Growing Patterns

Upside Down T Counting Squares The Z Circle Pattern

Upside Down T a. Draw the next two steps in the Upside Down T Pattern. b. How many total tiles (i.e., squares) are in step 5? Step 6? c. Make some observations about the Upside Down T Pattern that could help you describe larger steps. d. Sketch and describe two steps in the pattern that are larger than the 10 th step. e. Describe a method for finding the total number of tiles in the 50 th step. f. Write a rule to predict the total number of tiles for any step. Explain how your rule relates to the pattern. g. Write a different rule to predict the total number of tiles for any step. Explain how your rule related to the pattern. Step 1Step 2Step 3Step 4

Key Questions How did you decide to solve the problem? Tell me what you are doing. Share with me what you are doing. That looks interesting, tell me about it. How is your group keeping track of your information? What changes and what stays the same in each step in the pattern? Have you solved a similar problem? Can you find a different approach to this problem? Tell me why you are....

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Counting Squares a. Assume the pattern continues, and draw the next step in the pattern. b. How many white squares will be in the 5 th step? c. If there are 7 black squares in a row, how many white squares will there be? d. How many white squares will surround 50 black squares? e. How many black squares will be in the row if there are 100 white squares surrounding them? f. How many black squares will be in the row if there are 71 white squares surrounding them? g. Generalize the number of white and black squares for any step (write a rule, make an equation, state a fact using a step, and so on).

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The Z The first three figures in a pattern of tiles are shown below. a. Write an equation that could be used to define the number of tiles in any step in the pattern.

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Reflection Questions What were the major mathematics concepts involved in the problems? What would students gain from solving these types of problems? What would students gain from presenting their solutions? How could the problem be modified for your particular grade level?

Circle Pattern The first three figures in a pattern of tiles are shown below. a. Write an equation that could be used to define the area of any step in the pattern.

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FYI: Searching for Relationships Students should extend the pattern several more frames until they are sure they understand the pattern. Next students should create a chart or table to go with their diagram. The, students should look for a recursive relationship (tells how a pattern changes from any given frame to the next frame). Then students should look for a functional relationship (a rule that determines the number of elements in a frame from the frame number). Finally, students should write the frame number to frame relationship as a formula in terms of the frame number.