Conjecturing

As Ms. Cooper tells her students in the following vignette, a conjecture is a mathematical statement that you think might be true, based on what you know so far. Conjecturing is the process of making these statements without yet deciding if they are true. In this chapter, you’ll learn how to guide students through that process.

Conjectures can be general or specific. General conjectures often follow from generating cases and noticing patterns—when students state what those patterns are, they are on their way to making conjectures that generalize from a few cases to what is true for all cases. For some activities, conjectures will be more specific. For example, students might conjecture a solution to one example equation before they develop general methods for solving similar equations. We discuss both kinds of conjectures in this chapter.

A good conjecture should contain precise language with the purpose of making the conjecture clear to everyone—but it’s unlikely that students’ first attempts will come out this way. That’s why clarifying conjectures is important. We explain more about clarifying conjectures in this chapter.

As important as the process of conjecturing is, there are times when you will want to start out with a premade conjecture. We discuss that approach and when to take it.

Before reading the vignette, take some time to make conjectures about what you think is always true about coordinates of vertices of all rectangles with sides parallel to the axes or what might be true just for the category of special cases you made. Consider also false conjectures that students might make.

Ms. Cooper structured a whole-class discussion leading to a broad variety of conjectures based on the patterns students had found. Through this process, she addressed important content: Students were exploring the properties of lines that are parallel to the axes through making conjectures about the vertices of sides of rectangles. She was preparing students to use algebra to describe coordinates in subsequent lessons by making general conjectures in words.

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