Exploring the Laws of Logarithms

I was looking for a more student directed way of learning the Laws of Logarithms for Math 30-1 and 30-2 students in Alberta when I found a tweet by RobAnthony01 talking about his process.  I used his 140 characters and turned it into something I can give to my students to work on independently, with my guidance as necessary of course.

You can download my activity at my Teachers Pay Teachers store (for free).   You have to sign up for a free account but then you can download the file.  I've added other math products (for free) in there as well.  Of course, you can just download it using File==> Download as...  This is my first time embedding a google doc within blogger so we'll see how well it works.

It's not a very complicated activity but sure would be a heck of a lot more interesting than me saying "When adding two logs...do this.  When subtracting two logs...do this." 

Reflection App - Well sort of...

Yesterday, I discovered a cool new tool and just had to share it.  It's called Reflection and can be found on the Reflection App website.

What it works on:
Devices:  iPhone 4S, iPad 2, iPad 3,
Platforms:  PC (Windows XP or greater), Mac (OS X 10.6.8)
I was testing on a Windows XP using my iPhone 4S.

What does it do?  It's a projection "app".  It allows you to wirelessly project what is happening on your iPhone or iPad onto your computer.

Right now, if you want to show what's happening on your iPhone or iPad, you have a few choices.

1. You can AirPlay it to an Apple TV.  Not sure how many classrooms have that.  Mine didn't.  
2. You can also purchase the Apple dongle at a cost of about $35. I've used the dongle.  Honestly, I had trouble with it.  I would have to take my cover off because I couldn't get it to attach securely.  Even with the cover removed, the second I moved around it had a tendency to jiggle loose.  Then, if it had jiggled loose too often, it would stop projecting my iPad completely.  So frustrating.  Especially when you're doing a presentation about how great iPads can be if used properly.  Thirdly, I was tied down and limited to the length of my VGA cable.  Frustrating.
3. I've heard people talking about other apps that will project as well but have not researched them.  I was really looking for something that would allow a class of 20-30 iPads to project inexpensively. 

So far, this is my favorite...Let me tell you about it.

I installed the software on my laptop.  It was fast and easy, about a 9Mb download.   Then I went to my iPhone 4S, (make sure your laptop/desktop and iPhone/iPad are on the same network) accessed AirPlay and started projecting.  That's it.  There's nothing to install on the iPhone or iPad.  That means anyone on the same network as my computer can instantly project through my computer to the projector.  Awesome!

Wait!  You scream.  Does that mean ANYONE can tap into my computer and start projecting?  Well technically yes unless you take the following action.  You can set a password.  This way, you can either 1) turn it off at the beginning of class or 2) give it to your students for that class and then change it afterwards.  Again, easy-peasy to do.

What can you use it for?  Let students show videos they've found, work they've done, anything they've created, brainstorming they've completed in groups.  It even projects sound!  So, if they're showing a video, the sound automatically transfers with it.  I did notice a slight lag when I ended the YouTube video.  It showed up on the computer screen for about 5 seconds after I closed the YouTube App.  It will not display the video on your device at the same time, however.  Oh, and apparently multiple devices can connect to the computer at once.  The website states that it will start to slow down as you increase the number of connections.  I didn't bring my iPad to work today so I will have to test it and see how it goes. This spot here will be updated as soon as I have tested multiple devices.

As far as I can tell, only two things didn't work on it.  1) Facebook wouldn't play embedded videos or at least not the one I had tried.  2) Skype didn't transfer.  There may be other apps that don't work but so far so good with the ones that I've tried.

One other small glitch:  Once when I had told it to connect, I had this blog open in edit mode.  When it displayed my iPhone it was itty bitty on the screen.  I just canceled the airplay and reconnected and it was fine.  So far that has only happened once.

Cost:  The free version only allows you a total of 10 minutes time and that's it to explore the app but within that time, you can pretty much guarantee that you'll love it.  Full Disclosure Here:  Once I ran out of my free time, I contacted the company, explained who I was and that I work with teachers across Central Alberta sharing resources, activities, apps and ideas that they may find useful within their math classroom and then I very politely requested a free copy of their app.  I want to be able to actually SHOW people how easy this app is to use.  The current cost is $14.99 per license or $49.99 for 5.  Contact them for more than 20 licenses for another discount.

Check out the Reflection App website to watch the video and find the download file.  Let me know if there's anything serious that I missed when playing with the app - either good or bad.

---

Edited to add:  I was asked where I'd heard about the Reflection "app" and I couldn't remember at first but now I suspect it was because of this post.  Not sure why I would have kept reading about it since she states that only works on Macs but it definitely works on PC's now.  So, thank-you, Kathy, for pointing me in this direction!

