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Students 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 sequence.
Students 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 sequence. One special feature of this lesson is the use of the TI-92, which is used to generate the data. Students find both the explicit and recursive forms for the sequence. At the algebra I level, students do not necessarily need the formal notation, but are very capable of exploring the pattern, plotting the model, and describing how to generate the sequence. In addition to the lesson plan, the site includes ideas for assessment, teacher discussion, extensions of the lesson, additional resources, and a discussion of the mathematical content. The lesson plan is accompanied by video clips illustrating lesson procedures. The user should first locate the Squares Inside Squares lesson and then access the appropriate video clips at the PBS TeacherSource website. The video player necessary to view the video clips can be downloaded for free from the site. (author/pk)
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computer
graphing calculator
Internet connection
overhead display device |
| Adobe Acrobat Reader |
| This technology integrated lesson provides an opportunity for the introduction or reinforcement of the following technology skills: |
| Grades 9-12 |
| Explore Mathematical Concept |
structured activity
website |
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| Mathematics Academic Content Standards |
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| Patterns, Functions and Algebra Standard | |
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| Benchmarks (8 - 10) |
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| A. | Generalize and explain patterns and sequences in order to find the next term and the nth term. |
| C. | Translate information from one representation (words, table, graph or equation) to another representation of a relation or function. |
| D. | Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations. |
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| Benchmarks (11 - 12) |
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| C. | Use recursive functions to model and solve problems; e.g., home mortgages, annuities. |
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| Grade Level Indicators (Grade 8) |
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| 1. | Relate the various representations of a relationship; i.e., relate a table to graph, description and symbolic form. |
| 2. | Generalize patterns and sequences by describing how to find the nth term. |
| 7. | Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems. |
| 8. | Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems. |
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| Grade Level Indicators (Grade 9) |
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| 2. | Generalize patterns using functions or relationships (linear, quadratic and exponential), and freely translate among tabular, graphical and symbolic representations. |
| 3. | Describe problem situations (linear, quadratic and exponential) by using tabular, graphical and symbolic representations. |
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| Grade Level Indicators (Grade 10) |
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| 4. | Use algebraic representations and functions to describe and generalize geometric properties and relationships. |
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| Grade Level Indicators (Grade 11) |
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| 1. | Identify and describe problem situations involving an iterative process that can be represented as a recursive function; e.g., compound interest. |
| 2. | Translate a recursive function into a closed form expression or formula for the nth term to solve a problem situation involving an iterative process; e.g., find the value of an annuity after 7 years. |
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| Grade Level Indicators (Grade 12) |
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| 1. | Analyze the behavior of arithmetic and geometric sequences and series as the number of terms increases. |
| 2. | Translate between the numeric and symbolic form of a sequence or series. |
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| Mathematical Processes Standard | |
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| Benchmarks (8 - 10) |
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| D. | Apply reasoning processes and skills to construct logical verifications or counter-examples to test conjectures and to justify and defend algorithms and solutions. |
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| Benchmarks (11 - 12) |
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| B. | Construct logical verifications or counter-examples to test conjectures and to justify or refute algorithms and solutions to problems. |
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| Principles and Standards for School Mathematics |
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| Algebra Standard | |
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| Understand patterns, relations, and functions |
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| Expectations (9 - 12) |
| generalize patterns using explicitly defined and recursively defined functions; |
| understand relations and functions and select, convert flexibly among, and use various representations for them; |
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| Represent and analyze mathematical situations and structures using algebraic symbols |
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| Expectations (9 - 12) |
| generalize patterns using explicitly defined and recursively defined functions; |
| understand relations and functions and select, convert flexibly among, and use various representations for them; |
| use symbolic algebra to represent and explain mathematical relationships; |
| use a variety of symbolic representations, including recursive and parametric equations, for functions and relations; |
| judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. |
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| Use mathematical models to represent and understand quantitative relationships |
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| Expectations (9 - 12) |
| generalize patterns using explicitly defined and recursively defined functions; |
| understand relations and functions and select, convert flexibly among, and use various representations for them; |
| use symbolic algebra to represent and explain mathematical relationships; |
| use a variety of symbolic representations, including recursive and parametric equations, for functions and relations; |
| judge the meaning, utility, and reasonableness of the results of symbol manipulations, including those carried out by technology. |
| use symbolic expressions, including iterative and recursive forms, to represent relationships arising from various contexts; |
| draw reasonable conclusions about a situation being modeled. |
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| Geometry Standard | |
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| Use visualization, spatial reasoning, and geometric modeling to solve problems |
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| Expectations (9 - 12) |
| use geometric models to gain insights into, and answer questions in, other areas of mathematics; |
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| Reasoning and Proof Standard | |
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| Make and investigate mathematical conjectures |
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| RESOURCE TYPE |
| Instructional Resource |
| PRACTICE LEVEL |
| Best Practice |
| STANDARDS ALIGNMENT |
| Grades 8 - 12 |
| TOPICS |
Mathematics --
Algebra;
Patterns, sequences;
Reasoning and Proof;
Technology |
| FOUND IN |
| COR |
| KEYWORDS |
recursion;
geometric sequence;
TI-92;
graphing calculator;
area of a square |
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Publisher: Public Broadcasting Service
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