Saturday, October 20, 2012

Fraction Charts

My husband thought I was nuts.  For two nights in a row, as we sat on the couch watching Netflix episodes of the Walking Dead, I was tracing itty bitty fraction circle segments onto colored paper.  For the next two nights I watched while cutting out itty bitty fraction circles from the paper.  My husband thought I'd finally gone too far.  This was crazier than the February phonogram dictionary episode.  He said, "I can't wait to see what you are going to do with those.  It had better be good."  I told him he should be quiet or I was going to hand him a scissors.

I want to toot Jessica's horn over at Keys of the Universe (do I sound midwestern or what?).  Her albums are the only albums I've seen that actually show you what these look like rather than just refer to them.  This is what made it possible to actually make these.

They probably would look best on white paper.  However, I wanted to follow Jessica's suggestion to use the same colors as our bead materials to facilitate reading the charts faster.  The manila paper allows for the visibility of the white sevenths.  I cut one inch of the length of 12x18 paper so that the paper would fit in our portfolios.  It was really tight, both lengthwise and widthwise, to get all of the information on the paper.

charts 1 and 2, fraction circles shown in bead bar color scheme

charts 3 and 4, equivalences

chart five, more equivalences

charts 7 (top) and 6 (bottom), addition, like-denominators

chart 10 (left) and 9 (right), subtraction with like-denominators, addition with unlike-denominators

chart 11, subtraction, unlike-denominators

chart 12 (right), multiplication by whole number
chart 15 (left) multiplication of a fraction by another fraction

chart 13 (top) multiplication by a whole number
chart 14 (bottom) multiplication of a fraction by a fraction

charts 16 and 17, division by a whole number

charts 18 and 19, division of a fraction by a fraction and group division

1. Job Well Done! I don't have Jessica's Album so I never knew what they looked like and could never find a source to show me. Thank you so much! I knew I was holding out on new presentations for Fractions for a good reason:) This post is certain to help me with Ken and DJ. Thank you again for sharing.

2. Wow!!@!!! Wow!!!, Thanks sooo much for this post, I was holding my princess on the fractions families... Love this idea!!! Thanks again!!

3. Amazing job! Thanks for sharing.

4. I'm so thankful you shared these!

5. Great Job! You are an amazingly committed mummy...and you are a great teacher too!

6. Beautiful! I can just imagine the looks from your husband ;)

I like the manilla paper idea myself, actually. For white on white, I outlined with a sharp pencil; the manilla seems to help everything stand out better though!

7. Great job ... so helpful for any Montessori elementary homeschoolers! Thanks for linking up with Montessori Monday. I featured your post at the Living Montessori Now Facebook page and pinned it to my Montessori Elementary Activities and Ideas Board at http://pinterest.com/debchitwood/montessori-elementary-activities-and-ideas/

8. WOW WOW WOW! Thanks so very much for sharing these! I really had no real idea of what I needed to have on these charts! I wish there was an easier way to make them! I am pretty sure my hubbie is going to hate this too! ;) Do you have a list of the rules you used to put on the function charts? Thanks again! It looks like I am going to be busy again! :)

9. Stephanie,

The idea is to provide a chart for each of the ideas presented in the fractions album page(s) - the essential concept at hand. So you can definitely make all of them, but it is also ok to just make a few as a concept needs reinforcing but others don't need the extra visual.

I made my second set with my son working on his own fraction work - it was during a time that I really needed to "not" be available to him every second but also reinforce that this is school/work time. And there were 2-3 I did not make (can't remember which ones). He ended up making his own samples of those ones :)

10. Wow! So impressive! You did an amazing job, thank you so much for sharing!

11. Thank you everyone! I really needed some pats on the back after all that!

12. Is there an easier way to make these? My mom e-mailed me when she saw this post and said I should have had her make all of the segments on her Cricut machine. I don't know how those machines work, but that might be a way to go.

13. Stephanie,

Did you click on the individual pictures? If you do, they should blow up to a readable size. Let me know if it works.

