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Ve c t o rThe Official Journal of the BC Association of Mathematics Teachers

Spring 2015 • Volume 56• Issue 1

ON THE COVER: Helena Kim, in grade 10 at Magee Secondary, drew this piece for a geometry project that sought to combine math and art (see La Haye and Naested’s Winter 2014 article). She was given many examples (mostly online) of art that had utilized geometrical objects and was requested to create something unique using geometical figures.

Ve c t o rThe Official Journal of the BC Association of Mathematics Teachers

Spring 2015 Volume 56

Issue 1

2014–2015 BCAMT EXECUTIVE

BCAMT EXECUTIVE

Ron Coleborn, President Math & Science Helping Teacher K-12 Burnaby School District [email protected]

Chris Becker, Past President Princess Margaret Secondary School [email protected]

Michael Pruner, Vice President Argyle Secondary School [email protected]

Colin McLellan, Secretary McNair Secondary School [email protected]

Debbie Loo, Treasurer Burnaby South Secondary School [email protected]

Brad Epp, Membership Chair South Kamloops Secondary School [email protected]

Colin McLellan, Listserv Manager McNair Secondary School

VECTOR EDITORS

Sean Chorney Simon Fraser University [email protected]

Peter Liljedahl Simon Fraser University [email protected]

NCTM REPRESENTATIVE

Marc Garneau Numeracy Helping Teacher K-12 Surrey School District [email protected]

ELEMENTARY REPRESENTATIVES

Deanna Brajcich Sooke School District [email protected]

Deanna Lightbody Instructional Services, District Teacher K-8 Langley School District [email protected]

Jennifer Barker Richmond School District [email protected]

Sandra Ball Numeracy Helping Teacher K-12 Surrey School District [email protected]

Selina Millar Numeracy Helping Teacher K-12 Surrey School District [email protected]

SECONDARY REPRESENTATIVES

Amos Lee Burnaby South Secondary School [email protected]

Robert Sidley Burnaby Mountain Secondary School [email protected]

Christine Younghusband Sunshine Coast School District [email protected]

INDEPENDENT SCHOOL REPRESENTATIVE

Richard DeMerchant St. Michaels University School [email protected]

NOTICE TO CONTRIBUTORSWe invite contributions to Vector from all members of the mathematics education community in British Columbia. We will give priority to suitable materials written by BC authors on BC curriculum items. In some instances, we may publish articles written by persons outside the province if the materials are of particular interest in BC.

Submit articles by email to the editors listed above. Authors should also include a short biographical statement of 40 words or less.

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The editors reserve the right to edit for clarity, brevity and grammar.

MEMBERSHIP ENQUIRIESIf you have questions membership status or have a change of address, please contact Brad Epp, Membership Chair: [email protected]

2014/15 MEMBERSHIP RATES $40 + GST (BCTF Member) $20 + GST Student (full time university only) $58.50 + GST Subscription (non-BCTF)

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Ve c t o r The Official Journal of the BCAssociation of Mathematics Teachers

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President’s Message

Math Websites

Problem Sets

BCAMT News

Mathematical Exploration

Pi and the Great Pyramid

By Blair Yochim

Research Report

Mathematical Habits of Mind in the New Curriculum

By Sandra Hughes

Issues in Math Education

Making Trigonometry More than SOHCAHTOA

By Dan Woelders

Implementation

Using Desmos as a Dynamic Interface to Explore Reciprocal Functions

By Chris Tsang

Book Review

The Math Olympian

Reviewed By Michele Roblin

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25 31

9 17

President’smessage

Vector • Spring 2015 6

Hi Everyone,

As we approach the latter months of the school year, the BCAMT executive is continuing to organize conferences, provide support for math education, and meet with teachers across the province. I am very appreciative of all the time volunteers provide towards these initiatives. The desire to support teachers and improve mathematics education is alive and well.

These days, the curriculum is at the forefront of many peoples’ minds. Writing teams will be meeting between February and April to revise the draft K–9 math curriculum and create a draft 10–12 math curriculum. Along with

resources, professional development for teachers, and assessment, curriculum is a key component to a quality education program. I encourage everyone to be active in the upcoming review process. As new drafts become available, I will be sure to inform members through the BCAMT listserv. Implementation dates are still unknown but speculation suggests the 10–12 curriculum will be ready for September 2017 (and K–9 perhaps the year before).

The teaching and learning of ‘basic facts’ in K-7 mathematics education is often a hot topic of debate. As a result, the BCAMT is releasing a pamphlet on this subject to help teachers and parents with the discussion. It will be available on our website soon, and we hope you find it very useful. Please note that the pamphlet on the website will contain much more information than the printed version.

For the past three years I have had the opportunity to work closely with many K-12 teachers. In so doing, I have seen great benefits in emphasizing what students ‘can do’ versus what they ‘cannot do’. Examining what students ‘can do’, helps teachers create learning opportunities, provides feedback that leads to understanding and instills confidence in students. The resulting growth mindset for students is very powerful.

Views on mathematics education can be very polarized. Determining the balance between students having time to explore unfamiliar problems and students practicing ‘standard content’ is not an easy task. Teaching is complex. One can generally find teacher stories and research articles that both promote and dismiss various claims. Personally, I try to keep an open mind and as much as possible, look very closely at information provided from those with both K-12 teaching experience and years of teacher observations in classroom settings. Jo Boaler and Peter Liljedahl are two such people that come to mind.

Despite some strong opposing opinions on math education, I often find there is common ground. In the end, we want students to enjoy math, be good at problem solving, be able to communicate their thoughts well and have a strong base of knowledge. To accomplish this


goal, a great deal of effort has been spent over the years discussing items such as scope and sequence of math topics, and different ways to introduce math concepts. Recently, however, the discourse has begun to include a lot of conversation on items such as learning strategies, and methods for developing a ‘thinking classroom’. I believe both types of conversations are necessary, and I am excited to see teachers trying to develop their ‘teacher craft’ in both areas.

Finally, I would like to remind you that the BCAMT is hosting the 2015 Northwest Mathematics Conference in Whistler on October 22-24. Many outstanding speakers such as Ron Lancaster, Simon Singh, Robert Kaplinsky, Egan Chernoff and Carole Fullerton will be presenting. To further pique your interest, the conference committee has secured an incredible price for the two featured hotels: The Fairmont Chateau Whistler and The Westin Resort and Spa. At both hotels, rooms will be $139/night + taxes. However, I recommend you book early before the room block is filled. For more details on the conference, check out www.bcamt.ca/nw2015.

As always, please contact us with your ideas and concerns. Thank you and enjoy the rest of your school year. I hope you continue to build good memories.

Ron Coleborn

BCAMT President

Mathematicalexploration


Pi and the Great PyramidBy Blair YochimBlair Yochim passionately specializes in the middle and high school mathematic and science subjects and currently teaches high school special needs students at Mediated Learning Academy in Coquitlam. He received “Teacher of the Year 2006/2007 Awards” from a large local tutoring referral agency and for more than a decade was very active in the BC Government/Science World program called “Scientists and Innovators in the Schools (SIS)” by presenting to over 7,000 students throughout BC. His previous career was a professional (electrical) engineer. He is the inventor of a patent on data encryption and is interested in puzzles, magic, astronomy, space travel, philosophy, vegetarianism, social, environmental and animal issues.

The Great Pyramid of Giza in Cairo, Egypt (also called Cheops Pyramid or Pyramid of Khufu) was finished about 2560BC, over 4500 years ago. As the tallest human-made structure for over 3800 years, it is one of the wonders of the ancient world and remains,

to this day, a thought-provoking enigma. Throughout history, it has attracted the attention of archaeologists because of the immensity of the ancient building, its construction and historical significance as well as numerologists who perhaps make false claims.

The value of the mathematical constant pi (represented with the Greek letter π) seems to have been designed into the Great Pyramid to a value of about 3.1419. This value of π was not rediscovered with such accuracy until about 2000 years later. So how did the Egyptians know or use an approximate value of π?! Well, in fact the Egyptians may have incorporated π inadvertently, but not accidentally, into their Great Pyramid with “great” precision without even an awareness of the number. The explanation does not involve numerology or questionable pseudo-scientific Pyramidology but instead involves scientific logic and high school mathematics. Here’s how...

First what is π?

A refresher course: The distance through the middle of a circle is called the diameter. The distance or perimeter around a circle is called the circumference. The circumference around the circle is about 3 times longer than the diameter across the circle, no matter what size of circle is being used. This number is called π and is actually 3.14159265... (considering the length limitations of Vector articles, I’ve chosen not to write the whole thing out) and goes on forever as an irrational transcendental number. This number π is probably one of the simplest to understand mathematical constants, the most used, and probably the most studied.


