Teaching Fractions with Cuisenaire Rods
In this post I'm continuing my series on Gattegno's Mathematics Textbook 1 learning to Teach fractions with Cuisenaire rods. We're in chapter 4 and this post covers exercises 19-24, which focuses on taking the awareness the student gained using color values for the rods to fractions using number values for the rods . We're right in the middle of a giant section on fractions and multiplication inside the number 5. I've chosen to end this post at exercise 23 because Gattegno introduces fractions like 3/2 in exercise 24.
APPLYING PAST KNOWLEDGE TO NEW SITUATIONS
One of the things I love about Gattegno is that we are always working with the same material: Cuisenaire rods; and we are always working with the same familiar structures: trains, patterns, mats and staircases. Once the student learns to set up these structures, which we worked on in chapter 2, the mind is free to focus on new awarenesses. Every time we introduce something new, we go back to what is familiar first.
If you aren't familiar with Gattegno, that word awareness is probably looks odd. What Gattegno means by awareness is something that you have become conscious of, or something you noticed that you didn't know about or notice before. Education happens in the arena of awareness whereas drill happens in the arena of memory. The two are not the same.
Now we have little Sam and Suzy and they have been working with rods: they are comfortable manipulating them. In fact, they spent a great deal of time working on fractions with Cuisenaire rods using letter names for the rods in the last chapter. They know the names for 1/2, 1/4, and 1/5 all the way up to 1/10. Our students also know, from the last couple sessions, that 2 x 2 = 2 + 2 = 4. Now we'll apply past knowledge to a new situation.
In exercise 19, we are taking what we know about multiplication and relating it to the fractions work with rods we did in chapter 3. We're not immediately transferring that knowledge to work with numbers. My 6 year old, when we first approached fractions with numbers, did this instinctively. When I asked what half of 4 would be, his response was, "a red rod."
Exercise 19 also starts with the word 'since'. That word lets the student know that what follows is part of the nature of things. We can call the red rod 'r' or we can call it '2'. If we call the red rod '2' then it follows that 2 x 2 = 4 means the same thing as two red rods equals a purple rod. Then that means that a red rod is 1/2 a purple rod.
What we are assuring them is the same thing we assured them in the previous sections: just because we change the name of the rod from red to the number '2' - addition, subtraction, multiplication and fractions still work the same way as they did before. All we have done is change the language. Gattegno provides a group of questions for the student as a refresher on how we use mathematical language.
Refresher Questions for Fractions with Cuisenaire
- How many reds make a purple rod?
- How many whites make a red?
- How many TIMES is a red rod needed to make a pink?
- How many TIMES is a white rod needed to make a red?
- Which rod is half of red?
- Which rod is 1/2 of pink?
MAKING THE TRANSITION
Now that we have reviewed fractions with colors, Gattegno makes a seamless transition to numbers. Sam and Suzy do not have out their rods. The following questions are posed to them:
- What is half of two?
- What is half of four?
- One is what of two?
- Two is what of four?
If the student is writing, the student should write the answers to those questions in addition to answering orally. If the student is an emerging writer then student can use math cards (there's a link at the bottom of this post) or dictate.
Exercise 21, the student reads various math statements using addition, multiplication and fractions. It's important that the students is able to read these with ease. Young children have minds that flit from one thing to another. Teaching them to read statements makes them focus and pay attention. Plus, as the student makes progress in mathematics we want the mind free to think about the math and the symbols on the page. Reading mathematical statements should become automatic.
Independence for the Emerging Writer
P hates anything that has to do with a pencil or a pen. Sometimes a act as a writer for him. Sometimes he uses math cards. A link to the math cards file can be found at the bottom of this post.
MASTERING FRACTIONS WITH CUISENAIRE RODS
In exercise 22 the student is reading written math and completing equations. When we do this, I ask P. to either read the entire equation aloud or make the whole thing with math cards. Speaking or writing entire statements is part of learning and becoming comfortable with the language of mathematics. Notice though, that we read the statements then we read and answer. We are taking baby steps towards our goal.
Exercise 23 is interesting. The textbook has you asking what appear to be very obvious questions.
- If the white rod is half of red, show me the other half of red?
- Place the two white rods end to end. Is there a rod equal to the halves of the red rod?
- If the red rod is half of purple, show me the other half of purple.
- Place the two halves end to end. Is there a rod that is equal to the halves of the purple rod?
I was going to skip this exercise, but I know better. Gattegno has this in there for a reason. And my gut instinct was correct - it wasn't a no-brainer for him. When I asked P. the show the other half of the red rod he was confused. Thought about it for a minute, laughed, and then found the rod. He didn't need to place the rods end to end he knew which rod was equivalent. I am constantly amazed that Gattegno knows where these little glitches are going to be. It's not that P didn't know that if 1/2 is white the other half is white, it's that he hadn't thought about it. When asked to think about it, some kind of awareness happened for him. I love watching those happen. That is probably my greatest joy in teaching math is all the "Aha" moments I get. Pick up your rods and teach some fractions with Cuisenaire rods.
If you would like a copy of the math cards you see in the image above you can download them here.