Wednesday, March 4, 2015

Living Math Book

Last week we happened across a great "living math" book because it was recommended supplemental reading in the The Story of the World history program.  In the story, One Grain Of Rice: A Mathematical Folktale, a young girl is suggests as a reward that she receive one grain of rice.  When this is declared inadequate she suggests that receive rice for thirty days and that each day the quantity she receives shall be double the quantity of the day before.  She receives one grain the first day, two the second, four the third, eight the fourth, and so on.  After reading this.  The boys suggest that we see if the book as accurate as well as determine the how much rice she received on the last day and in total.  (I had abstained from reading them the last page that had a chart with all of that information.)

I played the role of administrative assistant.  The boys took turns creating and solving the equations while I did the writing.   A child who is a more persistent writer than either of mine could do this completely on their own.

It was difficult to add the longer categories, such as the units, at the end.  The dot board would have been good here if it were set up for numbers bigger than 99,999.  We penciled in intermediate sums along the edge of the paper.

Wednesday, February 25, 2015

Work of Air: Prelude to the Winds, Demonstration Three

work of air, prelude to the winds

I got this demonstration to work and on the first try!  Unfortunately the video I took was one of the worst videos in the history of videos so I'll just share the picture.  This is demonstration 3 in the section of the AMI geography album (KotU) titled "The Work of Air:  Prelude to the Winds." 

Note to self: If you are going to keep posting pictures of science experiments on a cookie sheet, use one of the shiny cookie sheets.  

In this presentation we want to show that when hot air rises a void is created and other air will be drawn in to fill that void.  We review that hot air rises with the visual of the smoke from a stick of incense.  We used lavender.  I held the incense like a sparkler and demonstrated that the smoke rises straight up from the stick no matter what position I hold it in.  In hindsight, you may not want to train your child to wave burning incense sticks around like a sparkler. Next we need a sturdy tube with a door.  I made mine from a manila file folder and some tape.  A teal light candle at the bottom of the tube will heat the air so that it rises.   When the incense is placed near enough to the door the smoke suddenly stops traveling vertically and makes a very impressive horizontal jump toward the open door and smoke starts coming out of the top.  We definitely walked over to our fireplace immediately after this and made the connection between this experiment and how our fireplace and chimney work.

The script for this demonstration in the KotU album worked beautifully.  I appreciated having simple sentences provided to model.  I would have gotten wrapped up in my own tongue trying to explain this one.  The directions call for clay for holding the incense stick. I didn't need it until I needed my hands free to take a picture.  Perhaps the child would place the stick into the clay in several different positions rather than treat the incense like a sparkler.

On a side note, I have attempted Demonstration One another ten different times with the addition of things like duct tape with no luck.

Tuesday, February 24, 2015

Division Paper

This paper shows the progression of Kal-El's long division skills the past few weeks.  The equation in color at the top left of the page is the very last equation he ever completed with the racks and tubes while only recording the quotient.  It became clear that he could go, and needed to go, much more quickly through the racks and tubes work than he had been.  So directly below that equation you will two more which he completed with the racks and tubes recording both the quotient and the intermediate remainders.    

A little lower on the page and centered you'll find a few more equations he completed.  In addition to a couple of random multiplication equations there are a couple of division equations.  These were done without the racks and tubes, just on paper.  Here again he was recording the quotient and intermediate remainders.  

These equations were going so well that I decided he could move on to recording the quotient, the "number of beads used" (Does anyone know the name of this step?  Intermediate products?),  and the intermediate remainders.  We did a couple on the racks and tubes but Kal-El immediately wanted OFF the racks and tubes.  This was possible because his multiplication is so solid.  Therefore, he doesn't need to to count beads to find the products.  In fact, he really dislikes the carrying that you have to do to record the products with the bead.  What is recorded  on the bottom right-hand corner of the paper is his first equation purely on paper. 

This is a close-up of the equation itself, but scattered around the larger piece of paper you'll find the multiplication equations he used to find the answer.  This paper isn't organized in a way that could be used to turn in work at a school, but was fine for us at home because it's his work.  It's not being turned in for assessment.  However, when he started to add ANOTHER division equation to this paper I couldn't take it anymore and stepped in.  I made him use another piece of paper.  I'm all for conserving resources, but this was getting ridiculous.  

