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!

 

 

Adventures in Lucy Calkins' Units of Study

This summer my major purchase for personal professional development was Lucy Calkins' Units of Study in Writing.  I spent a delightful summer with Lucy.  Reading her book was like having a conversation with a great mentor teacher.  Many people comment that her ten-plus page lesson plans are a nightmare to get through.  I would agree with that, but just because the lesson plans are this long doesn't mean you have to do all of it.  I would much rather have much more than I need than have to make up my own stuff.  Most of my lessons to my students end up being "distilled essence of Lucy" rather than a verbatim recitation of her lessons. 
Today's lesson was definitely one of those essence ones.  The lesson was about making characters come alive by describing both their internal and external traits.  To liven things up a bit, I introduced my students to one of my favorite music videos on You Tube:  Julian Smith's Reading a Book.  In it, the singer acts all tough and mean when people interrupt him while reading.  His tough exterior is totally at odds with his zest for reading.   The kids got the juxtaposition between internal and external traits immediately. It was a fun introduction to the lesson and I saw several students who tried the technique, including one of my most reluctant writers.

One of the biggest frustrations I have teaching Writing is that students seem to so very rarely take the advice we given them in our lessons.  It's absolutely unheard of in a Math class for students not to follow the algorithm the teacher provides, yet in Writing "doing your own thing" is rampant.  Today I tried to combat that by, at the end of Writing, having each child get out a highlighter.  They traded daybooks with a partner and highlighted evidence of today's lesson on character traits (if they could find it).  Overall the class was very honest in their highlighting and this gave me a quick way to judge what students could still use some help in this area.

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.