Measurements of the Great Pyramid :

There have been many publications of differing measurements of the Great Pyramid throughout the last few hundred years by various historical investigators. It is difficult now to get precise measurements of the original Great Pyramid because the outer white limestone layer and the top capstone of the Great Pyramid were removed for other construction projects thousands of years ago and other building blocks have been damaged by factors such as looting, erosion and dynamite (yes, an English explorer Richard Vyse in the 1800s used dynamite to search for entrances to the Great Pyramid and wasn’t the only one to do so.) Also the measurements depend upon the measurement technology used and the professionalism of the archaeology applied. For example, the unit of length used (the cubit) has not been consistent throughout history. So the measurements vary slightly according to the sources. I have, however, tried to consider these variations and be as accurate as possible in this article.

The Egyptians used a unit of length measurement called a royal cubit which is the distance from the elbow to the extended middle finger. One royal cubit is about half a metre or more precisely it is 52.37 cm. Historically, there have been some other cubit units used, but they do not apply to the Great Pyramid.

The capstone white limestone blocks are missing from the top peak of the Great Pyramid. Originally the height was about 146.64 m or 280 cubits high. (One source claims 146.59 m.)

The lengths of the four sides of the base of the Great Pyramid vary from 230.25 m to 230.45 m or about 440 cubits long. (230.25 m, 230.36 m, 230.39 m, 230.45 m). Notice that they are all surprisingly close to about 230.35 ± .10 m, which is within 10 cm over a distance of 230 m. Remarkable!

If you take half of the perimeter of the Great Pyramid’s base

(230.25+230.36+230.39+230.45=921.45m 921.45m ÷ 2 = 460.725m)

and divide it by its height, the result is very close to π.

(460.725m ÷ 146.64m = 3.1419 π=3.14159...)

CIRCUMFERENCE ~3.14 X DIAMETER


(Or alternatively if you use the cubit estimate measurements: 440 cubits x 2 ÷ 280 cubits = 22/7 = 3.1429)

There is an Egyptian “Rhind Papyrus” written around 1850BC that states π as having a value of (16/9)2 = 3.16049 ... So how did the Egyptians design their Great Pyramid with a better approximation of π about 700 years earlier? Such accuracy with π wasn’t achieved again until Archimedes during the 3rd century BC to obtain π=3.14163.

An explanation of how the Egyptians built their Great Pyramid using π without knowing π:

The key to the mystery of how the Egyptians incorporated π was derived by thinking of another mystery of the Great Pyramid. How were they able to make the lengths of all four of the sides of the Great Pyramid so precise (within 10 cm) to each other over a distance of 230 m? They could not have used a rope to measure each side of the Great Pyramid to that accuracy because a rope will stretch depending upon the tension and the temperature, especially one 230 m long. They could not have used a long metal rod because a rod will also stretch depending on an inconsistent temperature. Besides, they did not have the means to manufacture a metal rod that long. Another possibility might have been a chain, but a chain would have had flexible links. Other modern surveying and measuring technologies did not exist at that time.

Since the four sides of the Great Pyramid are so similar in length, researchers started to wonder how they did that. I think some Japanese researchers were the first to realize the answer a decade ago or so. The only technology that existed that would give the Egyptians the precision they needed to match all the four side lengths was ... wait for it ... a trundle wheel. If π was involved, it insinuated that a wheel, which has π built-in, was somehow involved. A trundle wheel is simply a wheel that is rolled a number of times in order to measure distance. It turns out (pun intended) that a distance is surprisingly very accurately measured with a trundle wheel (small sand or gravel particles are negligible as compared with the size of the curvature of a large trundle wheel when it runs over such obstacles). It is a technology that is commonly used even today on sports fields and by surveyors.

It is thought that the Egyptians made a trundle wheel carved out of rock with a diameter of exactly 1 cubit (but I don’t think archaeologists have found this trundle wheel rock yet to prove this theory). Then on flat ground they rolled the trundle wheel rock exactly 140 times around for each of the four sides of the Great Pyramid. Therefore, the wheel was rolled a total of 280 times around from one corner of the Great Pyramid to the farthest corner on the other side. This gave them precise locations of the corners of the Great Pyramid. From there they started to pile the massive blocks to build the pyramid until they obtained a height of 280 cubits (using the same number 280 as the number of trundle wheel turns from opposite corners of the base). They couldn’t easily roll the trundle wheel straight up, so instead they counted the same number of cubits to obtain the height.

So they counted 280 turns of the trundle wheel rock from one corner to the opposite corner and measured 280 cubits high from the ground to the peak. Because they used a trundle wheel, which has the value of π built into it with the relationship between the circumference and the diameter, the value of π was automatically built into the Great Pyramid without the Egyptians knowing the value of π!


But as a mathematician, I must report more details to this story:

Even with a base with four exact measured sides, the Egyptians could have made a base in the form of a rhombus, not a square with 90° corner angles. According to one source, the difference in the distances between the base’s diagonal opposite corners is within 17 cm meaning they indeed obtained corner angles very close to 90°. Another source states that the corner angles are 90° 3’ 2”, 89° 59’ 58”, 89° 56’ 27”, 90° 0’ 33” which is an average error of less than 0.03° or 2’ (reminder: 1° degree of an angle has 60’ minutes and each of those minutes has 60” seconds). So how did the Egyptians obtain near-perfect 90° angles on the corners? The obvious answer would be that they measured the diagonals to be matching lengths or used the Pythagorean Theorem, which if carefully measured on the base perhaps with a trial & error method, produced 90° angles on the corners.

In about 1800BC the Babylonians (located in today’s location of Iraq) produced the “Plimpton 322” clay tablet that gave some “Pythagorean Triplet” sets of whole numbers (right-angled triangles with sides that are integers like the 3-4-5 triangle). This was well in advance of Pythagoras in 500BC who now bears his name to the famous formula a2 + b2 = c2. But evidence has been claimed to have been found within the Great Pyramid that the Egyptians may have known about some Pythagorean triangles. According to one source they called such triangles “holy triangles”. It is questionable though if the Egyptians at the time knew some Pythagorean Triplets because of the lack of accepted written historical evidence.

Suppose that the Great Pyramid designers used a Pythagorean Triplet. One question might be which one would they have used? Could they have used the 140 turns of each side of the Pyramid to obtain a diagonal with an integer value?

But wait! If you apply the Pythagorean Theorem, you obtain 197.99 turns for the diagonal hypotenuse, not an integer value, but VERY close. Perhaps then, the Egyptians didn’t know about Pythagorean Triplets, but instead knew what I call “Pythagorean Isosceles Triplets” which are close to being integers. Upon reflection, of course there never will be exact integer Pythagorean Isosceles Triplets because the hypotenuse will always be √2 times the triangle side and √2 is irrational.

This made me wonder what other Pythagorean Isosceles Triplets exist, so I wrote a spreadsheet program that lists how close the hypotenuse of each set of potential triplets was from an integer. I found more triplets. Here are the closest Pythagorean Isosceles Triplets:


Side (turns)

Side (turns)

Hypotenuse (turns)

Error (turns)

12 12 16.971 0.029

29 29 41.012 0.012

41 41 57.983 0.017

70 70 98.995 0.005

99 99 140.007 0.007

111 111 156.978 0.012

128 128 181.019 0.019

140 140 197.990 0.010

169 169 239.002 0.002

There is a pattern with the triplets. For example, if the 140-140-198 triplet is close, so will the triplet which is half the size, if the original hypotenuse is an even number. A length double the size will also be close. So 70-70-99 is another good possibility. It seems that the Great Pyramid designers had some options on the size of their pyramid. (Also, notice that the Egyptians could’ve used a larger triplet combination like 169-169-239, which would have made the Great Pyramid even bigger and probably not feasible to build because each dimension would be 20% larger with a 75% increase in the quantity of material needed to build it. Not to say the current size seems at all “feasible” to build.)

Using the 70-70-99 option which has the smallest error in the range used, perhaps they used this design:

Interestingly, notice that 140/99 = 1.4141... which is close to √2 = 1.4142 so were the Egyptians aware of this number? At least one numerologist has made this discovery that the square root of two is also built-in to the Great Pyramid, but this is the case with all of the Pythagorean Isosceles Triplets, and of course mathematicians will know that these patterns will naturally always be built-in with a square based pyramid.

