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How to Succeed in Math 141
How to Succeed in Math 141
Math 141 UNIT #1 Assignments Fall 2002
(25 pts) Billstein work The individual problem sets 1--6 are due on . You will be assigned to grade one of the sets; the graded papers are due back the next class period . NO LATE WORK. 5 pts are assigned for satisfactorily completely your grading duties.
Practice
9-1: 4,6
9-2: 5, 8
9-3: 2, 3, 6, 9, 12, 14, 24
------12-1: 4, 5, 18Brain Teaser pg. 683
12-2: 2, 4, 5
12-4: 2ac, 5a, 6a, 7ac
12-5: 2a, 4 ------10-1: 2, 12
10-2: 1, 3
“3pt” Problems (10 pts)
------set 19-1: 8, 9ab, 23
9-2: 2, 4, 5abcfgh, 12
9-3: 4, 5, 7bd, 8, 11, 18
------set 212-1: 1b, 3a, 6bd, 8, 10 (use any tools), 11, 13, 16, 17
12-2: 1, 4fhln, 9, 10abc
12-4: 2bd, 5b, 6b, 7bd
------set 310-1: 2, 6, 8, 9, 11a
10-2: 3, 9, 10, 18, 21
10-3: 9, 11, 14, 15
“6pt” Problems (10 pts)
------set 49-1: 21, lab activity pg. 475
9-3: 15, 16, 20
------set 512-1: 1 8
12-2: 1 3 12-4: 3, 13
------set 610-1: 14, 27
10-2: 5, 11, 16
10-3: 1 0
(15 pts) Dolan work Projects are due as assigned in class. LATE WORK LOSES 10% per day. Extra credit can be handed in any time up to the first Friday after the exam. Practice5-6 Square Fractions9-1 What’s the Angle?9-3 Triangle properties --
Angles
Class work9-4 Pattern Blocks10-1 Triangle Properties --
Sides10-2 To Be or Not to Be
Congruent12-4 Reflections
(part of Mira project)
Projects12-13 Tessellations
(10 pts)
Extra Credit 9-2 Inside or Outside?9-9 Map Coloring 12-11 Cut it Out
(50 pts) Supplement work Projects are due as assigned in class. LATE WORK LOSES 10% per day. Practicedefinitions
Class workMira (10 pts)Other pages will be discusses in class before assignments are made.
Projects Puzzles: tangram (10 pts) pentagon (10 pts) hexagon (10 pts)
(10 pts) Class log: attendence, participation, self-evaluation
(100 pts) Exam -
REVIEW
Number patterns Section 1-1 Activity 1-1 Activity 1-4 Activity 1-5
Square root Section 6-6
Proportional reasoning: scaling Section 5-4 pg 285 and #10, 14, 18, 23, 28, 29,
Area model Section pg 96 example 2-15 and #13
Algebra Section1-3
Tangram Activity 5-6
UNDERSTANDING AND CONSTRUCTING BASIC GEOMETRIC SHAPES
Basics
Objects:
given: points, lines, rays, line, segments, angles Section 9-1
defined: polygons, triangles, quadrilaterals, circles Section 9-2
Notions:
congruent objects Section 9-2
transformations Section 12-1, 12-2
Activity 12-1 Activity 12-2 Activity 12-4 Activity 12-11
constructions: throughout
parallel axiom Section 9-3
Tools for constructions
Diagrams throughout
Measuring
rulers for Lengths Section 5-1 #18, Section 11-1 Activity 11-1
protractors for Angles Section 9-1
Geoboards & Grid paper throughout Section 10-5
Pattern blocks Activity 9-4 (similarity) Activity 10-4 Activity 12-1
Tracing paper throughout
Folding throughout
straight lines
perpendicular lines Section 9-1 lab activity pg 475
bisecting angles and lengths
Mira
lines of reflection Section 12-2 Activity 12-4
symmetries Section 12-3
compass & straightedge Section 10-1
circles
SSS congruencies
Angles
Basics Section 9-1 Activity 9-1
Sum of angles, etc Section 9-3 Activity 9-3
Finding angles without measuring
tessellating Activity 9-4
auxiliary lines Section 9-3
theorems Section 9-3
algebra Section 9-3
Triangles
Triangle inequality Section 11-1 Activity 9-1 Activity 10-1
Classification Section 9-2 Table 9-6
Tessellating the plane & sum of angles Section 12-5 Section 9-3 Activity 9-3
Congruencies Section 10-1 10-2 Activity 10-2
Polygons Section 9-2
Sum of angles Section 9-3
Regular Section 9-3
Tessellating the plane Section 9-3 Activity 12-13
Quadrilaterals
Definitions Section 9-2 Table 9-6
Classification by side length & parallel sides.
