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ladder division

How In the World Do You Teach Rectangular Arrays and Division in 4th Grade?

Well, with large numbers this is something that my fellow colleagues  did not feel comfortable teaching.  And when that happens…who steps in??  None other than The Mathemagician…ta-da!  (which is me of course, but shhh don’t tell anyone!!)…I’ve been off for a few days as I write this, which makes me a little sillier than normal–and probably slightly more interesting! So, on with my lesson!  Now just remember when I post these pictures it is not a beautiful, I spent weeks preparing, colorful, lesson.  This is a practical lesson anyone could use whether you are savvy with a computer or not.  (I may turn this lesson into something more aesthetically pleasing later on.) The part that stumped the teachers was the fact that the standard says “up to 4-digit dividends”.

CCSS.Math.Content.4.NBT.B.6 Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

I had to think myself about how to teach this before I spouted off a lesson idea.  (I know I am supposed to be the math expert in my building, but, honestly, I have to look up what exactly common core is expecting before I assume I know what a standard is asking students to do.)  I looked at North Carolina Unpacked–my go to document for what common core expects…

Screen Shot 2013-11-30 at 2.25.01 PM

 

North Carolina shows arrays being broken into smaller arrays similar to ladder division.  Since these students had already been studying ladder division, I thought that they would have an easy time relating arrays to ladder division.  After reading the North Carolina document, I decided to create a large array with base ten blocks and have students find the missing side, but then I realized that students would just count the missing side.  I needed something better, but what?  Then I thought about a training I had been to which had arrays with part of one side covered up.  Students had to figure out a missing side .  However, these type of lessons  were for double digit numbers.
Here is what I decided to do to push students towards solving arrays with large dividends.  I used cm grid paper and cut out different sized arrays.  I cut out a very small one at first to use for modeling and discussion.     I told the kids that the principal had asked me to set up the auditorium with chairs for the 5th grade graduation.  Each cm square represents a chair.  Then I explained that I had spilled coffee all over my seating plans, and I didn’t know what I was going to do.  I told them I needed their help to figure out how many rows there were so I could recreate my plans.
I started  with this one.  Students had no problems figuring out the missing dimension in this array since they knew their facts.Array1
All I did to accomplish this spill effect was to cut out pieces of colored paper in blob shapes and tape them to the array before setting it on the copier.  In this case, the large blob was actually red, but it printed out black on the copier.
Then I showed an array that was larger with a 3-digit dividend.  This array actually had a blob on it, too, but I removed it so that we could discuss different students’ thinking after students figured out the missing dimension.  The blue marker shows my recording of the students’ thinking.
Array2
Then I gave students a much larger array to figure out a missing dimension on their own.  Several students tried to count how many squares there were underneath the coffee spill instead of using more efficient methods.  As you can see, this student drew lines on top of the blob to figure out how many rows there were.  She arrived at 22 rows.  (there were 24)
LargeArray3
This paper above although wrong, was a valuable piece of thinking to tie everyone else’s work together at the end when I had students share because it was so basic.  ( I typically don’t write on students’  work, but since the students were a bit shy about sharing  I asked her if I could write on the work to help other kids make connections.)
LargeArray1.
The array above shows a student who is thinking of each row as a group of 12.  Rather than multiply he chose to add 12 repeatedly.  He realized that 12×12 made 144 and that adding 144 twice would give him 288.  Even though this student is oh so close he isn’t explicitly saying that there were 24 rows of 12 chairs.
I’m still looking for someone who is thinking with groups of 10 so that I can relate this to the ladder method for division.LargeArray2
Here is yet another student who is thinking in multiplication with equal rows.  Surprisingly to me, the students are more comfortable with 12×12=144 than thinking with groups of 10 to break off parts of the array.  The above students is a GT (gifted) student and he had 144 + 144 on his paper for the longest time because he recognized the relationship among 288 and 144.  However, it wasn’t until we had closing discussion that he labeled his other dimension with 24 and columns with 24′s.
Let’s pause here for a brief  teacher reflection moment by Ms. K (soft music playing)next time I prepare an activity like this, I will make sure that I cover up enough rows so that the only rows that show are a group of 10 rows.  This will hopefully push students towards thinking about ladder division.
Ok, finally, someone who thought about the ladder method for division!!!!!!!!! (picture me running through a grassy field to meet this paper).  This student’s paper will let me tie all the other students’ papers together in closing discussion and to relate arrays to everything that the class has been discussing for weeks!!!! (woooooohoooo!)
LargeArray5
This lesson went successfully.  Now for the next lesson, I plan on giving the students another large 3 digit dividend array that is covered after row 10.   After this the next steps are giving students 4 digit dividends and/or rectangles with no lines.
Just so you know…one of the kids asked me if I really did spill coffee on the seating plans.  (so cute) I said, “What do you think?” :)

