## Wednesday, October 30, 2013

### Division with Bows: Two-Digit Division with the Golden Beads

Today Kal-El finished the memorization of multiplication sequence from the Montessori primary albums.  He has already begun the "Multiplication" thread in the elementary albums.  I had decided that I wanted to start the "Division" thread when he has completed the memorization of division primary sequence.  That is why you may have seen my post on the "division dance" Kal-El did when he finished his multiplication board earlier today.  That thread presupposes that the child has had a complete primary experience.  However, I made a conscious decision to skip two- and three-digit division with both the golden beads and the stamp game and save it as remedial preparation to use immediately before beginning the elementary division sequence.  There were a couple of reasons for this decision one of which was the following sentence in my AMI primary math album on the "Division with Bows" page that says:  "the child may be confused by this."  We were already nearing the end of "primary" and hadn't gotten to the stamp game yet. I decided that rather than risk getting stuck on the golden beads for even longer while we struggle with two- and three-digit division, I would rather move on and get through the four operations with the stamp game.  My intention now is to use the golden beads and stamp game and tackle two- and three-digit division as a warm-up for elementary test-tube division.

If you are looking for this work in your own albums take note that it would be in your PRIMARY album and it could be going by any one of several monikers.  It could be simply called "two-digit division" such as in the Karen Tyler album.  Montessori Research and Development calls two-digit division "decurian" division but they might be the only group of people to refer to division in that way.  It is spelled "decurian" in the album but I've only familiar with the word as "decurion."  If you are looking up information on this work on the web you'll have better luck if you spell it "decurion" as well.  (Why decurian/decurion?  It's because they are referring to the bows or skittles as "soldiers" and in Ancient Rome "decurion" would be "an officer in charge of ten men.  Likewise a "centurion" would have been "an officer in charge of 100 men." Anyway, that's my imperfect understanding of things.) AMI albums call this "Division with Bows."

I  took a LOT of pictures as we did this work and will talk about it at length because I have found that there is really very little information out on the web about Montessori two-digit division with the golden beads.  One of the few places I've seen it done is over at Making Montessori Ours.  It's a fun post, however, she is using the R&D albums and that album has the teacher do the portion of the presentation in which the "ten" tray is redistributed to show that each unit of the ten receives the amount of the quotient (likely because it's a BIG work that certainly would go faster if the teacher did it instead of the child...but that is the case with most things, isn't it?).  That is probably why Cherine didn't photograph that part, or maybe Xander didn't need that part (he's wicked smart).  The AMI album I use has the child do that and I haven't really seen that part shown anywhere.  A really great, easy-to-understand step-by-step of this kind of division (but still not the final redistribution) is available in this article by Peggy Bennett and Audrey Rule.  I highly recommend it if after reading your album presentation you feel like you still don't know what to "say" to your child (I did).  The example they give is of three-digit divisor division but you'll understand the procedure then for both.  The purpose of this blog post is to show a real homeschooling family actually doing "Division with Bows" as seen in a traditional AMI album, all the way down to the final redistribution.  This post is NOT a replacement for reading the presentation in your album...if I gave every nitty-gritty detail along with everything else I had to say this post would be too long.  For that reason, it might be confusing to actually try to learn how to do "division with bow" JUST from this blog post.

As I said, AMI calls this "Division with Bows."  Bows?  Really? Bows?  My boys weren't going to stand for "bows."  I checked my Karen Tyler album and she uses necklaces.  Oh, boy....I mean GIRL.  The easiest thing to do would probably have been to just pull out the place-value skittles from the stamp game (as they do in the Bennett/Rule article). I, instead, found myself standing in the dollar store staring at the little hair bows on barrettes thinking "no way!"  In the meantime the boys were standing next to me with pleading expressions on their faces holding packages of ninjas like these:

The package they were holding just happened to hold a GREEN and a BLUE ninja (not red and yellow as in the photo).  The image must have triggered my subconscious memory of Cherine using Hero Factory Men because I came home with many packages of blue and green ninjas.

Here is my set-up just as Kal-El is beginning to fill the rug with the dividend. We have all of the wooden number cards that we need, some trays (there will be more!), and two bowls of ninjas.

The equation that we were to do was 5664/12.

I gave Kal-El a tray and the large number cards for the dividend and he filled it from the shelves.

After he brought me the beads for the dividend we decomposed the divisor and discussed the role of the ninjas and the trays.  The green ninjas represent the units of the divisor and the blue ninjas represent the tens of the divisor.  For a divisor of twelve we needed three trays, one blue ninja, and two green.  Kal-El understands about "sub-contractors" so we said that the green ninjas were self-starters who did their own work and would collect their own paychecks.  The blue ninja is master ninja and subcontracts work to ten "apprentice" ninjas.  (We left out the part where the blue ninja in reality would take a portion of the proceeds before passing on paychecks to the "apprentice ninjas.")

