More math

I didn't think about skip counting as a method of learning math facts until I started homeschooling. Even then I didn't think of it much until it was time to teach Sparkle her math facts. Then I saw so many math programs using skip counting, so I bought into the idea of skip counting. I taught Sparkle skip counting and drilled her in skip counting.

Then I discovered that skip counting is slwo and error prone. It takes too long to skip count up to the higher numbers, and it too error prone because it is too easy to get numbers mixed up. Was the next number 42, 48, or 49? Does 45, 54, or 56 come next? Yuck. I find when I'm skip counting, I'm actually using my memorized multiplication tables in reverse.

So it's no more skip counting for me. Instead I'm going to teach Sparkle tricks and shortcuts for memorizing math facts. My favorite trick is for one of the hardest to remember facts: 56=7x8 (5678).

Division with fractions

This is going to be another of those rambling posts.

Homeschooling has given me a whole new perspective on grammar and elementary math. Before homeschooling I didn't think much about either subject. Now I adore these subjects. Too bad my kids don't share my fascination with either subject.

This morning I had a wonderful math moment.

To back up a bit. ....
I've been learning about math through teaching Sparkle and reading some math blogs. I read how treating multiplication like repeated addition falls apart when multiplying fractions. I've read elsewhere how many students (and many teachers) don't really understand dividing by a fraction. They might be able to do the procedure (don't question why, just invert and multiply), but cannot create an accurate word problem that reflects dividing by a fraction.

Most explanations of division of fractions that I've encountered explain division by a fraction with a lengthy discussion of division as the inverse of multiplication. It makes sense as I read it, but I didn't internalize it well enough to explain it to someone else.

Then a few days ago I was flipping through Sparkle's new math book. Dividing by a fraction is taught less than halfway through the new book. I was amazed. If you use this series of math books at the recommended levels, students would be encountering these problems halfway into the first semester of third grade! Sparkle will be older when she gets to it, but I still think that she will be on the young side for the concept. However, what really impressed me was that the concept was presented in a way that made total sense to me. I can really see a child can understand these concepts after working through these math books.

The problem (1/2) divided by (1/4) is expressed as the question "How may 1/4ths in 1/2?" It's the same way of phrasing (8 divided by 4) as "How many 4s in 8?"

My wonderful math moment was vaguely related.

Another tricky subject is subtraction of negative numbers.
I'm probably going to use incorrect terminology here, but I don't care.

Adding and subtracting are opposites.
A number plus its opposite equals the identity number for addition.
The identify number for addition is zero. (A number plus zero is itself.)
For example, the opposite of 5 is -5.
Subtracting a positive number is the same as adding the negative number.
5 - 5 = 5 + (-5)
The opposites cancel each other out. Swap subtracting with adding, Swap a number with its opposite.
So, subtracting a negative number is the same as adding the positive number because two opposites cancel each other out and everything stays balanced.

Similarly, multiplication and division are opposites.
The identity number for multiplication is one. A number times one is itself.
A number times its opposite equals one.
The opposite of a fraction would be the inverted fraction.
Inverting and multiplying to dividing by a fraction also involves two opposites. The inverted fraction is an opposite and changing dividing to multiplying is also an opposite.


Sorry if you've read this. It probably doesn't make much sense. I'm just typing this to try to capture my beautiful math moment for myself so I'll remember it later.

Poetry and Prose by Sparkle

Here is a poem by Sparkle. The assignment was to write a poem for the Renaissance Festival. She wrote this poem fairly quickly. I guess she felt inspired that day. Her first draft was almost three times as long, but the Renaissance Festival contest only accepts poems up to 20 lines long. So, we cut apart the stanzas and played around with them until we got this arrangement. Then Sparkle did a few more edits - mostly capitalization, punctuation, and a few minor word changes. The first draft was done on one day and the revisions on a second day.

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The Fairies

Flying with the birds,
Sitting on a snail,
Listening to it rain
On the water pail!

Hopping from a leaf,
Dancing on a flower,
Making rose buds bloom
With their special power.

Spinning in a garden
On their little toes!
Has anybody seen them?
Does anybody know?

The Fairies dance and sing.
The Fairies love to fly!
They watch it rain upon the pail
And see the world go by!



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Here is a piece of her prose writing. The assignment was the final essay from the end of The Paragraph Book 1. It took her several weeks and many, many, many drafts before we decided that it was good enough to stop working on it.

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How to Get, Peel, and Eat a Grapefruit

First, see if there are any grapefruit on the counter. If you can’t find one on the counter, go to the store to get a grapefruit. Come home from the store with the fruit. Put the grapefruit on the green kitchen table. You have the fruit.

Next, put your nails on the fruit. Dig your nails into the skin of the fruit. Make sure your nails don’t puncture the flesh of the fruit. Pull down with your nails to start peeling. You will see a hole in the skin. It is the start. Repeat until there is no skin. Rest because your fingers will be very tired! Then, you will see an underskin. It looks fuzzy and it’s whitish. Peel off the underskin. Look for any underskin on the fruit. It tastes bad. If there is still a little bit of underskin on it’s ok. You don’t have to peel all the underskin off. You have your peeled fruit.

