Montessori Fraction Charts

After the confusion that was the fraction charts last semester, I sat down and re-did them this week since my fraction section of the album was due.  It helped that I’ve actually seen and illustrated the whole fraction section of the album.  It made organizing and understanding why the charts were there.

I do want to note that my trainer said it’s better if the kids are making these charts.  So I don’t know whether or not I’m going to be printing these out for Thumper.  A few weeks ago, she made a multiplication table when she needed them.  I kind of want the same thing to happen with the fraction charts.  It’s no use as a reference if she doesn’t need them.

I have 22 fraction charts.  It’s based on the album and also copies of my trainer’s fraction charts (which were not complete and also were missing charts or had extra charts).  I also used these three sites as reference:

It’s two more than the first two websites, because of chart 8 and chart 11.  Their charts do not have these “statements”.  One is imbedded in another chart.  But since my album says so, I made mine that way.  There’s also a mysterious chart 22, which isn’t listed in the album itself.  I’m guessing it probably should belong within a multiplication chart, but the trainer didn’t have space for it.  I also made a change from the original charts so it said like and unlike denominators.

  1. Introduction to Quantity, Symbol, and Language
  • Chart 1 – Fraction as circular units divided into pieces from 1 to 5.
  • Chart 2 – Fraction as circular units divided into pieces from 6 to 10.
  • Children may make square charts.
  1. Equivalence
  • Chart 3 – Equivalence for 1/2 fractions
  • Chart 4 – Equivalence for 1/3 fractions
  • Chart 5 – Equivalence for 1/4 and 1/5 fractions
  1. Addition
  • Chart 6 – Addition of fractions with like denominator, ending with whole number
  • Chart 7 – Addition of fractions with like denominator, ending with fractions
  • Chart 8 – Statement: How to add and subtract a fraction with like denominators
  • Chart 9 – Addition of two numbers with unlike denominators
  • Chart 10 – Addition of three numbers with unlike denominators
  • Chart 11 – Statement: How to add and subtraction a fraction with unlike denominators
  1. Subtraction
  • Chart 12 – Subtraction with like denominators
  • Chart 13 – Subtraction with unlike denominators
  1. Multiplication
  • Chart 14 – Fraction times whole number
  • Chart 15 – Fraction times whole number plus rule
  • Chart 16 – Fraction times whole number and fraction times fraction
  • Chart 17 – Fraction times a fraction plus rule
  1. Division
  • Chart 18 – Fraction divided by whole number
  • Chart 19 – Fraction divided by whole number plus rule
  • Chart 20 – Group division
  • Chart 21 – Fraction divided by fraction

Chart 22 – Extra chart on multiplication

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Positive (or Addition) Snake Game

Age: 4.75yrs

positive snake gameToday, Astroboy chose a work that he had been introduced to, but didn’t want to work on, almost 3 months ago, at the beginning of January.  This time I changed my presentation format.  I’ve been letting him take over when he shows interest during a presentation over the past few months.  But recently I’d began to wonder if this is why he shows no interest in the work itself after the presentation.

So this time around, I showed him how to put his hands on his lap (it’s not in the writeup) and told him that I got to go first and then it would be his turn.  I did a classic presentation where I didn’t speak much and moved slowly but surely.  A minute in, he told me he wanted to trade the golden beads and we can take turns.  I reminded him that I would show it to him first.  Thankfully he did not throw a fit and let me continue.

Mark the spot where you count to 10, exchange that for 10-bar (gold) and place black/white bead bars for leftover beads.

Mark the spot where you count to 10, exchange that for 10-bar (gold) and place black/white bead bars for leftover beads.

The positive snake game comes after you’ve worked on the addition strip board.  You lay out some colored beads to form a “rainbow snake” and through counting, you exchange every 10 beads for one golden bead 10-bar, and whatever is left over with some black/white beads.  Directly, all they’re doing is counting.  We only show the child how to count 1 by 1.  However, the child will inevitably start recognizing the bead bars by their colors and start to skip when they count.  For example, you could have an 8 bar and 9 bar, and instead of saying, “1, 2, 3, ….” they would count “8, 9, 10, 11…”  Indirectly they’re learning how to add where the sum is greater than 10.

