This blog post covers exercises 3 & 4 in Gattegno Mathematics Textbook 1, chapter 3. This is where Gattegno first introduces the student to the word ‘relation’. This concept is critical not just for these particular exercises, but for all of math. If you’ve paid attention to anything I’ve posted in the last year or so, you know that ‘relationships’ is one of my favorite words. It’s not really, but in the context of teaching children mathematics, they won’t get anything unless they understand this one thing. Oh, they might learn some facts, but they won’t get the big picture of mathematics because ‘relationships’ are the big picture.

Activities 3 and 4 are exercises in building staircases. We did this in chapter two. The last time we did it, the students became aware that there is something called a staircase, that the rods can be ordered smallest to largest or largest to smallest, and that there is an order to the rods. If they were astute, they might have noticed that each **successive** rod is larger by the same length. If they didn’t, that’s fine because we are going to measure that distance with a rod. Then we will ask the students to **express** the **relation** between two **successive** rods.

By ‘successive’ we mean things that follow or things in a row; consecutive. By ‘relation’ we mean how one thing or idea is connected to another thing or idea. When we say ‘express,’ we mean how we talk about or how we write about the relationships. Please note that where I quote Gattegno directly I will use his letter patterns for the rods. Otherwise, I will use the one we use at home. To see what we do, you can look here.

### Understanding Addition and Subtraction As Relationships

When we make a staircase using all the colors, one of the things that bring these rods into a relationship is that each successive rod is one white rod larger than the rod before it. Those relationships can be verbally expressed by saying that a white plus a white is equivalent to a red, and a red plus a white is equivalent to a green. To express those relationships in written form we write w + w = r and r + w = g.