Factoring Numbers With Cuisenaire Rods
This blog post continues our work in Gattegno Textbook 1. We are on page 51, exercise 50 if you would like to follow along. For a free copy, click on the image of Mathematics Textbook 1. This post covers products, crosses and factoring numbers. We'll also be making a spot in our math notebooks for a Table Of Factors.
Exercise 50 is the student's first introduction to the rectangle or area model. Since we are in the middle of the study of the number 6, we're going to ask the student to make 2 rectangles. One rectangle should contain all red rods to represent 2 x 3 and the other should contain all light green rods to represent 3 x 2.
What Does The Student SEE?
Once the student has completed the task we're going to let them study the two structures and observe what the student sees. If you remember my last post we discussed the patterns Gattegno uses for teaching. One of those patterns is SEE, THINK, WONDER.
What Might Your Student See?
There is a lot of math in the image above. Here's some things your student might notice:
- There is a red rectangle and a light green rectangle.
- There are three red rods and two light green rods.
- The light green rectangle and the red rectangle look like they are the same size.
- Three red rods are the same as two light green rods.
- 3 x 2 is the same as 2 x 3
- 2 + 2 + 2 = 3 + 3
What if your student doesn't notice that the two rectangles are the same size?
You're going to need to force the student to notice. This doesn't mean grabbing the child's head and demanding that they look. Rather, it means making it so visually obvious, that the student must see that they are the same. Ask the student to place the green rectangle on top of the red one. What do they notice?
We want them to notice that the rectangles are the same size. The red rectangle is a red rod high and a green rod long. The green one is a green high and a red long - depending on how you have arranged them.
Thinking About Factoring Numbers
Once the student notices the dimensions of the rectangle we will explain that the rectangles can be replaced by a cross. Gattegno tells the student which rods to choose, but P was able to figure this out on his own. I asked the following: "If I wanted to make a cross that would tell you how I made the rectangle, which rods would you use and why?"
What I found interesting is that P couldn't tell me immediately why he picked one rod from each rectangle. He just thought it was the right thing to do. I had him measure the rectangle by length and width again and then he could tell me what each rod represented.
- A cross that is made with red over light green we read it 2 x 3.
- A cross that is made light green over red we will read it 3 x 2.
Factoring Numbers and Arbitrary Information
We are going to introduce two new definitions with this exercise: product and factors. If you have the Hands-On Learning with Gattegno book the general information section covers arbitrary and necessary information. Only after we have made crosses should we introduce the terms factor and product. Factors are numbers you multiply to get another number. When we are factoring numbers, we are looking for all the factor combinations for a particular number.
Say: The product of 2 x 3 is equivalent to 6. What is the product of 3 x 2 equivalent to?
Model the language and allow them to copy what you say.
Say: Two and three are factors of 6. One and what number would also be factors of 6?
Factoring Numbers - Keeping a Table of Factors
Just because Gattegno stops this exercise at the number 6 doesn't mean you have to. This is where the WONDER part of the SEE, THINK, WONDER pattern comes in. What happens to the factors if we put two red rectangles together? How do you think we would make that cross? What if we wanted to make factors for 8 or 9, what would that look like? How would we know when we had them all? What about the number 5? How many different rectangles can we make for the number 5?
While we explore all of these ideas, we need to start a Table of Factors and place it towards the back of your math notebook. The reason we do this is because there are a bunch of really fun patterns to discover on The Table of Factors. Plus, we can deduce divisibility rules for various numbers by looking for specific patterns in the factors.
You can study whatever numbers you like in whatever order, but it's important that the information entered onto the table be in order. Otherwise, searching for patterns will be nearly impossible. We don't enter factors every time we play math, but when we study integers, factoring numbers is always part of that study. You can download a copy of the Table of Factors notebook page here.
If you are interested in learning more about Gattegno and his way of doing math, you can connect with us on FB here. If you aren't sure about anything, and teaching math seems very overwhelming to you, get a copy of Hands-On Learning with Gattegno where I walk you through Gattegno's Textbook 1. And sign-up and join the Gattegno Study Group. We meet on Mondays. Not every Monday, because I have a life and stuff. We play rods and math games. Bring some tea and a sense of humor. Don't worry about not knowing all the right answers.