The High School Math Project —Focus on Algebra
Squares Inside Squares
(Sequences)
Objective
Students use the TI-92 to generate a sequence by determining the areas of squares inscribed in squares. They then write this sequence using the recursive form and the explicit form for the sequence.
Overview of the Lesson
In this lesson, students explore another geometric pattern as they investigate the pattern determined by the areas of squares inscribed in squares formed by joining the midpoints of the sides of the previous square. The sequence generated is a simple geometric one. One special feature of this lesson is the use of the TI-92, which is used to generate the data. Students generate both the explicit and recursive forms for the sequence. At the algebra I level, students do not need the formal
notation, but algebra I students are very capable of exploring the pattern, plotting the model, and describing how to generate the sequence.
Materials
• TI-92 graphing calculators and overhead unit
• TI-83 graphing calculators and overhead unit
• overhead projector
• Squares Inside Squares activity sheets
• Sequences Using the TI-92 Calculator activity sheets
• Sequences Using the TI-83 Calculator activity sheets
• Writing Sequences in Recursive and Explicit Form activity sheets
Procedure
1. Introduction: The video teacher introduces the lesson by discussing two simple sequences. By questioning the students and drawing on their
knowledge base, he discusses how to write a sequence, and how to indicate the nth term of a sequence. You can introduce the lesson in a similar way, tailoring it to your students' experience.
2. Demonstration of Working with the TI-92: First demonstrate the steps
involved in creating the geometric pattern on the blackboard. Draw a square, mark the midpoints of the sides, and then join these midpoints to create another square. After describing this pattern on the blackboard, demonstrate doing this on the TI-92 by creating a macro. After this ask the students to do it with their own calculators.
3. Data Collection: Have students work in pairs to create the design and gather data. Please note that the TI-92 calculator instructions are on the video immediately following the lesson.
4. Discussion of the Data: Have some students record their data on the board and discuss the patterns in the data including the accuracy of the data. Help students to notice that although different groups have different areas for the first square, the second square always has an area equal to half of the first square.
5. Generate the Recursive Model for this Pattern: Notice that if we want to find the area of the 10th square, we could not do this with the calculator because the calculator must be able to draw the figure in order to determine the area. The 10th square would be too small. Ask students to write a rule for this pattern.
Help them see that they need to use their initial areas, and then each successive area is determined by taking half of the previous area. Students should use the TI-92 sequence mode to write a function that will generate the sequence. In order for all the students to be using the same sequence, they should use 64 as the first term of the sequence. After checking to make sure that they are able to use the calculator to generate the sequence {64, 32, 16, 8 ... }, students must modify the TI-92 equation by changing the initial value or ui1 so that it is the first value of their original sequence. Now, when they examine the graph and the table of values, they find that it matches their original
sequence.
6. Determine an Explicit Model for the Relationship: Have students explore the shape of the graph and use what they already know about exponential models
7. More Practice: Have students work in groups to determine the recursive model, the explicit model, the next two terms, and the 10th term for the
following sequences. This activity serves as practice and review for the lesson.
a) {2, 4, 6, 8, 10, ...}
b) {4, 11, 18, 25, 32, ...}
c) {1, 1, 2, 3, 5, 8, 13, ...}
d) {64, 96, 144, 216, 324, 486, ...}
e) {125, 120, 115, 110, 105, ...}
Assessment
By carefully listening to student responses you can determine how well students understand the lesson that is being taught. A teacher is often able to pick up indications of strengths and weaknesses with both new material and previously taught topics. In this video lesson students had to remember and apply what they knew about exponential models in order to determine an explicit model for the sequence. Without that background the students would not have been able to generalize the formula.
The review at the end of class offers an excellent way for a teacher to find out how well the class understands this lesson. For some students, this sort of summary and practice is very important because it allows them one more opportunity to discuss the problems with their classmates and get feedback on their thinking.
Extensions & Adaptations
• Ask the students how the Squares Inside Squares problem would change if instead of putting the second polygon inside the first, it was constructed outside the first square with the vertices of the original square being the midpoints of the outside square. Direct the students to write a recursive and an explicit formula for each of the sequences.