Assessment - Killing the Overkill

Students Can Demonstrate Understanding with a Shorter Assessment Piece

I've been working on changing my assessments.  Marian Small has been a huge influence on this ever since I attended her session on the High School Math Institute.  Now, every time that I am creating a new assessment, I search "Marian Small" +topic in google to help generate some ideas rather than always recreating my own.  I always try to change the question so that I'm not just stealing her ideas, especially since I share so many of the things I create with the teachers around Central Alberta during Assessment sessions.  However, I certainly use her framework as it makes the questions so much more interesting and checks for a deeper understanding.

When I present a session on assessment, one of the first slides I show asks

What Does... Solve 34 on a test tell us about the student?

The inevitable response is that "They can operate a calculator".  It terrifies me that when I first started teaching, I would strongly populate my exams with questions like this...ones that tested their ability to operate a calculator.   Of course, I would have to have many questions like this to test whether they could do it in multiple settings.  So, my students were answering a bunch of repetitive questions and I was stuck marking a bunch of repetitive questions.  Of course, it was easy because a simple answer key worked but what did it tell me about my students' understanding?  Pretty much nothing.

Once I began my assessment journey, I realized how easy it would be to change these questions to better get at their true understanding.  Yes, it makes my answer key pretty much useless.  Yes, it might take me extra time to mark the question.  However, I was then able to cut down 20-50 questions to just a few deep questions so it balances out in the end.

Giving up a little bit of control during the assessment process allows students to demonstrate their knowledge more fully.

I wanted to know if students truly understood that 3⁴ is really 3x3x3x3.  Why not let them pick the numbers they used?  By wording the question carefully, I could prevent "easy outs".  This is how I changed the first question:  (The bold portions would not be bolded on the test.  They're bolded so you can see how I tried to ensure I didn't get situations like 1¹
.  The others are samples of how I would assess the other outcomes.



Question 1:
Choose a base greater than one:  ____________


Choose an exponent greater than the base:  ______________

Create a visual representation for the power you produced.


Question 2:
You simplified an expression.  The result was (3/4)^1/2.  What could the original expression have been if you were using the following operations?
(a^m)(aⁿ) = _____________________

a^m/aⁿ = _______________________

(a^m)ⁿ = _______________________



Question 3:
Choose one of the following operations:               (ab)^m     OR     (a/b)ⁿ

Choose your own bases and exponents, however "m" and "n" can NOT be a whole number.

Explain TWO ways to determine the result without using any "rules".




Question 4: (From Marian Small)
Write an expression that you would likely use three power laws to simplify.  Use a variety of exponents.

Simplify it.




Question 5:
Jamie states that a⁰ is a.  Pat says that it is 1.  Jessie says that it is 0.  Casey says that none of them are correct.  Who is right and how would you convince everyone else without saying "The rule is..."

---

Like I said, when I first started teaching, I would overkill on questions.  I believed an assessment was supposed to take all class which meant I would have to put a LOT of questions on it.  Then, of course, when you have a class of 16-30 students, you had to MARK all those questions.  I would now use this assessment to check students understanding in Math 9:

Side Note #1:  These types of questions are excellent discussion starters, review questions, exit slips, "homework checks" and formative and summative assessment questions. 

Side Note #2:  Students should NOT see this style of question for the first time on a final assessment!  They should experience it ahead of time.  I would worry that they would feel like they were being "tricked" otherwise.

I've considered including the outcomes on the assessment to give students and parents a better idea of students' success within each outcome.  I'm trying to decide between two formats:

Option 1:
I could attach the table included below to their paper after the assessment.  I would highlight the questions they were successful (or not successful) at.  

Number
1 Q1 Q5 Q4	Demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents by Representing repeated multiplication, using powers Using patterns to show that a power with an exponent of zero is equal to one Solving problems involving powers
2 Q2 Q2 Q2 Q3 Q3 Q4 All	Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents (am)(an) = am+n am÷an = am-n (am)n =amn (ab)m=ambn (a/b)n=an/bn, b ≠ 0

OR, I could add a column to the assessment (something like below) but then I would have to make sure to blank out the pieces that give answers.

