Unit 1, Lesson 2
Constructing Patterns
Integrated Math 1
2.1
Warm-up
Materials
To Copy (from Blackline Masters)
6–12 Blank Math Community Chart
Activity Narrative
This is the first Math Talk activity in the course. See the Launch for extended instructions for facilitating this activity successfully.
This Math Talk focuses on justifying claims based on a geometric figure. It encourages students to think about relationships and to rely on definitions of figures to mentally solve problems. The understanding elicited here will be helpful later in the lesson when students write and follow instructions to create figures.
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).
Launch
This is the first time students do the Math Talk instructional routine in this course, so it is important to explain how it works before starting.
Explain that a Math Talk has four problems, revealed one at a time. For each problem, students have a minute to quietly think and are to give a signal when they have an answer and a strategy. The teacher then selects students (likely 2–3, given limited time) to share different strategies, and might ask questions such as “Who thought about it in a different way?” The teacher then records the responses for all to see, and might ask clarifying questions about the strategies before revealing the next problem.
Consider establishing a small, discreet hand signal that students can display when 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 if 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.
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneHere are 2 circles with centers and .
Based on the diagram, decide whether each statement is true. Be prepared to share your reasoning.
- The length of segment is equal to the length of segment .
- Triangle is equilateral.
- .
- .
Activity Synthesis
To involve more students in the conversation, consider asking:
- “Who can restate ’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 ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____, so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Advances: Speaking, Representing
Advances: Speaking, Representing
Math Community
After the Warm-up, tell students that today is the start of planning the type of mathematical community they want to be a part of for this school year. The start of this work will take several weeks as the class gets to know one another, reflects on past classroom experiences, and shares their hopes for the year.
Display and read aloud the question “What do you think it should look like and sound like to do math together as a mathematical community?” Give students 2 minutes of quiet think time and then 1–2 minutes to share with a partner. Ask students to record their thoughts on sticky notes and then place the notes on the sheet of chart paper. Thank students for sharing their thoughts and tell them that the sticky notes will be collected into a class chart and used at the start of the next discussion.
After the lesson is complete, review the sticky notes to identify themes. Make a Math Community Chart to display in the classroom. See the blackline master Blank Math Community Chart for one way to set up this chart. Depending on resources and wall space, this may look like a chart paper hung on the wall, a regular sheet of paper to display using a document camera, or a digital version that can be projected. Add the identified themes from the students’ sticky notes to the student section of the “Doing Math” column of the chart.
2.2
Activity
Instructional Routines
Construct ItMaterials
To Gather
Geometry toolkits (HS)Activity Narrative
In this activity, students create an original design. 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.
Launch
Display these sample patterns for all to see:
Tell students that each of these patterns starts from the construction of a regular hexagon, though the students don’t have to limit themselves to these kinds of patterns. 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 GeoGebra Constructions Tool, recommend that they begin by drawing a circle and radius.