February 28, 2015 Honolulu, HI  HCTM Spring Conference
Raffling off one of our books.
Workshop Activities
Lynda Vaughan from the good old days!
HCTM Spring conference
Saturday, February 28, 2015
Kaiser High School
Raffling off one of our books.
Workshop Activities
Lynda Vaughan from the good old days!
HCTM Spring conference
Saturday, February 28, 2015
Kaiser High School
"PIrate Geometry"
Download HCTM Pirate Geometry 9.14.2
Saturday, September 13, 2014
8:30 am
Participants will use transformations and polyominoes to solve puzzles in a
Pirate treasure hunt setting. We will explore rectangular, polar and 3D
coordinates. Participants will learn how rotation transformations are used
to create a Grille Cipher. Then use that skill to decipher secret messages
to locate pirate buried treasure. Fun activities to celebrate International
Talk LIke A Pirate Day!
Ahoy Teachers!
Now there is an official day for you to celebrate your secret pirate passion and have a fun educational day with your students.
September 19 is International Talk Like A Pirate Day. ITLAPD was started by Mark Summers and John Baur and is celebrated all over the world. From Mark:
Coming at the beginning of the school year, Talk Like a Pirate Day allows the students to see their teacher's playful side. This should not be viewed as a day wasted on silly nonsense, but as a ‘teachable moment.’ Be creative, take risks and above all, have fun.
Go to their website and find all things pirate along with some curriculum for celebrating ITLAPD in the classroom. Of course, Pirate Math fits in perfectly with this momentous day. We will be posting Pirate Math puzzles on Facebook the month before September 19, 2014.
This service is available to schools at no additional cost upon adoption.
The following books are available in Digital eBooks and Teacher Resources.

Copyright © 2015 Kendall Hunt Publishing
At the California Mathematics Counsel Asilomar 2013, I had the pleasure of watching a presentation by Alex Bega, "Flipping the Secondary Math Classroom".
Alex is currently in her fifth year of teaching at Saint Francis High School. She has taught Precalculus for the past three years and is currently on her second year of teaching in a flipped classroom style. Alex taught K3 music and 68 computer technology before beginning her career as a 912 mathematics teacher.
Alex offers the following categories to help teachers implement the Flipped Classroom: Why Flip, Technology, Results, and Pros and Cons. She breaks down her formula for making and using video in her classroom with clear and field tested procedures.Thank you Alex for your great presentation. You can contact Alex Bega at alexbega@sfhs.com
If you'd like to use the Discovering Geometry® Videos to try out the flipped classroom, try it here!
DG43.7.3 Incentro (Free Sample) from Michael Serra on Vimeo.
Bien venidos a la suscripción de Videos Discovering Geometry que está diseñada para ser utilizado junto con el aclamado libro, Discovering Geometry. Las investigaciones se demuestran por Michael con conjeturas de terminadas abiertas para que el estudiante complete. Suscripción de Video Discovering Geometry ® puede utilizarse como un tutorial para el nuevo y sustitutos y para los alumnos que son educados en casa o haciendo el aprendizaje independiente. Ideal para usar junto con el libro de texto, especialmente en el ambiente "clase voltiada". Echale un vistazo a la muestra libre de Discovering Geometry serie de vídeo y el trailer.
Bien venidos a la suscripción de Videos Discovering Geometry que está diseñada para ser utilizado junto con el aclamado libro, Discovering Geometry. Las investigaciones se demuestran por Michael con conjeturas de terminadas abiertas para que el estudiante complete. Suscripción de Video Discovering Geometry ® puede utilizarse como un tutorial para el nuevo y sustitutos y para los alumnos que son educados en casa o haciendo el aprendizaje independiente. Ideal para usar junto con el libro de texto, especialmente en el ambiente "clase voltiada". Echa un vistazo a la muestra libre de Discovering Geometry serie de vídeo y el trailer.
Proof in Discovering Geometry pdf
Geometry has traditionally been the course in which a deductive structure is first taught and formalized. This is tradition, but not good pedagogy. In a subject that is highly visual we ask students to deny what they see until they have performed this mysterious ritual called a twocolumn proof.
