SPRING 2019 Volume 6 Issue 1 · I continue to come back to Simon Sinek’s 2009 book, Start with Why: How Great Leaders Inspire Everyone to Take Action. His three concentric circle

SPRING 2019

Volume 60

Issue 1

We can observe almost anything in the world around us with wonder.

Elysia Dubland

Contents Spring 2019 | Volume 62 | Issue 1

IN EVERY ISSUE

President’s Message

Problem Sets

Book Review

Math Links

BCAMT

07

42

45

46

48

8 13

20

09 Mathematics and the Imagination

By Elysia Dubland

12 Blending Geometry and Financial Literacy

By Audrey Venner

14 Changing Assessment Within a Changing Curriculum

By Michael Pruner

21 Rethinking Place Value

By Adam Fox

26 The Evolution of Fun With Functions

By Michèle Roblin

28 A Function of Freedom and Constraints

By Chris Hunter

30 Using Benford’s Law in the Classroom

By Timothy Sibbald and Tiberius Veres

34 Math Challengers

By Joshua Keshet and Dave Ellis

Errata • Fall 2018 issue of VectorThere were some errors in the printing of the article “Accessing and Addressing Student Calculus Readiness” by Kseniya Garaschuk. These have been resolved in the online version, which can be found at https://www.bcamt.ca/communication/vector/current-issue/.In the section “Results,” subsection “Student performance,” (p. 37) the graph is covering up the text, so the first paragraph is inaccessible.In the section “Results,” half of the subsection “Common themes” (p. 37) is missing. Questions 4, 5 and the common themes of student strengths and weaknesses are not included.The pie charts (p. 38) are difficult to read because they were printed in black and white.The references are missing. The editors would like to apologize to Kseniya Garaschuk for these errors.

The views expressed in each Vector article are those of its author(s), and not necessarily those of the editors or of the British Columbia Association of Mathematics Teachers.

Articles appearing in Vector may not be reprinted without the explicit written permission of the editors. Once written permission is obtained, credit must be given to the author(s) and to Vector, citing the year, volume number, issue number and page numbers.

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. Authors must also include a short biographical statement of 55 words or less.

Articles must be in Microsoft Word. All diagrams must be in TIFF, GIF, JPEG, BMP, or PICT formats. Photographs must be high print quality (min. 300 dpi).

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

Notice to Advertisers Vector is published two times a year: spring and fall. Circulation is approximately 1400 members in BC, across Canada and in other countries around the world.

Advertising printed in Vector may be of various sizes, and all materials must be camera ready.

Usable page size is 6.75 x 10 inches.

Deadline for Submissions Spring: FEB 1 (for peer review, December 1) Fall: SEPT 1 (for peer review, July 1

Advertising Rates Per Issue$300 Full Page $160 Half Page $90 Quarter Page

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

2019/20 Membership Rates $40 + GST (BCTF Member) $20 + GST Student (full time university only) $65.52 + GST Subscription (non-BCTF)

Cover Art: "Although it has been 35 years, I remember the conics assignment as being a novel departure from the usual rituals and routines of my high school math classroom. I thought my teacher had presented us with a fun and effective way to consolidate our learning of quadratic relations, equations and inequalities. I liked the feeling of creating something–even though I was really just re-creating something. I already loved math, but this assignment deepened the romance for me–even though all of the students in the class were involved in the same 'affair!'" — Michèle Roblin

BCAMT EXECUTIVEDeanna Brajcich, PresidentSooke School [email protected]

Susan Robinson, Vice PresidentGulf Islands School [email protected]

Michael Pruner, Past PresidentNorth Vancouver School [email protected]

Colin McLellan, Secretary and Listserve ManagerRichmond School [email protected]

Jen Carter, TreasurerVernon School [email protected]

Brad Epp, Membership ChairKamloops School [email protected]

ELEMENTARY REPRESENTATIVESJennifer Barker, Surrey School [email protected]

Adam Fox, North Vancouver School [email protected]

Debbie Nelson, Comox Valley School [email protected]

Alex Sabell, Surrey School [email protected]

SECONDARY REPRESENTATIVESRon Coleborn, Surrey School [email protected]

Josh Giesbrecht, Abbotsford School [email protected]

Chris Hunter, Surrey School [email protected]

Minnie Liu, Vancouver School [email protected]

Amanda Russett, Kamploos School [email protected]

POST-SECONDARY REPRESENTATIVESPeter Liljedahl, [email protected]

Christine Younghusband, [email protected]

NCTM AND NCSM REPRESENTATIVEMarc Garneau, Surrey School [email protected]

INDEPENDENT SCHOOLS REPRESENTATIVESDarian Allan, West [email protected]

Richard de Merchant, [email protected]

VECTOR EDITORSSean Chorney, [email protected]

Susan Robinson, Gulf Islands School District [email protected]

Vector • SPRING 20195

Contributors

Ray Appel Ray Appel has taught grades 2-8 been a mathematics/science helping teacher, spoken across Canada and the US, and has authored mathematics curriculum. He is currently retired, delving deep into reading, painting, drawing, walking, website design, a bit of math consulting, home building and more! His website, zapple.ca is used by thousands of educators worldwide.

Sandra Ball Sandra Ball taught in the Surrey School District for 40 years with a major focus on primary education and is currently enjoying retirement. Sandra offers a variety of professional development opportunities for classroom and support teachers in the area of numeracy, provincially and internationally. She believes all children are capable learners whose strengths, interests and passions need to be nurtured.

Elysia Dubland Elysia Dubland teaches secondary mathematics in the Surrey School District. She holds a masters in mathematics education from SFU, where her research focused on homework, autonomy, and self-assessment. She is passionate about building thinking classrooms and helping her students enjoy mathematics.

Dave Ellis Dave Ellis is a retired Vancouver secondary school mathematics teacher and department head. He was a long-serving member of the BCAMT Executive. Presently, he is a volunteer member of the Math Challengers Committee serving as Regional Liaison.

Adam Fox Adam Fox teaches kindergarten in the North Vancouver School District. Prior to becoming a teacher, he was an IT consultant in England. He recently completed his masters in Numeracy at Simon Fraser University.

Josh Giesbrecht Josh Giesbrecht is a secondary mathematics and digital media teacher at Abbotsford School of Integrated Arts. He has past career experience in programming and is easily fascinated by shiny blinking lights.

Chris Hunter As a numeracy helping teacher in Surrey, Chris Hunter collaboratively works with-and learns from-teachers of mathematics from Kindergarten to Calculus. He tweets at @ChrisHunter36 and blogs at reflectionsinthewhy.wordpress.com.

Joshua Keshet Joshua Keshet is a mathematician and a corporate executive. He is also the Academic Coordinator for the Canadian Math Challengers Society.

Michael Pruner Michael Pruner is a high school mathematics teacher from North Vancouver, BC and the past president of the BCAMT. Michael is also a PhD student at Simon Fraser University in Mathematics Education. Throughout his career, he has taught many levels of mathematics and recently discovered a passion for developing Thinking Classrooms. Michael believes that mathematics is a social endeavour and is best learned through collaborative activity.

Michèle RoblinMichèle Roblin lives in Squamish, BC, where she has been teaching mathematics at Howe Sound Secondary since 1994. She served on the executive of the BCAMT from 2006 to 2012. Her current interests include teaching Spanish through storytelling, and how the ritual and romance of storytelling might influence her teaching of mathematics.

Timothy SibbaldTimothy Sibbald is an associate professor with the Schulich School of Education

at Nipissing University in North Bay, Ontario. He teaches in the pre-service program and the graduate programs with a focus on mathematics instruction. He is also the editor of the Ontario Mathematics Gazette

Audrey VennerAudrey Venner is a Grade 6/7 French Immersion teacher in Burnaby. She strongly believes in empowering students and strives to give them choice in their learning. She also prioritizes curiosity and joy over grades and achievement, which contrasts with her own childhood experience in France’s education system.

Tiberius VeresTiberius Veres has been a teacher candidate in the Schulich School of Education at Nipissing University. He is pursuing a career change with an aim to move from a large urban area to a small city. This coincides with his changing careers from corporate banks in Toronto to pursuing teaching in British Columbia.

David WeesDavid Wees is a mathematics education teacher and consultant with 25 years of educational experience and has a masters in educational technology. David currently works remotely from his island paradise of Denman Island, BC.


President’s Message

Welcome to our Spring 2019 Vector issue. Inside you will read articles that will confirm, challenge and inspire your teaching ideas and learning environments. I would like to thank all of the authors who have taken the time to share their expertise and experiences.

Math for All: Teachers and StudentsAs the debate continues between mathematics teaching philosophies, it is important to remember what educators truly want for their students: to be engaged, to feel success and to be included. “I do, we do, you do,” “discovery-math,” “guided discovery-math,” “Reggio-inspired inquiry,” “math workshop;” no matter what we call our personal approach, it is our duty to ensure all students have the opportunity to be excited about mathematics. Teaching is about providing a balanced experience that includes the instruction of skills, but also extends thinking through problem-solving and the communication of ideas. Just as educators learn more through collaborating about new concepts, students do as well.

With the curriculum shift about to be fully implemented K-12, a shift away from lists of Ministry recommended textbooks, and the ease with which teachers are able to search for classroom materials on the internet, it is important to remember that a resource only partly guides what we offer our students. For the most part, a textbook, problem-set or worksheet is not the critical variable for success: it is how the students engage with it. The province has a plethora of new teachers, who may be struggling to stay afloat; a resource can be a life raft. We should all be examining the purpose of introducing any task in our classrooms. What are the learning goals? Do students have the opportunity to discuss their predictions, their processes and their results about any learning task presented by a resource? Teachers can turn any worksheet into an opportunity to demonstrate curricular and core competencies. We need to model this for those in the beginning years of teaching.

We, as educators, learn more effectively when we understand the purpose of

our teaching: our students do as well. I continue to come back to Simon Sinek’s 2009 book, Start with Why:

How Great Leaders Inspire Everyone to Take Action. His three concentric

circle model, the “Golden Circle,” guides my goals for teaching which will then inform and support my

students’ learning. What are the goals I want my students to aim for? How will I help them get there and what might they do to get there? And lastly, what is the purpose of attaining this goal? Essentially, this connects my students

more deeply to the learning and understanding, and improves retention of new knowledge. Most

effectively, I prompt the students to discuss and find their own “why.” It is an open model which allows

teachers to adjust and refine as they see necessary. This transcends any resource or any particular mathematical teaching approach and allows mathematics to be experienced by all.

With grade 11 and 12 teachers moving into the new curriculum in September, the curricular competencies will become more explicit in their classrooms. Assessment will have to look different as well. Peter Liljedahl drew a model for the BCAMT executive in February that provides teachers an alternative way to consider

assessment in their classrooms. Since assessment should not always be an event but an on-going process, balancing what happens in a classroom can be tricky. Looking at his model, in which quadrant would your assessment fall? Are there opportunities to shift some of your assessment into the other quadrants? What types of student self-assessment could be placed in each quadrant? Could you use this model to balance your instructional and assessment practices?

Math for All: BCAMT Fall Conference October, 2019The fall conference will be at Guildford Park Secondary in Surrey on October 25, 2019. Our theme is “Math for All,” which is the goal of all teachers in their classrooms. This “always-sold-out” event will feature educators from around our province. Come and join in discussion with your colleagues from across the province! Let’s get together and strive to reach everyone in our classes and inspire them to love mathematics as much as we do.


Adding to our event is a Reggio-inspired mathematics mini-conference facilitated by Janice Novakowski, Sandra Ball and other BC teachers. Explore how this approach can inspire and engage your students. If you are interested, please select this option at registration.

We need you: proposals now being accepted: BCAMT’s fall conference showcases the expertise and growth in our BC classrooms. Share with others the amazing things you are doing! Perhaps you are doing something innovative, maybe you have a specific skill set in teaching mathematics, or perhaps you have an engaging lesson you would like to demonstrate. Register now or submit your speaker application by visiting our website: https://www.bcamt.ca/fall2019/

We will come to you!The BCAMT executive continues to support teachers across the province in numerous ways: hosting conferences, supporting districts, facilitating meetings and supporting curriculum. If you are interested in one or more of our executive travelling to your area, please fill out the chapter request form, which can be found under the Professional Learning menu tab on the BCAMT website (https://www.bcamt.ca/).

Graduation Numeracy Assessment UpdateAs some of you are aware, effective July 2019, the Ministry is changing the grade in which students are required to take the GNA: grade ten. Previously, students could take it in either grade ten, eleven, or twelve. Of course, this changes a great deal about assessment and creates potential challenges for students. Each school is implementing the changes differently. Some schools are insisting all grade 10 students write this year. Other schools are focusing on grade 11s, but also allow grade 10s to write. These decisions may affect student’s confidence and affect their future because the results of the assessment will appear on their official transcripts. The BCAMT executive has written a letter to the Ministry expressing our concerns. The Ministry has not responded (yet).

New Mathematics ElectivesComing this fall, many schools will be offering the new mathematics electives: Geometry 12, Statistics 12, History of Mathematics 11, and/or Computer Science 12. Curricula for these courses were written by educational specialists, many of whom were BC teachers, to ensure an authentic mathematical experience for teachers and students. The BCAMT executive is very interested to know which of these courses your school is offering. Please take thirty seconds to fill out this anonymous google form: https://goo.gl/oRQrAf

BCAMT Grants AwardedEvery year the BCAMT grant committee considers grant applications from BCAMT members around the province. This year, we are proud and excited to award the following grants: RECIPIENTS INITIATIVE

Burnaby School District #41

Donna Morgan

Shift in practice: book club and collaborative lesson develop-ment and teaching

Kamloops-Thompson School District #73

Katie McCormak

Technology and robotics in the mathematics classroom: lesson development

Prince George School District #57

Marie Fanshaw & Tamara Deford

Exploring Open Ended Questions: Marian Small book club focusing on problem solving

Coast Mountain School District #83

Hazelton Area Teachers Math Group

Stacey Brown

Book club: improving mathemat-ics instruction

BCAMT CollaborationWe have a new Listserv channel: The Thinking ClassroomLooking for a more specific listserv experience? The BCAMT has now created a listserv channel focused on implementing Thinking Classroom experiences into classrooms. The conversation has already been rich with questions, experiences and tips from those who are actively working within the student-based model that Thinking Classrooms supports.

