of your homework assignment, not on the activity pages Growing Patterns-Multiple
ID: 3185468 • Letter: O
Question
of your homework assignment, not on the activity pages Growing Patterns-Multiple Representations of Functions Resource: http://www.visualpatterns.org/gallery.htm a. Describe how you see the shapes growing in the figure below. Describe what the 100th stage 3. looks like. Make specific connections between your verbal description and the pictures. Determine a rule for the perimeter of stage n for the pattern in part (a). Determine a rule for the area of stage n in part (a). Note: For parts (b) and (c), describe why your algebraic rule makes sense in the context of the geometry of the growing pattern. Draw sketches and link your sketch to an algebraic representation for stage n. b. c.Explanation / Answer
a) Noticing the pattern carefully, i have come to a conclusion that each side is increasing by one box in every step ,i.e., both row and column are increasing by one row and one column respectively.
In other words, left and right sides of the shape are increasing by one box in every step and top and bottom sides are also increasing by one box in every step.
From the given pictures we have,
In step I : left and right sides of the shape has 1 box each and top and bottom sides of the shape has 2 boxes each.
In step II : left and right sides of the shape has 2 boxes each and top and bottom sides of the shape has 3 boxes each.
In step III : left and right sides of the shape has 3 box each and top and bottom sides of the shape has 4 boxes each.
In 100th stage, left and right sides have a length of 100 units and top and bottom sides have a length of 101 units ,i.e., there must be 102 rows and 103 columns but the four corner cells must be deleted.
(If we consider the length of each box is 1 unit)
b) Rule for the perimeter of stage n for the pattern is = 2*[(n+2)+((n+1)+2)] units
= 2*[2n+5] units
= 4n+10 units
In step I : Putting n = 1 we have, perimeter = 14 units.
In step II : Putting n = 2 we have, perimeter = 18 units.
In step III : Putting n = 3 we have, perimeter = 22 units.
c) Rule for the area of stage n for the pattern is = (n+2)*((n+1)+2) square units
= (n+2)(n+3) square units
= (n2+5n+6) square units
In step I : Putting n = 1 we have, area = 12 square units.
In step II : Putting n = 1 we have, area = 20 square units.
In step III : Putting n = 1 we have, area = 30 square units.
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