Sidewalk Math: Circle Fractions, Division and Multiples

Okay, so we've walked past these two covers about a million times but today I saw them in a completely different light.

This winter/spring we've been doing a LOT with number multiples and conceptualizing multiplication and division.  Last week my mind moved toward the inevitable: fractions.  Although shivers go down my spine every time I think about fractions I'm still resolved to figure it all out for the sake of my seven year old, if not myself.  It's been sitting in the back of my mind so I guess that's why this cover caught my eye and brought me to a dead stop.

Can you see it?  Fractions and multiples!

And, a little further on, this beauty: an 8-star and some fractions!

What I really want to know is who designs these?  I want that job.  To see some other cool round things we found on a walk last spring, go read my post Channeling Tana Hoban: Juxtaposition Edition.   Now that was one amazing day for circles!

All-in-One? Using Shapes to Explore Number & Algebra Concepts

I'll start right out by saying that I'm pretty sure this was not the right activity at the right time for my 7.5 year old darling girl, but I did learn a lot about where her mathematical thinking is right now, which is always helpful.

In the last one and a half years my inquiry into elementary math education has kept pace with her math learning.  We've discovered so much together and it's been an incredible learning process for both of us.  Lately, though, it seems like the big picture concepts have clicked for me but as I try to move forward myself I end up rushing her.  The following activity is a case in point, but I still think it has merit for some child, somewhere!  Here's how it played out:

Back in November I bought a travel set of attribute blocks.  We haven't done much with them yet, but after looking through the little activity booklet that came with it I found an activity that piqued my interest.  It had to do with figuring out how many blocks of a certain shape (or combination of shapes) you would need to have to get a total number of sides.  It looked vaguely algebraic to me but was presented as a mental math activity.  So, I thought I'd create my own version of the activity on paper to make it a little easier to follow and to visually reinforce the differences between shapes.

This is the first worksheet I made.  In the first example I labeled the triangle with a 1 (meaning one group of three sides) and she had to figure out how many more hexagons made it add up to a total of 15 sides.  The second problem also had one shape already labeled, but in the final two problems I left it to her to figure out how many of both shapes.  You can see little pencil marks around the shapes at the bottom where she counted the sides one by one and then made notes for herself.

She was not completely happy with this activity (and was in a bad mood, distracted by whether the word 'futzy' was an insult or not).  Grumpy or not I think it really stretched her capacity in a good way, well enough for me to try again.  In the second iteration I asked her to write the total number of sides under each shape which I think really helped.  It was easier for her this time around.  You also might notice that I rephrased the question a little.

Then the holidays interceded with math learning.  Over that time, though, I did some thinking about how perhaps this kind of activity could be used to reinforce the concepts of multiples and the commutative property.  For example, 4 three-sided shapes (triangles) have the same number of edges as 3 four-sided shapes (squares or rectangles).  I also wanted to continue to stretch her idea of what the equal sign means; not necessarily a result, but a relationship -- various expressions of the same idea.

Here is the third activity.  In it I intentionally grew the numbers from 6 to 12 to 24 to 48. 

This time her strategy right out of the gates was skip counting whole groups of sides to work toward her answer instead of counting individual edges.  This means to me that somehow in the last three weeks her brain has begun to 'group' with more facility. I think this because, in the same time period, she has also experienced a huge jump in her reading abilities -- from having to sound out familiar words as if they were new every time to simply looking at a word and knowing what it says.  The math concept of 'grouping' and the reading concept of 'chunking' are essentially the same skill -- smaller items grouped into a larger whole. I saw that click into gear today with my daughter as she went to skip counting unbidden.

Anyhow, she moved through this last activity fairly quickly until the last two problems.  After it was finally over she proclaimed, "That was hard!  I hated it!  Forty-eight is such a big number!!"

That proclamation was revealing to me -- at this point in the game she's got facility with multiples of 0,1, 2, 3, 4, 5, 10 and 11.  The larger numbers are still a lot of work in terms of multiplication.  Being able to decompose a number like 48 was just too much at this moment in time. Ultimately, I think it's a call to put on my own brakes and, instead of trying to rush us forward, really dig into the mysteries of number composition and decomposition.  I know numbers are my weak point, so this will be good for me personally as well. 