BCAMT Listserv: Nothing better than asking a BC Teacher!Don’t forget, one of the best places to get ideas or ask for guidance is our own BCAMT listserv! With over a thousand members, emails are answered quickly, and a variety of perspectives are provided.

There are two convenient ways to sign up:1. Email Colin McLellan at [email protected]. Visit the BCTF website (https://bctf.ca/forms/PSA-EmailLists.aspx)

I wish you all a fabulous end of your year. With report cards, graduation and the Graduation Numeracy Assessment, the end of the year is exciting but also stressful. Remember you are experiencing the same anxiety your students: take care of yourself. Always choose things that challenge, invigorate and allow you to enjoy the mathematics you teach.



Mathematics and the Imaginationby Elysia Dubland

Shuffle a deck of cards. Do it again. Do it one more time. You have just made history… probably. Wait a minute… what!?!?

Consider the number of possible orders of a deck of cards. There are 52 choices for the top card, 51 choices for the second card, 50 choices for the third card, and so on. Thus, the total number of possible orders is 52! (52 factorial), which is equal to 52 × 51 × 50… × 3 × 2 × 1. This is roughly equal to 8.0658 × 1067. That is a huge number. In fact, it’s an astronomically monstrous number. This means that the number of possible orders of 52 cards is so incredibly large that most of them have never been assembled. Seriously. Throughout the entire history of the world.

If you find this hard to believe, you are likely not alone. It is incredibly hard for the human mind to even begin to comprehend a number of this magnitude. Luckily, the YouTube video, “Math Magic” (2016), describes Scott Czepiel’s shocking imagery which was designed to help the viewer understand just how large 52! actually is. He asks you to imagine setting a timer to have 52! seconds on it. You then stand on the equator and after waiting a billion years, take a single step. You wait another billion years to take your next step. Continue in this manner until you have circled the Earth. Then remove a single drop of water from the Pacific Ocean. Continue circling the Earth and removing single drops of water. Once you have completely emptied the Pacific Ocean, put one regular white sheet of paper on the ground. Refill the ocean and repeat the entire process. Continue until your stack of papers reaches the sun. At this point, the number of seconds left on the timer is seconds – essentially, the same number of seconds as when you began! And what’s more, if you did the entire process one thousand more times, you would still be only one third of the way done!

Czepiel (n.d.) offers another way to pass the remaining time. This time you randomly deal five cards until you deal yourself a royal flush; you then buy a single lottery ticket and repeat the royal flush–lottery ticket process until you actually win the lottery. When you win the lottery, throw one grain of sand into the Grand Canyon. Once the Grand Canyon is full of sand, remove one ounce (28 grams) of rock from Mount Everest. Once Mount Everest is gone, repeat the whole process 256 more times and only then will the

timer have reached zero seconds! (Stevens, 2016).

I have shown this YouTube video to vastly different audiences, including Pre-calculus 12 students, math 9 remedial students, and mathematics teachers. The reactions are very similar. Laughter, gasps of surprise, and murmurs of disbelief ripple through the crowd. And without fail, a lively and engaging discussion follows.

In fact, my math 9 remedial students turned the discussion into a mathematical exploration that lasted almost the entire class. As they considered all of the card shuffling that, for hundreds of years, has happened continuously around the world in casinos, game halls, and private homes, they simply could not fathom how it could be possible that most of the deck orders have yet to be created, despite Czepiel’s vivid imagery. So, we set out on a quest to estimate how many card shuffles have happened since the beginning of time. This started to seem a little daunting and so the students narrowed it to estimating how many shuffles have happened in casinos, as that might be one of the largest contributors to total shuffles. The process involved finding out when cards were invented; researching how many casinos exist in the world; and estimating numerous quantities, including how many dealers per casino, how many shuffles per night per dealer, and how many hours worked per dealer each year. Despite the fact that many assumptions were made and our estimates may have been inaccurate, the exercise led to a number that could possibly represent total casino shuffles and which was also glaringly miniscule when compared to 52! And throughout the entire process, almost every student in the room was actively engaged, giving their opinions, researching facts on their phones, and crunching numbers on their calculators.

I will now mention that this was a very challenging class. It was filled with students with mathematics learning disabilities, behavioral designations, and a heightened apathy towards learning in general. I had tried numerous lesson plans and techniques to engage them in the curriculum, with very limited success. And then I showed them this video, and they shocked me. In fact, the most difficult student in the class was a key instigator and leader of the discussion. Why did this happen? Why did my most difficult students excitedly engage in a mathematics


lesson involving permutations, probability, scientific notation, estimation, and arithmetic?

One word. Imagination. The video captured their imaginations. Both the extraordinary, make-believe visualizations of 52! and the disbelief surrounding being able to arrange a deck of cards into a never-seen-before order provided a captivating arena for the students’ imaginations.

Imagination is not typically associated with mathematics. I certainly never heard it mentioned during any of my grade school and undergraduate mathematics courses. Nor have I heard it mentioned in conversations with colleagues. And yet imagination plays a starring role in many school subjects such as English, history, and the visual arts, and it seems to be making increasingly more frequent guest appearances in the sciences. But in the minds of many, mathematics is water and the imagination is oil; they simply do not mix.

However, certain education researchers would disagree. Founded by Kieran Egan, the Imaginative Education Research Group (IERG) at Simon Fraser University did extensive work on the importance of the imagination in all education, including mathematics, which led to an educative approach they call Imaginative Education (IE) (2018c). The Centre For Imagination In Research, Culture & Education (CIRCE), is now continuing this work at www.circesfu.ca. The IERG researchers point out that

(a)ll the knowledge in the curriculum is a product of someone’s hopes, fears, passions, or ingenuity. If we want students to learn that knowledge in a manner that will make it meaningful and memorable, then we need to bring it to life for them in the context of those hopes, fears, passions, or ingenuity. The great agent that will allow us to achieve this routinely in everyday classrooms is the imagination (2018c, para. 4).

In their book, Imagination and the Engaged Learning: Cognitive Tools for the Classroom, Egan and Judson (2016) push back against the common view that rationality and the imagination are distinct, with the former having its place in the academic subjects and the latter belonging to the arts. They assert that the imagination has a role in all learning, defining it as “the capacity to think of things as possibly being so; it is the source of invention, novelty, and generativity; it is not distinct from rationality but is rather a capacity that greatly enriches rational thinking; and it has an equal role in successfully learning academic subjects as engaging in arts activities” (Egan & Judson, 2016, p. 4).

Egan and Judson provide cognitive tools that can be used in the classroom to invoke the use of students’ imaginations. For elementary students these include stories, binary opposites, mental imagery, metaphor and humour. For secondary students these include extremes of reality, the heroic narrative, hopes, fears, passions and ingenuity, evoking wonder and changing contexts (2016). Despite the separation, however, I believe the tools can certainly be used for either age group. For example, mental imagery can be used with secondary students (as seen with my grade 9 students), while evoking wonder can be used with primary students. The tools are powerful; I highly recommend reading the book or checking out http://ierg.ca to learn about all of them. The remainder of this article will focus on mental imagery, extremes of reality, and evoking wonder in the mathematics classroom.

Affective mental imagery involves invoking images (as opposed to pictures) in the mind that cause students to feel something (Egan & Judson, 2016; IERG, 2018a). In the example, with my grade nine students, the mental imagery caused by the shocking descriptions given to help understand the magnitude 52! likely produced feelings of awe, incredulousness and possibly mirth. As IERG puts it, “images can be very powerful communicators of meaning… [and can often] carry more imaginative and memorable force than can the concept” (2018a, para. 1). If I had simply told my students that 52! is an incredibly large number, the lesson’s impact would surely have been deflated.

Extremes of reality are a cognitive tool that involves using weird and exotic examples from the world around us to engage students in the curriculum. For example, when teaching about the water cycle in science, one could explore the most interesting places that water might go as it is recycled, including sinking to the depths of the Marianas trench, being trapped in a glacier for thousands of years, floating in clouds around Mount Everest, or being swallowed into the body of a big blue whale (IERG, 2018b). In mathematics, it seems to me that extremes of reality can take on two dimensions. First, we can find historical events related to mathematics; for example, when introducing irrational numbers, we might tell the tale of how shocking they were to the Pythagoreans and how Hippasus was supposedly drowned at sea for discovering them.

However, there are also more abstract extremes of reality—weird and exotic concepts—directly within mathematics. These are what I use the most within my classroom. These could include extremely small and large numbers, the concept of infinity, fractals, Gabriel’s Horn, or even imaginary numbers, just to name a few. We can then take these concepts and go on fascinating thought experiments. A classic example is Hilbert’s Hotel, which beautifully exposes the peculiarity


of infinity with a hotel that, regardless of being completely full, can always manage to accommodate more guests. In the card-shuffling example, the fact that most deck orders have never been created is definitely an extreme of reality. Yet 52! itself is also an extreme of reality, as it is a gargantuan number. Czepiel then leads us on an incredible thought experiment to help us understand 52! as the extreme of reality in mathematics that it is. Problems that make use of mathematical extremes of reality can deeply engage students’ imaginations; some further examples will be explored below.

Finally, sense of wonder is a cognitive tool with which the reader will likely be more familiar. As Egan and Judson put it, “[w]onder seems to exist halfway between magic and the mundane… The sense of wonder focuses us on the real world, not the world of magic, but it enlivens the mundane features of the world by investing them with some of the sparkle of magic” (2016, p. 101). We can observe almost anything in the world around us with wonder. As IRGE points out, “[w]onder can be an engine of intellectual inquiry… Stimulating wonder energizes the literate mind” (2018d, para. 1). For example, we might wonder how many trees there on earth or why the grass is green. In mathematics, this sense of wonder can lead to some very interesting problems indeed. My grade nines wondered how many card shuffles have happened in casinos since the beginning of time, which led to a rich problem solving session. Perhaps in a math eight class, we might wonder how many people it would take to encircle the earth at the equator if they stood side by side and held hands. Or in a math 10 class we might wonder how long our hair would be if we had never cut it since being born and what its growth rate would be.

I would personally add another dimension to sense of wonder. As the last two examples above demonstrate, we can wonder about and mathematically explore situations in our world that have an element of make-believe to them, in that the situation may not be very plausible in the real world (i.e. very few people in the world have never cut their hair, and unfortunately humans cannot walk on water). I would push back against the notion that education in general and mathematics in particular must always be directly related to the real world. Many of the make-believe problems I use have resulted in high levels of engagement and rich mathematical learning. And simply put, they are a lot of fun to think about! As Eugenia Cheng puts it in her book, Beyond Infinity: An Expedition to the Outer Limits of Mathematics, “[m]athematics suffers a strange burden of being required to be useful. This is not a burden placed on poetry or music or football” (p. 11, 2017). Why shouldn’t mathematics be done purely for fun?

Thus, in addition to “I wonder,” I would add “what if?” What if we could fold a piece of paper in half as many times as we wanted…

how many folds would it take for the thickness of the paper to reach the moon? Or, what if the moon WAS made of cheese… how many pizzas could we make? (I seem to like problems about the moon!) These and problems like them make use of the three cognitive tools; students’ sense of wonder is stimulated, affective mental imagery is generated, and abstract mathematical extremes of reality are explored. For example, the paper-folding problem results in a great deal of wonder within students; I usually observe a classroom of bewildered and intrigued students when I first pose the problem and students eagerly offer their guesses. The problem invokes vivid mental imagery and leads the students down a most interesting thought experiment that demonstrates the shocking power of exponential growth, a mathematical extreme of reality.

So, as IERG and CIRCE have effectively argued, engaging students’ imaginations is truly a powerful method for teaching mathematics in a way that is memorable, meaningful, and exciting. Affective mental imagery, extremes of reality, and a sense of wonder are powerful cognitive tools that can be utilized through rich problems and activities that involve both real-world and make-believe situations. I have personally witnessed the magic that can happen when my students’ imaginations are captivated, and I strive to continue this on a daily basis. Feel free to email me at [email protected] if you would like some additional resources. Now, go make some history and shuffle a deck of cards!

References

Cheng, E. (2017). Beyond infinity: An expedition to the outer limits of mathematics. New York, NY: Basic Books.

Czepiel, S. (n.d.). 52 factorial. Retrieved from https://czep.net/weblog/52cards.html

Egan, K., & Judson, G. (2016). Imagination and the engaged learner: Cognitive tools for the classroom. New York, NY: Teachers College Press.

Imaginative Education Research Group. (2018). Affective mental imagery. Retrieved from http://ierg.ca/teacher-resources/teacher-tips/affective-mental-imagery/

Imaginative Education Research Group. (2018). Extreme of experience and limits of reality. Retrieved from http://ierg.ca/teacher-resources/teacher-tips/extreme-of-experience-and-limits-of-reality/

Imaginative Education Research Group. (2018). Imaginative education. Retrieved from http://ierg.ca

Imaginative Education Research Group. (2018). Sense of wonder. Retrieved from http://ierg.ca/teacher-resources/teacher-tips/sense-of-wonder/

Stevens, M. (2016). Math magic. Retrieved from https://www.youtube.com/watch?v=ObiqJzfyACM


Blending Geometry and Financial Literacyby Audrey Venner

In my classroom, I’m all about creating or implementing mathematical activities that reflect the outside world. To me, mathematics does not come from a textbook, but rather is embedded in everyday life situations.

At the start of a geometry unit with my grade 6/7 class, I was inspired by this photo tweeted by Dan Meyer, a former high school math teacher from Oakland, California (#iteachmath–Jan 4, 2019, photo by Jenna Laib, K-8 Math specialist from Boston, Massachusetts). It gave me the idea to simulate a bathroom reflooring problem in my class.

SHAPE PRICE

Yellow hexagon $6.50

Red trapeze $4

Blue diamond $2.50

Orange square $1

Green triangle $1.50

Beige diamond $1.75

The Situation

First, I created a “store.” In the store was a box full of shapes (like the ones in the above photo) for purchase with a chart of the corresponding prices on the whiteboard. I gave each pair of students a large hexagon outline on a piece of paper (later referred to as the “base”), a transparent plastic sleeve, and a whiteboard marker.With the framework established, I explained that they had to tile their base, based on these four parameters: A. They had to estimate how many of each shape they wanted by

only looking at the samples and the size of the base in front of them. No touching before buying!

