Martha’s Carpeting Task - Carnegie Learning



5 Practices for Leading Productive Math Discussions in ClassMary Kay SteinUniversity of PittsburghIn preparation for the Webinar, please complete the tasks on pages 2 and 5 and read the vignettes on pages 3, 4 and 6-7The handouts in this packet are drawn from Smith, M.S., & Stein, M.K. (2011). 5 Practices for Orchestrating Productive Mathematics Discussions. Reston, VA: National Council of Teachers of Mathematics and from Stein, M.K., Smith, M.S., Henningsen, M.A., & Silver, E.A. (2009). Implementing standards-based mathematics instruction: A casebook for professional development (Second Edition). New York, NY: Teachers College Press.Martha’s Carpeting TaskMartha was recarpeting her bedroom which was 15 feet long and 10 feet wide. How many square feet of carpeting will she need to purchase? Stein, Smith, Hennigsen, and Silver, 2000Fencing TaskMs. Brown's class will raise rabbits for their spring science fair. They have 24 feet of fencing with which to build a rectangular rabbit pen to keep the rabbits.a.If Ms. Brown's students want their rabbits to have as much room as possible, how long would each of the sides of the pen be?b.How long would each of the sides of the pen be if they had only 16 feet of fencing?c.How would you go about determining the pen with the most room for any amount of fencing? Organize your works so that someone else who reads it will understand it. Stein, Smith, Hennigsen, and Silver, 2000Vignette 1 – Mr. PatrickStudents in Mr. Patrick’s have just started a unit on measuring two- and three-dimensional figures. Following an efficient check of the previous night’s homework, Mr. Patrick begins the lesson with a definition of terms and a review of formulas for finding the area and perimeter of rectangles. Mr. Patrick then demonstrates how to compute the perimeter and area of two rectangles on the blackboard-one with a length of 15 inches and a width of 7 inches, the other having four sides, each labeled 50 yards in length. Afterward, Mr. Patrick assigns 20 similar problems from the students’ textbook. As students work individually at their desks, applying the formulas to the labeled diagrams, he walks around the room. Most of his assistance falls into one of two categories: (1) help with two-digit multiplication and (2) reminders of which formula to use for area versus perimeter. As the period draws to a close, Mr. Patrick tells the students to finish working on the 20 problems for homework and to do a word problem (Martha’s Carpeting Task).Vignette 2 – Mrs. FoxThe seventh grade students in Mrs. Fox’s class are also working on a unit that involves measuring two- and three-dimensional figures. As students walk into the room, Mrs. Fox directs their attention to the task displayed on the overhead projector (The Fencing Task) and asks them to begin to work immediately in their small groups. She tells the students that they will have the entire period to work on this task and reminds them that, as usual, they may quietly get whatever paper, tools, or manipulatives they need to complete the task.Mrs. Fox walks around the room as students begin their work, stopping at different groups to listen in on their conversations and to provide support as needed. She notes that students started out by describing an assortment of pen configurations that could be built with 24 feet of fencing. As they kept coming up with new configurations, they realized they needed to keep track of the shapes they had already tried. This led them to construct a table that identified the dimensions of each configuration along with its area. Eventually, by looking for patterns across many configurations, students arrived at a conjecture regarding the shape that produced the largest areas, and then tested that conjecture with a different amount of fencing. During this time Mrs. Fox circulated among the groups asking such questions as “How do you know you have all of the possible pen configurations?”, “Which pen has the most room?”, and “Do you see a pattern?” These questions led students to see the need to organize their data, make conjectures, and test them out. As the period drew to a close, none of the groups have completed the task but most are well on their way to discovering that a square would enclose the greatest amount of area for any given amount of fencing. All students are deeply engaged with the task and are actively talking to their partners about how to justify, organize, and communicate their thinking. For homework, Mrs. Fox asks students to summarize what they have learned so far from their exploration and what they want to continue to work on in the next class.Vignette 3 – Ms. JonesMs. Jones also gave her seventh-grade students the Fencing Task. As she walked around the room, monitoring the groups as they worked, she was dismayed to see that they were not making much progress – some students were already off-task and many others were complaining that the task was too difficult. Not knowing where to begin, the students began to urge her to give them some help. Wanting them to feel successful and stay engaged, Ms. Jones pointed out to the students that the problem involved finding the area of all the rectangles that had a perimeter of 24. She told her students that they needed to make a chart of all possibilities, starting with a 1 x 11, and then find the area for each using the formula area = length x width. Students immediately set to work on the task and by the end of the class they had concluded that the 6 x 6 rectangle would yield the greatest area for 24 feet of fence and that the 4 x 4 would yield the greatest area for 16 feet of fence. As they began to work on part c of the task, they again began to press the teacher for additional assistance on how to proceed. At this point, Ms. Jones called the class back together and asked them what pen would give the most area for 36 feet of fence? When no one responded, she suggested that they divide 36 by 4 and see if that would work. She then asked students to find the dimensions of a pen that would give them the maximum area for 5 different lengths of fencing and to see if dividing by 4 would work for all of them.Leaves and Caterpillar TaskA fourth-grade class needs five leaves each day to feed its 2 caterpillars. How many leaves would the students need each day for 12 caterpillars?Use drawings, words, or numbers to show how you got your answer.Leaves and Caterpillars: The Case of David CraneStudents in Mr. Crane’s fourth-grade class were solving the following problem: “A fourth-grade class needs five leaves each day to feed its 2 caterpillars. How many leaves would the students need each day for 12 caterpillars?” Mr. Crane told his students that they could solve the problem any way they wanted, but he emphasized that they needed to be able to explain how they got their answer and why it worked.As students worked in pairs to solve the problem, Mr. Crane walked around the room making sure that students were on task and making progress on the problem. He was pleased to see that students were using lots of different approaches to the problem – making tables, drawing pictures, and, in some cases, writing explanations.He noticed that two pairs of students had gotten wrong answers as shown below.222313515684500Darnell and MarcusMissy and Kate2268855-13144500Mr. Crane wasn’t too concerned about the incorrect responses, however, since he felt that once several correct solution strategies were presented, these students would see what they did wrong and have new strategies for solving similar problems in the future.When most students were finished, Mr. Crane called the class together to discuss the problem. He began the discussion by asking for volunteers to share their solutions and strategies, being careful to avoid calling on the students with incorrect solutions. Over the course of the next 15 minutes, first Kyra, then Jason, Jamal, Melissa, Martin, and Janine volunteered to present the solutions to the task that they and their partners had created. Their solutions are shown on the back.During each presentation, Mr. Crane made sure to ask each presenter questions that helped the student to clarify and justify the work. He concluded the class by telling students that the problem could be solved in many different ways and now, when they solved a problem like this, they could pick the way they liked best because all the ways gave the same answer. Janine’s Work55245405765222251905Kyra’s Work397510267970-463557620Jamal’s Work-45720451485-5969031750Martin’s Work217170258445-4635517145Jason’s Work191770320040-3873545720Melissa’s WorkMonitoring ToolStrategyWho and WhatOrderUnit Rate--Find the number of leaves eaten by one caterpillar and multiply by 12 or add the amount for one 12 times Scale Factor--Find that the number of caterpillars (12) is 6 times the original amount (2) so the number of leaves (30) must be 6 times the original amount (5)Scaling Up--Increasing the number of leaves and caterpillars by continuing to add 5 to the leaves and 2 to the caterpillar until you reach the desired number of caterpillars Additive--Find that the number of caterpillars has increased by 10 (2 + 10 = 12) so the number of leaves must also increase by 10 (5 + 10 = 15) OTHER ................
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