Big Kid Pattern Block Puzzles

Initially, I wanted to figure out how to use polyominoes in a game-like 3rd/4th grade setting.  Got some help from my Twitter feed and the brainstorming brought me to this:

I'll write more when I have a chance to do the full activity sequence with kids (including building the 1-through 4-ominoes with unifix cubes) but, as always, one thing leads to another and I started wondering if I could make something similar using pattern blocks.

Turns out I can. And, it turns out it was an enjoyable challenge for my daughter, who is somewhat of a skeptic when it comes to trying out my new ideas.  Here's what I did:

I made an isometric dot grid.

I created some closed figures, some very open and balanced, some less regular. The goal of these puzzles is to fill in the entire shape using as few pieces as possible. My daughter did it by herself, but I could see it being an enjoyable, conversation-inducing project for a team of two.

I tried it out myself and thought, "This is too easy." Then I had the thought to ask my eight year old help me figure out if this was something that might work with kids in her class. Lucky for me, she consented.

The first puzzle resulted in a very interesting sequence.  I showed her how the hexagon, trapezoid, rhombus and triangle all fit onto the isomorphic dot grid.  I indicated the goal of using as few pieces as possible.  Here was her first attempt.  She used sixteen pieces.

To initiate a second try I wondered if she could fill in the space using less pieces.  As she looked at her first solution I wondered aloud if there were any places where a larger shape could take the place of the smaller ones.  This time she reduced the number of shapes to eleven!

Again I wondered if she could use less shapes, and on her third try she brought it down to nine!

I wondered how she would do with a less balanced, less regular shape. Here's the first try, eleven pieces.  In the process she observed that using a trapezoid and a triangle together covers the same area as two rhombi and uses the same number of pieces.

"This is a fun puzzle...and challenging!" she gushed.  Second try, eleven pieces.

 Still motivated. Third try, eleven pieces. "Wait, I think I see something!"

In the end, this may be a system with a short shelf life -- once you 'get it' you get it, but I love that it's very open ended and allows for multiple right answers (11 pieces, three ways, for example).  It's also a really friendly way to encourage perseverance in problem solving, spark discussions about composition and decomposition of units, and, heavy on my mind these days, a great way to ease into fractions.

I know this looks somewhat like a traditional pattern block activity but hopefully you can see that it has a different focus. At the very least it's telling that my kid (who has never been into 'solving' the already designed pattern block images) was really into this puzzle format. The only structure here is the outline of the puzzle shape. No need to give any hints by subdividing it in any way.  That would ruin the fun!

If you make puzzles of your own, I'd love to see them.  I'm going to keep working on a few more outlines and if I come up with some good ones, I'll be sure to share them.

Colorful Math: Area, Multiplication and Square Numbers Edition

Two things our house has a lot of: math and color.

 

We eat colorful, mathy breakfasts.  (Isn't this hexagon egg-citing?)

And then we work on our colorful and original attributes matching game.  The attributes include a multitude of choices in the following categories: shape, color and pattern.  We ask: How can I make the pair exactly the same?  How can I make the next pair different?  How are my designs similar to each other?

After a bath and second breakfast, we learn about square units and determine the size and area of each picture.  But why stop there when there are colored pencils around?  I wanted to move on but quickly concede that not only is color is a great addition for highlighting structure within the areas in question, but basic math gets a whole lot more fun to do!

At which point I think: Wow, I've got some graph paper...how about tomorrow we look into square numbers?

And how do I start that lesson?  Pick your seven favorite colors!

We do the first six numbers and the child, unbidden, suddenly looks over her work and says "Wow, they get bigger!  Oh, and there's a pattern!"  We figure out the specifics and then use that observation to predict the seven square.  I know there is lingering confusion about how exactly that number is made but, as you know, tomorrow is another day and I think I have an idea...involving colored pencils, of course.

Half A Cake

"Hey, tomorrow's your half birthday!  I know, let's make a cake to celebrate!"

Turns out the recipe called for 1 1/2 cups of flour.  Since our cake was ingredient free due to multiple food allergies, we used even parts of two gluten free flours.

"Okay so we need one and one half cups of flour.  How many halves in a whole?"

"Two!"

"Okay, so here is the 1/2 cup measuring cup.  One scoop of the buckwheat, one of the rice... Now, we need 1/2 cup more flour, using both our flours.  What's half of 1/2?"

"One-fourth!"

"How'd you know that?"

"Oh, I just knew..."
..............
Later in the morning, at the library.

"Hey look Mama!  The [really big, red tiled] square is made up of smaller squares!"

"Very cool.  How many small squares make up the big square?"

"Three, six...nine!"