Using triangle congruencies Section 10-2
Properties by diagonals etc Section 10-2 table 10-1Activity 10-8
SIMILARITY
*Section 12-3 Activity 10-8
Similar triangles and Scaling Section 10-4 Activity 10-4 Activity 10-5
Applications:
Converting units Section 11-1
Indirect measurement Section 10-4 Activity 10-6
Slope Section 10-5
Trigonometry, part I no references in either text
MEASUREMENT
Linear measurement: perimeter and circumference Section 11-1 Activity 11-2
Area measurement Section 11-1Activity 11-3 Activity 11-4
Concepts and units Activity 11-3 Activity 11-11
Geoboard Activity 11-5
formulas Activity 11-6 Activity 11-7 Activity 11-8
polygons
circles and sectors of circles
Scaling and converting units
Pythagorean Theorem Section 11-3 Activity 11-9 Activity 11-10
Trigonometry, part II no references in either text
Comparing perimeter to area
3-D VISUALIZATION AND MEASUREMENT
Classifying and building 3-D objects Section 9-4
Puzzles in 3-D
Surface Area Section 11-4 Activity 11-12
concept and units
application of area
Two formulas
Volume Section 11-5
concept and units Activity 11-13
Three formulas
Scaling and converting units
Comparing surface area to volume
NCTM Standards for Middle Grades 6-8Math Content Areas: Geometry, Measurement, and Data Analysis & Probability
Specific Standards for K-12 mathematics instructional programs are listed in bold face.The bulleted items are specific expectations for 6-8 grade students.
CONTENT: GEOMETRY
Analyze characteristics and properties of two- and three-dimensional geometric shapes and develop mathematical arguments about geometric relationships
• precisely describe, classify, and understand relationships among types of two- and three-
dimensional objects using their defining properties; • understand relationships among the angles, side lengths, perimeters, areas, and volumes of similar
objects; • create and critique inductive and deductive arguments concerning geometric ideas and
relationships, such as congruence, similarity, and the Pythagorean relationship.
Specify locations and describe spatial relationships using coordinate geometry and other representational systems
• use coordinate geometry to represent and examine the properties of geometric shapes; • use coordinate geometry to examine special geometric shapes, such as regular polygons or those
with pairs of parallel or perpendicular sides.
Apply transformations and use symmetry to analyze mathematical situations
• describe sizes, positions, and orientations of shapes under informal transformations such as flips, turns, slides, and scaling;
• examine the congruence, similarity, and line or rotational symmetry of objects using transformations.
Use visualization, spatial reasoning, and geometric modeling to solve problems
• draw geometric objects with specified properties, such as side lengths or angle measures; • use two-dimensional representations of three-dimensional objects to visualize and solve problems
such as those involving surface area and volume; • use visual tools such as networks to represent and solve problems; • use geometric models to represent and explain numerical and algebraic relationships; • recognize and apply geometric ideas and relationships in areas outside the mathematics classroom,
such as art, science, and everyday life.
CONTENT AREA: MEASUREMENT
Understand measurable attributes of objects and the units, systems, and processes of measurement
• understand both metric and customary systems of measurement; • understand relationships among units and convert from one unit to another within the same
system; • understand, select, and use units of appropriate size and type to measure angles, perimeter, area,
surface area, and volume.
Source: http://standards.nctm.org/document/chapter 6/index.htm
CONTENT AREA: MEASUREMENT (contd)
Apply appropriate techniques, tools, and formulas to determine measurements
• use common benchmarks to select appropriate methods for estimating measurements; • select and apply techniques and tools to accurately find length, area, volume, and angle measures
to appropriate levels of precision; • develop and use formulas to determine the circumference of circles and the area of triangles,
parallelograms, trapezoids, and circles and develop strategies to find the area of more-complex shapes;
• develop strategies to determine the surface area and volume of selected prisms, pyramids, and cylinders;
• solve problems involving scale factors, using ratio and proportion; • solve simple problems involving rates and derived measurements for such attributes as velocity and
density.
