Multiple Ways of Seeing

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Students will be able to determine the number of triangles in a composite figure.

Big Idea

Students will engage in structured, collaborative small group and whole group discussions to see how multiple ways of seeing can deepen the class' collective understanding.


10 minutes


Since I use this activity on the first day of school, it is important for me to get students talking to each other in a safe, structured way where every student is expected to talk--this helps me to get students to see their role in forming a mathematical learning community.  Additionally, because students can view the diagram in multiple ways, they must naturally talk in a precise way to convince others of what and how they are seeing (MP3 and MP6).

I display a figure that contains many triangles and ask students to use the Think-Pair-Group-Share structure for sharing out their ideas.  This will essentially be the first opportunity to introduce a group


Think: Show the whole class either this image or this image, both of which require students to decompose the figure to answer how many triangles they see.  Make sure that students can justify their answer, ensuring that there is a "because" in their explanation.

Pair: With the person next to you, starting with the first birthdate of the year, share/explain the # of triangles you see

Group: Starting with the person with the longest hair, share what your partner said.

Share: As a group, come to an agreement about the number of triangles and be prepared to share with the whole-class.




Multiple Ways of Seeing Whole Class Discussion Video Narrative

2 minutes

Share and Debrief

15 minutes


I call on one person from each group to share the number of triangles their group has agreed on--no explanation is needed at this time.   Since anyone should be able to explain the group's thinking if they truly shared their ideas, it is safe to call on anyone in the group and not just those students who want to volunteer. By upholding this norm, I've seen that I can keep groups accountable for making sure everyone is involved and engaged in the group discussion. As groups report out I record results (on the whiteboard) in a table format so that all the groups can see how they and others viewed the figure.

Group 1

Group 2

Group 3

Group 4

Group 5

Group 6

Group 7

Group 8

Group 9












My goal in the debrief discussion is for students to understand that there are multiple ways of approaching any problem.  I want students to see that listening to others' ideas offers problem solving strategies, new perspectives, and a deeper understanding of the problem.  I also want to create opportunities for students to give credit to others in their group for sharing their insight. Some prompting questions I might use are:

    • What strategies did you use when looking for triangles?  (Expect answers like drawing, numbering, and tallying.)
    • How did you try to keep track of what you had counted and still needed to count?
    • How did other students in your group help you to understand the problem better?





10 minutes

Because this is the first day of school, it's important to begin getting to know the students in a variety of ways, which includes everything from getting a sense of students' views on math, information about themselves as people and as students, as well as their views on homework.

I like to assign a Mathography assignment as well as a Homework Survey to get to know students better.