B. They could only come to the store twice but could buy as much as they wanted each time.

C. They could not return materials after buying them. D. They had an unlimited amount of virtual money, but had to

spend the least amount to win.

As we got started some groups came up to me, the store keeper, and said what they wanted to buy by announcing the calculated total cost. I recorded their expenditures on the board for everyone to see. Some students were fast at deciding what to buy, while others took their time. Some used their marker to draw possible arrangements for their floor, while some came up quickly to make their first purchase by buying one hexagon to have a reference point for the rest of their construction.

As the activity continued, I noticed that the difference in pace from group to group was becoming quite significant. For those groups who finished quickly, I asked them to find out what the winning combination would be (the cheapest they could have spent while still covering the base) if they could start over, now having a better grasp on the parameters.

When all students had created their design, they participated in a gallery walk. Here are a few examples of what the finished floors looked like:


Real-Life Applications

Supply and demandI had 10 groups of two that morning, and by the time the eighth group came, there were no more triangles available. However, no one had bought squares yet. So I paused and I asked the class: “What would happen in a real-life situation if a store ran out of one item, but overstocked something else?” We all agreed that the triangles would be advertised as “sold out” while squares would likely go on sale. So I decided to put a “50% off” tag on the squares.

It’s a trap!With the new discount in place, one of the last two groups–who had not yet bought their tiles–decided to buy squares now that they were so cheap. They hoped to spend less money and maybe win the game, despite having not been first to finish. Towards the end, I noticed that this group was struggling to finish. I paused the game again and asked the students if they could see the link between what happened and what happens in real life. We discussed the fact that sometimes sales are made to be very appealing even if you don’t really need the items. In this case, squares were on sale, but they were not easy to work with in a hexagon-shaped base. So no matter how cheap they were, they were not the best purchase. It led to a very interesting and important discussion about marketing strategies which made the students reflect on the “dangers” of impulse purchases.

Bargaining, bartering, and a second-hand economy, oh my!Some groups had accidentally bought a few too many hexagons and asked me if they could sell them to other groups, since they were not allowed to return them to the store. I paused the class again and asked the students if this would be possible in real life. Some students shared that they knew of various platforms to sell or trade second-hand products, such as Craigslist, Bunz (a trading app), or Marketplace on Facebook. In addition, we recognized that there are no rules for picking a price when selling something second-hand. After this discussion, the group approached another to sell their

extra hexagons for $6.25 each (a $0.25 discount). This allowed them to correct their over-purchase while motivating the other group to buy from them. One person’s trash is another person’s treasure!

The price to pay for originalityAt the end, we noticed that some groups had designed a floor with no yellow hexagons, opting for many smaller pieces and, as a result, their bill was higher. We pointed out how sometimes, in real life, if you are going for a unique design, you might want to put a bit more money into a project. So, yes, they had spent more, but their design was unique.

Other Possible Extensions

After this activity I was reflecting on how many different ideas we had touched on during this lesson. But of course, we didn’t get them all. If we did it again, we could:

• Discuss the price of each shape and point out the fact that, even though some shapes are half the size of others, they don’t always cost half the price in a hardware store. For example, there are six cuts needed to create a hexagon, but four to create a trapezoid even though it’s half the size of the former;

• Ask the students to come up with the most expensive way to cover the base;

• Challenge them to find the combination of tiles that would make them spend a set amount of money ($100 for example);

• Challenge them to have at least one of each available shape in their plan.

Looking back, I feel that the richness in this activity came from not only the deep mathematical and practical thinking the students were doing, but also from all the discussions that emerged from the teachable moments. Isn’t that what mathematics should be all about?


When planning my assessment strategies for the 2018-2019 school year, there were a few issues that I wanted to address. I wanted to develop a plan that recognized the curricular competencies and not just the content. I wanted to move away from a points system where everything is recorded as a score and tallied in weighted bins. And I wanted to create a system that was efficient and effective in informing students where they are and where they are going in their learning. Working with a colleague at my school, we discussed possibilities around how to keep some of what we were already doing with assessment and how to create some new instruments that suited these goals. Before I share exactly where we landed, I would like to first describe why I felt that my assessment practices needed to change.

Years ago, assessment in my mathematics classes was comprised almost entirely of test writing. My students would write quizzes within chapters, tests at the end of the chapters and a final exam at the end of the year. Quizzes were the formative assessments for students to see how well they were understanding the new content, and tests (including the final exam) were the summative assessments for student learning. Points were awarded throughout the year and collected in weighted bins. Here is an example of the weighted bins from my 2013-2014 Mathematics 10 course outline:

FINAL MARK

WORK HABITS HOMEWORK

30% Quizzes50% Tests20% Final Provincial Exam

Figure 1: Final Mark calculation from 2013 Mathematics 10 course outline.

I have many concerns with respect to this dated system of assessment. Quizzes here were intended to be a type of formative assessment that informed the teacher and the student on how well

students were learning the material. This was very ineffective. Students would focus on the mark they received and not the mistakes that they made, and so students rarely learned from their quizzes. Moreover, students who performed poorly on their quizzes and then showed excellent understanding on their chapter tests would end up with lower than deserved grades in the course, because the lower quiz points were still affecting their overall grade. I was also concerned with the perceived objectivity of these assessments. I was not so certain that a student scoring 26 out of 30 actually knew more than a student scoring 24.5, and yet these students were separated by letter grades on a report card (Romagnano, 2000). The biggest concern that I have with this system is that everything that was assigned value in this mathematics class was related to test writing. As a mathematics educator, I value so much more in the development of my students.

Thankfully, we are now working under a new curriculum from the Ministry that also places value on other aspects of a student’s development within a mathematics classroom. This new curriculum motivated me to re-think how and what my students were learning in my classroom. I always thought highly of the seven “mathematical processes” that were emphasized in the old WNCP curriculum, but these processes were mere supporting features to the high and mighty content. In the new curriculum, the mathematical processes have evolved into curricular competencies, and more importantly, the curricular competencies are alongside the content as part of the Learning Standards for the curriculum. From the ministry’s web page on assessment: “Curriculum sets the learning standards that give focus to classroom instruction and assessment” (https://curriculum.gov.bc.ca/assessment); consequently, curricular

Changing Assessment within a Changing Curriculumby Michael Pruner

• Communication [C]

• Connections [CN]

• Mental Mathematics and Estimation [ME]

• Problem Solving [PS]

• Reasoning [R]

• Technology [T]

• Visualization [V]

Figure 2: Mathematical Processes from the 2008 WNCP curriculum.


competencies are now front and centre with the content as my focus for instruction and assessment.

What I will share with you now is the assessment system that my colleague and I have implemented for this 2018-2019 school year. It is not perfect–in fact it has a lot of room for improvement; it does however, approach my initial stated goals of placing value on the curricular competencies, moving away from a points system, and being an effective and efficient means for assessing student learning.

Within the new curriculum, the learning standards are made up of the curricular competencies and the content. Our assessment model has a similar breakdown; it contains tools for assessing development within the curricular competencies and the content.

Some of these tools are new and developed with purpose and others are old assessment tools that have been adjusted to suit our needs. Although each of these tools could probably be used to assess a variety of different aspects, we decided to assign each tool to a specific learning standard for simplicity. Below is the list of assessment tools that we are currently using in our classrooms matched with their corresponding learning standard. I have also included a detailed description of each assessment tool and student exemplars where appropriate. For context, I should point out that I teach in a linear timetable with 80-minute classes meeting on alternating days.

ASSESSMENT TOOL LEARNING STANDARDProblem Solving Term Assignments (New) Reasoning and Modelling

Problem Solving as part of test (New) Understanding and Solving

Unit Test (Adjusted) Content

Reflection Journals and Practice (New) Reflecting and Connecting

Group Quizzes (Adjusted) Communicating and Representing

Problem Solving Term Assignments

My 80-minute classes always begin with students solving a non-routine task like this one:

You have two colours of paint. In how many different ways can you paint the faces of a cube if each face is painted? Painted cubes are considered to be the same if you can rotate one cube so that it matches the other exactly.

I have shared these tasks in the past through Twitter (#weeklymathtasks) and the BCAMT listserv. I like to begin classes this way because it places a focus on collaborative problem solving (something that I truly value as a math educator), students tend to enjoy these types of problems, and these types of tasks help in developing and maintaining a thinking culture in the class. Within each of our three terms, I ask students to submit a problem-solving assignment based on one of these tasks. This is a take-home assignment where students are asked to demonstrate their mathematical reasoning, estimation, analysis and modelling, and curiosity. It is marked by me, using a rubric with letter grades A (extending), B (proficient) and C (developing). A copy of the rubric and criteria can be found here: https://bit.ly/2EZovKJ. This assessment tool is being used to measure progress under the curricular competency, Reasoning and Modelling (Analyzing). Each row in the rubric corresponds to a specific learning standard from the curricular competencies and is written in language that students and teachers can understand. Over the course of a year, each student will have three opportunities (one for each term) to demonstrate

Figure 3: Pre-Calculus 11 revised curriculum.


their progress in their Reasoning and Modelling within mathematics.

Reasoning and Modeling

• Develop thinking strategies to solve puzzles

and play games

• Explore, analyze, and apply mathematical ideas using

reason, technology, and other tools

• Estimate reasonably and demonstrate fluent, flexible,

and strategic thinking about number

• Model with mathematics in situational contexts

• Think creatively and with curiosity and wonder when

exploring problems

Figure 4: Reasoning and modeling curricular competencies.

Unit Tests and Problem Solving as Part of Test

To assess the Understanding and Solving category in the curricular competencies, we thought it would be a good idea to have this as part of the student’s normal unit test. Unit tests are the more traditional forms of assessment in our classrooms that most resemble our assessment from earlier days. We are now using this existing tool to serve two motivations in assessment. Unit tests are a good tool for assessing student knowledge of the content. We have four unit tests over the course of a year (each test covering about two chapters of study). Unit tests are made up of multiple-choice and free-response questions usually scored out of 20 points. These tests are re-writable if students show corrections and complete a small practice assignment. To keep

Understanding and Solving

• Develop, demonstrate, and apply conceptual

understanding of mathematical ideas through play,

story, inquiry, and problem solving

• Visualize to explore and illustrate mathematical

concepts and relationships

• Apply flexible and strategic approaches to solve

problems

• Solve problems with persistence and a positive

disposition

• Engage in problem-solving experiences connected with

place, story, cultural practices, and perspectives relevant

to local First Peoples communities, the local community,

and other cultures

Figure 5: Understanding and solving curricular competencies.

re-writes manageable from a teacher’s perspective, they take place for all classes during a single lunch hour once per month. We have also adjusted this tool slightly to include an assessment of the Understanding and Solving curricular competencies. We do this by including an open-ended non-routine task at the end of each test. Below are two examples of these tasks, one from our Mathematics 9 test covering rational numbers and scale factors, and the other from a Pre-Calculus 12 class covering exponential and logarithmic functions.

Here is a map of your school and neighborhood. Design Terry Fox walk/run that is about 3km.

Figure 6: Sample task from Math 9 Unit test

Figure 7: Sample task from Foundations 12 Unit test

We include the rubric for marking this final task in the test, so students can see what we are focussing on for this assessment. After grading the unit test, students receive the test back with a

Minimum wage in BC according to the year it changed. �


re-writes manageable from a teacher’s perspective, they take place for all classes during a single lunch hour once per month. We have also adjusted this tool slightly to include an assessment of the Understanding and Solving curricular competencies. We do this by including an open-ended non-routine task at the end of each test. Below are two examples of these tasks, one from our Mathematics 9 test covering rational numbers and scale factors, and the other from a Pre-Calculus 12 class covering exponential and logarithmic functions.

Here is a map of your school and neighborhood. Design Terry Fox walk/run that is about 3km.

Figure 6: Sample task from Math 9 Unit test

Figure 7: Sample task from Foundations 12 Unit test

We include the rubric for marking this final task in the test, so students can see what we are focussing on for this assessment. After grading the unit test, students receive the test back with a

content grade out of 20 and an understanding and solving letter grade (A, B, C or Incomplete). A copy of the rubric can be found here: https://bit.ly/2snX9pp. The description for this curricular competency includes: “Engage in problem-solving experiences connected with place, story, cultural practices, and perspectives relevant to local First Peoples communities, the local community, and other cultures.” To address this competency, we decided to design tasks that were connected to place or included cultural perspectives relevant to local First Peoples communities. This is a challenge to accomplish and a work in progress, as you can see from the two examples above. Over the course of a year, each student will have four opportunities (one for each test) to demonstrate their progress in their Understanding and Solving within mathematics.

Reflection Journals and Practice

Reflecting on learning and practice is a daily activity in our classrooms. At the end of each class, students are provided with a reflecting question that captures the main concept being taught and a set of questions to practice at home. During their home practice, students are expected to take time (20–30 minutes in total) to think and respond to the reflection question and then

Connecting and Reflecting

• Reflect on Mathematical thinking

• Connect mathematical concepts with each other, with

other areas, and with interests

• Use mistakes as opportunities to advance learning

• Incorporate First Peoples worldviews, perspectives,

knowledge, and practices to make connections with

mathematical concepts

Figure 8: Reasoning and modeling curricular competencies.

spend the remainder of their time practicing the mathematics. Having the advantage of digital technologies (we use Scholantis in our district), students upload a photograph of their practice and reflection response to their class portal. I do not have the time to check this daily; but, once every five or six classes, I view one of their uploads and make comments on the quality of their reflection and practice. Every two weeks, students complete a self-assessment of their Reflecting and connecting by completing this rubric: https://bit.ly/2AITKGa.

Figure 9: Self-Assessment for Reflecting and Connecting

When completing these self-assessments, I ask that students provide evidence if they think they are at a “B” or “A” level in any of the rows. This evidence can either be a detailed description or a photograph of their work. I quickly scan these self-assessments before awarding students their self-assigned grade. Most of the time, I agree with their assessment; however, sometimes there are a few that I need to adjust down due to lack of evidence or some that I need to adjust upward due to students being too hard on themselves. Each term, I have three to four of these assessments to look at before deciding on a term grade for this competency.

