Exploring the Role of Rounding and Estimating in Subtraction

All Things upper Elementary, a blog that I follow regularly recently had a post about teaching students to really develop problem-solving skills, rather than just learn rote algorithms.  See her fantastic post here.  This post really resonated with me as we are in the final days of our unit on subtraction.  Most of the kids can use the algorithm successfully to subtract multi-digit numbers.  Some cannot.  But even those that can use the algorithm don't really understand what they are doing.  To quote the ATUE:

"A written algorithm is meant to SHOW you have to think, NOT teach you how to think."

 

I realized that I needed to teach more thinking and less algorithm.  So today's freebie lesson plan is designed to focus students on using rounding and estimation to help them with subtraction.

 

Materials:  personal whiteboards, marker, and eraser for each child, the same for the teacher. 

Duration:  About 30 minutes (can be made longer or shorter)          Grade Level:     3-5

 

1.  Warm-Up:  Pose some subtraction problems using multiples of ten.  Start with problems like 90-70 = ? and end with problems like 8,000 - 5,000.

2.  Write your final problem in the warm up on the board (let's say it's 8,000 - 5,000).  Pose a related problem, leaving the first number the same, such as 8,000 - 4,875.  Have the students estimate a solution and show their answer.  If more than a few students get the answer wrong (it should still be 5,000), demonstrate how 4,875 rounds up to 5,000.  So the answer will be about the same as the original problem.  You can't just use the front of the number to estimate the answer.  You need to look at the whole number.

Do several more examples, coaching after each one as needed.  Once most of the class can successfully estimate the difference for each problem you pose, ask them to not end their estimates in zero. Pose problems in a sequence like this:

12,000 - 8,000 = (4,000)          12,000 - 7, 679 = (4,000)          12,000 - 7, 679 = (4,335)

Ask students to tell how they decided if the difference was more or less than their estimate.  If students do not come up with a method, show them how when the subtrahend is less than the estimate, the difference will be larger.  Pose several problems using this method. 

 

By using a lesson progression like this, you can help students think beyond the algorithm.  Enjoy!

 

 

Rounding Strategies: 3NBT1, 4NBT3, 5NBT4

     After a rough start, my leveled Math class is going smoothly at last.  My students have gotten into a routine of a whole-class lesson on Monday (with pre-test), differentiated instruction based on the pre-test on Tuesday, Wednesday, and Thursday, finishing with another whole-class lesson on Friday with a final, standards-based quiz on that day.  Most students are seeing growth as they graph their pre- and post-assessments.  With this leveled group, I have often found that the class doesn't usually fall into three groups, like I have seen with heterogeneous groups in the past.  Quite often, they only fall into two groups.  This has lead to some changes in my I Love Math groups.  Most of the time, I divide the class into two groups.  One works with me while the other works on their Ten Marks or Manga High assignments on the computer.  Then the switch.  When there is a preponderance of students who have mastered the pre-test, I will often include Wednesday as an additional whole-class lesson. 
     This past week we worked on rounding.  Rounding was one of those skills that no one in the class was even close to mastering, even after the first whole-class lesson.  I had to dig deeper in order to meet my students' needs.
     The first resources I used was Learn Zillion.  Using the Quick Code LZ525 you can see a great lesson that uses a number line to show how to round.  We used the following template:

original number:_____________________

 

 

____________________________________________________________________________________

 

 

__________                                                   __________                                                    __________

low benchmark                                                 midpoint                                                 high benchmark

 

     For each  number, they created an individual number line and place the original number as a dot on that number line.  This was a time consuming process, but after doing several together, the light bulb went on for many struggling students who could finally SEE what it means to round.

     After a differentiation day where I reinforced this method, I taught method number 2.  This I introduced as a "game" I called "Slap! Trap!".  I passed out a place value chart and a highlighter.  (Kids always get excited when the highlighters come out!)  Students used a pencil and the chart to write the number to round. 

  I announced: "Round to the nearest thousands!".  Students repeated: "Thousands, thousands, thousands, slap, TRAP!"  When they say slap, they use their left pointer finger to cover the digit in the given place.

 

 When they say trap, they color the digit in the next place to the right with the highlighter.  If the digit they "trapped" is 5 or more, the digit they "slapped" will go up by one.  If not, it will stay the same. 

 

All the digits to the right of the slapped/rounded digit turn into zeroes.  Any digits to the left of the slapped/rounded digit stay the same.  After the number line method, this seemed like a shortcut to many students.