Exponent Laws Assessment

Question 1: Choose a base greater than one:  ____________ Choose an exponent greater than the base:  ______________ Create a visual representation for the power you produced.	Demonstrate an understanding of powers with integral bases (excluding base 0) and whole number exponents
Question 2: You simplified an expression.  The result was (3/4)1/2.  What could the original expression have been if you were using the following operations? (am)(an) = _______________________ am/an = _______________________ (am)n = _______________________	Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents (am)(an) = ___ am÷an = ___ (am)n = ___
Question 3: Choose one of the following operations:               (ab)m     OR     (a/b)n Choose your own bases and exponents, however "m" and "n" can NOT be a whole number. Explain TWO ways to determine the result without using any "rules	Demonstrate an understanding of operations on powers with integral bases (excluding base 0) and whole number exponents (ab)m= ___ (a/b)n= ___
Question 4: (From Marian Small) Write an expression that you would likely use three power laws to simplify.  Use a variety of exponents. Simplify it.	Solving problems involving powers
Question 5: Jamie states that a0 is a.  Pat says that it is 1.  Jessie says that it is 0.  Casey says that none of them are correct.  Who is right and how would you convince everyone else without saying "The rule is...	Show that a power with an exponent of zero is equal to___

So there it is. A short assessment piece I could use to check my students' understanding of the power laws. What do you do in your classroom to assess students' understanding of the power laws? I would love to hear ideas for a performance assessment piece, other assessment questions (ie. a fabulous word problem), etc.  How would you share the connections between their responses and the outcomes (like I was attempting in option 1 and 2)?
 
Share and link in the comment section! 

Chomping Down with "Screen Chomp"

Free
Ipad

I absolutely adore Screen Chomp.  Any time anyone asks me for a great ipad app, it's the first one I share.  Why?  First of all, it's so easy my Kdg niece can use it without help.  The easiest way to use it is to just hit record and then draw on the screen and speak.  When finished, you'll have a perfect recording of what you drew and said!  You can even prepare the screen ahead of time by adding images if you want.  Preview the recording immediately after making it.  Then it's next to nothing to share that recording on twitter, facebook or just grab the link and share that!

What a fabulous tool for schools working with digital portfolios.

How would I use this?  Digital portfolios, assessment for learning, assessment of learning, students creating tutorials for school website, projects, and so on and so on...  Can you imagine having a student reading a passage of text and posting it beside how they did on the same text a month ago?  Students don't even have to be in the room to share their understanding or thinking.  Struggling student?  Watch them thinking through a math problem pre-written on the screen.  My brain goes into overload every time I think about how it can be used in the classroom. 

The second reason I absolutely love this program is that it is completely and utterly free!

Third, as long as you have a wifi connection, you don't have to worry about how to download that video.

So, if you don't add any other app to your ipad, make sure to add this one!

I'll Be Back...

next year in my role as CARC's Mathematics Lead Teacher/Facilitator.  Woot Woot!  I miss my students but look forward to working with K-12 Math teachers in Central Alberta again.  I had so much fun running around to different schools providing PD and modelling lessons.  However, I have a whole new respect for substitute teachers.  I had no idea how hard it would be!  I've never subbed a day in my life and this year I modelled lessons in classrooms.  The only difference was that I had to create my own lessons rather than following a predetermined one.  When I return to my classroom, I'm going to make sure to leave chocolate and let them know how much I appreciate their hardwork, especially those who have to come in to my -1 courses with little or no math background.

Summary for this year:

Favorite Session I offered:  Hands on and Virtual Manipulatives
Although this often became a contentious issue when dealing with high school -1 teachers, I really enjoyed sharing some of the ideas I have gathered over the years.  My favorite has always been "Using Algebra Tiles to teach Factoring Polynomials and Completing the Square".  You can find the information about this in my previous post. I also learned a lot from the teachers who were willing to share their ideas for using the manipulatives. Many teachers at the high school level are concerned about the lack of basic skills (such as multiplying in their heads) so I also created a lot of games for practicing basic skills at any level.


Favorite Learning:
When I started this job, I spent a day running around to schools meeting teachers, taking notes about sessions they would like, questions they had, etc.  By the end of the day, my hand was cramped and you couldn't read my handwriting.  (I wasn't hauling my laptop around and my phone was too small to take extended notes).  That weekend, I went out and bought the iPad 2 with bluetooth keyboard.  What a lifesaver!  Now, I can take notes and still read my writing.  I'm also a much faster typer than writer so I could keep up much better.

Just after I bought the iPad, I was asked by a school to come in and help with an Early Numeracy Intervention Program using iPads that they wanted to set up.  Perfect!  So, off I went to explore and download a variety of iPad apps for improving math skills.  Luckily, almost every developer I contacted was willing to give me free samples!  Woot woot!  You can find a list of my favorites on my delicious account. 

My Goal for next year:
To post more on my blog.  Not sure what happened this year.  I guess it had a lot to do with learning and becoming comfortable with my role.

I'm also going to work on posting as much as possible specifically about Alberta 30-1, 30-2 and 30-3 math curriculum. So, if you have something to share, I would love to see it! 

Well, I should probably get back to planning for Grade 12 math now.  Will see you again soon!