Group Quizzes

I started giving group quizzes to my classes a few years ago, and I was struck by the level and quality of engagement by my students. This is one of the few times where I would hear students arguing with one another about mathematics, and I found it to be very effective for student learning. These quizzes were typically shorter than a regular quiz with only one or two questions, and the questions were more open and complex. For example, instead of asking students to factor x2-x-6, in a group quiz, I would ask students to show all possible factorizations for x2+bx-6 with corresponding values for b. I eventually moved to a point where all quizzes are now written in group form. These quizzes are


Communication and Representing

• Explain and Justify mathematical ideas and decisions

in many ways

• Represent mathematical ideas in concrete, pictorial,

and symbolic forms

• Use mathematical vocabulary and language to

contribute to discussions in the classroom

• Take risks when offering ideas in classroom discourse

Figure 10: Communicating and representing curricular competencies.

either written in groups at whiteboards or at their tables, each having its own unique benefits. The highly collaborative nature of these quizzes aligned very nicely with making this a tool to measure student’s Communicating and Representing curricular competencies. I still mark, make comments and record each quiz using a 4-point scale: A, B, C or Incomplete, but these marks are used strictly to inform students on their progress and understanding of the concepts, they are not used in determining a reporting grade. At the end of each quiz, students complete a self-assessment that attends to how well they represented their mathematical ideas and how well they communicated within their groups: https://bit.ly/2Mb40LW. Each term, I have two to three of these assessments to review and decide on a term grade for the curricular competency heading: Communicating and Representing.

Figure 11: Self-assessment for Communicating and Representing


Reporting

After each term and at the end of the year, I am required to produce a summative letter grade for my grades 8 and 9 classes, and a summative percentage for my grades 10–12 classes. If you have been with me up until now, you will have noticed that I have a collection of letter grades (from the competency assessments) and percentages (from the content assessments on the unit tests). To meld all of these into a single result, we use an Excel spreadsheet for organizing all of this data. Here is an example of what a term report looks like for one of my Mathematics 9 students:

Now that the data is organized, I can use my professional judgement to determine a suitable letter grade to represent this student’s progress. This student has demonstrated B’s and A’s through her curricular competency assessments (note: the 0’s are a result of the spreadsheet not having a value to show, they do not represent “no achievement” and do not factor into my overall assessment). She

also demonstrated a C+ level of achievement on her Unit 1 test, so when I put these two assessments together, I decided on a B for her Term 1 report. For Grade 8 and 9, this is sufficient; however, for grades 10 – 12, I need to turn this into a percentage. My colleague and I decided on using cut scores to help with this conversation (see Figure 13).

For the student above, I would have given her a 75% if I was required to give a percentage. I see that her curricular competencies showing B’s and A’s melded with her 71.5% content work justifies a B for her term. I should also add that because this letter grade is a result of my professional and

subjective judgement, I also consider my experience of working with this student. This includes observations and conversations throughout the course of the term. It is freeing to not be bound by an accumulated percentage based on points earned when deciding on a student’s level of achievement in a class.

High A A Low A High B B C+ C Low C C-

98% 90% 86% 80% 75% 70% 65% 60% 55%

Figure 13: Grade ranges

Figure 12: Student term report


Summary

I feel that these assessment tools go a long way towards meeting my goals from the beginning of the year. They recognize the significance of the curricular competencies within the new curriculum, and they have transitioned away from a points system to a system that recognizes growth and requires professional judgment in reporting progress. Efficiency was my third goal, and I do believe that it has the potential to be an efficient assessment system. The two self-assessments were intended not only to provide opportunities for autonomy in student learning, but also to make the entire assessment system more efficient. Right now, I still find that I am spending too much time reviewing and adjusting my student’s self-assessments; and therefore, the efficiency is lower than intended. In reflection, perhaps this is something that I need to let go of. I think that I can be satisfied with the fact that my students are spending time thinking and writing through these assessments and this demonstrates to them that it is a valued part of my mathematics classes. Perhaps this is good enough, and I do not need to review them so carefully.

Thinking forward to next school year, I would like to further refine the competency assessments, and I would like to begin the transition to standards-based grading (SBG) in course content. I do believe that SBG is a better tool for providing information to students on where they are and where they are going with respect

to the content, but it was too much for us to take on for this year. I have always known the mathematics classroom to be a complex space of content intertwined with actions and social interactions to support learning. In this new system of assessment, I am seeing students also understanding the classroom as this same complex space.

The conversation in all of my classes has changed. Students are not focused on half points earned on quizzes and tests, nor are they asking about percentage bumps to improve overall grades. There is now more conversation around actual content learning and genuine efforts to improve in specific competencies. I have seen students go out of their way to provide multiple solutions or explain their thinking on group quizzes in order to provide an improved outcome on their communicating and representing self-assessment. By placing value on other aspects of learning in a mathematics classroom, students are now seeing the importance of the curricular competencies in their own learning of mathematics. Students are working to develop and improve in the curricular competencies and they are seeing first hand how this is helping them with learning the content.

Reference

Romagnano, L. (2001). The myth of objectivity in mathematics assessment. Mathematics Teacher, 94(1), 31-37.


Re-Thinking Place Value: The Balance of Metaphor and Metonymby Adam Fox

“Mr. Fox, do you know what one million plus one million is?” asks Jacob. I purposefully offer him an incorrect answer and hide a smile. After all, why should I rob this eager, smiling student of his sense of achievement at knowing and sharing the sum of these incomprehensibly large (to him) numbers. Jacob is five years old and a student in my kindergarten class, and I am hiding my smile because Jacob’s question is similar to others I have received from students in kindergarten, grade 1 and grade 2 during my brief career as a teacher. During this time, I have noticed how students are fascinated by counting “for fun,” particularly when using large numbers, when they will often challenge one another to count as high as possible or undertake calculations on large numbers.

Area of interestMy interest in exploring intransitive counting and place value, for which I will be using a Gattegno tens chart (I will discuss this later), stems from watching a video of Alf Coles undertaking a similar activity with students in England.

I have also been influenced by the article “Re-thinking Place Value: From Metaphor to Metonym” (2017) where Alf Coles and Nathalie Sinclair consider what it means to count intransitively, without the context of one-to-one correspondence with objects, and what it means to “know about place value” (p. 5). In the article, Coles and Sinclair make the ambiguous claim that “early learning . . . does not lay the ground for later development of mathematics” (Coles & Sinclair, p. 8). Coles and Sinclair’s statement raises the question: To what extent are students limited developmentally in the complexity of the work they are able to comprehend?

Some background on countingIt is difficult to imagine a time in a child’s school “career” when they are more enthusiastic and willing to learn than when they are in kindergarten. Students often get excited by the most basic of games and are bundles of inquisitive energy, and it is

an educator’s responsibility to nurture this energy and promote a love of learning. Kindergarten students also love to talk. Ask any kindergarten teacher about the longest period of silence they have observed in their class (apart from snack time!) and they will likely tell you it is shorter than the amount of time it takes to boil water in a kettle. This combination of a love of learning and a love of talking, whilst also being a predominantly “blank slate” regarding formal mathematics education, makes kindergarten students a model of reference for exploring a progressive approach to counting and learning place value.

Not so long ago there were debates between mathematicians, philosophers of mathematics and psychologists regarding the most effective way to teach number sense. The outcome was that using a cardinal approach, the counting of “things,” became the most commonplace method to teach counting (Coles & Sinclair, p. 4). It is believed this metaphorical method of counting helps students understand what number “is” by replacing one thing with another, for example, the number “3” becomes three buttons. In contrast, some, such as Peano and Dedekind argued the focus should be on the ordinality of numbers, that is the position of a number in a series and its relation to the numbers before and afterwards, with no regard to quantity. Indeed, Seidenberg suggests, “the ordered recitation of the list of number words long precedes, historically speaking, the more cardinal counting of things” (as cited in Coles & Sinclair, p. 4).

Coles and Sinclair’s article “Re-thinking Place Value: From Metaphor to Metonym” (2017) advocates for students to be taught place value in an ordinal way. Their research found using TouchCounts (an iPad app) and a Gattegno tens chart “powerful” due to the connections of symbols, sounds, names, touch and gestures afforded by these tools (Coles & Sinclair, p. 8). This is a view shared by Jan Van Den Brink in in his article “Acoustic Counting and Quantity Counting” (1984), where he discusses the correlation between acoustic counting, i.e. intransitive counting, movement and the “sound systematics of acoustic counting” (p. 3). Van Den Brink observed the improved ordinal counting


ability of students who counted rhythmically with a tambourine while using movement (slowing, quickening, stopping) to control counting (p. 2) and the analogous relationship between numbers, for example, one hundred, two hundred, three hundred. In addition, he noted the influence of acoustic counting on quantity counting, where there are three elements of counting that must be combined when counting objects, with a) and b) being fundamental in acoustic counting:

a) The movements necessary for pointing out or moving objects

b) The numerals that must be expressedc) The objects to be counted

(Mierkiewicz and Siegler, as cited in Van Den Brink, p. 8)

Van Den Brink concludes that “when taking all these factors into consideration it would seem that in the first place counting has to do with movement and sound and only in the second place with objects” (p. 3).

My explorationCounting debates between mathematicians and psychologists, such as, Gelman & Meck, Russell (in favour of cardinal counting), Peano, Dedekind, Coles and Sinclair (who support ordinal counting), are conspicuous due to the absence of a recommendation from one notable person, the child psychologist Jean Piaget. In contrast to his peers, Piaget favoured a balanced approach to number and combined the ordinal and cardinal (Coles & Sinclair, p. 4). It is my opinion that people should aim to live a balanced life, and that too much of one thing is unhealthy. My philosophy extends into mathematics where I believe, like Piaget, that focusing purely on ordinality or cardinality to teach number sense, while each focused approach has its merits, will

ultimately not be as successful as adopting a balanced approach by combing the ordinal and cardinal.

I raise the question, “What is an adequate balance of ordinal and cardinal methods for teaching young students number sense?”

To explore this question, I used two tools with my kindergarten class, a Gattegno tens chart and Dienes (base 10) blocks. All students first began to count intransitively using the Gattegno chart, with the Dienes blocks being introduced later as a natural evolution into the metamorphic. In addition to whole-class work, I also worked individually with select students. Before discussing my approach and the outcome in more detail, I describe the Gattegno chart and Dienes blocks, offer supplementary information regarding school location and classroom demographic, and discuss data collection.

Working with a Gattegno chartThe Gattegno tens chart displays numbers in rows and always includes the “units” row and would typically also display the “tens” and “hundreds” row. It is also possible to include decimal rows. Some Gattegno charts place the highest numbers at the top, while others place them at the bottom. Although there is possibly some merit in placing the higher numbers at the top of the chart, which would connect the notion of “up” with large numbers and “down” with small numbers, and could be helpful for students exploring decimals, I opted to have the higher numbers at the bottom of the chart to reinforce to my students, who are just learning to read, the concept of reading from left to right and top to bottom.

In my classroom, I used a Gattegno chart similar to the one below (see Figure 1), however, without decimal numbers, and I also had three blank rows underneath where I intended to place

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0

10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0

100.0 200.0 300.0 400.0 500.0 600.0 700.0 800.0 900.0

1,000.0 2,000.0 3,000.0 4,000.0 5,000.0 6,000.0 7,000.0 8,000.0 9,000.0

10,000.0 20,000.0 30,000.0 40,000.0 50,000.0 60,000.0 70,000.0 80,000.0 90,000.0

100,000.0z 200,000.0 300,000.0 400,000.0 500,000.0 600,000.0 700,000.0 800,000.0 900,000.0

Figure 1: An example of a Gattegno tens chart


the 1,000,000, 10,000,000 and 100,000,000 rows at a later date. This intentional omission created some very interesting and unintentional discussions, which I will recount later. In addition, I considered it important to include a comma to separate the thousands in order to aid number recognition, especially from a distance, while also facilitating choral chanting by splitting numbers into distinguishable components.

Working with Dienes blocksDienes blocks are a mathematical manipulative used by students to learn mathematical concepts including counting, number sense, place value, addition and subtraction. Generally, the blocks come in four sizes to indicate their place value: units (ones place), longs (tens place), flats (hundreds place) and blocks (thousands place).

Figure 2: Base 10 Dienes blocks

There are two notable differences between standard Dienes blocks and the blocks I used in my class: I wrote the corresponding number of zeroes on the blocks using a black marker–one zero on the tens blocks, two zeroes on the hundreds blocks and three zeroes on the thousands blocks–to reinforce the link between the ordinal and cardinal, and I taped ten one thousand blocks together to create a ten thousand block (not shown).

Figure 3: The Dienes blocks used for this project

Background information I teach at a public elementary school in a suburb of the District of North Vancouver. In my class of twenty students, eighteen have attended pre-school, with two of those having attended school in a different, non-English speaking, country. Six students do not have English as their first language at home, and three of those students receive in-school support as an English Language Learner (ELL).

The class is very active and requires frequent “brain breaks,” with five boys having difficulty self-regulating and maintaining focus for even short periods of time. There are no identified students with special needs, which is typical of a kindergarten classroom, as students have not been in school very long, however, one boy is possibly on the autism spectrum (high functioning) and three other boys will be referred to the school-based resource team for further assessment.

