Category: Math

Starting Montessori Multiplication

Age: 6

Skills: Multiplication memorization and 2 digit multiplication operation.

“Mama!  4 plus 4 plus 4 is 12!”

Astroboy told me of this discovery a few days ago.   I love watching Astroboy discover his math.  You can totally see how math progression, curriculum-wise, needs to go.

With Astroboy, there really isn’t forcing a child to learn math, he thinks about math a lot outside of homeschool time.  He likes to read random numbers he sees. he wants to know how many more stops till we reach our destination.  He likes to know what day of the week and count how many days until xx happens.

In any case, hearing his declaration told me that he was ready for multiplication, because multiplication is basically addition, simplified.  (He’s very proud and happy to tell his friend this discovery whenever he can.)  I’m guessing because he has recently (mostly) mastered addition memorization, he is now moving to multiplication when he’s thinking about numbers.

For our trip, I brought along both Bead Decanomial and Checkerboard Beads, in large ziplock bags.  The only difference between them is the lack of 10-beads.  I whipped those out today to review skip counting, which is a precursor to multiplication.

Last year, we did skip counting a few times.   This year, he knows how to skip count his 2s and some 3s, and knows more math.  So with the bead materials, I laid them out much like the squaring chains and asked him to count and record into his notebook.

I love my concrete math materials for this reason.  With the same material, I can change the difficultly level by adding more and more writing.  My math trainer stressed that it’s not that important to write in the beginning when working with math materials.  The important part is manipulating said material.

But, like I said, with Astroboy, I can always see how to use the math curriculum.  Last year, he wasn’t interested in writing.  This year, he can write a bit more so we added writing to the curriculum.

Since he didn’t really want to manually count the material, I went ahead and printed out a bunch of skip counting worksheets at 7-eleven.  Tonight, he didn’t want to go to sleep, but instead went through all 12 skip counting stamping game twice.

He was not as enamored with the Montessori skip counting worksheet, which is there for children to discover common multiples.

What he likes the most are the two Montessori multiplication apps we’re using right now: Reflex Math and Edoki’s Multiplication math game.  With Reflex Math, he’s mastering multiplication memorization.  I ask him to read what he sees in Chinese to help him start memorizing the table the Chinese way.

Edoki’s multiplication math has both a memorization game and an operations game.  The operations game has the child move stamps as many times as you’re multiplying, to really drive home the concept that multiplication is just adding.  There is also a memorization game that is a duplicate of the primary multiplication bead board.

Because we’re not homeschooling very seriously right now, we’re using these apps the most.  I’m hoping eventually we can whip out the concrete materials again.  I can see it with the way Thumper is approaching her higher level math.  Having those concrete materials makes it easier for her to visualize the harder, more abstract concepts easily.   I don’t want Astroboy to miss this step.

 

Getting Started with Montessori Fractions

Age: 6
Presentations: Putting Fractiosn in order from 1 to tenth, Matching Fractions From Whole to a Tenth

Note: Here are my other posts on our Fraction presentations.

Traditionally, Montessori fractions start in first grade.  Now that Astroboy is officially in elementary, it is time to visit the subject.

I showed Astroboy a bit of it along side Thumper our first year homeschooling.  I spent a lot of time making materials and we zoomed through nomenclature,  The AMI style albums were a bit thin on follow up and the children weren’t too interested in making booklets.  Following the presentations, I tried having Thumper look for equivalence, she disliked the repetitive work.

Because of all this, we kind of gave up on fractions altogether our second year.  This year, with Thumper starting state testing, it is time to go back and cover all the subjects we were supposed to be teaching but never get around to.

In my Montessori albums, you start Fractions with nomenclature.  In Montessori primary, this is done with the skittles.  Astroboy did this once or twice.  He wasn’t too interested.  In elementary, you start with the fraction insets, which I did not bring to Taiwan.

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Introduction to Decimal Fractions 小數

As usual, new school year brings a spate of new learning, until we all get tired and drop everything except reading and math memorization.

Last week, I introduced decimal fractions.  This is something that should be introduced in second grade, ending in third.  Though I’ve seen it introduced later in other AMS style albums.

Last year, I was very anxious and worried that I didn’t start on decimals or fractions.  But I’m finding that it was good to wait.  Since Thumper learned her multiplication/division operations last year, it is much easier to introduce decimals and fractions additions this year.

To a certain extent, on paper, decimals are very easy.  All you’re doing is manipulating numbers and learning where to place that little decimal point.  You don’t really have to understand why it works if you don’t really want to.

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Addition Snake Game

A funny name for a presentation no?

The addition snake game is a way to learn addition memorization, addition exchanges, number bonds, among other things.  Here is one way to present it:

How it works

Hopefully the video explains it so much better than what I’m about to describe.

