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Unit 3, Lesson 10

Summing Up

Algebra 2

10.1

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

Notice and Wonder

Materials

None

Activity Narrative

The purpose of this Warm-up is to elicit the idea that the number of triangles added at each iteration of the snowflake follows a pattern, which will be investigated further in the following activity. While students may notice and wonder many things about these images, the relationship between the total number of triangles at each iteration and the one before should be a focus of the discussion.

This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).

Launch

Display the table for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss with their partner the things they notice and wonder. Then follow with a whole-class discussion.

Activity

None

What do you notice? What do you wonder?

iteration	total number of triangles added since the first
0	0
1	3
2
3

Activity Synthesis

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. If possible, record the relevant reasoning on or near the table. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the pattern of the sums in the right column does not come up during the conversation, ask students to discuss where they see the pattern in the images.

10.2

Activity

Instructional Routines

None

Materials

None

Activity Narrative

The goal of this activity is for students to use what they have learned previously about the identity to derive the formula for the sum of the first terms of the geometric sequence, commonly known as the formula for the sum of a geometric series, by manipulating the structure (MP7) of the earlier identity.

Monitor for students who begin multiplying out in order to notice the pattern to share during the discussion.

10.3

Activity

Instructional Routines

MLR6: Three Reads

Materials

None

Activity Narrative

The goal of this activity is for students to use the formula for the sum of the first terms of a geometric sequence in a non-geometric context. Since students have only just learned the formula, it is important during the discussion to show both the sum for the total amount and the formula side by side.

This activity uses the Three Reads math language routine to advance reading and representing as students make sense of what is happening in the text.

Lesson Synthesis

Display the geometric sequence , , , , . . . for all to see and ask students what they would need to do to figure out the approximate sum of the first 30 terms of this sequence. After a brief quiet think time, select 2–3 students to suggest ideas for figuring out the sum. Focus on identifying the values of (the starting term) and (the common ratio) in order to use the sum formula .

Give students time to work out the sum of the first 30 terms. If needed, display the general form for all to see, and tell students that they may want to use this form to help accurately identify and . Conclude the discussion by selecting students to share their results, recording the formula for all to see. The sum of the first 30 terms is approximately 0.333333 . . . Students are likely to notice that this value appears to be the same as . Ask students if they think adding more than the first 50 terms of the sequence together will ever get a value larger than . Select 2–3 students to share their thinking. It may surprise students to learn that it will not, since for each term added the sum will be moved only closer to the value of , but never above it. To help students understand why the sum will never increase beyond , it may be helpful to rewrite the sum of the first 30 terms in the sequence in decimal form, , to see that each additional term increases only the precision of the sum. This topic, finite geometric series, is studied further in future math classes.

to access Cool-down.

Student Lesson Summary

Sometimes identities can help us see and write a pattern in a simpler form. Imagine a chessboard where 1 grain of rice is placed on the first square, 2 on the second, 4 on the third, and so on. How many grains of rice are on the 64-square chessboard? Trying to add up 64 numbers is difficult to do one at a time, especially because the first 20 squares have more than one million grains of rice on them! If we write out what this sum is, we have

If we rewrite this expression as , we have an expression similar to one we’ve seen before, .

In an earlier lesson, we showed that is equivalent to the simpler expression . Using this identity with and , we have

This means that the sum total of all the grains of rice is , or . More generally, for any geometric sequence starting at with a common ratio , the sum of the first terms is given by .

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Settings

Audience

Assessment Access

Lesson 10

Summing Up

Warm-up

Standards Alignment

Building On

Addressing

Building Toward

HSA-SSE.B.4

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Activity Synthesis

Activity

Instructional Routines

Materials

Activity Narrative

Activity

Instructional Routines

Materials

Activity Narrative

Lesson Synthesis

Student Lesson Summary

Student Response

Building on Student Thinking

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Launch

Activity

Student Response

Building on Student Thinking

Activity Synthesis

Standards Alignment

Building On

Addressing

HSA-APR.C

HSA-SSE.A.2

HSA-SSE.B.4

Building Toward

Standards Alignment

Building On

Addressing

HSA-SSE.B.4

Building Toward