spring 2018 volume 60 issue 1 - bc association of math ... › wp-content › uploads ›...
TRANSCRIPT
SPRING 2018
Volume 60
Issue 1
Letʼs DOLetʼs D
o
5.67
October 18-20, 2018 • www.bcamt.ca/nw2018
Check out our website for Registration, Call for Speakers, Hotel information, and more!
Tracy ZagerAnnie Fetter
Christina TondevoldGraham FletcherMichael FentonFawn Nguyen
& more!
British Columbia Association of Mathematics Teachers proudly presents...
57th Northwest Mathematics Conference
Contents
IN EVERY ISSUE
President’s Message
Book Review
Problem Sets
Math Links
BCAMT
Spring 2018 | Volume 60 | Issue 1
30
21
23
17
34
06
38
42
45
46
40 17
51
36
13
The Dragon Curve Fractal: Experiencing Mathematics through Exploration By Alana Underwood
Nine Points on Rich Math TasksBy Daniel Finkel
Alike and Different: Which One Doesn’t Belong? and MoreBy Chris Hunter
Mathematical Mindsets Workshop with Cathy WilliamsBy Susan Robinson
Areas and Averages in KinematicsBy Pouyan Khalili
Some Challenging Word and Algebraic Problems from Math Challengers CompetitionsBy Joshua Keshet and Dave Ellis
Reggio-Inspired Mathematics: Connecting to PlaceBy Janice Novakowski
08
Letʼs DOLetʼs D
o
5.67
October 18-20, 2018 • www.bcamt.ca/nw2018
Check out our website for Registration, Call for Speakers, Hotel information, and more!
Tracy ZagerAnnie Fetter
Christina TondevoldGraham FletcherMichael FentonFawn Nguyen
& more!
British Columbia Association of Mathematics Teachers proudly presents...
57th Northwest Mathematics Conference
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 40 words or less.
Articles must be in Microsoft Word (Mac or Windows). 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.
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]
2017/18 Membership Rates $40 + GST (BCTF Member) $20 + GST Student (full time university only) $65.52 + GST Subscription (non-BCTF)
INDEPENDENT SCHOOL REPRESENTATIVE Richard DeMerchant St. Michaels University School [email protected]
Darien Allan Collingwood School [email protected]
BCAMT EXECUTIVEDeanna Brajcich, President Sooke School District [email protected]
Michael Pruner, Past President Windsor Secondary School [email protected]
Marc Garneau, Vice-President Surrey School District [email protected]
Colin McLellan, Secretary and Listserv Manager McNair Secondary School [email protected]
Jen Carter, Treasurer Vernon School District [email protected]
Brad Epp, Membership Chair South Kamloops Secondary School [email protected]
VECTOR EDITORS Sean Chorney Simon Fraser University [email protected]
Susan Robinson Gulf Islands Secondary [email protected]
ELEMENTARY REPRESENTATIVESDeanna Lightbody K-8 Langley School District [email protected]
Jennifer Barker Surrey School District [email protected]
Debbie Nelson, Comox [email protected]
SECONDARY REPRESENTATIVESRon Coleborn Semiahmoo Secondary [email protected]
Minnie Liu Gladstone Secondary [email protected]
Chris Becker Princess Margaret Secondary [email protected]
Chris Hunter Surrey School District [email protected]
Robert Sidley Burnaby Mountain Secondary [email protected]
Josh Giesbrecht Abbotsford School District / UFV [email protected]
Amanda Russet, Kamloops [email protected]
NCTM and NCSM REPRESENTATIVE Marc Garneau Surrey School District [email protected]
POST-SECONDARY REPRESENTATIVEPeter Liljedahl Simon Fraser University [email protected]
Cover Art: The cover art is a dragon fractal made in Geometer's Sketchpad (GSP). In GSP the fractal is interactive so it can be "dragged" through a continuous set of dilations. It was made by rotating two perpendicular line segments six times about its ever-changing endpoint.
Images on page 3, 12, 20 and 37, taken by Susan Robinson on a recent trip to New York.
Vector • Spring 2018 5
Contributors
Alana UnderwoodAlana has been teaching for the past six years, gaining pedagogical experience in Mali, the Lower Mainland, French Guiana and Dawson Creek. She is currently teaching grade 4/5 in Coquitlam and completing a Master’s degree in Numeracy at SFU.
Sandra BallSandra works as the Inner-City Early Learning Helping Teacher in the Surrey School District. She has taught for over 30 years, with most of her experience focused on working with early learners.
Dan FinkelDan is the Founder and Director of Operations of Math for Love, a Seattle-based organization de-voted to transforming how mathematics is taught and learned. Dan works with schools, develops curriculum, leads teacher workshops, and gives talks on mathematics and education nationally and internationally. The math games he co-created with his wife, Katherine Cook, have won a total of 18 awards. They include Prime Climb, the beautiful, colourful, mathematical board game, and Tiny Polka Dot, the colourful math game for children.
Chris HunterChris has been teaching mathematics in Surrey for twenty years. Chris collaboratively works with—and learns from–teachers of mathematics, K-12. Prior to becoming a Numeracy Helping Teacher, Chris was a secondary math teacher and department head. Chris tweets at @ChrisHunter and blogs at reflectionsinthewhy.wordpress.com
Susan RobinsonSusan is a mathematics teacher/curriculum coordinator in the Gulf Islands, who is struggling and learning alongside her students as she works towards changing the culture regarding as-sessment in a mathematics classroom.
Pouyan KhaliliAfter completing an honours BSc in mathematical physics and a BEd in secondary mathematics, Pouyan completed an MSc in secondary mathematics at SFU. He teaches physics at Heritage Woods Secondary in Port Moody. He is interested in the mathematical needs in the physics classroom and is currently editing a workbook he has written for Physics 12.
Joshua KeshetJoshua is a mathematician and a corporate executive. He is also the Academic Coordinator for the Canadian Math Challengers Society.
Dave EllisDave is a retired Vancouver secondary school mathematics teacher and department head, a former BCAMT Executive member and a Member-at-Large of the Math Challengers committee.
Janice NovakowskiJanice is a District Teacher Consultant for the Richmond School District and facilitates the BCAMT Reggio-Inspired Mathematics Collaborative Inquiry Project with educators from multiple BC school districts.
Shiqi XuShiqi is a grade 11 student at Sutherland Secondary School. If Shiqi is not writing a mathematics contest or attending a mathematics workshop, she can sometimes be spotted running between her numerous volunteer commitments or extra-curricular activities. She is currently taking a course on discrete mathematics at Capilano University.
Jen BarkerJen is a Numeracy Helping Teacher with the Surrey School District. She has a passion for mathematics and believes mathematics should be meaningful and engaging, and develop conceptual understanding. She has taught Grades K–7, and worked as a Faculty Advisor and Instructor at the University of British Columbia. She has her Masters in Educational Technology with a focus in mathematics.
Vector • Spring 2018 66
President’s Message
It is exciting to reflect on the ongoing events and changes
occurring in mathematics education in BC. From the implementation of the new Grade 10-12 Curriculum to the piloting
of the Graduation Numeracy Assessment, teachers are
exploring new and exciting pedagogies to improving student
experiences and relationships with mathematics.
The BCAMT’s mandate is to promote excellence in mathematics teaching. One way we can do this is by presenting awards and grants to teachers we recognize as innovative and exemplary. This year, the Outstanding Elementary Teacher Award was presented to Kristy Doornbos from Peace River South for her work with inquiry based learning strategies. Jennifer Kirkey, from West Kelowna, received both the Outstanding Secondary Teacher Award and the Ivan L. Johnson Memorial Award for her leadership in teacher professional development:. The BCMAT will financially support Jennifer to go to the NCTM Annual Conference in Washington DC. Lastly, Selina Millar was awarded the BCAMT Service Award for her many years of leadership and professional learning for teachers across BC.
Recognizing our colleagues for their amazing work in and out of the classroom is important. Perhaps you know someone who deserves recognition or even a chance to go to the NCTM Annual conference paid for by the BCAMT. Please visit our website at http://www.bcamt.ca/awards-recognition/, or see page 48 in this issue.
Another avenue for growth for both students and teachers occurs through BCAMT grants. This year we were proud to support four group initiatives: Reggio-inspired mathematics, Spatial Reasoning, Math Challengers and High Yield Routines. All groups will share their learning and inspire others by writing an article for our Vector journal.
Reaching out to those teachers who are leaders in mathematics
in our province is an important mandate of our association. The BCAMT provides an informative networking opportunity at our annual Leadership Summit. Our next summit will be early next fall. If you have not received any invites through emails sent to your district, please let me know and I will make sure the invite reaches you for the Leadership Summit 2018. Please share this opportunity with your colleagues.
Don’t forget, there are many ways the BCAMT wish to connect with you or provide a way for you to connect with others in our mathematics community. The Listserv continues to be provide a dynamic discussion, with over 1000 members on the list. Our Twitter account keeps up-to-date with current events and, of course, we are always available by email at [email protected]. We continue to update and improve our website with new resources and relevant information about BC mathematics education.
