Observing Student Reasoning and Proof
 Introduction | Sums of Numbers | Problem Reflection #1 | Products of Numbers | Problem Reflection #2 | Classroom Practice | Observe a Classroom | Your Journal

Kaylee's fourth-grade class is in the midst of a problem called Products of Numbers -- an investigation into properties of numbers and relationships between numbers. Each small group was assigned a number and asked to look for patterns and relationships between the sum of that number added to itself (that is, doubled numbers) and the product of that number multiplied by itself (that is, square numbers). After drawing several consecutive squares on grid paper (1 by 1, 2 by 2, 3 by 3, and so on), Kaylee's group noticed a relationship between the drawings:

Following is another teacher-student dialogue for you to consider. As you read, focus on how the teacher's questions help the students formulate and think through their assertions. As noted previously, this exchange is just one possible approach; upper elementary school students will come up with a variety of ways to tackle this problem.

Kaylee: We put the squares in order. We saw that the next square is always one square wider and one more square down.

Teacher: Please show us and tell us again what you mean.

Kaylee: This is the 3-by-3 square; I'll make it red. Look at how I can mark off the same 3 by 3 inside the 4-by-4 square. See, each row of little red squares has one blue square on the end. Then, in the bigger square, we colored one blue square below each column of red squares.

Teacher: Can someone state a conjecture about Kaylee's group's observations?

Sarah: When you have squares in order, the next-bigger square always has one more row and one more column.

Teacher: Can we add something about what we mean by "squares in order" to be especially clear?

Kaylee: Use the counting numbers in order. For example, after you make a 6-by-6 square and find out it has 36 small unit squares, you need to make a 7-by-7 square. You make the squares with seven rows and seven square units in each row, like when you make them with colored tiles.

Teacher: So, we saw what happens for 3-by-3 and 4-by-4 squares. How can we show that it's always true for any whole number?

Sarah: I can show it by making a red square, like Kaylee's. Look, every group can make a square, whatever their number. If you had to switch to the square made with the next counting number, it'd be like your first square, but made with one more row and one more column because the next number is 1 more. (She draws a red square and then a blue one around it, and points to one more row and column between the edge of the red square and the blue.)

Teacher: Talk with your partner about Sarah's argument. Who thinks that she has made a convincing argument?

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