Lesson 2

This is the first math talk activity in the course. See the launch for extended instructions for facilitating this activity successfully.

The purpose of this Math Talk is to elicit strategies and understandings students have for justifying claims based on a geometric figure. For example, students may notice that the two circles have the same radius, and use that fact to explain why segment \(CD\) is cut into three congruent segments. These understandings help students develop fluency and will be helpful with future topics such as proofs and formal reasoning.

When students notice that two circles have the same radius and use that fact to reason about other distances, they notice and make use of structure (MP7). Students also construct viable arguments and critique the reasoning of others when they explain why a statement about the given diagram is true or participate in questioning another student’s explanation (MP3).

This is the first time students do the math talk instructional routine, so it is important to explain how it works before starting.

Explain the math talk routine: one problem is displayed at a time. For each problem, students are given a few minutes to quietly think and give a signal when they have an answer and a strategy. The teacher selects students to share different strategies for each problem, and might ask questions like “Who thought about it a different way?” The teacher records students’ explanations for all to see. Students might be asked to provide more details about why they decided to approach a problem a certain way. It may not be possible to share every possible strategy given the limited time; the teacher may only gather two or three distinctive strategies per problem. 

Consider establishing a small, discreet hand signal that students can display to indicate they have an answer they can support with reasoning. This signal could be a thumbs-up, a certain number of fingers that tells the number of responses they have, or another subtle signal. This is a quick way to see whether the students have had enough time to think about the problem. It also keeps students from being distracted or rushed by hands being raised around the class.

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:

• “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
• “Did anyone have the same strategy but would explain it differently?”
• “Did anyone solve the problem in a different way?”
• “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
• “Do you agree or disagree? Why?”

In this activity, students create an original design with sixfold or threefold symmetry. As they create their design, they record the steps they took with their compass and straightedge. In the next activity, students will trade their instructions with a partner and attempt to recreate the partner’s design by following the written steps. Their instructions need to be precise enough that someone else can replicate their pattern based on the description.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

The Are You Ready for More? problem contains a video created by Samira Mian, used with permission.

Display these sample patterns for all to see:

Tell students that each of these patterns starts from the construction of a regular hexagon, though they don’t have to limit themselves to this kind of pattern. As long as they use straightedge and compass moves, and record their moves so that someone else can understand them, they can make any pattern they choose. Share some helpful tips:

• Larger constructions are easier to accurately recreate.
• If any part of the construction involves freehand drawing rather than straightedge and compass moves, it won’t be possible to recreate it precisely.

For students using the digital Constructions tool, recommend that students begin by drawing a circle and radius.

If students struggle to start direct them to the display of construction moves to remind them of their options.

Ask students to predict how well their pattern will be reproduced based on the instructions they gave.

The purpose of this activity is for students to follow precise instructions for reproducing a pattern and analyze what kinds of instructions are clear and precise, and what kinds of instructions are ambiguous or hard to follow. Identify students who use words like circle, line, line segment, point, or label figures with letters.

Making dynamic geometry software available gives students an opportunity to choose appropriate tools strategically (MP5).

Invite students to trade instructions with a partner. It would be best if partners haven't already seen the final design.

For students using the digital Constructions tool, recommend that students begin by drawing a circle and radius.

Display several pairs of original patterns next to recreations. Consider asking:

• “What was one thing about the instructions that made them clear and easy to understand?” (If the instructions were step by step with labels.)
• “What was one thing about the instructions that could have been more precise?” (Using vocabulary words or labeled points.)