Kal-El's next job was to distribute the dividend to the ninjas.  The blue ninja gets ten times what a single green ninja gets, or the green ninja gets 1/10th what the blue ninja gets...however you prefer to think about it.  This is where the Bennett/Rule article helped me out.  When I read the presentation in MBH I was immediately stuck.  We have trained the child in math to "always start with units unless we divide, then we start with thousands."  However, immediately you don't have enough thousands.  If you follow the starting with thousands "procedure" you would immediately begin exchanging all of the thousands for 100 squares but that's not what you want to do.  If you were going to do that, you might as well just get out twelve tray from the get-go and not do category division at all. I warn you, it will be just as unwieldy a project as the redistribution we do at the end will be...just wait.  The Bennett/Rule three-digit divisor example is obviously a different equation that what we were doing, but it gave really good advice on how to talk to the child:

As in paper and pencil division, the student should always begin with the highest-magnitude place value of the dividend.  For our example problem, this is the thousands place.  There are two-thousand cubes in the dividend.  Take one cube and give it to the red centurion (the highest place-value of the divisor).  Ask students, "If I give 1000 to a centurion who collects for 10 people, what is each person really getting?  The response should be that each person should get 10.  Explain that in division everything is always fair and even.  Each person (but not necessarily each skittle), in the end, must get the exact same amount.  Now point to one of the blue decurions.  "If the centurion, who collects for 100 people, received a 1000-cube, what should I give to this decurion who collects for ten people, so that every person will get the same amount--so that each person gets ten?"  The response should be that the decurion gets a 100-flat because a hundred divided among ten people is ten.  So give each of the decurions a 100-flat.  Then move to one of the green foot soldiers.  "If we have been giving base-ten blocks so that every soldier gets ten, what should we give a foot soldier?"  The response should be ten.  Give each foot soldier a ten-rod.
So, in our equation, Kal-El began by giving each green ninja a 100-square when the blue ninja received a 1000-cube.

When the blue ninja receives a 100-square, the green ninjas received 10-bars.

Kal-El counted up the beads on each of the green ninja's trays to find the quotient.  However, we weren't done.  This was our very first time accomplishing division this way and just one time we want to really prove to the child that the blue ninja has enough beads to distribute 472 beads to his ten apprentice-ninjas.

We set up ten empty trays in a circle around Kal-El.  He sat in the middle with the blue ninja's tray and distributed the beads to all of the apprentice ninjas.  Putting a green ninja on each tray was his idea, the MBH instructions I had did not say to put bows on the trays.  So, don't feel like you have to have enough bows or ninjas to do this.

You WILL need a lot of beads!  To do the "proof" at one point I think we needed 47 100-squares.  I had to use all of my wooden hundreds (including stealing the hundreds BACK from my original two ninja's trays) and two of my golden 100's off of the bead chain cabinet.

You need a lot of ten-bars too!  At one point I think we had more than 70 in circulation.  I had to empty my decanomial box AND my elementary snake game box to get them.

If you were wondering if this was too tedious for my elementary-aged kid, just check out that facial expression.

In the end each tray held 472 beads and Kal-El verified each and every one!  I offered help verify the ten trays and he said he wanted to do it himself!  The good news is that we don't need to do this again.  I am glad that we did it.  Kal-El admitted that he really didn't believe the blue ninjas tray had enough beads to give 472 to each of the apprentices.  He said early on in the verification, "So, our hypothesis is that each tray will have 472 in the end?"  He kept referring to it as a "hypothesis" and said point-blank that he didn't think the blue tray held enough beads for that to be true.

We will practice division this way, with the ninjas and golden beads but without the verification step, until he can do the work solidly without my help.  Then we will move on to the same work but with the stamp game.

In case you are wondering what this might all look like more traditionally and with bows, here is a photo of my MBH album page (clicking on it should make it larger):

1. Oh my goodness! I'm totally lost! Will need to take some time to digest! Thanks for sharing this! I know I would need it one day!

1. Oh dear :( The purpose of the blog post is so that people can see images of someone doing what their albums outline. I am NO GOOD at replacing albums. Read it in your album, THEN look at the blog post again. If you are still stuck, read the article I linked to. Tons of images, tons of clear explanations.

2. Thanks! I guess I'm lost 'cause I haven't reach this stage yet with my kid. But I reckon this post will be useful when I eventually hit this phase!

