Task: Will and Latisha s Tile Problem Algebra I

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1 Tennessee Department of Education Task: Will and Latisha s Tile Problem Algebra I In math class, Will and Latisha were challenged to create their own pattern with tiles. Latisha built the first three arrangements of their pattern as shown below. Both Will and Latisha decided to investigate continuing the pattern. Arrangement 1 Arrangement 2 Arrangement 3 A. Will and Latisha wondered how many tiles would be needed to build each of the arrangements below. If the pattern continued, determine how many tiles would be needed for each arrangement and show how you determined your answer. 5 th Arrangement 10 th Arrangement 101 st Arrangement B. They decided that it would be much more efficient if they could use what they know to generalize a way to find the number of tiles needed for any arrangement. Write a function rule that could be used to find the number of tiles needed to build any arrangement in this pattern. Show how your function rule relates to the arrangements. C. Will claims that there is an arrangement in the pattern that would take 56,646 tiles to build. Is he correct? Use mathematics to justify your answer.

3 Explore Phase Possible Solution Paths Part A. 5 th Arrangement: th Arrangement: st Arrangement: 10,406 Some students may just build with tiles or draw the arrangement on graph paper and count the tiles. (5 th and 10 th ) Students may use various calculations based on how they visualize the arrangements. These concrete calculations can lead to Part B (the general case). Part B. Below are some possible ways that students may be thinking about their function rule. 1. Partitioned: (n + 1) n = number of arrangement 2 Assessing and Advancing Questions Build the next arrangement. Build the next arrangement (5 th ). How did you know to build the arrangement that way? Describe how you found the number of tiles. What do you notice about the relationship between the arrangement number and the number of tiles in the arrangement? How could you use what you know about the 5 th and 10 th arrangements to find the number of tiles for the 20 th arrangement? 50 th arrangement? 101 st arrangement? What do you notice is changing and what is staying the same about each arrangement? Show me that your expression works for the known arrangements. Where is the (n + 1) 2 in your arrangement? Where is the 2? How could you partition the arrangements differently? What expression would represent this? How does your new expression relate to your first expression?

4 2. Partitioned: n 2 + 2(n + 1) + 1 n= number of arrangement Show me that your expression works for the known arrangements. Where is the n 2 in your arrangement? Where is the 2(n + 1)? Where is the 1? How could you partition the arrangements differently? What expression would represent this? How does your new expression relate to your first expression? 3. Partitioned: n 2 + 2n + 3 n = number of arrangement Show me that your expression works for the known arrangements. Where is the n 2 in your arrangement? Where is the 2n? Where is the 3? How could you partition the arrangements differently? What expression would represent this? How does your new expression relate to your first expression?

6 Part C. Below are several methods that students may use to solve the problem. 1. Completing the Square Note the connections to some of the representations. Why did you decide to use Completing the Square? Talk me through how you solved the problem. Given the scenario, which value for n would make How could you work the problem a different way? How would you know which method to use? 2. Quadratic Formula Why did you decide to use the Quadratic Formula? What would be the values for a, b, and c? Talk me through how you solved the problem. Given the scenario, which value for n would make How could you work the problem a different way? How would you know which method to use?

7 3. Some may use the generated function rule of (n + 1) and solve by Taking the Square Root. Why did you decide to use this form of the equation? Talk me through how you solved the problem. Given the scenario, which value for n would make How could you work the problem a different way? How would you know which method to use? 4. Graph or Table finding the x intercepts (roots) n 2 + 2n + 3 = 56,646 n 2 + 2n 56,643 = 0 Why did you decide to use graphing to solve this Or, why did you decide to use a table to solve this Why did you decide to use this equation? How did you use your calculator to solve this Show me the process. What does the x value represent in the graph/table? What does the x value represent in the scenario? Given the scenario, which value of x would make What if you decided to use x 2 + 2x = 56,643? a. How would this change the graph/table? b. How are the graphs related? How could you work the problem another way? How would you know which method to use?

8 5. Graphing or Table finding the intersection n 2 + 2n + 3 = 56,646 n 2 + 2n = 56,643 Why did you decide to use graphing to solve this Or, why did you decide to use a table to solve this Why did you decide to use this equation? How did you use your calculator to solve this Show me the process. What does the x value represent in the graph/table? What does the x value represent in the scenario? Given the scenario, which value of x would make What if you decided to use x 2 + 2x 56,643 = 0? c. How would this change the graph/table? d. How are the graphs related? How could you work the problem another way? How would you know which method to use? Possible Student Misconceptions 1. The function rule that students generate may work for a specific arrangement but not in the general case. Have students test their expression with the known (built) arrangements. Does your function rule work? Why or why not? Advancing Question: Have students show how each part of the expression relates to the arrangement drawing.

9 What does the variable represent in your expression? 2. Students may let the variable represent the number of tiles in the arrangement, not the arrangement number. 3. Students may approach the sequence as recursive. Although this is a way of representing a sequence, the intent of this task is to generate an explicit form. Entry/Extensions If students can t get started. Advancing Question: How could you define your variable in a way that would allow you to find the number of tiles in the 20 th arrangement? 40 th arrangement? 101 st arrangement? What does the variable represent in your expression? Why did you choose to represent the sequence with this approach? What limitations does this recursive approach have in this particular Advancing Question: How could you generate a function rule that does not rely on needing to know the number of tiles in the prior arrangement? Assessing and Advancing Questions What is the question asking? Have students draw or build the next few arrangements and describe how they determined the arrangement. Have students mark/show what each part of the arrangement represents from their expression. Have students test their expression with the known (built) arrangements. Does your expression work? Why or why not?

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