There are two expectations in our Grade 9 Academic curriculum that I felt I never did a great job of addressing and I know many other teachers feel the same. They state that students will:
pose questions about geometric relationships, investigate them, and present their findings, using a variety of mathematical forms
illustrate a statement about a geometric property by demonstrating the statement with multiple examples, or deny the statement on the basis of a counter-example, with or without the use of dynamic geometry software
Really, they seem like quite fantastic expectations. They provide an opportunity for some great open exploration. Unfortunately, I think they are often left to the end of the course and aren’t given much attention because they don’t seem to be that necessary for future courses. I have to agree that the actual “content” of these expectations may not be critical for future math courses. However, I would have to argue that the “processes” in these expectations; the ability to pose questions, explore them and be able to prove or disprove ideas is indeed critical for future math courses and the very thing that many of us hope our students can do.
With that in mind, a grade 9 teacher let us “borrow” her students to see how they might deal with these expectations.
We started the lesson by talking about the word “conjecture”. None of the students in the class were able to provide a definition so we gave them a few clues (posted a couple of tweets that had the word in it) and had them talk with a partner about what the word might mean. After some time, a student suggest it means “a guess or a prediction”.
I made a conjecture that “all grade 9 students in this school have a cell phone” and asked whether they thought my conjecture was true. A student quickly put his hand up and said “no”. I asked him how he knew and he smiled slyly and said “because I don’t have one…” Which led to a discussion about the usefulness of a counterexample.
Students created a sketch of a parallelogram on GSP. I told them a student had a conjecture that the angles that were diagonally opposite each other were equal. We discussed what we would need to do to “prove” this conjecture and then the students proceeded to make the necessary measurements. (Note: I realize that testing many cases on GSP may not be considered a true “proof” in the academic sense, but for our purposes we decided it would be good enough for now).
The next task students drew the diagonals of the parallelogram and were asked to come up with their own conjectures.
This is how the conversation went:
S: The opposite angles, where the diagonals intersect will be equal.
T: Great! Does anybody have a different conjecture about this diagram?
S: The two diagonals meet in the middle of the parallelogram.
T: What do you mean by middle?
S: The center of the parallelogram.
T: Taylor, you seem to have an idea…
S: The sum of the angles where the diagonals intersect will be 360°
T: Wes, you have an idea:
S: The triangles that are opposite each other are congruent.
T: What do you mean by the triangles opposite each other?
S: The one on the left and the one on the right.
T: And can you explain what you mean by congruent because some people may have forgotten that word.
S: Perfect mirror images of each other.
We recorded all of the conjectures that students proposed and then they set out trying to prove them. We found that as students were trying to prove their conjectures they started to make new conjectures based on some of the measurements that they were making.
Students were then given some additional sketches to create and then make their own conjectures about. See the file here: making conjectures
In general, students were quite engaged. I think mainly because they had a stake in what they were doing. It was “their” conjecture so they wanted to prove that they were right. It reminded me that we need to give more ownership of the mathematical ideas to the students and allow them to be mathematicians as much as possible!