In order to train him to record his work in a more teacher-friendly manner (in case I'm abducted by aliens and my husband sends him to school) I "invented" a new specialty paper for him to use for long division.  I am attracted to specialty papers like raccoons to shiny objects.  Now, I don't really think I "invented" this.  I'm sure I've seen it somewhere at some point in my life.  But, the fact stands that I don't know where or when and have no idea what to call it or how to find it.  

He has been using 1/4 cm graph paper for most of his math, but at his developmental level the squares were fine for recording simple numbers but too small for recording regroupings.  I bumped him up to the 1 cm squares.  I have to figure out what I'm going to do for the really big equations with four-digit divisors and seven-digit dividends, but this will do for now.  I was really happy to see that he had written "yay" (in cursive no less) after his quotient came out to match the back of his equation card.

Me Too has started the racks and tubes as well, but has been choosing to hold off work on them until he's finished the bingo board.  I though he would be done last week but I forgot that we do all the extensions listed in the MRD math album for these.  I had scanned the Montessori by Hand album to refresh my memory and she only lists one activity for this.  Me Too remembered.

This reminds me that I made some much needed room on my iPad last week by DELETING all of my digital primary albums.  I still have them on my desktop but no longer need them on my mobile device.  That really felt like a landmark.

I have a couple of "big" posts in the works.  Those of you who have been reading me for years probably guessed that already due to the slower rate of posting lately.  That always happens to me when I'm putting together big ones.  I have given the science area of our classroom a much needed makeover.  I am working on putting together a post on our science equipment storage, chemical storage, and science experiment resources.  I also have been slowly working on the posts that readers have requested regarding art and Spanish.  I don't know how long it will take, but they are in the works.

Friday, February 20, 2015

Yet Another Failed Experiment

I am really on a roll when it comes to failing at the traditional Montessori geography demonstrations.

This is the first experiment in the "Work of Air" section of the geography album.  It is  supposed to show that air takes up space.  When you pour the water quickly from the tumbler into the funnel that is sealed to a bottle with plasticine most of the water should remain in the funnel.  I tried this at least fifty times.  Fifty shades of failure.  I tried several different bottles.  I also tried a different funnel. The directions said that the funnel end should be a straight-across end rather than angled.  I didn't fail any harder with the angled funnel than I did with the straight funnel.  I also tried several types of clay. I occasionally got the water to go in a little more slowly and bubble a bit, but never stay in the funnel.  

You can read more about this experiment over at Frugal Fun for Boys.  Sarah is apparently a little handier than I am.

I couldn't find any videos of this working.  My suspicious mind says, "That is because no one can get it to work."

Oh.  And one of the boys took a drink out of the glass before I took my pictures.  So, now there is a dirty lip print on the drinking glass.  Even the photo is a "fail."

Tuesday, February 10, 2015

School Days

It may have taken three months to do the division bead board, but Me Too finished the division finger board in three days.  He took a break most of last week, but will likely start and finish the division bingo board this week.

One of our Spanish lessons was about flags.  This got Me Too interested in flags and continent boxes again.  He colored many flag pages.  You can print them at Enchanted Learning or, as we did, from Crayola.

The boys enjoyed some new items in their Central American souvenir collection.  A friend recently sent us some very neat things from Mexico such as this Aztec calendar stone.

A little bobble-headed turtle....

We're not so sure what this is.  My gut says Chichen Itza, but it has be reshaped into something of a boat.  Maybe it's a Chichen Itza pencil holder?  The boys have been laying their pencils across the back so they stop rolling off the table while we work. Our house is not very level.

This all prompted more digging in the North America box.  We have a separate box for the U.S.A., so this box holds our items from other North American countries.

All that digging around in the North American box prompted him to dig around in the Antarctica box as well.  He reread a lot about McMurdo station.  He is very enamored with this set of real scientist's dog tags.  I think he's been wearing this on his wrist for a week.

He was so absorbed in exploring these slides that he was super surprised when the preliminary flash went off on the camera.

Kal-El was inspired by his Story of the World listening.  We've been reading about the Mesopotamians.  Here Kal-El has made "mud bricks" our of modeling clay.

And then, he stacked them into a structure.

We've been doing a lot of Story of the World inspired reading.  It is finding our way into our other work.  One of Kal-El's sentence constructions with the grammar arrows this week was "Hammurabi rides his bike."  Kal-El takes pictures of all of his grammar sentences with our little point and shoot, but he decided to take a video of that sentence.  Videos are a lot of work to post, so no go.

Kal-El is still plugging along through our sentence analysis materials.  Here is a sentence he did take a picture of.