I think, even more indirectly, they’re learning how many different ways you can add up to 10.  I’ve seen a few kindergarten worksheets and they usually have problems that ask ___ + ___ = 10 and want you to give them a few possibilities.  Or they ask you to list the different ways you can add up to 10 (1+9, 2+2+2+2+2).  This second question leaves it open ended and allows the child to see just how many ways you can add up to a number.

Anyways, isn’t it neat that this all gets worked into a simple exercise whose direct aim is just counting?


First, lay out black/white bead stairs

Of course, being a child, Astroboy didn’t just want to do what I showed him to do.  He wanted to make a super long snake that fills up the WHOLE length of the rug.  I was able to leave him after the presentation (yay!) and he exchanged a few himself.  But then it was lunch time and he interrupted himself, cleaned everything up and put it away.  (Another first!)  I felt so sad I had to interrupt his work but we really had to go to the library before swim practice today as I had lots of overdue books.

Arrange color bead stairs into a super long snake and start exchanging

Arrange color bead stairs into a super long snake and start exchanging

Tonight, he pulled it all out again, set it up and tried again.  Unfortunately, this time it was the end of the night and he could not really concentrate.  He called me over every 5 minutes to work with him.  I finally stopped him, left the work as is, and put everyone to bed.

Hopefully we will be able to finish our long rainbow snake that is molting into a golden snake tomorrow.

More thoughts & notes on Montessori elementary math

My math class is chugging along.  Once again 1/3 of the album is due and I’m madly catching up after taking 2 weeks off to making Chinese materials.

Now that I’ve seen a lot of the presentations, I feel less anxious and less confused about the math scope and sequence.  Even though if you were to ask me how I ought to teach math, I still could not give you an answer, just like 3 months ago.  But, the only difference is, this time I can tell you why I can’t tell you.

I feel that AMI math training (or at least AMI flavored AMS training) takes a very big view.  Whereas AMS math training (at least the albums I’ve seen) takes the small, detailed view.  What I hear often from my teacher is, “it depends on the child’s interest,” “follow the child’s interest”, “the most important thing is to invoke the interest of the child.”   Yes, there is a general….path, because come on, you cannot really learn the more advanced stuff like squaring and cubing without knowing your addition and multiplication facts.  But, within that, there seems to be some leeway.

Because I’m just giving Thumper a bunch of different presentations whilst we work on our addition/multiplication memorization and multiplication operation, I see this at home.  When a presentation is at that perfect N+1 level, the level that engages her, makes her think, we can spend 1+ hour on it.   Case in point our Multiplication Checkerboard presentation, or our commutative property presentation.  Versus the stuff where she drags her feet and doesn’t want to do, like our attempts at fraction.

As my trainer have told us several times, she would start with a lesson plan for the year, and typically she ends up not following the plan.  Because the children will show her what they’re interested in.

Anyways, obviously saying this helps noone if they’re having the same problems I had, which is, “What do I present for the first year?!”  So here’s the super high overview I now have in my head.

Supposing your kid is in a super traditional program (not many are), then by age 6, they would have, after learning how to count, worked on memorization of +/-/*/÷ , big number operations (+/-/*/÷ 4 digits), been introduced to fractions, skipped counted, etc.  The first elementary year, you introduce them to digits up to 1,000,000.  You do a lot of multiplication of various kinds (commutative property, Least common multiple, greatest common multiples, multiples, etc), factors, continue with +/- probably of big numbers, fractions, intro to squaring and cubing.  And depending on how far you go, you could also introduce long division, divisability, and decimal fractions.

On a side note, squaring and cubing or learning the quadratic equation can start in earnest in third grade!  “Why do kids learn this?” my trainer asked today.  We gave her a bunch of answers like, “It’s good to learn it deeply rather than abstractly, starting from young age,” or “Kids learn better by manipulating materials,” or “it’s good to use multiple senses to learn a concept,” etc.  But the reason is rather simple: “Because kids find it interesting.”  It tickles me to no end hearing this answer.

In any case, look at how many things I could do!  Thumper ought to be interested in at least ONE of these things. And they all work on those basic skills (multiplication, addition etc) somewhere.  I’m still going to figure out my scope and sequence when I get a chance.  But I think it’s more for me.  Just so I have an idea in my head of how the various presentations are linked together.