Professional Learning
The BCAMT’s Fall Conference 2017 was a success. With a strong lineup of presenters, we offered professional learning opportunities for nearly 800 teachers. The best part of this conference was the hallway conversations occurring all over the school. Networking, strategizing, and of course, collaborating, are indirect benefits of coming to our conferences.
We also tried something new this year: we invited prominent mathematics researchers, Jo Boaler and Cathy Williams, to present a whole-day experience for our BC Mathematics teachers. Although Jo could not attend due to illness, Cathy provided teachers with an enthusiastic and inspiring journey through mathematical mindsets and deepening learning. We hope to be able to have Jo back another time.
I am excited to see the BCAMT Northwest Mathematics Conference coming together. This coming October 18-20, in beautiful Whistler, we will host thought-provoking speakers from around North America. Our Keynote Speaker is Tracy Zager, the author of Becoming the Math Teacher You Wish You’d Had: Ideas and Strategies from Vibrant Classrooms. Join us and meet fellow BC educators and mathematics specialists from around North America. Featured speakers will also include: Annie Fetter, Graham Fletcher, Michael Fenton, and Fawn Nguyen. Speaker proposals are still being considered for this event. If you are interested in sharing your expertise and passion, please complete a proposal on the website at http://bcamt.ca/nw2018/#proposals before May 4th.
Vector • Spring 2018 7
New Curriculum
As you are all probably aware, the mathematics grade 10-12 curriculum official document is complete, and is being hosted on our BCAMT website. It is has not been posted to the Ministry site so please go to www.bcamt.ca and share it with your colleagues. Those found on the Ministry site are still in draft form. Lisa Marshall, the new Mathematics Coordinator, has informed the BCAMT that the Pre-calculus 10 and Workplace Mathematics 10 will be on the curriculum website by May, and the grades 11-12 courses will be uploaded by July. Please submit any comments you may have to [email protected]. All feedback contributes to the continuing development and refinement of the curriculum. All BC schools are preparing to implement the new Grade 10 curriculum in 2018-19. The Grades 11 and 12 curriculum will remain available for an additional year of trial use in 2018-19, and will be implemented 2019-20. As stated by the Ministry of Education, “The new Grades 10-12 curriculum is intended to support both disciplinary and interdisciplinary learning, encourage locally-developed curriculum, and enable a variety of learning environments.”
The BCTF needs your feedback about the implementation of the K-9 new curriculum, and most importantly, the impact it is having on Grade 10-12 courses. Please share your experiences and recommendations with Grahame Rainey, Assistant Director of the Professional and Social Issues Division at the BCTF: [email protected]
In regard to assessment and reporting, the message is still very unclear. This has been a significant challenge for our members at all grade levels and we look forward to seeing some clarity from the Ministry. I have been told that key directions for K-9 Reporting Policy is to focus on “strength-based approaches,” with personalized communication about student progress. They are working towards “a structured policy with some flexibility” to respond to local needs. There are policy shifts under current discussion at the ministerial level, with a proposed communication plan to be developed. What this will look like is unknown.
Graduation Numeracy Assessment
The Graduation Numeracy Assessment pilot occurred in January 2018 with a sample of 9040 students. Unfortunately there were accessibility issues for those who have difficulty using technology.
The more significant problem, however, was the high reading level which many thought prevented students from showcasing their mathematics knowledge. Full implementation is fast approaching in June 2018; therefore, mathematics teachers are hoping changes will be coming soon.
Shona Peron from the Ministry has provided the following update:
The managed implementation of the Graduation Numeracy Assessment began this past January with a subset of students across the province participating. Full provincial implementation takes place this June. The ministry is in the process of compiling feedback and data from the managed implementation, and looks forward to sharing updates with the sector soon. This will include detailed information to support the full provincial administration in June. The ministry is continuing to work with post-secondary institutions as they work through their review and governance processes to ensure alignment with the changes in the K-12 sector. This includes their consideration of the Graduation Numeracy Assessment. Post-secondary institutions are at different places in terms of how they are proceeding, and in their readiness to communicate. It is anticipated that most institutions will publish their admissions and pre-requisite requirements for specific programs by Fall 2018.
As teachers, we recognize it is all about our students. From thinking classrooms to encouraging growth mindsets, an underlying theme is the inclusion of “productive struggle” in our classrooms. Our new curriculum attemtps to address this. Although not a new concept, it is becoming a focus of more and more teachers as they aim to improve the experience for students in the mathematics classroom. When students struggle with new concepts or procedures they will learn mathematics more meaningfully. Maintaining this rigour for all learners, at all levels, is a challenge but also an achievable goal.
I conclude this message with a new beginning. We have renewed our logo after thirty-five years! Although it is difficult to move on, we wanted something different so as to reflect important aspects including: British Columbia, mathematics, interconnectedness, as well as recognizing aboriginal ways of knowing. In collaboration with the certified aboriginal business Animikii Inc, we are so proud to welcome our new design. We hope you like it!
Sincerely, Deanna Brajcich
Vector • Spring 20188
The Dragon Curve Fractal: Experiencing Mathematics Through Explorationby Alana Underwood
I was recently introduced to fractals, and was intrigued by these patterns that are self-similar across different scales. These patterns, found in rivers, tree branches, blood vessels, snowflakes and paper folds, go infinitely deep with each layer being created by smaller and smaller iterations of itself. Fractal repetition creates seemingly chaotic and unpredictable patterns, but due to the consistency of the repetition of iterations, I felt that fractal patterns should somehow be systematic and predictable. I had never before encountered (or been interested in) a mathematical conundrum such as this one, but since becoming an educator, I have a new perspective for this subject and was eager to explore this phenomena mathematically.
Growing up, I was convinced that I was not a “math person” and I avoided the subject at all costs, so when I became a teacher, I was concerned about my ability to teach the subject effectively to my students. This worry motivated me to take my Master’s in Numeracy where I learned different strategies to help my students build a positive relationship with mathematics. To incorporate some of the strategies I’ve learned in my Master’s in my classroom, I have started to encourage my students to consider mathematics as a subject that they are able to experience and personally explore through questioning, technology, observation and reflection. As I did not learn mathematics in this style, I took my personal teaching philosophy to heart, and I decided to explore the fractal created by paper creases, the Dragon Curve, using these measures.
The Dragon Curve fractal is created by a pattern of seemingly random creases that point up (peaks) and creases that point down (troughs) (Figure 1). The haphazardness of the directions of creases, but the self-similar nature of fractals enticed me to find a regularity in the direction of creases. While exploring this problem, I ran into some barriers, but through physically folding paper, using the dynamic geometry program Geometer’s Sketchpad (GSP) and adjusting my approach based on the difficulties I encountered,
I was able to track the direction of creases in various ways and gain a thorough understanding of this fractal. Once explored, the direction of creases in paper folds are actually quite systematic and predictable.
My exploration of the Dragon Curve fractal was mainly guided by these three questions:
• When folding a strip of paper consistently in half (by bringing the right edge of the paper on top of the left edge, and then infinitely repeating this sequence of folding the paper in half by bringing the right edge of the paper on top of the left edge), what patterns of peaks and troughs will emerge?
• Will the pattern (of peaks and troughs) be constant and predictable? Why?
• Will I be able to create a formula to pre-determine if a crease will be either a peak or a trough?
Figure 2 When continually folding a piece of paper from right to left, there will be a pattern of creases that I classified as troughs and peaks. When pulling the strip of paper horizontally, I consider “troughs” the creases that point down and the “peaks” the creases that point up.
Figure 1 This picture (from Wikipedia) shows a strip of paper being folded in half over-and-over again by always bringing the right edge of the paper on top of the left edge. When the strip is unfolded, the Dragon Curve fractal is created when the creases are at 90-degree angles.
Vector • Spring 2018 9
I started my exploration of the Dragon Curve using a strip of paper. To create the Dragon Curve, I repeatedly folded a strip of paper in half by bringing the right edge of the paper on top of the left edge. When this strip of paper is unfolded and the creases are all at 90-degree angles, the Dragon Curve fractal becomes apparent. Pulling this strip of paper horizontally, creates an irregular pattern of creases that point up and creases that point down (Figure 2). I classified the creases that point up as peaks, and the creases that point down as troughs. The physical piece of paper helped me begin to see the pattern the troughs and peaks created throughout folds, because the creases were in a straight line. The difference between the troughs and the peaks were evident because they could be easily seen and felt.
To see which creases were created by specific iterations, I colour-coded the creases (Figure 3). In doing this, I noticed that the new creases (separate colours) always appeared at the midpoints between the previous creases, and if you look at the creases created by each iteration in isolation (focus on one colour), there is always an alternating pattern of peaks and troughs. Furthermore, there is a reflection on either side of the centre crease (which remains a trough created by the first iteration). After the first fold, due to the layers created by folding a strip of paper continuously in half, creases are created in pairs of a peak and a trough on top of each other when folded. Thus, when unfolded, they are found at the exact same location on either side of the center crease. The creases created by every iteration are in multiples of 2x+1 with the exponent increasing by one with every fold.
