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Homework for Workshop Participants Lesson Plan Deborah Loewenherg Ball, Professor, University of Michigan Lesson Plan Transcript Class Description 216 217 219 225

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HOMEWORK FOR WORKSHOP PARTICIPANTS August 7, 2000 (given out at the workshop} Problem ~ Suppose you have three kinds of coins in your pocket. One is worth ~ cent, one is worth 5 cents, and one is worth 10 cents. If you pull out three coins, what amounts of money might you have? Problem 2 Again, suppose you have three kinds of coins in your pocket and that one is worth cent, one is worth 5 cents, and one is worth 10 cents. If you pull out two coins, what amounts of money might you have? Problem 3 Next, suppose you have three kinds of coins in your pocket and that again one is worth ~ cent, one is worth 5 cents, and one is worth 10 cents. What if you pull out four coins? What amounts of money might you have? Problem 4 Now suppose you have four kinds of coins in your pocket and one is worth ~ cent, one is worth 5 cents, and the last is worth 25 cents. Suppose you pull out three coins. What amounts of money might you have? Problem 5 Which problem seems harder problem 3 or problem 4? Why? A P P E N D ~ X G

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LESSON PLAN Problem: ~ have pennies, nickels, and dimes in my pocket. If ~ puB three coins out, what amount of money could ~ have? This is the fourth day of the school year. Three purposes for the class today: 1. (a) To develop students' habits of searching out multiple solutions, establishing whether all solutions have been found. (b) To develop students/ability to produce a mathematical explanation. In this case, an explanation for a solution must establish that: (~) three coins were used, of these three types; (2) the amount of money produced is correct. 2. To communicate to the students what doing mathematics wall mean (e.g., explain solutions to the teacher and to one another; listen to, critique, and use other stu- dents' ideas; be accountable for their ideas). 3. To begin to learn about the students: Their addition (multiplication) skills. Their openness to multiple answers and solutions. Their strategies for finding solutions. How they keep track of their solutions. How skeptical they are that they are done and how they go about determining when they have completed the problem. How they work with concrete materials. Their disposition to confer, to consider others' ideas. Plan: 1. Set up the problem: Make sure students can read it. Review coins if necessary. 2. Allow individual work time (10+ minutes) Whole group discussion Elicit solutions Ask for explanations/justifications. Guide the construction of explanations as necessary. Guide students to attend to and respond to (evaluate) one another's solutions. Conclusion: Did we find all the solutions? How do we know? How can we prove that we have all the solutions? A P P E N D I X G

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Strategies they are likely to use: I. Pull out coins, three at a time, see what you have, if it doesn't repeat something you already have, then write it down. If you keep repeating, you must be done. 2. Think of combinations, make them concretely. Write them down. 3. Think of combinations and write them down. Different possible approaches to recording: 1. Write down nothing. 2. Write down total amounts of money (e.g. 2l, 12) but not the coins used to make them. 3. Write down addition to represent coins (e.g., 5 + ~ + ~ = 71. 4. Write down coins and amounts (e.g., ~ dime and ~ penny and ~ nickel = 16 cents). A P P E N D ~ X G

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LESSON PLAN TRANSCRIPT RecorcIs of Instruction: The Three-Coin Problem in the Thirc! Gracle September ]8, ]989 Spartan Village Elementary School Teacher: Mrs. Ball Seating Chart Christina ~ Ogechi ~ ~ | Charles || Safriman | | Kip | Pravin E3 L3 This was the fourth mathematics class of the school year. The class had been working on combination and permutation problems since the first day. The problem on the board stated: ~ have pennies, nickels, and dimes in my pocket. If pull three coins out, what amounts of money could ~ have? Ball: Can I ask somebody to read the problem on the boa rem David could you read its David: I have pennies nickels and dimes in my pocket. If... 1... pull three... coins... out... what... amount of money (pause) would Ball: Put a C. Could. David: . could I have. Ball: What problem is this similar tog (pause) Charles Charles: The one with the the coins. Ball: What number of coins did we work with last weeks Charles: Um, two. Ball: Two coins A P P E N D I X G

