Lesson 2 -- It Begins with Good Instruction

Differentiated Lesson Example

Grade 6 -- Mathematics

Concept: Patterns

Key principle: There are rules that govern patterns.

As a result of this learning experience, students should be able to:

• Know (facts) -- Exponents, exponential growth, particular number patterns (powers of 2, squared numbers)
• Understand (principles) -- Patterns can be generalized, and mathematicians can use symbolic language (algebra) to describe patterns efficiently and in general terms
• Do (skills) -- Organize a search for patterns using a t-table

Today's experience:

This learning experience is designed to engage students in problem-solving situations that involve searching for meaning in patterns and organizing information about the patterns to enable them to figure out the rules underlying the patterns.

The lesson begins with a whole class activity that includes the following assignments.

• Read aloud The King's Chessboard (a story in which a wise man solves a problem for his king and asks as his reward that he receive one grain of rice on the first square of a chessboard and that the amount be doubled every day for the next square until all squares have been used.) Stop throughout the story so students can use their calculators to figure out the number of rice grains that have accumulated.
• Discuss with students how mathematicians reading this story would have the urge to translate this information into numerical form and then play with it to see if there is something happening that can be described mathematically.
• Using data from the story, introduce the t-table as an organizing tool to help students think, recognize, and predict patterns. Students and teacher discuss the pattern in the table and write a verbal rule for the pattern. ("It doubles each time.")
• Introduce the idea of exponents as an efficient way to show the pattern. ("It doubles every time" becomes 2 squared). Discuss the idea that mathematicians want to be able to make a general statement about the pattern they see, turning verbal rules into symbolic language of algebra, and how the t-table helps you see the pattern more easily.
• Have students write an exit card to explain how the principle in The King's Chessboard works.

  The learning experience continues with a set of tiered activities using another problem to provide opportunities for students to look for meaning in patterns using t-tables as an organizing tool. Teachers assign students to one of three groups based on readiness. All three groups

• Solve the same problem.
• Must produce a t-table.
• Must write a description of the process they followed to discover the pattern.
• Must write a description of the pattern itself, using either words or symbols.

The Locker Problem

A school has 1,000 lockers and 1,000 students. The students decide to have fun one day, so they take turns opening and closing the lockers, according to the following plan.

• The first student opens every locker.
• The second student closes every second locker.
• The third student opens every third closed locker.
• The fourth student closes every fourth open locker.
• The students continue in this manner until all 1,000 students have had their turn.

When all the students are finished, which lockers remain open?

Students work in pairs or individually (their choice) on the assigned version of the Locker Problem.

Group 1:

• Two-color counters are provided for physically modeling the open/closed sequence.
• Students receive a graphic that lets them record the open/closed position for the first 16 lockers.
• Students receive a t-table with appropriate column headings for recording information about which lockers remain open.
• Students must write a description of what they did to discover the pattern and describe the pattern in words.

Group 2:

• Students receive a graphic so they can record the open/closed position for the first 16 lockers.
• Students must produce a t-table, and no format is provided.
• Students must write a description of their process and describe the pattern in words.

Group 3:

• Only the problem is provided (no graphic, no formatted t-table).
• Students must produce a t-table.
• Students must write a description of what they did to solve the problem and must provide the generalized, algebraic rule.

After students have completed the problem, the whole class discusses how they thought about solving the problem, the tools they used, and their varied ways of solving it. Then students select from one of two homework assignments -- one requires solving another problem that involves a pattern, t-table, and exponents, the other has students develop such a problem.

Note: Adapted from unpublished teacher materials provided by Carol Ann Tomlinson, University of Virginia.

Analysis

This activity is differentiated primarily on the concrete to abstract continuum and the more structured to more open continuum.

Group 1's task is a little more concrete than the other groups' tasks. Group 1 students are given the two-color counters that allow them to manipulate the opening and closing sequence, a t-table that is formatted with column headings, and a graphic for recording the open/closed sequence. The task is not all the way to the concrete end of the continuum, because the graphic only accommodates recording of 16 lockers. Students will need to think about the development of the pattern over 1,000 lockers.

Group 2's work is a little more abstract, because students don't have the counter manipulative or a formatted t-table.

Group 3's task is the most abstract of the three groups because students do not have the counters, graphic, or t-table provided, just the problem. This requires them to determine how to construct the t-table and how to identify the pattern. Group 3 must provide the algebraic rule rather than describe the pattern only in words. This requirement steps up the abstract level of the students' task.

The presence and absence of the counters, graphics, and formatted t-tables also vary the structure of the task for each group. Group 1's task is the most structured of the three, but it is not at the far end of the continuum because students don't receive the exact steps for solving the problem. Group 2's task is less structured through the absence of the counters and formatted t-tables. Group 3's task is the most open of the three because students must decide how they will approach finding the solution.

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