Figure 3 I colour-coded the peaks (Ps) and troughs (Ts) in the order that they appeared: red (first fold), orange (second fold), yellow (third fold), green (fourth fold) and blue (fifth fold) which allowed me to clearly see that although the GSP showed that each iteration is created by adding to the consecutive fold, troughs and peaks appear in between the creases made from previous folds. To make further observations, I created the Dragon Curve fractal on GSP–a dynamic geometry software (Figure 4). I created the Dragon Curve fractal on this program by first
Figure 4 Tx = the middle trough of each iteration. To better see the effect of each fold, I colour-coded every new iteration of troughs and peaks: red (first iteration), orange (second iteration), yellow (third iteration), green (fourth iteration), light blue (fifth iteration), dark blue (sixth iteration), light purple (seventh iteration), dark purple (eighth iteration) and dark green (ninth iteration).
drawing a single line. Then, I copied this iteration and rotated it 90-degrees. I continued the process of copying the entire iteration and rotating it 90-degrees nine times to create this Dragon Curve fractal. Although I only made nine iterations (virtual paper folds) on GSP, working with this technology allows an infinite amount to be created. This is an affordance that working with a physical piece of paper does not have, as it is impossible to fold a physical piece of paper more than five times before the paper becomes too thick and/or too narrow to fold again. Out of my nine iterations, I gave each one a different color to keep track of the virtual folds that I created. I determined that each “trough” created a right angle, and each peak created a 270-degree angle (Figure 5).
Figure 5 Marking the angles in the iterations made it clearer for me to see the pattern of troughs and peaks. In addition, I noticed that all of the squares are created with two peaks (270-degree angles).
Vector • Spring 201810
Although creating this fractal on GSP was simple, making observations about the “direction” each crease was “pointing” (peaks and troughs) proved to be much more difficult than when working with the piece of paper. To find the order of angles of the creases I had to consistently stay on one side of the Dragon Curve and follow the string of creases without crossing to the other side of the fractal. This was difficult, especially in larger iterations, as boxes were created, making it near impossible to know which direction I needed to follow in order to make observations about the pattern of troughs or peaks. In encountering these difficulties, I developed strategies that helped me follow the order of angles and track them correctly. As the iterations became bigger (with more boxes and thus a higher possibility of mixing up the order of angles in the Dragon Curve) I started with the center angle of each iteration (noticing that it is consistently a right angle), and I worked outwards from the center angle to the ends of the line of angles. Furthermore, to make the order of angles easier to follow, I widened the rotation of the iterations to 95-degrees. This eliminated the boxes that the 90-degree rotations created, making it easier for me to follow the line of angles and find a pattern of troughs and peaks (Figure 6). In copying the entire iteration nine times, and adding it to the end of the “string” of iterations in order to create the next iteration, GSP helped me see that the creases (angles) within each iteration are fixed and every consecutive iteration includes the entire previous iteration.
Figure 6 Creating the folds with 95-degree angles eliminated the boxes. Troughs became 95-degree angles and peaks became 265-degree angles.
I found it interesting that the physical piece of paper and GSP highlighted two different ways to create the pattern of troughs and peaks within the Dragon Curve fractal. The physical paper showed new creases were incorporated in between creases from previous
iterations. Contrarily, working with GSP showed the creases within each iteration were and that new creases do not disrupt previous iterations. Instead, the new creases appear in the centre of the entire pattern of troughs and peaks. Despite being contrasting processes, both resources create the same pattern of troughs and peaks. This inspired me to combine the observations I made from both sources in a grid, in the hopes of gaining a more complete understanding of the Dragon Curve fractal and answering my questions (Figure 7).
Figure 7 I color-coded the peaks (Ps) and troughs (Ts) in the order that they appeared: red (first fold), orange (second fold), yellow (third fold), green (fourth fold) and blue (fifth fold) like I did with the physical piece of paper.
Observations made about the Dragon Curve fractal from working with the grid:
• The formula to determine the number of creases that appear in each fold is 2(2 f-1)-1(with f being the number of the fold).
• The center crease of each iteration is 2 f-1. For example, on the fourth fold, there will be 2(24-1)-1=15 folds and the center crease is the 24-1=8th fold.
• The center crease is the first “point of symmetry” for the trough and peak pattern of creases that radiates from the center towards the edges of the paper.
• To the right side of the center crease, every 2 f-1-1. . .
(decreasing) will be a trough and reflected to the left side of the center crease, every 2 f-1-1. . . (decreasing) will be a peak.
# of fold # of creases
2(2 f-1)-1Center crease
2 f-1
1 1 1
2 3 2
3 7 4
4 15 8
5 31 16
Table 1 Summary of formulas.
Vector • Spring 2018 11
To highlight these observations, here is a break down of how I got to the 5th iteration from the grid in Figure 7.
• Red: The center crease is 24=16. Decrease the center crease by 21 (24-1=23) and work outwards from the center crease 8(23) creases.
• Orange: This will create twice as many orange creases (two creases) as the red crease (one crease) that was created by the previous iteration. The orange creases will be found at the midpoint between the edges of the paper and the red crease that was created by the previous iteration. When working towards the left of the paper 23 creases, the orange crease will be a peak. When working towards the right from the red crease 23 creases, the orange crease will be a trough. Using these two new orange creases as points of symmetry for the creases on either side of the center crease, decrease 23 by 21 (23-1 =22 ) and work outwards from the two orange creases four creases to the left, and four creases to the right.
• Yellow: There will be twice as many new (yellow) creases (four creases) compared to orange creases (two creases) that were created by the previous iteration. The new yellow creases will be found at the midpoint between the previous creases. Moving four creases to the right of the orange creases will be a yellow trough; moving four creases to the left will be a yellow peak. Like in the previous folds, decrease 22 by 21 (22-1 =21 ) and work outwards 21(2) creases from the yellow creases.
• Green: There will be twice as many new (green) creases (eight) than the creases created by the orange iteration (four). These new green creases will be found at the midpoint between the previous creases. When moving 21 creases to the left of the orange creases, the green creases will always be peaks. When moving 21 creases to the right from orange creases, the green creases will be troughs. Decrease 21 by 21 (21-1=20) and now, work outward from the green creases, 1 crease.
• Blue: There will be twice as many blue creases (16) than the creases created by the previous fold (four). These creases will be found at the midpoint of previous creases. When moving 20 to the left from green creases, the blue creases will be peaks; when moving 20 to the right from green creases, the blue creases will be troughs.
In breaking down the fifth iteration of this grid, I noticed that moving 2x creases to the right of the center creases always yields a trough, and moving 2x creases to the left of the center crease always yields a peak. In seeing this regularity, I predicted that because the Dragon Curve is a fractal, the pattern of troughs and peaks was
going to be self-similar throughout each iteration like a fractal. This inspired me to see if the pattern of troughs and peaks would fit into a flow chart where every branch of the flow chart breaks off into two new branches and those two new branches continually break off into two new branches … like a fractal (Figure 8). The pattern of peaks and troughs that this flow chart highlighted was very surprising to me.
Like the grid, I used the fifth iteration of paper folds to make the flow chart (Figure 7). I started with the center peak, and I noticed that in each layer of the flow chart (a new layer was created with each iteration), peaks were always found at the left (created by subtracting 2x from the previous iteration), and troughs were always found on the right (created by adding 2x to the previous iteration). I don’t know how I didn’t notice this regularity throughout the folds before I created this flow chart, or why I didn’t predict that a consistency like this would appear sooner. It was only when I worked with this flow chart that I discovered a simple method to determine if a specific crease in a certain iteration is a peak or a trough.
Figure 8 This is a small branch of the flow chart I created of creases within the Dragon Curve fractal. This flow chart, always branches into a a peak to the left and a trough to the right.
The formula:
To determine if a target crease is a trough or a peak, the number of the fold must be known and the center crease must be determined. Then, by only using 2x-1-1. . . (decreasing), add or subtract 2x-1-1. . . (decreasing) until you reach the target crease. If you have reached a number that is a below the target crease, you will add 2x-1-1. . .
(decreasing). If you have reached a number that is above the target crease, you will subtract 2x-1-1. . . (decreasing). If 20 is subtracted to reach the target crease, it will be a peak. If 20 is added to reach the
Vector • Spring 201812
target crease, it will be a trough.
Here are the steps I would take to determine if the 17th crease in the 5th iteration is a peak or a trough.
