*Scenario 1:*A teacher decides that she wants a math lesson to focus on two-digit by two-digit multiplication. She finds an appropriate problem for the students to work on. Although she knows that six or seven students still struggle with the concepts involved in multiplying by even a single-digit number, she presents the problem to all students, making sure that the struggling students receive help from herself or other students.

*Scenario 2:*A teacher is working on teaching fact families. He asks students to describe the fact family for 3 + 4. One student offers a response: 3 + 4, 4 + 3, 7 - 4, 7 - 3. The teacher records this on the board and checks that other students concur. The whole episode takes less than five minutes, only one student responded, and now the teacher needs to set up another activity.

## Two Beliefs That Need to Change

## An Idea Takes Root

## Open Questions

*multiply*is familiar, any student can contribute. Let's look at some student responses:

*Student 1*: I would choose 2 and 5 because I know that 2 × 5 = 10.

*Student 2:*I would choose 45 × 100 because 45 × 100 means 45 hundreds. That means you write 45 in the hundreds place in a place value chart, so it's actually 4,500.

*Student 3:*I would choose 4 × 9 because you could just double 2 × 9 to get 18 and then double that. Eighteen doubled is two 10s and two 8s, so that's 20 and 16, which is 36.

*Student 4:*I would choose 4 × 25 because I know 4 quarters make a dollar, and that's 100.

*Student 5:*I would choose 1 × 34,782 because if you multiply by 1, you don't have to do anything; it's just the other number.

That single question about multiplication ends up reinforcing a wide range of mathematics concepts: place value; working with money; and several properties of multiplication, including multiplying by 1 and the notion that multiplying by 4 is the same as doubling twice.

## Creating Open Questions

*Strategy 1: Start with the answer*. A teacher can take a straightforward question and present it backward. For example, instead of asking, "What is 23 + 38?" a teacher could say, "I added two numbers. The sum is 61. What numbers might I have added?"

- The area of a rectangle is 20 square inches. What might be its length and width?
- A 3D shape has 8 vertices. What might it look like? (Students might suggest a cube.)
- The 10th term in a pattern is 36. What might the 8th and 9th terms be? Describe the pattern. [Students might think 36 = 26 + 10, so the pattern might start 27 (26 + 1), 28 (26 + 2), and so on, with the students realizing that the 8th and 9th terms are 34 (26 + 8) and 35 (26 + 9).]

*Strategy 2: Ask for similarities and differences*. Asking students how two things are alike and how they are different can provide teachers with valuable assessment for learning information. A teacher might ask,

- How are the numbers 4 and 9 (or 350 and 550 or 100 and 1,000, and so on) alike? How are they different? (Students might point out whether the numbers are even or odd or divisible by numbers other than themselves.)
- How is the formula for the perimeter of a rectangle like the formula for its area? How is it different? (Students might indicate that both formulas involve using values for the length and width of the rectangle, but that one involves addition and the other doesn't.)
- How are these two patterns alike? How are they different?4, 8, 12, 16, 20,…4, 7, 10, 13, 16,…

*Strategy 3: Allow choice in the data provided*. Students are empowered by the opportunity to choose one or more numbers with which to work. For example, teachers might ask,

- Choose a number for the box on the left. What is the length of the hypotenuse of this right triangle?

- Choose a value for the fourth number in the series that follows and calculate the mean: 4, 5, 6, ___.
- A pattern starts at □, and you add ▵ each time. Choose values for □ and ▵. Will 40 be in your pattern? Explain.

*Strategy 4: Ask students to create a sentence*. Asking students to create a sentence using specific mathematics vocabulary is a good way to assess student understanding of the vocabulary and to foster creativity. A teacher might ask,

- Use the words
*even*,*more*, and*always*, and the number 10 in a sentence. (Students might say, "If you add 10 to an even number more than 10, the answer is always even and always has at least two digits.") - Use the words
*length, width*,*formula*, and the number 10 in a sentence. - Use the words
*increasing, decreasing, pattern*, and the number 18 in a sentence.

## Parallel Tasks

*different*levels of difficulty, thus taking into account the variation in student readiness.

## Creating Parallel Tasks

*Strategy 1: Let students choose between two problems*. The teacher might give students a choice between two problems at different levels of difficulty:

- Choice 1: There are 427 students in Tara's school in the morning. Ninety-nine of them left for a field trip. How many students are still in their classrooms? (The problem involves subtraction and is suitable for mental math calculations because 99 is so close to 100.)
- Choice 2: There are 61 students in 3rd grade. Nineteen of them are in the library. How many students are still in the classrooms? (This problem also uses subtraction and is suitable for mental math, but it involves smaller values for students who are not ready for work with 3-digit numbers.)

*Strategy 2: Pose common questions for all students to answer*. The teacher could ask

*all*students the following questions, no matter which task they completed:

- Before you calculated, could you tell whether the number of students left in the classrooms would be more or less than one-half of the total number of students? Explain.
- What operation did you or could you use to solve your problem? Why that one?
- Would it be easier to solve the problem if one more student had left the classroom? Why?
- How could you use mental math to solve your problem?
- How did you solve your problem? How many students are still in their classrooms?

## Meaningful—and Manageable

#### Open Questions for Every Grade

### Grade 1

The answer is 10. What might the question be?

How are 5 and 10 alike? How are they different?

Choose two numbers to add. What is the sum?

Create a sentence using the words and numbers

*and*,*more*,*5*, and*3*.

### Grade 4

The answer is ⅔. What might the question be?

How are 80 and 800 alike? How are they different?

Create a sentence using the words and numbers

*product*,*8*,*almost*, and*50*.The product of two numbers is almost 30. What might the two numbers be?

### Grade 8

The answer is 30π. What might the question be?

How are the formulas for the circumference and the area of a circle alike? How are they different?

Create a sentence using the words

*surface area, volume, greater*, and*300*.The sum of two integers is a negative integer very far from zero. What might the integers be?

### Grade 11

The answer is [see above]. What might the question be?

How are calculating powers and calculating logarithms alike? How are they different?

Create a sentence using the words

*irrational, repeating*,*4*, and*greater*.An irrational number is approximately 8. What might it be?