ObservationsThe first whole-class observation occurred in mid-February, approximately three weeks after I had placed a large Gattegno chart on the wall in my “calendar” area, an area near a carpet where students often sit to receive instruction. Surprisingly, no student had commented to me about the chart in this time. However, when we began the lesson, all the students were alert and keen to be introduced to something new. I started by asking the class what they noticed about the chart. Sarah said, “It goes 1, 1, 1, 1”, referring to the numbers in the first column all starting with a 1. Other students made similar statements for the other numbers. One student observed, “There is one zero, two zeroes, . . . five zeroes.” I pointed to the blank space reserved for the millions row and asked, “How many zeroes would be here?”, to which the class chanted, “Six zeroes.” I proceeded to point to the blank spaces for the ten-millions row and hundred-millions row, before working my way down imaginary rows all the way to the floor, with students increasing the number of zeroes by one each time. Interestingly, no student commented that the first row does not contain any zeroes. Brian noticed how the numbers, “Look like stairs.” Brian’s comment and the observation regarding the number of zeroes are pertinent because it shows students are attending to place value purely from the visual aesthetic of the numerals on the chart, which echoes Coles and Sinclair’s statement regarding students “attending to place value seemingly without concern for the actual size of the number involved” (Coles & Sinclair, p. 3). Moreover, the rows on the chart I intentionally left blank were responsible for a wonderful five-minute discussion of number:


Student A: There are numbers up there and not numbers down there.

Fox: Do you think there are numbers down there? [points to blank rows on Gattegno chart]

Student A: It’s as high as it goes.

Student B: If it’s not there, it doesn’t exist.

Student C: It just keeps going.

Fox: Do numbers ever stop?

Student C: No, they just keep going.

[Multiple students agreed. Jacob proudly informed the class that numbers do stop because he had counted to the last number!]

This conversation showed the broad range of thinking between my students. Having an awareness that numbers do not stop, aged four or five, is incredibly lucid. The view that “if it’s not there it doesn’t exist” can possibly be attributed to the student’s experience of counting using metaphors, when counting ceases if there are no more objects to be counted. Returning to the Gattegno chart, Iris said she noticed how the numbers, “Go 1, 2, 3 . . . 8, 9, 10 . . . then, 10, 20, 30 . . . 90 . . . then, 100, 200, 300 . . . 900.” I was shocked and amazed at her ability to recognize all the numbers to 900! When the students had finished sharing their observations of the Gattegno chart, I had the class chant the number I pointed at. I began at 3 (I don’t know why) and continued 4, 5, 6, 7, 8, 9. In the tens row I skipped from 10 to 60, which I pronounced “six-ty,” with a slight pause between “six” and “ty,” and an emphasis on the “ty.” I continued by pointing to 70, 80 and 90, before reversing direction and counting down to 10, each time emphasizing the “ty.” This was a deliberate act to introduce students to the repetitive, metonymic nature of counting before the pattern is disrupted by fifty, thirty, twenty and ten. I encouraged students to say “five-ty,” “three-ty,” “two-ty,” and “one-ty,” but many were aware of the correct names and I said, for example, “We can say ‘two-ty’, but we can also say ‘twenty.’” Throughout the observations I purposefully used both naming conventions as I could not be sure all students were aware of the correct terms. I continued my work with the Gattegno chart for five more classes, with the students chanting more complex numbers in expanded form, for example 5367 as I pointed at 5,000, then 300, followed by 60 and finally 7.

The Dienes blocks were introduced after I was confident in the ability of my students to intransitively count numbers on the Gattegno chart. I began by asking the class:

Fox: What does two zeroes mean?

Class: Hundred.

Fox: What number is this? [Places a Dienes flat with “00” written on it in front of the class]

Class: Hundred.

Fox: That’s right. Two zeroes mean a hundred. But how many of these are there? [Waves flat in air]

Class: One

Fox: What number is this? [Adds a second “flat”]

At this point the class shouted random numbers, so I addressed the question to Jason:

Jason: Two

Fox: Two what?

Jason: Er, thousand?

Fox: No, what does two zeroes mean?

Jason: Hundred. Two hundred.

Fox: What number is this? [Adds another “flat”]

The class began to recognize the pattern and join in as I repeatedly add flats. I repeated the process with “longs” (tens) and “blocks” (thousands), with the class continuing the counting pattern without any intervention from me. Finally, I created the number 241 using Dienes blocks and asked the class if they could tell me what number it is. With reminders from me to consider the amount of each type of Dienes block on display the class was able to correctly identify the number. This represents a fundamental shift from the rhythmic, intransitive and metonymic counting of previous observations to transitive and metaphoric counting using physical objects to represent number.

Ensuing lessons had the students taking it in turns with one student pointing to numbers on the Gattegno chart and another student creating the number using Dienes blocks. I also began to write the numbers on a whiteboard as the students pointed at them, first in an expanded form across the board, followed by a more traditional vertically stacked form (while modelling basic addition), and finally in standard form.

On one occasion, I wrote the numerals 165 and asked two students to count using the Gattegno chart and Dienes blocks. Both stood nervously in front of their peers. When Nancy pointed, unsure, at 1,000, Jacob responded, “No, that’s one thousand. Find the one hundred above it. Good, now find sixty. Yup. Now find five.” Jacob had read standard form number and had become the teacher.


Returning to school after a two-week spring break, I challenged Jacob and Mable to write the number 2144 in another way. After some confusion I encouraged them to consider numbers they had seen previously, before they were “pffft” (when “squishing” [as I told the students] the vertically aligned numbers for addition I had purposefully made a “Pffft” sound in an effort to associate a sound with the action). Both Jacob and Mable immediately began writing (see below).

Figure 6: Jacob (left image) and Mable (right image) expanding 2144

Jacob showed that, while the numbers are not aligned correctly, he could separate the numbers by place value. Unfortunately, Mabel’s work was erased before I could take a photograph, however, she wrote the number as seen above. She displayed an awareness of place value by staggering the numerals, and her thinking became apparent as, when asked to read the number, she correctly read, “two-thousand, one-hundred, forty-four.” Mable is astute in recognizing numbers are written from right to left, however, confusingly for new learners, we read numbers from left to right. It is also notable that she did not include any zeroes, despite showing an awareness of place value and the increase in size of each number.

FindingsIn the previous sections, I have described some of my experiences of working with kindergarten children, in which adopting and maintaining a balanced approach to counting, using both the metonymic and metaphoric in tandem, intimates that young children can acquire and develop a knowledge of number and place value that far outweighs current expectations.

I suggest an initial focus on intransitive counting and the metonymic, in a group setting, prior to the introduction of the metaphoric. It took four sessions of approximately ten minutes each for me to be confident of my class’s ability to count using the Gattegno tens chart, before introducing the Dienes base 10 blocks.

It was interesting to listen as the whole-class chanting transformed from a collective voice into a rhythmical, melodic tune. I consider my actions of pointing to the first numeral and dragging the pointer beneath the remaining numerals conducted and facilitated the chanting; however, I believe it is likely students would have developed their own rhythm in due course as they become more familiar and confident counters. Rhythmical counting, with a slight pause between first and last sounds, helped my students when moving from the metonymic to the metaphoric, a phenomenon Van Den Brink also observed in his students (p. 12).

This balanced approach to counting, using the Gattegno chart and Dienes blocks in tandem, was demonstrated by my students’ ability to effortlessly move between the metonymic and metaphoric with little regard to the subtle change in the nature of their counting.

Final ThoughtsThis project was fascinating. I was continually amazed by my students’ ability to learn new concepts quickly and demonstrate them in multiple ways. My students loved counting for, and quizzing, their parents, and I received lots of positive feedback from parents who were amazed by their child’s achievement.

I do wonder, and I am keen to see, how another group of students responds to a similar process. I look forward to refining the process and using both the Gattegno tens chart and Dienes blocks with my class next year.

References

Coles, A. (2014, February). Journeys on the Gattegno Tens Chart. Retrieved from NRICH: https://nrich.maths.org/10314

Coles, A., & Sinclair, N. (2017). Re-Thinking Place Value: From Metaphor to Metonym. For the Learning of Mathematics, 37(1), 3-8.

Van Den Brink, J. (1984). Acoustic Counting and Quantity Counting. For the Learning of Mathematics(Vol. 4. No. 2.), 2-13. Retrieved from http://www.jstor.org/stable/40247847


I have wrestled with how to use technology effectively in teaching and learning mathematics for almost three decades. There is no doubt in my mind that technology can enhance the teaching and learning of mathematics, deepen students’ understanding, and enrich their mathematical experience. However, as teachers, we also know that technology can be a double-edged sword. We have students who reach for their calculators, and other tools, when we feel they shouldn’t need to do so. In my efforts to nurture a healthy dependence on technology with my students, I haven’t always succeeded. However, I am often delighted by students’ insights and creativity which arise from their use of technology.

As do many high school mathematics teachers, I give my Precalculus 11 and 12 students the opportunity to deepen their understanding of transformations of functions and to demonstrate their understanding of transformations by having them create (or re-create) an image using Desmos or other graphing technology. The process of tweaking the equation of a function with instant feedback until the desired shape is achieved is conducive to learning about functions and the results are often very gratifying. Still, there are times when I want to revert to old-school practices of not allowing students to use technology for their “Fun with Functions” assignment–like the time I discovered that a few of my students had “borrowed” their ideas from the internet and practically copied the idea, function for function. There are so many Desmos creations out there, I have learned to be vigilant.

Every once in a while, I am wowed by a stu-dent’s creation. This year, one of my stu-dents created the im-age on the left. I was impressed with how topical it was—and by the amount of effort that went into it. This

work, entitled The Trumpster is by Dylan Matthews. www.desmos.com/calculator/4dv9tkztn6

Antoine Mouchet, an international stu-dent from Belgium, used an impressive number of functions to create his version of the famous bronze statue in Brussels, Manneken Pis, by

sculptor Jerôme Duquesnoy. If you go to the link, you will see that this one is animated!www.desmos.com/calculator/5xyg29gnaf

Joaquin Suarez used 179 functions to achieve a remarkably detailed fit to the image of “The Astronomer.”

www.desmos.com/calculator/b6nde4ha5q

Jenny Lie, international student from Norway, worked meticulously to achieve the detail she wanted on this beautiful image of a

bird in flight. www.desmos.com/calculator/ckb1iickql

Eliza Wilson’s Sydney Opera House is a sort of tribute to her Australian roots.

www.desmos.com/calculator/ns14zijv0h

The Evolution of Fun with Functionsby Michèle Roblin


And there’s something Roy Henry Vickers-esque about Sam Watt’s creation. www.desmos.com/calculator/cafagm4ouq

Technology has definitely allowed my students’ fun with functions to evolve over time. Years ago, when I had students create images by hand, they did not have the same ability to tweak their functions—and, if they did, it was certainly a much more time-consuming process. Below are some of the older submissions, done without the use of technology. They were impressive in their own right.

Michèle Shannon (c.1985)

I still remember a conics assignment I had to do when I was in high school in the mid-80s. Every student in the class was given the same list of equations and inequalities. I remember enjoying the assignment, but we all had the same finished product (shown above)–or, at least everyone who had done it right!While it is difficult to deny that technology allows for more fun and creativity in the math classroom, I remain uncertain as to which activity allows students to better their knowledge of functions and/or their ability to graph functions.

There is only one thing of which I am certain: “Old math teachers

never die—they just lose their functions.”


A Function of Freedom and Constraintsby Chris Hunter

Many teachers have been incorporating open questions into their math lessons. An open question is designed to elicit a variety of possible approaches or solutions. But what if an open question doesn’t elicit multiple approaches?

At the Northwest Mathematics Conference closing keynote, Nat Banting proposed a conjecture: “Shifting constraints triggers new mathematical possibilities.” Making an open question less open can lead—perhaps counterintuitively—to a more diverse and complex set of student responses. Nat’s closing keynote at “the Northwest” reminded me of an experience from last year in which this relationship between freedom and constraints played out in the classroom.

In June, a colleague invited me into his classroom to teach a mathematical modelling task—Desmos’ “Predicting Movie Ticket Prices”—in his Pre-calculus 12 class. Students had experienced exponential functions earlier in the course. We were curious about whether his students would apply what they knew about exponential functions to a task situated outside of an exponential functions unit—a task not having to do with textbook contexts of half-life, bacteria, or compound interest. They did. And they deepened their understanding of how change by a common ratio appears in exponential equations (vs. change by a common difference in linear equations). They did this within 45 minutes of a 75-minute class. So, my colleague let me try out another, less sexy, task—one adapted from Mathematical Assessment Resource Service (MARS). This task, like much of Pre-calculus 12, is about naked functions; no real-world context here.

Figure 1

The original MARS task is closed: two functions, one linear and one quadratic, each passing through four points (Figure 1). I wanted to open it up so I changed the prompt: “A set of functions pass through the points shown. What could the equations for the functions be?” Also, I removed one of the points—(5, 3)—to allow for different solutions of two functions. Again, the thinking is that open questions encourage a variety of approaches. And then, from fifteen pairs of students, fifteen identical solutions (Figure 2).

Figure 2

I anticipated this uniformity. I had lowered the floor but no one entered {y = 5, y = 7, y = 8, y = 9} to duck a linear-quadratic solution. I had raised the ceiling but no one wrestled with equations for sinusoidal or polynomial or radical or rational functions. This makes sense: the points scream linear and quadratic. To use Nat’s phrasing, they are sources of coherence. That is, how the points were arranged was familiar and recognizable to students, giving them a way to begin to make sense of and engage with the task. Students knew that a linear function contains points that “line up”; they knew that a parabola has symmetry at its vertex.

The freedom within my open question didn’t bring about new and diverse ideas. To support creativity—mathematical creativity!—I had to introduce a source of disruption, a constraint: “A set of nonlinear functions pass through the points shown. What could the equations for the functions be?” Not allowing linear relations jolted them out of thinking about lines; it sparked them to think about other possibilities, other functions from Pre-calculus 12. Students now generated a wider range of solutions.


A student could have used the linear nature of absolute value functions to get around my nonlinear constraint (Figure 3) but no one did.

Figure 3

Instead, some students picked up on the symmetry of two new possible parabolas (Figure 4).

Figure 4

Writing the equation of the second parabola—finding the parameters a and q—presented more of a problem.

Others bent the line; they saw the middle of its three points as the vertex of a cubic function that had been vertically stretched and reflected (Figure 5).

Figure 5

Some saw four compass points one unit north, east, south, and west of an imagined centre and wrote an equation of a circle (Figure 6). This led to a function-versus-not-a-function conversation—”Does that count?”–that would not have taken place without the constraint. Others saw a sine function that passed through three of these four points. There were also solutions that did not quite pass through the given points. Notice that the sine function in Figure 6 narrowly misses (1, 5) itself (but grazes the dot that represents it).

