### Tag: skip counting

Age: 6.5

Fall semester our Gap Year, Astroboy expressed an interest in multiplication.  When we came home, I dug out my multiplication card game out of my closet and we played games for a few months.

I tell people this is how Astroboy learned his multiplication.    Though that is not quite true, but that’s the next post.  However, I think this was a huge part of how he memorized multiplication table painlessly.

The game is called Speed! Multiplication.  It’s a skip counting game.  I love the cards because it is almost like a Montessori material with its bead chains printed on each hard.  If it were made in the same Montessori colors, it would be perfect.

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.

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。

Wow, I just looked over all the pics I took over the last month and we’ve been doing more work than even I remember.   These pics were taken mid July to first week of August.  I’m the most happy not because of the variety of work we did manage to do, but because the kids actually will spend most of the work period being really concentrated on their work.  Especially Astroboy.  By the end of last year, he could not sit still in the classroom.  Now, with zhuyin class 3 times a week, playdates the other 2 times, and maybe some work periods on the weekends, we’re actually not doing that many 3 hour work periods.  And yet they manage to have really good ones when we do.

What I Learned

One thing I learned from the tutoring class is how I need to really sit with the children when they work.  That and the 3 months break is helping them spend 2-3 hours actually working instead of running around.  I was going to say it’s also due to them being used to the work period routine, but given how the last week they are starting to resist work (because we have not been playing as much) I don’t think that’s the reason.

The other thing I’m seeing is that the children’s limit are about 1 to 1.5 hours of concentration time.  They naturally want a break after that.  This matches what I’ve observed of my own concentration time.  And lastly, I noticed that sometimes the kids do their best work at their 7pm homework time, after a day of playing outside.

Age: 7.5

Presentations: Multiples: Concept and Language of Multiples, Multiples: Common Multiples, Commutative Law of Multiplication: A Number x a Number, Commutative Law of Multiplication: A Sum x a Number

Last week, I did 3-4 math presentations related to multiplication.  I believe this is called 現買現賣 in Chinese.  I basically practiced on Thumper what I’d learned in class.  Partly because my teacher said these presentations should be done as soon as possible in Elementary, maybe the first few weeks/months of school?

I skipped the Wooden Hierarchical Material because I just don’t have time to make the material and while Thumper needs a review and could really use the visual, she’s been exposed to the concept already.  Instead, we did the following presentations:

• Multiples: Concept and Language of Multiples
• Multiples: Common Multiples
• Commutative Law of Multiplication: A Number x a Number
• Commutative Law of Multiplication: A Sum x a Number

These four are just part of two different write-ups.  But it seems that these are the most basic ones that can be grouped together because they require no writing from the child whatsoever.  After what my teacher said about working with the math manipulatives without writing, I’m now not as confused about what I’m supposed to be doing with these presentations.  As I dive into the materials, I can see that these same concepts in multiplication are reviewed over and over again, each time getting more abstract, so I don’t need to stay with work with one material for too long if it seems like Thumper gets it.  I would know she doesn’t get it when she’s stuck at understanding how to use a material.  For this reason, I did not “make” her to take out the beads again and try it herself at a later date after she said she wasn’t interested.

Multiples

The Language of Multiples presentations are about defining what a multiple is.  Like, 2,4,6,8. You lay out the short bead chains, label them, skip count for the experience and then give a definition.  This lesson took Thumper all of 5 minutes.  Astroboy was the one who actually counted for her and laid the number labels down on the rug, since he likes to count.

I laid out the short bead chains, have the child label them, and for each number say something like, “有幾個5在10裏面？（How many 5s are in 10.  剛剛好有兩個五在十裏面，而且沒有剩下。（There are exactly 2 5s in 10 with nothing left over）…..二十五是五的倍數 (25 is a multiple of 5)”  After repeating the “nothing left over a few times” (沒有剩下）, Thumper suddenly interjected and said, “Oh! What about the other numbers, do they have something left over?”  And she proceeded to try it out by skip counting numbers 4, 3, 2, 1 using the bead chain of 5 while I continued working with Astroboy in counting and labeling his bead chain of 6.  Thumper determined that for all numbers except 1, 25 is not their multiple.  I really liked this aha moment for her and I think is one of the strong arguments for using manipulatives.

I will make sure to say “nothing left over” as well taking about multiples in other lessons now.  I do want to note how it does not roll off my tongue translating the album and saying it in Chinese.  It is obviously not a daily conversation I hold with people.