CONTENT AREA: Data Analysis and Probability
Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer them
• formulate questions, design studies, and collect data about a characteristic shared by two populations or different characteristics within one population;
• select, create, and use appropriate graphical representations of data, including histograms, box plots, and scatterplots.
Select and use appropriate statistical methods to analyze data
• find, use, and interpret measures of center and spread, including mean and interquartile range; • discuss and understand the correspondence between data sets and their graphical
representations, especially histograms, stem-and-leaf plots, box plots, and scatterplots.
Develop and evaluate inferences and predictions that are based on data
• use observations about differences between two or more samples to make conjectures about the populations from which the samples were taken;
• make conjectures about possible relationships between two characteristics of a sample on the basis of scatterplots of the data and approximate lines of fit;
• use conjectures to formulate new questions and plan new studies to answer them.
Understand and apply basic concepts of probability
• understand and use appropriate terminology to describe complementary and mutually exclusive events;
• use proportionality and a basic understanding of probability to make and test conjectures about the results of experiments and simulations;
• compute probabilities for simple compound events, using such methods as organized lists, tree diagrams, and area models.
Source: http://standards.nctm.org/document/chapter 6/index.htm
Tangram Problems
1) Make a tangram set from a 3 inch by 3 inch square. Use a 3x5 note card so you can easily trace the shapes. Make four sets. Using different colors for each set is a good idea.
1
2
3 4
5
6
7
Coloring common lengths the same color can help understand the tangram. For one of your sets, color the longest sides that appear in any piece red. Color the next longest sides blue. The next, red. And the smallest lengths, blue
2) Arrange the tangram pieces to make a familiar shape like a house or an animal. Use all the pieces from your set and trace the figure here:
3) Arrange the pieces of one tangram set to make this figure.
Tangrams 1 Saunders 2002
4) Use the two smallest right isosceles triangles(pieces 4 and 6) to make a square? Use the same pieces to make a triangle? Describe the triangle. Use the same pieces to make a parallelogram? Is this parallelogram congruent to piece 7? Is it a rhombus?
5) In the previous problem you made a square using just 2 pieces from the set. Is it possible to make a square using just 3 pieces? 4 pieces? 5 pieces? 6 pieces? Show each possible square.
Tangrams 2 Saunders 2002
6) I made a fancy tangram set from wood. The square (piece 5) weighs 1 oz. How much does each of the other pieces weigh?
How much does the entire set weigh?
7) Look at all the pieces of your tangram set. How many different lengths of sides are there? List these lengths (in inches) from longest (a1) to shortest (a4):
measured length decimal equivalent exact value
(long red) a1=
(long blue) a2=
(short red) a3=
(short blue) a4=
Predict the next number in this sequence:
a5=
Tangrams 3 Saunders 2002
7
1or2 34
or 6
8) Using the lengths a1, a2, a3, a4, how many non-congruent equilateral triangles can you construct? Construct each one. Measure and label the angles in each one. Classify each one as right, acute or obtuse.
9) Using these lengths, how many non-congruent isosceles triangles can you construct? Construct each one that is not congruent to one of the tangram pieces. Measure and label the angles in each one. Classify each one as right, acute or obtuse
10) Using these lengths, how many non-congruent scalene triangles can you construct? Construct each one. Measure and label the angles in each one. Classify each one as right, acute or obtuse
Tangrams 4 Saunders 2002
Group Project -- enlarging the tangram set Names:
Discuss with your group how you would construct a tangram set such that the square (piece 5) measures 3 inches on each side. Record your method here:
There will be class discussion before you complete the next part.
Each person is in charge of bringing one or two or three of the pieces to class next session. It will also be good if we have extra pieces 4 and 6. Record here who is responsible for which pieces:
\
Tangrams 5 Saunders 2002
two each: 4, 6
two each: 4, 6
3, 7
4, 6, 5
2
1
personpiece(s)
Giant Tangrams Name:
We have made three different size tangram sets:
1) The first one is 3 inches by 3 inches.
2) Each table used the pieces from the first tangram set to make a second larger set.
3) Each table used their second tangram set to make pieces for a third, even larger, set.
In theory, we could keep going; that is, use the pieces from the third set to make an even larger fourth set and then use those pieces to make an even large fifth set . . .
Complete this table and answer the questions:
8
13 inches
third set
fourth set
first (original) set
fifth set
second set
length of the side(inches)
Number of original sets used
Suppose fifty people were working on the project of making the fifth tangram set in this sequence. How many of the original sets would each person need to make?