1. I would determine the center crease of the 5th iteration: 2 f-1(25-1=24=16).
2. Since 16 is below the target crease I need to add the next 2 x-1-1 . . . (decreasing)16+2 3=24.
3. Now, I am above the target crease, so I need to subtract the next 2 x-1-1 . . . (decreasing) 24-22= 20.
4. Since I am still above the target crease, I need to subtract the next 2 x-1-1 . . . (decreasing) 20-21=18
5. Since I am still above the target crease, I need to subtract the next 2 x-1-1 . . . (decreasing) 18-20=17.
6. My last equation was subtraction meaning the 17th crease of the 5th iteration will be a peak.
Some other examples of this process used to determine if a certain
20 is subtracted (peak) 20 is added (trough)
(5th fold) 25th crease 16+ 8+4-2-1=25
(6th fold) 56th crease 32-16+8+4-2-1 = 56
(5th fold) 17th crease 16+8-4-2-1 = 17
(4th fold) 2th crease 8-4-2=2
(5th fold) 19th crease 16+8-4-2+1=19
(3rd fold) 7th crease 4+2+1 = 7
(2nd fold) 3rd crease 2+1=3
(8th fold) 15th crease 128-64-32-16-8+4+2+1=15
crease is a trough or a peak (verified with the fractal on GSP).
In conclusion, creating my own questions, focusing on finding patterns and using different resources to support my exploration made this mathematical process very personal and engaging for me. Since I created the parameters of this project, it was suited to my level of understanding and it interested me. Due to this, I was intrinsically motivated to find the answers to my questions and develop a formula to determine the peaks and troughs within the Dragon Curve. By personally experiencing the method of learning that I am teaching in my classroom, I have developed a newfound appreciation for students learning mathematics as it is a subject that can be unique and engaging for every learner.
Bibliography
Dragon curve. (2017, November 06). Retrieved November 16, 2017, from https://en.wikipedia.org/wiki/Dragon_curve
Vector • Spring 2018 13
When it comes to possibilities for math class, I use many of the following words synonymously: puzzles, problems, tasks, explorations, questions, activities. Each has a different shade of meaning, and probably many of us use them differently. What is important though, is that students have a pathway to a rich learning experience that challenges, engages, and deepens their learning.
My goal in this article is to inquire into what makes mathematical tasks this sort of rich learning experience for students in the classroom. I advocate for encounters with beautiful, transformative mathematics that are accessible and appealing to students. Engagement in this kind of experience is precisely what I mean by richness.
The critical element of richness is that it describes what our students experience, rather than some inherent quality of the activity. I have had conversations about long division that were incredibly rich, counter to my expectations. The opposite is all too common as well: the task you think will be rich doesn’t work in the classroom, or even more confusingly, is amazing in one class but doesn't work in another.
Let's consider a problem that I believe has the capacity to be quite rich.
The problem: How many different ways can you fill in the blanks with positive integers to make this equation true?
8 = ________ + _________ + _________
Point 1. The richness of a task depends on how your students experience it.
One student’s rich task is another student’s tedious exercise, and another’s intractable problem. If a student doesn’t care about it, isn’t grabbed by it, finds it too hard, too easy, too pointless, then any richness that could potentially emerge from the problem never does. We need to be aware of the potential pathways to and beyond the problem.
It may be easy to come up with one solution to this problem, but for fifth graders, for example, it is probably difficult to come up with all the solutions so that students might check out, or require more prompting to really engage. The point is:
Point 2. The task is only rich if students are motivated to engage in it.
So, before we pose a question, we need to think about why it’s actually interesting. I’m a big fan of the inherent interest in mathematical problems, and from what I’ve seen, students in fifth grade and up are pretty interested in abstract counting problems, these “what are all the ways” type of problems.
However, we can motivate it with a simple warmup problem, around which we could lead a class discussion. Let’s imagine a dialog.
Warmup: Fill in the blanks to make this equation true.
8 = ________ + _________
Student 1: 5 + 3
Student 2: 7 + 1
Student 3: 8 + 0
Teacher: How many ways do you think there are to fill in the blanks?
Student 4: Infinity. Because I could do 7.5 + 0.5, or 7.6 + 0.4. As many different decimals as I want.
Teacher: That’s a good point. Ok, I’m going to change the problem a little so the answer isn’t infinity anymore.
Updated Warmup: How many ways can you fill in the blanks to make this equation true, using whole numbers?
Nine Points on Rich Math Tasksby Daniel Finkel
Vector • Spring 201814
Student 5: Can I use negative numbers?
Teacher: How would you use them?
Student 5: Well, -1 + 9, -2 + 10, -3 + 11.
Teacher: What do the rest of you think? Do these work?
Student 6: They all equal 8! There will still be infinitely many ways to solve it!
Teacher: I agree. I’m going to alter the problem again.
Updated Warmup: How many ways can you fill in the blanks to make this equation true, using positive whole numbers?
Student 3: But that won’t let me use 8 + 0.
Teacher: True. Let’s add this challenge for later. [Teacher writes on the corner on the board: “How will allowing 0 change the answers?”] For now everyone take a minute and figure out how many possibilities there are altogether.
[Students work in pairs or trios for a minute or so.]
Student 7: There are four different answers.
Student 8: I got seven.
Teacher: That’s strange. Could we list them all?
[Students 7 and 8 write their lists down on the board.]
Student 7: You shouldn’t include 5 + 3. It’s the same as 3 + 5.
Student 8: I think they’re different.
Teacher: This is a decision we should talk about.
Teacher: Something you both did that I like a lot was to organize your lists. I can see all the possibilities very clearly.
Teacher: Okay, two more warmups, and you can choose which one you want to try, or do both. How many ways are there to fill in these blanks to make each of the equations true, with all the same rules as before?
6 = ________ + _________
40 = ________ + _________
Let’s take a minute and look at what’s happening with this problem so far. The teacher is starting from a place that should be easy for everyone, and everyone is getting some initial successes as they get oriented on the kind of thinking this problem requires. The teacher is also choosing to open up the discussion to include what rules and conventions will make this problem the most interesting. Allowing non-integers and negative numbers will immediately lead to “infinity” as an answer; that’s less interesting (at this moment, for these students), so we hone in on what will make the problem interesting. Other issues, like whether 5 + 3 should be counted as distinct from 3 + 5, are something the teacher may know by virtue of playing around with the problem before.
Even though we haven’t gotten to our main problem yet, we have the students engaged and thinking, having initial successes, and clear on the technical details. We’ve essentially created a runway leading to the problem, that will allow them to fly once they really get it.
Point 3. Rich tasks should either be quickly accessible, or, students should have the tools to make them accessible.
The main thing is to get students thinking and engaged as quickly as possible. You can do this by posing the question right away, posing easier questions to build up to it, posing harder questions, or involving students in a conversation.
I often imagine our goal as helping students to change the step size they encounter when they work on a problem. In other words, when they reach a problem that feels like an impossible step up, I want them to have the tools to break it into a staircase (see Figure 1). In this case, the first step isn’t too hard. But if it were, I’d want them to try to get just a single answer to make 8 = ___ + ___. Or change the 8 to a 4.
Figure 1 An impossible step gets broken into manageable steps.
Vector • Spring 2018 15
On the other hand, if the step feels too small, I want them to think about how they could make it harder. How could we answer the question if the 8 were a 40? Is there a way to quickly solve the problem for any number?
Point 4. Rich tasks balance success and challenge.
This illustrate a balancing act: problems that are too hard or too easy won’t hold students’ attention for long. (They tend to call both types of problems “boring.”) Perfectly crafting the experience for all students isn’t realistic, but with rich problems, we don’t have to: they are adjustable by the students, and once they get engaged, they tend to find ways to strike this balance on their own.
Point 5. Rich tasks teach habits of mind and content simultaneously.
Of course, we can help them if we teach them habits of mind that allow them to take control of their experiences in tackling the problems. If students feel like they own the problem, then they can take liberties like changing the problems to make them simpler. This maneuver is actually a profoundly useful problem-solving technique: start simple, and slowly build the complexity back in. If you add in a table, chart, graph, or other organizational tool, you’ve got a recipe to overcome a lot of what’s going to be presented in a mathematics class.
Returning to the classroom example, we might expect students to notice that there are five ways to make 6 from two addends (1+5, 2+4, 3+3, 4+2, 5+1), and 39 ways to make forty. It might be worth a quick discussion of the general structure here (are there always n–1 ways to make n from two addends? Why?). Or, we might choose to go right into the main problem.
The problem: How many different ways can you fill in the blanks with positive integers to make this equation true?
8 = ________ + _________ + _________
All the habits of mind—i.e., starting simple, organizing your work—that came up before will be useful here too. But at this point, I’d expect most students to be ready for this much harder problem. The rules are clear, they’ve already had a success, they are ready for a greater challenge, and they (rightly) anticipate that there’s something cool to find out.
And now, we let them work. It depends on how much stamina they have built up working on problems of this kind, but I would expect that students could work on their own for 20 minutes to 40 minutes, with the teacher talking to pairs and trios to further their thinking.
Point 6. You need to try out rich tasks on your own before you give them to your students.
Seriously. Take a minute and solve this problem. How do you approach it? What’s your answer, and more importantly, what is your pathway towards your answer? We have to do the math we ask students to do, or we won’t have a personal experience of what the various pathways through a problem are, and how they feel.
