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GPS Alignment: Instructor Page
Week 4: Algebra - Patterns


Pascal's Patterns (Printable PDF)

                1                
              1   1              
            1   2   1            
          1   3   3   1          
        1   4   6   4   1        
      1   5   10   10   5   1      
    1   6   15   20   15   6   1    
  1   7   21   35   35   21   7   1  
1   8   28   56   70   56   28   8   1

What is the sum of each row? Find a way to describe the pattern of sums by comparing each row's sum to the corresponding row number.

GPS As seen in Problem Exploration

M6A2: Students will consider relationships between varying quantities.

a. Analyze and describe patterns arising from mathematical rules, tables, and graphs

 

The triangle is formed by the sums of the two numbers above any number. The sums of each row is a pattern that can be written as a function of the corresponding row number. The student must understand the relationship between varying numbers to complete this investigation.
 

M7A1: Students will represent and evaluate quantities using algebraic expressions.

a. Translate verbal phrases to algebraic expressions.

b. Simplify and evaluate algebraic expressions, using commutative, associative, and distributive properties as appropriate.
 

The sum of each row can be expressed as a pattern and an algebraic expression. The sum of each row is equal to 2^(n-1), where n is the number of the row.
 

M7A3: Students will understand relationships between two variables.

b. Represent, describe and analyze relations from tables, graphs, and formulas.

c. Describe how change in once variable affects the other variable.

The pattern that represents the sum of each row is an exponential function. The sum of each row is equal to 2^(n-1), where n is the number of the row. The data can be displayed in a spreadsheet, graphed in a spreadsheet and analyzed.

M8N1: Students will understand different representations of numbers including square roots, exponents, and scientific notation.

i. Simplify expressions containing integer exponents.
 

The pattern that represents the sum of each row can be written in exponential notation. The sum of each row is equal to 2^(n-1), where n is the number of the row.

M8A1:  Students will use algebra to represent, analyze, and solve problems.

b. Simplify and evaluate algebraic expressions.

c. Solve algebraic equations in one variable, including equations involving absolute values.
 

This problem can be explored algebraically, from the identification of the pattern in the triangle, the sum of each row, the pattern of the sums and then a generalized function to represent the sum.