### Early algebra ideas involving two variables, Clip 18 of 18: Problem 10 and the value of keeping secrets

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the final clip of the series of 18 for Early Algebra Ideas Involving Two Variables with the class of 6th grade students, researcher Robert B. Davis presents the table for problem 10 (see below) as a challenge for the group to consider for future sessions. The students begin to analyze the number patterns in the table. Brian, Romina, Ankur and Michele I. report the pattern of differences in the "triangle" column to Davis. The...

### Early algebra ideas involving two variables, Clip 14 of 18: Publishing the "code" for problem 6

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the fourteenth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, a number of the students present their equations for Problem 6 (printed below) to the researcher, Robert B. Davis. He notes that when mathematicians are satisfied with a "secret" they publish it to the community, suggests that this might be the time for the Table 6 equation...

### Early algebra ideas involving two variables, Clip 15 of 18: What about problem 7?

Robert B.) Robert Benjamin Davis, Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the fifteenth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, the researcher, Robert B. Davis, asks the students to think about the table for problem 7 (printed below). Several of the students immediately refer to their previous work on the quadratic equation for problem 6 and indicate that problem 7 is similar. They share their equations privately...

### Early algebra ideas involving two variables, Clip 11 of 18: Attempts at solving problem 6

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the eleventh of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, several of the students share their preliminary ideas with the researcher, Robert B. Davis, about the table for Problem 6, as printed below. After Ankur and Michele I. describe the function verbally, Davis asks them to write a symbolic equation to describe the relationship. Romina and Brian...

### Early algebra ideas involving two variables, Clip 12 of 18: Solutions for problem 6

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the twelfth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, researcher, Robert B. Davis notes again the importance of everyone having a chance to discover the "secrets" for themselves. When he asks the students to share their ideas only with him, several pairs of students point out general ideas that they have noticed to lead to a...

### Early algebra ideas involving two variables, Clip 13 of 18: Sharing ideas about problem 6

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the thirteenth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, the researcher, Robert B. Davis, suggests that students who have figured out Problem 6 might share some of the "secret." There is general discussion and students appear unwilling to listen to each other. After Michele I. and Ankur present an explanation about how each "triangle" value is...

### Early algebra ideas involving two variables, Clip 10 of 18: Mike's solution to problem 2

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the tenth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, the researcher Robert B. Davis asks Michael to share his solution to his choice of one of the first problems with the class. Michael copies Table 2, as printed below, onto the white board. He then explains the pattern of the differences in the "triangle" column and...

### Early algebra ideas involving two variables, Clip 7 of 18: Michael discovers the secret!

Robert Benjamin Davis, Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the seventh of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, the students work to find general ideas, or "secrets" that will help them in finding equations for the Truth Sets for Problems 2 through 5 printed below. While Milin encourages Michael, other students share their ideas with the researcher. Michele I., Ankur, Milin, Michael and Stephanie are...

### Early algebra ideas involving two variables, Clip 9 of 18: Problem 6 is different?

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the ninth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, the students note that the Truth Table for problem 6 does not conform to their general ideas for constructing the first 5 equations. Pairs of students share what they are observing about the table as presented below with researcher, Robert B. Davis. Romina, Brian, Michele I., and...

### Early algebra ideas involving two variables, Clip 8 of 18: Sharing secrets for tables 2 through 5

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the eighth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, individual students and pairs of students share general ideas ("secrets") about the equations that they are generating with researcher, Robert B. Davis. In particular they point out the importance of the "triangle" value associated with zero as the constant value for the equation. They also notice that...

### Early algebra ideas involving two variables, Clip 6 of 18: Work on guess my rule problems 2 through 5

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the sixth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, the students work together to find the equations for the Truth Sets for Problems 2 through 5 printed below. Students share their work with researcher, Robert Davis, who suggests whole class sharing. Several students ask for more time to discover the "secrets" for themselves. Michele I., Ankur,...

### Early algebra ideas involving two variables, Clip 5 of 18: Recap of day 1, moving from one to two variables

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the fifth of 18 clips from Early Algebra Ideas Involving Two Variables on the second of two consecutive classroom sessions with the class of 6th grade students, Researcher Robert B. Davis revisits the quadratic equation tasks that the students had done on the previous day. He reminds them about the last task that they had considered, completing the Truth Table (function table) for two variables for the equation below. Michele R. comes to the...

### Early algebra ideas involving two variables, Clip 4 of 18: Scientists and the nature of secrets

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
The fourth of 18 clips from Early Algebra Ideas Involving Two Variables begins the second of two consecutive classroom sessions with the class of 6th grade students. Researcher Robert B. Davis opens the session with a whole-group discussion about what scientists do and the nature of scientific secrets. The group discusses when and how scientific secrets are discovered and then consider how recognizing secrets are a part of building powerful mathematical ideas. Jeff and Matt...

### Early algebra ideas involving two variables, Clip 3 of 18: Introduction to guess my rule

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the third of 18 clips focusing on Early Algebra Ideas Involving Two Variables, when Researcher Robert Davis asks the students to explain what they understand to be the mathematical task, Milin responds that they are to find the equation for a Truth Set presented in table form. Davis challenges the class to study the Truth Set tables on the worksheets that he has just distributed and to write equations for each of them. The...

### Early algebra ideas involving two variables, Clip 2 of 18: Truth sets in a table

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the second of 18 clips focusing on Early Algebra Ideas Involving Two Variables, Researcher Robert Davis writes the equation printed below on the white board and asks the 6th grade students to think of values for the “box” and the “triangle” that would make the statement true. The children are asked to figure out each value for the "triangle" when “box” is evaluated for consecutive numbers beginning with zero. Each pair of numbers that...

