expressing it. Boaler emphasizes that this has been disproved many times and it's helpful to read some of the research to have for discussions with parents and administration. She also discusses Complex Instruction, which examines student engagement in relation to their (actual or perceived) status in a group and works to directly create equity and student accountability.

One thing I will take from this is more of an emphasis on the use of group roles. I used them when I taught middle school, but have moved away from them in high school. I want to give them another go this year with this framework to help and see how it works. I think it might be especially helpful in my standard level classes, where group work can be more of a struggle. 

I particularly loved this quote from the section on Complex Instruction...it was a huge part of my STEM group norms at my last school:

6. Assessments Shape Mindsets
Students label themselves by their test grades and this identity can be such an impediment to mathematical growth. In such a performance-centric world, Boaler encourages a complete reexamination of assessment practices. 

One strategy I loved was one a 20 year veteran teacher had shared with her. Students were told to answer as many questions on an assessment as they could, then draw a line across the page when things got too difficult. Any questions below this line could be answered with the help of a textbook. The work beneath the line then became the fodder for classroom discussion. This was more about assessing students in an effort to continue their learning journey- a true example of formative assessment. 

Boaler also discusses Assessment for Learning (in contrast to assessment of learning). This demands:

1. "Clear communicating to students what they have learned"
2. "Helping students become aware of where they are in their learning journey and where they need to reach"
3. "Giving students information on ways to close the gap between where they are now and where they need to be"

I love the idea from this section of generating "I can" statements for each unit of your course and having students use their as a self-assessment. If anyone has geometry or AP Calc AB resources for this, please feel free to share :)

Boaler gives lots of advice on grading and I'm still working on wrapping my head around the practicality of a lot of these. One that will particularly influence my grading scheme is the idea of a 10 point separation between letter grades A-D, then a 60 point separation to an F. She suggests using a point scale of 0-4 to keep grading more mathematically fair and I like this system for homework. 

7. Growth mindsets can be built by teachers

The final chapter of the book is strategies for growth mindset teaching and they are helping me shape my first day activities that I shared previously. In my Post It Note activity from a previous blog, I was still working on shaping the questions I wanted students to answer. Boaler gives a list of the positive norms she encourages in class and I am going to frame my questions around these.

http://www.youcubed.org/wp-content/uploads/Positive-Classroom-Norms2.pdf

Read more about here norms HERE. 

She also talked about participation quizzes, a method to encourage productive group work. This fits perfectly with my previous post about AP Calc study groups and I'll definitely be using it with them. Here's a summary of the strategy.  

Lastly, she talked about the way that teachers interact with students in the classroom. I want each of my students to truly believe I think they can learn and I want to make a conscious effort to give growth praise and help, not fixed praise and help. Praising my kids for their efforts, their specific strategies, and their failures will be more beneficial for them and provide me better information about my kiddos in the long run. It will make me a better aunt and (someday) a better parent, too. 

To anyone who hasn't read the book yet, I highly encourage it. It is a game-changer for anyone who feels like they're in a rut or wants some new ideas and it contains tons of research and strategies for your benefit, no matter the level you teach. 

Next, I'm diving into Make It Stick! My kids won't know what hit them this year! 

P.S. Anyone else feel like we need to start an MTBoS Book Club? Do some online book studies? 

#MTBoSBlaugust Day 9: Canvas Crash Course

I was lucky enough to pilot a blended learning course for my old district using the teacher reach model, where my class size was doubled and I saw each group every other day. On the days when I didn't see my kids, they had assignments through an online learning management system. Year 1, we settled on Schoology. I finally felt like I'd figured out all the kinks and started making tweaks to my course when....BAM...Year 2, the district switched Canvas. Now I've moved districts AND changed preps, so I'm on to LMS #3 in 3 years. 

So, in effort to save some of you a little time and a little headache, I've prepared some of my favorite Canvas tips. It's a lot less intuitive to set up than Schoology or Edmodo, but has a lot of capabilities that can be explored. This is by no means an exhaustive list, but I hope it will get some people started who might be in a situation like I was....trying to frantically piece together a whole course right before the school year starts! 

Tip #1- Make It Navigable

Canvas is an LMS with tons of options- great, great, great. This is a slippery slope, though, because with options come lots of ways to get lost. [Think....you start looking for one thing on Google and end up in the Wikipedia wormhole....that kind of lost] For example, a student can get to an assignment from the assignments tab, the modules tab, the upcoming reminders on their homepage, the gradebook, and more. Be cognizant of how you're teaching students to navigate to content. I opted to do this by creating buttons on my syllabus (which I made my homepage) that would take students directly where I wanted them to be. 

Here's a tutorial I take no credit for creating: Creating Buttons on Canvas

You are able to hide or show certain options on your side bar with ease and only give students access to what you want them to see. With younger student, you may want to use only use buttons and not give them the access to the left side bar at all. My homepage looked like this for my senior level Pre-Calculus class:

Another important element of using this with students was some direct instruction with how to use it. Other LMS's are much more intuitive for kids as they behave like the most common social media outlets. Canvas takes a bit more finesse to use correctly and teachers need to be patient with their students when they are still learning.  I used this Canvas Scavenger Hunt  (that my wonderful husband created and I modified) to get my kids started. 

Tip #2- Head Cheating Off at the Pass

Let's be real, when kids aren't being monitored some of them are going to try to cheat. There are tons of great activities online that can be used for practice or exploration, but unless your school has boundless money, chances are they aren't all things that kids will have a unique login to use. To try to combat this, if I ask my students to screenshot anything I require that they either:

a) have their email open in another tab

b) have their username in the corner (easy on Macs)

Could kids still do this and cheat? Yes. But the effort would be so much greater and if kids perceive that you're holding them accountable I have found they're less likely to try to take the easy way out. 