Point 7. There is more than one way to solve a rich task.
I recently posed this problem in a fifth grade class, and there were at least three ways students approached it, one of which I had never seen before.
Method 1: Choose the highest number you can first, and then systematically make a pattern working down.
There were variations on this too, like starting with the lowest number first. Either way, it yields a beautiful structure: a kind of triangle, organized by columns that each start with different numbers. We can count the number of ways to solve the problem now: 1 + 2 + 3 + 4 + 5 + 6 = 21 solutions. Moreover, this method feels like a generalization! For the students who solved it, we might ask them to replace the 8 by a 12.
Method 2: Start small, and work your way up. Students using this method found all the ways to add three positive integers to make 3 (there’s just one way: 1 + 1 + 1), and then 4, and on up. They ended up with a table:
Sum 3 4 5 6 7 8
Number of ways to make the sum with 3 addends
1 3 6 10 15 21
This is pretty nice! There’s a pattern in these numbers (lots, in fact), and students could use the pattern that emerges to make predictions about much larger numbers.
Vector • Spring 201816
Method 3: Account for rearrangements. This one surprised me. Students considered 8 as the sum of 6, 1, and 1. How many sums can we get from those addends? Three: 6 + 1 + 1, 1 + 6 + 1, and 1 + 1 + 6. What about 5, 2 and 1? In this case, there will be 6 possible rearrangements. It’s a subtler argument, and requires a bit more work, but it’s quite a beautiful approach.
There are many other approaches to this problem as well.
Point 8. Save time to wrap up.
There were a lot of things to share, and it’s good for students to have some time to try to explain what they figured out, and also to be aware of what we still don’t know about a problem. Do we have a way to solve the problem for any sum? How many ways could we solve 25 = __ + __ + __ using positive integers in the blank spaces? Why do the different approaches from before all give a right answer, and what does it tell us that they do? In method 3, it seems like every rearrangement is the sum of 3s or 6s, so it seems like the total should be divisible by 3; but in method 2, we saw that not every rearrangement was divisible by 3. Can anyone make sense of that? What if there were four blanks in our equation instead of 3?
The wrap up isn’t about knowing everything, but it’s important to know what you know, and what you don’t. I always like my students to leave with something to reflect on.
For me, introducing richness into mathematics is the raison d’être of mathematics class: to have rich experiences thinking about mathematics. But rich tasks don’t have to go perfectly, and they won’t. All you really need to get started is an idea for what a pathway through a problem might look like; if it was fun for you to think about, maybe it will be fun for your students too.
Point 9. Go in with an idea for the rich task, plus an easy version, plus an extension.
In this example, I have three levels in my head as class starts, involving counting all the ways to make an equation true by putting positive integers in the blanks.
The main problem: 8 = __ + __ + __
The easier version: 8 = __ + __
The extensions: 14 = __ + __ + __ OR 8 = __ + __ + __ + __
If I focus on whether or not my students are having a rich experience, I can make sure they have enough success to take on the challenge, have enough challenges to be engaged, and have the right amount of engagement for them to discover the beauty and richness of mathematics.
For more ideas about rich tasks, visit Dan's blog https://mathforlove.com/blog/, or follow him on Twitter @MathForLove.
Vector • Spring 2018 17
In the last few years, Which One Doesn’t Belong? (WODB?) has become popular among many mathematics teachers, from kindergarten to calculus. This rise in popularity is due, in large part, to Christopher Danielson’s shapes book (Danielson, 2016) and the single-serving website that it inspired (curated by Mary Bourassa). WODB? is a 5-to-10 minute instructional routine in which students are shown a set of four numbers, expressions, graphs, or shapes and asked the same two questions every time: “Which one doesn’t belong? Why?”
A few years ago, I created several WODB? sets1 of my own. I invite you to think about the set in Figure 1. Which one do you think doesn’t belong? Why?
Figure 1: Which one doesn’t belong?
You may have selected the spinner on the top right because it’s the only one with three, not four, possible outcomes. You may have
chosen the top left because it’s the only one with equally likely outcomes. You may have picked the bottom left because it’s the only one in which the probability of a single outcome (landing on orange) is greater than 1/2. You may have decided on the bottom right because it’s the only one in which the probability of landing on blue is not equal to 1/4. You may have come up with additional reasons. Notice that any one of the spinners could be the one that doesn’t belong. All answers are correct; what matters are the justifications.
This isn’t accidental; it takes purposeful planning to make sure that an important mathematical property sets each element apart. I can use a table to help me organize my thoughts. I can begin by listing four characteristics. For example:
3 possible outcomes
Equally likely outcomes
P(outcome A) > ½ P(outcome B) ≠ ¼
Next, I can assign each property to one and only one of the elements.
3 possible outcomes
Equally likely outcomes
P(outcome A) > ½
P(outcome B) ≠ ¼
Top Right ✓ ✗ ✗ ✗
Top Left ✗ ✓ ✗ ✗
Bottom Left ✗ ✗ ✓ ✗
Bottom Right ✗ ✗ ✗ ✓
Finally, I imagine elements with these properties, as well as the classroom discussion itself.
However, not all two-by-two arrays make good WODB? sets. Often, I painted myself into a corner when I strayed from thinking about four distinguishing features ahead of time. Adding the fourth element can change the reason why one—or more!—of the other
Alike and Different: Which One Doesn’t Belong? and Moreby Chris Hunter
1 https://reflectionsinthewhy.wordpress.com/2015/03/11/which-one-doesnt-belong/
Vector • Spring 201818
three doesn’t belong. For example, the set in Figure 2 didn’t make the cut:
Figure 2 Which one doesn’t belong? Hundreds/hundredths
The top right and bottom right are close to 100 (or 1); the top left and bottom right are greater than 100 (or 1); top left and top right have seven parts, or rods, of tens (or tenths); all involve seven parts in some way. There is an assumed answer to the question “Which one is 1?” in these noticings: a flat is 100 if we’re talking whole numbers and 1 if we’re talking decimals. But what if 1 is a flat in the top left and a rod in the bottom left? Now both represent 1.7. This flexibility was front and centre in my mind when I created this set.
Thus, even if this falls apart as a WODB? set, rather than scrap the hundreds/hundredths grids, I can modify my questions (Figure 3):
Figure 3 What’s the same? What’s different?
Another image that elicits discussion of equivalent fractions and place value (Figure 4), looking at 0.25 or one-quarter; 0.25 or two-tenths and five-hundreths.
Figure 4 What’s the same? What’s different?
I created a few more of these alike and different questions, including several variations on five subtract two (Figures 5 and 6):
Figure 5 Five-quarters subtract two-quarters or five root-two’s subtract two root-two’s.
Figure 6 Five 1001’s subtract two 1001’s or five (x + 1)’s subtract two (x + 1)’s.
Like Mary Bourassa, Brian Bushart created Same or Different?2 a website with plenty of crowd-sourced images meant to generate mathematical discussions. Both are excellent places to start; designing your own can be challenging at first.
Another question that I like is “Which two are most alike?” (Small, 2010). I like it because the focus is on sameness, and like WODB?, students must make and defend a decision. Also, this “solves” my painted-into-a-corner problem; there are fewer relationships between elements to consider.
2https://samedifferentimages.wordpress.com/
Vector • Spring 2018 19
Similar connections can be made in Figure 9:
Figure 9 Which two are the most alike?
This time, the first and second expression involve a quotative (or measurement) interpretation of division: groups of (–3) or 3x, not (–3) or 3x groups. The second expression is equivalent to the first if x equals –1.
What’s the reason for the second and third expressions to be paired together? Maybe this isn’t a good “Which two are most alike?” That’s okay. Asking “What’s the same? What’s different?” and including only two expressions gets me out of that jam. Again, it’s about inviting students to notice properties and justify their decisions, not being loyal to a particular question stem.
Which One Doesn’t Belong? is a powerful routine. First, all students can contribute to classroom discussions. Some students may challenge themselves to find at least one reason for excluding each element in a set; in fact, finding a reason for the fourth can often prove to be elusive. However, it is not necessary for each student to do so. A learner who can identify a single “imposter” can still participate in the routine; sometimes, their justification is
The numbers in the left and right images (see Figure 7) are less than 100 (if a dot is 1); the numbers in the centre and right can be expressed with 3 in the tens place; the left and centre image can both represent 43, depending on how we define 1. As with the hundreds grids above, this flexibility was front and centre in my mind when I created this set.
At the 2017 Northwest Mathematics Conference in Portland, my session3 addressed operations across the grades. A big idea ran through the workshop:
“The operations of addition, subtraction, multiplication, and division hold the same fundamental meanings no matter the domain in which they are applied.” (Small, 2010)
This big idea underlies the question in Figure 8:
Figure 8 Which two are most alike?
At first glance, the second and third expressions are most alike: both involve decimals. But the quotient in both the first and second is 20; in fact, if we multiply both 6 and 0.3 by 10 in the second, we get the first. The first and third likely involve a partitive (or sharing) interpretation of division4: 3 groups, not groups of 3. However, context often determines meaning.