But if the Egyptians used the above design using the 70-70-99 triplet, then they would’ve used exactly 99 turns of their trundle wheel for the diagonal which would’ve made the expected 90° right angle inaccurate because of the error in Pythagorean Isosceles Triplets. How close were they with the ~90° angle? Using the Cosine Law: c2 = a2 + b2 - 2abCos C


992 = 702 + 702 – 2 x 70 x 70 x Cos Cwe discover that the ~90° is actually C = 90.00585° which is slightly more than 90°. This would mean that the pyramid would be slightly indented half way on each side towards the middle of the pyramid because the angle in the middle of the pyramid would be double 90.00585° which is 180.0117°. This indent actually exists. Under the right particular lighting conditions, shadows show that the Great Pyramid is indeed indented in the middle on each side and apparently is the only pyramid to have this characteristic. So the above 70-70-99 design seems to confirm the hypothesis of the construction of the Great Pyramid.

According to one source, the actual indent on each side averages about 59cm (middle distances through centre reported to be 229.19m & 229.14m). However, one cannot easily directly measure the distance through the centre so this indent “measurement” might be questionable. How much would this expected indent be with the above 70-70-99 triplet? I calculate the indent on each side would be only about 1.2cm, which is much less than the reported 59cm.

Using a triangle formed by the indent with the error angle of 0.00585° and the hypotenuse length of 70 turns:

sin(0.00585°)=(indent) / (70 turns) (indent = 0.00715 turns)

0.00715 turns x π cubits / turn = 0.0225 cubits

0.0225 cubits x 52.37 cm / turn = 1.2 cm.


The above 70-70-99 design seems to have some merit because it explains how π was incorporated within the Great Pyramid with such precision without the Egyptians having to know the value of π or the Pythagorean Theorem and provides a suggested theory as to why the Great Pyramid has indented sides, which continues to be a mystery.

So, here was a logical mathematical explanation of how the Egyptians involved π (& √2) with precision in their Great Pyramid without any necessary knowledge of the digits of π or √2. Perhaps someday their 1 cubit diameter trundle wheel tool will be discovered which should support this hypothesis.

This has been an interesting investigation into the mysteries of the Great Pyramid: How did the Egyptians build the Great Pyramid with such precisely equal base sides with limited technologies? How did they involve a surprisingly accurate value of π and √2 in their design thousands of years before such precision was obtained? Why are the sides of the Great Pyramid indented? Using scientific logic and some mathematics, these puzzling questions which have existed for many years have been addressed with plausible theories.

It is historically interesting that the Egyptians possibly may have known of these approximate Pythagorean Isosceles Triplets, perhaps before they discovered exacting integer Pythagorean Triplets such as 3-4-5. But I would be interested in knowing if any of you have heard of the Pythagorean Isosceles Triplets elsewhere and any other applications of them. You may want to consider while you are teaching the Pythagorean Theorem to your own middle or high school classrooms to challenge your students to search for such Pythagorean Isosceles Triplets to find the closest set. This may be a means to show that the hypotenuse of an isosceles triangle will always be an irrational √2 multiple of the sides.

V

BC Association of Mathematics TeachersPresents...

10

-5

2

12

Ron Lancaster ~ Egan Chernoff ~ Simon SinghRon Lancaster ~ Egan Chernoff ~ Simon Singh

Fawn Nguyen Andrew Stadel

Robert Kaplinsky Geoff Krall

Carole Fullerton Janice Novakowski

Elham Kazemi Allison Hintz

Peter Liljedahl Kim Sutton

and many more!Whistler, BC

October 22–24, 2015

Issues in matheducation


Making Trigonometry More than SOHCAHTOABy Dan WoeldersDaniel Woelders is a math teacher and department head at Pacific Academy High School in Surrey, BC. Over the past 5 years, he has taught in the Abbotsford and Langley school districts in both a math and science classroom. He is currently in the process of completing his Masters of Education in Secondary Mathematics at Simon Fraser University and is an active blogger on the subject of mathematics and math education.

Every year that I have taught high school mathematics I have had the opportunity to introduce students to trigonometry. This is an opportunity show that trigonometry is not only a tool to solve triangles but also a different way of describing the slope of a

line. Despite my best efforts, students seem to complete the course and carry forth a limited understanding about trigonometric ratios. If asked what they understand about trigonometry they consistently answer, “SOHCAHTOA” and when prompted to explain what that means they reply, “Sine is opposite over adjacent”. No doubt this is a description of the sine ratio at a very basic level but in no means does it touch at the very heart of its purpose. Rarely do you hear a student describe sine as a relationship between an angle and the ratio of its opposite side to the adjacent side of any right triangle. While the argument could be made that they knew it but couldn’t explain it, this hardly seems likely given my follow up with students following the unit. Some of the following questions have been posed to students previously and were intended to get at their conceptual understanding of trigonometric ratios.

Could there be more than three ratios? Explain your reasoning.

Some of the more comedic responses include “because there is only three buttons on our calculator” or “yes, there are more than three ratios, sin-1, cos-1, and tan-1”. However, the most common response was there was only three ratios because there are only three sides or perhaps that there were only three angles. These are true statements in a roundabout way but they miss the idea that three sides can only be paired in three ways. If they saw the ratios in this way they might have some insight as to why there can actually be six ratios if you were to flip all three ratios upside down. Students might even argue why the sine ratio needs to be opposite/hypotenuse instead of hypotenuse/opposite. This may lead to the discovery of the graph of each of these functions.


Does the sine ratio have a maximum value? If so, what is it? Explain your reasoning.

Responses to this question varied in the past. Often students responded incorrectly that the sine ratio maxed out at 90 degrees because you can’t have a triangle with two 90 degree angles. Part of this, of course, is true, but students misunderstood the nature of the increasing sine ratio and often mistook it as a positive linear correlation. They assumed that the ratio increases as the angle increases. This becomes more evident in the next question.

If the cosine ratio of 60 degrees is 0.5, estimate the cosine ratio of 30 degrees? Describe how you arrived at your answer (this is done without the use of a calculator).

Not surprisingly most students tend to answer that the cosine ratio of 30 degrees would be 0.25. The same reason that students misunderstood maximum values for sine ratio resurfaces. Once again the assumption is that trigonometric ratios and angles form some sort of linear relation. This is potentially hazardous as student begin to interpret the periodic motion of the sine and cosine functions.

Estimate the angle that would have a tangent ratio of 2? Describe how you arrived at your answer.

The response to this question varied drastically, some relatively accurate, others dipped below 45 degrees. The most frequent inaccurate response was 2 degrees, suggesting that either the student had no tools to even venture a guess or they truly believe that ratio and angle were equivalent. The misunderstandings extend beyond the evidence that I have accumulated in post-unit surveys. It was not uncommon to see solutions to trigonometric statements that make you question whether students really understand anything about the concepts involved in trigonometry. The sin 60 = x/6 was often answered by stating the value of x in degrees (as opposed to a length unit).

This has bothered me for years, but I often dismissed it as the student’s fault. Year after year I taught trigonometry the same way, and year after year their grasp of the concepts was dismal. “Students are slow”, I excused. Now I wonder who the slow one is. The approach to the unit was a definition-based approach, filled with equations and algorithms, and the students were able to do exactly that. They could solve a triangle and do the algebra. The following is an example of a common approach I took to introducing trigonometry.

Now, this is not to say it is a terrible approach. It necessitates the use of trigonometry and would maybe accompany a real life example really well. However, this introduction was often followed by stating one of the three ratios. Right away the students experience trigonometry from a formula-based approach. It also begins with a foreign term like “cosine” and quickly defines it as opposite/adjacent. To the students cosine is not a ratio, it is a formula for solving

1410

x

14

30x

In the past we have been able to solve this triangle by using the Pythagoras equation. However, the triangle right does not have two sides that we need to find the third side. Instead, it gives us the angle. Furthermore, using Pythagoras doesn’t allow us to find the angles of the triangle on the left. Now, we can use trigonometry to find this information.


triangles. The first lesson consisted of getting them to memorize that tangent was opposite/adjacent, cosine was adjacent/hypotenuse, and sine was opposite/hypotenuse. I would then hand them a worksheet where they spent 30 minutes labelling triangles with the letters O, A, H and theta (another strange adjustment in their understanding of variables that I rarely explained). Historically the second lesson of the trigonometry unit really excited me. It was the lesson where I would get to tell them what that button on their calculator labelled “sin” did. “If you hit ‘sin’ followed by any angle you want, it will tell you the opposite/adjacent”, I would explain. In my attempt to teach trigonometry I deprived students the opportunity to experience the world of mathematics. In the book, “Developing Understanding of Geometry” (Sinclair & Pimm, 2012)1 , the authors identify four big ideas that should drive our approach to geometry. Using these big ideas, I will suggest the problems that emerge in the lesson described above and introduce a new approach that I have recently experimented with and the results of that approach.

Big Idea 1: Working with diagrams is central to geometric thinking.

