Respecting Others' Thinking: How Many Squares?

Since the Common Core mathematical practices require students to share their thinking as they construct viable arguments (MP3), I stress the importance of students respecting each other's thinking.  I use Number Talks because they give students the opportunity to talk about their thinking, critique others' thinking, make mathematical connections, and explain how their thinking changed.  Often times in Number Talks, students are initially "wrong" and have the opportunity to explain how their thinking has changed--this is a great opportunity for me, as the teacher, to explicitly value mistakes as a site for deepening and clarifying understanding.

Here are the tasks for today's Number Talk:

1. 51+39
2. 99 x 24
3. 103 x 16

Launching the task for How Many Squares? simply requires me drawing a 2x2 square on the whiteboard and asking students how many squares they see.  Students will quickly reply "four!" which garners me giving students a quizzical look.  Inevitably, students will see that the correct answer is five.

At this point, I pass out the How Many Squares? task card, which asks students to extend their thinking, make use of structure, and consider other cases (MP8).

I alert my students that I will use the Participation Quiz to run this lesson.  Essentially, I tell students that I will provide virtually no math help, which forces them to rely on themselves and each other to make sense of the problem.  During this time, I tell students that I will be eavesdropping on their groupwork, publicly typing and projecting my notes on how they work together.  One of the best outcomes of making these notes public is that students can immediately see what I value about their group's work, as well as other groups, and how their behaviors, questions, and strategies are helping them to understand the mathematics. Please see Participation Quiz Video to learn more about why and how I use Participation Quizzes in the classroom.

Debrief the Math:

1. I ask groups of students share out how they came up with a method for counting and keeping track of the number of squares (5-10 minutes)
   • I place emphasis on the value of having some kind of systematic approach for thinking about this problem
2. I ask for various generalizations for determining the number of squares in an n x nsquare (5 minutes)

• Various groups will have different ways of seeing patterns in this problem
• Students should listen to these groups' ideas, trying to make connections between them and see how they are related

In the past, I have seen some groups struggle with generalizing the pattern they see; I have found that posing a larger case, a 31x31 square, is often helpful for getting students to understand the relationship between the number of different-sized squares.

• Ultimately, students should be able to see that the number of squares in an n x n square can be written as f(n)= 1²+2²+3²+…+(n-1) ²+n², where n represents the side length of the square.

Debrief the Participation Quiz:

1. Going one group at a time, I display the notes I have taken, which highlight positive group behaviors, questions that pushed the group's thinking, effective strategies, and mathematical breakthroughs that were made
2. By debriefing how each group upheld the group norms, I show groups how and why they were able to make their understanding of the problem come to life.

In the How Many Triangles? homework assignment, I hope to assess the following student skills: 

• Using a systematic way to count
• Organizing their work and thinking
• Justifying their reasoning

How Many Triangles? also serves as the individual accountability piece of the lesson, asking students to, on their own, extend the thinking they did during class.