Unit 1, Lesson 11
Adding Up
Algebra 2
11.1
Warm-up
Standards Alignment
Building On
Addressing
Building Toward
HSA-SSE.B.4
Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.
Materials
NoneActivity Narrative
The purpose of this Math Talk is to elicit strategies and understandings students have for adding fractions. It encourages students to think about common denominators and to rely on what they know about factors of numbers to mentally solve problems. The strategies elicited here will be helpful later in the lesson when students add and subtract fractions.
Students look for and make use of structure (MP7) when they use the previous answer when they solve the next step.
Launch
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
NoneEvaluate mentally.
Activity Synthesis
The goal of this discussion is to review students’ strategies for adding fractions that do not have common denominators.
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use 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. Examples:
Advances: Speaking, Representing
- “First, I because .”
- “I noticed , so I .”
Advances: Speaking, Representing
11.2
Activity
Instructional Routines
MLR8: Discussion SupportsMaterials
Activity Narrative
The purpose of this activity is for students to represent a situation with a sequence in which it makes sense to add the terms of the sequence together in order to answer a question about the situation. This situation was chosen for its hands-on nature to help students make sense of why we would ever need to add up terms in a sequence. Students will continue this type of thinking in the following activity where they will work with a famous mathematical shape, the Koch Snowflake.
Launch
Arrange students in groups of 4. It may be helpful to assign one student to be “Tyler” to carry out the actions described in the task as a demonstration for the class. Students may find it useful to fold and unfold the paper first to have crease lines to follow when making the cuts. Provide groups access to paper and scissors.
Representation: Access for Perception. Read the Task Statement aloud. Students who both listen to and read the information will benefit from extra processing time.
Supports accessibility for: Language
Supports accessibility for: Language
11.3
Activity
Instructional Routines
NoneMaterials
NoneActivity Narrative
This activity is an opportunity to contrast when it does and does not make sense to sum a sequence. This task is relatively unscaffolded compared to previous lessons, giving students opportunities to make sense of the problem by, for example, drawing Step 2 or organizing the data in a table (MP1).
If you wish, share with students that this shape is known in mathematics as the Koch (pronounced “coke”) Snowflake.
Launch
Tell students to close their books or devices. Draw and label an equilateral triangle as Step 0. Next to it, draw and label Step 1, starting from an equilateral triangle, erasing the middle of each side, and drawing two additional segments popping out of each side (as shown in the Task Statement). Invite students to describe in their own words how the second triangle was drawn and to state how many sides Steps 0 and 1 have.
Arrange students in groups of 2–4. Give time for groups to work, and follow with a whole-class discussion.
Lesson Synthesis
Ask students to consider what it would mean to find the sum of the sequence defined by for . (The sum of the terms of appears to get closer and closer to the more terms we add together.) After some quiet work time, select students to share their thinking, and record their ideas for all to see. In particular, highlight any idea that connects back to the "Paper Trails" activity, in which the sum of the fractions each person has also appears to get closer and closer to the more times Tyler cuts up and passes out pieces from his original sheet of paper.
Student Lesson Summary
The sum of a sequence is the sum of its terms.
For example, suppose you were given \$1 on the first day, then \$2 the second day, then \$4 the third day, and it doubled each day for seven days. After finding each term of the sequence, you can find the sum:
For these seven days, the total amount of money is \$127. In a later unit, you will learn a method to find the sum of a geometric sequence more efficiently.