The Hundred-Face Challenge [C-rods and Constraints]

It was "Math with Malke" day in my daughter's 3/4 class! This activity was inspired by Simon Gregg's ongoing Cuisenaire Rod work with students; it was one recent conversation in particular about some "faces" that had shown up during one of his classes that got me thinking.

Because the kids' exposure to the C-rods was limited, I wanted to give the third and fourth graders a short intro to the rods before presenting the project.

I built a sequence of simple investigations that led up to the big challenge. Kids were split into (semi) random pairs; each group got a bag filled with a random amount of C-rods. I asked them to open the bags and find one of each color. 

While they were sorting, I noticed that the groups were naturally ordering the rods as they went through their bags. I paused the class to have everyone look around the room at how others had set out their 10 rods. We did a quick list of our noticings:

- 1 cube different between each

- Different sizes

- Each block is 1 number away from each other

- Looks like stairs
- Looks like a graph chart
- Looks like a sail
- Different colors

To give them some tactile experience with the lengths/amounts of the rods I had them pick up white, red, light green and yellow rods and put them behind their backs. They faced their partners and gave a series of commands: "Show me...green!" "Show me...white!"

As a final introduction before the face making I modeled adding the rods together, white to orange. Then came the challenge! 

Make a face that "adds up to 100" or as close as you can get. 

Constraint #1: Use only the rods in you bag. 

That's it. Off they went! Some groups added as they built and found a need for paper and pencil to keep track.

Some kids went immediately for the tens which made for easier counting.

Some kids built first, counted second and added or subtracted rods as needed.

Some kids just made awesome faces. Me: "Hmmm...that looks like it's more than 100. What are you going to do?" Kid: "I guess we'll take off the hair."

Some faces were closed (all rods touching), some were open.

I like this humble little guy:

Then it was time to pause for some more noticing:

- We used 10 rods to make 90 

- Some groups built on top of other blocks

- We changed plans a lot

- We had to add carefully

- Added by 10s

- We counted as we went along

- Everyone used at least one 10 rod

- Some people built and then counted

And the final challenge! Make a second 100 face different from the first. For example, if your first one was open, find a way to have all rods touch each other. If you had a closed face, make an open one.

Constraint #2: No matter what you do, the face needs to be balanced like a human face.  I illustrated this step by step on the board. [And, yes, I know that a human face is naturally balanced, but I wanted to make the idea explicit. It's one thing to see the balance/symmetry, it's another thing altogether to actually make it.]

When everyone was done with face #2 I said: 

"We’ll go around the circle and you can tell the group just a little bit about how you made this particular face.  What changed about your strategy from the first time you did it? Was there something you had to do differently, something you guys talked about that was a challenge while you made it? Or something that came to mind while you were building it."

Here are few of the conversations we had:

Group 1:
Girl: We started by just making a symmetry face and then counted them up and figured out we needed 24 more. So then we basically just added the hair with 24. Then it wasn’t perfectly symmetrical so we kind of made it the same on each side.

Me: And did it add all the way up to the hundred or did you get close?

Girl: It equals to the hundred.

Me (to boy): Is there anything else you want to add?

Boy: Well, no.

Me: I saw you put the pencil down [the middle of the face). Want to show them what you did? [He puts pen down the center.] Do you see that there’s a red on each side, an orange on each side?

Group 2:
Girl 1: At first we had a round face and we had eyes, a nose and mouth in it. But this time we made the background of the face ten each so we have 90 here and then we used twos to make the face.

Group 3:

Girl 1: Our first face was open and then we made another face but then Malke told us to use smaller pieces…

Me: [Laughing] Because why? Why did I give you that challenge?

Girl 1: Because we finished only using 10s, 9s…and the last face wasn’t as symmetrical. So we decided to do four browns in the middle and two blacks on either side. But then if you counted them up there was only 80 on the bottom.  So what we had to do is 4 + 4 is 8 plus 2 is 10 so we could only get up to 90 and if you let us use the 10 we could have made feet.

Me:  [Laughing] But that’s the thing, you have to work within the limits.

Group 4:

Boy 1: Well we started off again with a round face but then…

Boy 2: But then he was already making this other face on the ground and we just added some more to it.

