### 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 5 of 11: Solving equation four

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the fifth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the students work together to find the truth set for equation four on the worksheet recorded below. Milin suggests 2, substitutes it into the equation and evaluates the result. Davis encourages the class to do the calculation and, when they agree that the resulting equation is true, records 2 as one value in the truth set....

### Early algebra ideas involving one variable, Clip 6 of 11: Trying to complete truth sets for equations two through eight

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the sixth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the researcher, Robert B. Davis, challenges students to try to find more than one value for each of the open sentences printed below. As students respond, Davis keeps track of numbers that work by recording them in the bracketed truth set for the particular equation. Davis poses a question to the group about whether and when...

### 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...

### Early algebra ideas involving one variable, Clip 3 of 11: Introducing quadratic equations and solving equation one

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the third of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, researcher Robert B. Davis builds on the idea about equations with one variable and integers introduced in Clips One and Two. The class of 6th grade students is challenged to find the truth set for the open sentence printed below. Students respond and Davis keeps track of answers that work and those that don’t, based on...

### Early algebra ideas involving one variable, Clip 4 of 11: Working on equations two and three

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the fourth of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the students work in small groups to find and record solutions for the first two of a series of open sentences that have been presented as a worksheet by the researcher, Robert B. Davis. As the students test different values in the equations, Davis keeps track on the white board of answers that the students propose...

### Early algebra ideas involving one variable, Clip 2 of 11: Introducing integers with pebbles in the bag

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston, Amy Marie Martino & Thomas LaMonde Purdy
In the second of eleven clips from the first day of the Early Algebra Ideas 6th grade class sessions, the researcher, Robert B. Davis, models adding positive and negative integers to a class of 6th grade students by having the students add and remove small colored stones from a bag. For a given action, after a student gives the command: “Go”, a second student, responding to directions from the class, first adds a specific number...

### Early algebra ideas involving one variable, Clip 1 of 11: 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 Marie Martino
In the first of eleven clips from the first day of the Early Algebra Ideas sessions, the researcher, Robert B. Davis, introduces ideas about algebraic equations with one variable and truth sets to a class of 6th grade students. Davis begins with the first problem statement printed below to introduce ideas about algebraic sentences or equations with one variable that can be true, false or open. The concept of a variable is introduced by varying...

### Early algebra ideas involving two variables, Clip 17 of 18: Presenting ideas About problem 8

Davis, Robert B. (Robert Benjamin), Carolyn Alexander Maher, Alice S. Alston & Amy Marie Martino
In the seventeenth 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 present their ideas about problem 8, shown in the table below. At the suggestion of researcher, Robert B. Davis, Robert and Amy-Lynn describe the pattern of differences between consecutive values for the "triangle". Jeff theorizes about the pattern for subtraction in an equation....

• 2009
50
• 2010
13
• 2011
12
• 2012
9
• 2013
47
• 2014
460
• 2015
298
• 2016
40
• 2017
601
• 2018
73

• Film
1,603

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