### Fraction as number, an introduction, Clip 6 of 8: Problem posing for the class

Carolyn Alexander Maher
In the sixth of eight clips from this session, researcher Carolyn Maher asked the students to work with their partners to come up with questions to challenge the class. Alan was first to present his challenge. He asked the class to find the rod that would be called one if the red rod was called one fifth. Graham responded that the orange rod would be called one, and explained that five red rods were equal...

### Fraction as number, an introduction, Clip 5 of 8: What is the unit?

Carolyn Alexander Maher
In the fifth of eight clips from this session, researcher Carolyn Maher asked the class to give a number name for the red rod if the brown rod was called one. Danielle explained that it would be called one fourth and could be justified by placing four red rods next to the brown rod. The researcher then asked the students a related question, requiring them to name the brown rod if the red rod was...

### Fraction as number, an introduction, Clip 4 of 8: Establishing a unit for comparing length of rods

Carolyn Alexander Maher
In the fourth of eight clips from this session, researcher Carolyn Maher mentioned that she had heard a student suggest that the red rod be called one if the dark green rod were called one. She asked the class it that was possible. Erik said it was not possible and Michael agreed with him and explained that the green was bigger than the red and that it would take three of the reds to make...

### Fraction as number, an introduction, Clip 3 of 8: Permanent color names and flexible number names for rods

Carolyn Alexander Maher
In the third of eight clips from this session, researcher Carolyn Maher asked the students what number name they would give the red rod if the green rod was called one. Beth said that the red rod would be called one third and explained, “if you put three on them it makes one whole.” Next the researcher asked the students if the light green rod was one third as long as the blue rod. Jessica...

### Fraction as number, an introduction, Clip 2 of 8: Students model fraction problems

Carolyn Alexander Maher
In the second of eight clips from this session, researcher Carolyn Maher said, “Someone told me that the red rod is half as long as the yellow rod. What do you think?" Danielle responded that the statement was not true, and justified her solution. The researcher then asked the students if the purple rod was half as long as the black rod. Alan and Erik, as well as Meredith and Sarah said that it could...

### Fraction as number, an introduction, Clip 1 of 8: Assigning number names to rods

Carolyn Alexander Maher
This video was recorded during the first of many research sessions in a yearlong study conducted in a fourth grade classroom by researcher Carolyn Maher and colleagues. At the start of this session, students were introduced to Cuisenaire rods, which they would begin to use as tools for building models to represent fractions as numbers. In the first of eight clips from this session, the researcher posed the first task to the students. She said,...

### Early algebra, investigating linear functions, series 5 of 7, ladder problem, Clip 7 of 7: Combining a six step ladder with a two step ladder

John Francisco & Prashant V. Baldev
In the final clip of the series of seven from an after-school enrichment session in an urban middle school, Ariel, a 7th grade boy, revisits his method for solving a specific example of the Ladder Problem. Researcher John Francisco asks Ariel explain how he had predicted the number of rods for a ladder with 125 steps. The procedure that Ariel had articulated and followed for smaller numbers had the following steps: (1) Find the closest...

### Early algebra, investigating linear functions, series 5 of 7, ladder problem, Clip 6 of 7: How many rods for a ladder with 125 steps?

John Francisco & Prashant V. Baldev
In the sixth of seven clips from an after-school enrichment session in an urban middle school, Ariel, a 7th grade boy, continues to apply his method for solving specific examples of the Ladder Problem. Researcher John Francisco asks Ariel to use his rule for finding the number of rods for ladders with odd numbers of steps specifically for a ladder with 125 steps. The procedure that Ariel had articulated and followed for relatively small numbers...

### Early algebra, investigating linear functions, series 5 of 7, ladder problem, Clip 4 of 7: Predicting the number of rods for ladders with 80 and then 120 steps

John Francisco
In the fourth of seven clips from an after-school enrichment session in an urban middle school, Ariel, a 7th grade boy, continues his exploration of ideas about linear functions. Researcher John Francisco asks Ariel to test the rule he has proposed and written down for finding the number of rods in a ladder with an even number of steps. The procedure that Ariel had articulated and followed for relatively small numbers had the following steps:...

### Early algebra, investigating linear functions, series 5 of 7,ladder problem, Clip 3 of 7: Recording the procedures for ladders with odd and even numbers of steps

John Francisco & Prashant V. Baldev
In the third of seven clips from an after-school enrichment session in an urban middle school, two 7th grade boys, Ariel and James, are exploring ideas about linear functions. When Ariel explains his reasoning, researcher John Francisco asks him to write out the "rules" that he has developed for solving the ladder problem. Ariel has one procedure for a ladder with an odd number of steps, and a second procedure for a ladder with an...

