Making Comparisons With Cuisenaire

This post covers making comparisons as found in activities 5-9, of chapter two in Gattegno’s Mathematics Textbook 1. You can find all the posts in this series by clicking on Gattegno Textbook 1 in the categories box on the left. If you are new to this series or got lost on the web and ended up here, we are going through Caleb Gattegno’s math textbooks for use with Cuisenaire Rods. Gattegno is unique in his introduction of algebra before arithmetic to 5 – 7-year-old children. This blog series is to help teachers/parents use the books more effectively.

Gattengo has four activities under the heading of smaller/bigger, also included is the concept of equality or sameness.  He asks that the students compare the following:

For each of the three activities above, you are going to ask your students to tell you if the rod/s you are comparing are bigger, smaller or the same. After doing that a few times, you will ask your student to find a rod or rods that are bigger, smaller or equal to the length of rod/s  he already has.

How We Did This

Sometimes I would dump blocks on the table. Then I asked my son to close his eyes and pick a rod from the pile. The chosen block is the comparison rod. Now it is time to sort the blocks. Blocks of the same length went into the center pile. Larger blocks to the right, and smaller rods to the left. My son always uses his rod for measuring.

We also did all of the above exercises as part of one math lesson. I would make a train of 2 rods and ask my son to build one train that is smaller, one that is bigger, and one that is the same size. I might ask him to build a train of 3 rods that is smaller than a red and a green rod or a train of two rods equal to a black rod. You can take turns making trains and giving directions. Each direction adds a new constraint to the exercises: Make a train that is smaller than an orange and a green rod; it must contain a blue rod. Make a train larger than an orange and green rod, but you can’t use yellow.

You could assume that if the young student chooses the brown rod, which is 8 units long in both Mortensen and Cuisenaire, that she would immediately pick out the brown ones and place them in the center pile. I have never had a young student do this. The student will measure the brown against the brown to make sure. This is true even if we have done other exercises measuring the rods. Even if my student told me yesterday that all the brown rods are the same length, she would still measure today. And she will likely measure more than one brown rod. Some students will measure every brown rod. Similarly, the student will usually have to measure several yellow rods, and purple rods against the brown to be sure they are smaller as well as several blue to confirm that those are longer.

At this stage, Gattegno is using the blocks in a way that lets the student check their work. If the child can see two blocks in front of her, she will then take two additional blocks and place them side by side with the first two. She can see with her eyes which train is larger or smaller. The next step for us, though Gattegno doesn’t mention it, is to place a rod or two on the table and ask the student the following, “If I had a red and dark green would those be larger or smaller than the blue rod?” “What about a yellow and a purple?” If the student provides an answer, you would then ask him to check using blocks. This activity helps the students visualize quantity and makes mental math work easier down the road.

Once the student can easily find rods and combinations of rods that are the same as another group of rods, we will ask the child to “find the missing rods.” We will follow the progression as we did above. 

1. We take a single rod and a smaller rod. We place them side by side. We ask the student to find the missing rod that will allow the smaller rod to make same with the bigger rod.
2. We take two rods and make a train. We place one smaller rod side by side with it and ask the student to find the missing rod that will allow the smaller rod to make same with the train.

3. We add a little twist. We take a set of rods from the exercise above and mix them up. We ask the student which rods go with which to make a two trains the same size. In other words, the student needs to recreate the trains from acquired knowledge, memory, or by discovery. You may clarify, but you may not assist or ask questions. Just let the student figure it out. What happens if you put any two rods end to end?

Comparison War – A Variation on the Theme

Student and teacher each choose one or two rods with which to make a train and compare. If the trains are the same size, each must pick new rods. If the rods are not the same size, someone flips a coin. If the coin lands heads side up, the person with the longest train wins and all the rods belong to her. If the coin lands tails side up, the person with the shortest train wins and all the blocks belong to him. The person with the either the most blocks, this can be tested placing blocks side by side, or the person who can make the longest train wins.

What are We Doing Here?

1. Learning that quantity can be described as bigger, smaller, larger, greater, equal to or same.
2. There is language we use when comparing quantities.
3. Quantities can be the same even if all the parts aren’t exactly the same. Numbers are made up of other numbers.
4. A single quantity can be described multiple ways.  Red and a black are the same as a blue, red and black are also the same as purple and a yellow.
5. Descriptive words like bigger/smaller depend on context. An orange rod compared to a green and a white and a red train is bigger, even though there are more cars in the latter train.
6. Equal means same.
7. Laying groundwork for later work in addition, subtraction and algebra. Finding the missing rod is the same as finding “x” in a very simplified form.