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Homework for Workshop Participants
Lesson Plan
Deborah Loewenherg Ball, Professor, University of Michigan
Lesson Plan Transcript
Class Description
216
217
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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?
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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?
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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).
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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
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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.
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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
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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
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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.
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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.
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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.
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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.
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