Students build and analyze geometric growing patterns, determine rules, and write simple expressions to describe the growth.
Storyboard: This two-part interactive investigation of patterns encourages students to use visualization, spatial reasoning, and geometric modeling to build their own patterns or recreate them from given square arrays. As teachers are encouraged to reflect on implementing technology effectively, students are challenged to develop flexibility using multiple representations.
Use a classroom object to illustrate a growing pattern. Here are two possible activities:
Books on the Shelf: Begin the lesson with two books already on a shelf.
Explain that each day we will get three more books. Ask students to work with a partner. What will the shelf look like on Day 1? Day 2? Day 3? Day 5? Day 10? After sufficient time, allow partners to share their strategies for figuring this out. Engage class in a discussion about when we might reach a goal of 50 books.
Answers (Books on the Shelf):
Goal of 50 books on the shelf. At this point it is not important to figure out the exact answer of 16 days. The hope is that students will produce an estimate greater than 6 but less than 30, with reasoning to support their thinking.
Creating, Describing, and Analyzing Patterns: Begin by allowing students to explore patterns by engaging with technology. Direct them to the NCTM interactive activity, Creating, Describing and Analyzing Patterns. Prompt them to create pattern units of 2 to 5 squares and visualize how the grid will look when their pattern is repeated before it is displayed on the grid. If time allows, ask them to try these challenges:
Students explore growing patterns using the actual pattern and tables and determine a rule to tell what comes next.
Students continue to explore growing patterns and rules to determine what comes next. They analyze, describe, and justify their rules for naming patterns. Since students are likely to see growing patterns in a different way compared to their classmates, this is an opportunity to engage them in communicating about mathematics. This lesson requires students to explain correspondences among their verbal descriptions of the patterns, tables, and graphs that will help them eventually build an equation to solve the problem.
In this lesson, students use the idea of what comes next to determine the relationship between the pattern number and number of objects in the pattern (explicit rule).
Students explore a toothpick staircase problem to apply their skills of finding the rule to describe the relationship between corresponding terms.
Continue to explore relationships between terms by exploring a growing pattern that involves several rules.
Pose interesting, but more difficult-to-generalize growing patterns.
Each lesson has a formative assessment task and/or questions teachers should pose during class discussion to gauge students’ understanding of the concept. To revisit the idea presented in the Hook you can present the following scenario:
Day 1 - 8 cans
Day 2 - 16 cans
Day 5 - 40 cans
Day 10 - 80 cans
Goal of 100 cans: You will not reach the exact goal of 100 cans (on Day 12 there will be 96 cans and on Day 13 there will be 104 cans). However, the goal is reached on Day 13. Encourage students to consider how they could change the problem so that the result would be exactly 100 cans (either adding 4 or subtracting 4).
How can I express a growing pattern with multiple representations (picture, table, graph, formula)?
algebraic thinking, rules, formulas, patterns, algebra, growing patterns, generalizing
CCSS, Content Standards to Domain Level
CCSS, Standards for Mathematical Practices