How to “Exhaust the Relationships” in a Math Lesson

We spent the morning on a math game that we’ve come to call, “Exhausting the Relationships.” This time we pulled out some Cuisenaire Rods. And boy did they make for some fun work. Jerry Mortensen made dramatic improvements with the blocks no doubt. Placing the numbers on the blocks cleared the waters in a lot of areas, but numbered blocks muddy the waters in other areas. We’ll likely be using both types of blocks in the future.

What follows in nothing new, I’ve talked about this before in other posts. But what happened this morning is that I pulled out the little book, “Learning with Cuisenaire Rods” that came with my used set (one set of 74 blocks is NOT enough), and as per Cuisenaire’s recommendation, we used letters for the blocks and not the numbers. Then something magical happened. By removing the numerical value from the block we were no longer learning just facts, rather, the whole world of algebra opened up. Below you can see what we did with a preschooler while eating breakfast.

exhausting the relationships


Cuisenaire Rods from 1-10 are labeled as follows: w, r, g, p, y, d, k, b, e, and o. My husband hates the it looks like a zero. O stands for orange but it could be a tangerine color. So we tried t. That can look like an so we settled on an O with a line through it.  At the very top left you can see I have all the blocks lined up and I wrote the corresponding letter on each block except black. Then we got to work.

P’s favorite color is blue. So we pulled out the blue block. We noticed that the blue block (e) is the same as 9 white blocks (w). So we wrote that out. 

  • e=9w

We can also see that a blue one is the same as one brown one (b) and 1 white one. And that looks like this:

  • e = b+w

So, of course, the following can be determined from that as well:

  • e – w = b
  • e – b = w
  • w = 1/9 of e  or   w = 1/9 e
  • b = 8/9e
  • 4/8b + 1/2b + w = e
  • 1/2b + 1/2b + w = e
  • 1/4b + 1/4b + 1/2b + w = e
  • 2(1/4b) + 1/2b + w = e
  • e – 1/2 b = 1/4b + 1/4b + w

To the far left you can see we were playing a little with equivalent fractions once we got started with 1/4 and 1/2 b‘s. Where the c-rods are perfect – we can set the 1 to be anything. In  this case we said the orange block was our one. If O = 1 then y = 1/2 = 5/10 and r = 1/5 = 2/10 and w = 1/10.

Questions I asked: Can you find me two rods that make same with the orange rod? I didn’t choose which ones. He picked them. Now can you make me a train of all the same color rods that make same with the blue rod? What happens if I take a white from a blue, what do we have left? What happens if I take a brown from the blue what do we have left?  How many white rods make a brown rod? What else can you tell me about the blue, brown and white rods?

Mortensen and Math-U-See block users can easily do the same thing. Just flip the blocks on their sides and label them with letters corresponding to the colors. 10 = b, 9 = e (t for teal looks a like x and I used l for light blue), 8 = b, 7 = c or w, 6 = p, 5 = l, 4 = y, 3 = k (p is taken), 2 = theta, 1= g. 

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