Kal-El has started a new stage of checkerboard work.  He is now recording his partial products.  He can do this work abstractly on paper.  He works on long multiplication once or twice a week.  For every equation he does on a "material" he is allowed to do one just on paper.  He needs a little more practice before I feel comfortable that he has internalized the process.  He understands works like these instantly and it would be easy to not make him repeat. However, I find that he memorizes algorithms extremely quickly but it takes many repetitions with the concrete manipulatives to internalize the real process. I want  to make sure he understands the process and not just the algorithm so a minimal amount of work on the materials is still a requirement here.

We used to go to the library to take advantage of "Reading to Rover."  Now we have our own private Read to Truman program at home.  Truman will not be homeschooled.  He starts puppy classes on Thursday.

Me Too has moved on from scowling at division on the stamp game to beaming at division on the racks and tubes.  We are getting to this nice and early so hopefully he will enjoy it for a long time.

Kal-El is moving through the work on this same material rapidly.  He is a little old for it and doesn't like moving all the beads.  I sit next to him while he works and help him clear the boards to speed things up for him.  He demonstrated that he could do equations with four-digit divisors and remainders so I let him skip most of the equations in the box on his first pass through.  The top equation on the paper below is one of his last equations on that first pass.  The first level requires only recording the final answer.

It looks like we will be skipping most of the equations on the second level as well.  The second level requires that you record your remainders at each step and bring down the next place value on your paper.  What you are not yet doing at that stage is recording the number of beads used and subtracting to find the remainder.  However, the second equation on the paper above was one that Kal-El completed his second day of this while I was in a meeting with another Mom at the kitchen table.  He doesn't like the racks and tubes so he did it in his head. This means he was able, in his head, to picture the beads on the board, multiply the width and height of the bead rectangle or square produced, subtract that from what he had in his head and write down only the remainder.  Once we've tackled a few with larger divisors we'll go ahead and move onto the level that allows you to record that step.  Then he won't have to do that in his head, although apparently that poses no problem for him.  He has done many equations like this on paper this week.

Thursday, February 5, 2015

Tweaking the Bank Game Box

Last week I posted about our adventures with the B'omarr Monk Game or Montessori Bank Game.  After tweaking the game itself, it was time to tweak the box.  I bought our bank game from a friend.  I don't know where she bought it.  As it came, it needed laminating.  I noticed even before I laminated that the unit cards were nearly impossible to get out of their cubbies.  When I laminated I trimmed those cards a little smaller, but it was still ridiculous.  To add insult to injury, I had to trim after laminating because the cubbies are so tight.  So now as our chubby fingers struggled to take the unit cards out we were peeling the laminating film right off of our cards.  Maybe your box is sized properly.  However, if it isn't here is how you can tweak your box to fix this little problem

Inspired by the ribbons in some battery compartments, I cut some ribbon to the proper size and then super glued just the very edge down to the bottom edge of the smallest cubbies.  A staple or tack would be great here, but the wood on the bottom of my box is too thin to take one.

Then, when you load your unit cards into the box you can load them on top of the ribbon.  Use the extra ribbon to lift the unit cards out of their cubby.

Wednesday, February 4, 2015

Prelude to the Winds: Experiment 2, Warm Air Rises

I promise I practiced my presentation several times before giving it to the kids this time.  I could NOT get this to work until I changed the procedure significantly.  The photo above shows my interpretation of the instructions.  I have a knitting needle affixed to the tray with poster putty.  I cut a spiral from paper and balanced it with a pinhole on the tip of the knitting needle.  I put one tealight candle beneath.  NOTHING HAPPENED.  So, next I tried adding more candles.  Below is the video I took with four candles.

PRACTICALLY NOTHING happened.  The spiral wiggled a bit.  It is supposed to spin.

On the internet I found some science teachers doing this with a burner instead.   So then I amped things up A LOT and held it over our turbo gas burner.  The first time we did it, it not only spun but inverted itself and spun so we had a tornado.  Every time I tried to video this it wouldn't spin but did invert itself as you can see below.


Oh dear.  I had a lot of fun and the boys thought it was hilarious that I started things on fire multiple times while practicing.

What is beautiful about this presentation as it is in the albums is that the child can do it himself on a tray in the classroom.  Also, the child can really understand that it is the hot air causing the movement rather than something magical about the materials because they make their own spiral and assemble this themselves.  Once they understood the concept, I thought it might be neat to bring out something special like  the German candle pyramid I saw at Our Happy Homeschool or these Scandinavian Christmas Angel Chimes I bought.