Back to the class itself….

What I love about my math class is seeing how easy a lot of math concepts are when you manipulate materials rather than numbers.  You start seeing patterns (a state standard that is not explicitly taught in Montessori but is everywhere) in all the materials you’re laying out.  I remember a concept so much better after manipulating the materials with my hands.  There’s also something about seeing a concept illustrated visually.   It really drove home for me something I hear over and over, even in non-Montessori circles, that movement of the hands really helps with learning.

I’ll stop here.  Here are some more “aha!” notes I took from my class.

  • One of the bigger curriculum in elementary math is actually squaring and cubing.  The kids spend a lot of time on it.
  • My trainer didn’t like the higher fraction pieces they sell now a days.  By the time they get to these materials, they’re often too old for them.  Also, 1/11…1/20 is seldom used in equations due to lack of equivalence in many of the numbers.  The goal is to move away from concrete materials, so there’s no need to keep adding concrete materials if you don’t need it.
  • There is no need to do followup work if a child is not really interested. (What a relief to know this!)
  • I finally realized that lesson plans are just like regular schools in that you could plan for multiple concepts to be learned in one year in some sort of sequence.  But this does not mean you’re done.  You will come back again late, maybe the next year with same concept, but at a higher level abstraction.  It isn’t that I work on all fractions till I’m done.  Maybe I do fraction addition for a few weeks, then I move on to another concept.  I don’t have to concurrently run 5 different threads at once.  Whew!
  • The large bead frame and checkerboard are the same concept.  But kids don’t like repetition so this is a way for them to work on the same skill using different materials.
  • Presentations are kept really concise and without a lot of talking or extra info.  The point of these lessons aren’t necessarily to “teach”.  I’m the guide provoking a child’s interest in math.  The real learning comes when they’re manipulating the materials anyway.  So, unlike a teacher, I don’t need to tell them the secret, the trick, the how.  I need to make the presentation interesting so they will want to work with it after I’m done.  If I give all the secrets away, that may be one reason they don’t want/need to work with it after.
  • The stamp game has no 2 digit division.

Fieldtrip: Muirwoods 墨爾紅木公園

IMG_5319 copyWe took our relatives for a hike @ Muirwoods today.  We arrived at around between 9:15-9:30 and the parking lots were full already.  Thankfully a parking space was found about half mile from the entrance.  Not too bad at all for a Saturday.

Muirwoods is a National Park located in Marin, famous for its tall redwoods.  The paved path in the park are in a loop, with 4 bridges; offering you 4 ways to walk around the redwoods.  Today, we decided to hike the actual trail that branches off of the 4th bridge, the Hillside Trail.  With 2 kids and constant stops, it took us 3 hours total for the 2 mile loop.

With so many tourists, it was really nice to get off the main trail.  Thumper complained and asked when she can leave (“It’s so boring!”) after about an hour.  But we found things to look at and entertain her, including some made up story about a Little Thumper who lives in the woods with her family.  We examined banana slugs, wood stumps, and the red sorrels.  At times like this I resolve to go hiking once a week.  It does not please me at all my children think nature is boring.

I used the time to learn some new Chinese vocabulary.  They’re words I can’t even pronounce.  It’s only a few because at my advanced age I can’t even remember these after 5 minutes.  Hopefully we can use it again when we go again with visiting friends in 2 weeks.  AND I found a copy of the Muirwoods brochure in Chinese!

1.  Banana Slug 香蕉蛞蝓 (Xiāngjiāo kuòyú)


2.  Redwood Sorrel 紅木酸模 (Hóngmù suānmó)


The sorrel leaves fold down when they get direct sunlight.

3.  Lichen 青苔 (Qīngtái) or Moss 苔藓 (Táixiǎn)

IMG_5314 copy

Is this moss or lichen?

4.  Chipmunks 花栗鼠 (Huā lìshǔ)

Wish I’d gotten a pic of the cute chipmunks close to the cafe!

I need to get moving and make some vocab cards for the children before our next visit!  Maybe after my album.  (The ever due album!)