In the Common Multiples presentation you lay out a bunch of bead bars.  The kids find for themselves multiples that are common to each number.  For example, in the photo, she is laying out the multiples of 2 and 3 in a multiplication table format, then laying out the answer using your decanomial bead bars (bead bars #1-#10), then looking for all the common ones they have. (6, 12, etc).  I’m finding that Thumper still has a liking to moving beads around.  She got the concept pretty quick since she knows some of her multiplication tables.

Sorry I don’t have photos for the rest.  I will show you my album pics instead.

Commutative Laws of Multiplication

The Commutative Laws ones shows multiplication rule that it doesn’t matter what order you put the numbers in:

• a x b = b x a
• (a + b) x c = c x (a + b)

In this exercise, you lay out the multiplicand (被乘數) in bead bars horizontally, the multiplier (乘數）using a gray card, then you calculate, exchanging and laying the product out vertically.  You do the same thing in reverse to show that the answers are the same.  And my favorite part is the teacher saying, “I wonder if this will work with other numbers?”  I like the invitation to exploration rather than telling the child about the property and that’s it.

Thumper really lingered on the last presentation (A Sum x a Number).  We must have spent more than an hour on this.  She just kept pulling out numbers to try even though she kind of understood the concept already after the presentation.  Similar to the previous one, you’re using bead bars for the multiplier and gray cards for the multiplicand.  And just showing that the products are the same.  (a + b) x c = c x (a + b)

Even when Astroboy and I went upstairs to have lunch she was still figuring them out.  I loved that she found the topic interesting.  I couldn’t help myself (I know I know) and told her that there is a secret in the work.  I wanted her to figure out that you can add the two numbers together in order to get the sum and then multiply them by the multiplier.

It was really neat to watch how her brain is working out new knowledge.  After the first presentation she knew that the answers are the same.  But she still wanted to figure out the answer each time for each side.  I’m not sure why that is.  She got really busy making her own multiplication tables, using the Chinese version of Nienhuis-like multiplication table sheets I made for her 3 months ago.  It was so she can look up the answer instead of calculating in her head each time.  I didn’t ask her to do that.  And in hindsight I could have pointed her to the multiplication chart, which is a Montessori material.  But I just let her explore as she wanted.

She did figure out the secret to the “presentation”; that you can add the numbers together first.  Though this week I found out that you don’t really want them to figure this out because your’e teaching commutative law.  oops!)

If I go with what my teacher said, I would probably just ask her next week during our planning session to see if she wants to try working them again without me.  We are obviously a bit in this presentation, but it was still fun for Thumper.  I think you can start this at 6+, depending on what the child knows.

All in all a really good week on the math front.  Really glad I decided to splurge and take that math class.

Multiples in Chinese

Here are the vocabulary I needed to look up in order to present.  Chinese is much easier to understand new math terms because I don’t have to know Latin.

• Multiples – 倍數
• Common Multiples – 公倍數
• Multiplier – 乘數
• Multiplicand – 被乘數 (literally means the number being multiplied)

Finally splurged last week and purchased the bead cabinet.  I forgot to buy the arrows so now I’m busy making them.  This required some research into what’s included and it’s actually quite confusing.  So I thought I’ll write it down here for reference.

The bead cabinet contains the following.

1. Short Bead chains has square chains.  So 1, 2/2, 3/3/3, 4/4/4/4, etc.  The golden Bead Chain of 100 is also sold separately.
2. Bead squares  are square representations of the short bead chains #1-#10.   #10 is basically hundred square.
3. Long bead chains has are cube chains.  So 1, 2/2/2/2, 3/3/3/3/3/3/3/3/3, etc.  The golden Bead Chain of 1000 is also sold separately.
4. Bead cubes are cube representations of the long bead chains #1-#10.  #10 is basically thousand cube.

The reason I got confused was that I wanted to make the arrows myself.  I couldn’t bring myself to spent \$45 (including shipping) to buy the cheapest set from Kid Advance.  But in doing research, I found that most free printables don’t do it up to the standards of Nuinhuis.  The arrows have different widths to help the child.  The Nienhuis and Allison’s Montessori ones have cursive numerals printed.

In the Nienhuis catalog description it says:

• 1/4″ for units
• 1/2″ for multiples
• 3/4″ for squares
• 1″ wide for cubes

What are multiples???  Since I haven’t actually used the material, I had no idea.  Good thing Montessori Materials actually has printables.  I wish people make their numbers in cursive.  It’s very important to me because Astroboy constantly tells me that is not how 4 is written when I show it to him the cursive way.  It’s the little details like this that makes Nienhuis so expensive.

Combined with repeated staring of catalog pics, here is what I made: