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Text Box 8.3 A Math Happening 8c: The Turtle Race

Torty the turtle can travel 2/3 of a mile in 1 hour. How long will it take to travel 6 miles?

In a research lesson called the Turtle Race, teachers were asked to anticipate strategies students would use to solve the problem. In Japanese Lesson Study, the board plan from the lesson plan shows how the teacher expects they will select and sequence the sharing of the anticipated strategies. Typically, when a teacher plans the sequences of the different strategies, they have specific agenda in mind. For example, they may sequence the strategies based on concrete to abstract such as an algebraic lesson that starts off from strategies that use pictures to numbers, tables and then abstracting from the computation to formulate a rule. Or other times, it could be the very opposite to show how an algebraic approach would be more complicated and perhaps unnecessary because a pictorial approach is more efficient and elegant.

A major goal for this unit is that students conceptualize ways of thinking about division with fractions. In particular, we want students to consider how a common- denominator method for dividing fractions might be embedded within physical models of division situations. In this lesson specifically, students will begin to explore measurement contexts for dividing a whole number by a fraction. Our goal is for students to develop a way of solving the story problem presented in a way that makes sense to them. There are some key ideas that precede this work:

  • 1. Both sharing and measuring problems can be modeled by division
  • 2. In dealing with “leftovers,” we can ask either “what fraction of a group is left” or “what fraction of an item will be in each group”?
  • 3. A divided by B is the same as A/B

On the teachers’ plan, the strategy map showed several paths students could take. A popular solution they named the linear (number line) model would begin with an open number line marked from 0 to 6 miles. Students would mark off 2/3 mile at a time, and count how many “2/3s” it took to get to the end (9 hours). They hoped that student would keep counting on by 2/3s until they got to 6 or 18/3. Then to know how many hours, students count how many 2/3s they have in 6 (see Figure 8.8).

Using a number line

Figure 8.8 Using a number line.

Another model students might use would be 6 discrete circles or rectangles, since some students are accustomed to using fraction manipulatives to think about parts of a whole. These students are likely to use a similar strategy of marking of 2/3 at a time but using a region model.

Some students may convert 2/3 to approximately 66% since they have recently done work converting fractions into equivalent forms. This method will not be very useful here since the repeating decimal requires some rounding off and may not make the answer as apparent.

Some students may use a 2-column ratio table to keep track of both the hours and the miles

They considered that some students may use what they called reasoning up or down to notice that if 2/3 miles is 1 hour then 1/3 mile = 30 min and think back up to 1 mile = 90 min (or 3/2 hours or 1.5 hours) and multiply 3/2 x 6 miles. This way of thinking most closely resembles the traditional “invert-and-multiply” method.

Students may also use what they know about multiplication of fractions and think 2/3 x (some number of hours) = 6 miles. Students may use a guess and revise strategy to figure out what you could multiply by 2/3 to get 6. They called this is the missing factor method.

After drawing models, some students may reflect on their work as 18 thirds divided by 2 thirds = 9 thirds or 18/3 divided by 2/3 = 9.

When asked to write a number sentence to match their work, students may write several equations:

If someone finds 3/2 in the model, we may choose to ask the class to examine how this 3/2 relates to the 2/3 in the other methods. If symbolic notation seems difficult, we will listen for kids to articulate their answer as 9 groups of 2/3 (9 x 2/3). Again, this gives us an opportunity to relate the different equations and talk about the relationships between them. If someone identifies the problem as a division story, the student will explain how he/she saw the story that made him/her think it was division.

The teachers discussed that they would leave the number sentences using division as it was generated by students. Ultimately, they wanted to see students who could articulate their thinking of the 6 as 18/3 in order to begin to develop the idea that if the pieces are the same, we can divide the numbers as we would normally in whole- number division. The teachers were interested in what informal knowledge students would rely on. Some questions they planned to ask included as follows:

  • 1. What operation would you say you’re using to solve this problem?
  • 2. What number sentence can you write that matches your model?
  • 3. How does this problem compare to the cookie problem we worked on yesterday?
  • 4. What makes this a division problem? (or addition, etc., depending on what students say)
  • 5. Where is your answer in your model? In your number sentence?
  • 6. If students use “invert and multiply,” where is the 3/2 in your model?

They also wanted their students to demonstrate their current understanding in their problem-solving strategies and related written explanations. For this lesson, the goal was for students to think about the situation and use a method that made sense to them to solve it. Eventually, they would look to generalize methods and for efficiency, but the day’s agenda was to begin this process. After this lesson, they would follow up the lesson by investigating whether similar problems can also be solved by converting both numbers to a common denominator, or by examining a reciprocal relationship, and how to deal with problems whose answers don’t result in a whole number.

Using this strategy map, teachers embarked on their research lesson and were surprised by their students. They were surprised that more students didn’t draw a linear model given the context. They were also surprised that a boy who was typically not a strong student saw 3/2 in the problem. They noticed some students who were accustomed to working only with symbolic forms struggled more than others who were willing to draw a model.

This lesson reinforced the need to choose carefully who will share their work and to be thoughtful about questions we ask to support student thinking and connections. Teachers had many wonderings as they observed their lessons as they shared during the lesson debrief. They wondered, how would students handle “leftovers”? What if he traveled 9 hours and still had a little distance left to go? Which student methods are generalizable and which have specific purposes? What knowledge is necessary before beginning this work and what is gained as we go along?

Student work showing sensemaking

Figure 8.9 Student work showing sensemaking.

This student modeled discrete thirds and grouped every two together to represent each hour.

The difficult part was keeping track of not only the hours, but also the total miles.

These students found a way to keep track of both the hours and the miles. Teachers felt better prepared to navigate through the classroom discussion because they had participated in the act of anticipating solutions and creating a board plan consisting of their strategy map.

Keeping track of the change

Figure 8.10 Keeping track of the change.

Think about it!

What are all the everyday situations that ratios arc used? How can we expose them to our students? Share ideas with your colleagues.

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