I took a video so you could hear the gentle tinkling of the bells.


If you would like to see a neat series of YouTube videos showing different experiments that demonstrate that hot air rises, check out Jared Hottenstein's series.  He has several.

The spiral.

Tea Bag Take Off

 I LOVED watching the birthday candle hot air balloon.

He has several more experiments on his YouTube channel.

Friday, January 30, 2015

Montessori B'omarr Monk Game (er, Montessori Bank Game, Elementary)

The Montessori Bank Game can be confusing for a couple of reasons.  One reason is that it is an elementary work, but some primary teachers call the collective exercises with the golden beads the "bank game" as well.  I actually think the name suits the golden beads better.  Kal-El aptly stated, "This isn't like going to the bank at all."  Homeschooling families may wonder how they are going to do this with perhaps only one child or with the wider mix of ages that can naturally be present in family but not in a Montessori classroom.  In truth, this work can be done by anywhere from 1-6 children and 1-6 named jobs can be listed depending on which album you are using.  For example, MRD writes up the presentation as if it is being done by one child;  Mid America, two; KotU and Cultivating Dharma, three; and my actual bank game set from the manufacture has six job cards (Customer, Clerk, Bookkeeper, Banker, Controller, Cashier).

Basically the bank game is a long multiplication exercise that uses the same steps as the large bead frame except in this case the child uses cards and not an abacus.   If the large bead frame work can be done by one child, than the bank game work can be done by one child.  My understanding is that it was invented to give the child yet another way to practice long multiplication, particularly this time in a way that caters to the child's desire at this age to work in collaboration with other children.  Do not feel pressured to do this work with a particular number of children.  It can be done with ANY number from one to ?.  The take-away here is that this is a one person job being divided up into as many pieces as suit your purpose and you can divide it up in such a way that someone who is not actually capable of doing the complete work can still participate.  It's kind of like the way the government creates jobs.

So rather than being a group game only suitable in traditional Montessori classrooms, the elementary bank game is actually great for homeschoolers because you can recruit your primary kids who are able to read number cards and involve them right along with your eight year old who needs the work and your twelve year old who needs the review but would be too cool to do it if he wasn't helping his little brother.    Your child might not NEED another way to practice long multiplication so you might not need this work.  Your child might need it, but can do it alone.  Your child might need it and do it with you.  Your child might need it and do it with siblings or friends.  In our house, Me Too needs it but Kal-El does not.  Kal-El needs to learn to be a leader and work with others who are still learning so I pulled him in to practice anyway.  However, my kids really really dislike each other right now and can't get along.  So, in the end, Me Too is better off doing it with me or by himself.

 On a typical day in our school room we have two students and one guide.  So, we can play this game as a one, two, or three person game.  Once we started trying it out, Idecided that if it were going to be divided up it really makes the most sense for us as a two person job. One person is doing the calculating the other is doing the physical work of getting the cards from remote locations.  If I have a job, I would call it "referee."  I sit nearby and break up all of the arguments.  Now, getting cards from the "bank" can involve some exchanging and depending on the abilities of your runner they may or may not be able to do that.  In the case of our youngest student, Me Too, he is an old pro at doing long exchanges in his head and does it on his own.  If you had a younger child helping the person doing the calculations could just be more specific as to what to bring and/or the exchanging can be broken down into mores steps.

The boys weren't particularly interested in anything called "the bank game."  And, as Kal-El said, this is not much like going to the bank.  So after the boys refused the presentation a few times I renamed it.  In the Star Wars universe, some B'omarr Monks become so enlightened that they no longer required their bodies. Their enlightened brains are surgically removed from their bodies and transferred to nutrient-filled jars, freed from the distractions of life.  They sit on shelves meditating and pondering the infinite for centuries.  They rarely need to move around the monastery, but when they do their jar can be carried around by a special "spider droid" designed for this purpose.

You will have seen a B'omarr Monk in Star Wars, Return of the Jedi, at Jabba's palace.

So at our house when we play the "B'omarr Monk game" (Bank game) one child is the "brain" and the other is the "spider droid."  The brain stays at the working table and does all of the calculating.  The "spider droid" runs back and forth to the distant rugs holding the number cards and takes care of moving around the cards on the working table.  I replaced the sad little "clerk" and "cashier" cards from our bank game box with cards with the following pictures instead:

Spider Droid


If you are as crazy as we are and want to do the bank game this way, here is a link to a Google doc with all of these ready to print.  I threw the photo at the top of this post from the Star Wars movie on to the top of our Bank Game box as a label.