Basically you first lay out a black/white bead stair.  Then you create a rainbow snake out of various length bead bars.  Now you start counting up to 10, exchanging that for a 10 golden bead bar and using the black/white bead bars for the remaining numbers.  Start counting from the black/white bead bars and exchange for 10 golden bead bar.  You keep doing this until you’ve got a golden snake with a little rattle.

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Skip skip skip ….counting!

Horray for having a curriculum!  Sunday night I looked on my presentation calendar and realized I was supposed to present skip counting on Monday.  I hurriedly looked at the ideas from What Did We Do All Day and made my own set.  She also has a second post on a game you can play.  I didn’t even bother doing a bunch of research.  We ended up with about 9 ideas from her website.  I got both kids to work on them yesterday.

What’s Skip counting?

Skip counting is a state standard for Kindergarten (or it was last year).  It is the precursor to learning multiplication and comes after your child has mastered counting.  In Montessori, you show the kids how to count these short and long bead chains.  The short bead chains are squares of a number, (so for 9, you would be able to count to 81) and the long bead chain are cubes of a number.  But you don’t show the kids how to skip!  They’re supposed to arrive there on their own after getting tired of counting one by one.  Makes sense from a development point of view.  It is how you know that they’re ready to move on from counting.  Of course in practice I don’t know if it’s really true.

I want to emphasize this because if you teach the trick to skip too early, you could end up with a child who knows how to skip count but not know how to count well.  Knowing how to count is important because it helps the child know the relationship between two numbers.  It’s the foundation for all math.

I had one epiphany yesterday watching the kids skip count.  There are two aspects to multiplication.  One is learning your multiples, and the other is knowing the result when two numbers are multiplied together.  To me, they’re related but different.  So for example, the What Did We Do All Day activities are asking the kids to recite their multiples, for example, 3, 6, 9, 12, etc.  But that doesn’t tell me off the top of my head that 12 is 3×4.  What it tells me is that 12 is a multiple of 3.  Useful when you have to learn Common Multiples.

On the other side is learning your multiplications table.  This is what you need when you are doing equations like (1234 x 4321=?)  Multiplications table is pure boring memorization.  I don’t know of any activities, short of singing, that will make it more fun.  Whereas learning multiples there are a variety of activities that I see online.

Where the Kids Were

Last year Thumper got to memorizing 6 and then got stuck, could not remember multiples of 6,7,8,9.  I was going to “force” her to continue.  Hey, I remember standing next to my mom memorizing them when I was 7, she can do it too!  But thankfully I read Life of Fred math.  It basically split up what you would normally think of as a complete concept to learn, like learning to add up to 20 all at once, or learning multiplication table up to 9 all at once.  Rather, kids have difficulty the bigger the number so they could do well with the beginning numbers (addition up to 10, multiplication up to 5) and then need to wait a year for the rest. So I let it go.  This year Thumper is more willing to learn the rest of that multiplication table.

As for Astroboy, he knows his numbers up to 1000 for sure, 10000 sometimes, so we’d been working on counting the bead chains.  But I needed more variations.  I think the fact that Astroboy is now also adding small numbers together is another good indication that he is ready to figure out the next number in the sequence without counting.

What We Did

I looked through all of the link’s activities and printed them out.  I ended up with the following work:

  • 選一個數字。 可以丟骰子選。
  1. 數長的跟短的珠串
  2. 在一百板上每數到這個數字,用筆塗顏色,念它的乘法表出來。
  3. 把數字寫下來在空的一百板上,每遇到他的倍數,用新的一行。
  4. 在珠串復習紙上寫數字。
  5. 玩迷宮遊戲。
  6. 看電視,唱九九乘法表歌。
  7. Astroboy: 寫 數字在空的一百板上。
  8. 描寫數字。
  9. Thumper:把20個數字寫在筆記本。

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Math Cabinet for equations

Finally, after a whole year, I managed today to finish my equation slips and start filling my math cabinet.  I borrowed the idea from What Did We Do Today, as usual.

But, my math cabinet post isn’t really about math cabinets.

It’s really about autonomy.

I had a really hard time doing this last year because I was very confused on the concept of equation slips.  We weren’t really introduced to them in our training.  The equation slips I saw were so detailed.  For example, for addition, they would have static and dynamic, just like our training.  But then they would break it down into 2 digits + 2 digits, 3 digits + 2 digits, 4 digits + 4 digits, etc.  Some equation slips I’ve seen have those special cases like 0’s, 0’s in unit place, 0’s in tens place, etc.  I assumed that a child was to work through each step of these equations, which are increasing in difficulty much like how it’s presented in regular schools.  Totally different from the way I learned addition and subtraction in class.  Even multiplication, was split much finer than what we learned.

