## Learning Math: Geometry

# Classroom Case Studies, 6-8 Part A: Geometry and Reasoning (25 minutes)

## Session 10: 6-8, Part A

The study of geometry in high school is often associated with proof — usually an axiomatic development with a focus on two-column, statement-reason type proofs. Newer curricula based on the NCTM standards have reduced this emphasis on two-column proof in geometry in favor of a problem-solving approach. Even so, geometry remains an ideal way to approach reasoning and proof, and it can be started earlier than high school. **Note 2**

When viewing the video segment, keep the following questions in mind:

- How does the teacher incorporate geometric reasoning into the lesson?
- Where in the lesson are students learning new geometric content? What is that content?
- Where in the lesson are students drawing logical conclusions and thinking mathematically? How does the reasoning relate to the geometric content?
- Thinking back to the big ideas of this course, what are some geometric ideas these students are likely to encounter through their investigation of this situation?

Video Segment
In this video segment, sixth-grade students in Ms. Saenz’s class have, through data gathering, conjectured a form of the triangle inequality: Three lengths make a triangle if the sum of any two lengths is greater than the third length. The students, however, are unsure what happens when the sum is equal to the third side. Here, one student tries to explain why he thinks a triangle can’t be formed in this case. You can find this segment on the session video approximately 14 minutes and 7 seconds after the Annenberg Media logo. |

### Problem A1

Answer the questions you reflected on as you watched the video:

- How does the teacher incorporate geometric reasoning into the lesson?
- Where in the lesson are students learning new geometric content? What is that content?
- Where in the lesson are students drawing logical conclusions and thinking mathematically? How does the reasoning relate to the geometric content?
- Thinking back to the big ideas of this course, what are some geometric ideas these students are likely to encounter through their investigation of this situation?

### Problem A2

This lesson is not couched in a “real-world context.” Students are thinking about mathematical ideas in the abstract. What are the advantages and disadvantages of this kind of lesson? Are “mathematics only” lessons important in your classroom? What purpose do they, as opposed to contextualized lessons, serve? **Note 3**

### Problem A3

Ms. Saenz’s lesson was based on a lesson from Session 2 of this course. Discuss the ways Ms. Saenz’s lesson was similar to and different from the one in this course. What adaptations did she make and why?

### Notes

**Note 2**

Before examining specific problems at this grade level, you will watch with an eye toward geometric problem solving a teacher in her classroom who has also taken the course. The purpose in viewing the video is not to reflect on the teacher’s methods or teaching style, but to watch closely the way she and the lesson encourage students to tie together their geometric knowledge, intuition, and logical reasoning.

**Note 3**

This is a particularly good discussion to have with your colleagues. Everyone has different opinions and thoughts about the use of context in the mathematics classroom. Spend some time talking about not just what you think, but why you think it. Cite examples from your own experience instead of focusing on what you have heard others say.

### Solutions

**Problem A1**

Answers will vary. Some ideas: Students come up with the triangle inequality through data gathering and checking, but the teacher persists in asking them to explain why it should hold. That stretches students into not just the geometry content, but also the logical deductions. This course covered the triangle inequality, and several of the same ideas came up. One particularly interesting piece is at the beginning of the clip, when students use a geometric model to check if the sum of two lengths is greater than the third length. Many teachers rely on algebra and arithmetic more, and it’s nice to remember that many students do have a sense of geometric reasoning as well.

**Problem A2**

People have very different, and often very strong, opinions about the use of context in mathematics classrooms. It is important to present students with a variety of lessons. Students can be engaged by problems that are not context-based, as well as by those with real-world connections.

**Problem A3**

There were many adaptations. Here are some: The materials were different; Ms. Saenz used what was available at her school. The activity was more directed; students were asked to look particularly at sums of sides rather than to find whatever relationship they could.