Practical patterns

Teaching Pre K-8, Nov/Dec 2001 by Naylor, Michael

Activities to develop algebraic reasoning at every grade level.

Patterning activities are central to much of mathematics at all grade levels. This month, we have a trio of activities that will help children develop algebraic reasoning.

The Nines Board (Grades 2-5)

There are exciting patterns hidden inside our familiar counting system. Hunting for patterns on a nines board is fun, can motivate some rich classroom discussion and develops algebraic reasoning and place value concepts.

Preparation

Students should have graph paper with squares large enough for writing a two-digit number in each square. Two-- centimeter graph paper works great.

Action

Have your students draw a 9 x 11 rectangle (9 across and 11 up and down). Ask them to write the whole numbers from 1 to 99 in nine columns, so that 1-9 are in the first row, 10-18 are in the second row and so on as shown in the nines board on this page. Divide your students into groups and give them several minutes to find and record any patterns they find. After the initial burst of discovery has settled, have each group explain the patterns they found.

Discussion

Here are some of the patterns your students may find:

* Each number differs by 9 from the one above it.

* The last column contains multiples of 9.

* Numbers with the same last digit lie on the left-to-right diagonals.

* The sum of the digits of the numbers in a column is the same, with a few exceptions. For example, the numbers in column 7 are 16, 25, 34, etc., and 1 6=7 ,2 5 = 7, etc. The ninth number in that column, however, is 79, and 7 9 =(diagonal line through equal sign) 7. But, 7 9 = 16, and 1 6 = 7. Therefore, 7 is the "digital roof' of 79.

Extension: Digital Roots

When you add the digits of a number together and repeat the process on the sum until only one digit remains, this digit is called the digital root of the original number. For example, to find the digital root of 685, add 6 8 5 to get 19. Nineteen is not a one-digit number, so add 1 9 to get 10, then 1 0 to get 1. The digital root of 685 is 1.

Numbers on the nines board share digital roots with others in the same column because the numbers differ by 9 from those above or below it, and 9 = 10 - 1. In other words, to move down a column, you can add 1 to the tens place and subtract 1 from the ones place, which doesn't change the sum of the digits.

Super Nines Board

(Grades 5-8)

This board is so full of patterns and place value concepts that the act of constructing it teaches interesting properties of numbers.

Preparation

Students need two-centimeter graph paper.

Action

Have your students construct a 10 x 11 array of multiples of 9. Talk through the first row-- and-a-half, then let them finish on their own. Students should note shortcuts they take while completing the chart - many patterns will be noticed in the process. Don't make copies of the above chart for your students. Have them check their work with others in their group.

Discussion

As you move down a column, you'll find the following:

* The numbers have the same last digit (every shift in column is a change of 90, zero ones are being added).

* The tens digit decreases by one and the hundreds digit increases by one (a change of 90 is the same as adding 100 and taking away 10).

* The digital root of every number in the chart is 9.

* The numbers along the main diagonal from upper left to lower right are all multiples of 9 with a zero in the middle. How do the differences between these numbers relate to the digits?

Color Cube Patterns

(Grades K-1)

Children in the lower grades are learning the counting sequence and how number names relate to quantities. They're not ready to consider patterns in place value concepts. Here's a pattern activity using colors to develop algebraic reasoning.

Without allowing students to see your pattern, place colored cubes in a line, perhaps starting with red, blue, red, blue, etc. Place a cup over one cube, and ask students to guess what color is hidden underneath. Try various patterns, and try covering up more than one cube.

Children must be able to recognize the pattern and mentally repeat it to find what's missing. Looking for patterns and thinking about unknown elements in a sequence provide important foundations for powerful mathematical reasoning.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA and a Teaching Editor of Teaching K-8. E-mail: mnay lor@cc.wwu.edu