Students will be able to sort their Four Triangles figures given a particular criterion and give a working definition of their assigned criterion to other students, providing examples using their four-triangle figures.

Students will take risks and build a sense of community as they share working definitions of their criterion and show how their four-triangle figures exemplify that criterion with their peers.

30 minutes

Since the Common Core math practices require students to persevere in solving math problems (**MP1**) as well as construct viable arguments and critique the reasoning of others (**MP3**), it is essential to establish the classroom as a mathematical community of learners; ideally, in this community, students will take risks and support their peers in doing so. By explicitly talking about risk-taking, making connections to real-world and academic situations, and linking risk-taking to learning gains, the teacher can engage students in an engaging and relevant discussion that will build a sense of safety and trust.

I use an adaptation of a National School Reform Faculty protocol to work towards a supporting culture in my classroom. The activity, Zones of Comfort, Risk, and Danger, helps students focus on their feelings and become more aware of the feelings of others.

To prepare for the Zones activity , I draw three large concentric rings on the ground outside the classroom with chalk. Then, during class I take students outside and ask them to consider a set of situations ((these should be a mix of academic and non-academic situations):

- Eating food from the school cafeteria
- Swimming in a lake
- Traveling to a foreign country
- Working in groups
- Moving to a new school in the middle of a year
- Zip-lining
- Eating liver
- Getting called on in class without any warning
- Singing in the shower
- Dancing in front of a crowd
- Going to college
- Asking the teacher a question when you are confused
- Discussing religion, race, or sexuality with people you don't know well
- Presenting your work and/or teaching a math problem when you aren't 100% certain you are correct
- Receiving public praise or recognition
- Speaking in front of the entire school

As I pose each situation, I ask students to place themselves in the appropriate zone, having them consider which situations feel really comfortable, which seem to pose some kind of risk, but are still generally positive and which make them feel defensive or worried and wanting to retreat?

- Centermost ring: Danger Zone
- Middle ring: Risk Zone
- Outermost ring: Comfort Zone

Afterward, we have a **Debrief Discussion** using the following outline as a guide:

- What situations did you see many of your peers moving to the risk zone?
- What can we do to support each other in taking risks but making sure to keep people out of the danger zone?
- Why might we do an activity like this?
- Ideally, students will share out comments that support the notion of fostering a risk-taking community where all students can learn as much as possible.
- Additionally, students will hopefully see that many of their peers share their fears and concerns as well, diminishing their sense of isolation and feeling that "I'm the only one who seems to be scared of ________"
- End with summarizing each zone:
- Comfort Zone
- Where people feel no tension and have a good grasp of the situation.
- Danger Zone
- Where people feel full of defenses, fears, red lights, desire for escape, paralysis--we want to keep people out of here!
- Risk Zone, the most fertile place for learning
- Where people are willing to take some risks not knowing anything
- Where people want to learn and take the necessary risks to do so
- Where people open up to others with curiosity and interest
- Where people will consider options or ideas they hadn't thought about before

35 minutes

Ask the Resource Manager will get all materials for the group and may choose another person in the group to help him/her.

To start the sorting process, I pass out one set of Four-Triangle Criterion Cards to each group (number of sides, line symmetry, rotational symmetry, convex or concave), which contains the characteristic for sorting along with a definition for the characteristic (**MP7**).

I give each group about 10 minutes to sort their four-triangle figures and glue these figures to a poster. Ideally, I try to ensure that there are at least two groups who have each of the sorting criteria so that they can compare.

During this time, I circulate the room to check each group's progress. After about 10 minutes, I ask groups that worked on the same sorting characteristic to get together and discuss how they sorted, justify how/why they made certain decisions, and make adjustments if they need to. The goal is for every student in these groups to become experts on their characteristic for sorting, as each student will participate in a jigsaw* where he/she will present a definition for the characteristic and use four-triangle figures as examples. This will take about 5 minutes.

After like groups compare, I ask students to return to their original group where they will each make a mini-poster on centimeter graph paper that will serve as their finalized individual product to present to their jigsaw group. This will take about 15 minutes.

*see the next section of this lesson for jigsaw explanation.

12 minutes

I connect the sorting activity to Zones of Comfort, Risk, and Danger and explain why it is important for students to be able to talk to each other, explain ideas to one another, and ask questions to build and deepen each other's thinking. Then, I explain that the next activity, the Jigsaw, will give students the opportunity to take a small, but safe, risk as they teach their peers about their characteristic for sorting in a small group setting. Because I teach this lesson at the beginning of the year, students start getting used to seeing their role in building and maintaining a mathematical community of risk-taking learners.

I make new groups of 3-4 students, with each person representing a different characteristic for sorting the four-triangle figures. I have each student fold a piece of blank, white paper into quarters--each quarter will provide space for one of the four characteristics for sorting to record the shared explanations in the Jigsaw rounds.

10 minutes

**“What I learned from the Four Triangles Problem”**

It is essential to assess students' understanding of the Four Triangle problem, as well as the extent to which students learned from each other in the Jigsaw. In this open-ended exit ticket that features lots of student choice, students can express what they learned about the geometry content (number of sides, symmetries, concavity, etc.) as well as how they thought about the problem itself (organization, visualization, exhausting all possibilities, etc.). Additionally, asking students to write about what they learned gives the teacher an opportunity to assess students' writing skills.

Use at least 5 of the following terms to describe in complete sentences what you learned from the Four Triangles Problem. You may write about the experience of coming up with all of the four-triangle figures, how you sorted the figures, or what you learned from your peers in the jigsaw.

- Organization
- Visualization
- Exhaustion
- Sides
- Rotational symmetry
- Line symmetry
- Concave and convex