### Early algebra ideas about binomial expansion, Stephanie's interview seven of seven, Clip 4 of 7: Investigating numerical and algebraic representations for each case for the number of Unifix-cube towers 4-tall selecting from red and yellow cubes.

Carolyn Alexander Maher & M. Donna Weir
In the fourth clip in a series of seven from the seventh of seven interviews, 8th grader Stephanie and researcher Carolyn Maher investigate how to numerically represent the number of Unifix towers for n = 4 cubes tall, selecting from red and yellow cubes, for each case from r = 0 red cubes to r = 4 red cubes. Maher poses questions about the algebraic representation for each case and how it might be generalized...

### Early algebra ideas about binomial expansion, Stephanie's interview seven of seven, Clip 3 of 7: Investigating the algebraic generalization for building Unifix-cube towers n-tall with exactly two red cubes from towers with exactly one red cube.

Carolyn Alexander Maher & M. Donna Weir
In the third clip in a series of seven from the seventh of seven interviews, 8th grader Stephanie continues to investigate the construction of Unifix towers four-cubes tall, selecting from red and yellow cubes, when moving from towers with exactly one red cube to towers with exactly two red cubes. She works with researcher Carolyn Maher to begin to develop and explain a generalized formula based on her physical and numerical representations of the towers....

### Early algebra ideas about binomial expansion, Stephanie's interview seven of seven, Clip 2 of 7: Generating Unifix-cube towers 4-tall from exactly two to exactly three red cubes and from exactly three yellow cubes to the tower with all four yellow cubes.

Carolyn Alexander Maher & M. Donna Weir
In the second clip in a series of seven from the seventh of seven interviews, 8th grader Stephanie first predicts the number of Unifix-cube towers with exactly three red cubes to be generated from the six towers with two red and two yellow cubes. In response to questions from researchers Carolyn Maher and Donna Weir, Stephanie predicts and then builds twelve towers from the six and identifies the four sets of three duplicate towers to...

### Early algebra ideas about binomial expansion, Stephanie's interview seven of seven, Clip 1 of 7: Stephanie revisits generating Unifix-cube towers 4-tall from exactly one to exactly two red cubes.

Carolyn Alexander Maher & M. Donna Weir
In the first clip in a series of seven from the seventh of seven interviews, 8th grader Stephanie revisits her earlier exploration of how to generate Unifix-cube towers across cases with researchers Carolyn Maher and Donna Weir. Selecting from red and yellow cubes, Stephanie builds the four towers with exactly one red cube and explains that from each of the four, holding the single red cube constant, she can build three new towers with exactly...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 9 of 9: Hypothesizing about the algebraic terms in the expansion of (a+b) to the fourth power.

Carolyn Alexander Maher
In the final clip in the series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, researcher Carolyn Maher asks Stephanie to predict the topic for the next interview. When Stephanie responds that she expects they will be exploring (a+b) to the fourth power, the researcher asks whether there might be terms in the algebraic expansion that could be predicted based on her earlier...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 8 of 9: Issues related to a physical model for (a+b) cubed and the volume of the model for a=1 and b=2.

Carolyn Alexander Maher
In the eighth clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie revisits her physical model of (a+b) cubed. She identifies each piece of the model and points out that the pieces are in one-to-one correspondence with the terms of the algebraic expansion. When researcher Perl raises questions about the small cube, Maher agrees that the specificity of the...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 7 of 9: Explaining each piece of the geometric model of (a+b) cubed as it relates to the terms in the algebraic expansion.

Carolyn Alexander Maher
In the seventh clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie explains how each piece of her physical geometric model for (a+b) cubed relates to the terms of the algebraic expansion. She then answers questions about her model that are raised by researchers Teri Perl, Carolyn Maher and her classroom teacher. The problem as presented to Stephanie:Explain your...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 6 of 9: Explaining the algebraic and geometric representations of (a+b) squared and the algebraic expansion of (a+b) cubed.