"So, you wanna hear something interesting?  When you hear someone call a number 'square' that's what it means.  Look, nine squares literally make a 'square' number!  Isn't that cool?"

"Mmmm hmmm..."
.............
After lunch.

"Hey, it's time for cake!"

"YAY!!"

"But we can only eat half.  How would you cut it in half?"

[Girl gestures a line from the middle of one side to the middle of the opposite/parallel side.]

"Well, that's one way.  How else could you do it?"

[Girl pauses and then gestures diagonally from corner to corner.]

"Excellent!  But I have one more idea.  How many squares made up the big square at the library today?"

"Nine."

"So, I"m going to divide the cake into nine parts.  Here's one for you, and one for me. We'll have more later, after dinner."
...........
After dinner, and after one more piece each for the child and myself.  (Papa doesn't like chocolate, so no cake for him!)

"Mama, can I have a little more cake?"

"Hmmm, maybe.  We're only going to eat half of the cake -- let's see if there's any left....  How many pieces did we have to start?"

"Nine."

"What's half of nine?"

"Four and a half."

"So, together you and I have eaten four pieces.  There's half a piece left and you can have it!"

Ladies and gentlemen, I present to you: half a cake!

Totally Territorial: Cats, Maps, Area & Multiplication

I know it doesn't look like much, but there was a battle going on.  We rolled the dice (one six sided, the other twelve sided) and staked out our territory.  We're warrior cats, you know.  You didn't?  Have you read those cat clan books?  My kid hasn't -- too many kitties getting hurt makes her crazy -- but her best friend has.  They 'play' those stories every time they meet.  You should see them in their 'warrior training'.  It's crazy.

The minute we started rolling the dice I knew I needed to give this activity a special spin.  We had just finished a 'make fifteen' game where the last person to fill in a combination of fifteen wins.  There was no real strategy involved, hence her lack of buy-in.  In fact, she had been thoroughly unimpressed with it and didn't appear too eager to try something else. 

This new activity was basically built on the same model.  The goal was to see, after lots of dice rolling, who would come out with the most area colored in after the whole page was covered.  (For the life of me I cannot remember where I got this idea, but I know it was from a recent blog post.  If it's you I apologize for my lack of recall -- and please let me know so I can credit you!) 

Since success in this game was mostly left to chance, I knew I had to think quickly. "Let's stake out our clan territories and see who gets control of the most area!" I exclaimed.  Needless to say that's all the motivation she needed.  She decided almost immediately that each new piece of territory had some kind of real-life correspondence -- campus (the scene of our recent Tana Hoban adventure), our favorite hangouts, she wanted them all.  I pointed out that she could be strategic with where she filled in her boxes and, if you look at the first photo again, you'll see that she's surrounded parts of my blue territory so that I have to cross her land to get to other parts of mine.  Nice.

Then I realized that since she was thinking of real places we could probably do the same game on a real map!  Here's what we did later in the day...

We got a free map of our fair city at the bike shop.

With her watching, I took the yard stick and marked off a 1/2" grid over the entire city.  The grid was not to scale but was consistent across the entire map.  The area unit was 'one square'.

We looked at the map and found our landmarks -- hangouts, parks, campus, lakes, our neighborhood -- and at the map's key to find the roads, schools, bike and walking trails, etc.  Then we started strategizing about which areas we'd most like to 'have control over'.

We rolled the twelve sided die first to get our length (or width, whichever, depending on where it was best used strategically).  In those boxes we placed a dot using a permanent marker.  (We wouldn't be able to read the map if we colored everything over.)  Then we drew a line to show the border of that 'territory'.  Using the six-sided die we rolled the other side and figured out the total area of our newly acquired land.

I used the words area, length, and width as well as north, south, east and west during our conversations about what parts of the city she and I wanted to acquire and why.  Some of our reasons were strategic and others were more personal -- I got a lovely lake with the hiking trails, for example, and she got the big park with two awesome playgrounds.  I also modeled a lot of multiplicative thinking.  Even though we did use skip counting to compute the total area, I would point out groupings: "How many rows of seven do you need to fill in?  Second row, two sevens are fourteen.  Third row, three sevens are 21..." 

The strength of this game is that we know our city pretty well.  We've been to a lot of the parks, we move through town frequently on foot, on bikes, on buses, and in the car.  Our personal landmarks relate to where our friends live and work and go to school.  It really is our territory which is why the map was so real to her; it made sense to her despite being an abstract 2D visual representation.  Plus, the cat clan narrative is one she knows intimately, further increasing the emotional connection to the material.  Tomorrow we might play the game again and total up the red and blue areas to see who has the most.  But then again, given the engagement level today, it almost doesn't matter.  I think we've both already won.