Figure 6

I didn’t anticipate these “close enough” solutions. Students weren’t as constrained by “pass through” as I was, an interesting difference that would not have come up in the original task from MARS. Within each new solution, students remained motivated to capture the points using only two functions, as before.

With more time, I could have shifted constraints yet again: “A set of functions pass through the points shown. What could the equations for the functions be? (P.S. The graph of at least one of them has an asymptote.)” Again, this constraint could disrupt students’ thinking. Prior strategies would be made obsolete. This could have triggered exponential and logarithmic or rational functions: “What do I know about asymptotes?” Even without introducing this tweak to the problem, we noticed at least one student playing with rational functions at the end of class.

Nat reminds us that it is our job as teachers to observe and adapt to what’s happening with our learners in the present tense. I did not set out to write the one question with the perfect degree of openness the Goldilocks open question. Rather, this activity illustrates a process of tinkering with the question at hand in light of students’ responses.

Above, there’s evidence to support Nat’s conjecture. Despite its virtually limitless possibilities, my more open question didn’t cut it. To orient students to these possibilities, restrictions had to be placed. The student thinking—and conversations that I had hoped for—only emerged when freedom intersected with constraints.


While working for General Electric in the 1930s, Frank Benford followed up on an observation that tables of logarithms showed more wear and tear on the earliest pages than later ones. He considered other sources of numbers and showed that the first digits of numerical data from a wide range of circumstances demonstrate a pattern. Numbers that start with a digit of one occur about 30% of the time! In his exploration of this phenomena, he observed the pattern in data for the area of rivers, street addresses, numbers appearing in newspapers and Readers Digest, baseball statistics from 1936, power lost in air flow, even the number of pages in publications and number of footnotes (Benford, 1938). Subsequent exploration of the phenomenon has shown this awareness has practical use for detecting fraud and spotting unusual events in large collections of data. It is considered to be a general probability phenomenon.

In this article we explain why the phenomenon arises in fraud detection. We do this using domain and range considerations that are suited to pre-calculus courses. Our goal is to provide a mathematical explanation of Benford’s law for specific functions. This has implications for statistics but we do not examine the general application to statistical data. For teachers of statistics we aim to provide a foundational understanding so that they are comfortable exploring a significant opportunity.

Benford’s law is a model of the occurrence of the left-most digit in sequences of numbers. For example, in the sequence 3, 9, 27, 81, 243 the left-most digits are 3, 9, 2, 8, 2. In this case the digit 2 has occurred the most in this sequence. The practical value of Benford’s law arises in fraud detection where the growth of money over time allows one to examine the left-most digits as a histogram that Benford found should have a particular structure. In this article, we propose ways that the theory can be developed as a pre-calculus activity and then we show ways that it can be used in Grades 7 to 9 within data management.

Compound interest is exceedingly common with respect to financial transactions such as loans or investments. Graphically, this provides an exponential curve that shows how the value of the money increases with time. What Benford’s law highlights

is that, when the range of the exponential function is broken into segments defined by the first digit that the width of the corresponding domain is not uniform. Consider, as an example, the value of a dollar invested with an unreasonable but purposeful 16.6% annual interest rate: y = 1.166x A graph of the function for a 15-year period is shown in Figure 1. Note that the range of values is 1 ≤ y < 10, which means that different values of y reflect all possible left-most digits (i.e. y can begin with a 3 for example). This property is the reason for choosing an unrealistic interest rate.

In terms of understanding Benford’s law, consider the portion of the range where 1≤ y <2. All of these values of y have a left-most digit of 1. Similarly, 2 ≤ y <3 corresponds to the left-most digit of y being 2. Similarly, for the other digits 3, 4, …, 9. Note that the digit zero is not considered. The key concept for Benford’s law is that the intervals in the range correspond to the left-most digit and the corresponding intervals in the domain are the length of time that a particular left-most digit should occur. This is shown for the digit 2 in

Using Benford’s Law in the Classroomby Timothy Sibbald and Tiberius Veres

Figure 1: Value of a dollar with 16.6% interest

Figure 2: Determining the interval of periods when the digit 2 occurs.


Figure 2 and means that the size of the domain corresponding to a specific interval in the range is the length of time that the value will have a specific left-most digit. For example, 1≤ y <2 is the condition so that the left-most digit of the value of a dollar is 1. The corresponding domain is 0 ≤ x < 4.51, which means that for approximately 4.5 years the left-most digit of the value of the investment will be 1. Similarly, 2 ≤ y <3 in Figure 2, will give the duration of time the leading digit of the value is 2.

Determination of the domain from the constraints on the range is an expectation within pre-calculus. Students can interpret the concept graphically where Figure 1 shows that when y = 1, x = 0, and when y = 2, x is approximately 4.5. This shows that the occurrence of a first digit of 1 for the value will occur for 4.5 time periods out of the 15 time periods on the x-axis. That is about 4.5/15 or 30% of the time. The value y = 3 corresponds to x being approximately 7. So, a first digit of 2 in the value will occur about (7-4.5)/15 or 17% of the time.

Having estimated graphically, students in pre-calculus can then apply algebraic approaches to firm up the values. Since y = 1.166x the inverse function is ( )logx y.1 166= and the exact values can be determined. The occurrence of a first digit of 1 requires that 1 ≤ y <2 and that occurs when ( ) ( )log logx1 2. .1 166 1 1661 1 . This gives a domain width of ( ) ( ) .log log2 1 4 51. .1 166 1 166- = where the entire domain has a width of 15. Similarly, the domain intervals for other left-most digits of the value can be calculated. Values are shown in Table 1.

Digit Domain width(out of 15)

Occurrence(%)

1 4.51 30.12 2.64 17.63 1.87 12.54 1.45 9.75 1.19 7.96 1.00 6.77 0.87 5.88 0.77 5.19 0.69 4.6

Table 1: First digit occurrence rates

The use of Benford’s law for detecting fraud is based on the notion that these occurrence rates should arise in data that reflects compound-interest growth. For example, when interest is reported on income tax, the statistics of the first digits should

reflect Benford’s law. This can be instructional for students because it suggests that many investors would have held their investments for different periods of time, but collectively this implies they would be uniformly distributed on the x-axis and that leads to Benford’s law when the first digit of the range is considered for a scenario based on exponential growth.

In our times it may seem odd that we set the interest rate at 16.6%. This was purposefully chosen to give a value of 10 when 15 periods of time had passed. This was by design so that only one interval in the range corresponded to each first digit value. In realistic situations the values are not ideal and effects arise because the values in the range do not cover all digits equally. Consider, for example if 1 ≤ y < 11. In this instance the first digit of 1 occurs when 1 ≤ y < 2 and when 10 ≤ y < 11. While the former still has a domain width of 4.51 (see Table 1) the latter is ( ) ( )log log11 10. .1 166 1 166- , which adds a domain width of 0.62. Note that the overall domain width is no longer 15 periods as the alteration of the maximum value of y has increased the overall number of periods to 15.62. So the occurrence of a first digit of 1 is (4.51 + 0.62)/15.62 or 33%. The other digits will be scaled because of the increased domain size. For the first digit being 2 the occurrence is 2.64/15.62 or 17%.

Pragmatically, for fraud detection the concept is that financial data may have characteristics hidden in specific digits. If one models the characteristics and examines how much it may vary because domains or ranges may not match the ideal then there is potential for this to detect fraud. Discussion of the bigger picture is relevant for the classroom and highlights that the modeling used the range to determine the domain. That can be done for any function, but for the financial sector the primary focus is exponential growth associated with compound interest. It could be investigated for annuities, mortgages, or other financial calculations. Furthermore, within a statistics course there is an opportunity for students to investigate the phenomena using statistics applied to financial data.

Going beyond Benford

Can the concept responsible for Benford’s law be generalized? Is it feasible to say anything about the second digit from the left in a series of numbers? Or the third, fourth, or any other digit? Consider the second digit from the left using Figure 1 and the conceptual approach. The key is to set up the circumstances. A second digit of 2 in the range arises in multiple intervals on the y-axis. It can arise when 1.2 ≤ y < 1.3, 2.2 ≤ y < 2.3, … , or 9.2 ≤ y < 9.3 and the occurrence entails determining the corresponding domain intervals.


As was done earlier, each range interval is converted to a domain interval. The first three are shown and the pattern continues for all nine intervals.

. . ( . ) ( . ) .

. . ( . ) ( . ) .

. . ( . ) ( . ) .

log log

log log

log log

y

y

y

1 2 1 3 1 3 1 2 0 521

2 2 2 3 2 3 2 2 0 289

3 2 3 3 3 3 3 2 0 200

. .

. .

. .

1 166 1 166

1 166 1 166

1 166 1 166

(

(

(

# #

# #

# #

- =

- =

- =

The calculation is a teachable moment for students learning about logarithms. The issue is that the particular logarithms are not convenient on scientific calculators and the following manipulation is useful for the calculation:

( . ) ( . ) ( .. )

( . )

( .. )

log log loglog

log1 3 1 2 1 2

1 31 1661 21 3

. . .1 166 1 166 1 16610

10

- = =

The values for each domain interval are summed to calculate the total portion of the domain that will give a second digit of 2. The same process is used for any other second digit. In the case of a second digit of 2 the sum is 1.632 and arises 10.88% of the time. The occurrence for each second digit is given in Table 2.

The percentage occurrence for the first digit varies from 30.1% down to 4.6%. However, the occurrence of the second digit only varies from 12% down to 8.5%. There is not as much distinction with the second digit as there was with the first digit, however this is a teachable moment for the statistics course. For fraud detection, the second digit will not be as discriminating or effective as the first digit generally. However, if there is enough data for distinction of second digits (which may simply imply more data is necessary) it provides an opportunity for fraud detection that is independent of the first digit.

Digit Domain width(out of 15)

Occurrence (%)

0 1.80 11.971 1.71 11.392 1.63 10.883 1.56 10.434 1.50 10.035 1.45 9.676 1.40 9.347 1.34 9.048 1.31 8.769 1.28 8.50

Table 2: Second digit occurrence rates

A challenge for enrichment is for students to consider the third digit. The process is essentially the same as the first two digits. However, it entails one hundred intervals in the range. These lead to one hundred intervals in the domain and summing those necessitates the use of technology. It is suited to enrichment using a spreadsheet to ease the burden of calculation. Should you decide to use this for enrichment be advised that the end result shows that the third digit should have 10% occurrence for each digit—that is, the third digit is uniformly distributed. Perhaps it would be worthwhile to have students make a list of 100 fraudulent values before engaging in this enrichment. This will enrich the aspect of fraud detection.

A Benford Twist for Pre-CalculusThe impetus for Benford’s law is its use in finance. However, the concept is richly connected to the use of the domain, range, and the inverse function. Within pre-calculus there is a requirement to explore different families of functions. The possibilities to develop Benford’s law for different families of functions are enormous. To give a flavour, we work through the first digit scenario for a quadratic. We follow this by some comments regarding possibilities for the classroom.

Digit Domain width(out of )

Occurrence (%)

0 0.316 31.61 0.131 13.12 0.101 10.13 0.085 8.54 0.075 7.55 0.067 6.76 0.062 6.27 0.058 5.88 0.054 5.49 0.051 5.1

Table 3: First digit occurrence for a quadratic

Consider the quadratic y x2= with y0 10# # . A first digit of zero arises when y0 1# # and, using the inverse, a square root function, y0 1# # . Similarly, y1 21# arises from y1 21# . Generally for digit “d” it is required that d y d 11# + and that arises from d x d 11# + . The length of the entire domain is 10 and that provides the occurrence to be determined. The occurrence is outlined in Table 3.


Comparison of Table 1 and Table 3 shows a notable difference in the occurrence of digits of “1”, “2,” and “3.” This implies that the Benford approach is different for quadratics than exponential functions. It might be feasible to use the difference to look at first digits in a set of data and tell whether it is quadratic or exponential. However, more investigation is needed as one example is not sufficient to be certain—perhaps you were looking for an open-ended challenge?

Digit Domain width(out of 1.57)

Occurrence (%)

0 0.100 6.41 0.101 6.42 0.103 6.63 0.107 6.84 0.112 7.15 0.120 7.66 0.132 8.47 0.152 9.78 0.192 12.39 0.451 28.7

Table 4: First digit occurrence for a sine

As a second example consider ( )siny x10= with x0 21#r . This

could be done in degrees, but pre-calculus includes expectations around radians. The results for this example are shown in Table 4. It is notable that these results have several characteristics that are different than the exponential and quadratic functions. It is also

notable that the occurrence sequence of 6.4%, 6.4%, and 6.6% reflect the linear behavior of a sine function when the argument is close to zero.

Numerically this example might be simpler in degrees because 90° is easily used as a value. On the other hand, it might be interesting to offer students the possibility that Benford’s law would give the same occurrence rates if a horizontal stretch was applied. This would allow siny x10 2

r= R W to simplify the calculation since the range of all possible digits corresponds to a domain of x0 11# . Will this give the same results? Another opportunity for a classroom investigation, but in case you wonder, no, it does not alter the results.

ConclusionBenford’s law is a curious opportunity for pre-calculus and statistics courses. In this article the underlying conceptual basis is provided and it is well positioned to support instruction about domains, ranges, and inverse functions. The application to other families of functions provides an unexplored opportunity for innovative teachers. As well, in the statistical spirit that inspired Benford, there are many potential opportunities for students to explore its application to data associated with a wide range of phenomena.

Reference

Benford, F. (1938). The law of anomalous numbers. Proceedings of the American Philosophical Society, 78(4), 551–572.


Some Geometric Problems from 2014 and 2015 Math Challengers Competitionsby Joshua Keshet and Dave Ellis

Math Challengers is a mathematics competition for BC students in Grades 8 to 10 (or lower). During February 2018, school teams competed in regional tournaments (at Okanagan College, Camosun College, University of the Fraser Valley, and SFU). A Provincial final tournament was held in April (at UBC). In 2018 Math Challengers involved more than 1000 students from more than 70 schools.