Boy 1: One of the newer things was we took away the triangular nose and  put in these two [white blocks] and we added these red things.

Me: Does it add up to 100?

Boy 2: Mmm hmm. We got hair and you can [to other boy – can you hand me your pen? Thank you…laying the pen vertically down the center] it actually does go down and there are two teeth and two red things on either side.

The conversation reveal thinking about sameness and differences and emergent thinking about symmetry and balance. The teacher and I were both extremely happy about the activity, engagement and conversations in the individual teams and the class as a whole.

Next time I want to be make the constraints well, more constraining, like: "use only rods one through six."  I think this would also be a good to do again with this same group so we can deepen and extend the idea of balance.

Making and Playing Math with Kids: Day One

This was Day One of hopefully many more in which I get to play and make math with kids while simultaneously fulfilling my parent volunteer hours in my daughter's 3rd/4th grade classroom.

Group #1 | 20 minutes

"So, here's something.  Math isn't just about numbers, it's about finding patterns. If you get one rod of each color I wonder what kinds of patterns you'll find?"

"Can you describe what you made?"

"Well, the underneath one goes biggest to smallest and so I reversed it on the top."

"How many whites make a red?
"Two..."
"So, if white is worth one, how much is green?"
"What do you mean?"
"Well, if red is two whites, and you add another white, how many is that?"
"Oh! Three! Green is three!"

"What kinds of patterns or designs can we make using only the first five rods: white, red, green purple, yellow?"

"Your design looks like it's the same on both sides. Can you tell me more about how you made it?"

"Well, I started in the middle and built out."

"Hey everyone, come here! What do you notice about the pattern that is showing up on this design?" 

"I see those white blocks in every corner."

"How do you describe that line running from corner to corner?"

"Diagonal!"

"Okay, one minute left. Time to clean up!"

"Awww...I just gotta put down one more block..."

What's awesome is that these designs were made by the kids whose relationships with written numerals are not happy ones and yet, it's not for a lack of ability in noticing patterns or structure.  At the end of this session they all left with smiles on their faces, saying "That was fun!" We will continue with colorful and visual math because it's fun and I am fairly confident that this process, in the end, can be helpful in shoring up that bridge between mathematical meaning and symbols/abstraction.  p.s. This bin of Cuisenaire rods is now on their game shelf for free-time play with the following label:

Group #2 | 20 minutes

Group #2 came in buzzing. Their teacher had given them the two different multiplication charts I had laminated for every student and they couldn't stop talking about the archetype times tables from crebobby.com.  (See my post Marvelously Math-y Mondays for all the details and links).

When they settled down I ran them through a slow game of Speed! which is a super fun game created by Highhill Homeschool. I'll write about it more later but my biggest observation was that just because kids know how to skip count, doesn't necessarily mean they can quickly think "4 more" or "4 less" than any number in that sequence. This game seems to support more flexible thinking about multiples and is now on their free-time activity shelf as well.  As a group they had experience with 2- through 5-speed but I hope they will persevere into the higher numbers. If not, I'll make something happen the next time we meet. Yay for challenges!

Back in the classroom, I guess the enthusiasm for the archetype times tables had become unbearable and their teacher had to hand out the 'make your own' sheets I created.  His biggest goal/hope, I think, was for them to become more familiar with how a grid works in combining/multiplying numbers.

I think this make-your-own thing might have helped a lot in terms of learning to track rows and columns!

Energy and enthusiasm during math time! It makes a heart glad.

Design Build Experiment Play

What to do with a whole bunch of unused Cuisenaire rods?  Put them on the activity shelf at your kid's school, clearly labeled for play, that's what! 

 

 

 

 

 

A Useful and Beautiful Division

Me: Oh no!!  Four sets of Cuisenaire rods are all jumbled up!  I've got to get them sorted out before the next Math in Your Feet Family Night!  Do you think you could help me? 

Kid: How should we start? 

Me: I've set out four bags, one for each set of rods.  We need to divide up the rods equally so there is one full set in each bag.

Kid: (Finding all the orange rods.)

Me: "How many orange rods are there all together?"

Kid: "Twenty one."

Me: "How many orange rods go in each pile?"

Kid: (Starting with four in each group, then adding one more to each, one remains.)