Finally, during my math class, my trainer tells me that students should be encouraged to generate their own math problems, she was mostly against pre-printed equation slips.  She said that they will hit those special cases during their work somehow and if I weren’t sure I could just do a quick assessment before allowing them to move on.  I had a Montessori teacher friend who talked about how her students decided to write a huge long string of numbers and divide them.  I saw the pic.  It must have been 30+ digits long.  They ran out of paper and had to tape more and more paper on the first sheet in order to solve the equation.  When I hear stories like that, I see how not having worksheets can really spark a child’s imagination in what they may want to try.   The elementary child really love big things, and trying things that you did not talk about during your presentation.

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Numerical Decanomial with Paper

Age: 7.75

Date: June 2, 2015

Presentation: Numerical Decanomial with Paper Rectangles and Squares

After the Decanomial layout we did last week (or was it the week before?  The days blur…), Thumper is doing the numerical layout this week.

You can watch a video of how it’s done on youtube:

You’re basically doing the decanomial layout using paper.  Now, when I was making my Cultivating Dharma album, I got confused by the writeup because it was not very clear.  I had to cross reference with the video and other write-ups to come up with my current version.  In the video, you will see that the papers are all the same size.  But after doing some research I thought using graph paper and having sizes that are equivalent to their actual multiplication size (1×1, …10×10) is better.

You have 10 envelopes, on the outside labeled Decanomial 1, 2, 3, etc and one labeled Squares 1-10.  On the inside, you have two sets of numbers for each decanomial:

  • Decanomial 1: 2, 3, 4, 5, 6, 7, 8, 9, 10
  • Decanomial 2: 6, 8, 10, 12, 14, 16, 18, 20
  • Decanomial 3: 12, 15, 18, 21, 24, 27, 30
  • etc.

Basically it has the numbers for that decanomial, assuming you haven’t used it already in the previous one.  For example, decanomial 1 is 1×1, 1×2, 1×3, etc.  “Decanomial 2” is 2×1, 2×2, 2×3, 2×4, etc.  But since 2×1=2 was already in decanomial 1, and 2×2=4 is in the “Squares 1-10” envelope, you don’t need to include these.

In the write up, you have the child lay out diagonally the squares 1-10 first, then you build Decanomial 1, 2, 3, etc.  You can talk about the multiplication table, to pick up the tickets in random and place them etc.

What We Did

Given how old Thumper is, and my own laziness, I really did not follow the presentation.  I basically showed her my write up and said, we’re making this table, which is a paper representation of the bead layout you did last week.  I showed her how you would mark off squares and rectangles on the graph paper and cut them out; reminded he she needed to write the value of the rectangle on the paper; that she does NOT have to mark and cut in order.  She could very well do 1×1, 1×2, 2×1, 1×3, 3×1, etc.  I know she knows half of her multiplication table so there is no need for order for us as part of learning process.  I also told her she would glue these rectangles on our Ikea roll.

IMG_6195
Work in progress

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十項式排列 Decanomial Layout

Age: 4.75 & 7.75

Presentations:

  • Tables Layout
  • Decanomial Layout: Finding Squares
  • Adjusted Decanomial: Commutative Law
  • Adjusted Decanomial: Tower of Jewels
  • Stacking the cubes

Even though I’ve shown Thumper presentations from squaring & cubing, and squares and cubes, we actually never finished some of the introductory exercises in squaring and cubing.  After reading a blog post about it from What Did We Do All Day, I finally found time to do it with the children on Sunday. What Did We Do All Day has a really great post about the exercises and also the variations within the different albums.  I don’t have the NAMC or Montessori R&D albums, though I do have another really detailed AMS album I used as reference when creating my math album this semester.  Ultimately, since my goal was just to show the children these exercises without accompanying written work, I went with  my album.  Partly because I feel I don’t have a strong grasp of how to implement followup work, but also partly because I just wanted Thumper to have the physical experience without all the written work, which often discourages her from working. IMG_6144

These presentations for me are just exercises that are fun and arouses the children’s interest in the squaring and cubing material, all the while giving them a sensorial experience of what it means to square and to cube.  I took all the related exercises together and we just flowed with it.  The whole thing took about 2-4 hours, with lots of breaks in between.  In the classroom, you would do this with at least two children.  I think 2 is a good number.  Anymore and the layouts get messed up way too easily.

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Introduction to Squares and Square Root – 平方與平方根

Age: 7.5

Presentation: Transformation of a Square, Passing from One Square to Another, Concept and Notation of Square Roots, Extracting a square root for numbers less than 225.

After our introduction presentation about a month ago to squaring and cubing, I gave some more to Thumper on Monday, as part of my studying for my oral exam.  Maybe a terrible reason to give presentations, but, I’m discovering, we’re actually at exactly the right age for learning how to make squares.  She really loved the material and learning.  There were many, “aha!  I know why” moments.  I wish everyone could learn their squares this early!