• Have students use the TI-92 to explore inscribing many other regular
polygons. What pattern can be found in the ratio used to generate the areas of the sequence of inscribed polygons. What happens to this ratio as the number of sides increases from 3, 4, 5, up to n sides?
• Challenge your students to determine an explicit formula for the Fibonacci sequence, or to do research to find out if someone else has determined an
expression for the general term of this famous sequence. They may be surprised to find Binet’s Formula, Fn = 1
5
1+ 5 2
n
− 1 5
1− 5 2
n
.
• Some students may recognize the golden ratio in Binet’s Formula, 1+ 5 2 . Have students divide each term of the Fibonacci sequence by its preceding term and look at the sequence of numbers obtained. Have the students express the golden ratio as a decimal approximation correct to six decimal places.
The students may be surprised to see that
nlim→ ∞Fn+1
Fn =1+ 5 2 .
Mathematically Speaking
The study of sequences affords the opportunity to reinforce the idea that there are many correct ways to express the same relationship. In the video, students determined a recursive model and several explicit models. All of the models generated the exact same sequence, and all were correct. Students need to be encouraged to express answers in a variety of forms. They also need to be encouraged to check different answer forms to make sure that they indeed do represent the same model. As students continue their study of mathematics, their ability to determine if two different looking expressions are equivalent will become even more important.
The following problem offers the opportunity for students to review ideas of recursive and explicit sequences as well as quadratics, finite differences, graphing, and data
analysis. Have students determine many different ways of expressing the relationship between the number of straight slices in a chocolate cake and the maximum number of pieces produced for that number of slices.
Number of Slices (n) 1 2 3 4
Maximum Number of Pieces (tn) 2 4 7 11
Recursive Form Explicit Form t1 =1, tn =tn−1+n
tn = 1 2n2+ 1
2n+1
Tips From Ellen
Using Technology
Changes in technology continually result in growth and changes in the discipline of mathematics itself. The abacus, the slide rule, and the graphing calculator each not only helped to solve the problems of their time, but helped to shape the problems that presented themselves for consideration.
Technology is neither the evil some fear nor the panacea some hope for, but rather a tool to be used thoughtfully. Following are some things to think about as we use technology in the classroom.
Practice first. Become comfortable with using not just the immediate function or program you will use in the next class, but any related operations as well. Make sure that you know exactly which steps to follow so that you can give explicit directions to students. Learn to recognize and interpret screens so that you can easily trouble- shoot when you or students make an error or become lost.
Provide students with sufficient practice so that they, too, can become comfortable with the technology. Use talk aloud techniques to articulate what you are doing and why; encourage students to do the same. Make sure you move quickly from directed practice ("next push this button") to independent practice.
Ensure that all students have access to technology. Provide enough graphing calculators or computers so that every student has one; if students must share, structure sharing so that students take turns having a hands-on experience. Groups of students can support each other and facilitate learning, but as with all educational outcomes, articulate the importance of individual understanding and provide opportunities for students to work alone before checking with others.
Do not use technology to substitute one meaningless mechanical exercise for
another. For instance, graphing an equation is not the desired end, but the means to a richer understanding of functions. Avoid substituting rote pushing of buttons on calculators for rote plotting of points on paper. Graphing calculators can enable us to complete pages of meaningless graphs quickly, or can be used to eliminate the
drudgery that previously kept students from a higher plane of understanding. Talk about the purpose for graphical displays, and create problems that use them as analytical tools rather than as answers unto themselves.
Talk explicitly about technology as a tool, and when it is most useful. Assigning problems for which there are multiple paths to solutions and having students share their problem solving strategies both within and across groups is one way for
students to learn and value efficient methods.
Discuss the limitations of technology and processes to deal with them. Talk explicitly about adjusting the scale on graphs, zooming in and tracing for detail, zooming out for the big picture, issues of rounding, the meaning of error messages, and so on.