Our Chinese library collection

Finally!  Today, I went through the mess that was 巧虎 (Chiao Hu) and had been bugging me for the last 3 years.  I don’t think I’ve actually really looked at our Chiao Hu magazines for 2-3 years.  It gets tossed into a box every month and Thumper digs it up and plays with it, strewing all her mess around before I put a stop by stuffing it into an even bigger box and then hiding it.

I love my collection even if I don’t read the books.  Hmm….if I personally had a huge stash of Chinese books to go through, I think I would be in heaven.

I’m planning to add to the collection this summer with more age appropriate reading material.  I’ve got a long list saved up of books to explore and buy that I need to wade through.

In my ideal world, the bookshelf would be actually tagged and sorted by subjects.  Maybe after the kids go off to college. 😉

Here’s what’s on the bookshelf. Sorry the pic is kind of crappy.  No time to fiddle with the light with a too-short cord.

1.  My small collection of Chinese books I read.  I love 侯文詠 and have all his books until the last few years.  There’s also a stack of 國語日報 that we’re not using.  It should take us to 3rd grade probably in content.

2.  From the left, my Chiao Hu teacher’s stuff.  More personal Chinese books that are really tall.  The 20% I couldn’t finish sorting.  School related catalogues: Rainbow Resource, Uline, etc.

3.  My Chiao Hu collection spanning 4.5 years, from 2 years old to half of second grade.

4.  “Readers”  Three collections from Hsin Yi.  Some are easy, some are hard for Thumper to read.  I like the series a lot because it has both non-fiction and fiction items.  It’s written by Chinese authors, so it does not have the easy language of translated works and I think super great to read to children.  I also have a few sets of really thin books for 0-3 years old.  The right 1/3 is board books for Astroboy.

5.  Left is a set of easy picture books for Astroboy about a turtle and another set about a bug, all translated books.  Then a set for Thumper called “I Love Martine”.  Then another set of 12 books from Hansen which has one story for each day of the year.

6.  Non-fiction Chinese books covering science, history, geography, etc.

7.  My 金庸 set for when one of my kids are old enough and have good enough Chinese to read.

8.  Teacher stuff.  Lots of workbooks, books from 僑務委員會,北美洲華語, catalogues, etc.

9.  Sagebooks, How to write, books about the Chinese language itself, dictionaries that we never use.  I even have a Math dictionary!

10.  Bridge books and lower elementary books for Thumper.

11.  Picture books for Astroboy.  Though Thumper loves them still.

12.  Two sets of encyclopedias, covering topics from biology, astronomy, botany, etc.  Both series follow the Montessori curriculum sequence and that’s why I bought them instead of the more often seen picture encyclopedias that kind of jump around.

Checkerboard Multiplication

IMG_5241 copyAge: 7.5

Presentation: Checkerboard multiplication, presentation #1, #2, #3.

I’ve been hitting Thumper with various presentations in my elementary math album.  Any presentation that should be at least introduced in first grade.  At my last training class, I finally asked my trainer about scope and sequence and she basically said that I can introduce a topic, study it a bit, drop this subject, and then come back again next year where I review and introduce a higher level of the same topic.  It’s a spiral curriculum.  This relieved me of feeling “How am I supposed to cover ALL this by the end of this year?!”  And freed me to just present all age-appropriate presentations to her now.

This week we worked on check board multiplication.  Checkerboard multiplication comes after the Large Bead Frame.  The checkerboard kind of mimics the multiplication process, except it breaks down the steps for you.  By multiplication process I mean, we take the unit of the multiplier and we multiply the multiplicand, then we take the tens and multiply the multiplicand, then we take the hundreds place of the multiplier and multiply the multiplicand, etc.

This entry may not make sense unless you’ve seen the presentation:

I introduced the checkerboard early in the week.  I was surprised at how long it took me to introduce the concept that a unit bead changes its value depending on where you place it in the checkerboard.  And we spent a LONG time with naming the numbers.  Montessori math is not designed for the Chinese math system.  She colored coded each place value green, blue, red.  So green 1, blue 10, red 100, green 1,000, blue 10,000, red 100,000.  Because these are the “families” (thousands family, millions family, billions family, etc).  But Chinese really puts the comma after 4 zeros.  Our counter really starts repeating after 10,000.  So really we would say 1,0000,0000 for 100,000,000.

