"I'm confused. How does figuring out the perimeter help us?""We don't know how to count the shaded parts.""We're stuck!"

## Why Is Struggle Important?

## Questions That Get Students Started

#### Figure 1. Math Task 1: Modeling Percent

**Find the percentage, decimal, and fractional amount of the shaded area in relation to the whole grid.**

*Source*: From

*Bridges in Mathematics Curriculum*(1st ed.) by Math Learning Center. Salem, OR: Author. Copyright Math Learning Center. Used with permission.

C<EMPH TYPE="5">ollin: We're stuck.M<EMPH TYPE="5">r. C<EMPH TYPE="5">raig: Whatdoyou understand?M<EMPH TYPE="5">arina: That we need to find out how many squares are shaded out of 100 because that will tell us the percent.C<EMPH TYPE="5">ollin: Yeah, but we don't know how to count the little shaded parts.M<EMPH TYPE="5">arina: (tracing her finger around the grey shaded portion) We think the shaded area is a square.M<EMPH TYPE="5">r. C<EMPH TYPE="5">raig: What do you know about finding the area of a square that might help you?

- What do you already know? What do you need to know?
- What do you understand so far?
- What parts of this problem make sense to you?
- What in this problem doesn't yet make sense to you?
- How might we decide which approach makes more sense?
- What are some math ideas we've worked on before that could help you with this new idea?

## Questions That Make the Mathematics Visible

- How are percentages, decimals, and fractions related?
- What do you know about finding the area of a square that might help you?
- How does that array relate to multiplication and division?

## Encouraging Student-to-Student Questioning

#### Figure 2. Math Task 2: Modeling Percent.

**Find the percentage, decimal, and fractional amount of the shaded area in relation to the whole grid.**

*Source*: From

*Bridges in Mathematics Curriculum*(1st ed.) by Math Learning Center. Salem, OR: Author. Copyright Math Learning Center. Used with permission.

H<EMPH TYPE="5">enry: I'm thinking it's 25 [small square units]A<EMPH TYPE="5">maya: I'm thinking it's 50.H<EMPH TYPE="5">enry: Let's count it.A<EMPH TYPE="5">maya: Remember, we have to try not to count [the units] one by one.H<EMPH TYPE="5">enry: How should we get started?[Amaya writes "10" on her paper]H<EMPH TYPE="5">enry: So the perimeter is 10? How does figuring out the perimeter help us?A<EMPH TYPE="5">maya: 10 is the number of square units on the edge of the grey part [tracing her finger around the edge of the shaded portion]. I counted 20 half squares, so it's 10 whole, but I don't think 10 is the perimeter—I think it's the total area of the half squares on the edge. What do you think?

## A Vibrant Pursuit

##### References

Kapur, M. (2010). Productive failure in mathematical problem solving. *Instructional Science, 38*(6), 523–550.

National Council of Teachers of Mathematics. (2014). *Principles to actions: Ensuring mathematical success for all*. Reston, VA: NCTM.

Smith, M. S., Bill, V., & Hughes, E. K. (2008). Thinking through a lesson: Successfully implementing high-level tasks. *Mathematics Teaching in the Middle School, 14*(3), 132–138.

Smith, M. S., & Stein, M. K. (1998). Selecting and creating mathematical tasks: From research to practice. *Mathematics Teaching in the Middle School, 3*(5), 344–350.

Stein, M. K., Smith, M. S., Henningsen, M. A., & Silver, E. A. (2009). *Implementing standards-based mathematics instruction: A casebook for professional development* (2nd ed.). New York: Teachers College Press.