Closure What have I learned?

Reflection and Synthesis

Closure Objective: The objective of chapter closure depends on which closure activities you choose. However, the two main objectives of each activity are for the students to reflect on w hat they have learned in the chapter, and for the teacher to assess students’ understanding informally.

Length of Activity: The time needed for chapter closure varies from class to class and depends on the needs of the students. While some form of closure is strongly recommended for all classes, teachers should decide if it will take the form of a full day’s lesson, part of a lesson, and/or a homework assignment. Six closure activities are described. The number of

activities assigned by teachers ranges from 1 to as many as 3 or 4. Decisions about closure activities should be made with regard to student needs and time constraints. If you choose to have students work on problems from the “What Have I Learned?” section, you should assign them to be done outside of class time, as they are intended for students to self-assess individually. As the year progresses, these should become a student responsibility. Note: It is neither expected nor recommended that you assign all six activities.

Materials: For teachers using closure activity #1 – “Team Brainstorm”:

• Poster paper and colored markers (optional)

For teachers using closure activity #2 – “Making Connections”:

• Chapter 2 Closure Resource Page: Concept Map Cards (2 pages in all), one set per team. If these cards are not cut apart ahead of time, you will also need to provide scissors.

• Poster paper, glue sticks, and colored markers (optional), one each per team

For teachers using closure activity #4 – “Summarizing My Understanding”:

• Chapter 2 Closure Resource Page: Sequence vs. Function GO (graphic organizer), one per student (optional, but especially helpful for English Language Learners)

 Team Brainstorm

A “Team Brainstorm” activity should include brainstorming of topics students focused on during this chapter as well as a discussion of how those topics connect to previous material. Follow the “Team

Brainstorm” with a class discussion about how the ideas connect to each other. Students can summarize the class discussion on a poster, which you can hang up and refer to later in the course. For more details about

 Making Connections (Concept Map)

“Making Connections” can help students see how the topics they have been studying relate to one another. To make a meaningful concept map, students will need teacher support. It is stronglyrecommended that you read the “MakingConnections” section of the course notes for chapter closure preceding Chapter 1 before doing this activity with the class. It is also recommended that you make your own concept map before having students make them.

Having cards available with the topics and vocabulary from the chapter can help students test out their ideas about how concepts connect by moving and grouping them before drawing the map. A resource page is provided for this purpose. Students could also brainstorm the vocabulary first and then write the words or phrases they came up with on sticky notes or index cards.

Above at right is an example of what a concept map could look like. The directions in the student text ask students to sketch an example for each idea or term. This example is intended merely as a visual reminder of the meaning. It would be a good idea to show a few examples to your class.

 Growth Over Time

In this part of closure, students are given the opportunity to demonstrate growth over time for several key ideas of this course by solving and explaining everything they know about one problem at the end of each chapter, up through Chapter 9. The same problems will reappear three times throughout the course, allowing students to add to what they know each time. These “Growth Over Time” closure problems should take students between 15 and 20 minutes, and they are ideal portfolio entries. (For more ideas about using portfolios, see the “Portfolios” section of the assessment resources at the end of this teacher guide.) If you are not having your students keep a full portfolio, it would be a good idea to set up a folder for “Growth Over Time” problems so students can enter their preliminary attempts and then have them available for reflection after completing the problem the last time.

The “Growth Over Time” problem in this chapter appears here as well as at the end of Chapters 5 and 8. After students have completed the

problem at the end of Chapter 8, they should be given the opportunity to compare their work from their three different attempts during the course and to reflect on what they have learned.

At this point, students should be able to make a table and graph the given function. They should describe its domain, range, asymptote, and

 Summarizing

My Understanding In this section, students are given the chance to summarize their understanding of one or more central topics of the chapter without being too narrowly guided. There are a number of ways that this can be done.

1. Chapter Summary: This would begin during class time. Ask teams to identify what they see as the bigideas of the chapter. Have teams share their ideas with the class and lead a discussion during which the class will agree on a short list (no more than five, but usually fewer). Be sure to allow students to formulate their own thoughts about what the big ideas are and to describe them in their own words. If students are landing on smaller details (like “rebound ratios” or “rabbit populations”), then ask questions such as, “What was most important in the chapter?” “Are these ideas related to or examples of other ideas?” and “Which of these ideas could be included as part of the description of a bigger idea?” When the class has decided on its list, instruct students to find (or create) and solve one or two problems for each big idea that shows their understanding of that idea. They should also explain what they know about the idea and/or what questions they still have. The big ideas for Chapter 2 will differ depending on how your class describes them. They may use some of the following descriptions.

• Arithmetic sequences • Geometric sequences • Sequences vs. functions

• Growth by multiplication vs. growth by addition • Equivalence

• Solving equations by rewriting

A summary can be presented as a straightforward written assignment, as described above, or it can be modified to be written in the style of:

• A storybook or magazine advertisement or article.

• A letter to a peer, parent, friend, or younger student. If possible, have the intended recipient read the letter and respond.

• A poster that thoroughly demonstrates understanding of a concept. • A PowerPoint presentation, web page, etc.

2. Mini-Summary: For this option, students could compare and contrast sequences and functions, showing their similarities and differences in multiple representations.

3. Graphic Organizer (primarily for English Language Learners):

For students who need extra English language support, the “Sequence vs. Functions GO” (graphic organizer) is available as a resource page. This

4. Writing a Chapter in a Math Book: For this option, the student acts as a textbook author and writes a chapter for his/her own math book. This could continue throughout the year, so that by the end of the year, each student has authored his/her own textbook. After this chapter, an appropriate topic for a math book might be “Sequences.”

 What Have I

Learned? This section gives students the opportunity to see if they can work with the current topics at the expected level. It also provides teachers with the opportunity to identify the topics with which students are struggling and to determine whether any topics need revisiting or emphasizing in later

work. Students should not spend class time on this. They should do it for

homework with the expectation of discussing the problems with their team in class the next day and formulating a short list of questions that no one in the team can answer. This activity can be followed by a whole- class question/answer session, or teachers can have teams exchange questions or send “scouts” to find guidance from other teams.

 How Am I

Thinking? This course has been designed to help students recognize five different ways of thinking they might use when they solve problems or try to understand a concept. This closure activity focuses on one of the five

Ways of Thinking:justifying. It asks students to learn about how they

think when they try to convince someone else whether an idea or solution

is true. This approach enables students to learn about how they think

when they defend an idea or approach. This “thinking about thinking” will probably seem new and perhaps strange to some students. It may be helpful for some classes to do this activity as a whole class, both to clarify what the Ways of Thinking are and to see how they connect to students’ work.

Chapter 2 Closure

What have I learned?

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