The van Hiele Model
Whenever there is discussion about proof in geometry we should first review the research of Dina van HieleGeldorf and her husband Pierre van Hiele. Their research resulted in the van Hiele model of geometric reasoning. Research by numerous mathematics educators including Usiskin 1982; Senk 1985; Burger and Shaughnessy 1986; Geddes and Fry 1988; Clements and Battista 1992; and more recently Battista 2007, support the accuracy of the van Hiele model. The model consists of five levels of geometric reasoning that students pass through from pre deductive visual thinking to an understanding of formal proof and mathematical structures.
Level 0: Visualization –students can identify rectangles by sight but squares are squares and are not seen as special rectangles.
Level 1: Analysis (descriptive) – students can identify properties of rectangles (by drawing, measuring, and making models) but cannot yet derive other properties from given.
Level 2: Informal Deduction – students can give an informal argument to justify that the figure is a rectangle from given properties.
Level 3: Formal Deduction – students are capable of creating original logical arguments.
Level 4: Rigor – students are capable of reasoning about mathematical structures (i.e. Euclidean vs. nonEuclidean geometries).
The van Hiele model asserts these levels are sequential and hierarchical. That is, students cannot move to the next higher level until they have successful mastered the previous level. Furthermore movement from a lower level to the next higher level depends more on content and pedagogy than on maturity and age.
The van Hiele research shows students aren’t ready for formal proof until they have had concrete experiences with geometric figures and have successfully mastered earlier levels of visual thinking. Research shows 70% of high school students enter geometry operating at level 0 or 1 on the van Hiele measure of geometric reasoning (Shaughnessy and Burger 1985; Senk 1989). Yet "traditional" geometry textbooks that begin with establishing postulates and proving theorems are expecting students to immediately begin their geometry experience at levels 34. When the teacher and the textbook are presenting geometry at van Hiele level 3 or higher, while the students are functioning at level 0 or 1, it should be no surprise that there is such a high failure rate in traditional geometry courses.
From Mathematician's Delight by English Mathematician W.W. Sawyer (19112008)
It would, I suppose, be quite possible to teach a deaf and dumb child to play the piano. When it played a wrong note, it would see the frown of its teacher, and try again. But it would obviously have no idea of what it was doing, or why anyone should devote hours to such an extraordinary exercise. It would have learnt an imitation of music. And it would fear the piano exactly as most students fear what is suppose to be mathematics.
Some geometry teachers claim they can successfully teach all of their geometry students how to create geometry proofs. Perhaps they only have those 30% that are entering geometry and functioning above van Hiele levels 0 and 1. I suspect however, they may also be teaching an "imitation geometry." Their students are trained to go through the motions of stating the theorem about to be proved (5 points), then stating the given (5 points), then stating the show (5 points), then drawing the diagram to the right (5 points), then drawing the big T (5 points), then putting the given information in the first few lines of the Tproof (515 points), then writing given to the right of each statement (515 points), then throwing in enough statements to garner enough points to get credit for the exercise without having any clue as to what he or she was doing or why anyone should devote hours to such an extraordinary exercise. They are doing imitation geometry.
Discovering Geometry, in its first edition, was an innovator in addressing students’ needs for gradual development of the deductive process. Discovering Geometry is the only high school geometry textbook on the market that is aligned with the van Hiele model and other research on geometric proof. We accept the vast majority of student are entering geometry at very low van Hiele levels of geometric reasoning and our goal, with careful deliberate scaffolding, is to move them to higher and higher levels of geometric reasoning as they progress through the course.
The Role and Function of Proof
Geometry student's consistent difficulties with understanding proofs (an international problem BTW) should not be solely attributed to their inability to reason but perhaps our inability to recognize there are many purposes for doing proofs and we have been stressing the wrong purposes for proof at inappropriate times. Professor Michael de Villiers' research on the role and function of proof identifies six basic roles for proof:
• Verification –to remove doubt, to convince someone of the truth of a statement
• Systematization –organize known results into a deductive system of postulates, definitions, and theorems
• Explanation –insight into why something is true
• Discovery –proof can occasionally lead to new unexpected results
• Communication –proof can create a forum for critical debate
• Intellectual Challenge –proof can be a testing ground for intellectual stamina and ingenuity
The function of proof in a high school geometry course has been mostly twofold: to remove doubt, to convince someone of the truth of a statement –verification and to establish geometry as a mathematical system –systematization.