Tip #3: Find Resources that Work for You

Blended learning can be a benefit for kids who don't like feeling pressured to work at an unnatural pace. Whether they work more slowly or more quickly, it's great to find resources that can adapt to an individual student. Some of these cost money, but some are free and can be used to differentiate or just for extra practice:

• ALEKS- Honestly, if your district pays for it, it's a blended learning dream. You can create assignments, allow kids to re-test as many times as you want and it will generate new questions, assign specific things to specific kids, and pre-assess easily. 
• Khan Academy- Not just videos, ya'll! The activities are great because they provide hints, instant feedback, and will keep going until students demonstrate mastery. This can be incredibly frustrating for kids, though. Make sure to assign things that are in their ZPD or you'll have some upset kiddos on your hands. 
• Quia- My kids used to beg me to post "Millionaire" activities. Again, you can require students to get a 100% or demonstrate mastery in some way and then screen shot it. Just make sure you play them first yourself; they sometimes have errors.

Top #4: Canvas is Your Test Prep Friend

Whether it's prepping for a quiz tomorrow or prepping for the AP Exam, I really love Canvas as a test prep tool. First of all, you can upload answer keys to "open" with certain requirements- either at a certain time or when a student has submitted a certain assignments. That way, kids can get feedback once they complete study guides, but are given some time to work without a key. 

And then, there's maybe my favorite trick of all. My AP kids need practice with timed multiple choice sections more than almost anything as we get closer to the test. I used Canvas for this exact purpose daily as we began our review. Here's why I love it:

• You can set a time limit. Kids see their individual time remaining and can learn to keep track of their pace. 
• Kids get instant feedback. You can even set up notes to display depending on a kids answer. For instance, if they got answer choice c because they integrated instead of differentiated, you can put in a note that will pop up saying "You used integration instead of differentiate here! Remember to think about....." 
• Multiple attempts are allowed. Great for mastery or doing test corrections. Can decide whether you want to take highest grade, average grade, last grade, etc. 
• Class data is easily available. As students were finishing, I'd pull up the class data summary (nameless, of course) and display it. We could see as a class where the struggles were and I could choose what to go over. We could also chart our progress and it helped my kids to be able to see how we were improving. 

An example of some quiz data (from blog.stephens.edu )

Tip #5: Math Type is Hard. Stop it. 

Yes, Canvas has an equation editor. It's actually a pretty good one. But let's be honest....that isn't going to create the graphs you need and it takes time. My secret for making quizzes quickly?

1) Use your favorite resources to find questions you love! I use:

• released state tests from my course AND the one above/below (if relevant)
• released AP tests
• http://www.jmap.org/ 
• http://www.problem-attic.com/

2) Screenshot each question and save individually as a picture. I have a folder for each unit and throw them in there so they're already sorted:

3) In the quiz editor, upload the image and make your answer choices A, B, C, D. BOOM! You're done. 

Tip #6: Walk Your Kids Through the Notification Preferences

My students would often complain that they didn't know when I posted an announcement when the course began. Canvas has an incredible amount of notification options, including texts, tweets, emails, and more. There is no reason you should need to use ANOTHER app or system to talk to kids if they have their settings on correctly. Here is a guide to Setting Notification Preferences to look through and then chat with your kids about. 

Tip #7: Explore the Other Cool Features (When You Have Time)

Canvas can do a whole lot more than what I've described here. While it might not be the LMS I would have chosen, it has a lot of perks and your district is paying for it. Why not get the most out of it? Here are some more options I've played with that were pretty cool:

• Re-assessing for mastery
• Curving feature
• Voice submissions and comments to students
• Group assignment features
• Link to PowerSchool or other grade book
• Parent accounts and contact
• Speedgrader

#MTBoSBlaugust Day 8: Geometry Scope and Sequence

First and foremost, I just have to tell you all that I'm continually amazed by math teachers in general and, more specifically, this community. Teaching is such a personal subject, something that we pour our hearts and souls into every day, and that can make us very vulnerable in our work. It would be totally natural for people to be private about their successes and their failures, as well as possessive about things we've created. But year after year, I continue to meet new people who are willing to share and help and it's helped make me even more willing to share and help. I feel privileged to be part of a community like this- one that I don't think many other subject areas have the pleasure experiencing. (My science-teaching husband is pretty much constantly jealous of the things I talk about people sharing and doing in the MTBoS :) )

http://www.gogeometry.com/geometry/word_cloud_geometry.html

Today I'm sharing what I've been pouring over all morning (between Olympic swimming heats, of course). I'm trying to piece together my big Geometry puzzle this year, as I've not taught Geometry since 2011 and I've never really taught with the resources of a textbook. My last district stopped buying them long before I ever worked there. Thankfully, in the true math teacher spirit, my amazing new PLC shared all their daily materials with me to get an idea of their pacing. I have tried to cross reference this with the standards and the sections where these things pop up in the text. We are using Holt McDougal Geometry (NY Edition). 

Geometry Course Outline

Any huge differences anyone sees with their scope or sequence? I am sure there are things I'll be revising, but I want to make sure I have a realistic big picture before I move any further. I'm still getting back in the Geometry saddle, so any and all advice and feedback are appreciate from the Geo pro's out there! 

P.S. No post yesterday, as my entire portable hard drive of a career's worth of teaching materials for grades 6-12 AND wedding pictures was seemingly corrupt for a short time. After rescuing the files, I needed a nap.