Adding Fractions - Different Denominators

Good ole fractions!  We finally dusted these bad boys off again.  Kui is pretty behind on fractions so I am making quite the effort to get her back into the swing of things.  I think I was initially scared off from these when I was looking in the album.  Not quite sure why.  Maybe it is the endless amount of charts that need to be made for the fraction work.  Who knows!  Regardless, I am back at it making more and more charts to be used for this work.  Kui was pretty excited to get a presentation today and worked very well.  We have been working upstairs this week because Bubs has been sick and quarantined upstairs.  I needed to stay close to hear him.  It worked out though.  We had plenty of space for all of the fraction pieces. 

 

Kui finished up the presentations for adding fractions with the same denominators awhile ago so we just took a minute to refresh the concept.  She picked up on the new concept pretty quick, however, we do need to dive back into the Least Common Multiple work so that she can be a little quicker in finding the common denominator.  We will do that tomorrow.

 

Here Kui is working with the first problem in the album.  I believe it was something simple like     1/2 + 1/4 =.  She needed to get the denominator in the same "family" like all halves or all fourths.  She looked at the fraction pieces and said, "How many fourths will fit in a half."  This should come very easy to the child at this point and she quickly found out that 2 fourths fits into a half.  So her equation changed to 2/4 + 1/4 = 3/4.  She could easily push all of the fourths together now since they were from the same family.  She worked on many simple examples like this today.

 

She also has this box of extra fraction pieces if she needs them.  Sometimes the fraction insets are not enough.  You can buy these plastic pieces here.  I believe I got mine from Montessori Outlet, but for some reason that link was not working.  They are a little pricey, but well worth it.  I never did end up finishing mine that I started cutting 3 years ago.  I just broke down and bought a set.  They are going to be more precise than my cutting skills anyway!

Once we went through a few examples and she got the hang of it, I introduced one of the charts to her.  This is a great visual to show how the fractions are changed to match the same family.  1/4 gets changed into 2/8.  It is very clear by the color coding on the chart.  Then you can add eighths with eighths.

 

She was very inspired by my beautiful drawing and coloring skills (NOT!!) and wanted to make her own chart.  Here it is below. 

 

It is actually a good review and practice for the child to do this.  It allows them to color which is Kui's all time favorite thing to do.

Here is another chart showing 3 addends and how we exchange along the way to see them all in the same family by the last row.  (Please remember that you can click on the picture to enlarge it.)  I realized half way through that all of my fraction pieces would not fit and I had to tape on a bottom piece.  Doesn't look the greatest, but gets the concept across.  I love this one because it also shows how the fraction can end up as 9/9 or 1.  This reminds the child to simplify the fraction when they are done. 

 



Here is Kui's attempt.  Can you tell her interest waned by the end?  It was approaching recess and her brothers were already heading out.  It is also a good reminder as to why making the charts is really the job of the teacher and not the child.  It is too labor intensive for the elementary child.  They have moved past this developmentally.  However, I did let her do it since I was trying to follow the child.

 

I also started making her some equations that she can do on her own for extra practice for each operation.  I still need to cut them out, but here is a picture for your reference.  Tomorrow we will probably touch on a refresher for Least Common Multiple before we move on to subtracting different denominators.  Plus, this teacher needs to get busy making the rest of the charts for this group!  It was a good reminder to check the state of the colored pencils every so often as well.  This child clearly needs a new set!  I was using a red pencil today that was the size of my pinkie.  Makes coloring nicely quite the challenge!

Fraction Match-up Game

First off, I apologize for not posting lately.  We have had (and still have) a long term visitor from overseas.  My husband's sister is visiting for a month or so and we are all transitioning with the new body in the classroom!  It is like VIP day everyday for "Auntie".  (VIP day stands for Very Important Person).  The Montessori school that the children attended did this once a year and the child was able to invite anyone that fit into that category, so it wasn't just limited to parents.  It could be grandparents, aunts, cousins, etc.  We have implemented that in our classroom a few times a year so that "daddy" can be a VIP.  It is hard for him to understand all the lingo I use until he sees it in action.  I think it also helps him to see where all of that money is going for materials!  The budget is always being pushed to the edge with materials.  There is always something to make or buy to enhance the children's experience in the classroom.

 

Ok now for the Fraction Match-up Game:

 



 

 

I have really been slacking with the idea of making so many fraction materials.  I started making an extra set of fractions up through tenths about a year and a half ago and have still not finished.  My hubby was cutting them out two Christmas breaks ago and then life happened and we moved and I couldn't tell you where they are.  So...needless to say I broke down and spent the money on these.  This sparked me to then get some fraction presentations together.  I took the information that My Boys Teacher had on her blog about her fraction cabinet.  She has the dream fraction cabinet that I think I will never have.  I can't find a storage cabinet like hers.  I am using envelopes right now for my presentations and that seems to work fine for Kui.  Maybe by the time T-Man is ready for them, the fraction section will look like hers.  Ok, enough jealousy for the moment. 