Me: So, five in each group, with a remainder of one?

Kid: Look, Mama.  Four times five is twenty.  I knew that.

Yeah, I know.  I know that addition, subtraction, multiplication and division are steadily working their way into her consciousness and her skill bank, and all of this primarily by delving deeply into doubles, halves and evens.  I'm not kidding when I say that doubles/halves/even-ness comes up practically every day, one way or another.  I think it's a great way for a six year old to explore and learn about numbers, geometry, and the meaning of math.  There's actually something quite comforting in our daily discoveries of balance and symmetry whether in number play, through observations of our physical environment, or in the things we create with our own hands (the attributes matching game, for example, or our recent exploration of circles). 

I also know this kind of sorting activity is pretty basic, but having lots and lots of experience with operations in as many different contexts as possible is always useful.  In this case it was extra useful 'cause I didn't have to do this chore myself and I was able to work in some math inquiry -- sorting, classifying, measurement, division and a chance to get reacquainted with the rods -- all at the same time!   

Number Discovery

In retrospect, this amazing discovery started with parental neglect and erroneous math.  I freely admit I was at fault, but it all worked out amazingly well in the end.

It was all precipitated by an innocent question, yelled from another room: "Mama, what is half of 38?"

I did a little mental math: "Sixteen!" I yelled back.  In my defense I was in the middle of something very important.  That's why I didn't insist she figure it out herself, you see. 

When we were finally in the same room it took me a minute.  I was actually fairly impressed with her thinking.  Even though it was all wrong, she still had logic and structure to her reasoning.  Can you figure out what her 'rule' was?

[And, yes, at six she still writes many things backward; sometimes, as is the case here, she even writes from left to right.  Ah, the growing brain!]

Luckily, I had my wits about me and simply said, "Cool, look at that!  Hey, let's check your work with the Cuisenaire rods!"  This was a brave move since, prior to this moment, we have done absolutely nothing with 'taking away' or 'difference' in any formal way let alone using the rods.  Fortunately, the taking away part was so wonderfully obvious in this visual/tactile realm that I had no problem explaining it and the girl got it right away.  During this process I also noticed that in the intervening couple months between our first major experience with Cuisenaire rods and today, her ability to visualize and attribute amounts to the rods has become second nature.

Here the total number was 31, take away nine.  It was so very satisfying to physically take away a number and literally see the difference. 

Sorry this is so blurry, but hopefully you can see that we started with 37 and took away 15.  I'm not completely sure why we didn't start with the original number in question, 38, but like I said, it all worked out in the end.  Just look at it!  It's beautiful, don't you think?  By the fourth equation I asked her if she could figure out what would come next.  She guessed right for the last five equations, but wanted to check her work with the rods each time anyhow.

Today I am basking in the joy of unexpected discoveries and a growing mind. 

What Do You Make of This? (Build What I Have, Redux)

In my recent post Conversational Math: Part Two I described a game that my daughter and I like to play with Cuisenaire rods called "Build What I Have".  It's really fun.  So fun, in fact, that my daughter asks for it every other day or so.
Here's a funny thing that happened today:

It was my turn to give directions.  To start, take a look at how my design looked when I was finished giving directions.  I should point out that there is only one layer of blocks laid out on the board we were using.  I should also point out that we hid our work from each other.  She is trying to replicate my design using only my spoken words as a guide.  Ironically, no gestures are allowed in this game!

Mine.

Here's how I started: "Take three light green rods and put them together so they are all touching and all perpendicular to the bottom of the board.  It should look like a square that is three units across.  Now add a fourth light green rod on top of the square.  It should be placed perpendicular to the other green rods and parallel to the bottom of the board.  It should look like a rectangle now."

So far so good!

Me: "Take an orange rod and place it on top of the green rectangle."
Kid: "Like a teter totter."
Me: "Yeah, I guess so! Now, take two white blocks and put them on top of each end of the orange rod" (etc.)

Then it was time for the big reveal...tah dah!

Hers.

Geez Louise, what happened?!?  Her orange rod IS on top of the light green rectangle, but it appears the kid built up while I built...up?  I suspect we were both right, but I am at a loss to explain why or how.  I think it is accurate to call the kid's work three dimensional, but so is mine because it is built with polyhedra and not 2D lines on a flat surface.