Many high school mathematics teachers seem to hold this naive view the main function of proof is to provide verification that a given statement is true. The role of proof as a means of verification is a useful method of verifying the "truth" of a proposition within a mathematical system, especially when coming across surprising (nonintuitive) results. However this view does not stand up to actual mathematician's experiences. Professor George Polya wrote, “When you have satisfied yourself that the theorem is true, you start proving it.” If the sole or primary purpose for doing proofs in a high school geometry course is to provide verification of the truth of a statement then students functioning below level 4 on the van Hiele scale will continue to question, or worse, disregard the process of proof.
If systematization is emphasized as a primary function of proof right from the start of a geometry course, the same poor results will persist. The van Hiele model tells us systematization requires the highest van Hiele level of geometric reasoning. Geometry textbooks that begin their geometry program with lists of definitions and postulates, and then begin doing twocolumn proofs are ignoring the research. Only an honors course or any class in which all students are finally operating at van Hiele level 4 would have any chance of success in a course that looks at geometry from the perspective of a mathematical system. Until some magic happens and all students beginning geometry enter the course functioning at van Hiele level 3, any attempt to create a mathematical system for a regular or informal geometry course is likely to continue to have major problems.
Students can acquire a very high degree of confidence in a conjecture arrived at by inductive methods but these methods may not explain why the conjecture is true. Here is where proof can come to the rescue. An example would be the inductive discovery that the sum of the measures of the three angles of a triangle is always 180°. A good inductive first approach would be to ask students to measure the three angles of a number of triangles thus gaining reasonable confidence the sum is indeed 180°. The same can be done with dynamic geometry software. Either investigative approach is a good first step because it is the first approach students would take. These inquiry approaches do give students confidence in their conjecture however they give no insight as to why the sum is always 180°. The investigation should be followed by a second investigation where they cut out the triangle and then tear off two of the angles and arrange them on both sides of the third angle to create a straight line. From this arrangement they can see the three angles create a line parallel to the third side. This visually explains what properties this conjecture is dependent upon and why the conjecture is true. This can also be done quickly with a patty paper investigation.
From The Role and Function of Proof in Mathematics by Michael de Villiers:
It is not a question of "making sure," but rather a question of "explaining why."
Explanation as the Primary Role of Proof
Using proof as a means of explaining why something is true is the most meaningful role proof can play in a high school geometry course. Asking why something is true, after performing investigations that have convinced students it is true, is a powerful 12 punch. Explaining why can be an effective tool regardless of a student's van Hiele level. This is the approach we take with Discovering Geometry.
In practically every lesson in the fourth edition of Discovering Geometry (DG4) students are asked to perform geometric investigations and then make their geometry conjectures. After performing their investigation and making their conjecture they are asked, "can you explain why?" For example, after their very first two investigations leading to geometric conjectures, the Linear Pair Conjecture and the Vertical Angles Conjecture, students are asked:
"Developing Proof You used inductive reasoning to discover both the Linear Pair Conjecture and the Vertical Angles Conjecture. Are they related? If you accept the Linear Pair Conjecture as true, can you use deductive reasoning to show that the Vertical Angles Conjecture must be true?"
We then ask them to work with their cooperative group members to develop a paragraph proof explaining why the conjecture is true then check their reasoning with ours.
Later after discovering the Triangle Sum Conjecture students are asked:
"Developing Proof The investigation may have convinced you that the Triangle Sum Conjecture is true, but can you explain why it is true for every triangle?"
We then direct them to look back at the arrangement of the three angles they tore off and reassembled forming a line. We ask, "how is the resulting line related to the original triangle?" This is their leadin to creating a paragraph proof explaining why their conjecture fits with what they have already discovered/proved about parallel lines.
The inductive and deductive reasoning in DG4 continues with investigating, conjecturing, and explaining why, from Chapter 2 through Chapter 12. It isn't until the last chapter, when there may be students ready for van Hiele level 4 reasoning, that we introduce geometry as a formal mathematical system.
Again from Mathematician's Delight by English Mathematician W.W. Sawyer (19112008)
The Great Pyramid was built in 3900 B.C. by rules based on practical experience: Euclid's system did not appear until 3,600 years later. It is quite unfair to expect children to start studying geometry in the form that Euclid gave it. One cannot leap 3,600 years of human effort so lightly! The best way to learn geometry is to follow the road which the human race originally followed: Do things, make things, notice things, arrange things, and only then –reason about things.
