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Students complete a triangular array by inserting numbers into the top row and adding pairs of numbers for subsequent rows to achieve a final sum at the bottom. After discovering that the position of a number in the top row affects the final sum, they are challenged to find five different numbers that lead to a sum of 100 in the bottom row.
Students complete a triangular array by inserting numbers into the top row and adding pairs of numbers for subsequent rows to achieve a final sum at the bottom. After discovering that the position of a number in the top row affects the final sum, they are challenged to find five different numbers that lead to a sum of 100 in the bottom row. They then analyze the diagram algebraically to determine a formula for the final result in terms of the starting numbers. Next, they compare the coefficients in their formula to the corresponding row of Pascal’s triangle and use that pattern to predict and complete a larger array with a specified end sum. An overhead transparency, activity sheet, template for Pascal's Triangle, and solutions are included. This mathematically rich problem was developed for the Ohio Resource Center to accompany the Mathematics Program Models for Ohio High Schools proposed by the Ohio Department of Education. (author/sw)
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| Mathematics Academic Content Standards |
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| Patterns, Functions and Algebra Standard |  |
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| Benchmarks (5 - 7) |
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| G. | Write, simplify and evaluate algebraic expressions. |
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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. |
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| Grade Level Indicators (Grade 7) |
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| 1. | Represent and analyze patterns, rules and functions with words, tables, graphs and simple variable expressions. |
| 7. | Justify that two forms of an algebraic expression are equivalent, and recognize when an expression is simplified; e.g., 4m = m + m + m + m or a * 5 + 4 = 5a + 4. |
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| Grade Level Indicators (Grade 8) |
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| 8. | Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve 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; |
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| Represent and analyze mathematical situations and structures using algebraic symbols |
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| Expectations (6 - 8) |
| recognize and generate equivalent forms for simple algebraic expressions and solve linear equations. |
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| Expectations (9 - 12) |
| generalize patterns using explicitly defined and recursively defined functions; |
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| RESOURCE TYPE |
| Rich Problem, Inquiry, or Exploration |
| STANDARDS ALIGNMENT |
| Grades 7 - 10 |
| TOPICS |
Mathematics -- Algebra; Patterns, sequences; Polynomials |
| KEYWORDS |
| Pascal's Triangle |
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Author: Steven P. Meiring Publisher: Ohio Resource Center
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