Ariel Constructing Linear Equations for "Guess My Rule" and the "Ladder" Problems

PurposesProfessional development activity; Student elaboration; Student engagement; Student model building; Reasoning; Representation
DescriptionThis analytic portrays a seventh grade student, Ariel, from Frank J. Hubbard Middle School in Plainfield, New Jersey, who works on several problems involving linear functions. Ariel is one of a number of middle school students who participated in the Informal Mathematical Learning project (IML). IML was an after-school, 3-Year National Science Foundation-funded longitudinal study (Award REC-0309062) conducted by the Robert B. Davis Institute For Learning at Rutgers University. This analytic provides evidence of student reasoning as Ariel explores different approaches to solving problems that link various algebraic concepts. Ariel’s engagement in the problems posed by the researchers in this analytic illustrates four Common Core State Standards (CCSS) For Mathematical Practice (CCSSI, 2010, p. 6-8):
1. Make sense of problems and persevere in solving them.
4. Model with mathematics.
7. Look for and make use of structure.
8. Look for and express regularity in repeated reasoning.

In this analytic, Researcher Arthur Powell first introduces Ariel and his seventh grade classmates to an activity called “Guess My Rule” (Alston & Davis, 1996, p. 14). He challenges students to solve several problems, each in the form of a truth table, which is made up of a number of coordinate pairs that fit a particular rule. Researcher John Francisco later presents Ariel with the “Ladder Problem,” (Alston & Davis, 1996, p. 15-16). This problem is analogous to the "Guess My Rule" activity; however this problem is modeled with Cuisenaire rods and values are not recorded in a truth table. The analytic concludes with Researcher Cecilia Arias interviewing Ariel one year later in the eighth grade, where he revisits the “Ladder Problem.” This analytic shows Ariel several times working with James, a classmate who was also part of the IML project.

Dr. Robert B. Davis (1992) of Rutgers University wrote: “Students are determined to understand, and they create their own ways of understanding” (p. 226). The Researchers seen in this analytic follow what Dr. Davis (1992) referred to as an “Emerging New View of Mathematics Education” by presenting Ariel with “problems or tasks” instead of starting with “mathematical ideas and then applying them” (p. 237). This “New View,” (Davis, 1992, p. 237) facilitated by the researchers as demonstrated in this analytic, can empower students to be fluent not only in the Common Core State Standards For Mathematical Content, but engaged in the Common Core State Standards For Mathematical Practice as well.

Both the "Guess My Rule" and "Ladder" problems are ideas of Dr. Robert B. Davis. Dr. Davis referenced the idea of box and triangle and the use of Cuisenaire rods in his work Discovery in Mathematics A Text For Teachers published in 1980, yet it was "In the 60’s when these materials were first published" (Davis, 1980, Preface section, para. 1).

The problems that Ariel is seen working on in this analytic are as follows:

The first "Guess My Rule" problem presented orally by Researcher Powell and the table developed by Researcher Powell and the students:

☐ ∆
5 13
3 7
6 16
8 22
4 10
0 -2

The "Guess My Rule" problems presented on worksheets:
Problem 1.
0 1
1 3
2 5
3 7
4 9
5 11

Problem 2.
0 5
1 7
2 9
3 11
4 13

The Ladder Problem

Problem posed orally to Ariel by Researcher Francisco:
"How many rods would there be for a ladder with ten steps?"

Worksheet wording of the problem presented to Ariel by Researcher Arias:
"A company makes ladders of different heights, from very short ones to very tall ones. The shortest ladder has only one rung (we could build a model of it with 5 light green Cuisenaire rods.)
A two-rung ladder could be modeled using 8 light green rods.
Build a rod model to represent a 3-rung ladder.
How many rods did you use?
How could you represent the number of rods needed if you were to build a ladder with any number of rungs?"

All problem statements and student work can be accessed through The Video Mosaic Collaborative;

The “Informal Mathematical Learning” Project, led by Carolyn Maher, Arthur Powell, and Keith Weber of Rutgers University is supported by the National Science Foundation ROLE Grant REC0309062. The views expressed in this analytic are those of the authors and not necessarily those of the National Science Foundation.


Alston, A. S., & Davis, R. B. (1996). Proceedings from ’96: Report of a Mathematics Education Seminar for Teachers and Educators, The Development Of Algebraic Ideas. Rio de Janeiro, Brazil.

Common Core State Standards Initiative. (2010). Common Core State Standards For Mathematics. Retrieved from:

Davis, R. B. (1980). Discovery in Mathematics A text for teachers. New Rochelle, NY: Cuisenaire Company of America Inc..

Davis, R. B. (1992). Understanding “Understanding”. Journal of Mathematical Behavior, 11, 226, 237.

Maher, C. A., Sigley, R., & Wilkinson, L. (2013). Tracing Ariel’s growth in algebraic reasoning: A case study. Psychology of Mathematics Education, 4, 219.
Created on2014-03-23T16:38:53-0400
Published on2015-06-16T17:14:00-0400
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