PATTERNS AND FIGURES

 

GOALS

1.        use and create dot patterns, number strips, or charts to visualize number sequences

2.        create and use recursive formulas to describe number sequences

3.        create and use expressions and direct formulas to describe number sequences

4.        understand arithmetic sequences

5.        understand the sequence of square numbers

6.        combine (add or subtract) number sequences and the corresponding expressions

7.        use Euler’s formula as an example of combining sequences

8.        justify equivalent expressions

9.        use triangular and rectangular numbers as examples of describing sequences

10.     use visual models to extend understanding of equivalent expressions and formulas

11.     generalize a concrete number sequence to an expression that describes the sequence

12.     use formulas and expressions to describe patterns and sequences in realistic situations

 

SECTION SUMMARIES

 

SECTION A: PATTERNS

1.       Direct formulas are used to find any term in a sequence, provided that you know its position in the sequence.

2.       Recursive formulas are used to find the next term in a sequence using the current term.

 

SECTION B:  ARITHMETIC SEQUENCES

1.       Arithmetic sequences are formed by repeatedly adding or subtracting a number at each step.

2.  The general expression for any arithmetic sequence is a+bn, where are a is the starting number, b is the step-by-step increase or decrease, and n is the pattern number.

3.   The general expression for any recursive formula for any arithmetic sequence is NEXT = CURRENT + k, where k is the constant increase or decrease.

4.  Remember that an expression has no equal sign, as opposed to a formula.

 

SECTION C:  COMBINING STRIPS

1.        We will be adding or subtracting the corresponding numbers in two number sequences.  The sum (or difference) of the expressions is equivalent to the new expression.

2.         V – E + F = 2 is Euler’s formula, which describing the relationship between the vertices, faces, and edges of any simple, closed polyhedron (a geometrical solid formed using flat faces).

 

SECTION D:  SQUARE NUMBERS

1.       The square numbers are 1,4,9,16,25,….  The differences between the square numbers increases by a constant amount of 2.

2.       The area model demonstrates that (n+1)2 does not equal n2+ 12, and helps us to understand that (n+1) 2 = n2 + 2n + 1.

 

SECTION E:  TESSELLATING WITH TRIANGLES

1.    When n black tiles are used to form each edge of an equilateral triangle, the total number of black and white triangles used in the tessellation is n2.

 

SECTION F:  TRIANGULAR  NUMBERS

1.    The number of black tiles in the tessellated triangles forms the triangular numbers 1,3,6,10,…, and the rectangular numbers 2,6,12,20,… are half the triangular numbers.

2.    The number of dots in a rectangle with the dimensions n by n+1 is (n)(n+1).  The formula for the triangular numbers is (˝)(n)(n+1).

 

 SECTION F:  CHOREOGRAPHY

1.    This section brings together direct formulas, number strips, and triangular numbers in the context of choreography.