I am thinking about making a quilt that would look just like one of these sets. Would the fifth set be the right size? Explain your answer.
Tangrams 6 Saunders 2002
Other Tangram Questions
1) How many “different” squares can you tessellate using one tangram set? Make a list of the lengths of the sides.
2) How many “different” right isosceles triangles can you tessellate? Make a list the lengths of the legs and hypothenuse.
3) How many “different” triangles can you tessellate using any number of tangram sets?
4) Can you tessellate a rhombus? Explain.
5) Can you tessellate a equilateral triangle?
6) How many “different” trapezoids can you tessellate using one set?
7) How many “different” convex quadrilaterals can you tessellate with one set?
8)
Tangrams 7 Saunders 2002
Area with Tangrams
1) In any tangram set, if the area of the square piece is one square unit, what is the area of the whole set? What is the area of each piece?
7
1or2 34
or 6
2) Find the exact area of each square in the giant tangram puzzle. Use inches and square inches for units and give exact values, i.e. you will need to use 2 .
3) Can you make a square from tangram pieces that has side length equal to red + blue?
4) Using as many of the tangram pieces as you like,
Make two figures that have the same area but different perimeters.
Make two figures that have the different areas but the same perimeter.
Show the figures on the reverse side of this paper.
5) Make a tangram set such that the square (piece 5) is one square inch. What is the area of theentire tangram?
Tangrams 8 Saunders 2002
Dealing with Infinity
12
14
18
116
132
164
1
1
2
1
2
1
2
1
2
1
2
1
2
1
21
1 2 3 4 5 5
+ + + + + + ⋅⋅ ⋅ =
+ + + + + +⋅ ⋅ ⋅+ + ⋅ ⋅ ⋅ =
OR
n
Tangrams 9 Saunders 2002
WHAT IS A POLYGON?
These are polygons:
Sketch two more polygons:
These are NOT polygons:Sketch two more figures that are NOT polygons:
In your own words, describe a polygon:
Look up the definition of a polygon in the text book or any other source. How does it compare with your description?
Definition Polygon Saunders 2002
WHAT IS CONVEX?
These figures are convex:
Sketch two more convex figures:
These figures are NOT convex:
Sketch two more figures that are not convex
In your own words, describe a convex figure:
Look up the definition of convex in the text book or any other source. How does it compare with your description?
Definition Convex Saunders 2002
The van Hiele Theory
In the late 1950's in the Netherlands, two mathematics teachers, Pierre van Hiele and Dieke van Hiele-Geldof, husband and wife, put forth a theory of development in geometry based on their own teaching and research. They observed that in learning geometry, students seem to progress through a sequence of five reasoning levels, from wholistic thinking to analytical thinking to rigorous mathematical deduction. The van Hieles described the five levels of reasoning in the following way.
LEVEL 0 (Recognition) A child who is reasoning at level 0 recognizes certain shapes wholistically without paying attention to their component parts. For example a rectangle may be recognized because it looks "like a door" and not because it has four straight sides and four right angles. At level 0 some relevant attributes of a shape, such as straightness of sides, might be ignored by a child, and irrelevant attributes, such as the orientation of the figure on the page might be stressed.
LEVEL 1 (Analysis) At this level, the child focuses analytically on the component parts of a figure, such as its sides and angles. Component parts and their attributes are used to describe and characterize figures. Relevant attributes are understood and are differentiated from irrelevant attributes. For example a child who is reasoning analytically would say that a square has four "equal" sides and four "square" corners. The child also knows that turning a square on the page does not affect its squareness.
A child thinking analytically might not believe that a figure can belong to several general classes and have several names. For example a square is also a rectangle since a rectangle has 4 sides and 4 square corners, but a child reasoning analytically may object, thinking that square and rectangle are entirely separate types even though they share many attributes.
LEVEL 2 (Relationships) There are two general types of thinking at this level. First, a child understands abstract relationships among figures. For example a rhombus is a 4 sided figure with equal sides and a rectangle is a 4 sided figure with square corners. A child who is reasoning at level 2 realizes that a square is both a rhombus and a rectangle since a square has 4 equal sides and 4 square corners. Second, at level 2 a child can use deduction to justify observations made at level 1.
LEVEL 3 (Deduction) Reasoning at this level includes the study of geometry as a formal mathematical system. A child who reasons at level 3 understands the notions of mathematical postulates and theorems and can write formal proofs of theorems.