Do Your Kids Need a Dawning Division Moment?

After sitting in a week’s worth of meetings about Common Core instruction and best math practices, I feel as if my way of instructing students is gradually morphing into a new beast, which I am not sure I fully can describe yet.  In the midst of all of this, I went back to the fifth graders I have been working with on the day after these meetings.  I decided to try out some of the teacher practices I had been learning.  Sidenote:  This set of fifth graders isn’t yet proficient at long division or ladder division for that matter.  I posed a long division problem to the students, but instead of giving it to them in “naked numbers” alone, I gave it to them in word problem form as well.  In reflecting, I should have started out with the word problem alone and let the students figure out that they were supposed to divide.  After watching the students struggle with solving using ladder division or the traditional long division algorithm, I encouraged the  students make equal groups to see if that helped them.  One student in particular that I worked closely with (whose work is pictured below) managed to find a correct answer using equal grouping, however he could not find his mistake when he used ladder division.  Another student…one I might add who typically doesn’t have much perseverance to solve problems…related completely to the contextual situation and persevered through to solve the problem to the end.  To get to my MAJOR DIVISION REVELATION, let’s look at this student work below:

One student's incorrect ladder division and solving with equal grouping correctly.

 

The same student's equal grouping and the relationship to the correct ladder division which I showed him. Notice how 20, 30, 10, and 5 are the same in the circles and to the side of the division problem.

So, you ask, what is the big AH-HA?  If you notice that the numbers in the student’s circles (equal groups) are the same numbers to the side of the division problem (or the ladder).  The numbers to the side of the ladder are actually the chunks that students would naturally break off if they were naturally dividing cookies among people etc.!  Now I know according to my PD classes that I shouldn’t have just told and shown the student this relationship, but I should have allowed him to figure it out.  I was just a little too excited to hold in my own personal discovery I suppose.

Another discovery I had in experimenting with having students discover their own strategies, I learned that students actually do better and persevere more when the problem is in a context familiar to them and it isn’t just a naked numbers problem.  Having a contextual situation actually gives the students an entry point into the problem, and so they don’t give up as quickly.  I have always thought backwards.  Students MUST UNDERSTAND THE NAKED NUMBERS FIRST and THEN they can SOLVE HARD WORD PROBLEMS.  But in reality after my experiment with these fifth graders, they were much better at solving this division problem in their own way when they had a context.  I have had a complete paradigm shift, and you were all here to see it  above– that is.

 

Are Your Students Struggling with Traditional Long Division?

Generally it takes students a few weeks initially to learn division– especially long division with multiple digits.  Marilyn Burns’ book Extending Division has several helpful hints within her lessons.  In place of the traditional method for division, she shows how to teach students with the ladder method in which students divide a dividend with multiples of ten rather than break the dividend apart like the traditional method.  The ladder method helps students understand numbers more effectively because they are estimating and thinking of the dividend as a whole. See this example to gain a better understanding of the ladder method.

 

Add all of the numbers to the right (the ladder) to find the quotient.