### Early algebra ideas involving two variables, Clip 1 of 18: Open sentences that can be made true or false with legal or illegal substitutions

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Martino
This first of 18 clips focusing on Early Algebra Ideas Involving Two Variables occurred toward the end of the first of two consecutive 6th grade class sessions. Researcher, Robert B. Davis, builds on earlier exploration with a single variable to introduce algebraic equations with two variables. He uses the symbols of square (referred to as a “box”) and a “triangle” as to represent the variables. Using the equation printed below, the class discusses ways to...

### A28, Night Session, Pascal's Identity (presentation view), Grade 11, May 12, 1999, raw footage

Carolyn Alexander Maher, Ralph S. Pantozzi, Regina D. Kiczek & Elena Perrone Steencken
In this full-session, raw footage video, students have come to school in the evening for a night session. The group, made up of Jeff, Michael and Romina begin discussing the coefficients of the binomial expansion, specifically (a+b) to the 10th power. In attempting to explain why 45 is the coefficient of the third term in this expansion, the students refer to counting how many 10-tall towers have exactly two cubes of a specific color. As...

### A73, Revisiting construction of large models to compare fractions (classroom view), Grade 4, October 8, 1993, raw footage

Amy Marie Martino
Amy Martino began the session by asking the students to discuss the task that they had worked on during the previous two sessions: Which is larger, two thirds or three fourths, and by how much? Several students offered conjectures of how to generate multiple models with Cuisenaire rods that could accommodate both thirds and fourths. Then, the students worked to reconstruct the models that they had found for this task in the previous session. One...

### Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 10 of 10: Generating the exponents in the expansion of (a + b) to the seventh power.

Carolyn Alexander Maher
In the final clip in a series of ten from the fifth of seven interviews, 8th grader Stephanie continues her exploration of Early Algebraic Ideas about Binomial Expansion. Researcher Carolyn Maher asks Stephanie to use the relationships that she has recognized between Pascal's Triangle and both unifix-cube towers and expansions of the binomial (a + b) to predict the exponents for each term of (a + b) to the 7th power.The problems as presented to...

### Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 8 of 10: Rebuilding Pascal's Triangle using n choose r notation.

Carolyn Alexander Maher
In the eighth clip in a series of ten from the fifth of seven interviews, 8th grader Stephanie continues her exploration of Early Algebraic Ideas about Binomial Expansion. Stephanie is asked by the researcher, Carolyn Maher, to recreate Pascal's Triangle with each numerical value replaced by the appropriate combinatorics term. She examines how the entries in each row can be generated from the row before and begins to develop a general rule for representing the...

### Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 9 of 10: Relating coefficients of the binomial expansion to Pascal's Triangle and to unifix cube towers, selecting from two colors.

Carolyn Alexander Maher
In the ninth clip in a series of ten from the fifth of seven interviews, 8th grader Stephanie continues her exploration of Early Algebraic Ideas about Binomial Expansion. Researcher Carolyn Maher refers to written work that Stephanie had completed in an earlier interview where she had recorded the binomial (a + b) expanded to successive powers from one to five. Stephanie maps the coefficients for each expansion to the corresponding row of Pascal's Triangle and...

### Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 6 of 10: Connecting the combinatorics notation for tower choices when selecting from two colors to the first 5 rows of Pascal's Triangle.

Carolyn Alexander Maher
In the sixth clip in a series of ten from the fifth of seven interviews, 8th grader Stephanie continues her exploration of Early Algebraic Ideas about Binomial Expansion. Researcher Carolyn Maher asks Stephanie to represent towers one-cube tall using the combinatorics notation introduced earlier for n choose r, and records this in a form that is the first row of Pascal's Triangle. As Stephanie continues to develop this correspondence for towers 2 and 3-cubes tall,...

### Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 7 of 10: Continuing investigation of Pascal's Triangle to generate rows 5 and 6 and calculate the totals for each row.

Carolyn Alexander Maher
In the seventh clip in a series of ten from the fifth of seven interviews, 8th grader Stephanie continues her exploration of Early Algebraic Ideas about Binomial Expansion. Stephanie is asked by the researcher, Carolyn Maher, to use the addition pattern that she has recognized in the first four rows of her triangle representation of the terms to predict the numbers in row 5 and to check that the total is consistent with the total...

### Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 4 of 10: Investigating the "doubling pattern" for unifix towers of increasing heights.

Carolyn Alexander Maher
In the fourth clip in a series of ten from the fifth of seven interviews, 8th grader Stephanie continues her exploration of Early Algebraic Ideas about Binomial Expansion. Using combinatorics notation, researcher Carolyn Maher records as Stephanie reviews the total number of towers 4-tall selecting from red and yellow cubes. In each of the five cases for n equal height 4 and r equal the number of red cubes from zero to 4, Stephanie indicates...

### Early algebra ideas about binomial expansion, Stephanie's interview five of seven, Clip 1 of 10: Combinatorics notation for selecting a particular number of items from a total of 4.

Carolyn Alexander Maher & Elena Perrone Steencken
In the first clip in a series of ten from the fifth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, the researcher, Carolyn Maher, introduces Stephanie to mathematical notation to represent the number of ways to select a specific number of objects from a set. Maher asks Stephanie to recall an earlier problem task, in which she was asked to determine the possible number of unifix "towers", four...

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