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…
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.
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.
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)
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.)
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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.
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!)
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?” 🙂
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