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Unit 1, Lesson 4

Using Technology to Work with Sequences (Optional)

Algebra 2

4.1

Warm-up

10 mins

Where Does It Live?

Standards Alignment

Building On

Addressing

Building Toward

Instructional Routines

None

Materials

To Gather

Spreadsheet technology

Activity Narrative

In this activity, students use a spreadsheet to write, read, and evaluate expressions. The spreadsheet allows students to perform quick calculations, such as adding, subtracting, multiplying, or raising a number to a power using the values that they have added into cells. This Warm-up directly prepares students for the following activities in which they will study the structure of sequences using technology.

Launch

Spreadsheet technology is needed for every student. If students are using the digital version of the materials, show them how to open the GeoGebra Spreadsheet in Math Tools.

Tell students that, in a spreadsheet, each “cell” has an address given by its column and row. For example, cell B3 is in the second column and the third row.

Arrange students in groups of 2. It is recommended that students not be expected to write down responses to the questions in this task, but rather discuss them with their partner.

Activity

None

Student Task Statement

Follow your teacher’s instructions to open a blank spreadsheet.

Type the following in each cell:

In A1, type “2”

In A2, type “3”

In A3, type “-10”

In A4, type “1/5”

In B1, type “=A1+A2”

In B2, type “=A3*A4”

In B3, type “=B1+333”

In B4, type “=abs(B2)”

Look at the numbers that appear in B1, B2, B3, and B4 after you press enter. Where did these numbers come from?
Experiment with typing some different values in A1, A2, A3, and A4. Describe what happens.
Experiment with typing some new formulas in new cells.
1. Can you figure out how to raise a number to a power?
2. What happens if you forget to start a formula with the = symbol?

Student Response

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Activity Synthesis

The goal of this discussion is to make sure students understand the basics of entering numbers and performing calculations in a spreadsheet. Invite students to share what they noticed when they changed the values in A1, A2, A3, and A4, and highlight the connections between the cells.

If time allows, students can challenge each other to guess a formula. For example, one student puts some numbers into column A and a secret formula into column B. The partner changes the numbers in A to see what happens in the cell with the formula and tries to guess the formula. (They have to agree not to peek at the answer by clicking on the formula cell.)

4.2

Activity

10 mins

Fill Down

Instructional Routines

Extend It

MLR2: Collect and Display

4.3

Activity

15 mins

Plot Some Points

Instructional Routines

Fit It

Materials

Lesson Synthesis

The goal of this lesson is for students to consider what advantages technology offers when working with sequences. Here are some questions for discussion:

“How do you create a table of values using technology?” (You type the input data and then use an output rule to calculate all the outputs.)
“What are some advantages of using technology to find a rule for a sequence and extend the number of its terms?” (The advantage of having the technology calculate the outputs is that I don’t need to repeat the computations by hand and I can use fill down to generate many terms of the sequence quickly.)
“When do you think it is appropriate to use technology to represent data or to calculate statistics?” (Using technology to calculate a table of values makes sense in most situations because the terms are calculated recursively. So, to get to the 100th term, we need to do 100 calculations. The chance for making a mistake while calculating by hand makes using technology a good choice.)

to access Cool-down.

Student Lesson Summary

A spreadsheet is very useful for creating new terms in a sequence. For example, for an arithmetic sequence that starts with 5 and has a rate of change of 2, you can type the formula in this image. “A1” refers to the contents of cell A1. It's like an address for the cell.

<p>A spreadsheet with rows 1 to 3 and column A. A 1 contains 5. A 2 contains "equals A 1 plus 2" and has a blue border. A 3 is blank.</p>

After pressing Enter, you can click cell A2 and drag the corner down to show more terms. This move is called fill down.

<p>A spreadsheet with rows 1 to 7 and column A. A 1 contains 5. A 2, 7. A 3, 9. A 4, 11. A 5, 13. A 6, 15. A 7, 17. A 2 through A 7 are highlighted with a blue border.</p>

If you navigate to cell B2 and type “=A2-A1,” the result will be 2. If you fill down from cell B2, all the cells will say 2. This is a way we can tell that the sequence is arithmetic, and its rate of change is 2. If, in B2 you type “=A2/A1” instead, the result will be 1.4. If you fill down from cell B2, all of the cells will contain different values. Since the sequence does not have a growth factor, we can tell the sequence is not geometric.

Technology can also be used to plot the terms of a sequence as a function of the term number. The term numbers and term values can be entered in a table, and these pairs of numbers are plotted as points in the plane.

You can also enter a rule for the function in the table header and automatically generate new terms. For example, here is a table and graph representing an arithmetic sequence that starts at -2 for and has a rate of change of 2:

<p>Two column table and a discrete function graph.</p>

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Launch

Spreadsheet technology is needed for every student. Arrange students in groups of 2.

MLR2 Collect and Display. Circulate to listen for and collect the language that students use as they used the spreadsheet to identify the type of sequence and the rate of change or growth factor for each sequence in the last two questions. On a visible display, record words and phrases, such as “First I type each value of the sequence into a cell,” “In the column next to the sequence, I found the quotient between terms,” or “The difference between each term and the next is always the same, so the sequence must be arithmetic.” Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading

Activity

None

Student Task Statement

Open a blank spreadsheet. In A1, type “10” and hit the Enter key.
1. In A2, type “=A1+3” and hit Enter.
2. Click once on cell A2 to highlight it. See the little + in the lower-right corner? Click and drag it down to highlight several rows in that column, and then let go. (this is known as “fill down”). Describe what happens.
3. What simple thing can you edit so that column A shows the sequence 12, 15, 18, . . . ?
4. What simple things can you edit so that column A shows the sequence 12, 11, 10, . . . ?
In B1, type “16” and hit Enter.
1. In B2, type “=B1*0.5” and hit Enter.
2. Click on cell B2 and fill down. Describe what happens.
3. What simple thing can you edit so that column B shows the sequence 10, 5, 2.5, . . . ?
4. What simple things can you edit so that column B shows the sequence 10, 30, 90, . . . ?
In column C, starting at C1 and going down, type these terms of a geometric sequence: 700, 70, 7, 0.7, 0.07
1. Type “=C2/C1” in cell D2. What is the result?
2. What is the meaning of the result?
3. Click on cell D2 and fill down. What happens?
In column E, starting at E1 and going down, type these terms of an arithmetic sequence: 7, 10.5, 14, 17.5
1. Type “=E2-E1” in cell F2. What is the result?
2. What is the meaning of the result?
3. Click on cell F2 and fill down. What happens?
Use the spreadsheet to decide whether the sequence 8, 12, 18, 27, 40.5 is arithmetic or geometric, and find its rate of change or growth factor.
Use the spreadsheet to decide whether the sequence 50, 42.1, 34.2, 26.3 is arithmetic or geometric, and find its rate of change or growth factor.

Student Response

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Building on Student Thinking

Are You Ready for More?

Launch

Keep students in the same groups. It is recommended that students not be expected to write down answers to the questions, but rather to check in with their partner until they know how to use the tool.

Graphing technology is needed for every student. Instruct students on how to create a new table using the graphing technology available in your classroom. These instructions use the Desmos graphing calculator.

Open a Desmos graphing calculator window either from Math Tools in the digital version of the student materials or by navigating a browser to desmos.com/calculator.
- Show students that, in the expression list, you can type something like “,” and a point is plotted.
- What if we want to plot several points efficiently? We can make a table. Find the “+” button in the expression list, and select “table.”
- Start typing numbers under and . Note that these pairs are plotted as points in the coordinate plane. Give students time to complete all parts of the first question.
To make the values in one column a function of the other:
- In a new table, replace the with an expression in terms of , for example, (to write “” in Desmos, type “x_1”).
- Type 0 in the first row under , and then 1 in the next row. If you then keep pressing Enter, the column should automatically increment by 1, and the other column should automatically compute the output.
- Show students how to zoom the graphing window out to see more points, either using the “–” button in the upper right or scrolling the mouse.

Representation: Internalize Comprehension. Begin with a physical demonstration of using the Desmos graphing calculator (or other graphing technology) to support connections between new situations and prior understandings. Consider using these prompts: Directions for instructing students how to create a new table using this technology are in the Activity Narrative and should be made accessible to students.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

Activity

None

Student Task Statement

Open a graphing utility, and follow your teacher’s instructions to create a new table with 2 columns. Learn how the 2 numbers in each row can be plotted as points in the coordinate plane.
1. Change the numbers in the table so that all of the plotted points lie along a diagonal line with a positive slope.
2. Change the numbers in the table so that all of the plotted points lie along a horizontal line.
3. Change the numbers in the table so that the graph created does not represent a function.
Follow your teacher’s instructions to make one column a function of the other.
1. Change the expression in the second column so that the plotted points lie on a line with a different steepness.
2. Change the expression in the second column so that the plotted points do not lie on a line.
3. Change the table so that some of the points are plotted in Quadrant II of the graph (the upper-left quadrant).

Student Response

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Lesson 4

Using Technology to Work with Sequences (Optional)

Warm-up

Where Does It Live?

Standards Alignment

HSF-IF.C

Instructional Routines

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Activity Synthesis

Activity

Fill Down

Instructional Routines

Activity

Plot Some Points

Instructional Routines

Materials

Lesson Synthesis

Student Lesson Summary

Have feedback on the curriculum?

Building on Student Thinking

Are You Ready for More?

Materials

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Activity Narrative

Launch

Activity

Student Task Statement

Student Response

Building on Student Thinking

Are You Ready for More?

Activity Synthesis

Standards Alignment

HSF-LE.A.1

HSF-IF.C

Standards Alignment

HSA-REI.D.10

HSF-IF.C