Once deductive arguments began to sprout up around Ancient Greece, it still took awhile for the process to be accepted. From the first deductive arguments by Thales to Euclid's Elements, it took over 300 years. We owe it to our students to give them time to move up through the van Hiele levels so they come to understand the role proof plays in mathematics.
Contents
Chapter 1 Warm Up Puzzles
1.1 Color Network Puzzles
1.2 Square Route Puzzles
1.3 Container Puzzles
1.4 Sequential Movement Puzzles
1.5 Cryptarithm Puzzles
1.6 Robot Programming Puzzles
Chapter 2 The Game of Racetrack
2.1 Racetrack
2.2 Mathematical Connections: Vectors
Chapter 3 Movement Puzzles
3.1 Coin Puzzles
3.2 Sliding Block Puzzles
3.3 Queen Bee Puzzles
3.4 Mini Robot Puzzles
3.5 Mathematical Connections: Archimedean Tilings
Chapter 4 Tour Puzzles
4.1 Rook’s Tour Puzzles
4.2 King’s Tour Puzzles
4.3 Knight’s Tour Puzzles
4.4 Mathematical Connections: Euler Paths
Chapter 5 Magic Square Puzzles
5.1 Introduction to Magic Squares
5.2 3x3 Magic Squares
5.3 4x4 Magic Squares
5.4 5x5 Magic Squares
5.5 6x6 and 7x7 Magic Squares
5.6 8x8 Magic Squares
5.7 12x12 Magic Squares
Chapter 6 Sequential Reasoning and Algebra
6.1 Algebraic Magic Square Puzzles
6.2 Squaring the Square Puzzles
6.3 Number Chase Puzzles
Chapter 7 Sequential Reasoning and Geometry
7.1 Angle Chase Puzzles
Appendices
A.1 Racetracks
A.2 Playing Boards
A.3 Properties of Algebra
A.4 Properties of Geometry
A.5 Hints
A.6 Answers
A.7 Proofs
Sample Sequential Reasoning Puzzles
Color Network Puzzle
Each circle is to be colored Red (R), Yellow (Y), or Blue (B). However no two vertices of a triangle are permitted to be the same color. What must be the color of the circle marked with a question mark?
Tour Puzzles
A knowledge of chess is not necessary in order to solve these puzzles. A rook moves horizontally or vertically, but not diagonally. Fill in the missing numbers 1 49 for puzzle 14. For puzzle 20, fill in the missing numbers to generate a continuous path in each square.
Rook's Tour Puzzle 14 h Rook's Tour Puzzle 20
Mathematical Connections: Euler Paths
If all the edges of a network, called a graph in Graph Theory, can be covered without retracing any edge, then the graph has an Euler path. If all edges of a graph can be covered without repetition and you can return to the starting point, then the graph has an Euler circuit.
Magic Squares
A Magic Square is a square array of distinct integers such that the numbers in any row, column, or main diagonals, have the same sum (called the magic sum). The magic square below is a normal 4x4 magic square.
A magic square puzzle is an incomplete magic square. Complete the 5x5 magic square. You will need to calculate the magic sum first.
Magic Square Puzzle 36
Angle Chase Puzzles
Students rely on their recall of geometry properties (and can review them in Appendix 4), not their protractor, when solving Angle Chase puzzles. The puzzle below uses properties of parallels, vertical angles, isosceles triangle properties, and polygon sum properties. Find the measures of all the angles in the Angle Chase puzzle below.
Summary
This article describes four different sequential reasoning puzzles (and each puzzle has many variations). These puzzles, played as part of your school mathematics curriculum, make great classroom openers, end of period thought provokers, or for students to work on at home during school breaks. The puzzles in Smart Moves are wonderful for cooperative problem solving activities in the classroom. You can project a worksheet onto a screen as the class solves the puzzle together. I encourage parents to participate in these games and puzzles with the entire family. Sequential reasoning puzzles can provide a setting for children and grandparents to work on problem solving and mental exercises together. Try them. I think you’ll see the “reason” for including them in your lesson plans.
P. 8.  Robot Puzzle 1 instruction, change the last sentence
“Write a program with 12 or fewer steps” to “Write a program with 14 or fewer steps.”
P. 176.  RP Puzzle 1 answer should be:
F8 TR TR F2 TR F4 TL F4 TR TR F5 TL F3 TL
P. 179. – MR Puzzle 3 answer should be:
1 → 4 ↓ →
Copyright © Michael Serra 2011