2. Thank you so much for this post. I really appreciated that quote from Bennett/Rule. I am in a similar boat, right behind yours...with T. He did a small bit of primary unit division last year, but now this year I can see that we may come upon racks and tubes sooner than I had thought. I know he didn't do any double digit division, yet, so we'll have to swing back to this lesson before moving on.
When I was originally exploring this path, I looked at the YouTube video Mathematics Decurion Test Tube Division, by IslandVillage and was kind of confused, then I read about group division and was a little less confused. And then I did practicing with the stamp game. But this post solidified it for me. Thanks.
The part that is interesting to me is the "proof" part. That isn't something I ever learned when I was a child, it was kind of glossed over. I am amazed that the Montessori-trained child can understand this in the back of their mind because of their work with the base-ten hierarchical materials. I love that you both proved that the Decurion does in fact get 10 times what the foot soldier gets.
And ninjas are very near and dear in this house too. We may need to find some ninjas too when we start.
Oh, another way to "explain" this could be, to parallel the lesson with our "tradition" long division most of us were pulled through in elementary school. We are making groups of 12. (one ten and two units) So say, how many times does 12 go into 144? Traditionally 12 goes into the 140 first...once, with 2 left over, and we place a one in the tens answer place, (meaning we can make (10) groups of 12 out of 140) and then pick up the unit 4, add this to the remainder and see how many times you can make groups of 12 out of 24....(2 times, and this goes into the units place, making the answer 12.) does that make sense?

3. Yes, that makes sense :)

I'm so glad that you appreciated that quote. I wanted to copy and paste it into the blog post and it WOULDN'T copy. Also, my computer wouldn't cooperate and let me have two browser windows open so that I could type from the article. I had to keep clicking back and forth typing in as much as I could remember at the time. I kept thinking "It would be easier to just not include this quote." LOL!

4. MBT,

I am totally copying you on this one! When we get to two-digit division, I will be bringing out the ninjas, too! How many ninjas of each color do you recommend getting?

Cristina

1. The albums should say, but 9 of each because that is as many as could be in that category (well, if you want to do this work with a 4-digit divisor, you'd want a few extra green ;) ).

2. My albums DIDN'T say how many actually. They just said "a bowl of blue bows and a bowl of green bows."

I bought nine pairs for the reason Jessica stated. I wished I had three more when I set out the ten trays because Kal-El wanted a green ninja on each tray. I took the original two green ninjas back off of their trays and we were still short one. I could have bought just one extra set and had enough for the ten trays if I still took the originals, but it felt wrong taking the original ninjas. The presentation does NOT call for "bows" on the redistribution trays however, so nine should be more than enough. If I couldn't find nine packages I don't think I would have even cared. We aren't going to do this work THAT many times and I could just stick to slightly smaller divisors rather than "90-something" divisors.

My albums (MBH) only do this work with two digit divisors. I noticed that Cherine went ahead and did three-digit divisors. I don't intend to do them, and DON'T plan to do four-digit divisors. However, just watch, next week you'll see a post...."Kal-El wondered if we could do 3- and 4- digit divisors" and there I'll be back at the dollar store. They did sell red ninjas too :)

5. THANK YOU, THANK YOU!! I had this on my week schedule to present....I gave some rounds to the work!!!!LOL but with this AMAZING explanation.... I will try!!!LOL but of course..... WITHOUT NINJAS!!!lol you know.... girly girly!!!LOL Thank you!!!! hugs!!

1. Maybe the ninjas could wear frilly dresses?

6. Thsoe ninjas are TOO cool! ;)

1. Oh yeah. They keep escaping from school. There were ninja armies marching across the living room carpet yesterday.

7. Awesome post, and great timing for us! Thanks so much.

1. Great! I'm glad to hear that other people are doing this work instead of skipping it. I feel less alone :)

8. Great job!! This was a very long work for Xander! He found the distribution slow going, but using Hero Factory definately jazzed things up for him. The Ninjas are awesome:)) It is great to see you doing this work - I was feeling a little alone <3

1. Thanks Cherine! Thanks to you, Xander, and Hero Factory the boys were saved from torment by hair bow. I'm so glad that there are a few excellent bloggers like you out there taking pictures and writing about elementary Montessori. It makes the lessons so much less intimidating when I've seen someone else do it. I feel more like I'm on a well-traveled path (which is the reality) then blazing a new one. LOVE your blog.

9. Guess what we got from the dollar store today? Guess which lesson T is getting next week? I got 10 of the green and the blue, the same exact kind no less, but I think that I'll get a couple extra for that final distribution proof, and maybe a few red ones. Guess we are lucky in this regard that our classroom is on another level and things tend not to migrate elsewhere. Thanks again for this post...so very, very, helpful.