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Teaching Math: A Video Library, 9-12

Staircase Problem

Content Standards
Functions, Algebra, Discrete Mathematics
Process Standards
Reasoning, Problem Solving, Communication
School: City on a Hill Public Charter School; 65 students
Location: Boston, Massachusetts
Teacher: Jesse Solomon
Years Teaching: 4
Students in Classroom: 17
Grade: 9
Staircase Problem
Video Overview
Students are challenged to identify a pattern and then find a rule that determines the pattern, in this case the relationship between the number of steps in a staircase and the number of blocks needed to build them. Working in groups, students use small paper squares to represent steps, and record their data and questions in tables they create on chart paper. At the end of class, Mr. Solomon asks students to write their strategies and conclusions for homework.

The Class Challenge
The drawings below illustrate staircases with one step, two steps, and three steps.

6

3

4

5

1

1

2

1

2

3

Students use the drawings to solve the following problems:
How many blocks are needed to make a staircase with 50 steps? 100 steps? n steps?

Course and Curriculum
City on a Hill, a public charter school founded in 1995, is state–funded and operated by a board of supervisors. Most students have not taken algebra prior to this required course, which integrates geometry and algebra. Mr. Solomon uses this and similar lessons throughout the year to teach students how to generalize from arithmetic to algebra and how to use multiple representations and approaches to solve a problem. In this lesson, one student recalled an earlier activity and asked about using a one–step or a two–step rule, terms the students created for situations solved with one rule and situations solved with two rules. To solve the Staircase Problem, students should see the first staircase as 1 block, the second staircase as 1 + 2 blocks, the third staircase as 1 + 2 + 3 blocks, and continue until they recognize the pattern. In previous units, students learned how to estimate and measure, find patterns, and evaluate expressions. They also studied the distributive property, congruence and symmetry, powers, scientific notation, and angle relationships. The following day, students used their data to define rules and find relationships between the height of the staircase and the number of blocks in the staircase. In future units, students worked with equations, square roots and cube roots, data, matrices, graphs, measures of central tendency, histograms, stem–and–leaf plots, and representation of functions in graphs.

A Pre–Viewing Exploration for Teacher Workshops
What Happens in the Locker Problem?
Objective: To think about the process of finding and using patterns to solve mathematical problems.
Materials: One set of 20 index cards marked with the numbers 1–20 for each pair.
Activity
Solve this problem in pairs.

As 900 students enter a school, they pass lockers that are numbered from 1 to 900. The first student opens every locker; the second student closes every second locker; the third student changes the position of every third locker (by opening the closed lockers and closing the open lockers); and the fourth student changes the position of every fourth locker. This pattern continues for all 900 students. Which lockers are open after all students enter the school?

Turn your first five cards face up to represent 5 lockers. Open and close the lockers for five students. Which lockers are left open?
Which lockers are left open when you have 10 lockers and students? 15? 20?
What conjecture can you make about a rule for 900 lockers and students? n lockers and students?
Exploring Content

Justify your generalization with mathematics.
What mathematical topics would be helpful for students to know before using patterns to establish rules for functions?

Exploring Teaching Issues

Where would you use a problem like this in your curriculum?
What kinds of questions do you ask students to assess their knowledge of mathematical content and their ability to understand a problem-solving process?

Exploration Answers
Activity

1 and 4.
Number of
Lockers Open

10 1, 4, 9

15 1, 4, 9

20 1, 4, 9, 16

In both situations, the lockers whose numbers are square remain open.

Exploring Content

Each perfect square has an odd number of factors, while other numbers have an even number of factors.
An even number of actions results in a closed locker; an odd number of actions results in an open locker. If the locker number has an odd number of factors, such as 9, the locker will be opened, closed, and opened again. This locker will be opened at the end of 900 passes. Factors of numbers are pairs, except for square numbers; thus square numbers have an odd number of factors. Lockers with square numbers will remain open.

Functions, including what a function is, where it comes from, and that functions can represent different situations; and sequences, both arithmetic and geometric.

Topics for Discussion
What would you expect from students' writing about their understanding?

What algebraic and geometric concepts did students use in their discussions?
Based on the students' discussion, what instructional decisions would you make?
How do students benefit from writing their mathematical thinking as they work on a problem?
What and when do you ask students to write about mathematics?
How do Mr. Solomon's questions promote student reasoning?

What questions would you ask to help students identify the relationship between the staircase height and the number of blocks needed to build the staircase and then to generalize it symbolically?
How did students use Mr. Solomon's suggestion that they think in terms of a square?
Students attempted to develop two rules: (1) a recursive rule, which yields the number of blocks needed for a given height if you know the number of blocks needed for the previous height; and (2) a rule for finding the number of blocks needed for a single height. How does Mr. Solomon challenge students to understand the difference between these rules? Is one rule easier to use? Explain.
What questions could Mr. Solomon ask the following day to help students discover functional relationships and make predictions based on their data?
How can you help students balance collecting data with formulating a rule?

Why do you think some students tried to find a rule before collecting the data and finding patterns?
How would you ensure students spend enough time gathering data and understand why that time is necessary?
The correct answer to the Staircase Problem is S(n) = n(n+1)/2, with n representing the number of steps and S(n) representing the total number of blocks. How could students verify this rule?
Discuss the similarities and differences between the mathematical structure of arithmetic and algebraic systems. What is gained, lost, and retained in moving from one system to the other?