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Can somebody give one example of amount of money that you could haven Sheaf Shea: Ball: Shea: Ball: Shea: Ball: Lin: Ball: Lin: Ball: Like you, like, you could have um Could you speak up just a littler I'm not sure that Bernadette can hear you. You could have like, five you could like, pull out one of each of them and you could like, you could get um, sixteen. Okay, one penny, one nickel, and one dime. Is that three coins Uh huh. And he says that's sixteen cents all together. What Jo other people think about thatch Link I agree with him. How would you get sixteen cents Um, one Jime would be ten cents, a nickel, a nickel, ten cent Wait, a dime is ten cents and then a nickel is five cents and it's fifteen cents and if it's a penny it will be sixteen cents. Okay. I would like you to work on this for a few minutes and see what different combinations you can figure out. And use the money if you want, to to help you, and then aher a little bit we will stop and talk about it together. First I would like you to work on it a little bit alone and see what you can come up with by yourself. Make sure you have the whole problem copied. And then write Jown whatever you need to write Jown in your notebook to help you remember what you're figuring out. The students worked on the problem alone or with the people seated near them. After they had been working for about 12 minutes, Mrs. Ball called the class together for a brief discussion. Lin: Ball: Shea: Ball: Lin: Shea: Ball: Shea: Ball: I have a question to ask. Does anybody have a prediction of how many solutions they think they will find for this problems I'm not that... Sheaf How about around ten. Excuse me just a second. Ogechi and Lin can you hear Sheaf Yeah. Around ten. Around ten. Is there a reason why you predicted tend Because um, I'm not sure. A P P E N D I X G

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Ball: Any, any different predictions Link Lin: Nine. Ball: Anybody else Wicks Mick: I've found nine. Ball: You come up with nine already Mick has already found nine. How are you going to know when you have all the solutions (pause) Any ideas Howwould you know if you had found them ally Limb Liz: You would start um Ball: Excuse me just a second, Kip and Pravin and Safriman- Right now Liz is talking and I would like you to be able to hear her. Liz: You would start doing the ones that you've already done over. Ball: Pravin could you hear Limb Pravin: Yes. Ball: What did she says Pravin: (shakes heaJ) Ball: Excuse men Can you say it one more time I'm sorry. Pravin listen to Liz, okays Speak a little louder. Liz: Okay. Um, you would start you would use the, the same ones over again. Ball: You would use the same one over again Anybody have any other ideas of how you would know if you had gotten them alla Is there any other way to tells (pause) Shekira Shekira: When we confer with somebody and if they have the same answers as you. Shekira: If they Jon't, then you Jon't have all the answers and you need to write it down, then you have all of them. Ball: I have a small concern, right now. I would like everybody to out their coins down and their pens down. . . . . I would like everybody to put their coins down and their pens down for a moment. One of the things that's very important is that if one person in the class talking other people need to listen, because people are saying things that can help you think about the problems. Shekira said some interesting things, and so did Liz, but lots of people were not listening to them. I know it's because you're finding more solutions yourself. But, one of the things that I would really like you to, to see you Joing is to listening. ...to be listening really hard when someone else has an idea, because it might help you with your thinking. IS A P P E N D I X G

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I m sure a lot of people didn't hear Shekira. And I had to ask Liz to say, what she said three times. Stop for a moment now, and listen hard to Shekira and see what you think about what she is saying. Could you say it one more timed Shekira: Well, when you think you're done, confer with somebody else at your table and if they have the same answers as you Jo then Then, you know you have all of them and if you Jon't then you write it down on your paper. Or they write the ones they have that you have. So Liz and Shekira gave us two different ideas. Liz said when you start repeating yourself in the ideas you come up with, you think you probably have them all. Shekira saiJ...Shekira added something to it. Shekira said when you think you have them all, you could confer with somebody else at your table and see if they found any that you haven't found. Are there any other ways to know if you have all the answers Micah It's not about the answers. So, it's another comments Yeah. Whets Um, I think it's ten because I just came up with one more. Okay, well, Mick just came up with one more. He's up to ten possible Ball: Mick: Ball: Mick: Ball: Mick: Ball: answers. Take a few more minutes to work on it, and then I'd like to spend a few minutes talking about what you've come up with. The children returned to working on the problem individually and in small groups. After a few minutes, Mrs. Ball called them together again. Ball: Let's stop for a moment. Put your coins down. It's easy to tell when people have stopped with the coins, because the coins make a lot of noise. Put the coins down and put the pens down for a moment. You're going to want your pen though, because while we have while we discuss the problem, if somebody brings up an answer that you didn't find, you might like to record it in your notebook. We have one answer recorded on the board, one solution. Who would like to share another solution that they came up withy Safriman~ Safriman: Twelve. Ball: Okay, twelve cents. Could somebody tell how Safriman got twelve cents A P P E N D I X G