Hmmm....any ideas?

Conversational Math: Part Two

In trying to capitalize on the kid's penchant for 'talking math' I recently decided to try a game with her that I found in the booklet that came with our set of Cuisenaire Rods. 

The game is called Build What I Have.  One person describes a design they are making with their rods and others try and reproduce that design by listening closely.  One of the main points in this game is to introduce and/or reinforce math vocabulary.

The suggested age range for this activity is 2nd-8th grade; even though the kid is a young six I knew we could still get something out of it.  I decided that, to start, I would capitalize on concepts she already knew (parallel, points, edges, top, bottom, sides, etc.) and introduce some new ideas (perpendicular, horizontal, vertical). 

The rest we'd muddle through somehow, I figured, but she did surprise me by knowing her lefts and rights.  "We've been doing that in ballet class, Mama," she stated mater-of-factly.  Fabulous.

To start, we hid our designs from each other.


This is the first design.  I led and she followed, trying to make her design match mine by following my instructions. I started by saying: "Lay your blue rod parallel to the bottom of our work surface."  She already knows the concept of parallel really well, often times finding and noting examples of parallel lines when we are out and about.  "Then," I continued, "take your green rod and place in perpendicular [holding the rod in the air] up and down like this, and place the end in the middle of the blue rod."  Success!  Our designs matched!

This is the game she led.  To start she told me to put my orange rod parallel to the bottom of the workspace, but about an inch up.  The second orange rod was to be 'a couple' inches above the first one, but when she told me to put the blue rods on the sides to 'make a rectangle' I clarified the distance.  "Looks more like three or four inches, to me," I said.  I asked her to clarify the placement of the blue rods -- do they go on the outside ends of the orange rods, or inside?  Notice that this design is mostly made up of parallel lines, a concept she is most familiar with.

This is the second design I led.  I said, "Take your three light green rods and put them so they are together and vertical, up and down, in your workspace...Oh look!  They make a nice little cube!"  At first she thought she needed a fourth one to make it a square, but I clarified and said we're not making the outline of a square, but a solid shape.  When we revealed our designs to each other we saw some differences! 

This is how she recreated my instructions.  The white cubes are essentially in the right areas, but I had actually challenged her to put each white block 'point to point' with each corner of the light green square.  The dark green rods are essentially in the correct place; I knew that was somewhat complicated to execute.  And, I just noticed, the light green rods are horizontal, not vertical.

This is the last design in our session, which she led.  Perfect!  She wanted to use a bunch of different rods, but everything is still parallel here.
 
Here is what I find fascinating:  

My daughter's designs were much simpler today than normal and I think it might be because she had to describe what she was doing as she built them.  There is an equivalent experience that I find to be true in my work with 4th and 5th graders as well.  Often times I tell those kids that they are doing complex mathematics in their bodies and grade-level math on the page; they understand more math in their bodies than they can communicate through words or symbols.  Sometimes it is impossible for them to notate their Jump Patterns because they are just too complex for their current stage of symbolic mastery.

Often kids can do, know, and understand way more than they can communicate symbolically.  If we only judge a kid by her output on paper, we're not really seeing the whole child.  There are many ways represent comprehension: we need to listen and watch carefully for other indications of understanding as well. 

It wasn't too long ago when I brought the word 'parallel' into my daughter's universe.  It will be exciting to observe her body and conversations show me she's 'got' the concepts of perpendicular, horizontal and vertical. 

Survival Math -or- How Cuisenaire Rods Contribute to Independent Living



From South to North: Our House, Bloomington, Park,
Nashville, Columbus (IN), Ohio (see the archway?)

Darling girl is an early riser and a morning learner.  Every day she rises, recites her list of questions and plans, and sets it all in motion.  This morning the dear child drew herself a map.

Soon after, she was really, really frightfully intent on writing number sentences.  All before my first cup of caffeine.  Normally I'd just let her chatter away and let it all wash over me (it was only 7am after all) but today the tide was just too strong.  I didn't really understand why there was such a need to do all this before breakfast but sometimes you just need to act first and ask questions later. 