Carolyn Alexander Maher
In the sixth clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie explains her algebraic and geometric representations for (a+b) squared to researcher Carolyn Maher, three visiting researchers and Stephanie's classroom teacher. After completing this explanation, she builds on her algebraic solution for (a+b) squared to explain her algebraic expansion of (a+b) cubed. The problem as presented to Stephanie:Explain...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 5 of 9: Building (a+b) cubed and identifying the pieces.

Carolyn Alexander Maher
In the fifth clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie uses the small and large cube and 6 rectangular prisms provided to construct a cube with side (a+b). She identifies each of the pieces as it relates to her algebraic expansion of (a+b) cubed to researcher Carolyn Maher. The problem as presented to Stephanie:How could you build...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 4 of 9: Building the first layer of (a+b) cubed.

Carolyn Alexander Maher
In the fourth clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, researcher Carolyn Maher challenges Stephanie to consider how a packet including a small and large cube and 6 other rectangular prisms might be useful in constructing a model of (a+b) cubed. Stephanie recognizes that four of the pieces can be arranged to replicate her drawing of (a+b) squared...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 3 of 9: Describing the volume of (a+b) cubed using math manipulatives.

Carolyn Alexander Maher
In the third clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, researcher Carolyn Maher presents Stephanie with base-ten materials, algebra blocks and other rectangular prisms; then asks her how she might use any of the manipulatives to explain volume in general and the volume of cube with side (a+b) in particular. Stephanie uses the small cubic unit and the...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 2 of 9: Reviewing the algebraic expansion of (a+b) cubed.

Carolyn Alexander Maher
In the second clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie describes to researcher Carolyn Maher her algebraic expansion of (a+b) cubed. Stephanie first rewrites the expression as (a+b)(a+b)(a+b), then replaces two factors of (a+b) with the quantity (a squared + 2ab + b squared) and uses the distributive property to complete the expansion which she then simplifies.The...

### Early algebra ideas about binomial expansion, Stephanie's interview four of seven, Clip 1 of 9: Explaining that (a+b) squared = (a squared + 2ab + b squared), algebraically and geometrically

Carolyn Alexander Maher
In the first clip in a series of nine from the fourth of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie explains the conclusions that she has justified in earlier interviews. In response to the researcher, Carolyn Maher, Stephanie firsts writes out the expanded form of (a+b) squared. She then explains that she had used geometric drawings to illustrate the meaning of the square with side a, first...

### Early algebra ideas about binomial expansion, Stephanie's interview three of seven, Clip 7 of 7: Testing the algebraic expansion of (a+b) cubed with numbers and beginning to imagine a physical model

Carolyn Alexander Maher
In the final seventh clip from the third of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie checks her completed symbolic expansion for (a+b) cubed by letting a=3 and b=7. She substitutes the values into each term of the expression, calculates the values for each and adds to find a total of 1000. When researcher Carolyn Maher asks if she is convinced that her expression is the same...

### Early algebra ideas about binomial expansion, Stephanie's interview three of seven, Clip 6 of 7: Expanding (a+b) cubed algebraically.

Carolyn Alexander Maher
In the sixth clip in a series of seven from the third of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie uses her numerical representation of the expansion of (3+7) cubed and develops a parallel expansion for (a+b) cubed. She uses the distributive property to write out each term and then simplifies and collects like terms, explaining each step of the process to researcher Carolyn Maher. The problem...

### Early algebra ideas about binomial expansion, Stephanie's interview three of seven, Clip 5 of 7: Testing the partial symbolic expansion of the cube of (a+b) for a=3 and b=7.

Carolyn Alexander Maher
In the fifth clip in a series of seven from the third of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie is asked by researcher Carolyn Maher to represent the expansion of the quantity (a+b). After Stephanie rewrites the expression (a+b) cubed, first as (a+b)(a+b)(a+b) and then as (a+b)( a squared + 2ab + b squared), she tests her work by letting a=3 and b=7, uses the distributive...