LEVEL 4 (Axiomatic) The study of geometry at level 4 is highly abstract and does not necessarily involve concrete or pictorial models. At this level the postulates or axioms themselves become the object of intense rigorous scrutiny. This level of study is not suitable for elementary, middle school or even most high school students, but is usually the level of study in geometry courses in college.
Source: http://nunic.nu.edu/~frosamon/history/mike.html
Some Characteristic Indicators of the van Hiele Levels
LEVEL 0 (Recognition)
*primarily visual understanding of shapes*not actively aware of properties of figures
LEVEL 1 (Analysis)
*reference to visual prototypes to characterize shapes (doors, balls, building, etc.)*inclusion of irrelevant attributes when identifying shapes such as orientation, or color, or size*inability to conceive of an infinite variety of shapes
LEVEL 2 (Ordering)
*explicit reference to component parts when identifying shapes, sides, angles, diagonals, etc.*single attribute sorting of shapes, neglect of other attributes*prohibiting class inclusions of shapes*application of a "litany" of necessary properties when defining a shape, instead of sufficient properties
LEVEL 3 (Deduction)
*form complete definitions, i.e. sufficient conditions, for types of shapes*ability to use equivalent forms of definitions*explicit use of if - then statements, ability to form informal deductive arguments*confusion on the role of theorems and axioms
LEVEL 4 (Rigor)
*understanding of the importance of precision in dealing with foundations and interrelationships between structures
Source: http://jwilson.coe.uga.edu/EMT668/EMT668.Folders.F97/Nipper/EMAT6690/Curriculum/vanHiele.html
x
a
b
Explain why:
x = a+b
Using Algebra in Angle problems
Angle Theorems Saunders 2002
The exterior angle of a triangle is equal tothe sum of the opposite angles.
R
PQ
S
RS ⊥ SQPS ⊥ RQm PRS( )∠ = °37
Find x.
x
C
A
B
D
What is the measure of <BCD?
35˚
21˚
90°
a
b
Explain why angles a and b are complementary, that is a+b=90˚.
Auxiliary lines can help to solve problems.
132˚
81˚
66˚
x
Auxiliary Lines Saunders 2002
20°
43°x
Find x.
A B
CD
Polygon ABCD is a parallelogram.that is, opposite sides are parallel.
Show that opposite angles are equal.
Find x.
48°
?
A
B
C
D
E
F
GH
I
A regular 9-gon ABCDEFGHI is shown. Is the triangle ACE obtuse, acute, or right?
A square and a regular pentagonare shown. Find the indicated angle:
If the sketch is drawn to scaleyou might measure to find theanswer.
Which of these two sketches is drawn to scale?
In any case, you should be ableto solve both problemswithout measuring
Two Problems with Regular Polygons
Polygon Angles Saunders 2002
120°
110°
77°
a
b
c
Find the measures of angles a, b, c without using a protractor.
Is the drawing to scale? Check by measuring each given angle.
If the drawing is to scale, you can also check your answers by measuring.
Finding angles
Finding Angle s Saunders 2002
AB || CGBC || JDHG ⊥ CGED EF≅m ABC( )∠ = °126
Find the measure of angle x
BA
C
J
D
E
F G
H
x
Using the Mira 1
1) Learning about reflections: Do the first two pages of Dolan Activity 12-4, pages 281-282
2) Bisect ∠ A .
A
4) Quadrisect ∠ B
B
6) Use the Mira to show i) this is an isoceles triangle ii) the angles opposite the equal sides are equal
G F
C D
5) Draw lines to quadrisect GF
3) Bisect CD
Using the Mira 2 -- lines of symmetry
7) Draw all the lines of symmetry in each figure.
8) How many lines of symmetry are there in a circle?
Using the mira, find the center of this circle:
Using the Mira 2 Saunders 2002
10) A line segment from the vertex of a triangle to the opposite side, which is perpendicular to the opposite side is called an altitude of the triangle. A triangle has three altitudes. An altitude does not necessarily lie inside the triangle. Draw all three altitudes for the triangle:
9) For each case, draw a line through the point P which is perpendicular to l . You may need to extend the line.
Using the Mira 3 -- Perpendicular and Parallel lines
l l
P
P
P
l
Using the Mira 3 Saunders 2002
11) Use the Mira to draw a line through the point P that is parallel to the line l .
