Task:Use the pattern below to answer the following questions. Please show all work.
- Draw the next two steps in the pattern.
- Describe how you knew how to draw Step 5.
- How many total tiles are in Step 5? Step 6?
- How can you determine the number of tiles in Step 25?
- Write a rule to predict the total number of tiles for any step. Show how your rule relates to the pattern.
- Try to think of a different rule that you can use to predict the total number of tiles for any step. Show how your rule relates to the pattern.
Alignment to Content Standards: / Alignment to Practice Standards:
Mathematical Goals:
In addition to their potential to help students develop algebraic reasoning, pattern tasks may also be a powerful tool in helping establish classroom norms and practices at the beginning of the school year. All students, regardless of prior knowledge and experiences, can explore pattern tasks. In addition, students have the opportunity to participate in a learning community.
There are usually many different, yet equivalent, ways of expressing the relationship between two variables in a pattern task. This addressesseveral foundational concepts to solving equations including exploring the equivalence of expressions and simplifying expressions to show equivalence. In addition, solving equations in multiple ways also encourages students to become flexible in their thinking.
Relationships between the quantities in pattern tasks can be expressed using symbols, tables, and graphs, as well as words. Thus, pattern tasks give students opportunities to make connections among representations – a key component in developing an understanding of function.(Mathematics Teaching in the Middle School, Smith, Hillen, Catania, August 2007)
It will be valuable to revisit this task as you begin to explore sequences (recursive and explicit strategies). Students can also use this pattern to explore the growth rate of a linear pattern and explore this growth represented in a table, graph, and equation.
Prerequisite Knowledge:
What definitions, concepts, or ideas do students need to know to begin to work on the task? / Accessing Prior Knowledge:
What questions will you ask to help students access their prior knowledge and relevant life and cultural experiences?
Resources/Tools:
What resources or tools will students have to use in their work that will give them entry into, and help them reason through, the task? / Structure:
How will the students work—independently, in small groups, or in pairs—to explore this task? How long will they work individually or in small groups or pairs? Will students be partnered in a specific way? If so, in what way?
Questions that Assess and Advance Student Learning:
How will you help a group get started on the task? How will you focus student thinking on the key mathematical ideas in the task? How will you assess student understanding of key mathematical ideas, problem-solving strategies, or the representations? How will you advance student understanding of the mathematical ideas? How will you encourage all students to share their thinking with others or to assess their understanding of their peers’ ideas?
Reminder: The questions you ask should keep the task at a high level, not push kids toward a particular solution path, turning the task into an opportunity to practice procedures.
Task Solutions:
What are all the ways the task can be solved? / Anticipated Student Solutions:
Which of these methods do you think your students will use?
Select:
What solution paths do you want to have shared during the class discussion? / Sequence:
In what order will the solutions be presented? Why? / Connect:
What specific questions will you ask so that students: Make sense of the mathematical ideas you want them to learn? Expand on, debate, question, the solutions being shared? Make connections among the different strategies that are presented? Look for patterns? Begin to form generalizations?
Anticipating Misconceptions:
What misconceptions might students have? What errors might students make? How will you address the errors?
Extensions of the Task:
What will you do tomorrow that will build on this lesson?
Mathematical Pattern Task: Use the pattern below to answer the following questions. Please show all work.
- Draw the next two steps in the pattern to the right of Step 4.
- Describe how you knew how to draw Step 5.
- How many total tiles are in Step 5? Step 6?
- How can you determine the number of tiles in Step 25?
- Write a rule to predict the total number of tiles for any step. Show how your rule relates to the pattern.
- Try to think of a different rule that you can use to predict the total number of tiles for any step. Show how your rule relates to the pattern.