756 Works

Continuing to explore fraction comparisons, Clip 5 of 7: Michael and Brian compare two third and three fourths

Amy Marie Martino
In the fifth clip Michael and Brian extended their model using the orange and red train to show thirds and twelfths in addition to fourths. They showed that the difference between two thirds and three fourths was one twelfth, and repeated their justifications to a visitor and researcher Amy Martino on separate occasions.

Comparing fractions and evaluating models that represent solutions, Clip 3 of 8: Sharing Alan's candy bar model

Carolyn Alexander Maher &
In the third clip Alan presented another model to show the difference between one half and one third. He built a model using one dark green rod, two light green rods, three red rods, and six white rods, and showed that the difference between one half, or a light green rod, and one third, a red rod, is the length of the white rod, which he named one sixth. Jessica challenged Alan’s solution by saying...

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

, 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...

Revisiting construction of large models to compare fractions, Clip 1 of 5: Which is larger, two thirds or three fourths, and by how much?

Amy Marie Martino
In the first of five clips from this classroom session, the researcher, Amy Martino, provided the students with an opportunity to briefly discuss a task that they had been working on during the previous session, which was comparing two thirds and three fourths. Andrew and Jessica described how they had attempted to make large models and use trains for thirds and fourths. The researcher asked the students if they had all recorded their large models,...

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...

Reviewing rod relationships and the candy bar problem, Clip 3 of 6: What is the number name for red when the yellow and light green rod is two? A whole class discussion

Carolyn Alexander Maher
In the third clip from this classroom session, during the whole class discussion, Sarah and Audra presented their solution to the class. Audra explained that they called the red rod one fourth when the train of yellow and light green was called one, and that the red rod would be called one and one fourth when that train was called two. Researcher Carolyn Maher asked Audra to explain her other solution. Audra built a model...

Reviewing rod relationships and the candy bar problem, Clip 4 of 6: Switching units, candy bar metaphor

Amy Marie Martino, Carolyn Alexander Maher & Thomas LaMonde Purdy
In the fourth clip from this classroom session, researcher Carolyn Maher led a whole class discussion. She told the class that she gave each of the two adults who were in the room half a chocolate bar each and that one of the adults said that what she had done was unfair. After asking the class what she might have done that was unfair, she showed the class that she gave one adult half of...

Discovering equivalent fractions and introducing fraction notation, Clip 5 of 5: Compare one half and two thirds, establishing equivalence

Carolyn Alexander Maher
In the fifth clip, researcher Carolyn Maher called the class together. At the overhead, Erin, Jackie, and Jessica had built the six and twelve centimeter models. The researcher asked the class to provide the solution to the problem, and they answered in unison that two thirds was larger than one half by one sixth. The researcher then asked if anyone had built a model that gave another solution. Meredith indicated that she had, and she...

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...

Equine behavior: behavior test, stepstool test, Brisa

Test conducted as part of the Young Horse Teaching & Research Program of the Equine Science Center, Rutgers University, November, 2009. Subject was RU Brisa, a 6-month-old weanling draft cross born May, 2009. Test conducted by Dr. Sarah L. Ralston, with Elyse Conway, at the George H. Cook Campus, Rutgers University. Duration of test: 53 seconds. Overall score on a scale of 1-4 (4 is highest compliance):

Night session, Pascal's identity, clip 1 of 7: thinking about the meaning of combinatorics notation

Carolyn Alexander Maher & Ralph S. Pantozzi
This is the first of seven clips from the night session. In it, Jeff, Michael, and Romina discuss 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. Note: The 10-tall towers problem is an instance of the n-tall towers...

Intermittent Chemoprevention of Colon Cancer

David E. Axelrod
This video shows the advantage of intermittent pulse schedules compared to constant schedules, for the chemoprevention of colon cancer in a computer simulation of cell dynamics in human colon crypts.

Discovering equivalent fractions and introducing fraction notation, Clip 2 of 5: David and Meredith compare one half and two thirds

Amy Marie Martino & Carolyn Alexander Maher
In the second clip, researcher Carolyn Maher asked the students to build a model to show which is bigger, two thirds or one half, and by how much. The students worked with their partners to solve the problem. David and Meredith worked together, and Meredith built a model using a dark green rod, two light green rods, and three red rods. David replicated the model, and they each lined up six white rods against their...

Comparing fractions and evaluating models that represent solutions, Clip 2 of 8: Comparing one half and one third, Jessica and Erik

Carolyn Alexander Maher
In the second clip from this session, researcher Carolyn Maher asked the students if they felt confident to justify their solution to the problem: Which is larger, one half or one third, and by how much? Jessica built a model of an orange and red train, two dark green rods, three purple rods, and six red rods at the overhead. She named the rods one, one half, and one third, and one sixth, respectively. She...

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 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 &
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 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 &
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...

Garshunography: Terminology and properties

George Anton Kiraz
George A. Kiraz is the director of the Beth Mardutho Research Library and president and co-founder of the Gorgias Press, both located in Piscataway, NJ. He is introduced by conference chair, Jack Darakjy.

Fraction problems, Sharing and number lines, Clip 3 of 3: Sharing strategies

Carolyn Alexander Maher
Researcher Carolyn Maher leads a whole class discussion after the students' exploration of the problem: Which is larger, one fourth or one ninth, and by how much? Several students described their conjectures and attempts at solving the problem, and James, Erin, Beth, and Jackie described the models that they had built to solve the problem. The researcher closed the session by noting that the difference between one half and one third, a problem that they...

Reviewing rod relationships and the candy bar problem, Clip 2 of 6: What is the number name for red when the yellow and light green rod is two? Brian and Jacquelyn

Carolyn Alexander Maher
In the second clip from this classroom session, researcher Carolyn Maher posed a challenge. She asked the students to name the red rod if a train of yellow and light green was called two. The researcher then added a second task to name the red rod if the yellow and light green train would be called one. Jacquelyn and Brian C. worked together on this task. Jacquelyn built two identical models to represent the problems,...

The infinite number line, Clip 1 of 4: Naming points on the number line

Carolyn Alexander Maher
Researcher Carolyn Maher began the session in clip 1 by stating that the class was discussing rulers at the end of the prior session. The researcher pointed out that rulers may be constructed differently and then used Jessica’s ruler to illustrate. The researcher continued by asking the students if they recalled Alan’s idea when they were talking about a twelve inches rod. A show of hands indicated that students in the class responded affirmatively. The...

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)...

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...

Early algebra ideas involving one variable, Clip 5 of 11: Solving equation four

, 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....

The infinite number line, Clip 4 of 4: Placing fractions and mixed numbers on the number line

Carolyn Alexander Maher
In the fourth clip researcher Carolyn Maher asked the students to each come up and take a turn placing some fractions on the number line. She called on Gregory to come up first. He placed a one half on the number line half way between zero and one. Laura then came up to the projector and placed one fourth midway between the zero and one half. Brian commented that she should put it on the...

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