Cuisenaire Rod Staircases – All About Counting

Gattegno's Textbook 1

We're making Cuisenaire Rod staircases again in connection with numbers 1-10. We're in chapter 4 of the Gattegno Mathematics Textbook 1 series, which you can get here. Flip to page 60. 

Gattegno is rounding out the study of 1-10 with the staircases. As always, every time I go through his work and have to write one of these blog posts or talk about it in the Gattegno Study Groups, I see anew how stinking brilliant this man was. Each time, I get a better grasp of where he is taking us.

Not everything I write comes from the textbooks. Part of what I am writing comes from reading - not using as a textbook - but reading other Gattegno related resources like the 

Handbook of Activities which you can download free along with other materials  on the resources page.

Using Cuisenaire Rod Staircases to Order Numbers

Let's have the student's build a staircase from smallest to largest using all the rods. When they are done the staircase should look like this:

Using Cuisenaire Rods to build number sense. From multiplication to logarithms.

Gattegno uses addition and subtraction to note the location of the rod in the staircase.  If w = 1 then 5 + 4 = 9 means that we are on the 5th step of the staircase and we jump 4 more steps to end on the 9th step. Nine is 4 steps after 5 in the staircase. We could also say that 9 is larger than 5 by 4 steps. 

9 = 10 - 1  tells us that nine is smaller than ten by one and it comes right before 10 in the staircase. 8 = 10 - 2 tells us that 8 is two steps before 10 in the staircase. This shouldn't be anything new to your student as we did this before in chapter 3. I'm not exactly sure where Gattegno is going with this, but he feels this kind of thing is important as it has come up twice and he spends a fair amount of time on it. Since he hasn't disappointed me this far, I will follow his lead.

Using Cuisenaire Rods to build number sense. Addition, subtraction, multiplication, exponents and logarithms.

We spent a lot of time this week on staircases and scaling them in different ways. We needed a way to communicate what kind of staircase we are talking about in our math notebooks. We settled on the following:

We call this a Staircase That Counts +2( stc+2). What is dark green in a Staircase That Counts +2? In order to write this efficiently we settled on: stc+2 d = ___? That just means: in a Staircase the Counts +2 dark green = what step? The 3rd step. What is stc+2 4?
Using Cuisenaire to build number sense. Addition, subtraction, multiplication, exponents and logarithms.

We call this a Staircase That Counts +3( stc+3). What is 6 in a stc+3? Or stc+3 6= ____.

In a Staircase That Counts +3 6  is the 2nd step. stc+3 6 = 2. 

Now consider that stc+3 ro (red/orange) = 4. In a stc+3 12 – 6 = 6. This tells us that in a Staircase that Counts +3, there are 2 steps between 12 and 6. We know this because there are two 3’s in 6.

We could also say that 6 comes 2 steps before 12 in a stc+3.

Using Cuirsenare to build number sense. Addition, subtraction, multiplication and logarithms.
We call this a Staircase That Counts +5( stc+5). What is 10 in a stc+5? Or stc+5 10 = ___. In a Staircase That Counts +5, ten is the 2nd step. stc+5 10 = 2. What about stc+5 15? What could you say about 20 – 10 in a stc+5?

What Do We Notice About Staircases that Count +n

As we build these we want to take time to notice what's going on with our Cuisenaire Rod Staircases. For the sake of clarity, the +n above is shorthand for + some number, or +n, whatever you want n to be.

If we work out the staircases for + 2 starting with 2 (0+2), you'll notice that you are working out the 2 times tables. The 4th step is 4 x 2. The value of the 20th step will be 20 x 2 or 40. The difference between 8 and 40 in a Staircase That Counts + 2 is 32 or 16 steps up the staircase. What if we compare a Staircase That Counts +2 and a Staircase That Counts +4?

If you haven't noticed yet, we've leaped way past numbers 1-10. But we are still studying how numbers 1-10 behave as steps in a staircase or when we multiply 1-10 with other numbers. 

Cuisenaire Rod Staircases That Count xn

Occasionally, I make connections between areas of math and what I'm doing with Gattegno. That is what started our little Staircases That Count project. One of Jerry Mortensen's 5 basic principles of mathematics is that math is the study of numbers and the only thing we can do with numbers is count. 

There is that moment when you realize that all of elementary math and elementary algebra is just fancy counting - that's it. If you sit around playing with rods, and noticing stuff, you are going to learn to do a lot of fancy counting. 

Let's build some more Staircases That Count but this time instead of by +n we will make it by xn. Instead of + 2 we will change it to x2 . I chose x2 because these staircases grow very quickly.

Using Cuisenaire Rods to build numbers sense. Addition, subtraction, multiplication, exponents and logarithms.

In the above image we note that in a Staircase That Counts x2, the difference between 16 and 4 is 2 steps.

What is the value of the 5th step?

What about stcx2 8?

Can you predict what  stcx2 128 will be?

Things to Notice About Staircases that Count xn

In +n staircases, we noticed that the staircase represents the multiplication table of n. What happens when we change the sign from +n to xn?  

In our picture, the first step is 2 because 2 x 1 = 2 because we are counting by x2. Step two is 2 x 2 = 4. Step three is 4 x 2 = 8. Step 4 is 8 x 2 = 16. Step 5 is 16 X 2 = 32.  

We can write this another way:

  • edit
    Step 1= 2
  • edit
    Step 2= 2 x 2
  • edit
    Step 3 = 2 x 2 x 2
  • edit
    Step 4 = 2 x 2 x 2 x 2
  • edit
    Step 5 = 2 x 2 x 2 x 2 x 2

Can we write this last one another way?

  • Step 1 = 21
  • Step 2 =22
  • Step 3 = 23
  • Step 4 =24
  • Step 5 =25

What is stcx28?  We said it is 3.  We counted up three steps and there is the brown rod. We could just as easily write that log2(8)= 3.

That is a logarithm. That thing that you  never really got in high school (if you were me, I only speak for myself, maybe you did actually get it.)

This logarithm asks where is 8 on the staircase if we are counting by x2? Eight is the 3rd step.  Wasn't that easy? Why didn't I get to play rods when I was in high school? I have 8, how many x2's do I need to get to 8? That would be 3 of the x2's. 

Make a staircase that counts by x3. What is the value of the 3rd step? Where does 81 fall? What step will 729 be? How many steps after 81 is 729? 

Why Play Staircases That Count Games

So what are we doing in these little exercises? A Lot. First of all, we are freeing the mind from being constrained by +1. Plus one, as we have seen, is the foundation of all counting. But, there are a lot more ways to count. And those ways of counting are pretty fun.

You can count by square numbers and end up with polynomials, you can count by +9 and end up with the 9 times table. But there is even more to discover if you stop and smell the factors. 

We don't talk about logarithms in our house. We are still playing Staircases That Count. We count by +2, +4, x3, x10. That x10 one is a bit of a bugger. You run out of rods very quickly. We like to watch the staircases grow. That is fascinating all by itself. 

Arithmophobia No More