This interactive provides supported practice in thinking about Responsive Actions during the course of a lesson.

Possible Responsive Actions are summarized by the following four actions:

• Gather more evidence
• Provide further instruction
• Provide formative feedback
• Move on

The interactive provides a scenario of a lesson, with responses and sample thinking from a selection of students from the class. You have the opportunity to consider what responsive actions might be appropriate, on the individual, small group, and whole class levels. When you make a choice, our reasoning about that choice will be provided. (Note that disagreement does not mean you are wrong; rather, it means we have another perspective that you should consider, if you haven't already.)

For more about Responsive Actions, see Chapter 3, "Gathering, Interpreting, and Acting on Evidence," in Bringing Math Students into the Formative Assessment Equation: Tools and Strategies for the Middle Grades.

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Step 1. As part of a lesson on comparing fractions less than 1, Ms. Harriman wanted to see how well her students met the success criterion, Given two fractions, I can write another fraction between them. She gave her students a series of problems to answer on their white boards. She had them draw lines to create 6 areas so they could keep their answers to discuss after all 6 problems were done.

Here's the series of problems. Try them yourself before continuing.

Press the "next" button at right when you're ready to continue.

Step 2. Before you look at some student work, read through these common methods for finding a fraction between two fractions. Consider how students might use them incorrectly. (Other methods are possible.)

• Use benchmark fractions. In some cases, comparing the given fractions to known benchmarks can provide an answer. For example, many students know 3/4 is a benchmark fraction between 1/2 and 1.
• Rewrite using common denominators. This strategy allows students to more easily see the given numbers' relative sizes. In a case like 1/2 and 4/5 (using common denominators, 5/10 and 8/10), a fraction between can then be easy to pick by using the same denominator and choosing a numerator between the existing numerators (in the example, 6 or 7 would work).
• Make the pieces smaller. When the fractions differ only by one "piece", such as 3/5 and 4/5, students might imagine further partitioning the pieces to make smaller pieces that are still equal in size to each other. Using 3/5 and 4/5, they might partition the whole into 10ths, so 3/5 and 4/5 become 6/10 and 8/10, respectively. Then they can choose a new numerator between the existing numerators (7/10).
• Make one a little smaller or larger. Using this strategy, students will alter one of the fractions to "nudge" it closer to the other. For example, to find a fraction between 1/4 and 1/2, students may say that 1/4 can be made a little larger by subtracting one from the denominator, making it 1/3. This strategy sometimes works, but it comes with a warning: you have to be sure the little smaller/larger isn't too much of a change, or the final result will be smaller (or larger) than both fractions rather than between them.

Step 3. After completing the series of problems, the students discussed their methods in their small groups. Ms. Harriman circulated among them, listening both for common strategies that didn't work (or didn't always work) and interesting correct strategies.

The following tables give example responses from three groups, along with what she overheard from those students as she circulated. (For your convenience, correct answers are colored green and in normal type; incorrect answers are red and italic. Note that this does not necessarily reflect accuracy or flaws in thinking, only whether the answer is correctly between the given fractions.)

In the next step, you can focus on individual students, groups, or the class as a whole (as represented by this selection of students). For now, just take a moment to read over the responses and get a sense of the class from this subset of groups.

Step 4. Explore the tabs below (Individual Students, Small Groups, and Whole Class). In each, you have the opportunity to look at the student responses and decide what responsive action you think is most appropriate.

Remember, as part of a lesson on comparing fractions less than 1, Ms. Harriman was checking on how well her students met the success criterion, Given two fractions, I can write another fraction between them.