 

I started Kui off with this fun match-up game to just get her excited about fractions again.  It was very simple to print off, color-code, and put together.  She now has a new excitement for fractions and has been willingly (as in choosing them herself) doing presentations for adding fractions.  We are definitely behind since she is still doing same denominators, but at least that part of her planner is not growing dust anymore like her geometry! (That is another story altogether!)

 

The game has four columns: Word Form, Number Form, Fraction of a Group, and Fraction of a Whole.  She then matches each picture up under the corresponding column. 

 

Here is what they look like all matched up.

Then she can turn them over to see if the color codes are the same for each set. 

 

This game has really proven to be something that has worked for her.  I am pretty hard core when it comes to her math and I tend to be a little rigid.  I was hesitant to introduce this to her because I thought it was too easy and she would just keep choosing it.  However, the fact that is was a bit of a challenge, but not too much actually encouraged her and now she does not find fractions so overwhelming.  That teaches me!  Duh....Montessori IS all about following the child right?!

Decanomial - Sensorial and Bead

THE DECANOMIAL......dun dun dun......

 

Coincidentally, I showed Bubs the sensorial decanomial in the same week that Kui and I got to the bead decanomial (Well really we got to it because we finally got the beads in the mail!) 

 

I presented this to Bubs after I had shown him the short bead chains and he had worked with them for awhile.  He is quite aware of the different colors for each number.  For example, red is one, four is yellow, etc.  So when I started showing him this material he immediately pointed to the bead cabinet and said, "Mommy, it looks just like that!"  His shrill little excited voice encouraged me greatly in my homeschooling journey because all of these materials are quite time consuming to make.  Yes, there are quite a few materials that can be bought, but there are also quite a few that need to be made by the teacher (a.k.a. myself).  A great example of my hand made material can be seen in the third picture below.  Anyone see the wavy rectangles?!  Ha  Those got jammed in the laminator and when I tried to reprint them, my ink cartridge ran out and the blue squares looked more like a sky blue.  Needless to say that I wasn't going out to get ink any time soon and every day is precious in the wonderful world of Montessori learning.  So.....I decided to go ahead and present to him even though the materials were somewhat less than perfect.  (I do plan on fixing these, but for this presentation, it had to do.) 

 

I also need to make a storage container for all of these little rectangles.  Not sure where I will find that.....

Those wavy rectangles just add character right?!  This material is truly one of a kind.

 

Now for Kui's bead decanomial.  Wow, I definitely did not allow enough of the morning to complete this one!  I mean how long can it take to lay beads out in rows?  Clearly over an hour.  Ok maybe we had a few distractions like the table being bumped and beads messed up, T-man discovering that he can reach the top corners of the table and grab beads, snack time, etc.  Kui finally completed it though.  She was very proud of herself. 

Working diligently

 

In the picture above, Kui has already switched out her loose squares with the fixes squares on the diagonal.  If you click on the picture it will become larger.  Then you can see them in the top left hand corner and going diagonal to the bottom right hand corner.  Then she switched out the beads using the commutative law of multiplication.  2 x 3 is the same as 3 x 2 for example.  This is what gives you the pattern of the decanomial.

 

Her finished decanomial.  Phew!  It was a lot of work, but she discovered that it was the same thing that Bubs was working on and he thought that was pretty cool.  He looks up to his big sis and the fact that they were doing something so similar was cool to him.

Decimal Fractions - Part 1

 

 

Yay!  We have finally started something official for math this year: DECIMAL FRACTIONS.  I am shocked that I actually have my act together enough for this.  It isn't "easy peasy lemon squeezy" like my daughter goes around saying.  The tricky part to all of this Montessori homeschooling business is that I have to first understand it, figure out how to do it, and then be able to present it to the kiddos in a way that makes sense!  Thanks be to God for You Tube and all of those teachers who like to video themselves doing these presentations.  Just when I think no one in their right mind would video themselves talking about how to read decimal fractions, what do ya know?  There are at least 100,000 videos on the topic.  Go figure! 

We have moved along pretty quickly with the first few initial presentations.  Simply explaining that whole numbers are a unit multiplied by 10 to get 10, 10 x 10 is 100, 100 x 10 is 1,000, and so on.  Then, if you take a unit and divide it by 10, you get a tenth.  You have to almost spit on your child to emphasize the "th" at the end.  (Ok, how many people just tried to say tenth to see how much spit you produce?!)  Then if you divide 10 x 10 you get a hundredth.  A hundredth divided by 10 is a thousandth, and so on. 