While students certainly worked on questions that included diagrams, they rarely had to produce one themselves. They were not required to describe triangles or ratios, and thus, the language was only necessary because they needed to know which number to plug into the formula.

Big Idea 2: Geometry is about working with variance and invariance, despite appearing to be about theorems.

What I missed in my lessons is the opportunity to see the degree of invariance involved in right angle trigonometry. When a triangle gets bigger, the sides change. When the angle changes, the sides change. There is a lot of change occurring in the triangle. What is special are the things that do not change, the invariance. However, instead of allowing my students to experience variance and search for invariance, I told them what the invariance was. It wasn’t even special to them. This is the danger of a lecture-based approach to the curriculum, students do not experience the treasure found in invariance. To the student everything is invariance. Students learn the invariance, not by discovery but by memorization. It is not surprising that students cannot remember a simple thing like factoring or area of a triangle; they never construct it and therefore rebuilding those ideas becomes impossible. The best they can hope for is a mnemonic device to trigger their memory (examples, “FOIL”, “SOHCAHTOA”, “adds to b, multiplies to c”).

Big Idea 3: Working with and on definitions is central to geometry.

Students in the lessons above were never required to work with a definition of sine. They utilized sine ratio, but they didn’t work with or on it. Compare it to being given a hammer, told that it is for driving nails and then asked to drive nails for a few hours. Chances are that you would get really good at driving nails but you miss out on all the other things that hammer could do. This is how students use trigonometric ratios. Show them how it works and then let them do as the teacher does for a few hours. For years I started with a definition, often from a textbook, and left no room for students to contribute to the building of the definition. We were working from a definition, not towards it.

Big Idea 4: A written proof is the endpoint of the process of proving.

This is perhaps a missed opportunity teachers often make in the classroom. You would never exempt an English student from writing a conclusion to their essay or a chemistry student from writing their experimental conclusion. However, it is rare that a math student will get the

1Sinclair, N., & Pimm, D. (2012). Developing essential understanding of geometry for teaching mathematics in grades 9-12. Reston, VA: National Council of Teachers of Mathematics.


opportunity to search, discover, conjecture, test and then formalize/generalize their conjecture as a written proof. This rarely occurred in my classroom let alone in my trigonometry lessons. Students believed that the ratios worked for all triangles because I told them that they work in all triangles, end of story, no evidence needed.

So, after years of teaching trigonometry I realized that I was not effectively getting students to grasp the concept of ratio. Despite repetitive practice with algorithms they seem to not understand what they were calculating. I had a sense that if any of my unit plans needed to change it would be this one. Thus, I tossed away my lessons and started from scratch.

One new approach I had to teaching trigonometry was piggy-backing it off of a topic that I had felt was rather concrete. The month previous we had dedicated our time to linear functions. More specifically we looked at the arithmetic nature of linear functions and the meaning of slope. We spent days shifting, describing, drawing, and calculating slope. My students experienced slope. If I were to give them a particular ratio, most could picture it in their mind and sketch it out with relative precision. I felt this was an opportune time to begin asking them if they could do the same with angle. And so, I gave them an angle and asked them to visualize it and then draw it as accurately as possible. I realized that they probably had an even greater ability to visualize angles than they did slopes and thus, the first trigonometric ratio, tangent, had already been introduced and understood. I still wanted them to play around with the relationship between angle and slope ratio so I introduced it by diagram. My initial plan was to have them create the slope and angle but without Geometer’s Sketchpad I realized this would be impossible. My options were to have them graph it on paper or have them play with it on Sketchpad Explorer. The decision was really between the benefits of construction versus the benefits of dynamic geometry and variance. Emphasizing Big Idea 2, I chose the latter and constructed this sketch for them to use in Sketchpad Explorer.

There were three things I hoped to accomplish by introducing students to this exploration. First, I hoped that students would recognize that steepness could be described in two ways. This related to Big Idea 3 and more specifically addresses Essential Understanding 3a, “Geometric objects can have different definitions. Some are better than others, and their worth depends


both on context and values”. Here steepness could be described in two ways: slope described the steepness relative to the horizontal axis and angle described it relative to any other line. Both valuable concepts and both worthy of discussion in various contexts. Second, I hoped to show them that we were working with triangles all along. Rise and run create the legs of a right triangle, and it does not matter where you place those legs or how big they are, the slope remains constant. In other words, I wanted them to see the invariance amongst the variance. Third, I wanted to develop the invariance between angle and slope; no matter where you placed B, slope was unique to that angle. This invariance allows us to use slope as a tool to derive angle and make predictions, as was the case in question 7. Students began to make connections that I had hoped. I overheard students use phrases like, “so 35.15°= 0.70”. I knew once that other ratios were introduced students would need to correct this logic and notation.

The second part of my lesson was targeted to address Big Idea 2 and 3 and to necessitate a change of language. Below is the sketch that students worked through in small groups.

The conversation that ensued was encouraging. While there were students who needed a lot of direction on where to look, many dismissed the “slope” measurement that GSP calculated and instead focused on “RISE/RUN”, noting that it stayed the same until the angle changed. Students also began to argue over the term “rise” and “run”, suggesting that it was no longer a “rise” or a “run”. While there was no consensus on what the names should be, one student did offer the terms “opposite” and “beside”. This was a good start and I did not address it until I felt the correct language was necessary.


This, of course, gives rise to the third side of the triangle that some students had already begun to address. While there were a few students who already knew the name of the third side, none of them had thought about using it in a ratio like we did with opposite and adjacent. For that purpose I created this lesson.

Once again I wanted to draw students’ attention to the value of the ratio and how that could help determine the approximate angle. Questions 4 and 5 readdress the predictable nature of the ratio and also helps students narrow their scope to look at one side at time.

If the instructions were followed correctly students should be able to determine that when you keep adjacent constant and increase opposite and hypotenuse, the angle increases. Perhaps more surprising for students is that when you keep opposite constant and increase adjacent and hypotenuse, the angle actually decreases. This is something students rarely have the opportunity to see, which is why most students couldn’t tell you why the cosine ratio increases as the angle decreases. This sketch was followed by more instructions that would explicitly have them determine the ratios.


The use of Geometer’s Sketchpad throughout these activities gave students access to the nature of the ratios and allowed them to spend more time to get a feel for how the ratios are associated with the angles. These tasks were followed by a bit more terminology as students felt it necessary to give names to these ratios so that they could remember what each one described. However, before I shared “cosine” and “sine” I had students explain what the ratios told them about the triangle. Students commented on the idea that any ratio could be used if you knew two of the sides and employed the Pythagoras theorem. In a way, they suggested, all of the ratios are part of a family. They also commented that the opposite/hypotenuse ratio went from 0 to 1 and depending on the value you could estimate the angle. Meanwhile, adjacent/hypotenuse ranged from 1 to 0 and could also be used to determine the angle. I suggested that perhaps it would be easier to use a table that told us all these values instead of trying to memorize them. And that is precisely what we did. For three days students used nothing but trigonometry ratio tables to determine angle, and a calculator was later introduced as a device that could store these tables for us.

As we came to the end of our unit, I have seen some students slip into the routine of “SOHCAHTOA”, but there has been a number of difference in the students’ approach to solving trigonometric problems. One difference I observed was that students understood the range of values for each of the ratios. They understand why cosine and sine have a maximum but tangent does not. The second difference was that students began looking for patterns. They manipulate the triangles and look for clues that will help them determine future scenarios (I even saw a student start his own table of ratios). Thirdly, trigonometry became more geometrical and less algebraic. I was not interested in whether students could solve a ‘fill-in-the-blank’ algorithm. My questions were geared to estimation and patterning. Finally, students seem to enjoy a personalized math experience. Trigonometry became something different than the rest of the curriculum. They controlled the triangle, they controlled the values, and they controlled the experience. While there were adaptations I would make, this new approach to trigonometry contributes great value to not only the students’ understanding of trigonometric ratios but more importantly an understanding about the nature of mathematics.

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“Here is my fractions homework. I didn’t get it, so I only did 1/7 of it.”

Researchreport

Vector•Spring2015 25

Mathematical Habits of Mind in the New CurriculumBy Sandra HughesSandra currently teaches mathematics and science. She is also working toward her Masters in Mathematics Education at Simon Fraser University. Her previous teaching experiences range from the intermediate grades to secondary mathematics in both the public and private sectors.

Under the ‘What’s New?’ heading in the draft for the new BC mathematics curriculum, we read “a focus on developing mathematical habits of mind and encouraging students to wonder how mathematicians think and work” (BC Ministry of Education, 2013).

This statement refers to habits of mind. But what are mathematical habits of mind and why are they now important to public school math education? To understand the impetus that drives this necessity of change, it is important to briefly examine current mathematics practices and their effectiveness, and then consider the plausibility of teaching mathematical thinking and what that might look like.