Montessori elementary bank game in a homeschool

Here is our school room set up to start the game.  We have two and do the actual work at our black table.

This is our rug with the white cards that are used for the products.

This is our rug that holds the colored cards for the multiplicand and the grey cards for the multiplier.

Our work table happens to already have a category chart under the glass that we use with the stamp game.  The boys like to use that chart for this as well.  When they get numbers bigger than 9000 they just pretend there are more columns to the left.  This was one of our first equations and we started with one-digit multipliers.  We progressed from one-digit, to two- and three- digit multipliers the very first day.  To the left you can see the the decomposed multiplicand and the grey multiplier card that has been moved alongside.  I'll save walking you through an equation with a two- or three-digit multiplier for another day.  What I will do right now is describe the division of labor.

We switch jobs among the three of us.  Most of the time we play as a two player game with me and one child, not both.  Me Too likes all of the jobs.  Kal-El would prefer to always be the brain.  That makes sense due to his age.  He is old enough to be uninterested in moving a bunch of little pieces around. 

This is one work for which I felt no desire to have a pack of preprinted equations. After they had an example of each of the different sizes of multipliers, I set them free to invent their own equations.  So, when we play the game, The Brain invents the equation and asks the The Droid to bring him the colored cards he needs for the multiplicand.  Then, he asks him to bring him the grey cards he needs for the multiplier.  Next, he asks The Droid to "do the magic slide" with the multiplicand cards so he can verify the number.  Then, he asks The Droid to decompose the multiplier.  We don't have a set  person responsible for moving the grey cards around as needed during the game.  Often I do it if I'm playing the referee because I'm sitting right next to the cards.  However, either The Brain or The Droid could be the one to move the cards.  Technically The Droid should do it because The Brain has no arms.

The Brain continues his work by doing the calculations and asking The Droid to bring him the product cards he needs.  This is where there is the most opportunity to vary the difficulty of the task to follow the needs of your Droid.  Me Too is brilliant at sums and exchanging.  It creeps me out he's so good.  So, he clearly calculates his own sums and decides when exchanging needs to happen on his own.  He usually figures out the chain of exchanges for multiple categories at the rug and we never see multiple product cards in a category on the working table. I'll have an "900" card on the table and send him off for a "200" card and he'll come back with three place values worth of cards because he has already summed the hundreds and understood the chain reaction of exchanges across several place values. If your Droid has trouble exchanging you'll want to have them bring the product cards for each layer of the multiplicand to the table and do each exchange one place value at a time.  Often times the product card you need for a new layer is already on the table and the Droid needs to figure out why it's missing and find the sum of the existing card and the missing card so that the correct product card can be placed on the table.  You don't have to say a word.  They kids will find their own solutions.  If your Droid is super young or super new to this your Brain can go ahead and tell them when to exchange or even what cards to exchange for what.  It depends on your kids. Me Too's main problem on the large bead frame is realizing that something like "14 ten-thousands" is the same as "one hundred forty thousand."  So, I've trained Kal-El when he is The Brain to always ask Me Too when he is The Droid for "14 ten-thousands" (just an example) so that Me Too has to figure out which cards that means.  I've already seen a drastic improvement on the large bead frame as a result.

I don't know if Kal-El is as quick at the exchange chain as Me Too or not.  I suspect not.  As I said, Kal-El always wants to be the brain.  I am perfectly happy to be the droid and, unlike Me Too, all I do is bring him things.  When he plays with me, Kal-El has to find all the sums and exchanges and tell me what to bring.  In essence, he is doing all the work except physically fetching things.  That works out perfectly because that's the only part he doesn't like.

Me Too says that the Bank Game is easier than the large bead frame because there is less writing, you never have to cross out a decomposed multiplicand, and it is easier to split the zero(s) from the multiplier and move them to the decomposed multiplicand.

Tuesday, January 27, 2015

Common Multiples

This is the second exercise in the KotU album for common multiples.  Kal-El is laying out the multiples of two, three, and four horizontally with their products vertically below them.

It is really easy for the child to see which products each series has in common.  There is a nice script in the album for explaining what a common multiple is.

Next, the child identifies the common multiples.  We used the number tiles for the common multiple. Each equation and its product are set out in a row and enclosed in parentheses.

I snapped this photo when he was nearly finished.  I think he still had the number 18 to find.

Next week we'll work on finding the common multiples  of two-digit numbers (such as 12 and 16).