Division with Cuisnaire Rods
This blog post we're finishing up fractions inside 5 in chapter 4 of Gattegno's Mathematics Textbook 1. This is the longest section in this chapter. Since most of it is redundant, this post will also cover exercises 40 and 41 on division. If you've been wandering around the internet and found yourself here by accident, welcome. You can find all my posts on this text here. You can find the book here. I suggest you grab the book, it's free, and follow along, as not much of this will make sense without it.
In my last post on fractions, we covered 1/2, 1/3 and improper fractions. Gattegno addresses one quarter and one fifth in nearly the same way as he did 1/2, and 1/3 so I don't want to repeat myself in this blog post. The exercises covered in this way are 30-39 of chapter 4. If you need a refresher on those exercises, you can go back and read the previous post.
There are a couple of exercises in this section that are new to us that I do want to cover. Recently, I wrote about an exercise I did with P. using a staircase. Towards the bottom of that post you will see the following image :
Exercise 37 is nearly this very exercise without the staircase. Gattegno wants the student to understand that the red rod is 2/3 of green, but it is 1/3 of dark green. The rod has a value in relationship to some other rod. It doesn't have a value all on its own. Light green is 3x white, but 1/3 of blue. You can build a staircase like we did above or do it from memory.
Exercise 38 is interesting. Using only the 5 smallest rods, find a pair of rods, not necessarily the same rods, that show the relationship: 1/2, 1/3, 3/5, 4/4 and so on. Gattegno has a list of fractions he wants the student to express but any fractions from half to fifth will work. Don't forget to include improper fractions with numerators as large as five.
I thought, for certain, that P. would struggle with this exercise but he didn't; he rather liked the game. We took turns going back and forth with him choosing a fraction and me finding it. Then me asking for a fraction and he had to find it. We did both proper and improper fractions.
Intro to Division With Cuisenaire Rods
In exercises 40 and 41, Gattegno is doing a brief introduction to division. We've been doing division the whole time we've been working on fractions and multiplication. The only thing we are adding is the language and symbols that will allow a student to express division both orally and in writing.
The question Gattegno asks the student is "How many red rods placed end to end make a yellow rod?" This is a question that's been asked since the 2nd chapter. The student should know the answer without pulling out the rods.
The student should have been practicing problems like 5 = 2 x 2 + 1 for the last few weeks. If she has, she already knows how to write it, or at least have her parent scribe it for her.
Making the Transition from Multiplication to Division with Cuisenaire Rods
Now we are going to make the transition from multiplication to division: two twos and one are the same as five. How many twos are in five?
Gattegno uses the word makes instead of 'is the same as'. I've chosen to use the latter as it is important that the student realize that 2 x 2 +1 do not magically turn into five, but rather they are 5. 2 x 2 + 1 is another way of saying 5. It is the same thing.
When we ask how many twos are in five or threes are in four we are asking a division question.
Writing Division Statements
Gattegno offers two ways to write the division problem.
We are using the ÷ symbol only. That is personal preference as I don't intend to introduce long division to P in the traditional sense, so we have no need for the other version. Gattegno writes long division in the book differently than I've ever seen it written. Parents should take note and use whatever is traditional where you live, especially if students will be taking standardized tests.
Gattegno also offers the language for remainder. How many threes are in four. There is one and one over (left over), or one remainder.
If you've been paying attention to the exercises, by now you have noted that we are using the same structures over and over again. What has changed is the language and how we talk about the relationships. If I want to know what five 5's are equivalent to that is the language of multiplication. If I want to know how many fives are in 25, that is the language of division. However, the Cuisenaire Rod structure is the same for both relationships.
Next blog post, I'll be covering exercise 44 of chapter 4 in great detail, and just touching a bit of42 and 43. As we move ahead, I won't be covering all the exercises in detail. Many of the exercises are repeats or variations on exercises we've already covered. As Gattegno introduces something new, I'll cover that, but otherwise, we have covered the concepts behind each exercise, the only thing that changes is the numbers.
If you are new to Caleb Gattegno and want to learn more, my resources page has all kinds of information on it. You can join our Facebook group here. The Facebook group is not just for Gattegno users, we have Math-U-See, Mortensen, and people just using the rods to models problems. Come hang out with us and learn from some really great parents, tutors and teachers.