### Early algebra ideas about binomial expansion, Stephanie's interview three of seven, Clip 4 of 7: Beginning to explore volume as it compares to area

Carolyn Alexander Maher
In the fourth clip in a series of seven from the third of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie refers to the Base-10 squared materials to explain the difference between area and volume to the researcher, Carolyn Maher. They extend an earlier problem modeling the square of (a+b) by considering various numbers, a and b, for which the sum of the two numbers is 10. Stephanie...

### Early algebra ideas about binomial expansion, Stephanie's interview three of seven, Clip 3 of 7: Modeling the square of (a+b) for the case of a=3 and b=7.

Carolyn Alexander Maher
In the third clip in a series of seven from the third of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, the researcher, Carolyn Maher, asks about the square of (a+b) for the particular case when a=3 and b=7. Referring to the symbolic expansion that she had figured out earlier, Stephanie calculates the value of each part of the expression after substituting 3 and 7 for each "a" and...

### Early algebra ideas about binomial expansion, Stephanie's interview three of seven, Clip 2 of 7: Explaining the meaning of area of a square with concrete materials

Carolyn Alexander Maher
In the second clip in a series of seven from the third of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, the researcher, Carolyn Maher, asks how she would explain her ideas about area of squares to her younger sister. When Stephanie indicates that she would revisit basic ideas about units and square units, Maher offers her base-ten blocks from which Stephanie selects and uses a cubic unit and...

### Early algebra ideas about binomial expansion, Stephanie's interview three of seven, Clip 1 of 7: Revisiting earlier ideas about the square of the quantity (a + b)

Carolyn Alexander Maher
In the first clip in a series of seven from the third of seven interviews in which 8th grader Stephanie explores Early Algebraic Ideas about Binomial Expansion, Stephanie arrives with several pages of notes that she had written about her work in the first two interviews. When the researcher, Carolyn Maher, asks Stephanie to review her work, she explains that the focus of the earlier sessions had been to determine what was (in expanded form)...

### Early algebra ideas involving one variable, Clip 11 of 11: Are there impossible equations?

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In this final clip of the series of eleven from the first day of the Early Algebra Ideas 6th grade class sessions that focus on solving equations with one variable, the students and the researcher are revisiting the important ideas, or "secrets", that have helped them to find numbers that will make the equations true. When researcher Robert B. Davis challenges the group to solve the first equation printed below, the students use their "secret...

### Early algebra ideas involving one variable, Clip 10 of 11: Owning the secrets

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the tenth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions,all but three of the students claim to know the general "secrets" for finding values that make a given equation true. Researcher Robert B. Davis challenges the three students, Michele I., Jeff and Michael to solve the equation below. When they find one value that works, they are satisfied. Finally, Ankur articulates the general rules about adding...

### Early algebra ideas involving one variable, Clip 9 of 11: Finding a second secret

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the ninth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the students continue to work on equations and figure out the general ideas for recognizing values that will make the statements true. The researcher, Robert B. Davis, first poses equations from problem 9 below and students quickly offer 5 and 6 as solutions. By the end of the discussion, two explanations for "secrets" for finding the...

### Early algebra ideas involving one variable, Clip 8 of 11: Losing the "secret"

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the eighth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the researcher, Robert B. Davis, challenges the students to find values for the equation printed below. Students suggest a number of the factors of 60. However, when each factor is tested by substituting it into the equation and evaluating the result, none of the factors makes a true statement. Various students suggest that the "secret idea"...

### Early algebra ideas involving one variable, Clip 7 of 11: Beginning to recognize "secrets"

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the seventh of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the students look for patterns for finding a solution to the problems. As solutions are identified for each of the equations below, students posit that they “know why.” A discussion of the appropriateness of telling or keeping a "secret” to them follows. Students then whisper their reasoning to Davis. Jeff, Brian, Bobby, Ankur, Matt, Milin, and...

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