Epilogue: After drafting this post this afternoon and then leaving to let it sit for a while I ran across the multiplication card game called Snap it Up which I found a month or so ago while at Goodwill (read about moreof my thrifted math here!).  I decided to give it a try and what do you know?  It was fun for both of us!  One interesting observation was that when I said 'what's x times y'  she'd give me a blank look but when I said 'what are two fives...' or 'how many tens make eighty' she totally got it.  I love it when the math stars align for us like this.  It happens a lot, actually, but I am grateful each and every time. 

Weaving Inverse Operations, Multiples & Frieze Patterns

It's been a super exciting few days in mathematical weaving land here at our house.  This weekend I figured out some ways to facilitate basic paper weaving and grid exploration for the youngers, riffing off Patrick Honner's Moebius Noodles guest post.   Last night I did a little more searching for how others have managed the logistics of paper weaving and found this fabulous example of warp management which, in turn, inspired some incredibly productive inquiry on my part. Ultimately, all this will turn into something I can do with my seven year old but, for now, let me show you what I did!

Here's how I started the morning, with multiple colors of paper sliced down to 1/2" strips.  Previously, I had tried my hand at weaving with 1" and 3/4" strips, but both were a bit too chunky for my tastes (although they're perfect for the young ones).  My first attempt at the new approach started with gluing the top of each strip down onto a piece of paper, with a space between each consecutive strip of paper.

Below is some experimentation with keeping the weft (horizontal) snug, in yellow, and a little looser, in green.  I love the look, but for a child's mathematical inquiry, I think it's better to keep both the warp and the weft snug, so the final design is as mathematically accurate as possible.

Since I now had some flexibility with the number of vertical strips I started wondering if the number would affect the ultimate design.  Curious about working with threes I started with a warp of nine strips. At this point I was just playing around to find a design I liked.  There's a basic reflection from top to bottom and left to right in each design.

Here is another multiple of three, a six-strip warp -- a frieze pattern, I think!  The design is made possible because of the two-color warp.

It's also where I started of thinking about inverse operations.  Weaving technique requires the use of some combination of overs and unders.  Row 1, from right to left: [3-over, 1-under, 1-over, 1-under].  Row 2 & 3: both the inverse of Row 1, but with different colors.  Row 4: repeating Row 1, but in a different color.  Then repeat!  Even now it seems like magic.  I can't believe I figured this one out.

Why do inverse operations matter?  From what I've read, and the number work I've done with my daughter, I know you can't really fully understand addition until you also understand subtraction.  Same for multiplication and division.  Add/subtract and multiply/divide are each two sides of same process.  It seems simple to our adult brains but it can be a very hard concept for a child to grasp fully.  I'm thinking that a focus on creating a weaving algorithm and it's inverse might really be a supportive numeracy effort. 

Here's another multiple of 3 warp:

After this I tried a warp of four strips.  Each woven 'unit' is made up of three horizontal yellow strips.  I love how the red and the yellow are rotations of each other from left to right, and reflections of each other from top to bottom. 

Also, notice that the second unit is the inverse of the first.  Instead of having the first yellow strip go 1 under, 3 over, the second unit starts 3 over, 1 under.  Interestingly, I usually weave from right to left, but when faced with an inverse, I found myself weaving from left to right.  I did it every time.  It was quite fascinating to watch myself in this process, which is why I suspect this would be really great for kids.  There's so much to learn and understand as you work to create a visually pleasing design. 

Here is a multiple of four, an 8-strip warp.  In this case, it's a basic over/under weave, but the two colored warp gives it some real variety.  I love this one.

Another four multiple, this time putting together the previous two ideas: the weaving algorithm of the first and the two colored warp of the second.  I wasn't trimming the edges on my designs at this point, which I think affects how your eye sees the patterns.

Now I was curious to see what it would look like all one color.  I think using green in both warp and weft brings out the structure in the weaving in a whole new way.  I like the green one untrimmed.

And finally, onto multiples of five.  This one is nice...

But this one is by far my favorite!

The first row is double strips that go [1 under, 2 over, 1 under, 1 over] and the second row is its inverse.  So simple yet very effective.  A ten strip warp is interesting because it's comprised of two, five-strip units.  A nice juxtaposition of odd and even.

It was such a fun day!  I think I understand the symmetries and the inverse weaving at this point.  I'm not sure how or if building a warp out of multiples affects things.  If you have any insights or see anything in my post that is mathematically shaky please feel free to correct me.  And, if I can clarify anything, please do let me know if you have questions.

p.s. I've got a new Facebook page where I'll be sharing links to cool math activities I find and some other things I'm doing with math, making, dance and rhythm.  Hope to see you there!