Chapter 1 follow-upCheck digit division by 9:Usually the sum of the digits are divisible by 9

Mod 9 check digit scheme:Usually the last digit is congruent mod 9 to the sum of the previous digits.

Congruent mod 9Some examples:

22≡4 mod 9 because 9|22-4

19≡1 mod 9 because 9|19-1

30≡3 mod 9 because 9|30-3

Congruent mod 9Find the missing digit using mod 9

73?11

The sum of 7+3+d3+1≡1 mod 9

11+d3≡1 mod 9

Congruent mod 9Find the missing digit using mod 9

73?11The sum of 7+3+d3+1≡1 mod 911+d3≡1 mod 9

The missing digit must be 8 because 19≡1 mod 9

Chapter 2Shapes in Our Lives

Tilings Symmetry, Rigid Motions, and Escher

Patterns Fibonacci Numbers and the Golden Mean

Tilings

• Repeated polygons with no gaps

Tilings

Regular Tessellation

Regular Tessellations

• Triangles• Hexagons• Squares• Do any other regular polygons

tessellate?

Regular Tessellations

• Investigate the interior angle measures of a regular polygon

Sum of the measures of a triangle = 180 degrees

Regular Polygons

• Investigate the interior angle measures of a regular polygon

Sum of the measures of a triangle = 180 degreesWhat is the sum of the measures of the interior angles of a square, hexagon?

Regular Polygons

Sum of interior angles of square = (4-2) 180 = 360

Sum of interior angles of a hexagon = (6-2)180 = 720

Regular Polygons

Sum of interior angles of any polygon

(n-2) 180

n=number of sides

Regular Polygons

Sum of interior angles of any polygon(n-2) 180 n=number of sides

Measure of each interior angle in a regular polygon = (n-2)180/n

Regular Polygons

Measure of each interior angle in a regular polygon = (n-2)180/nMeasure of each angle in a regular triangle = 180/3

square = 360/4 =

regular hexagon = 720/ 6 =

Regular Polygons

Measure of each interior angle in a regular polygon =

(n-2)180/n

Measure of each angle in a regular octagon =

Regular Polygons

Measure of each interior angle in a regular polygon =

(n-2)180/n

Measure of each angle in a regular octagon =(8-2)180/8 = 135 degrees

Regular Polygons

Why do these 3 shapes tessellate and other regular polygons don’t?

Regular Tessellations

Look at the point where the triangle vertices meet.

What is the sum of the angle measure?

Regular Tessellations

What is the sum of the angles at the point where the hexagons meet?

Semiregular Tessellations

• A tessellation that uses two or more different types of regular polygons.

• See poster in classroom for explanation

Escher Tessellations

• See pg 85 in textbook-more in section 2.2

• See posters in classroom

Pythagorean Theorem

a2 +b2 = c2

Pythagorean Theorem

• Find the length of the missing side:

5

12

hypotenuse

Section 2.1 assignment

• Pg79 (3,5,33,35,43)• And the following project:• A presentation to include

2 photos of a tessellations 1 regular tessellation drawing using any

medium 1 semiregular tessellation drawing using

any medium

Symmetry, Rigid Motion, and Escher Patterns

• Symmetry Line of symmetry

Symmetry, Rigid Motion, and Escher Patterns

Line of symmetry

Rotational symmetry

Symmetry, Rigid Motion, and Escher Patterns

• Rigid Motion or• • Isometry

• “same measure”

• Translation

Symmetry, Rigid Motion, and Escher Patterns

• Glide reflection footprints

Symmetry, Rigid Motion, and Escher Patterns

• Glide reflection

Symmetry, Rigid Motion, and Escher Patterns

• Escher Patterns - how to make one on pg 99-100

• Use patty paper to draw an Escher design that will tessellate

Symmetry, Rigid Motion, and Escher Patterns

• Section 2.2 Assignment pg 102 (3,13,15,33,34,45)

• An original Escher creation from a square- directions are on pg 100. Tessellate several copies of your design

• An original Escher creation that uses rotation (start with an equilateral triangle) - directions are on pg 107. Tessellate several copies of your design

Fibonacci Numbers and the Golden Mean

• 1,1,2,3,5,8,13,21,34,55, ____,____,____

• This is called the Fibonacci Sequence

Fibonacci Sequence

• The Fibonacci sequence is generated by recursion - each number in the sequence is found by using previous numbers.

• fn = fn-1 + fn-2 and

• f1 = 1 and f2 = 1

Fibonacci Sequence

• The Fibonacci Sequence occurs often in nature:

• http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

• Also, see examples in text on pgs112-118

Geometric Recursion

• Figures can be built by repeating some rule or set of rules.

• For example:

Geometric Recursion

Sierpinski gasket

The Golden Ratio

• Look at the sequence of ratios of pairs of successive Fibonacci numbers:

€

11

, 21

, 32

.53

,85

,138

,2113

,...

1,2,1.5,1.66,1.6,1.625,1.615384,...

1+ 52

=1.61803

The Golden Ratio

• The golden ratio has figured prominently in art and architecture.

The Golden Ratio

• The golden ratio has • figured prominently in • art and architecture.

Section 2.3 assignment

• Pg 125 (1,3,11,13,27,28,31) and• Research Leonardo DaVinci’s use of

the Golden Ratio. Include an explanation of what you find. This explanation may be a written paragraph and/or a drawing that includes an explanation.