She had already started her 'math lesson' by writing down 1+1=2.  Then 2+2, 3+3, 5+5.  At some point she said, "Every time I add, the numbers get bigger!"  The answer to 5+5 was doubled, which was doubled again.  80+80 gave her some pause, but by then I had drunk my tea and decided we might as well get out those Cuisenaire rods again. 

I wasn't completely sure if she was ready to move on to labeling each rod with a unit number or, for that matter, if I really understood how to use the rods to represent addition, or if we were going in the 'right' sequence of learning with these things.  But why let any of that stop me?

I said, "First, can you just put the rods in order from smallest to biggest?"  Done.  "Okay, let's look at your first number sentence, one plus one.  Here are two of the white blocks.  Which block is the same length as two white blocks?"

She picks up the red rod.  I say, "Okay, how many whites makes a red?  So when you write 'two plus two' this is what it looks like.  Which rod equals four whites or two reds?"  She pulls the purple rod to match.

 

 Then I had a bright idea:  "Can you figure out how many white blocks make an orange?  What other ways can you combine the rods to make ten?"  Girl interrupts my line of thought, "I want to make a line of a hundred!" She knows that ten tens make one hundred so she starts stacking them together to make sure she has ten of them before she lines them up.  I persist, "How many more ways can you make a ten?"

 

She wanted to see how long a line the ten tens would make when lined up end to end. 

This is how long it was and there are apparently many different ways to combine the rods to make ten.  We didn't get to all of them, I'm sure.  Then she wanted to see what a line 200 units long would be...



"Look Mama!  I'm 110 units long!"

Overall, my biggest takeaway from this morning is that, although she has enough questions to keep us busy for now, I need to do some more research on the best way to use Cuisenaire rods -- quick!

Later in the day I did find out why she was so intent on doing math this morning.  It's actually a secret so if you see her don't let on that you know.  "Mama," she said, "You know why I've been wanting a longer math lesson and have been so interested in maps and money?  Because T. [a good friend] and I were wondering what would happen if you [the mommies] stopped loving us and those things would be important to know."

Apparently, she and her great friend T. have hatched a sort of secret survival plan for an undefined point in the future when "our mommies don't love us any more."  They've decided they need to know how to do math, figure out how to work with money, and know how to survive in the outdoors.  After I assured her repeatedly that I will ALWAYS love her, we went to the library to get some books about how to identify and cook with edible plants. 

She said, "Don't tell T. that I told you.  And even if you never stop loving me, it's still helpful information to know."

"I won't tell," I replied, "But it's probably good that I do know so I can help you find the information you need. Learning all this will be really helpful when you grow up and live on your own."

"But I'm going to live with you forever."

White...Red...Light Green...Purple...Cuisenaire Rods!

I pulled out the cuisenaire rods yesterday.  The set had been sitting on a shelf for almost a year.  If you don't know what they are the best I can say right now is that, essentially, they are unit blocks of specific colors designed to (at least at first) help kids develop number sense as well as strengthen conceptual understanding of addition, subtraction, multiplication and division. 

The girl immediately remembered the long ago time when we played a 'trick' of putting the first four (white, red, light green, purple) behind our backs.  One of us would tell the other which color to bring forward, and every time we'd be right!   I can tell I'm going to do a little more research on how to use these things, but I know from what I've seen so far is that the first thing you do is focus on the attributes of each color in relationship to each other before assigning a unit number.  Here's how things developed today:



Girl: "I'm going to put them biggest to smallest..."  Mama: "Say it with me!  Orange, blue, brown, black...  Okay now, with our eyes closed!  Orange, blue, brown...Yay we did it!  Now smallest to biggest, white, red, light green, purple..."



Mama: "Hey look!  A blue with a white makes an orange.  Oh, and a red with a black makes an orange, too!"  Girl: "Let me do it."

Girl: "It's divided here."  Mama: "Oh, I see it, on the diagonal.  Let's put a block down to show where it is."

Girl: "I want to pull them apart."  Mama: "Look, they're in the same order!"

Girl: "Now I want to make a design."

What she figured out is that to make this design work you had to skip colors: orange + 2, blue, (skip brown), black, (skip green), yellow, etc.

So many more things to do with these -- I can't wait!!