Ohio Resource Center
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Content Supports - Activities and rich problems
Predicting Patterned Behavior
Discipline
Mathematics
Grades
9, 10, 11, 12
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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. 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)


Common Core State Standards for Mathematics
Standards for Mathematical Practice
CCSS.Math.Practice.MP1 
Make sense of problems and persevere in solving them.
CCSS.Math.Practice.MP2 
Reason abstractly and quantitatively.
CCSS.Math.Practice.MP7 
Look for and make use of structure.
High School - Algebra
Seeing Structure in Expressions
Interpret the structure of expressions
HSA-SSE.A.1 
Interpret expressions that represent a quantity in terms of its context.
  1. Interpret parts of an expression, such as terms, factors, and coefficients.
  2. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.
HSA-SSE.A.2 
Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – (y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).
Ohio Mathematics Academic Content Standards (2001)
Patterns, Functions and Algebra Standard
Benchmarks (5–7)
G.
Write, simplify and evaluate algebraic expressions.
Benchmarks (8–10)
A.
Generalize and explain patterns and sequences in order to find the next term and the nth term.
Grade Level Indicators (Grade 7)
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.
Grade Level Indicators (Grade 8)
8.
Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems.
Principles and Standards for School Mathematics
Algebra Standard
Understand patterns, relations, and functions
Expectations (9–12)
generalize patterns using explicitly defined and recursively defined functions;
Represent and analyze mathematical situations and structures using algebraic symbols
Expectations (6–8)
recognize and generate equivalent forms for simple algebraic expressions and solve linear equations.
Expectations (9–12)
generalize patterns using explicitly defined and recursively defined functions;