Fractions Squares: A DIY Math manipulative to teach Common Core standards

I have really been inspired this week by my fellow bloggers out there.  It just amazes me how one short post with a quick idea can get my creative juice flowing again!  It made me realized I should post more often, even if I only have one small item to share.
My students are finishing up with adding and subtracting fractions this week with a little performance task making trail mix using fractional amounts.  Next week we tackle multiplying fractions! I know that multiplying fractions is much easier than adding and subtracting them, but the Common Core standards are very clear and specific that students need to not just have computational accuracy, they need to understand the process.  I had to do a lot of reading up on this because, quite frankly, I was only taught to multiply fractions using a rote algorithm, so I never learned this important step.
      In books like Curriculum Based Assessment: Fractions by Michael Battista, the author uses squares to show the process of scaling.  I wanted to create a manipulative that students could use over and over again to demonstrate the process of scaling and truly understand that multiplying a fraction by another fraction results in a smaller number than your started with (which is, frankly, counter-intuitive to students' experience multiplying whole numbers).  The resulting do-it-yourself manipulative (which I called fraction squares) is explained in the You Tube video above.

Masterful Mini-Lessons

So when life gives you lemons...
This was a particularly tough week for me professionally.  I had a big term paper due in my Master's class.  At the same time I realized I had to make a major "course correction" in Math.  The students were just not getting the material and I knew that they wouldn't be ready for the test I scheduled on Friday.  So I spent several days scrambling to get more (and different) material to teach the topic in a new way and to fill an additional week of instruction.
At the same time I realized that I am unhappy with the quality of my reading and writing mini-lessons.  I am way too task-oriented.  I do too much telling them what to do and not nearly enough time showing them how to do it.
Squeezed in between all the professional reading I have do to for my Master's, I am also reading Lucy Culkins' A Guide to the Writing Workshop Grades 3-5.  This book is a breath of fresh air.  Her words really resonate with me.  It is not about doing more, it is about setting up our instruction and instructional time purposefully, and choosing our instructional language carefully.  I find myself actually quoting from the book when I conference with my students! Her book really gets you into the heads of intermediate grade students.  I realized, reading her book, how I could "beef up" my mini-lessons in both Reading and Writing.  So I created some scaffolding to get me started: a template to help me develop my lessons with care and thought.  I hope it is useful to you.  Please leave a comment if it is.

What Students Think of Questions

     A few days ago I had an interesting experience while questioning my students.  It started in Math class.  I had the following problem on the board:
Jack says 2/5 + 2/5 = 4/10.                                                                Jill says 2/5 + 2/5 is 4/5.
     I had the students sign their names under what they thought was the correct answer.  The first to sign up was one of my brightest students in Math.  She (correctly) chose Jill's answer. Other students rapidly signed up under Jill's answer.  Lastly, my behavior-disordered student signed his name under Jack's answer.  Suddenly, like a leak in a dike, one child after another moved their name to Jack's side, until about half the class had chosen the correct answer and half the incorrect answer.  I gave each child a personal whiteboard and asked them to find two ways to prove their answer was correct.  If this changed their mind, they were welcome to move their name.  More shuffling of names ensued.  Finally, I paired up the students that answered "Jack" with the students that favored "Jill" and had them explain their reasoning to their partner.  They could change their answer if they wish.  After one last shuffle, only three students remained on the wrong side.
     I asked the students, particularly the ones that changed answers several time, to examine why they changed their ideas.  What convinced them?  What failed to convince them?
     Later that same day (with a different group of students), I asked the class to evaluate the activities we had done over the past four days.  The students have participated in a "Mission to Mars" simulation in order to help them understand the journey of Lewis and Clark.  However, I have never mentioned Lewis and Clark to the students, nor told them why we were simulating a Mission to Mars.  I asked the class:  The past few days, have we been doing a Science or a Social Studies unit?  I used this question in order to determine if they understand that Social Studies is the study of human experience.  At first vote, everyone agreed it was a Social Studies unit.  As I questioned students about the reason for their choices, they began to doubt their choice.  More and more students became convinced that they were beginning a Science unit.
     Before the class became thoroughly confused, I told them about the similar experience I had had that day in Math.  That how asking students to clarify and explain their work caused them to question their answer, until many had gone back and forth several times and were very confused.  I felt on the verge of understanding something about critical about using questioning itself.  I asked my students:  Why do teachers ask you questions in school?  Why do we ask you the reasons for your choices?  Many students said that, when the teacher did that, they began to doubt their answer.  They assumed, if questioned, that the answer was wrong.  When they couldn't explain how or why they knew, that confirmed for them that their original "gut answer" was incorrect.  I emphasized to them that, when a teacher asks you to "prove it", he or she is trying to tap into your memory and logic skills that led to that (often correct)  "gut answer".  In fact, more often than not, if a teacher asks you to prove it, he or she means you are right!
     It is so interesting to me how teachers and students perceive things differently.  Is this the result of a lack of questioning?  Should I add more questioning into my teaching?  Or is this indicative of the age of the pupils or the impact of their home environment?  I would love to hear comments from you!