Finally, peel off the membrane around the fruit segments. Hold a grapefruit segment up to your mouth. Do this to start eating the fruit. Bite the fruit segment and chew, and swallow. Catch any falling juice, it’s good, and spit any seeds out into a bucket or trash can. Repeat the steps. There are baby seeds too so watch out! You ate your grapefruit.

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To better appreciate the work in the above composition, here are her original frame paragraph and her first expanded draft.

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Original Frame
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How to Peel a Grapefruit

First, get a grapefruit from the counter or store. Next, get your nails in the fruit. You do it to start peeling. Then, pull down with your nails. Make sure you don’t bend your nails. Then, repeat until there is no skin left. Peel of some of the stringy skin too. Finally, eat your grapefruit.

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Original Expanded Draft
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How to Peel a Grapefruit

First, see if there are any grapefruit on the counter. Next, go to the store to get a grapefruit. You do this if you can’t find one on the counter.Then, come home from the store.Don’t forget the grapefruit. Then, put the grapefruit on the table.Put it on the green kitchen table. Finally, you have the fruit.

First, put your nails on the fruit. Next, dig your nails into the skin. Make sure your nails don’t puncture the fruit. Then, pull down with your nails. You do this to start peeling. Then, you will see a hole. It is the start. Finally, rest.

First, reapeat until there is no skin. Next, you will see a underskin. Peel off underskin. Then, look for any look like banana skin on the fruit.It tastes bad.Then, if there is still a little bit of skin on it’s ok. You don’t have to peel all the skin off.Finally, you have your peeled fruit.

First, peel of the clear skin.Next, hold the grapefruit up to your mouth. Do this to start eating.Then, bite the fruit and chew and swallow. Catch any falling juice it’s good and spit any seeds out.Then, reapeat the steps. There are baby seeds too so watch out! Finally you ate your grapefruit.

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Tucked in like Pippi

Our current bedtime read-aloud is Pippi Longstocking. Tonight I read from the chapter where Pippi gets into bed and puts her feet on her pillow so she can wiggle her toes.

Moments later, when the girls crawled into bed, both of them tried to imitate Pippi.

Sparkle stuck her feet on the pillow and lay on top of her sheets. When I burst into laughter at her silliness, she started to wiggle her toes.

Glitter wasn't as clear on the concept. She moved her pillow and stuck it at the foot of her bed. Then she put her feet at the head of the bed. So her pillow was where her feet normally would be, and her feet were where her pillow would normally be. However, her head was still on the pillow. Glitter also managed to fold the top sheet backwards on itself and over her body. She is still short enough that the sheet covered her.

Crazy cute kids. I tucked them in and said good night.

Math, Music, and Siblings

Glitter is almost done with kindergarten math (RightStart Level A). I plan on having her do the last lesson of her book tomorrow. Then I'll move her on to the next book (RightStart Level B), even though she is still in kindergarten.

Sparkle's math is coming along more slowly. She is almost done with the second grade Miquon books. I expect her to finish them by the end of the month. She is also almost done with the first grade word problems book. I expect her to finish it by the end of the semester. Then hopefully she'll be ready to move on to a third grade math textbook. After all, she is in third grade.

I am trying to prepare myself that Glitter may eventually outpace her older sister in math. I doubt that she will surpass Sparkle's ability with rote calculations, but she might catch up conceptually.

Similarly, Glitter has passed her sister in the Suzuki aspects of their piano lessons. Glitter practices piano even when she doesn't have to. As a result, Glitter learned their most recent Suzuki song before Sparkle. (Okay, she got her right-hand sticker first only by a matter of minutes, but still, that's impressive given the age difference, and the fact that Glitter got no formal instruction on the song.)

Glitter has a passion for math and music that Sparkle doesn't have. I hope that doesn't discourage Sparkle. Sparkle's talent for drawing and spelling hasn't frustrated Glitter. However, it's natural for an older sibling to do something better than you. It's something else when the ages are flipped.

Unschooling Writing

I wish that I could unschool writing.
Sparkle comes up with all sorts of writing projects for herself.

She created a mini-book about the plants and animals at the local botanical gardens. Each two page spread has an illustration of the plant or animal and one or two sentences describing it. She begged to stay at the gardens for a few more minutes in the sweltering heat so that she could finish the book.

She found a blank picture study notebooking page that I had given up using with her. She copied a painting and wrote a nice little paragraph about it, making some thoughtful connections with her life. Then she found another page and drew an original picture and wrote another paragraph explaining the picture.

She is in the process of creating another mini-book with monsters from Wander's D&D book. She copied the picture of the monster and some of the text describing it -- in beautiful cursive. Sparkle kept balking when I asked her to use cursive in school work, so I finally let her print everything. Her handwriting for school varies from legible to fairly neat, but hardly every as well formed as in this little booklet. She dislikes copywork and is annoyed when she makes mistakes in dictation for school. She discovered her own mistakes when reviewing her work on this booklet and happily corrected it.

However, when it comes to me teaching Sparkle how to write, it's a struggle. She has a hard time figuring out what to say. Her handwriting is sloppy. She wants to know the bare minimum she has to do to get by. On the other hand, I know that she is having fun with her own writing projects because I have taught her the underlying skills, and I need to continue doing so.