Infinity

An acquaintance recommended a site full of "math in daily life" videos.  Lately, math around here has become a little dull.  SpiderGirl is practising multiplication tables; BatBoy hasn't had anything new for awhile; so the inspiration was welcome.

"The Infinite Life of Pi" caught BatBoy's imagination.

He calls these his "infinite drawings."  He made one that looked just like a visual representation of Fibonacci's sequence too.  (It was a series of adjacent squares.)  From there, we went on to explore ways infinity shows up.  His drawings reminded me of fractals, so I found some videos to show him.

And of course, the Ted-Ed had a video on fractals too.

Finally, we looked at what happens when we allow long division to go into decimals instead writing a remainder or changing to a mixed fraction.

Operations in Fractions

BatBoy understands fractions in the concrete sense, as equal parts of a whole.  We haven't really touched on concepts like finding common denominators or multiplying fractions.  Instead, he has spent time and energy really getting to know a few basic fractions and how they "work together."  As he has learned to add and subtract, and then multiply and divide, he has been able to apply these operations to these basic fractions (halves, thirds, quarters, eighths, and occassionally, sixths).  Using his visual understanding, he has solved problems involving adding or subtracting fractions of the same "family" (halves, quarters, and eighths; or thirds and sixths) including their mixed numeral relatives.  He is able to find answers with negative values, such as the solution to 1/8 - 1/4.  Today, he was playing with the sequence of dividing numbers by two.  Beginning with 8, he divided by 2, again and again, until he got to 1/16 and couldn't (his words) "multiply 16 by 2" without paper.

SpiderGirl plays with concepts less and so, seems less drawn to mathematics than BatBoy.  But given problems, she solves them quickly.  She gets mental blocks when she is anxious.  If she thinks that there is a "right" way to solve a problem (a way that she is less than confident with), or that I expect a particular answer, solved in a particular way, she freezes and claims she has no idea.  But, if the problem is presented in a low pressure environment, she can shine brightly indeed.  SpiderGirl understands fractions visually.  Even when fractions are not of the same "family," she can puzzle it out using manipulatives.  She used manipulatives to figure out to find common denominators in order to add or subtract.  When she gets more comfortable with multiplication, factors, and multiples, I have every confidence she will gain a more methodical way of finding common denominators.   When manipulatives are not available, SpiderGirl is also to present reasonable guesses to problems involving fractions, decimals (to hundredths), and percents.  And she is learning that there is value in her ability to estimate.

When It Rains, It Pours.

Have I mentioned that BatBoy likes math?  Yep, I think he does.  He likes math games, he likes manipulatives, and he loves worksheets.  What?  Who loves worksheets??  Seriously, this boy drags them around like a security blanket.  He brings them everywhere we go.  Whenever he can spare a minute, whether he's waiting for his sister to finish her class or riding in the car or hanging out at home, he works on his worksheets.  He even drags the worksheets to bed and refuses to sleep until he is done what he is working on.  Between his obsession and his willingness to brush off and learn from errors, BatBoy is gaining arithmetic skills at a rapid rate.

What Have We Learned This Year?

As I complete weekly reports throughout the year, I ocassionally purposely slip in a little math, just for the record.  But really, this year, I had been feeling that we hadn't done a ton of math.  SpiderGirl never mastered the multiplication tables, worked much with area and volume, or advanced in fractions, all things that had been on her learning plan.  BatBoy has been exploring many mathematical concepts, but still, not much to go into a report.  So I thought.  When I began working on their final reports for our Distributed Learning school, I realized just how much they have learned this year.

Levels of Infinity

One of the things I like about Life of Fred is that the exercises leave a lot of room for control by the user.  Not only can we skip questions; we can take a question and travel down that line of thought as far as we would like to go.


In Butterflies, Dr. Schmidt tells us that the set of whole numbers is infinite.  That was easily reworded to ask (since we're in the "Your Turn to Play" section anyway), "How many whole numbers are there?"  SpiderGirl picked a big number.  Then I added 1.  BatBoy jumped up and responded with an even bigger number.  So I added 1.  BatBoy threw out a VERY big number.  Before I could respond (and I could see she had been thinking while BatBoy was jumping) SpiderGirl blurted out, "They go on forever!"  

Avocado's Numbers

The sample booklet of my workbook, Avocado's Numbers, arrived in the mail yesterday.  I was so excited, and I think my excitement must have rubbed off on SpiderGirl and BatBoy.  They spent an hour or so go through the first few pages.

The Little Things

Most of what we do when it comes to learning math is not blog material. Not every task contains an 'ah-ha!' moment; not every activity is new or surprising. This post lays out some of the little things we do to support mathematical thinking and learning, and some things that we don't do yet, but are in the works.

Jumping Number Line

A few weeks ago, a good friend of mine introduced me to an iPad/iPhone app called Motion Math Zoom.  This game is a highly visual, intuitive exploration of the number line.  With help from cute animal pictures, the player gets a good sense of scale when looking at pieces of the line, marked off by ones, tens, hundreds, thousands, thousandths, hundredths, and tenths.  The scale can be changed with a simple finger pinch.

I was impressed with this game, and a little sad that it didn't include fractions in fraction form. So one day, needing the kids to have something to do while I cooked dinner, I used masking tape to mark off a number line along the tiles on our kitchen/dining room floor.

Fractions Manipulatives

Although I have always used paper cut-outs for exploring fractions, there is something to be said for having something that is a bit more durable in a manipulative.  One of my friends suggested cutting out foam sheets from a dollar store.  I was very happy with the results.

Integers Before Subtraction?

The learning consultant to whom I report SpiderGirl's learning activities seemed rather impressed that she was learning about integers in Kindergarten.  It's understandable, I suppose, given that integers usually aren't broached until middle school here.  After all, I haven't told anyone else, lest they think I am "pushing" her.

I offered the concept upon request for something new in math to explore.  We had already introduced simple fractions.  This seemed to be the next logical concept to introduce.  Yet, SpiderGirl hadn't yet mastered subtraction or time or grouping.  What would make me think of such a thing?

Was there method to my madness?