Simply put, mathematical habits of mind are the thinking processes that mathematicians employ as they solve or create mathematics. They are mathematical behaviours of thought. Mathematical habits of mind are based on the general habits of mind work of Costa and Kallick (1982) who state “the critical attribute of intelligent human beings is not only having information, but also knowing how to act on it. A ‘habit of mind’ means having a disposition toward behaving intelligently when confronted with problems, the answers to which are not immediately known” (p. 2). From the list of 16 general habits of mind such as persistence, thinking flexibly and finding humour, the Educational Development Center (EDC) mathematics curriculum development team of Cuoco, Goldenberg and Mark developed mathematical habits of mind in 1992 in response to reform practices of the time that appeared to be more traditional than progressive.

It is in this context that mathematical habits of mind must be viewed. As a curriculum topic it provides fuel for both traditional and reform arguments. Mathematics education today can be loosely classified as either traditional or reform. For ease of discussion and reference in this paper, traditional mathematics describes a direct instruction, teacher-driven classroom where students take notes and complete questions based on lecture examples. Reform math is student-centered and inquiry-based and emphasizes conceptual understanding. It can have descriptors such as problem-based learning, experiential learning or whole math. The descriptors of traditional and reform apply to math education in many countries.


The results of the 2012 Programme for International Student Assessment (PISA) by the Organization for Economic Co-operation and Development (OECD) is one yardstick that has both Canadians and Americans concerned. In mathematics, Canadians fell out of the top ten to thirteenth position, while the Americans placed much further below. For one measurement Canada claimed 16.4% and the USA had 8.8% share of the top performers in mathematics compared to the top country, China, at 55.4%. These indicators, combined with such statements as in a recent Globe and Mail article that claims research shows “math matters more than reading...both for academic success and future job prospects” continue to focus attention on the problems in math education. The OECD Secretary-General Angel Gurria states on the welcoming page of PISA, “More and more countries are looking beyond their own borders for evidence of the most successful and efficient policies and practices. Indeed, in a global economy, success is no longer measured against national standards alone...”(Anderssen, 2014, p. 2) (OECD, 2013, p.1,5)

With this in mind, reform math supporters in North America are looking outward. While traditional supporters agree change is required in math education, they are not convinced that the standards and practices of other countries can be applied here. They view change from topic reorganization and more practice (Cuoco, Goldenberg & Mark, 2010). To contrast, the Dutch have changed their approach to curriculum, and the Realistic Mathematics Education has become their standard. Its goals are of fostering understanding, reasoning and exposition. With this new curriculum, students “puzzle out the steps without being given answers or formulas (in) an environment that emphasizes process rather than product” (Case, 2005, p. 379). As a result, they have moved up to tenth position on the 2012 PISA. Like the Americans, and unlike the Canadians, the Dutch change is organized by a national unit out of the Freudenthal Institute. Currently in the USA, the Common Core State Standards (CCSS) are being implemented nationwide while in Canada every province is responsible for its own curriculum development. But if the Dutch example is promising it drives home the need for consistency. To implement this approach to curriculum and see results has taken 30 years. If the math wars continue to divide educators in North America and cause focus to change back and forth between traditional and reform, it will be difficult to progress forward.

In one of his seminal articles Alan Schoenfeld (1987) discusses why work in metacognition, or thinking about your own thinking, is important to math education. One research study he conducted contrasted the problem solving approach of high school students to a mathematician. The graphs revealed that the “mathematician spent the vast majority of his time thinking rather than doing ...” (p. 194). In addition, the mathematician was able to solve the problem and many students could not, and they knew a lot more geometry than the mathematician. This point is also a concern for many educators, mathematicians and scientists: students are not able to use the math they are ‘learning’ in school. In addition, it appears that unproductive mathematical habits of mind are developing among students. For example, persistence is a mathematical habits of mind that is essential for students when faced with a complex problem. However, many students expect that all mathematical problems can be solved in 10 minutes or less (Schoenfeld, 1988, 2009). It is argued that developing productive mathematical habits of mind are important to all students from those that will engage in advanced mathematical learning to those that will not pursue a future involving mathematics (King, 2013; Cuoco, 1996, 2010).

Mathematical habits of mind is a ‘new’ feature in the BC math curriculum but not necessarily a new idea. John Dewey wrote about ‘habits of thought’ in 1933 (in Costa & Kallick, 2008). In a 1924 publication of Mathematics Teacher, Elsie Johnson wrote an article entitled ‘Teaching Pupils the Conscious Use of a Technique of Thinking.” In it she argued that teachers were obligated to train their students to think better and if students were trained in the ‘new method’ they would be more creative when problem solving and would recognize incorrect results


more quickly. She was concerned when teachers gave students the technique for solving a problem because the techniques became purely mechanical and they did not represent all the stages involved in solving a problem. In other words, the complete act of thinking was not represented.

More recently, in Focus in High School Mathematics: Reasoning and Sense Making, the authors discuss mathematical reasoning habits which they define as “a productive way of thinking that becomes common in the processes of mathematical inquiry and sense making” (NCTM, 2009, p. 9). The CCSS uses ‘mathematical practices’ to describe how a student should be able to think and act mathematically. Still other authors link mathematical habits of mind to ‘flexible thinking’ (Weiss, 2012). One support website charts the different phrasing used to describe mathematical practices, mathematical habits of mind and the essential elements of project-based learning to show the connections and similarities between them (Kiem, 2013). The point is that these examples all focus on the thinking processes in math. Reformers are trying to shift the current sole emphasis from the products and results of mathematical work and convey that how mathematicians think is equally important as their results.

What would mathematical habits of mind look like in a math classroom? In the article “Folding Corners of the Habits of Mind,” Peter Wiles (2013) describes a reasoning approach to origami that gives students, and in this article the students are preservice teachers, the opportunity to explore geometrical habits of mind. In a very simple activity called ‘Turned Up Point’ (TUP), a corner of a square of paper is folded up to a decided point in the square. The teachers then examined the shape created by the fold and noted whether it was a triangle or a quadrilateral. The ultimate goal was to predict the shape of the flap depending on the choice of point location. This exploratory activity had a low floor for both paper folding and mathematical knowledge. However, it provided access to four key geometrical habits of mind: 1) Reasoning with relationships 2) Generalizing geometric ideas 3) Investigating invariants and 4) Balancing explorations and reflections. For example, the teachers had to decide when to stop collecting data and make conjectures, which required that they reflect on their data. This is the fourth geometrical habits of mind. Ultimately, this activity was a study of invariance because the teachers had to identify what stayed the same in order to make a conjecture about the location of the point.

This short article and simple activity were surprisingly rich in ideas. Once the teachers discovered the solution to TUP, the exploration was not finished because they were encouraged to consider how the solution generalized. That simple request wasn’t easy for the teachers as there are many layers to generalizing. For example, they had to decide what features of the problem itself would generalize as well as what parts of the solution would generalize to a new figure, such as a rectangle. Would there be a circular solution as well? This task focused on geometrical habits of mind but also allowed for “exploration of important geometric ideas [and] how to test ideas, make conjectures, pose new questions and feel the thrill of uncovering relationships”(p. 213). Mathematical content was also being addressed.

Mathematical habits of mind can and should be fostered in elementary school. In “An algebraic-habits-of-mind perspective on elementary school” (Goldenberg, 2010) the author provide examples of how young children think abstractly and have an intuitive sense of, for example, invariance. When two hands hold up five and two fingers and then cross over to change sides, children understand that the quantity has been preserved. This article also qualifies that though the abstraction of algebra is possible in young children, formal operations do not develop until approximately 12 years of age.

Agreeing that thinking through reasoning should occur in elementary schools, Martin and Kasmer (2010) stress the move away from memorized algorithms toward personalized


knowledge. This article provides a good example of both the potential power and the potential confusion behind organizing a curriculum around thinking. The authors write “...where students were encouraged to develop their own strategy for subtraction...” (p. 289). Taken alone, this phrase is fodder for traditionalists and parents. No substance is evident, and ‘what if ’ the student does not construct an effective subtraction strategy? For many, a guaranteed concrete outcome is important. The next sentence “...students had to consider the problem’s relevant mathematical concepts in order to proceed with determining a plausible strategy to solve the problem” can also be misconstrued and considered problematic. Understanding and thinking are what math classrooms should promote, but those against reform classrooms will argue that the direction here is not clear enough for children and possibly even teachers.