### Early algebra, investigating linear functions, series 5 of 7, ladder problem, Clip 2 of 7: Predicting the rods for odd and even numbers of steps

John Francisco & Prashant V. Baldev
In the second of seven clips from an after-school enrichment session in an urban middle school, two 7th grade boys, Ariel and James, are exploring ideas about linear functions. In responding to questions from researcher John Francisco, Ariel develops procedures for calculating the number of rods needed to construct a ladder with an even or an odd number of steps. Researcher Francisco poses the question: "How many rods are needed to build a ladder with...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 11 of 11: Extending ideas about generating towers 4-cubes tall selecting from green and blue cubes to towers 5-cubes tall with exactly 1 green cube and connecting to Fermat and Pascal

Carolyn Alexander Maher & Speiser, R. (Robert)
In the final clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie is challenged by researchers Carolyn Maher and Robert Speiser to extend her ideas about how to build the Unifix-cube towers across cases for towers 4-cubes tall for particular cases when selecting from blue and green cubes to towers that are 5-cubes tall. She describes how the five towers, 5-cubes tall with exactly one green cube, would generate...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 10 of 11: Developing mathematical expressions for generating the number of towers 4-cubes tall selecting from green and blue cubes for exactly 2 green cubes, exactly 3 green cubes, and for 4 green cubes

Carolyn Alexander Maher & Speiser, R. (Robert)
In the tenth clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie works with researchers Carolyn Maher and Robert Speiser to synthesize her ideas about how to build the Unifix-cube towers across cases and to describe mathematically how she has generated the number of towers 4-cubes tall for particular cases when selecting from blue and green cubes. She explains how the number of towers for each case -- exactly...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 9 of 11: Beginning to generalize mathematical expressions to generate cases for towers 4-cubes tall selecting from green and blue cubes

Carolyn Alexander Maher & Speiser, R. (Robert)
n the ninth clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie is asked by researchers Carolyn Maher and Robert Speiser to describe mathematically how she has generated sets of Unifix-cube towers 4-cubes tall across cases. She first considers the case for all green cubes when generated from the set of cubes with three green cubes and one blue cube and determines that the single resulting tower could be...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 8 of 11: Working backwards from towers 4-cubes tall selecting from blue and green cubes, with exactly two green cubes, to towers with exactly one green cube

Carolyn Alexander Maher & Speiser, R. (Robert)
In the eighth clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie is asked by researchers Carolyn Maher and Robert Speiser to revisit her set of six Unifix cube towers 4-cubes tall each with exactly two green and two blue cubes built selecting from green and blue cubes. The researchers ask Stephanie to determine which towers, 4-cubes tall, with exactly one green cube could have generated each of the...

### Early Algebra Ideas About Binomial Expansion, Stephanie's Interview Six of Seven: Clip 7 of 11: Generating towers 4-cubes tall, selecting from blue and green cubes, from towers with exactly one green cube to towers with exactly two green cubes.

Carolyn Alexander Maher & Speiser, R. (Robert)
In the seventh clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie is asked by researchers Carolyn Maher and Robert Speiser to consider how the Unifix-cube towers 4-cubes tall, selecting from green and blue cubes, that correspond to row four of Pascal's Triangle (given rows from 0 to n), could be generated horizontally. Taking each of the four towers with exactly one green cube, Stephanie generates three new towers...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 6 of 11: Generating towers 3-cubes tall, selecting from blue and green cubes, from towers with exactly one blue cube to towers with exactly two blue cubes

Carolyn Alexander Maher & Speiser, R. (Robert)
In the sixth clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie is challenged by researchers Carolyn Maher and Robert Speiser to consider how the eight Unifix-cube towers, selecting from green and blue cubes, that correspond to the third row of Pascal's Triangle (given rows from 0 to n), could be generated horizontally. Taking each of the three towers that have exactly one blue cube, Stephanie generates two new...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 5 of 11: Developing the correspondence between towers 4-cubes tall, selecting from two colors, Pascal's Triangle and the symbolic algebraic expansion of (a+b) to the 4th power

