Improper Fractions With Cuisenaire Rods

This blog post in the next in the series on Gattegno's Mathematics Textbook 1. We are in chapter 4, beginning with exercise 24. This section is part of a long section on multiplication and fractions inside the number 5. Exercise 24 introduces the concept of improper fractions. For those new to the blog, we'll be working improper fractions with Cuisenaire Rods. If you don't have a copy of the book, it is available free here. Download it and follow along.  You'll find all my Gattegno Mathematics Textbook 1 posts here.

In my last post, there was an exercise that I nearly skipped because it seemed so obvious. If you go to that post, you'll find my comments in the last real paragraph of the post. The questions Gattegno posed are these: what is half the red rod? and what is the other half of the red rod? I didn't skip the question and I'm glad I didn't. The reason for the question became obvious when we began work on exercise 24. 


Improper fractions with Cuisenaire Rods Half

In exercise 24, Gattegno pulls out that word since again. He's asking the student to take what they know, previous knowledge, and apply it to the new situation. These moments are very important. They fall under habits of mind and require the student to think. We do not want to interfere with the student's thought process. So resist any temptation to over explain this. 

The student has touched 3/2 of the red rod. The student knows that 3/2 of the red rod is the green rod. The student heard the language and produced the rod that is equivalent to 3/2 of the red rod. Now the student will learn to write 3/2 of the red rod and several relationships connected to it. Those relationships follow because if  a is true and b is true then it follows that c must also be true. 

Writing improper fraction relationships with Cuisenaire Rods

If the student struggles to answer  "What is 5 halves of 2?", back up and cover previous exercises again or consider laying this lesson down and coming back to it a different day. 

​Exercise 26 provides enough practice for the student to determine whether he or she has gained understanding.  There is no harm in the student using the rods for this exercise, but the student really needs to be doing oral work. If the student needs to use the rods, it shows rod dependence, not understanding, or it is a sign of an auditory processing problem. P. probably has both rod dependence and a processing problem. Long ago, I should have insisted on a bit of oral word every day, but I didn't. We solved this by playing games outside of the "math lesson". While driving in the car or some other time where there isn't a pressure to perform. We started with a game we call "guess my rod" or "guess my train". I make a set of clues and he must guess the rod from those clues. If he guesses correctly, he imagines a rod or train and provides me with clues. No rods involved.  Then we moved to addition and subtraction statements for  "guess my rod", then I added fraction questions. It didn't take long to break P of rod dependence, but it is better to not develop it in the first place. We are not at the point that P. can imagine a mat and write statements based on the mat, but we are getting there. 

Exercise 27 is a repeat of exercise 24 using thirds for improper fractions instead of halves. ​How many white rods cover the light green rod? We call those thirds. 

Improper Fractions with Cuisenarie Rods Thirds

​Gattegno sneaks in a couple more questions in exercise 27.  Is there a rod that is equal to 4/3 of the light green rod? Is there a rod equal to 5/3 of the light green rod? The students are becoming aware that numbers have all kinds of names. If you spent a lot of time in chapter 3, this is not new information. But if you breezed through chapter 3, I would encourage you to not breeze quickly though this chapter.  The rods can represent colors, and numbers. Numbers can be made with addition statements, fraction statements, multiplication and subtraction statements. The same number can be 1/2 of one number and 1/3 of another. 5 is five white rods, 5/3 of a light green and 5 halves of the number 2. This is an important awareness.

Gattegno stops at 5/3 of the light green rod, but there is no rule that says you must stop at 5/3 of the light green rod. Let  your student play and play with the idea of improper fractions. Make sure when the student provides an answer the answer is in improper fractions with Cuisenaire Rods, for instance, five thirds of the light green rod. Five thirds doesn't have meaning by itself, we need to know five thirds of what. 

Measuring with Improper Fractions and Cuisenaire Rods
  • Measure the table in thirds of the light green rod.
  • Measure a mini-fig in thirds of the light green rod.
  • How many thirds of a light green rod are in 2 orange rods?
  • How many light green rods are in two orange rods?
  • How many more thirds of light green do you need to make a train of light green rods equivalent to a train of 2 orange rods? 
  • Without measuring, can you say how many halves of the red rod is the mini-fig?


Gattegno is introducing the idea of the math composition in exercise 28. If your student isn't producing math compositions, now is the time to start. We began math compositions over a year ago, but that doesn't mean your student is the same as my student. Obviously, Gattegno waited until this moment to put compositions into the curriculum. What is a math composition? A composition can be written for a mat or, as in the current example, it is written for the number 2. ​


2 = 1 + 1 = 1/3 x 3 + 1/3 x 3 = 2/3 x 3 = 4 - 2 =  5 - 3 = 5 - (3/3 x 3) 

Every effort of the child to produce a composition should be met with approval, even if it is no more than a couple statements. As the student feels more comfortable with writing and producing math, the compositions will grow in complexity. This is where a teacher learns what the child actually understands. 

The second part of exercise 28 contains practice problems where the student is asked to complete the equation. Exercise 29 is similar to exercise 15, where the teacher reads 2 statements and the student must determine which statement is bigger. For example: which is larger half of two or one third of three. We do these types of problems in the car while driving or while cleaning house. I ask them of P and he will ask them of me. We make them up as we go. He has become fairly good at this. He can do half of most even numbers up to 20 and a third of multiples of three up to 18. We haven't formally covered these numbers in class time, he figured it out on his own.

Next post I'll be covering quarters, fifths and completing this section on multiplication and fractions.  If you are interested in using Cuisenaire Rods to teach math or perhaps you have some Math-U-See or Mortensen Blocks collecting dust and you'd like to use those. Then come join us over in our Facebook Group. It is full of parents and teachers and tutors who are learning to use base ten blocks to teach math. Some of them know what they are doing, most of us are just learning and seeking help. Everyone is very helpful. I don't mean to brag, but I really like these guys. 

Arithmophobia No More