Now that mathematical habits of mind, or any phrase that describes mathematical thinking processes, have become important parts of curriculum documents, educators and involved parties need to be clear on the meanings and outcomes of this goal. While the EDC team states that habits of mind are a cohesive structure for organizing math curriculum, all articles surveyed here agree that a problem solving environment is necessary to foster mathematical habits of mind. One other important theme that emerged from these articles is the need for teachers to step away from the norm of the traditional role if mathematical thinking is to develop in classrooms. However, until teachers are engaged in habits of mind activities, such as TUP described earlier, they may not realize that thinking skills and mathematics can emerge from these activities. Problem solving activities and the awareness of mathematical habits of mind can deepen not just students’ understanding of mathematical concepts and processes, but also teachers’, and can provide for personal growth. As this investigation has revealed, there are many activities and resources like these articles, to textbooks designed by the EDC team, that are available to support teachers.

With the current cognitive research that challenges our traditional practices combined with the belief that students should think for themselves in school so that they can think for themselves in the future, thinking should be a given in a mathematics classroom. In the words of Alan Schoenfeld: “One can’t simply turn the clock back; too much is known about mathematical thinking and learning” (2004, p. 30).

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REFERENCES

Anderssen, E. (2014). Why the war over math is distracting and futile. Globe and mail. Retrieved from http://www.theglobeandmail.com/news/national/education/why-the-war-over-math-is-distracting-and-futile/article17178295/z

B.C. Ministry of Education. (2013). Transforming curriculum and assessment. Retrieved from https://curriculum.gov.bc.ca/

Case, Robert. (2005) The Dutch Revolution in Secondary School Mathematics. Mathematics Teacher 98(6), 374-384.

Costa, A., & Kallick, B. (2008). Learning and Leading with Habits of Mind: 16 Essential characteristics for success. Retrieved from http://2011ilead.wikispaces.com/file/view/Learning+&+Leading+with+Habits+of+Mind.pdf

_____. (1982). The 16 habits of mind identified by Costa and Kallick. Retrieved from http://www.ccsnh.edu/sites/default/files/content/documents/CCSNH%20MLC%20HABITS%20OF%20MIND%20COSTA-KALLICK%20DESCRIPTION%201-8-10.pdf

Cuoco, A., Goldenberg, P., & Mark, J. (2010). Organizing a Curriculum around Mathematical Habits of Mind. Mathematics Teacher, 103(9), 682-688.

_____. (1996). Habits of mind: An organizing principle for mathematics curricula. Journal of Mathematical Behavior 15, 375-402.


Goldenberg, P., Mark, J., Cuoco, A. (2010). An algebraic-habits-of-mind perspective on elementary school. Teaching Children Mathematics, 16(9), 548-556.

Johnson, E. (1924) Teaching Pupils The Conscious Use Of a Technique Of Thinking. The Mathematics Teacher, 17(4), 191-201.

Kiem, J. (2013). Standards for Mathematical Practice = Habits of Mind = Project-Based Learning. Retrieved from http://ocmbocesis.wordpress.com/2013/10/10/standards-for-mathematical-practice-habits-of-mindproject-based-learning/

King, K. (2013). Mathematical habits of mind. Retrieved from http://horizonsaftermath.blogspot.ca/2013_04_01_archive.html

Mark, J., Cuoco, A., Goldenberg, P., & Sword, S. (2010) Developing Mathematical Habits of Mind. Mathematics Teaching In The Middle School, 15(9), 505-508.

Martin, G., & Kasmer, L. (2010) Reasoning & Sense Making. Teaching Children Mathematics 16(5), 284-291.

NCTM. (2009). Focus in High School Mathematics: Reasoning and Sense Making. Retrieved from http://education.illinois.edu/smallurban/chancellorsacademy/documents/ReasoningandSenseMaking.pdf

OECD. (2013). PISA 2012 Results in Focus: What 15-year-olds know and what they can do with what they know. Retrieved from http://www.oecd.org/pisa/keyfindings/pisa-2012-results-overview.pdf

Schoenfeld, A. (2009). An Essay Review of the NCTM High School Curriculum Project. Mathematics Teacher 103(3), 168-171.

_____. (2004). Math Wars. Retrieved from http://www.math.cornell.edu/~henderson/courses/EdMath-F04/MathWars.pdf

_____. (1988). When good teaching leads to bad results: The disasters of “well-taught” mathematics courses. Educational Psychologist, 23(2), 145-166.

_____. (1987). Cognitive Science and Mathematics Education. New Jersey: Lawrence Erlbaum Associates, Inc.

Weiss, M. & Moore-Russo, D. (2012). Thinking like a mathematician. Mathematics Teacher 106(4), 269-273.

Wiles, P. (2013) Folding Corners of the Habits of Mind. Mathematics Teaching In The Middle School 19(4), 208-213.

FACULT Y OF EDUCATION

GR ADUATE PROGR AMS

CURRICULUM & INSTRUCTION: NUMERACY (M.Ed.)This two year cohort-based M.Ed. program is designed for elementary and middle school teachers who wish to examine multiple perspectives on teaching and learning of mathematics, explore interconnection between mathematics and numeracy, and enhance their own personal problem solving skills.

Designed around the definition of numeracy as ‘mathematics in action’, the program is based on the principle that mathematics is best learned, and hence, best taught through an emphasis on ‘doing’ mathematics.

www.sfu.ca/education/gs/degreediploma/masters/ ci-numeracy2014/

We are currently accepting expressions of interest for a September 2016 Program Start Date.

Please indicate your location and interest in the link below. The location showing the most interest will be chosen.

http://bit.do./numeracy2016

Please spread the word about this opportunity to friends and colleagues who may be interested.

Implementation


Using Desmos as a Dynamic Interface to Explore Reciprocal FunctionsBy Chris TsangChristopher Tsang is completing a Masters of Secondary Mathematics Education at Simon Fraser University and is currently a secondary mathematics teacher at Magee Secondary School in Vancouver, BC. His current interest in education includes the epistemology of learning through technology and teaching mathematics through problem solving.

Inspiration

Seymour Papert describes his childhood experience of playing with gears to mental models that benefited him in the subsequent years in school. He shares this opportunity in constructing LOGO. LOGO, a visual programming language, allows a student to freely explore and play with how a turtle moves around in a two dimensional space. It is through this construction and play that a student may make connections to learning across other areas in school and life (Papert, 1993). Like LOGO, Desmos, allows for visual exploration of how functions move by manipulating the sliders that control constants and/or coefficients.

Taking advantage of the increased opportunities for manipulation, exploration and observation, the user takes an active approach in the construction of knowledge allowing students to achieve these abstraction processes (Laborde, 2001). This use of information technology is widely available in our schools, but the experimental approach of mathematics has only recently been implemented in pure mathematics in more recent years (Borwein & Bailey, 2003).

Design

The BC Ministry of Education’s common curriculum framework includes seven processes that are critical aspects of the students learning, performing, and understanding mathematics.

Students are expected to

• use communication in order to learn and express their understanding,• make connections among mathematical ideas, other concepts in mathematics, everyday

experiences and other disciplines,• demonstrate fluency with mental mathematics and estimation,• develop and apply new mathematical knowledge through problem solving,• develop mathematical reasoning,


• select and use technology as a tool for learning and solving problems, and• develop visualization skills to assist in processing information, making connections and

solving problems.

In particular to the topic of graphing and analyzing reciprocal functions, the relevant processes in this learning outcome as prescribed by the Ministry are connections, reasoning, technology and visualization (Ministry of Education, British Columbia, 2008).

This lesson on reciprocal functions is designed as a three part activity beginning with a table of values using Desmos, then experimenting with graphs of linear reciprocal functions with sliders, and finally extending the experimental approach to graphs of quadratic reciprocal functions with sliders. This will be my first attempt at using Desmos sliders as a class exploration activity. Students will also be given a worksheet [Figure 1] for the transference of knowledge and for pencil and paper graphing then using Desmos as a feedback tool. In all three activities, students can access prepared Desmos activities from weblinks shared on my blog.

The first activity is designed to use table of values. The goal is for students to construct an understanding of the transformational behaviors of coordinates from y=f(x) to y=1/f(x). Working with the function y=x as the base function, the students are instructed to manually enter the y-coordinates of y=1/x. As students enter their output values they will understand why and how the y-values change. They will notice through observations that there are invariant points when y=±1 and observe the occurrence of vertical and horizontal asymptotes [Figure 2]. Upon the completion of the graphing task, they are expected to transfer their plots to the handout.

The use of technology is a substitute for pencil and paper graphing on a Cartesian grid. Although similar outcomes could be achieved using only pencil and paper, the purpose of using Desmos is to facilitate a quicker and more accurate estimate of rational coordinates.