Carolyn Alexander Maher & Speiser, R. (Robert)
In the fifth clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie responds to questions from researchers Carolyn Maher and Robert Speiser about the correspondence between towers 4-cubes tall, selecting from blue and green cubes, and the symbolic binomial expansion of (a+b) to the 4th power. After developing the correspondence for Unifix-cube towers with no blue cubes, exactly one blue cube and exactly three blue cubes, Stephanie refers to...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 4 of 11: Developing the correspondence among towers, selecting from two colors, Pascal's Triangle, and the symbolic algebraic expansions of (a+b) squared and (a+b) cubed

Carolyn Alexander Maher & Speiser, R. (Robert)
In the fourth clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie remembers that she had figured out the expanded algebraic expressions for (a+b) for powers up to 6. When asked by researchers Carolyn Maher and Robert Speiser to connect these expressions to Pascal's Triangle and Unifix-cube towers that she had built by selecting from two colors, Stephanie developed a correspondence between the variables, a and b, and the...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 3 of 11: Comparing towers, selecting from two colors, built inductively and corresponding to the addition rule of Pascal's Triangle

Carolyn Alexander Maher & Speiser, R. (Robert)
In the third clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie continues a discussion of ideas about binomial expansion with researchers Carolyn Maher and Robert Speiser. With continuing reference to combinatorics notation, Stephanie organizes the towers of successive heights, selecting from two colors, to show how they grow inductively, row by row. She then writes the first several rows of Pascal's Triangle onto her paper and places the...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 2 of 11: Stephanie rebuilds Unifix towers 1-cube, 2-cubes, 3-cubes and 4-cubes tall, selecting from two colors

Carolyn Alexander Maher & Speiser, R. (Robert)
In the second clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie continues her exploration of early algebraic ideas about binomial expansion with researchers Carolyn Maher and Robert Speiser. After reviewing the meaning of the combinatorics notation, Stephanie begins to logically build sets of Unifix towers of different heights when selecting from green and blue cubes. She asserts that there would be exactly two towers that are one-cube tall...

### Early algebra ideas about binomial expansion, Stephanie's interview six of seven, Clip 1 of 11: Stephanie revisits combinatorics notation for building towers

Carolyn Alexander Maher & Speiser, R. (Robert)
In the first clip in a series of eleven from the sixth of seven interviews, 8th grader Stephanie revisits her earlier exploration of particular algebraic ideas about binomial expansion with researchers Carolyn Maher and Robert Speiser. Stephanie reports that, in Interview Five, she had worked with Maher to use mathematical notation for selecting r items from a set with number n. In responding to questions from Speiser, Stephanie first writes the symbolic notation for selecting...

### Early algebra ideas about binomial expansion, Stephanie's interview seven of seven, Clip 7 of 7: Finding a general direct "rule" for the number of Unifix-cube towers, selecting from two colors, for any height.

Carolyn Alexander Maher, M. Donna Weir & Steven Maher
In the final clip in a series of seven from the seventh of seven interviews, 8th grader Stephanie responds to questions from researcher Carolyn Maher about problems that she and her classmates had encountered in the past, particularly when working with researcher Robert B. Davis. Specifically, they discussed investigating the Tower of Hanoi problem and finding a general rule for predicting the number of Unifix-cube towers selecting from two colors for particular heights. Finally, Stephanie...

### Early algebra ideas about binomial expansion, Stephanie's interview seven of seven, Clip 6 of 7: Making sense of the Pascal Triangle addition rule with Unifix cube towers .

Carolyn Alexander Maher & M. Donna Weir
In the sixth clip in a series of seven from the seventh of seven interviews, 8th grader Stephanie first revisits her investigation of the number of duplicates that would be produced when building Unifix-cube towers 4-tall with three red cubes from the six towers with two red and two yellow cubes. Researcher Carolyn Maher asks Stephanie to justify her conclusion that there would be four sets of three duplicates. Maher then challenges Stephanie to use...

### Early algebra ideas about binomial expansion, Stephanie's interview seven of seven, Clip 5 of 7: Developing numerical representations for each case for the number of Unifix-cube towers 5-tall and 6-tall selecting from red and yellow cubes.

Carolyn Alexander Maher & M. Donna Weir
In the fifth clip in a series of seven from the seventh of seven interviews, 8th grader Stephanie and researcher Carolyn Maher extend their earlier investigation of how to represent the number of Unifix cube towers for cases from no red cubes to four red cubes for towers 4-cubes tall to towers 5-tall and then towers 6-tall, still selecting from red and yellow cubes.For each case, Stephanie builds or imagines the actual towers and records...

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