051809 Cadre2 Complete

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?

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?

© 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 I’ve got to make sure that my thinking is right. I didn’t like it nearly as much as the one on the side because the side, 14 over 35, that’s 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 one’s gotta be more milky.

P: I think it’s 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. It’s 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, here’s 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 there’s less milk. This one has © 2008, University of Michigan

more milk and there’s more altogether, so then this ones would be the milky one, which is B anyway. P: But it’s 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 don’t 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 what’s interesting is, if we didn’t 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, it’s 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?

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 your evidence?

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

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

ADDING RATIOS Sharing Pizza – Classroom Scenario on Day 1

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.

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?

Teacher: We can actually represent this situation by using ratio notation. (She wrote 4 : 10 + 3 : 8 = 7 : 18 on the board.) Now, let’s discuss why this statement makes sense.

Students solved the Sharing Pizza, and then the teacher posed the following question.

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.

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 can’t 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

Sharing Pizza – Classroom Scenario on Day 2

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 of the rule for combining ratios? 2. How would you respond if some of your students suggest that it makes sense to use 3 : 8 + 2 : 5 = 5 : 13 to figure out the combined ratio of pizza to students?

Features of High Quality Professional Development (Smith, 2001 )

Cadre II Teachers’ rankings on April 17, 2009 Professional development should:

1

Have students’ learning as the ultimate goal

20

2

Support the ongoing work of teaching

3

4

5

6

1

1

1

3

4

5

1

7

3

Be grounded in mathematics content

1

7

1

3

3

4

Model and reflect the pedagogy of good instruction

1

3

5

4

1

5

2

2

1

3

6

Create some disequilibrium for teachers

8

9

2

3

2

2

10 Mean Median Mode

2

1.39

1

1

6.09

5

5

4.48

4

2

1

1

4.61

4

3& 6

1

3

7

7.61

7

10

4.91

5

3

Encourage teacher collaboration

3

5

3

4

2

2

2

2

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

2

1

4

3

4

7.74

8

10

Continue over the course of a teacher’s career

1

1

7