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**Connection Patterns & Functions.2**

Mosaics Connection Patterns & Functions.2

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**When have you solved a problem with an equation or graph?**

Learning Target Connecting Patterns & Functions Target 3a I can write and graph equations and use them to solve problems. When have you solved a problem with an equation or graph?

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**Launch How do you see the pattern growing?**

What do you think we will do with this pattern?

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**Describe the situation**

Reuben’s Mosaic Reuben learned in art class that a mosaic is made by arranging small pieces of colored material (such as glass or tile) to create a design. Reuben created a mosaic using tiles, then decided on a growing pattern and created a second and third mosaic. He continued his pattern my building additional mosaics. He counted the number of tiles in each mosaic and the represented the values in multiple ways. He thinks he sees a relationship between the mosaic number and the total number of tiles in the mosaic. PTT Read Describe the situation to a partner.

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Reuben’s Mosaic Represent Reuben’s set of values from the mosaics problem in at least three ways, including a general function rule, to determine the number of tiles in any mosaic. Write a description of how your rule is related to the mosaic picture. Include a description of what is constant and what is changing as tiles are added.

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Reuben’s Mosaic 3. How many tiles would be in the ______ mosaic? Use two different representations to show how you determined your answer. 4. Would there be a mosaic in Reuben’s set that uses exactly ______ tiles? Explain your reasoning using at least two representations. Be prepared to share with the class.

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Reuben’s Mosaic Use what you learned in the discussion to answer the questions. Describe all of the quantities that can be used for any input of the mosaic. 6. Describe all of the quantities that can be used for any output of the mosaic. (It may be easier to find quantities that cannot be used.)

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Function Notation We can use function notation to describe the relationship between the mosaic number and the number of tiles in that mosaic. In Reuben’s mosaic r(1) = 5 means Mosaic 1 is made with 5 tiles. What do you think r (2) = 8 means? What do you think r(5) means? Find r(5) = ___.

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**Function Notation What do you think r(n) = 32means?**

Find the value of n such that r(n) = 32. Fill in the blanks: r(n) is the _______ (input or output) which counts _________. n is the _______ (input or output) which counts _________.

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**Reuben’s Mosaic In Reuben’s mosaic there are two tiles in the center.**

8. How would the function rule change if the center contained 4 tiles instead? Explain your reasoning using at least 2 representations.

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A New Mosaic 9. For the function equation t(n) = 2 + 4n describe in words and/or diagrams the first two mosaics and a rule for the general mosaic.

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**More Mosaics For each of the mosaics:**

f(n) = n B) g(n) = 4n + 3 C) g(n) = 4n + 6 10. Draw the first three mosaics. 11. Make a table. 12. Graph all three functions on the same axis. 13. Describe any similarities among two or three of the mosaics. 14. Describe any difference between two or three of the mosaics.

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You have been looking at geometric patterns and ways in which those patterns can be represented with x→y tables, graphs, and equations. In Lesson 4.1.2,

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