Making Numberblocks

We have two students in our pre-K classes, Y in Star Class and G in Moon Class, both five years old now, who are fanatical about fairly large Numberblocks numbers. "I love rectangles!" G exclaimed the other day!

I've blogged about this series before and I - along with all our students - think it's brilliant! There's a page I go to to access all the episodes. As my students are between three and five years old, we're leaning towards the small numbers.

At the start of the year G's parents told me he really loved the series, and moreover, knew the square numbers and 'Step Squad' numbers (Numberblocks-speak for triangle numbers). I waited, but he didn't seem to be showing it much in class. It was really in December that he discovered a fellow enthusiast in Y, that he started to talk about it.

Most of what happens in Star and Moon classes is voluntary - the students choose which play they engage in. We adults put out things that we think will interest and engage them, and we also have a couple of 'meetings' a day with our classes, but for most of the day the students choose from what's provided.

We've noticed that influence is a really important factor in what the students try out, and what they persist in. I wrote a blog post about this - 'Copycats'. So it's to be expected that it would be social factors that would really bring the enthusiasm out into the open.

At first they were asking to see pictures of Numberblocks numbers on the Internet. Then they were making them with Multilink cubes.

On the 19th December, G made these:

That's 24, seen as two twelves, 18, seen as two nines, and 27, seen as three nines.

On the same day, Y was telling Estelle a story to write down, all about Numberblocks.

And also putting together 30, 40 and 48:

These are regular Numberblocks colours. 30 is yellow because 3 is yellow, 40 is green because four is green. And eight is pink.

There followed a flurry of number-building, which continued in the new year.

We allowed them to display their creations.

There's always a mathematical structure in these. For instance, here's 64 as a cube:

There's the sixty, which should be purple really, but we don't have many purple cubes, and the four, which is green.

A new kind of story began:

We act out lots of our stories - and this one was a challenge - but we did it! A whole series of similar 'times table' stories followed from both students.

A concern though. These students are outliers. Would there be any way for the 'copycat' thing to happen? These two were so deep into their number inquiry - would there be a way for the others to access what they were doing? I was giving a lot of time to them- a pleasure for me - but I wanted to be giving that time to more students.

Luckily - the answer seems to be... yes.

W, who is only telling brief stories, told me this:

Ar., one of our three-year olds, told me this one:

G and Y were pressing on... with 125, as a cube. You can see it's structure here:

and 49, as a square:

One way to spread the goodness was to put out the Unifix cube stairs. They seem to always get filled with the Numberblocks colours.

Our furry versions help too!

Then we hit gold. I put out squared paper and black pens. Somehow it was a lot easier to draw them than make them.

Al. drew this one:

Students were enjoying just drawing the grids - five or six new ones joined in, some of them just enjoying reproducing the grid:

An. wrote this story:

We scanned the drawings and let the students colour in digitally. And now there seems to be momentum building, with lots of them engaging in some way. 

W. drew this one - I helped him start off as he's done hardly any drawing this year:

An. did these:

K was very pleased with this. Though he needed me to tell him the size of the rectangles, it was G and Y who advised on the colouring:

I'm excited about the contagion and I'm hopeful that we'll find more ways to build bridges to allow the student-to-student influence that happens in our classes to do its thing.

I'll keep you posted.

Stress

One of the great things about my STEAM coach role this year is that I get time to research new approaches to familiar units of inquiry. Last year I taught the Grade 3 Structures unit, but felt I wanted to explore the ideas of load and stress a little more scientifically.

The Structures unit of inquiry is part of the How the World Works transdisciplinary theme, with these as the lines of inquiry:
• Properties of building materials
• Considerations to take into account when building a structure
• Structures used for building
• Relationship between structure, design and function.

Why are we seeing so many circles, triangles and squares in structures? Which one is strongest in which ways?

One idea I came across this year was the simple idea of using a sheet of A4 paper to construct three simple towers, one triangular, one square and one circular in cross section. How many books would these hold up?

We made predictions first:

The G3 students really got stuck into this investigation. Here, in G3P:

The results were similar in the two classes. With twelve groups getting similar results, we think the results are quite reliable.

And then, the big question is why? Why is the cylinder tower so much stronger? We don't know the answer, but here are some of our ideas. Maybe more than one of them is correct?

The following week, a mother of one of our Grade 3 students, Mrs N, who's a stress engineer, came into school and talked to us briefly about what being a stress engineer involves.

