## Students Discussing 3a+5b Proofs

This video was taken in a mathematics for liberal arts class with 15 honors students. It met two times a week for 75 minutes each.

This video was taken in a mathematics for liberal arts class with 15 honors students. It met two times a week for 75 minutes each.

Students have been exploring the 3a+5b problem: Determine all possible values generated by the Diophantine equation 3a+5b when a,b≥0 (see the blog 3a+5b Proofs which contains a complete video of her explanation).

Even though the proof is still incomplete, I comment that the proof idea is beautiful. Next, I ask: Does it make sense to you? Are there questions? Normally I would give wait time here but since class is almost done, I use a show of hands to monitor understanding.

For homework, students were trying to find an equation that connects two whole numbers, their least common multiple and their greatest common factor.

The video begins with one student sharing her findings. I revoice and represent her thinking. In her explanation, the student has made a distinction between the larger and the smaller number. Even though I know that that is unnecessary, I go along with her thinking. Another student asks "Does it matter?" and again I give that question back to the class without evaluating.

Students are investigating Salsa Rueda dancing. The question is how to predict if you will dance with every leader/follower given n pairs and only dancing dame k (Salsa Rueda chapter in the Dance Book). Prior to the scene in the video, I asked a student to share her conjecture with the class because she had done the greatest number of examples and I knew that her work would be a great resource for the class. We see the student explain her conjecture.

The students are trying to prove why the finger trick of determining the multiples of 9 works. We see one student explain her thinking. I ask if someone can repeat her reasoning. As the second student revoices, I silently represent her thinking on the board. Notice that I do not evaluate in either case whether the thinking is correct or complete. Instead I continue with a new question that connects this idea back to the original problem.