The next two activities take advantage of the Desmos’ slider graphing technology. In these two activities, the graphs of a linear function in slope-intercept form and its corresponding reciprocal transformation and vertical asymptote Figure 1: Student worksheet for recording work


are displayed (Figure 3). The sliders allow manipulation of the slope and y-intercept, a concept that students should be familiar with from previously learned mathematics. This manipulation, experimentation and observational approach to learning by the students would help them construct an understanding of the abstract mathematics (Laborde, 2001). The purpose of this activity is for students to play with the sliders to observe the changes and invariances between the two functions. Keeping as true as possible to Papertian style, the goal is not for students to be instructed on the formal rules of reciprocal tranformation but rather develop and construct an understanding of how y=f(x) and y=1/f(x) moves together in a Cartesian plane through play.

After given time to explore, students will be instructed to graph using pencil and paper the function y=2x+5 and its reciprocal function. If they have developed an understanding of the behavior of the transformations, the pencil and paper graphing will reinforce that. A blank graphing space is provided for students to problem pose a function of their own.

The third and final Desmos activity extends the activity from linear reciprocals to reciprocals

Figure 2: Screenshot of Desmos table of values activity. Points on y=x are prepared for the student. Some points of y=1/x have been entered manually

Figure 3: Screenshot of Sliders activity of linear reciprocal transformations


of quadratic functions (Figure 4). The quadratic function presented is in vertex form so that the translations and expansions/compressions of the original function are transparent. The focus then is for the students to observe how the reciprocal function changes as the original quadratic changes. Similar to linear reciprocals, following the sliders activity, students are presented opportunities to graph quadratic equations and its corresponding reciprocal function using pencil and paper. I would expect students to use Desmos as a feedback tool in this portion of the lesson.

The slider technology allows for the family of graphs to be viewed as a continuous family of functions rather than a discontinuous series of exemplars that highlights various cases that students may encounter. The movement of simple transformations of the quadratic function simultaneously with each of its reciprocal transformation necessitates a high degree of detail that is impractical using traditional graphing methods.

As described by Borwein and Bailey (2003), experimental mathematics is the methodology of performing mathematics that includes the use of computing technology for the following tasks

• gaining insight and intuition,• discovering new patterns and relationships,• using graphical displays to suggest underlying mathematical principles,• testing and especially falsifying conjectures,• exploring a possible results to see if it is worth formal proof,• suggesting approaches for formal proof,• replacing lengthy hand derivations with computer-based derivations, and• confirming analytically derived results.

The application of Desmos as designed allows opportunities for students to demonstrate all of the above uses except to explore results for the purpose of formal proof or how to approach a formal proof, as the objective of the lesson does not involve proofs. Due to the unstructured component of play, the degree that students use Desmos for testing and falsifying conjectures and confirming analytically derived results varies.

Figure 4: Screenshot of Sliders activity of quadratic reciprocal transformations


Observations and Insight

The first activity is a substitution use of graphing functions on Desmos, as opposed to traditional pencil and paper. I did not anticipate the simplistic nature of this activity to produce a varied pace between groups, which was further amplified by the technological issues in logging into computers. The time needed for the groups to arrive at the final outcome of the first activity made it difficult to level the class through discussions of their observations on the relations between y=f(x) and y=1/f(x) .

Another observation from this first activity is a high occurrence of student disengagement. In most groups, one student was doing most or all of the data entry without discussion of the behavior of the points with participation from other group members. One hypothesis is that the simplistic nature of this activity did not create any curiosity from the students.

As students progressed towards the next two exploration activities with Desmos’ sliders, two students expressed that they did not know what to look for. Keeping to the essence of Papert, this activity was designed to be unstructured. The students were asked to play with the sliders and make observations on the comparisons between y=f(x) and y=1/f(x). Even though these groups were manipulating the sliders, they were unsure as to what outcome I would want them to discover. Nathalie Sinclair characterizes student behavior in unstructured activities as one of three possibilities: path-finder, track-taker or floater (Sinclair, 2001). The track-taking behavior observed from these students indicate that they would prefer to have an established pathway that gets them to the final result. In contrast, the path-finders would demonstrate comfort in navigating through the complex relationship of a reciprocal transformation.

Time and technological resource constraints limited the amount of time that the students could explore and experiment. As much as I resisted, the unstructured student activity evolved into a teacher centered lesson. It was clear to me that students would not be able to achieve the intended outcome in the time that we had. Following the teacher led instruction, the students were instructed to continue their exploration at home, and a follow up lesson summary was planned for the next class.

VREFERENCES

Borwein, J., & Bailey, D. (2003). Mathematics by Experiment: Plausible Reasoning inthe 21st Century.

Goldenberg, P. E. (2000). Thinking (and Talking about Technology in Math Classrooms. Education Development Centre, Inc.

Laborde, C. (2001). Integration of Technology in the Design of Geometric Tasks with Cabri-Geometry.International Journal of Computers for Mathematical Learning, 283-317.

Papert, S. (1993). Mindstorms: Children, Computers, and Powerful Ideas (2nd ed.). New York: Basic Books.

Sinclair, N. (2001, March). Aesthetics “Is” Relevant. For the Learning of Mathematics, 21(1), 25-32.

The Minister of Education, British Columbia. (2008, January). Curriculum Documents. Retrieved March 18, 2014, from Ministry of Education: http://www.bced.gov.bc.ca/irp/pdfs/mathematics/WNCPmath1012/2008math1012wncp_intro.pdf

Thurston, W. P. (1994, April). On Proof and Progress in Mathematics. American Mathematical Society, 30(2), 161-177.

Bookreview


Dr. Richard Hoshino’sfirst novel, The Math Olympian, is about

a teenage girl from a small town in Nova Scotia who discovers a BIG passion for mathematical problem-solving. She commits herself to the goal of representing her country at the International Mathematical Olympiad. Through hard work, grit and the support of a number of mentors, Bethany Macdonald trains to become a world-champion “mathlete.” She overcomes adversity on her remarkable journey as she develops confidence and creativity in problem-solving and in life.

Just as Bethany discovers the splendour and the essence of mathematics, so will many readers of this book. One of its many themes is that mathematics is a subject that’s not about memorizing formulas, but rather about problem-solving. The book will certainly appeal to “mathletes” as they are more likely to relate to Bethany’s fascination with mathematics, along with her struggles and ambitions. It may have an added appeal to girls as Bethany shatters the stereotype that only boys can excel in math.

The Math OlympianReviewed By Michèle Roblin

Michèle Roblin teaches mathematics and Spanish at Howe Sound Secondary. She has been living and teaching in Squamish since 1994. She is a past president of the BCAMT.


But don’t think that this novel would only appeal to aspiring young female “mathletes” and their coaches. While those involved in Math Olympiad or other math contests will be able to identify with Bethany’s experiences and would gain some useful insights into problem solving techniques, the book also offers the less-inclined (or even some reluctant math students) more than a glimpse into beauty of mathematics. The thrill of detecting patterns and discovering connections and the satisfaction of writing elegant solutions and experiencing precious “Aha” moments could appeal to a variety of readers with wide-ranging backgrounds in mathematics.

A unique novel with a unique format, The Math Olympian incorporates actual Math Olympiad problems throughout the story. Five problems, along with their solutions, are woven into the story and the central character’s thought processes unfold before the reader’s eyes. Diagrams and equations decorate the text without detracting from the story or distracting the reader from the story. (Well, some readers might be distracted by the problems, but in a good way!) Dialogue drives the plot, develops the characters and hooks the reader, yet the mathematics proves stronger than the narrative. While not many people uninterested in mathematics would find Bethany’s story truly compelling, the blending of the narrative and the mathematics is the novelty of the book.

The Math Olympian can and does provide the basis for a problem-solving course in mathematics. In fact, the author uses the book as a textbook with his students at Quest University. I suspect it offers his students a richness often lacking in people’s experiences of learning about mathematics from a textbook.

The novel may have autobiographical elements as Dr. Hoshino competed in the International Mathematical Olympiad (IMO) in 1996 in Mumbai, India, earning a silver medal for Canada. He was also a coach and trainer for Canada’s Math Olympiad team.

Through Bethany, The Math Olympian gives mathematics a human face—a face that is hardly dull. This novel presents mathematics through Bethany’s eyes. Many readers, not only female high school students with a passion for mathematics, would find this book not only very readable but interesting and inspiring.

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MORE ABOUT THE AUTHOR

Dr. Hoshino currently teaches mathematics at Quest University in Squamish, BC. Prior to joining the faculty at Quest, he worked as a research scientist for the Government of Canada developing math-based solutions to improve security and efficiency for Canada Border Services Agency. As a post-doctoral fellow in Tokyo, he optimized the schedule for the Japanese Professional Baseball League, greatly reducing the League’s travel costs and carbon footprint. When he is not inspiring his students at Quest, he can sometimes be found playing community basketball at Totem Hall in Squamish.