We also looked at some transparent plastic being pushed or pulled between two polarised filters.

Rainbow patterns start to appear. And the more the plastic is stressed, the more rainbows there are.

In pairs, we wrote about what we noticed and what we wondered.

We shared some of our observations and questions together:

Mrs N showed us a sponge, which you can squash. Engineers call this squashing 'compression'.

She pulled a hair tie too, to illustrate stretching, or as engineers call it, 'tension'.

She also drew a rectangle on a pool noodle:

When the pool noodle was bent, we could see what was happening with the rectangle:

This has started to give us more idea about what was going on inside those three towers.

We asked Mrs N about becoming an engineer. What do you need to be good at? She said that she had always enjoyed maths. She was also really curious about how things work and used to like to take things to pieces!

Mrs N wrote later to give her impression of the lesson:

"They were very clever and they could imagine what was happening in the sample without need of anybody to tell them. 

I am amazed by the learning methodology at the school, letting kids think and solve the problems by themselves with just a guidance of the teacher instead of teachers/adults lecturing them.  I am sure this has a positive effect on their learning."

There are still questions though. What is it that makes the cylinder pillar stronger? Would several smaller pillars be smaller than one? What if the pillars were covering exactly the same area? If we have time, we may investigate further.

Quadrilateral Sets - the lesson

In my last post I shared an idea for a lesson. So, how did it go?

First of all, I checked that we understood the idea of a set. Showing this,

students responded that it was a set of animals. When I asked them to be more specific, they observed that it was a set of rainforest animals. I tried another, this time with some of the girls' names, and again it seemed to be straightforward; the students at first said 'girls' and then 'girls in our class'. So onto the sets of shapes:

1. Students look in pairs at what the members of each set have in common . What are its defining features?

 

We paired up and got looking and annotating.

2. Students share what they think

While this was going on, Estelle, Laura and I circulated and asked children to talk to us about what they had noticed. It was good to hear them attributing some of the observations to their partner, and set us up for asking for hands up if they could say something their partner had noticed. Almost everyone did this, while I scribed some of the observations on the whiteboard. 

I was pleased when K said, "E saw something really cool..." and made sure I emphasised not just the listening but also the appreciation K was giving.

There was plenty noticed. Some things I'd hoped for didn't come up; no-one pointed out the parallel lines in the shapes in the bottom right. No-one saw the features of kites in the top right. But there was lots else. A lot of students commented on the fact that the shapes could be split into triangles. And quite a few talked about how some shapes were like squares that were distorted.

That was quite a lot of articulating our thinking together, so I called it a day, and we played a few games of Aggression to finish off (which involves thinking of its own kind).

Today I returned to it. Rather than try to clarify or introduced terms, I decided to just go with what the students had noticed.

One of the things was that the four-sided shapes could be split into triangles. We looked at that using Geogebra.

"Do you think all four-sided shapes can be split like this?' - everyone agreed with

 this generalisation.

The other thing that had stood out was their way of seeing parallelograms as "twisted", or "tilted" squares. Again using GeoGebra, I showed how a square could be viewed in a 3D space and by changing the point of view seen as different shapes.

It was as if, I said, we were turning a square in the sunlight and seeing different shadows. Someone asked if we could do that, so at the beginning of break time we took down some squares and spent a couple of minutes drawing shadows before the other classes came down:

I've not really thought  of teaching projective geometry before, but the lesson gave me the impression that the students understood the idea more readily than they did the categorisation of quadrilaterals. Perhaps something to return to?

3. After we've got a thorough feel for the four sets, we do the normal Which One Doesn't Belong?

Knowing that the picture was more complex than normal, I didn't push this for too long; in fact I stopped after four observations:

And there's "squares and rectangles! "No rectangle." Strawberries and fruit.

Where next? Afterwards I talked with Estelle about how the lesson went. We both thought the quality of the students' listening and reporting to the class while they were working in pairs was great. How to build on that, and get them listening even better and building on each others ideas, that will be part of our project. I'm looking forward to reading, adopting and adapting some of the thinking moves suggested in Making Thinking Visible.

It's given me a picture of which ideas about these shapes are within reach of and appealed to the students. Aside from the projective geometry and splitting into triangles, which I'm not sure how I would approach, the top left set seemed the most approachable, with the idea of right angles featuring. I wonder if we should return to Mondrian with this lovely Mondrian geometry tool, and use the idea of perpendicular lines to construct digital Mondrians?

What would you do next?