Figure 7 Which two are most alike?
4https://reflectionsinthewhy.wordpress.com/2017/10/11/dividing-by-decimals-fractions-ham-ribs/
3https://reflectionsinthewhy.wordpress.com/nwmc2017/
Vector • Spring 201820
unique, adding to the collective understanding of the classroom community of mathematicians.
Further, WODB? is powerful because it engages students themselves in the “doing” of mathematics or, here in BC, curricular competencies. WODB? is most strongly connected to “use mathematical language to contribute to mathematical discussions” and “explain and justify mathematical ideas and decisions.” In addition, by their nature, many WODB? sets encourage students to “visualize … mathematical concepts” (https://curriculum.gov.bc.ca/curriculum/mathematics).
Finally, WODB? provides elementary and secondary math teachers with a common framework for facilitating mathematical–and professional!–conversations. WODB? can be implemented in any classroom, be it one in which seven-year-olds talk about attributes of 2D shapes, or one in which seventeen-year-olds argue over rational functions. By having our students experience WODB? throughout K-12, meaningful mathematical discourse in our classrooms may even become…routine.
Additional examples can be found in the original blog post: https://reflectionsinthewhy.wordpress.com/2017/11/01/alike-different-which-one-doesnt-belong-more/.
Websites of interest: https://samedifferentimages.wordpress.com https://talkingmathwithkids.com http://wodb.ca
References: Danielson, Christopher. (2016). Which one doesn’t belong? A shapes book. Portland, ME: Stenhouse.
Small, Marian. (2010). More good questions: great ways to differentiate secondary mathematics instruction. New York, NY: Teachers College.
Chris tweets at @ChrisHunter and blogs at reflectionsinthewhy.wordpress.com.
Vector • Spring 2018 21
On a snowy Friday in February, about 300 educators gathered in Richmond for the Mathematical Mindsets workshop hosted by the BCAMT. Over a year in the planning stages, educators from across the province had been looking forward to hearing from Jo Boaler and Cathy Williams, cofounders of youcubed. With less than 48 hours until the start of the workshop, it was announced that Jo was ill and could not travel to Canada. Cancelling the workshop was not an option, and so it was decided to carry on with Cathy as lead presenter.
And what a day. Cathy had everyone engaged right from the beginning, asking us to reflect on our own relationship with mathematics, giving us problems to ponder in groups, and sharing insights from neuroscience research into how we learn mathematics. There was time to reflect as individuals, and some much valued time to have conversations with colleagues at our tables.
Jo and Cathy’s work on youcubed.org is dedicated rich mathematical learning experiences that builds confidence and a positive outlook on mathematics. “All students can learn mathematics to high levels, and teaching that is based on this principle dramatically increases students’ mathematics achievement” (youcubed.org). Their work is based on mindset research focused on mathematics, and they have created resources that supports our redesigned curriculum with a focus on big ideas, and students developing a solid foundation in conceptual understanding, problem solving abilities, computational and procedural skills. According to Cathy, “If we pull the right tasks for the right big ideas, the mathematics will fall out. We need to stop the laundry lists of things students ought to do." This was a call to action to move away from a focus on procedures, and ensure that all students have the opportunity to develop their conceptual understanding.
To believe that all students are capable mathematics learners requires us to shift our own relationship with mathematics.
What mathematics are we doing? What is our own mathematical identity? How do we create a mathematical culture in our schools? How do students bump into mathematics outside of the classroom? We need to do more mathematics together. And if we’re not doing mathematics, we should be looking at student work together.
Throughout the day, Cathy kept bringing us back to mathematics with problems (see problem set on page 44) for us to work through at our tables. We visualized, we played, and we investigated, using pictures as a starting point rather than rushing into symbolic notation. We experienced productive practice through a problem that mathematicians are still trying to figure out (the Collatz Conjecture). Cathy reminded us that mathematicians don’t have the answers to all the problems they are working on, and that there are no answer keys. She encouraged us to approach mathematics problems with our students that we didn’t know the answers to, and reminded us that in a community of mathematics learners no one person needs to know all of the answers.
With a goal of making research available to teachers, Cathy touched on the neuroscience of how to teach mathematics well for student engagement and achievement. Some of the highlights of day included the importance of finger perception, the importance of the plasticity of the human brain and how we can rewire children’s brains to develop mathematical thinking, and the brain’s ability to compress concepts but not procedures.
We were briefly joined by Jo in the afternoon via a video call. Jo was visibly unwell, but held things together long enough to answer a few questions. She shared with us what she considers to be the most important teaching strategy: giving students the mindset message that when they are struggling, their struggle is what causes their brain to grow. Cathy followed this with the advice that we shouldn’t pull our kids out from the "pit", that we all need to get into the pit for some struggle. “We need to have kids struggle and lean into the struggle, because one day things will be hard.”
Mathematical Mindsets Workshop with Cathy Williamsby Susan Robinson
Vector • Spring 201822
“Sending a short note to extend my thanks for an inspiring, thought-provoking day today . . . Cathy Williams was a tremendous presenter! Definitely not a “substitute” for Jo Boaler. I laughed. I teared up. I felt glad to be present for the reflections, the conversations, the learning.”
–Doug David, District Curriculum Support Teacher, SD71
All in all, Cathy pulled together a tremendous day of professional learning and personal growth, overcoming what might have been disappointment at Jo’s absence with her enthusiasm, expertise and emotional impact. Many thanks to Rhona Soutar from Campbell River, whose energy and determination were the driving force behind the idea of the day, and to the organizing committee for all their efforts at ensuring everything ran smoothly. Despite the snow piling up outside, and the messages of school closures and teachers being sent home, the vast majority of the audience remained in their seats right to the end.
At times, Ray’s biggest handicap was his passion for math.
I’m determining the target line for an equidistant putt along a 30 degree arc around the ball. the point where these lines coincide is the place to aim for.
I’m going slightly uphill so, “Y” = the ratio of the hit, “x” is the variable of the slope up, “b” is the constant. So, I simply multiply the ratio of the hit/
roll for the given slope by the distance to the hole.
I MISSED! How is that possible? I was so focused!
Vector • Spring 2018 23
What follows is an approach that represents numbers geometrically to connect the concept of areas and averages in kinematics. The approach will lead us to an intuitive perspective on the multiplication operation as an area calculation. Areas under functions will then be associated with function averages in kinematics to reveal some remarkable results. The use of calculus is purposefully omitted to show that significant progress can be made by geometrizing numbers and extending the concept of the multiplication operation.
Let us entertain a geometrical view of numbers, one that identifies numbers as measurements. For instance, the number 3 would be identified as 3 units of measure relative to a single unit of measure. Thus 3 can represent the total length of 3 strings each of length 1, collected end-to-end. Recall that when two numbers are multiplied together, the repeated addition method states that multiplication is an operation that produces a single number copied the number of times that it’s being multiplied by, i.e. 2 x 3 = 2 + 2 + 2 = 6. Thus 2 multiplied by 3 is the sum of 3 copies of the number 2. Multiplication can also be interpreted as a transformation of two 1-dimensional objects into a single 2-dimensional object.
When two numbers (viewed as two 1-dimensional measurements) are multiplied together, they span another dimension. Geometrically this means that one of the numbers represents the length of a rectangle and the second number represents the width of the same rectangle. Thus, when multiplied together the two numbers span the area of the rectangle. In terms of the example above, consider:
Figure 1 Area Spanned by the Multiplication of Two Measurements
In this approach, the geometrical representation of multiplication is presented as an area calculation. One interesting application of this arises when the concept of the average of a quantity is considered. Recall a defining property of an average value of a set X = {x1, x2, ..., xn}:
Re-arranging, we can produce the familiar average value formula:
(1)
Areas and Averages in Kinematicsby Pouyan Khalili
Vector • Spring 201824
For instance, consider the set of numbers {x1, x2, x3, x4} = {1, 4, 6, 7}, Eqn (1) yields:
The result is exactly what we expect. For instance, if the elements in the set X are dollars earned by each person in a group of 4, then each person earns an average of 4.5 dollars. Now, a geometrical interpretation of Eqn (1) means that if numbers are represented as measurements, then the product n. xavg can be interpreted as the area spanned by measure n and measure xavg. In other words, if we take n as the width of all the (unit length) rectangles added together and xavg as the average height of those rectangles, then we can say that the product n . xavg is the total area of all n rectangles. For instance, n. (the average of set X) is simply 4 x 4.5 = 18, as shown in Fig. 2:
Figure 2 Area Spanned by the Average of set X
Figure 2 shows that there are four elements each of unit length, and on average 4.5 units in height. That is equivalent to Area = (1 x 1) + (4 x 1) + (6 x 1) + (7 x 1) = 18, which is the area spanned by each element represented with height (given by the magnitude of the element) and width (given by the unit length), as shown in Fig. 3:
Figure 3 Total Area Spanned
Thus, the total area spanned by the unit length and each element in a set is equivalent to the average of the set multiplied by the number of elements in the set.
