Trains of One Color
We have reached Trains of One Color in Gattegno’s Mathematics Textbook 1. We’re covering exercises 20-30 of chapter 2. This post is part of a series of posts on using Caleb Gattegno’s work to teach with Cuisenaire Rods. You can find the others by clicking here.
Where are we headed with trains of one color? What can we be thinking about as our students work through this set of exercises? Our work with trains of one color begins preparation for work in multiplication, division, and fractions, LCM, and GCF. We are always working on developing our understanding of quantity. If you want to learn with your students, you need to play along or do these activities at another time.
In the last post, I mentioned that Gattegno followed a predictable pattern when introducing new material. This group of exercises is no different. The pattern is familiar: find the trains of one color for a single rod, find the rod for a train of one color, compare a train of one color to another train of one color, compare a train of one color to multiple trains of one color.
There are ten activities that Gattegno uses to work on these activities. But there are actually only four; it comes to ten because he is giving examples with particular colors. Again, colors are arbitrary. Do them all, do them more than once. These exercises are experiments for the children. We aren’t going to be dictating how to do it or moving rods for them. We are presenting challenges and asking them to meet the challenge. Then we can talk about what they found. Our students will have an easier time later if they are given enough time to manipulate and experiment with the rods now
Trains of One Color for One Rod
You can follow Gattegno’s recommendation or come up with your own games to play for this activity. As soon as the student grasps what we are doing, and that shouldn’t be a problem since we’ve been working on equality and building trains for some time, we build mats for each rod. In the beginning, this will take some time. We can first build a mat for one or two rods and move to something else. When we first start out, there is a lot of measuring going on. The student is collecting and organizing information. This is a good thing. Let them do it.
Once we’ve built our mats, we spend some time observing stuff. Some blocks have mats with only a train of white rods in them, and some blocks have several trains of one color in them. Not every rod can be split into two equal pieces. If your student doesn’t understand 1/2, you don’t need to introduce it now. I asked my son to count the number of rods in each train. When I brought this up, I asked, “Find the dark green rod. If you made a train of light green equal to the dark green, how many light green rods would you need? If I have an blue rod, how many light green rods do you need to make a blue rod?”
The student should first do the discovery with the rods. Then the parent/teacher should show the rod and the student should be able to pick out the rods that make trains of one color. Next, the student should be able to do it from memory with no rods visible. Additionally, the student should be able to make a complete pattern of trains of one color for each of the rods. Then s/he should be able to pick out the rods that will make trains of one color when the teacher holds up a rod. Last, the student will be able to do this from memory.
When we compare multiple rods to one rod, we are doing the above in reverse. Which rod contains a train of 4 red rods? The answer to that is brown. In the first exercise we found the trains contained in the rods, now we will find the rod that equals the train. Do you see how we are prepping for division, multiplication, and fractions with these exercises? In the first exercise, we are looking for factors, and we are partitioning the block into equal parts. In the second exercise, we have the parts, and we are looking for the whole. We are multiplying.
When going from a train/s to a rod, sometimes I built mats and sometimes just a train. In the beginning, children measure. They all measure. Even if you think your student has a pattern memorized, they will still measure. These are the experiments students need to conduct as they acquire information and gain confidence. Gattegno doesn’t suggest that the student memorizes the trains for each mat. However, we went over these enough times that they were internalized. The easiest way to have your students internalize the mats is to ask him/her to read them.
Comparing Trains of One Color
When we are comparing trains to trains we’re attempting to discover three different types of information.
- If we line up two trains of one color, can they be made the same length? The answer is aways yes, but you may not have enough rods. That is ok. Right now, we’re just becoming aware that trains of two lengths may or may not end up in the same place. 1) Do they line up?
- Next, if we line up multiple trains of one color, the various trains will meet at different points or not at all, which depends on the number of blocks used and which color trains are chosen. 2) Where do they line up?
- Lastly, if you have a red train of 10 rods and a yellow train of 4 rods and an orange train of 2 rods, do they meet up at more than one spot? 3) How many times do they meet up?
We can do this by building three trains of one color or 8 trains of one color. It doesn’t really matter. Let your students choose the colors. Grab a handful of bricks and build those into trains of one color. Pick the train colors out of a bag. The important part about this is letting the students have time to explore the trains and to create an awareness of the patterns and non-patterns. Make a note of the things the students observe. Ask questions. Build along with your students and share your own observations.