14. One of my elementary trainers highly touted a type of scissors that when you cut a shape, it takes a little extra; this allows for the whole fraction to be "traced" in light pencil where you want your pieces to go, and place your cut pieces in perfect place with just a little gap (because of the extra bit taken off).

But that doesn't cover all the pieces used. So I just traced and cut mine with regular scissors ;)

Maybe when you have the elementary material section up, we can swap more ideas ;) no pressure!

15. I just made the equivalence charts using Word and Excel and I had a question while making those. I see that you made only equivalences with one piece of a fraction, 1/2, 1/3, etc. Do we have to show all possibilities like 2/3 = 4/6, etc, with more than one piece. I see that there are presentations in my album for working with the child on more than one fraction piece, but do we have to make the charts too? May be this is the time when the Fraction Research Charts (like the one in Nienhus) will be useful so the child can see how the other combinations are reduced.

16. If you are going to have the fraction charts, it is best to have a chart that shows each concept, but doesn't necessarily show "everything" so there is still room for exploration.

And... not all the children need all the charts. I DO think that at home, it is nice to make them during the children's work time, at a time when they need the extra fraction practice but are reticent to repetition for whatever reason. Then they get some of that "classroom experience" of seeing someone else's work and being inspired by it.

Just my own two cents ;)

17. Mommy to the Princesses,

My impression of the fraction charts is that it is ONE visual reminder of a "rule." It is not an exhaustive display of fractions. My book of fraction charts is like a little "dictionary" of fraction operations. You look up a word in a dictionary, read it's meaning, and read ONE sentence with the word in it. They don't list every sentence in the world with the word in it. In my mind they are for increasing the child's independence. When they return to an operation after some time has passed, they don't have to come to the teacher to ask how to do it if they have forgotten...they can SEE how it's done by looking at an example laid out with paper fraction pieces.

What albums are you using? They should be giving you a clue as to how to use your fraction charts. I have FOUR fractions albums right now. And ALL of them basically say the same thing...give the presentation, practice a few problems with the child, SHOW them the fraction chart so they know it exists for their reference, have them complete a packet of loose equations on their own.

18. Mommy to the Princesses,

Oops, I wanted to speak to what you said about the fraction research charts. I was thinking of using them as a control chart to check reductions when they are doing equations in their fraction notebook...just like the control chart is used with the addition boards. I don't think you really need them during the actual equivalence presentations. They are like worksheets and it is probably better for the child to do the work with the material.

19. Thanks MBT and Jessica!

I see your point, the charts are reference and not an exhaustive list. Regarding the research sheets, I meant those to be used as a control later when the child is working out the problems and not during presentation. Agreed! At present I am looking at Cultivating Dharma and Barb Jens for Math. I don't have Jessica's yet, no budget right now. Thanks again for taking the time to help me out.

20. And the presentation in those albums don't talk about using the charts during presentation, but do talk about doing equivalences using more than one piece eventually. My original question rose out of that!

21. Cultivating Dharma does refer to charts occasionally. It is usually in the extensions section and says, "The circle charts can be left in the room after the lesson as a reference." He isn't very invested in the charts and has the child making booklets and equivalence charts instead.

The equivalences using more than one piece is simply the last step of the equivalence work and doesn't need it's own chart.

The Barb Jens album puts the charts into the presentation steps. For example, the presentations for Multiplication of fractions by a whole number step nine says "show chart 14." Those offhand statements are referring to the fraction charts.

22. Hey, I'm in this right now. Tedious to put together. I seriously don't understand how you didn't go insane.
On the flip side of tedious, or maybe the exact same side, T has me doing homemade pin maps!!! How DID you not go insane making yours? :)

1. Abbie - when my son was ready for some of the pin maps, I made him make them ;) I'm that mean ;)

He printed out some outline maps we had from Geography Matters (Uncle Josh's or something like that - we have the version that has blank maps, with dots for cities, etc for the child to label) --- mounted them on corkboard (with a push pin in each corner). He wrote out all the names he needed to learn on strips of paper, using regular clear tape to fold over the label and wrap around the pin (I'm not describing that well - I hope it makes sense). He bought some of his own sewing pins after he took off with mine.