Vector • Spring 2018 25
Extending this approach to functions, we propose that the height of our previous rectangles becomes the function value, and the widths become the individual widths of those rectangles, Δx, typically taken as the length of the total interval divided by the number of rectangles (to create an even distribution of rectangle widths).
Recall that the average value of a function f(x) is the sum S (the area under the graph of the function) divided by the length of the interval [a,b] of interest:
(2)
Equation (2) is the function generalization of the average formula given in Eqn (1), where the xi terms are now function values fi and the widths (which were the unit) are now Δx. Like Fig. 2 and Fig. 3 it can be shown that the area S under the curve f(x) is favg . (b-a):
Although the statement above requires integral calculus, progress can still be made without explicitly using limits, differentiation or integration.
Vector • Spring 201826
Let us consider how the area formulation could play a role in vector kinematics, the branch of physics concerned with the most basic observations of motion. The basic quantity of motion is velocity, defined as the spatial displacement traversed in an interval of time. The definition of instantaneous velocity, vins , is given as the slope of the tangent line on the displacement vs. time graph of the motion, as shown in Figure 4. Furthermore, the definition of average velocity is given as the slope of the secant line, as shown in Fig. 5:
Figure 4 Slope of the tangent line at point X Figure 5 Slope of the secant line between
yields vins at point X two points yields vavg between those points
The slope of the tangent line yields the instantaneous velocity:
(3)
The slope of the secant line yields the average velocity:
(4)
The product of velocity with time yields displacement (a well-known fact). Given those defining properties of velocity, consider the motion of an object undergoing constant motion, i.e. constant velocity. A simple velocity versus time graph of this uniform motion is shown in Fig. 6:
Vector • Spring 2018 27
Figure 6 Uniform Motion
Notice that in uniform motion vins = vavg , because the tangent line and the secant line are identical on the displacement versus time graph. In other words, every rectangle in Fig. 2 has the same constant height. Furthermore, we can say that velocity multiplied by time is not only the displacement, as in Eqn (4), but by Eqn (2) also the area of the rectangle under the velocity function. Thus, in all uniform motion, the total displacement is given by the following: Δd = vins . Δt = vavg . Δt, as in the velocity versus time graph of Fig 6. In other words, in the constant linear case (uniform motion), the average of the velocity function given by Eqn (2) yields:
(5)
Although Eqn (5) was derived for a constant velocity function, a simple application of calculus shows that the area under even more general graphs of velocity versus time yields total displacement.
Another special case worth considering is constant acceleration, or uniform accelerated motion. In this case, the shape of the velocity versus time graph is either a straight horizontal line (uniform motion, acceleration is zero, covered previously), or a diagonal line (acceleration is constant but not zero). Since we are dealing with linear graphs, the areas we need to compute are simple; they are rectangles and/or triangles. For instance, consider this uniform accelerated motion:
Figure 6 Area Under Uniform Accelerated Motion
Vector • Spring 201828
The area under the graph of Fig 6 has been split into the area of the rectangle A1, and the area of the triangle A2. The total area under the graph is given by A = A1 + A2 :
By Eqn (2):
(6)
The fact that the average value of occurs midpoint in velocity is evident since it is a linear function.
Note: Equation (6) is ordinarily taught as the average velocity formula, unfortunately the important condition in which it holds true is sometimes omitted. To find this important condition, consider uniform motion with two different velocities:
We apply the definition of average velocity, Eqn (4):
is equal to
if and only if
Therefore, for two uniform motions Eqn (6) holds true if and only if the time intervals for each motion are identical. For general motion this is obviously not always the case, therefore Eqn (6) is a very special case of average velocity.
Extending this idea, a motion split in to two uniformly accelerated intervals of time:
Vector • Spring 2018 29
Again, it follows that the same condition holds true:
if and only if
Thus, we find that it must be the case that the acceleration is uniform, and the time-intervals are equivalent, for Eqn (6) to hold true, otherwise we must resort to Eqn (4).
A simple example that shows the failure of Eqn (6):
The beach is X kilometers away. You drove half-way to it at a rate of 60 km/hr. How fast would you have to drive back to have gone the whole trip at an average speed of 120 km/hr?
As deceptively simple as that may problem may appear to be (many would say (60 + vf ) / 2 = 120 implies that vf = 180), the answer is of course that no physically allowed travelling speed (i.e. less than infinity) on the ride back will yield an average speed of 120 km/hr for the entire trip. This example shows that Eqn (6) is only valid under certain conditions, outlined above, and that in all other scenarios the more general Eqn (4) must be used to yield the correct result.
To recap, the measurement approach to numbers was shown to provide insight into the geometrical nature of the multiplication operation and its connection with the average value of a function in kinematics. This treatment may be fruitful for the learner who is developing their conceptual understanding of the geometrical representations of kinematical function averages.
Vector • Spring 201830
Math Challengers is a mathematics competition for BC students in Grades 8 to 10 (and sometimes earlier grades). In 2017, students competed as school teams in regional tournaments (at Okanagan College, Camosun College, University of the Fraser Valley, SFU, and UNBC) in February, and in a provincial tournament (at UBC) in April. The past year Math Challengers involved approximately 750 students from 42 grade 8, schools and 49 grade 9 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 school first half of the 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 best performing 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/
We hereby provide nine problems dealing with basic algebra, numbers, and probability. Some are phrased as word problems. Though, the problems may look rather difficult at first glance, the reader may find them easier once understanding exactly what to look for based on the information provided. We strongly suggest trying to solve the problems without looking at the solutions, and then compare both work and answers. You may find that problems can be solved in a number of different ways. We encourage students and teachers to discuss their solutions in group meetings, 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 algebraic concepts. If either
teachers or students are interested in providing feedback, they are encouraged to contact the first author at [email protected].
Problem 1:
In the game Lucky 7, you roll a fair die a few times and try to reach a total sum of 7 on your rolls. There is only one rule: “if on roll n you got a certain number k, and on roll n+1 you get a number equal to or larger than k, then the game is over after roll n+1”. A few valid sequences in the game are (1,1), (1,3), (5,4,3,3), (6,5,5), (6,5,4,3,2,1,5). Please note that the maximum number of rolls until the game is over, is 7. What is the probability of reaching a total sum of 7 when the game is over? Examples of sequences where the total sum of 7 is reached are (1,6) and (2,1,4). Note that the sequence (6,1) is not a valid sequence since you still have to roll a third time. Express the answer as a common fraction.
Solution:
The solution is quite straightfoward as there are only a few winning sequences, and they are easily found.
(a) The only 3 winning sequences of two rolls are: (1,6), (2,5), and (3,4); each one of them has a probability of 1
6× 16= 136
(b) Similarly, there are only five winning sequence of three rolls: (5,1,1), (4,1,2), (3,1,3),(3,2,2) and (2,1,4); each with probability of 16× 16× 16= 1216
.
(c) Finally, the only way to win with four rolls is with the sequence
(3,2,1,1), with probability of (16)4 = 1
1296, and, also, there is no
winning sequence of more than four rolls.
Some Challenging Word and Algebraic Problems from Math Challengers Competitionsby Joshua Keshet and Dave Ellis
Vector • Spring 2018 31
Adding up the probabilities:336
+ 5216
+ 11296
= 108 + 30 +11296
= 1391296
Problem 2:
The five students on the team that won the Provincial Math Challengers competition decided to celebrate the event with a gift exchange party. The rule is that each of the five students is to give one gift to one of the other four students so that each student is to give exactly one gift and is to receive exactly one gift. An example of such gift exchange is “A gives to B, B gives to C, C gives to D, D gives to E, and E gives to A”. How many ways are there to do the gift exchange? (Note that “A gives to B, B gives to C, C gives to D, D gives to A” is not a possible gift exchange because E can neither give nor receive a gift.)
Solution:
We have to divide all valid options, into groups. One group (L5), is when all five students are part of one “loop,” similar to the example given in the question. To find the number of such possible “loops” consider the fact that the first student (A) can select any one of the other 4 students to give the gift to, and then that selected student can only select one of the remaining 3 students, and so on. Thus, the number of loops in group L5 is 4 × 3× 2 ×1 , or 24.
What other groups of valid options are there? Clearly, “loops” of four students are not valid because they contradict the rule (the fifth student has to give a gift to self–just as in the note in the question). Thus, the only other remaining group is where three students make a “loop” of 3 (L3), and the remaining 2 students make a “loop” of two (otherwise, at least one student must give a gift to themself). Suppose that students A, B, and C are the members of the “loop” of three, then there are two ways for them to exchange gifts in that loop: (a) A gives to B, B gives to C, and C gives to A; and (b) A gives to C, C gives to B, and B gives to A. Note that the number of ways to select 3 students out of 5 students is:
5!3!× 2!
