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COFFEE PROBLEM

Consider the following two coffee mixes.

In mix A the ratio of milk to coffee is 2 : 3. In mix B the ratio of milk to coffee is 3 : 4.

Which recipe will make coffee that is the most milky? Solve the problem in two

different ways. In one of these ways, solve the problem without using fractions,

percentages, or decimals. Use a diagram to show why your strategy works. Record

your solution and diagram on chart paper.

Ben said he used percentages to figure out which mix will be the most milky. See

his work below. Does his strategy make sense?


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2008, University of Michigan

Amanda disagrees. She claims that they are equally milky because they each have

one more unit of coffee than milk. Does her thinking make sense?

Mix A: Mix B:

What might you ask or what comment would you make in response to these student

strategies?


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2008, University of Michigan

FACILITATING PROFESSIONAL DEVELOPMENT

Coffee Problem Transcript

T: I think that is just something for us to think about as teachers.

Sam (P1), you had a comment?

P1: Well at first, looking at the bottom one, I now Ive got to

make sure that my thinking is right. I didnt like it nearly as

much as the one on the side because the side, 14 over 35,

thats easy, your denominators is the P1e. But, looking at

this, if you remember the label, then you can remember that

mixture A is close, but you can say that it has a little bit less

milk and more coffee, and B has more milk and less coffee.

So just thinking of it that way you can kind of think, this ones

gotta be more milky.

P: I think its nice just when you have a common feature. You

either need to have a common milk or a common coffee or a

mixture. Something needs to be in common so you can

justify our reasoning. Its easier, especially for kids. You need

to make the logic, whether its coffee and milk or milk and

chocolate and your making chocolate milk, whatever it is. But

I mean to justify it, I think you have something in common.

P: Well, heres a, sorry Like these two right here are always

[unintelligible] multiply them both by four and three so that I

got to the twelve. But here, so this one has less coffee and

has a less concoction but theres less milk. This one has

more milk and theres more altogether, so then this ones

would be the milky one, which is B anyway.

P: But its harder to see.

T: But I think that Sam was saying that with mixture A you have

less milk but more coffee Compared to mixture B, we have

more milk and less coffee. Which one is going to be more

milky? Can we reason about that too?

Ps: [Many are talking at once]

T: So we dont always have to have the common numbers or the

common units, but it is usually easier to justify also. But we

can reason about the other strategy as well.

T: But whats interesting is, if we didnt have 14 to 21, 15 to 20

can we compare mixtures if the ratios were given like 2 to 3, 3

to 4 in the way that we compare 14 to 21, 15 to 20? Can we

do that kind of comparison with the original ratios?

T: You know what I am trying to get at? In the original ratios, 2

to 3, 3 to 4 there is more coffee. So I cannot decide which

one is going to be more milky. But with the other two ratios

that we came up with 14 to 21 and 15 to 20, its easy to see

that in one case we have less milk but more coffee.

T: Do you see how different those two cases, those situations

are?


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ANALYZING PD FACILITATION

Reflection as a Tool to Improve Facilitation Skills

1. What do you notice in this video clip?

2. What do participants seem to understand about the mathematics? What is yourevidence?

3. What was effective about this exchange? What could have been done to make thisexchange more effective?

4. What do you notice about the norms that have been established?


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ADDING RATIOS

Sharing PizzaClassroom Scenario on Day 1

You are going to eat pizza with your friends. When you arrive you see

that the restaurant has set up one large table that seats 10 people, and

a small table that seats 8 people. The server places four pizzas on the

large table and three pizzas on the small table. At each table, everyone

shares the pizzas equally. Which table should you sit at if you are really

hungry and want to eat as much pizza as possible?

Students solved the Sharing Pizza, and then the teacher posed the

following question.

Teacher: If you were to combine the two tables, what will be the ratio of

pizza to people? (After working on this in pairs for a while, they

had a whole group discussion.)

Carla: When you combine the two tables, there will be 7 pizzas and 18

people. Therefore, the ratio of pizza to people will be 7 : 18.

Rob: I have the number sentence for what you just did. (He wrote the

following statement on the board: 4/10 + 3/8 = 7/18.)

Students: That cant be right.

Teacher: Why not?

Jack: Both 4/10 and 3/8 are close to a half. When you put them

together you have to be close to 1. But 7/18 is not even a half.

At this point the teacher invited the whole group to think about the

meaning of each fraction in the context of the problem. They decided

that 4/10 of a pizza is the amount each person gets at Table 1 and 3/8

of a pizza is the amount each person gets at Table 2. Therefore, the

sum of these two fractions gives us the amount of pizza two people

(one from each table) get, which is different from the ratio of pizza to

people when the two tables are combined. Then the teacher introduced

the following notation.

Teacher: We can actually represent this situation by using ratio notation.

(She wrote

4 : 10 + 3 : 8 = 7 : 18 on the board.) Now, lets discuss why this

statement makes sense.

Sharing PizzaClassroom Scenario on Day 2

You are going to eat pizza with your friends at the fourth-grade party. In

one fourth-grade classroom the ratio of pizza to students is 3 : 8. In the

other fourth-grade classroom there were 2 pizzas for every 5 students.

The teacher posted this problem on the board and she wanted students

to investigate the generalizabilitiy of the rule for combining ratios.

Teacher: Yesterday we were talking about joining tables and we found

the combined ratio of pizza to people. Can we do the same

thing here? Can we find the ratio of pizza to students if the pizza

and students are combined in one classroom?

Analyzing the Mathematics & Connecting to Practice

1. What conclusions might students make about the generalizability ofthe rule for combining ratios?

2. How would you respond if some of your students suggest that itmakes sense to use 3 : 8 + 2 : 5 = 5 : 13 to figure out the

combined ratio of pizza to students?


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Cadre II Teachers rankings on April 17, 2009

Professional development should: 1 2 3 4 5 6 7 8 9 10 Mean Median Mode

Have students learning as the ultimate goal 20 1 1 1 1.39 1 1

Support the ongoing work of teaching 3 4 5 1 3 2 3 2 6.09 5 5

Be grounded in mathematics content 1 7 1 3 3 4 2 2 4.48 4 2

Model and reflect the pedagogy of good instruction 1 3 5 4 1 5 2 1 1 4.61 4 3 & 6

Create some disequilibrium for teachers 2 1 3 6 1 3 7 7.61 7 10

Encourage teacher collaboration 3 5 3 4 2 2 2 2 4.91 5 3

Take into account teachers contexts 2 1 1 1 3 2 6 2 5 7.13 8 8

Make use of the knowledge and expertise of teachers 5 1 6 2 2 3 2 1 1 5.00 4 4

Be sustained and cohesive 3 3 1 4 1 1 5 5 5.96 6 8 & 9

Continue over the course of a teachers career 1 1 2 1 4 3 4 7 7.74 8 10

Features of High QualityProfessional Development

(Smith, 2001)

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