Learning Numbers 1-10 Review
This blog post continues our work in Gattegno's Textbook 1. In the last post, I finished up with the number 6. At that point, we'd covered all the exercise variations that Gattegno uses for learning numbers 1-10 which takes us to exercise 66 on page 62. Get a free copy of the book here.
In this blog post, I want to review those exercises and throw out a few variations. Learning the numbers 1-10 is not really about learning numbers in the same way we would think about it in a traditional math setting.
What Gattegno proposes is the study of numbers and structures which is a completely different way of thinking about mathematics education.
When most students are introduced to numbers 1-10, they learn to recognize that a group of objects of a certain quantity is some number. A group of 7 objects is 7. Our students will be studying a host of structures connected to numbers 1-10. Those structures include addition, subtraction, multiplication, division, fractions, mixed and equivalent expressions. And if studied properly, they will enjoy the challenge.
Exercises For Learning Numbers and Number Study
Fortunately, the same exercises are repeated over and over again. Someone asked me recently where we are in the book. My response is that I rarely use the book, except when I have to write blog posts. Gattegno works in patterns which I wrote about here. Once I understand the patterns and types of exercises he is using, I follow P's lead. We both keep math notebooks and I pay attention to his observations and that is how I know what to work on next.
These are the exercises for working with numbers 1-10.
1. Measuring Against the White Rod: This exercise demonstrates to the child that numbers are made of +1. One = +1, Two = + 1 + 1, Three = +1 +1 +1 and so on. Three means we have 3 of the +1's grouped together. It's just fancy counting. Measuring against the white also facilitates the study of fractions.
2. Measuring Against The Red: This exercise is glossed over a bit in Gattegno. When we measure against the red, we are determining if the number is odd or even. Which is one of the more important things you can know about a number. We worked a lot with odd and even when P. was little and I think he has this down. But I know when I think P. has something down, if I change the context, a little bit, I find his understanding is incomplete.
3. Creating Single Color Trains of an equivalent length: trains of a single color lend themselves to the study of multiplication, division and fractions. Don't forget to keep track of factors in your math notebook.
4. Saying And Writing Fractions: Fractions as operators is covered extensively in Gattegno's textbook 1. Fractions are no harder for P. than addition and way easier than subtraction.
5. Compliment Study: Compliments are any two rods that are equivalent to another rod. We use staircases for compliment study.
6. Complete Patterns for a Single Rod: Decomposing numbers into a complete pattern is an excellent exercise for organizing thought, developing number sense and discovering patterns. Lacy has this free exercise based on finding complete patters or permutations.
7. Measuring Any Rod Against Another Rod: After measuring each rod against the white and discovering the number name, we can study fractions, division and subtraction. Red is 2/3 of light green. There is 1 and 1/2 reds in light green. The red and the light green have 2/3 of light green in common - the difference is 1/3 of green. We can also call that difference 1.
8.Creating and Writing Equivalent Expressions - Compositions: Compositions are the crown on top of all this glory. The composition is where they take what they know and generate their own math. Your students haven't mastered math until they understand it enough to create math with it. Learning numbers turns into a study of structures. We aren't memorizing individual facts, but rather, understanding the whole integrated system.
Equivalent expressions are the foundation of our compositions. Equivalent expressions are just a variation of the substitution game. Instead of replacing a single number, the student replaces and entire expression. Example:
9 = 10 - 1 = 45 / 5 = 3 x 3 = 1/5 x 45 = 1/3 x 27 = 9/90 x 90 = (1/2 x 6) + (4/2) + (1/20 x 100)-1
There are lots of ways to do this. Lacy over at Play, Discover, Learn 24/7 posted several activities based on equivalent expressions. She has added various constraints to what the student can use to write the expression. The constraints help the student become more creative and think.
I asked her for a copy, turns out she has a packet of nearly 40 of them. She has limited the composition to 8 lines, which is just about right for beginning students. Once your student is generating these fairly quickly, just keep the tasks and write in your math notebook. I love that you can change the number every time. Which means instead of 42 pages, you have an infinite number of variations.
I would laminate these and use wet erase markers. Then you only have to print them once. You can buy these dirt cheap (no - I don't get compensation for writing this) at her TPT's site here.
The ways to study numbers is not limited to what Gattegno has written in the textbook. It is only limited by your own imagination.
Learning Numbers with Number Building
One of our favorite activities for learning numbers and number study is what we call - "Number Building". I know, I need to get more creative with my titles. Here's how you play: Pick any three rods, preferably from a bin with eyes closed. Now starting at the number 1, how many numbers can you generate with the rods using the four basic operations and fractions. If your kids can do squares throw them in too. Which rods are required to make the first 10 numbers? Which rods generate the most numbers 1-20. Which numbers are good for generating the most numbers 20-50.
Wait you say, we are only on numbers 1-10. Well, once we hit 5 we quit doing numbers in order. We just play. Gattegno didn't write the books for you to follow the exercises one right after the other.
This closes out the study of numbers 1-10, sort of. Gattegno is taking us back to staircase work in the next section. We're looking forward to it. Staircase work has some great opportunities for discovering of all kinds of patterns.
If you'd like to learn more about Gattegno and how to use you Cuisenaire Rods you can join us in the Facebook Group here. You can also download my teacher's guide - Hands-On Learning with Gattegno and you'll get directions for joining both the Gattegno Study Groups we have going on right now.