Defining Reasoning and Proof
 Introduction | Pattern Recognition | Repeating Patterns | Reasoning About Repeating Patterns | Growing Patterns | Classification | Your Journal

Although most patterning experiences for young students will focus on repeating patterns, students can also begin to visualize and talk about growing patterns in the early grades.

A linear growing pattern is a pattern that increases or decreases by a constant difference. For example:

 In this linear growing pattern, each row is one greater than the previous row. Kindergarteners reinforce their understanding and ability to reason about counting numbers through examples of growing patterns.

Let's take a look at a classroom experience with another growing pattern.

Teacher: What do you notice about these houses?
Emile: There are squares and triangles.
Martha: They get bigger each time.

Teacher: Describe what you mean by "bigger."
Martha: There are more squares and triangles with each house.
Fred: I see a pattern. Each time there is another house, there is one more square and one more triangle.

Teacher: What do you think the fifth house will look like?
Emile: I think it will be one bigger than House 4.

Teacher: Can you show me?
Emile uses pattern blocks to build the fifth house:

Teacher: Describe your house to me.
Emile: It has five squares and five triangles.

Teacher: What about the next house?

The students continue to build and describe the houses until Martha notices another pattern.

Martha: I see something else: The number of the house is the number of triangles and squares.

Teacher: So, what do you think House 20 will look like?
Martha: It will have 20 triangles and 20 squares.

Teacher: How do you know that's correct?
Martha: There's a pattern. See, House 1 has one square and one triangle, House 2 has two squares and two triangles, House 3 has three squares and three triangles, and it just goes on and on so that the number of the house is the same as the number of squares and triangles.

Teacher: What do you think House 100 would look like?
Emile: I think it would have 100 squares and 100 triangles -- but we don't have enough blocks to build that one!

Teacher: Do you think if you knew how many squares and triangles there were, you could figure out what house it is? Let's try. I have a house with 10 squares and 10 triangles.
Fred: It must be House 10, because the number of squares and triangles is the same as the house number. This is just like finding the number of squares and triangles, only backward.

A great deal of reasoning is occurring in this activity. The children begin by describing the pattern and extending it with physical materials. They make conjectures about the pattern and predict the number of tiles in elements that come later in the series. Finally, they make generalizations about the number of squares and the number of triangles for any house, based on the patterns they have discovered. Throughout the entire activity, the children explain their reasoning. The teacher challenges their thinking by asking probing questions and extending the activity.

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