The Math Olympian

by Dr. Richard Hoshino Friesen Press, 2015 ISBN: 978-1460258736

Mathwebsites


Inside Mathematics provides professional development resources, tasks and challenging problem solving opportunities for students.

The Math Reasoning Inventory wesbite has free resources such as interviews that can be used as diagnostics for what your students understand or do not understand about mathematics.

WWW.INSIDEMATHEMATICS.ORG

WWW.MATHREASONINGIVENTORY.COM

Spring 2015 Math WebsitesLinks selected and described by David Wees (www.davidwees.com)


The Centre for Education in Mathematics and Computing website has resources to enrol your students in challenging CEMC math contests and provides past contests from a variety of different grade levels that are ideal for challenging students.

All of the K–12 problems at Open Middle have a defined beginning and a single correct answer, but there are a variety of different ways students can approach the problem, hence an open middle.

The collection of nearly 400 problems at Math Arguments 180 can help spur discussions and arguments in your classroom.

WWW.MATHARGUMENTS180.BLOGSPOT.COM

WWW.OPENMIDDLE.COM

WWW.CEMC.UWATERLOO.CA/CONTESTS

Problemsets


KINDERGARTEN TO GRADE 1

Read Shapes That Roll by Karen Nagel to introduce students to the various 3D shapes that are in our world.

Encourage the students to explore the attributes of 3D shapes.

Have the students make a ramp using a flat surface and a variety of 3D objects to explore.

Ask:

1. Which 3D shapes roll?2. Which ones slide?3. Which ones stack?

Encourage the students to make predictions and explain their thinking before they experiment with the various objects.

Ask them to try all the various sides of the shape and see if it makes a difference i.e. a can shaped object (cylinder) slides on its ends but rolls on its sides.

Discuss how the different attributes helped or stopped the shapes from rolling, sliding or stacking. Have them give examples from their discoveries i.e. ‘round like a ball’ (spheres) objects don’t stack.

Spring 2015 Problem SetsProblem sets compiled by Sandra Ball and Kevin Wells


GRADE 2 TO GRADE 3

GRADE 4 TO GRADE 5

Allow students time to explore the attributes of various 3D shapes. Have them identify the faces, edges and vertices of the 3D shapes. Present various problems for them to solve:

1. If you had 3 cones, 2 cylinders and asphere, how many faces would you have?How do you know?

2. You have 1 cube and your friend has 4cylinders. Who has more faces? How doyou know?

3. I have some objects and in total Icounted 8 faces. What might the objectsbe? Explain your thinking.

4. I have a collection of objects that have 7 faces and a point. What shapes could they be?Explain your thinking.

1. Complete Figure 1.2. Complete Figure 2.3. With a partner, make up a similar challenge for each other.4. Add a third row of boxes and make up a new challenge for each other.

Figure 1 Figure 2


GRADE 8 TO GRADE 9

A farmer uses a circular water sprayer to irrigate a square shaped field. The diameter of the watered circles is 20 metres.

1. How could you calculate the totalarea that is that is left unwatered(shaded)? What value did you findfor this area?

2. Show how you would calculatethe area of the central shaded areabetween the circles. How manydifferent ways can you find to dothis? Share your methods andanswer with the class.

3. A new circle is watered in the spacebetween the four circles so that ittouches each of the other circles.Determine the area of this newcircle.

4. What could be the next question inthis problem?

A group of people at a party (you decide how many) are playing a mystery game in which each person is given a unique piece of information which they need to share. They can only share this with one person at a time. You want to have an idea of how many conversations are necessary before everyone has heard all of the clues so that the game doesn’t go on too long.

1. How would you start this problem? What do you think you need to know first?

2. Can you start in a simple way and then make it increasingly challenging? What strategiesdo you need to use to avoid unnecessary conversations? What would change if you wantedto make sure there were the most number of conversations?

3. How could you check your answers are correct? What would be the best way?

GRADE 6 TO GRADE 7


GRADE 10 TO GRADE 11

There is a game being introduced at a charity event in which you are asked to pick balls from two bags. Bag one contains an equal mix of blue, red, and yellow balls. Bag two has equal numbers of just two colours of ball. To win, you must draw the same colour ball from each bag.

1. What are the possible outcomes of this game?

2. What are your chances of winning a prize?

3. Do you think this is a fair game? Explain how you made your decision. What do you thinkmakes for the fairest game at such an event? Should there be a 50-50 chance of winning?

4. How do you think you could change the rules of the game to improve it? What would bethe possible outcomes of your new game?

Miles is staring out to sea and wonders how far away the horizon is...

1. Above is a two-dimensional,not-to-scale diagram of Milesstanding on the earth. Mark thearea on the earth’s surface thatMiles can see. (Miles can seea point if there is a line fromthat point to his eye, which isnot blocked by the earth. Use astraightedge.)

2. Label the horizon (the farthestpoint that Miles can see) “B.”Label Miles’s eye “A” and thecenter of the earth “C.” Drawin triangle ABC. What is angleABC? Make it clear how youfound this angle.

3. Assume that Miles is 1 m. tall and the earth’s radius is 6,000 km. (6,000,000 m). How farcan he see?

4. Consider a planet of radius r and a person of height h; develop a formula for the distanceto the horizon in terms of the r and h.

5. A ship with a mast 10 metres tall sails off in a line directly away from Miles until he canjust no longer see it. What can you find out about the ship’s position?

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GRADE 11 TO GRADE 12

Adapted from: www.jrmf.org/problems.php

bcamtnews


BCAMT Awards InformationThe BCAMT sponsors awards in three categories to celebrate outstanding achievements of its members. Winners are honoured at a BCAMT conference and receive a commemorative plaque.

AWARDS & CRITERIAOutstanding Teacher Awards (Elementary; Secondary; New Teacher with less than 5 years teaching experience)

• shows evidence of significant positive impacts on students, staff and parents

• has initiated innovative and effective programs in their classroom, school, district, orprovince (teacher research, technology, active learning, assessment, etc.)

• has and continues to demonstrate excellence in teaching mathematics regularly inBritish Columbia (teaching style, knowledge of the curriculum, current curriculumtrends, etc.)

• has made significant contributions to mathematics education at the district orprovincial levels over several years (workshops, seminars, conferences, communityprojects, curriculum development, publishing, etc.)

• is not a current member of the BCAMT Executive

Ivan L. Johnson Memorial Award

The Ivan L. Johnson Memorial Award is awarded in honour of long-time BCAMT executive member Ivan Johnson. Ivan donated money to the BCAMT for an award in which the recipient will receive significant funding to cover costs of attending the NCTM Annual Conference.

• inspires teachers to try new ideas that improve the quality of mathematics education

• consistently seeks ways to innovate practices in the math classroom

• actively engages in professional dialogue involving mathematics pedagogy

• is not a current member of the BCAMT Executive

Note: Nominees for the BCAMT Outstanding Teacher Awards will automatically be considered for this award. Previous winners of BCAMT Outstanding Teacher Awards may also be nominated.


Service Award

• has provided extraordinary service to mathematics education as an active member of theBCAMT for a significant period of time

SELECTION PROCESS

• all nominations are reviewed by the BCAMT Awards committee (consisting of a minimumof five previous award recipients) who recommend the recipients to the BCAMT Executivefor ratification

• each nomination is considered for two years, after which time the application can bere-submitted with updated information

HOW TO NOMINATESend all required documents listed below in an envelope to:

BCAMT Awards c/o Dave Ellis 2086 Newport Avenue Vancouver, BC V5P 2H8

DEADLINE: MAY 8, 2015

REQUIRED DOCUMENTATION

• a completed nomination form (one person per form)

• nominee’s curriculum vitae which demonstrates evidence of teaching, contribution,innovation, professional involvement and impact

• nominator’s summary (one page only) explaining concisely the reasons for the nomination

• two letters of support (one page each) with concise information about how the nomineefulfills the criteria.

BCAMT Awards Nomination FormPLEASE CHECK THE APPROPRIATE CATEGORY:

NOMINEE INFORMATION

Nominee’s Name

School/District

Address

City/Town Postal Code

Work Phone Home Phone

Email Address

NOMINATOR INFORMATION

First Nominator’s Name

Second Nominator’s Name

School/District

Address

City/Town Postal Code

Work Phone Home Phone

Email Address

Signature of Nominee Signature of Nominator

□ Outstanding Elementary Teacher□ Outstanding Secondary Teacher

□ Outstanding New Teacher□ BCAMT Service Award

□ Honourary Lifetime Membership