Resources
Dalton, LeRoy C. Algebra in the Real World. Palo Alto, CA: Dale Seymour Publications, 1983.
Internet location: http://www.ti.com
This site offers various calculator programs, CBL programs for the calculator, lesson activities, and resources for teachers that can be downloaded from the Web site. There is a great deal of material available. There are also links to other mathematics sites. If you have a Graph Link, you can download the TI-92 program so that you can capture TI-92 screen dumps to include in other documents, such as tests and instruction sheets.
Squares Inside Squares
TI-92 Instructions
Open a New Geometry Session
1. Press ON 2. Press APPS
3. Select 8: Geometry 4. Select 3: New
5. Give the new folder the name
"square."
6. Press ENTER twice to get to the geometry screen.
Construct a Square
1. Press F3
2. Select 5: Regular Polygon 3.
3. Use the cursor pad to move the pencil close to the center of the screen.
4. Drag the pencil out to open up a circle.
5. Press ENTER to get a chord on the circle.
6. Move one end of the chord until the number in the center is 4.
7. Press ENTER and a square will be drawn on the screen.
Construct a Square Inside a Square
1. Press F4 .
2. Select 3: Midpoint.
3. Move the pencil using the cursor pad until "MIDPOINT OF THIS SIDE OF THE POLYGON" appears on the screen.
4. Continue to put midpoints on each side in this same manor.
5. Press F3 .
6. Select 4: Polygon.
7. Move the cursor to the midpoint of a side and when "THIS POINT" appears on the screen, Press ENTER .
8. Repeat this until all of the midpoints have been selected and the square has been completed.
Create a Macro
1. Press F4 .
2. Select 6: Macro Construction.
3. Select 2: Initial Objects.
4. Move the pointer to select the larger regular polygon. Press ENTER when
"THIS REGULAR POLYGON" indicates the correct figure.
5. Press ENTER and it appears drawn with a broken line indicating that it is the one that is activated.
6. Press F4 .
7. Select 6: Macro Construction.
8. Select 3: Final Objects.
9. Move the pointer to the polygon formed by joining the midpoints of the larger polygon.
10. Press ENTER to select this polygon.
11. Press F4 .
12. Select 6: Macro Construction.
13. Select 4: Define Macro.
14. Give it the name "squ."
15. Give it an object name "squ."
16. Press ENTER twice.
17. Name the macro "squ."
Draw More Squares Inside Squares
1. Press F4 .
2. Select 6: Macro Construction.
3. Select 1: Execute Macro.
4. "Squ" is highlighted so press ENTER .
5. "THIS POLYGON" is highlighted so press ENTER .
6. Another square should now be drawn inside.
7. Repeat this to draw more squares inside squares.
Find the Area of the Squares
1. Press F6.
2. Select 2: Area.
3. Move the pointer until "THIS REGULAR POLYGON" appears on the screen.
4. Press ENTER to select the square.
5. The area should appear on the screen.
6. Point to the area with the pencil and then use the "hand" tool to move the number to the right and up.
7. Continue to get the areas of the remaining squares. Remember to lock the "hand" tool by holding it down as you press the cursor pad. To release the "hand" tool, press ENTER.
The procedure above can be used to create and determine the areas of other inscribed polygons. Here are the areas of inscribed pentagons.
The procedure was applied to hexagons and the following areas were
determined.
The procedure was also applied to decagons and the following areas were determined.
Sequences Using the TI-92 Calculator
Defining a Sequence Graphing a Sequence
Using a Sequence to Generate a Table of Values
Defining a Sequence
1. Press the MODE key to Set Graph mode to SEQUENCE.
2. Define the sequence on Y = Editor using the following variables:
n term number.
u1 sequence function (like Y1)
up through u99 may be defined.
ui1 initial value to be used as the first term.
u1(n – 1) the previous term.
u1(n – 2) the second previous term.
Note that you can define a sequence using the recursive mode or the explicit mode.
Notice that for this example, u1 has been defined in recursive form, and u2 has been defined in explicit form. Both sequences represent the same set of numbers.