P
l
A
B
C
D
O
P Q
R T
S
D
O GP
In each figure, two triangles appear to be congruent. Write down the apparent congruency.Justify the congruency OR draw another picture given the same information showing noncongruent triangles.
Being Careful about Congruencies
A
BC
M
N
TS
D
∠ =ABC 90o. BD AC⊥ Quadrilaterals, AMNC and BCTS, are squares.
Find two pairs of congruent triangles, write the congruencies and justify your conclusions.
Sometimes, congruent triangles need to be located within complicated arrangements. This example is from Euclid’s proof of the Pythagorean theorem.
Careful Congruencies Saunders 2002
PQ RT
QS ST
PQS
||
≅≅∆
∠ ≅ ∠∠ ≅ ∠
≅
OPG OGP
DPO DGO
DOG∆
AO DO
BO CO
OBA
≅
≅≅∆
H
Parallelogram Rhombus Rectangle Square Kite Isosceles Trapezoid Trapezoid
Side Properties
4 sides
At least 1 pair
of parallel sides
2 pairs of
parallel sides
All sides congruent
At least 1 pair
of opposite sides congruent
2 pairs of opposite
sides congruent
Angle Properties
Sum of angles = 360°
All angles are right angles.
Opposite angles
are congruent.
Adjacent angles
are supplementary.
Quadrilateral Properties -- Sides and AnglesPut an X in the box if the given property always applies to the given type of quadrilateral.
Quadrilateral Chart 1 Saunders 2002
Parallelogram Rhombus Rectangle Square Kite Isosceles Trapezoid Trapezoid
Diagonal Properties
At least 1 diagonal forms
2 congruent triangles
Both diagonals form
2 congruent triangles.
Diagonals
bisect each other.
Diagonals
are congruent.
Diagonals bisect
the vertex angles.
Diagonals
are perpendicular
Diagonals form 4
congruent triangles.
Symmetric Properties
Number of lines
of symmetry
Rotational symmetry
(smallest positive angle)
Quadrilateral Properties -- Diagonals and SymmetryPut an X in the box if the given property always applies to the given type of quadrilateral..
Quadrilateral Chart 2 Saunders 2002
Using Algebra1)
B C
D
A
In this figure AB AD= and BD DC= .
What is ∠ A ?
2)
A
B
CD
In this figure AD DB DC= = .
Show that ∠ = ∠A C12
3) The measure of the interior angle of a regular polygon is 160˚ -- how many sides does this polygon have?
Kurt measures the interior angle of a regular polygon to be 150˚ -- Explain to Kurt why this is not possible.
4) Make a triangle ∆ABC such that ∠ A is twice ∠ B and ∠ B is twice ∠ C .
Is it possible to make a triangle ∆DEF such that m DE( ) is twice m EF( ) and m EF( ) is twice m DF( )?
Using Algebra Saunders 2002
Hexagon Puzzle -- Make a puzzle project
First, make the puzzle pieces:
Cut the equilateral triangle (Use the size given here--four inch sides) into pieces -- you should make at least 4 but no more than 8 pieces. There should be at least three non-congruent polygons in your puzzle. Use only the angles 30˚, 60˚ , 90˚, and 120˚.
Describe each of the pieces geometrically. Label all angles with the angle measure. You should know them exactly.
Second, make up some the puzzle questions:
Here are some suggestions: Can you make larger, similar versions of each of your pieces? Can you make any shapes in more than one way? Can you make squares? rectangles? Parallelograms? Can you make a hexagon? If you had more copies of the puzzle, could you make a hexagon?
. Third, test it:
Give your puzzle pieces to a friend and see if they can reconstruct the hexagon. To make it easier, give them the outline of the hexagon to start. To make it harder, don’t give them the outline.
Finally, package your puzzle Provide a container for the pieces, outline of puzzle, suggested puzzles and an answer sheet. Make two extra copies of the puzzle pieces for a class project.
Hexagon Puzzle Saunders 2002
Pentagons and the Golden Ratio
1) Draw all diagonals in this regular pentagon
2) Label the different size lengths with the letters a1, a2, a3, a4 -- from smallest to largest.
3) Measure the lengths and list them here. Is this a geometric sequence? If so, what is the common ratio?
What is the next number in the sequence, a5?
a1 =
a2 =
a3 = a4 =
a5 =
4) Compute: a1 + a2 = a2 + a3 = a3 + a4 =
Golden Ratio Saunders 2002