The competition builds skills, promotes strategic problem solving, and exposes students to some complex problems that require creativity and persistence in order to be solved. Students have opportunities to exchange mathematical ideas through the competition. Teachers and volunteers (including former student participants) prepare competitors during the first half of the school year prior to the competitions, as part of in-class instruction, and as an extracurricular activity. The highest scoring schools and individuals from the regional competitions advance to the provincial competition, where the top scoring school teams and individuals in each grade are recognized with medals and trophies.

For more information about Math Challengers, go to https://www.egbc.ca/Math-Challengers/.

In this issue of Vector, we provide seven problems dealing with various aspects of geometry:

Problem 1 involves measurements of 3-dimensional bodies and their volumes. Problem 2 deals with relations between certain equilateral triangles and circles. Problems 3 and 4 involve basic analytic geometry. Problem 5 deals with relations between right triangles and certain circumscribed circles. Problem 6 involves analytic geometry and points in the plane with integer coordinates. The solution involves other aspects of mathematics such as probability and subsets. Problem 7 involves finding the areas of regions inside triangles based on similarities.

The reader should solve the problems without looking at the solutions, and then compare both work and answers. Problems can be solved in different ways. We encourage students and teachers to discuss their solutions in group settings, which will provide insight into using various techniques and concepts in solving math problems. These discussions will extend the horizons to a much greater understanding of mathematical concepts. If either teachers or students are interested in providing feedback, they are encouraged to contact the first author at [email protected].

Problem 1

What is the ratio of the volume of a sphere to the volume of a cube when the cube is the largest possible cube that can fit entirely inside the sphere? (Volume of a sphere is V r3

4s

3r= where r is the radius of the sphere). Express the answer as decimal correct

to two decimal places.


Solution

Let a be an edge of the cube. The largest cube that can fit in a sphere is a cube whose largest diagonal, dc , is also a diameter of the sphere.

Using the Pythagorean Theorem to find the diagonal, d f , of each of the faces of the cube:

a a d f2 2 2+ = . Thus, d a a a 2f

2 2= + = .

The diagonal of the cube, dc , which is also the diameter of the sphere, satisfies:

d d ac f2 2 2= + . Thus, d a a a a 3f

2 2 2= + + = .

Thus, ra23

= .

The volume of the cube V ac3= .

The volume of the sphere ( )Va

a a34

23

3 84 3 3

23

33 3 3

## #

r r r= = = .

Therefore, .VV

aa

23

232 72

c

s3

3

.r r= = .

Problem 2

The diameter, AB , of a circle is one of the sides of an equilateral triangle, ABCT . What fraction

of the area of ABCT is inside the circle? Express the answer as K

M Nr+ , where M, N, and

K are integers, and N has no square factor greater than 1.

Solution

Add the centre of the circle, D, to the figure. Name the two other intersecting points of the triangle with the circle as E and F, and draw line segments DE and DF.

DA and DE are radii. So, DA = DE and DEA DAE\ \= Since ABCT is equilateral it follows that DAE 60\ = c and therefore DEA 60\ = c . Therefore, ADE 60\ = c as well, and DAET is an equilateral triangle.

Using the same arguments for DBFT , it follows that BDF 60\ = c. ADE EDF FDB 180\ \ \+ + = c , so, EDF 180 60 60 60\ = - - =c c c c .

The area of the part of ABCT that is inside the circle is the sum of the areas of DAET , DBFT , and the area of the 60c sector of the circle bounded by DE and DF. Let r be the radius of the circle. Side AB of ABCT is a diameter. So, AB=2.


The height, h, of ABCT , using the Pythagorean Theorem, is: ( )h r r r r2 3 32 2 2= - = = . The area of ABCT , AT , is

Ar r

r22 3

3T2#

= = .

DAET and DBFT are equilateral, and, thus, the value of their sides is r, and the value of their heights is h r2 23

= . So, the areas

of each of DAET and DBFT is r432

.

The area of the 60c sector that is bounded by DE and DF is given by r r36060

62

2

#rr= .

The area of the part of ABCT which is inside the circle, AI , is: ( )Ar r r2 43

6 236I

2 22#

r r= + = + .

Therefore, ( ) ( ) ( )

AA

r

r

3236

3236

3 3

3 236

323

63

6 3

6 23

63

189 3

T

1

2

2

# #

r r r r rr

=+

=+

=+

=+

=+

=+

.

Problem 3

The following are the coordinates of the vertices of hexagon ABCDEF:

( , ), ( , ), ( , ), ( , ), ( , ),A B C D E2 0 6 6 2 6 4 3 2 4 4 3 2 0 3 2= = - = + = + =

and ( , ) .F 0 2= There are points inside the hexagon which are located furthest away

from any point on the boundary of the hexagon. Find the smallest y-coordinate of any such point. Express the answer as decimal correct to two decimal places.

Solution

The difference between the x-coordinate of point B and the x-coordinate of point A is 6 2- . The difference between the y-coordinate of point B and the y-coordinate of point

A is 6 2- . So, for line segment AB, the rate of change in the x-direction is the same as the rate of change in the y-direction. Thus,

the angle between segment AB and the x-axis is 45c .

The difference between the x-coordinate of point D and the x-coordinate of point E is 4. The difference between the y-coordinate of point D and the y-coordinate of point E is ( )4 3 2 3 2 4+ - = . So, for line segment DE, the rate of change in the x-direction

is the same as the rate of change in the y-direction. Thus, the line that goes through points D and E is also at an angle of 45c to the

x-axis.

Therefore, AB and DE are parallel.


Line AB intersects the x-axis at point A where x 2= . Point E is on the y-axis where

y 3 2= , so the y-intersect of line DE is y 3 2= . The x-intersect of Line DE (not

shown) is in the negative direction of the x-axis and is at the same distance from (0,0) as the y-intersect of that line. So, the x-intersect of DE is x 3 2= - .

Points inside the hexagon which are furthest away from both line AB and line DE must be located on a line parallel to both line AB and line DE at exactly half way in between. Add this line to the figure and identify some points on it. The x-intersect of this line is exactly half way between the x-intersects of line AB and line DE. Therefore, the value of the x-intersect of that line is ( )2

2 3 22

-= -

. The y-coordinate of point F is 2 . So F is at the same distance from (0,0) as is the x-intersect of the line

which is furthest away from both line AB and line DE. So, F is on that line. Therefore, line FG is the line containing the points furthest away from lines AB and DE. The distance, d, between lines AB and DE is the length of segments connecting lines AB and DE which are at a 45c angle to the x

and y axes.

F is on line FG, which is half way between line AB and line DE. So, the length of segment AF is d2 .

Using Pythagorean Theorem, we get: ( )d2 2 2 2 2 42 2 2= + = + = . So,

d2 2= .

All points, (x,y), inside the hexagon satisfy, thus, that the distance, h(x,y), from point (x,y) to the nearest point on the boundary of the hexagon satisfies ( , )h x y0 2# # .

Three of the sides of the hexagon, EF, BC, and CD, are parallel to either the x-axis or the y-axis. For each of these three sides there is exactly one point located on segment FG that is at a distance of d

2 2= units away from that side. Let H,I, and J be these points,

respectively (H and I are added to the figure). The value of all the x-coordinate of points on EF is x = 0. So, the x-coordinate of H is x = 0+2 = 2. All points on FG, satisfy y x 2= + . So, ( , )H 2 2 2= + .

The value of the x-coordinate of all points on BC is x=6, so the x-coordinate of I is x=6-2=4. So, ( , )I 4 4 2= + .

The value of the y-coordinate of all points on CD is y 4 3 2= + . So, the y-coordinate of J is ( )y 4 3 2 2 2 3 2= + - = + ,

and for this point x2 3 2 2+ = + .

So, (( ) , ) ( , )J 2 3 2 2 2 3 2 2 2 2 2 3 2= + - + = + + .

The distance from J to Line BC is: ( ) .2 2 2 6 2 2 4 4 2 2 1 17 21.+ - = - = - . So, point J is located at distance of


less than d2 2= from the exterior of the hexagon. Thus, the points (x,y) that are located on segment FG and satisfy h(x,y) = 2

must be at a distance of at least 2 from both segments EF and BC, and, thus, must all be located on Segment HI. The smallest value of such y-coordinate is the value of the y-coordinate of ( , )H 2 2 2= + . Therefore, . .y 2 2 1 41 3 41.= + + =

Problem 4

Find the largest x-coordinate of any point of the type described in Problem 3. Express the answer as decimal correct to two decimal places.

Solution

Using the conclusion of Problem 3, the point (x,y) is located on Segment HI. The largest value of the x-coordinate of any such point is the value of the x-coordinate of point I, which is the endpoint of this segment. The value of the x-coordinate of point ( , )I 4 4 2= +

is: x = 4 = 4.00 (as specified in the instructions of the format for the required answer).

Problem 5

The triangle below, ABCT , is right-angled. The two circles with centres at F and G have radius 2 and 1 respectively, and the circles and the triangle are tangent to each other as shown. D and E are points of intersections of the circles and AC. What is the length of the hypotenuse, BC, of the triangle? Express the answer as decimal, correct to two decimal places.

Solution

Draw a line segment from C to F. Let the intersection points of the circles with AB and BC be H, I, and L. Let BH = x, and draw lines segments FH, FI, and GL.

Since AC and BC are tangent to the larger circle centered at F, then radii FD and FI are perpendicular to AC and BC respectively.

Consider CFDT and CFIT . These triangles are congruent because CF = CF, FD = FI = 2, and FDC FIC 90\ \= = c . Thus, CD = CI, and DCF ICF\ \= . Thus, CF is a bisector of ACB\ .

AC and BC are also tangent to the smaller circle centered at G, so by using the same argument for ACB\ , it follows that CG is the bisector of angle ACB\ . Thus, G is on line CF.

CDFT and CEGT are both right triangles with the same angles and, thus, are similar triangles. DF = 2, and EG = 1. Thus,

EGDF

CGCF

12 = = , so CF = 2CG .

CG + FG = CF = 2CG, so CG = FG.


FG = 2+1 (the sum of the two radii), so CG = FG = 3.

Thus, CF = 2FG = 6.

Using Pythagorean theorem for :CDF CD DF CF2 2 2T + = .

DF=2 and CF=6, so CD 2 62 2 2+ = .

CD 36 4 32= - = .

Also, ADFH is a square, so AD=AH=2.

Let BH=x. BA and BC are tangent to the circle centered at F. Thus, BI=BH=x.

Using Pythagorean theorem for :ABC AC AB BC2 2 2T + = .

AD=AH=2, and CD CI 32= = .

Therefore, AC AD DC 2 32= + = + , AB=AH+BH=2+x , and BC BI CI x 32= + = + , so we have

( ) ( )

( ) ( )

x x

x x x x

x x

x x

x

x

32 2 2 32

32 4 32 4 4 4 32 2 32

4 32 8 4 2 32

2 32 4 4 32 8

2 32 2 2 2 32 4

32 22 32 4

2 2 2

2 2

+ + + = +

+ + + + + = + +

+ + =

- = +

- = +

=-+

Q V

Using the values of x and CI 32= : BC x 32

32 22 32 4

32= + =-++ .

Simplify: ( )

.BC32 2

2 32 4 32 32 232 2

2 32 4 32 2 3232 236 9 84.=

-+ + -

=-

+ + -=

- .

So, the length of the hypotenuse, BC, correct to two decimal places is 9.84.


Problem 6

A point in the plane is chosen at random from all points with integer coordinates (u,v) such that u1 9# # and v1 9# # . What is the probability that the absolute value of the difference between the x-coordinate and the y-coordinate of the point is greater than 2? Express the answer as a common fraction in lowest terms.

Solution

There are 81 points (u,v) that satisfy u1 9# # and v1 9# # . Instead of directly finding the number of points that satisfy | |u v 2$- , one can first

find the number of points that satisfy | |u v 31- .

Case 1: | |u v 0- = There are nine points where u v= , namely (0,0), (1,1), • • •, (9,9)

Case 2:

| |u v 1- = . The points that satisfy this are (1,2), • • •, (8,9), (2,1), • • •, (9,8), or a total of

16 points.

Case 3: | |u v 2- = . The points that satisfy this are (1,3), • • •, (7,9), (3,1), • • •, (9,7),, or a total of 14 points.

The total number of points included in Cases 1 to 3 is: 9+16+14=39. The total number of points that satisfy | |u v 22- can be calculated as the total number of points, (i.e. 81), minus the number of

points that satisfy | |u v 31- , (i.e. 39).

Thus, 81-39=42.

Therefore, the probability that a point satisfies | |u v 22- is 8142

2714= .

Problem 7

Side AB of ABCT is divided into four segments of equal length, AD, DE , EF, and FG. G and H are on BC and the segments AB, DG, and EH are parallel. What is the ratio of the combined area of the two shaded triangles to the area of the unshaded part of ABCT ? Express the answer as a common fraction in lowest terms.


Solution

Define the following: aT is the area of ABCT , aT is the total area of the two shaded regions, and is the total area of the three

unshaded regions.

Since EF=FC (given), EFHT and FCHT have the same base. EFHT and FCHT also have the same height (the height

measured from the H vertex). Therefore, the area of EFHT is the same as the area of FCHT .

Let a1 be the area of the shaded triangle EFHT . Let a2 be the area of ECHT . So, a a a a22 1 1 1= + = .

Use the condition that AB DG EH' ' : a) CAB CDG CEH\ \ \= =

b) CBA CGD CHE\ \ \= = , and

c) ACB\ is common to the three triangles, ACBT , DCGT , and ECHT .

Therefore, ACBT , DCGT , and ECHT are similar triangles. AD = DE = EF = FC, so, AC = 4EF = 2EC. Given that AC = 2EC, and that ACBT and ECHT are similar, it follows that

AB = 2EH, BC = 2HC , and the ratio of the heights of the two similar triangles is 2. The area of ACBT , aT , is thus, four times the

area of ECHT , :a a a a a4 4 2 8T2 2 1 1#= = = .