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David: Ball: Sarah: Ball: Sarah: Ball: Sarah: Ball: Liz: Ball: What coins did Safriman use to get twelve centsC Who thinks they known David c Um. Safriman used dime um, and two pennies. What do other people think about what David said. David said he thinks Safriman used two pennies and one dime. What do other people think about thatc SaraRc I agree. Can you prove that, that's rights Yeah, because it's three, three coins. Three coins. How can you prove that that's twelve centsC Becauseten and two is twelve and that's three coins. Any commentsC Lizc I agree with that. Anybody disagree with its Okay, another solutions Another possible way to do this problems No, I just asked you. Somebody differentC Bernadettec Bernadette: I got seven cents. Ball: Who thinks they know how Bernadette got seven centsC Kjpc Kip: One nickel and two pennies. Ball: Christinac What do you think about what Kip solids Christina: I agree with him. Ball: Can you prove that, that's seven centsC Christina: Yeah. Because um, a nickel and two pennies is a nickel is five cents and two pennies will add to seven cents. Ball: Comments from anybody elsec Anybody disagree with thisc _1 1 1 1 1 ~ ~ okay, do we have another solutions How about Ogechi, do you have something different . In your notebooks Ogechi: Thirty cents. Ball: How mucks Ogechi: Thirty Ball: Ogechi, do you want to tell us how you got it or should we ask other people to figure it outs Ogechi: Three dimes. Ball: Okay, three dimes. Jillian, what do you think about what Ogechi solids Jillian: I think she's right. Ball: Why do you think she's rights Jillian: Because ten plus ten is twenty and plus another ten is thirty. Ball: Any comments from anybody elsec David c David: Twenty five. A P P E N D I X G

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Ball: David: Ball: David: Ball: David: Ball: Sarah: Ball: Shea: Ball: Shea: Ball: Shea: Ball: Shea: Ball: Oh, you're already giving another ones Uh huh. Did you agree with billions Um, yeah. Okay, what is your solutions Fifteen. Fifteen. Who thinks they know how David got fifteen cents Sarah Um, three nickels. Shea, what Jo you think about thatch I'm not sure. Why are you not surer What are you figuring outs Oh. Shea, how much is one nickels Uh, five cents. And two nickels are ten cents, and three nickels are fifteen. So Do you agree with this then or disagreed Um, agree. We're going to stop. I would like everyone to look over this way for a moment. When we start math tomorrow, we're going to continue with this problem a little bit longer. I have a question to ask before we stop and a comment to make. I'll make my comment first and then ask my question. My comment was, I thought people did a better job just now listening to each other's solutions and giving each other time. Did you notice that when somebody was figuring something out people weren't going uh,uh,uh or interrupting. People were listening and thinking about whether it made sense. Did you notice thatch And did you also notice that people were explaining why it made sense. Like people would say it's thirty cents because, Jillian said, because ten plus ten is twenty and ten more is thirty. Was that a good explanations It was a good explanation because it helped us understand why that answer made sense. Mrs. Ball asked the students about a task she had posed to them the day before. After hearing brief reports from several students, she ended the class. A P P E N D ~ X G

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CLASS DESCRIPTION English How Long At Name Gender Race Country Proficiency This School' Benny M White Ethiopia fluent 3years Bernaclette F White Canacla native speaker just started Charles M Asian Taiwan/Canacla fluent 3years Christina F African-American U.S.A. native speaker 12 months Davicl M Asian Inclonesia developing 3years Ira F White Inclonesia developing 5 months Jillian F White U.S.A. native speaker 3years Kip M African Black Kenya fluent 3years Lin F Asian Taiwan fluent 2 years Liz F White U.S.A. native speaker 3 years Marta F Latina Nicaragua beginning just started Mick M White U.S.A. native speaker 2 years Ogechi F African Black Nigeria fair 3years Pravin M White Nepal beginning 5 months Rania F White Egypt good 3 years Safriman M Asian Inclonesia developing 12 months Sarah F White U.S.A. native speaker 2 years Shea M White U.S.A. native speaker 2 years Shekira F African-American U.S.A. native speaker just started 'NOTE: This column reflects the length of time the child had been in this school as of 9/89. No one had been in this class longer than a few clays. A P P E N D I X G

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CLASS DESCRIPTION English How Long At Name Gender Race Country Proficiency This School' Benny M White Ethiopia fluent 3years Bernaclette F White Canacla native speaker just started Charles M Asian Taiwan/Canacla fluent 3years Christina F African-American U.S.A. native speaker 12 months Davicl M Asian Inclonesia developing 3years Ira F White Inclonesia developing 5 months Jillian F White U.S.A. native speaker 3years Kip M African Black Kenya fluent 3years Lin F Asian Taiwan fluent 2 years Liz F White U.S.A. native speaker 3 years Marta F Latina Nicaragua beginning just started Mick M White U.S.A. native speaker 2 years Ogechi F African Black Nigeria fair 3years Pravin M White Nepal beginning 5 months Rania F White Egypt good 3 years Safriman M Asian Inclonesia developing 12 months Sarah F White U.S.A. native speaker 2 years Shea M White U.S.A. native speaker 2 years Shekira F African-American U.S.A. native speaker just started 'NOTE: This column reflects the length of time the child had been in this school as of 9/89. No one had been in this class longer than a few clays. A P P E N D I X G