Of late, I bought the Nienhuis flags (blank ones), but I like that he did his own work. He seems to have learned a lot more from it :)

2. Jessica, you know, I was thinking just the same thing as you these past week T and I got into this project together. I noted a tiny bit about it on my blog, but this project really came from T. HE was the one who wanted to make them, use them, whatever. He is doing most of the research, or well started. He started writing down all the cities, mountains, islands, and water forms, and states he wants to study. I wasn't sure about the technical part of the project and how much of that he could manage. Plus, I wanted this to ultimately be a classroom material, i.e. durable enough, that others could use it too. So I did the map cropping, (I got a digital wall sized relief map) printing, dotting, outlining, and tiny-flag making. We are doing something a little silly and that is putting colored nail polish on the heads of our pins so that they are the right color. T wanted to do that, but I was pretty sure that the nail polish was going to go everywhere. Maybe that would be a job for S since she paints her own nails. :)

I completely understand what you are talking about when the child is involved with the making process. They do get so much out if it all. Planning, perspective, and fine motor are all there. And the pride they get from viewing their finished product is awesome!

And I should have just bought the blank flags.:) T was too eager to get moving.

3. Am I missing something? Aren't those flags around a dollar apiece?

4. Scratch that, aren't those flags around \$1.65 apiece?

23. Packs of 10 are \$6.20 (so 62 cents each - plus shipping); but that is why I don't purchase the pins for our home pin maps. I purchased the Nienhuis ones for use with our Catechesis of the Good Shepherd atrium pin maps - limited number needed ;)

24. How many charts do you have? My album says 21 but I only see 19 here?

1. There are 21 pictured above. Count again.

25. Oops. I didn't realize the 10 times I counted that the last picture was printed twice. You are (of course) right there are 19 charts. I double-checked my albums and there are only 19 charts in my albums. I don't know which albums you have, but the KotU albums are the only ones that bother to give images of the charts. The other albums are grossly negligent in this aspect. MRD refers to fraction charts by number with no description as if you could buy them, but best I can tell they are NOT included with their expensive fractions curriculum. Also, fractions are divided across two MRD albums. The first album lists 17 charts in Appendix a. Album one only covers up through like-denominator work. The second album does not appear to have a list of charts at all.

26. so the more I think about it the more I like color coding the fraction pieces. But if it's so much easier, why don't they just color code the metal fraction pieces?

1. It is about isolation of concept: the metal fraction circles are all the same color to tell the child that a "unit" (the unit beads on teh bead cabinet are red) has been split into equal pieces.

By the time they get to working with fractions that these charts are handy for, they totally get that concept. Now we are looking at operations, and it is "easier" to see the numbers (fractions) being used when color-coded.

27. Okay, So I finally got my hands on some photocopies of these fraction charts. And nowhere did I see a chart for 1/4 * 2 = 2/4 = 1/2 with the strange equation that had me arguing w/ my husband for 2 hours and confounded my ideas of fractions and math.

However, I did see another chart which made so much sense and I think actually applies to this chart. It has no illustrations, just notes:

If we multiply or divide the numerator and denominator of a fraction by the same number, the value remains the same.

For example, 2/4 = 1/2
If the numerator and denominator of the first fraction are both divided by 2, we obtain second.....

For example: 1/2 = 2/4
If the numerator and denominator of the first fraction are both multiplied by 2, we obtain the second 1/2 x 2/2 = 2/4.

Huzzah! this makes so much more sense to me. And I think illustrates the point the other chart was making? Because I have never seen that second rule of the original chart used anywhere in teaching fraction operation and I didn't understand why we would teach kids that.

28. Is that the symbol for equivalence? I thought it was three lines?

1. It is the symbol for congruence, a type of equivalence.