 

The picture above is the "pinwheel" shape that is created when you write out the whole numbers and decimals with the cards.  I didn't get a picture of it, but if you stick a pin in the middle of the unit, you are able to spin it around, thus showing that everything revolves around the unit.  (Sorry about the glare in the photo.)

 

 

We jumped right into working with the decimal board which Kui really seems to enjoy.  She is using the piece from her chess set to show that the unit is King.  This reminds her to either multiply or divide depending on which way she is moving away from the unit.  If she is moving to the left, she would multiply to get a whole number.  If she is moving to the right, she would divide to get a decimal fraction.  Side note:  Don't mind the tray of beer bottles in the background.  She was working with those earlier to create different pitches with varying amounts of water.  Yes, #herdaddrinks

 

Here is a simple problem using the decimal squares and then locating the corresponding card.  She has 8 decimal squares in the ten-thousandths column and has found the 8 ten-thousandths card.

 

We did quite a few more examples of the simple problems and then moved into problems that involved exchanging.  For example, if she had 12 millionths, she would exchange 10 of those for 1 hundred thousandth and have 2 millionths remain on the board.  So, it is getting used to the idea of moving to the left to exchange quantities (which is a lower amount if you think of it as a whole number).  So far, so good though.  She enjoys it and keeps pulling it off of the shelf.

 

Stay tuned for adding and subtracting decimal fractions!



Racks and tubes (long division)

 

Kui has been working on racks and tubes quite a bit lately.  She is now doing a two digit divisor pretty well. She doesn't have her multiplication down solid yet, so it slows her down a little bit and she ends up counting the beads.  So, I am finding that she gets discouraged easily when the answer comes up wrong, but it is usually because she counted incorrectly.  Not to worry mon frere!! We have ramped up the checkerboard, flat bead frame, bank game, and multiplication word problems to help get those facts learned.  She's not too thrilled about it, but I think it is a fabulous way to review. 

We have also been working on writing her work on paper correctly.  She tends to get caught up on the subtraction work in between.  There is also the confusion with counting up how many beads she has used up, especially with the ten board.  For those not familiar, the ten board (blue) and the unit board (green) are for the two digit divisor.  The child evenly distributes the beads between the two boards.  Then she adds up the beads to find out how many were "used" up.  Then that number is written on the top line as part of the quotient.  The child can then check their answer by seeing how many beads are left in the individual cups.  The control of error is built in (unless you don't quite subtract properly, then you might get messed up) which was the case with Kui a few times.  We did go back on another day to a one digit divisor to boost her self esteem a little.  She was encouraged by the fact that she COULD get those problems correct. 

 

She is now working on doing the two digit without having any help from her mama (aka teacher).  Hopefully soon we will be working into abstraction and only doing it on paper.  I need to go back to the album pages to see how to correctly present that.  Maybe there isn't a formal way and we will have to wing it!  Stay tuned.

Game 1- Squares and cubes

My daughter is currently working her way through the squares and cubes chapter in the math album.  She has already finished with the notation of squares and cubes presentations and has started the first set of games.  Game 1 has the child fetch a square and a cube for each number 1-10.  Then the child puts the squares in a pyramid shape starting with the 10 square on the bottom working up to the 1 square at the top.  Doing the same for the cubes, but creating a tower instead of a pyramid.

I had practiced this presentation ahead of time to make sure I had all of the squares and cubes memorized to make it at least SEEM like I knew what I was doing.  Clearly I didn't have to use the cube of 9 with accounting (my pre baby life) and definitely not while being a toddler teacher.  But to my daughter, I know "everything" so I couldn't let her down.  Not to mention that my husband doesn't help the cause when he is telling her to ask her teacher when she doesn't know something!

She was thrilled to finally be able to stack the material after so many times of me telling her that was not what we were doing today.  So I skipped the formalities of the presentation and let her get to stacking. Then I asked her one of the great "Wonder" questions.  (We use a lot of those in the atrium to spark a child's imagination).  I asked her, "I wonder what the total of this pyramid is?"  She was excited to get started figuring it out.  Phew!  I thought she might just roll her eyes at me because it actually involved math.  She's clearly not a math fan.

The next step is to write out the squares on paper.  She started with 10 squared and went down to 1 squared.  She wrote answers in a column to be able to add.  She totaled it at the bottom to find the number of beads in the pyramid.

 

Then I asked her the total for the tower.  This was a little trickier and she had to use a side piece of paper to figure out the cubes she didn't know.  Little did she know that this work had some built in addition and long multiplication for her to practice!  Gotta love Montessori.