     After instruction, some students gravitated to one method; some to the other.  In the end, all students benefitted from learning both methods.

Week 2 Update: I Love Math

     Those of you that know me well know my passion for differentiated groupings that I call "I Love Math" groups.  I Love Math is a structure  that divides the class into a high, average, and low group and cycles those groups through three different stations.  Once the students get the rotations (which are always in the same order no matter what group the child starts with), things run like clockwork.  In the past I have always had an aide, Title One teacher, or special education teacher in my room to help ensure things go smoothly.  That way two of the groups have adult supervision.
     This year, for the first time, our grade has been required to level for Math.  Since I was in the fifth grade at the time, I could not attend the meetings where they assigned teachers to the different levels.  I have one of the average groups.  Two of our classes (mine and the high group) do not have a support person in the classroom for Math.  The other six specialist are divided among the bottom three classes.  This is my first year trying I Love Math groups by myself.  I knew it would be a real test of the structure that I love so much.  If I can do it alone, then there is hope for other teachers who are solo practitioners.  And yes, even a leveled Math group has enough differences in it to warrant differentiated instruction.
     First of I decided that I needed to have I Love Math groups a lot more often.  In the past I have done them regularly (once or twice a week), but only as needed. I know that the more I do them, the better the students get at it.  I decided that this year I would have a whole class lesson on Monday which would end with an ungraded pre-test.  This would be used to create differentiated groups Tuesday through Thursday.  Friday will be another whole class lesson and post-test.
     I also knew that I needed to use technology to help me keep track of the progress of my students in the same way that another person used to  do.  Luckily I found Ten Marks, a free site where I can assign specific skill lessons for my students. 
     After one week of following this plan the results have been fairly good.  Technology has been the biggest challenge.  Our wireless internet is not always reliable and the students are not used to logging in to the computers and finding websites in a timely manner.  I hope that time and experience will help the students complete this part of the task more quickly and easily.  If not, I may have to alter the activities or the order.  I wish I had enough computers for all of the students to start on them and then move to other activities, but the most I can get are 8.
     Great ideas are worth the time it takes to make them successful.  I'm up for the challenge!

A Common Core Pacing Guide

This summer I have embarked on a number of curriculum projects, mostly around the Common Core State Standards (CCSS).  For Math I wanted to increase the amount of differentiation and technology as well as firmly ground my teaching in the Common Core.  The first thing I noticed when I counted up the standards was that there was the same number of standards as full weeks in the school year.  That means that my students will have to master one standard a week!  At first that thought was quite daunting.  My next thought was this is an opportunity to have a very organized curriculum based on a single standard each week.  This is how I decided to organize and pace each week: 

   At the top of each week/page is the unit title, which part of the unit. vocabulary, skills, and the Common Core standard.

Monday will be a whole class, non-differentiated lesson straight from my Math program (Everyday Math).  At the end of the Math lesson, I will give a short assessment on the standard of the week.  For this part of the weekly plan I didn't reinvent the wheel.  Teachers Pay Teachers has several short, one page per standard bundles that are specific to my grade level. I will use these to pre-test my students and put them in differentiated groups for the next three days of the week.

Tuesday, Wednesday, and Thursdays are differentiated instruction days.  Students will participate in rotational groups I call "I Love Math" groups.  For more information on I Love Math Groups, listen to this podcast or watch this video (the first of 5 parts) on Youtube.  Technology will help make differentiation easier.  I will use the Ten Marks website on Tuesdays and the Manga High website on Thursday.  Both websites allow me to assign lessons/activities and also have fun games and activities for students who finish early.  Scoot Pad is a similar website (click the names for links).

  Friday we are back to another whole class lesson from Everyday Math.  One Friday I will give the post-test (a different "one-pager" from TpT) and this will be graded. 

So that's my curriculum pacing guide for next year.  I haven't offered this as a product on TpT because it is personal to my situation- a labor of love, if you will.  However, if you would like a free digital copy to tweak for you own use, email me at cjones@sau61.org.

Additional EDM/CCSS Alignment Resources:

First, the "official" CCSS alignment from EDM, then the more realistic version from New York. 

http://www.everydaymath.com/downloads/Brochures/EMandtheCCSS_Brochure.pdf

http://schools.nyc.gov/Academics/CommonCoreLibrary/CommonCoreClassroom/Mathematics/default.htmligning