# Numbers In A Series – Another Round of Staircases

One of last things in Gattegno Mathematics Textbook 1 is a section on numbers in a series. You can get a free copy online here. If you've been playing with staircases, you'll know that all we are doing is extending staircase work to 20.

On page 93, Gattegno introduces the words increasing and decreasing order. We write numerals from 1 to 20 and state those are in increasing order. What do we notice about increasing order?

We can then write numerals 20 to 1 and state those are in decreasing order. What do we notice about decreasing order?

We want to offer students the opportunity to put various strings of numbers in increasing and decreasing order. These numbers need not be consecutive. P. and I spent about 10 minutes on this exercise.

### Numbers in a Series: Ordinal Numbers

If your student hasn't been introduced to ordinal numbers yet, now is the time to do it. I hadn't specifically worked with ordinal numbers yet, but I wasn't surprised that P. already understood this. He sort of felt that this exercise was babyish. I did have him put some random numbers in increasing order and then name the first, second, third . . . numbers. I wanted to make sure that he understood that the first numeral in a series wasn't necessarily the numeral 1.

Likewise, we can write 20, 18, 16, 11, 8, 6, 3, 1 and the numeral one is the last digit. He had no trouble with this either.

### Numbers in a Series: Beyond the Obvious

Gattegno has this way of taking things over the top. We don't just write addends up to ten, we try to figure out all possible combinations of numbers by making a mat. It's like math on steroids.

Therefore, one ought not be surprised that writing numerals in increasing and decreasing order is just the first step to thinking about increasing and decreasing order.  Once the student has worked with numerals, we will then work with expressions.

I found that taking various expressions for different numbers from our number studies is helpful. We have many of these written down on expression cards and stored in a plastic container for use with different games. I can easily grab a handful and have P arrange them in increasing and decreasing order. You can download the template here to make your own.

### Numbers in a Series: Counting by a Common Difference

Student's were introduced to staircases back in chapter 2. They've measured the distance between steps, they varied the distance between steps and wrote statements of addition and subtraction using both letter names for the rods and number names. Now we are working with numbers in series, which is exactly the same thing, only this time, there are no rods and students are working with symbols only.

The task is to write the numbers 1- 20. Starting at one, count by two and write all the numbers the student gets in order. Then repeat going down the series starting at 19.

We will want to do this again starting at 2 going forward and backward. We'll repeat the activity counting by a common difference of three and four starting at different numbers, such as one, two and three.

We also created the following table on the white board for the number four and compared the numbers in the columns to the different series of four that we wrote down. Then we made some observations about them.

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### Numbers in a Series: Finding the Common Difference

The last task on numbers in a series is to have the student note the difference between each number in a series.

For instance:   1    4     7    10     13     16    19 .  What is the common difference between each numeral?

We did several variations of the above exercise. Other than for differences of 5 and 10, I didn't extend the series beyond 20.

The next post will be my last on Gattegno Textbook 1.  If  you want to learn how to use Cuisenaire Rods to teach your student math or would like to connect with others who are using base ten blocks, you can join us in our ​Facebook Group  here