IMG_6024
Finding the 2 squares in the big square

Transformation of a Square is the first presentation.  Here is where you hold up a 10 square, and using rubber hands, divide it up so that it has 2 perfect squares and 2 rectangles.  For our example, we divided it into (4+6)2.  Then you talk about what you just did.  This 10 got split up into a 4 and a 6 and 2(4×6) rectangles.  All this gives you: 10 = 4 + 2(4×6) + 6.

IMG_6028
Had to give a bored Astroboy a task to do.

Thumper tried it herself with other squares.  She has not quite figure out that the equation uses only 2 numbers.  So you can predict what the equation is even without looking at the concrete square.  But I was so good, I did not tell her.  Of course, being an elementary child, she wanted to try this “division on a 1-square and 2-square, which doesn’t play as nicely.  After playing with the squares, we wrote down the equation on our whiteboard and made some mental math calculations to verify that it all adds up.  The next exercise in the presentation had her drawing her own squares.  She drew a 17×17 square.  But was unable to write out the equation correctly.

IMG_6031
Passing from 5-square to 6-square

We then moved on to Passing from One Square to Another, which she immediately wanted to do when she was playing with the squares during the first presentation.  Because the first thing she did when trying to see what squares fit into the big 10 was to try to use the other bead squares to reconstruct.  In this presentation, you first try to go from a 4 to a 5, write the equation out.  Then you go from 4 to a 9.  Because she had learned how to transform a square, she knew how to write the equation out.  9 = 4 + 2(4+5) + 52.

IMG_6036
Passing from 5 square to 9 square

Technically I really shouldn’t have continued on with square root.  But I wanted to practice.  So I just gave her a short intro with the Concept and Notation of Square Roots and Extracting a square root for numbers less than 225, since those just use her knowledge of what a square is.

In these two presentations, you basically learn what a square root is, and then you try to figure out the square root of a number by using pegs.  See how many perfect squares you can build before you run out of pegs.  For example, we started with trying to find the square root of 14.  You do it by first building 1 square, then building it into a 2 square, then 3 square.  You have to stop after 3 square, with 5 as a remainder.

I stopped after that because she needs more practice, especially with her not being too familiar with multiplication yet.  And the next steps starts getting into algebraic expressions, using and instead of numbers.  I think that’s a bit too early for her.

But isn’t the material and concept fun and easy to learn?  I never thought about squares and square roots this way, but it makes so much sense when you are playing with concrete materials.

I find the children get math in Chinese because of the way we count.  BUT, they get confused by the place values because it’s not the same as in English.  So we will continue to teach math in Chinese, with English thrown in when I introduce nomenclature.

Chinese Nomenclature

Here are some words you will need to know to teach squaring and cubing in Chinese.

  • square – 平方
  • cube – 立方
  • square root –  平方根
  • cube root – 立方根
  • radical – 根號
  • radicand – 開方數

Binomial Cube 二項式正方體

binomial cube1Earlier this week, Astroboy worked on the binomial cube.

I’d tried to present this material to him before but he wasn’t interested.  This time, it was just laying around on the mat because I had to take pictures for my class.  Now he was interested.  I feel like evidence is building that often kids are more interested in materials when it’s just laying around, enticing them, rather than when I choose a material to present to him.  Not sure what this translates to when it comes to the classroom.  Do I now need to really focus on making all materials enticing? (which is really hard when it comes to math, most of the materials are always covered!)

Binomial in Chinese

Since I’m focusing somewhat on Chinese, I think I’m going to research all Chinese names related to the materials.  Bionomial cube is called 二項式正方體 in Chinese.  Cube is 立方體.  Another word for cube is 方塊.  I think the first one is the math term and second one daily use.  Other vocabulary you may have to use is 正方體 (square cube), 長方體 (rectangular cube).  Though in our english presentation, we call it prism.  I’m guessing (with only a little googling) it’s because we consider prisms as something that reflect light in Chinese?

The binomial cube in a primary classroom?

So what’s the binomial cube?  It basically shows you the equation (1+2)^2 as a concrete material.  In the primary classroom it’s part of the Sensorial Curriculum, things that cultivate your senses; in this case your visual sense.  It’s NOT in the math curriculum.

But that is totally the brilliance of Montessori.  The kids are going to see this again in Elementary when they learn squaring and cubing in earnest in 3rd-5th grade.  And if they have attended a Montessori preschool, they would have been familiar with the material, in how to handle and construct them.  And the kids don’t really know any different.  It’s just blocks to them that you have to construct a certain way in order for it to fit back in the box.  It’s a puzzle.

The cubes are color coded as a control of error: all red, all blue, blue-black, red-black.  When your construct it, the same color should kiss each other.  In Elementary, the red is labeled ‘a’, and the blue ‘b’.  So a red cube is a^3, a red-black rectangle is a^2b.  It all works.

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