Graphing a Sequence
3. Select which defined sequences to graph by selecting F4 . A small check mark appears in the left- hand margin of the selected items.
Do not select initial values.
Highlight the sequence and then press F4 to select it, or press F4 key again to deselect it. Notice that in step 2 both sequences are
selected, but in this screen only u1 has the check mark indicating that it is selected.
4. Press F6 to select the display style for the sequence.
5. Define the viewing window by pressing the diamond key and the window key.
6. If necessary, change the graph format by pressing F1 9.
7. Graph the selected sequences by pressing diamond key and GRAPH .
Generating a Table of Values
8. Press diamond table set [TblSet] to set the table parameters to:
tblStart = 1
∆tbl = 1
Independent = AUTO
9. Press diamond Window to set the Window variables to make sure that nmin has the same value as
tblStart.
10. Press diamond [TABLE] to display the table of values.
Sequences Using the TI-83 Calculator
Defining a Sequence Graphing a Sequence
Using a Sequence to Generate a Table of Values
Defining a Sequence
1. Press the MODE key to set graph mode to sequence SEQ.
2. Define the sequence on Y = Editor using the following variables:
n term number.
u, v, w sequence functions (like Y1, Y2, and Y3)
u is above the 7, v is above the 8, and w is above the 9.
u(nMin) initial value to be used as the first term. This is used only with a recursive sequence.
u(n – 1) the previous term.
u(n – 2) the second previous term.
Note that you can define a sequence using the recursive mode or the explicit mode.
Notice that for this example, u has been defined in recursive form, and v has been defined in explicit form. Both
sequences represent the same set of numbers.
Graphing a Sequence
3. Select which defined sequences to graph by making sure that the equal sign is selected or darkened. In sequence mode, the TI-83 graphs only the selected sequence when both the u(n) and u(nMin) are highlighted.
The icons to the left of u(n), v(n), and w(n) represent the graph style of each sequence. The default in sequence mode is dot which shows discrete values.
4. Define the viewing window by pressing the
WINDOW key.
5. Change the graph format if necessary by pressing
2nd FORMAT .
6. Graph the selected sequences by pressing GRAPH .
Generating a Table of Values
8. Press 2nd TBLSET to set the table parameters to:
TblStart = 1
9. Press WINDOW to set the window variables so that nMin has the same value as TblStart.
10. Press 2nd TABLE to display the table of values.
Writing Sequences in Recursive and Explicit Form
Directions: For each of the following sequences.
a. Write the recursive form of the sequence.
b. Give the explicit form of the sequence.
c. Give the next two terms of the sequence.
d. Give the 10th term of the sequence.
1. {2, 4, 6, 8, 10, ...}
a. b.
c. d.
2. {4, 11, 18, 25, 32, ...}
a. b.
c. d.
3. {1, 1, 2, 3, 5, 8, 13, ...}
a. b.
c. d.
4. {64, 96, 144, 216, 324, 486, ...}
a. b.
c. d.
5. {125, 120, 115, 110, 105, ...}
a. b.
c. d.
Writing Sequences in Recursive and Explicit Form
Selected Answers
1. {2, 4, 6, 8, 10, ...}
a. t1 = 2 b. tn = 2n tn = tn - 1 + 2
c. 12, 14 d. 20
2. {4, 11, 18, 25, 32, ...}
a. t1 = 4 b. tn = 4 + 7(n -1) tn = tn - 1 + 7
c. 39, 46 d. 67
3. {1, 1, 2, 3, 5, 8, 13, ...}
a. t1 = 1, t2 = 1 b. see extensions tn = tn - 1 + tn - 2
c. 21, 34 d. 55
4. {64, 96, 144, 216, 324, 486, ...}
a. t1 = 64 b. tn =64 3 2
n−1
tn = 3 2tn−1
c. 729, 1093.5 d. 2460.375 5. {125, 120, 115, 110, 105, ...}
a. t1 = 125 b. tn =125 –5(n -1) tn = tn - 1 – 5
c. 100, 95 d. 80