Using the same arguments as before: DC EC23= , so the values of sides and heights of DCGT are 2

3 the values of sides and

heights of ECHT , and the area of DCGT , a3 , is ( )23

492 = the area of ECHT , a2 So, a a a a4

949 2 4

9 23 2 1 1# #= = =

Using the same arguments as before: GC HC GH23 3= = . So, HC GH2= . GHET and HCET share the same height from

point E, and HC=2GH. So, the area of GHET is half the area of HCET . The area of HCET is a a22 1= , so the area of

GHET is a a22 1

1= . Thus, the area of GCET , which is the combined areas of GHET and HCET , is equal to a a a2 31 1 1+ =

The area of DGCT , a3 , is the combined areas of DGET and GCET . So the area of the shaded triangle DGET is equal to

a a a a a3 29 3 2

33 1 1 1 1- = - =

.

The combined shaded regions are DGET and EFHT so a a a a23

25

s 1 1 1= + = .

Therefore, a a a a a a8 2

5211

u T s 1 1 1= - = - = , and aa

a

a

21125

115

u

s

u

1

= = .


Three-Act Tasks: An Approach to Engage Students in Problem Solving.

Problem solving is an important part of mathematics. Often primary teachers present a story problem and ask students to find a solution. Students will look for key words to help solve the problem. Students might see problem solving as a task to following directions and get the right answer quickly.

The main goal of problem solving includes making sense of story problems, developing a range of strategies, and reaching accurate solutions. A crucial component that is often left out is the modeling of mathematics. Modeling mathematics involves identifying mathematical problems in our world, gathering information and determining which details will help find a solution, and developing and revising mathematical models of the situation. These models might include using manipulatives, pictures, numbers and words to represent the quantities and mathematical relationships.

Three-Act Tasks provide a structure that is specially designed to engage students in mathematical modeling. Graham Fletcher (2016) has created 3-Act Tasks images or videos that support modeling, his website https://gf letchy.com/3-act-lessons/provides a number of tasks that can be used from kindergarten through grade 7.

Example of the 3-Act Task Structure (using the Humpty Dumpty task by Graham Fletcher)

Act 1

• Teacher presents an image or video (intended to hook the students);

• Teacher asks the students what they notice and wonder;• Students share and discuss what they notice and wonder

including mathematical features of the situation;• Students estimate a solution and give their reasoning;• Students decide on a mathematical question to answer

(e.g. How many eggs broke?).

Act 2

• Teacher provides information that prompts the students to think about information they need to solve the problem;

• Teacher reveals the information;

• Students use various modeling strategies to solve the problem:

- Using manipulatives- Drawing pictures or

diagrams- Using numbers- Using words

Act 3

• Students share their strategies and solutions;

• Teacher may compare and connect students’ strategies and solutions;

• Teacher reveals the answer (as an image or a video).

Tips for Getting Started

• Choose a task or make your own;• Avoid the temptation to rush students. Make sure they have

enough opportunities to apply their own mathematical thinking to make sense and solve the problem;

• Move students carefully toward mathematical thinking. Try to have the students focus on the math instead of the objects. Ask: What questions might a mathematician ask?

• Ensure that the students have the stamina to complete the three acts. If not, have the students take a break between the acts;

Problem SetsContributed by Sandra Ball and Michael Pruner

“What information do you need to know to

answer our question? “

How many eggs were in the egg carton?

There were 9 eggs inthe carton to begin

How many eggs didn’t break?


• The 3-act structure will become a routine that can be repeated with different tasks;

• Go slowly. This is often a new structure for students and they may be confused about what they need to do;

• Have fun and enjoy the modeling of mathematical thinking of your students.

Source:

Fletcher, Graham. (2016). “Modeling with Mathematics through Three-Act Tasks.”

Lomax, Kendra; Alfonzo, Kristin; Dietz, Sarah; Kleyman, Ellen; Elham Kazemi. (2017) “Trying Three-Act Tasks with Primary Students.”

Problem #1: Number Sandwich

In this arrangement there is one number sandwiched between the “1” cards, two numbers sandwiched between the “2” cards, but only one number sandwich between the “3” cards.

Is it possible to make a complete sandwich with one number between the “1” cards, two numbers between the “2” cards, and three numbers between the “3” cards?

Is there more than one way to arrange the numbers? Can you make a complete sandwich with 1, 1, 2, 2, 3, 3, and 4, 4? Is there more than one way to arrange the numbers this time? Explore other number sandwiches.

Source: https://nrich.maths.org/sandwiches

Problem #2: Step Up

Source: http://www.playwithyourmath.com.s3-website-us-east-1.amazonaws.com/

Problem #3: Seventeen Puzzle

Each of the integers from 1 to 9 is to be placed in one of the circles in the figure so that the sum of the integers along each side of the figure is 17.

How many unique solutions can you find?Can you solve an “18” puzzle?Can you solve a “19” puzzle?

Source: https://brilliant.org/

Three-Act Tasks provide a structure that is specially designed to engage students

in mathematical modeling.


Problem #4

A farmer owns a rectangular piece of land. The land is divided into four rectangular pieces known as region A, region B, region C and region D.

A C

B D

One day the farmer was asked, “What is the area of your land?” The farmer replied:

I will only tell you that the area of region B is 200 m2 larger than region A; the area of region C is 400 m2 larger than region B; and the area of region D is 800 m2 larger than the area of region C.

What is the area of the farmer’s land?

Source: Sriraman, B., & English, L. (Eds.). (2009). Theories of mathematics education: Seeking new frontiers. Springer Science & Business Media

Problem #5: Promenade

Promenade is the set P of all natural numbers from 1 to 25 together with the following four functions:

S (x) = x + 5 for x ∈ P, x ≤20N (x) = x - 5 for x ∈ P, x >5E (x) = x + 1 for x ∈ P, x mod 5≠0W (x) = x - 1 for x ∈ P, x mod 5≠0

Any composition of the above functions is called a stroll. We say that stroll S leads from a to b if an only if S(a)=b.

Examples:

S(2) = 7(S ∘ E)(2) = S(E(2)) = S(3) = 8(E ∘ N ∘ N)(23) = 14

Find a stroll that leads from 13 to 21. Find a stroll that leads from 3 to 17. What numbers can be reached from 9 without using N and E? What numbers can be reached from 9 using S and E ONLY?

Source: Sfard, A. (1991). On the dual nature of mathematical conceptions: Reflections on processes and objects as different sides of the same coin. Educational studies in mathematics, 22(1), 1-36.


book review

Sync by Steven Strogatz Reviewed by Josh Giesbrecht

Sync is a fascinating look into the phenomenon of synchronicity as modeled by coupled oscillators: sets of individual, periodic timers that nudge their own clock a little farther ahead when another timer goes off. Strogatz has an accessible writing style that presents “sync” in a simple and fresh way without getting into centuries of theoretical or analytical foundation. Sync is open to a wide range of readers from high school students to professional mathematicians.

The opening hook of the book uses an example of blinking fireflies. In most cases, fireflies blink with a personal rhythm, identifying themselves to potential mates in a personal display of light. But on some riverbanks in Southeast Asia, fireflies have been known to begin blinking in sync with each other, collectively. Their waves of light flow across the rolling ground like strange rivers of living energy. No orchestrator controls the timing of this behaviour; the synchronization forms out of the sum of synchronicity across much of the population.

Strogatz explores this phenomenon of similar self-organizing behaviours across a diverse range of real-world connections: from the three-dimensional synchronization of pacemaker cells within our own heart, to electrical grids, and to pendulums aligning with each other in a seemingly supernatural display. Strogatz also shares his personal stories about connecting with other researchers in the field, referring to them as energetic and independent thinkers who shaped his own career.

One thing that fascinated me about the phenomenon of sync is that this area of study, which can be generalized as a study of coupled oscillators, is highly experimental, empirical work. Strogatz and other researchers in the field spend notable

amounts of effort running simulations of coupled oscillators in different configurations. Sometimes they see patterns that can then be proven theoretically; other times they discover trends that theory isn’t yet powerful enough to predict analytically.

What’s great about this as a reader is that these are relatively easy simulations to get started on. A student (or teacher) who knows some coding can begin by creating an array or list of “oscillator” objects. Each oscillator can be modeled by a timer that gets “nudged” when another signal goes off, and blinks or otherwise signals when it reaches its assigned time limit. This is all it takes to recreate the fireflies yourself and begin experimenting! (A great pre-coded starting point for experimentation is Nicky Case’s interactive fireflies simulation, https://ncase.me/fireflies/, which was inspired by this book.)

Sync is a great book designed to inspire new research. Strogatz mentions how he himself became interested in the topic

of coupled oscillators when he stumbled across a similar book written by an early pioneer in the field (Arthur T. Winfree), and makes no secret that he is hoping this book will have a similar effect on others. Strogatz offers a glimpse into the making of a new field of applied mathematics. The study of coupled oscillators is still in the early stages and its applications are both diverse and widespread.

This book is also a great example of how curiosity and inquiry in things that seem trivial such as strange looking fireflies can have life-changing and possibly even life-saving connections. If studying blinking lights can help us understand the neurology of sleep or find ways to treat heart arrhythmia, perhaps readers of this book will come to value their own mathematical curiosity as a vital and necessary part of life.


https://gfletchy.com/3-act-lessons

This collection of Three-Act Tasks are designed for use in K to 7 classrooms. If you aren’t familiar with the Three-Act Task structure, search for Dan Meyer’s posts on the “Three Acts of a Mathematical Story.”

https://diagnosticquestions.com

These questions are designed to help teachers learn how their students understand mathematical ideas. The questions are aligned to the mathematics expectations in the United Kingdom, but many of the questions are still likely useful with the BC curriculum.

https://curriculum.newvisions.org/math

This open source curriculum resource has tasks and resources for three high school courses. Although the courses and units are aligned to the US Common Core standards, the tasks themselves are aligned nicely to the BC curriculum competencies. Disclaimer: I am one of the main developers of this curriculum.

https://www.illustrativemathematics.org

This site is the launching place for the open source Illustrative Mathematics middle school curriculum and will soon contain links to the new high school and elementary school mathematics curriculum resources currently being developed. Although the courses and units are aligned to the US Common Core standards, the tasks themselves are aligned nicely to the BC curriculum competencies.

https://www.retrievalpractice.org

As the site says, “When we think about learning, we typically focus on getting information into students’ heads. What if, instead, we focus on getting information out of students’ heads?” This website contains links to research and practical strategies to support retrieval practice which is when students work to remember information learned and build their memories of that information.

Links selected and described by David Wees: http://davidwees.com

Previous websites can be viewed here: http://davidwees.com/m/mathwebsites

Math Links

Spring 2019 Mathematics Websites



AWARDS & CRITERIA

Outstanding Teacher Awards (Elementary; Secondary; New Teacher with less than five 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, or province (teacher research,

technology, active learning, assessment, etc.);• has and continues to demonstrate excellence in teaching mathematics regularly in British Columbia (teaching style,

knowledge of the curriculum, current curriculum trends, etc.);• has made contributions to mathematics education at the school, district or provincial levels (eg. workshops,

seminars, conferences, community projects, curriculum development, publishing, etc.);• is not a current member of the BCAMT Executive.

Service Award• has provided extraordinary service to mathematics education as an active member of the BCAMT for a significant

period of time.

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 mathematics classroom;• actively engages in professional dialogue involving mathematics pedagogy;• is not a current member of the BCAMT Executive, but is a member of the BCTF.

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. Recipients of this award are expected to contribute an article to Vector.

Teachers Awards InformationThe BCAMT sponsors awards in three categories (Outstanding Teacher, Ivan L. Johnson Memorial, and Service) to celebrate outstanding achievements of its members. Winners are honoured at a BCAMT conference and receive a commemorative plaque.


SELECTION PROCESS

• All nominations are reviewed by the BCAMT Awards committee (consisting of a minimum of five previous award recipients) who recommend the recipients to the BCAMT Executive for ratification;

• Each nomination is considered for two years, after which time the application can be re-submitted with updated information.

HOW TO NOMINATE

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 nominee fulfills the criteria.

Send all required documents listed below in an envelope to:

BCAMT Awards c/o Michael Pruner 2680 Standish Drive North Vancouver, BC V7H 1N1

Deadline: May 31, 2019


The BCAMT offers grants to fund initiatives throughout BC that support quality mathematics education. To be eligible, proposals must meet the Goals and Objectives of the BCAMT:

1. Professional Development: to promote excellence in mathematics education throughout the province by promoting professional development in all aspects of mathematics education;

2. Curriculum: to promote the development and implementation of sound curriculum and the selection of appropriate resources;

3. Communication/Public Relations: Promote excellence in mathematics education throughout the province by promoting good communication with members, other educators, Ministry of Education, parents, students and the community;

4. Membership: Promote excellence in mathematics education throughout the province and be applicable to many teachers throughout British Columbia.

The BCAMT values the sharing of ideas and requests that successful applicants submit a short report of their initiatives with its highlights for publication in a future edition of Vector or in our newsletter.

All mailed applications must be postmarked no later than November 30, 2018. Applicants will be informed about funding after approval by the BCAMT Executive. Successful applicants may wish to re-apply for funding each year but are not guaranteed continued support.

This grant is not meant for individual professional development.

Complete this form and be sure to include:

• Whether you are a BCAMT member (priority is given to members);• A rationale for funding request (maximum of 2 pages);

Ř Details of your initiative Ř How your initiative fits within the the goals and objectives of the BCAMT Ř Which of the following areas your grant applies to: Professional Learning, Numeracy, Curriculum, Teaching,

Assessment Ř Your target audience: BCAMT members, teachers, students, community, etc Ř Type of initiative: Workshop, Research, New Local Specialist Association, Outreach, etc. Include information

about location and dates• Detailed budget: List expenses, other funding, etc.; • Request for funding: $_____________ (maximum $2000).

Grant Application 2019-2020

Name of Initiative:

Applicant’s Name:

Email Address:

School / District:

Mailing Address:

City: Postal Code:

Daytime Phone: Evening Phone:

Send the completed application by mail or email to:

Brad Epp Chair, Funding Application Committee

BC Association of Mathematics Teachers #51 – 383 Columbia Street West, Kamloops BC V2C 1K5

Email: [email protected]

Grant deadline: November 30, 2019

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Documents

SPRING 2019 Volume 6 Issue 1 · I continue to come back to Simon Sinek’s 2009 book, Start with Why: How Great Leaders Inspire Everyone to Take Action. His three concentric circle