It’s all very complicated.  We tried to make a cheat sheet noting the place value names into Chinese.  That didn’t quite work since the checkerboard itself puts the commas in the “wrong” place for Chinese.  So finally I just taught her how to say it properly in English.  I’m going to have her “read” her answer when she writes by putting the comma in the right place first.  Otherwise I think it’s a lost cause.  I recite the place value names myself every time I’m looking at numbers and trying to say what it is in Mandarin.

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Keeping a pet snake in the classroom


We have a pet in the classroom.  She is a snake and her name is Viscountess Pudding Edwina, or Pudding for short. Pudding is a Rosy Boa.  I would never have kept a snake before. But I met a friend who keeps many reptiles as pets and their great enthusiasm for them rubbed off on me. I was still a bit hesitant with the idea of a snake as a pet until we visited the Vivarium in Berkeley, which sells all things reptilian or amphibian.

At the Vivarium, I learned that the rosy boa only grows up to 3 feet so they fit my idea that pets need to be a bit cute. When we first got her she was a teeny tiny thing, or, as we say in Chinese, 小布丁點 (xiao bu ding dian), hence her name Pudding, which is the English translation of 布丁.   Baba added the Edwina part.  I love her name since it’s bilingual just like the two kids. I’m really big on having bilingual names.

It’s been a whole year since we’ve had Pudding. We purchased her with the Christmas and birthday money the kids have received since birth.  So it’s technically the kids’ pet.  You would think that would make the kids be a bit more responsible, but alas, not really.  I still have to remind Thumper sometimes to feed the snake.  But after about a year of feeding it alongside her father, she now has the whole routine down and can feed Pudding all by herself.  And Astroboy, who was not interested with feeding, now wants to help as well.

Pudding requires one pinkie or fuzzy mouse a week, plus a bit of water every month or so.  We had to buy a cage, thermometer, shavings, a warming lamp and heat pad.   I think the whole startup cost was probably $300-$350, with Pudding the biggest cost.  A bag of fuzzy costs $11 bucks and lasts you 2.5 months.  And the shavings cost $5 and lasts 6 months to a year.  All in all Pudding is a very low maintenance pet, both in time and money.

It’s utterly fascinating to watch Pudding feed.  Here’s a video on youtube from someone:

I love having a snake for a first pet for several reasons.

  • As I mentioned, low maintenance in time and money.  I don’t like nagging the children to feed the pet daily.  Once a week is good.   And I wasn’t ready to another “child” in the house to take care of.  Best thing, if you forget to feed her for a few days, it’s okay!
  • No need to deal with poop as compared with dog/cat/chickens.  We just scoop out the poop once every few weeks, when we remember.
  • She’s not a big pet and doesn’t walk around the house, thereby requiring some vacuuming of hair or constant cleaning of something.  I don’t have to worry about her eating random furnitures or things she shouldn’t eat.
  • She can be handled by the children as long as they don’t abuse her.  Other than cats and dogs, many Montessori pets cannot be handled much by kids.  Thumper got nipped by Pudding once because she was playing with her while getting ready to be fed.  It just surprised her and she learned not to do it again.
  • She sheds her skin!  The kids get to observe her.  I didn’t realize how much they really learn from living with a pet until we went to the Academy of Sciences last semester and we sat through a presentation on snakes.
  • Baba loves reptiles.  Perfect!  He definitely helps with the feeding and petting.
  • It helps the children to like reptiles.  A snake is usually something that teachers, children, and many adults say “ewwww” to.  I really like how Thumper that isn’t afraid of touching them but at the same time has a healthy respect and knows that not all snakes are touchable.

Now that I know we can handle a pet and the issues I may run into (constant nagging), I’m ready for some chickens and cats/dogs.

So what’s a Montessori pet?  In the Montessori classroom, you typically keep the 5 types of vertebrates as pets (birds, reptiles, amphibians, mammals, and fish).   Maybe invertebrates as well.  So children really have hands-on experience caring for animals and observing them.  In my ideal classroom I would like to have some chickens (bird), keep bees (invertebrates), and a dog or cat (mammal).  Not sure about I want to keep an amphibian yet.  And of course, most classrooms don’t have mammals as pets, though I’ve heard that some 3-6 classrooms do.