= 4 × 52
= 10 . So, the total number of “loops” of 3 is: 10 × 2 = 20 . Also, note that once a “loop” of three is selected, the other two students are also, automatically, selected to become a “loop” of two. Thus, the number of total valid option in group L3 is 20.
Thus, the total number of ways for the gift exchange is by adding the numbers of the valid options in groups L5 and L3: 24 + 20 = 44 .
Problem 3:
Two fair dice are rolled. What is the average value of the non-negative difference between the numbers showing on the two dice? Express the answer as a common fraction.
Solution:
Consider the 36 possible outcomes of the roll of two dice: there are two possible outcomes where the difference between the dice is 5 (specifically (1,6) and (6,1)). There are four possible outcomes where the difference is 4, six possible outcomes where the difference is 3, eight possible outcomes where the difference is 2, ten possible outcomes where the difference is 1, and six possible outcomes where the difference between the dice is 0 (both dice show the same number).
So, to get the average non-negative difference we need to sum up and divide by 36.
Sum up: 2 × 5 + 4 × 4 + 6 × 3+ 8 × 2 +10 ×1+ 6 × 0 = 10 +16 +18 +16 +10 + 0 = 70
Divide the sum by 36 to get the average value: 7036
= 3518
.
Problem 4:
The number 360 000 has 105 positive factors. (Note that the numbers 1 and 360 000 are two of these factors.) How many of these 105 factors are divisible by 4?
Solution:
First, let us check that 360 000 does indeed have 105 factors by expressing it as a product of primes: 360000 = 26 × 32 × 54 . The number 26 has seven factors, 32 has three factors, and 54 has five factors. Since each factor of 360 000 can be expressed as l ×m × n where l is one of the seven factors of 26 , m is one of the three factors of 32 , and n is one of the five factors of 54 , then 360 000 has 7 × 3× 5 factors, or 105 factors.
Of the 105 factors of 360 000, we need to find all factors that are in the form of 4k where k is an integer. Since 4k divides 360 000 then k divides
3600004 , or k divides 90 000. In other words, k is a
factor of 90 000. So, the equivalent, but simplified, question is to find how many factors does 90 000 have?
Vector • Spring 201832
Using the same method as for finding the number of factors of 360 000, we get: 90000 = 24 × 32 × 54 . The number 24 has five factors, 32 has three factors, and 54 has five factors. So, each factor of 90 000 can be expressed as l ×m × n where l is one of the five factors of 24 , m is one of the three factors of 32 , and n is one of the five factors of 54 . Thus, 90,000 has 5 × 3× 5 factors, or 75 factors. So, 75 factors of the 105 factors of 360 000 are divisible by 4.
Problem 5:
Ten people, Alan and Beti and eight others, are divided at random into two groups, one with four people and the other with six people. What is the probability that Alan and Beti end up in the same group? Express the answer as a common fraction.
Solution:
Note that once we determine who will be in the group of four, we also decide by this, that the remaining six people are in the group of six. Thus, all possible way to divide ten people into two groups, one with four people, and the other with six is: 10!4!× 6!
= 7 × 8 × 9 ×102 × 3× 4
= 2 × 3× 5 × 7 = 210 .
If Alan and Beti are both in the group of four, then the number of ways to select the other two people of the remaining eight people for that group of 4 is: 8!
2!× 6!= 7 × 8
2= 2 × 2 × 7 = 28 .
Similarly, if Alan and Beti are both in the group of six then the number of ways to select the other four people of the remaining eight people for that group of six is 8!
4!× 4!= 5 × 6 × 7 × 8
2 × 3× 4= 2 × 5 × 7 = 70 .
Sum up and divide: 28 + 70210
= 98210
= 715
.
Problem 6:
A box contains N marbles, of which two are white and N − 2 are black. You know that if you take out three marbles at random from the box, the probability that exactly two of them are white is 1
210.
What is the value of N ?
Solution:
The number of ways to pick three individual marbles out of a box of N marbles is N!
(N − 3)!× 3!.
The number of ways that exactly two of those individual marbles are white is the same as multiplying two numbers: (a) the number of ways to pick two white marbles out of a box of two white marbles, and (b) the number of ways to pick one black marble out of a box of N − 2 black marbles. Or, in algebraic terms: 2!
2!× (N − 2)!(N − 3)!×1!
= 1× (N − 2) = N − 2 .
It is given in the question that the probability of getting exactly two
white marbles and one black marble is 1210 . Thus,
N − 2N!
(N − 3)!× 3!
= 1210
This is an equation for N, for which we need a solution in the form
of a positive integer. Simplify and solve:1210
= 3!× (N − 2)N × (N −1)× (N − 2)
= 6N × (N −1) .
Thus, N × (N −1) = 210 × 6 .
Since the 210 × 6 = 1260 ≈ 35.5 , we can simply check if N = 36is a solution (which it is). Obviously, there is only one positive solution to this equation.
Another way is to solve the quadratic equation N 2 − N −1260 = 0 , which has two solutions: N = 36 , and N = −35 . Eliminating the negative solution, we conclude that N = 36 .
Problem 7:
What is the smallest possible value of n2 −17n +100 as n ranges over all integers?
Solution:
The function x2 −17x +100 is a parabola (the general form of a parabola is y = ax2 + bx + c ). In the case that a >0, the smallest possible value of this function is where x = −b
2a. So, if a = 1 and
b = −17 , the smallest possible value of the parabola is given by x = 17
2= 8.5 . Since a parabola is a symmetric function, the values
for x = 8 and x = 9 will both give the smallest value of this parabola for integer values of . Calculate (from the symmetry we can do it for either x = 8 or x = 9 , and get the same result):
x = 8 :82 −17 × 8 +100 = 64 −136 +100 = 28 , or,x = 9 :92 −17 × 9 +100 = 81−153+100 = 28 .
Thus, for this function, both n = 8 and n = 9 give the smallest possible value of 28 when ranges over the integers.
Vector • Spring 2018 33
Problem 8:
You throw a fair coin six times. The total number of heads you got is less than five. What is the probability that the total number of heads is less than the total number of tails? Express the answer as a common fraction.
Solution:
The information that we have is that the total number of heads in six throws is one of the following five numbers: 0, 1, 2, 3, or 4. Call this event A. This determines that the event AC (in words: “not A”) is when the total number of heads obtained is either five or six. If P(A) is the probability to get event A, and P(AC) is the probability to get event AC, then, P(A)+P(AC)=1. We will use this equation because it is easier to find P(AC).
The total number of possible distinct outcomes out of six throws is 26 , or 64. Note that each of these distinct outcomes has the same probability of 1
64. One example of such outcome is TTHTHT , and
in this particular example, there are two heads and four tails.
There is only one possibility in getting six heads, namely HHHHHH , (or, in words, six heads in a row, where each throw is heads).
There are six possibilities in getting exactly five heads because a throw of tails has to appear in exactly one of the six throws.
Thus, the number of ways in getting either five or six throws of heads, namely all possible outcomes of event AC , is: 1+ 6 = 7 .
Thus, the number of ways in getting less than five heads (all possible outcomes of event A ) is: 64 − 7 = 57 (each with probability of 1
64).
Getting a fewer number of heads than the number of tails means that the total number of heads can be either 0, 1, or 2.
Using the argument of similarity in determining probabilities, it is clear that getting either zero or one heads has the same probability as getting either five or six heads, so both of these events have a total of seven distinct outcomes (each with probability of 1
64), as
we calculated earlier for the latter.
Calculating the number of ways to get exactly two heads requires a bit more arithmetic. We need to get heads in the following two distinct throws, (and tails in all other throws): “1 and 2”, “1 and 3”,
“1 and 4”, “1 and 5”, “1 and 6”, “2 and 3”, “2 and 4”, and so on, up to and including “5 and 6”. Summing it up we get: 5 + 4 + 3+ 2 +1= 15 , each with probability of 1
64.
So, the total number of ways to get fewer heads than tails is: 7 +15 = 22 .
Since each of the distinct outcomes has the same probability, then the probability to get fewer heads than tails (a total of 22 outcomes) given the condition that the total number of heads is less than five (a total of 57 outcomes) is 22
57.
Vector • Spring 201834
This cross-district collaborative inquiry project is in its third year, and includes educators from ten districts throughout British Columbia. Educators involved in the project are active on Twitter, using the hashtag #BCAMTreggio, and a collaborative blog is a place to curate ideas, publications, presentations, instructional resources, and a photo gallery. The foundation for the professional inquiry is based on the principles and practices that have been developed in collaboration with educators in the group with expertise and familiarity with the centres in Reggio Emilia, Italy, as well as collaborating with Reggio-inspired schools.
Where does math live here?
As we continue to consider ways to grow and sustain this project, we make connections between these principles and practices and aspects of British Columbia’s redesigned curriculum. In considering the core competencies and the First Peoples Principles of Learning, as well as specific references to place across many areas of the curriculum, we thought an area we could be responsive to in our professional learning was thinking about place-based pedagogy in mathematics. Inspired by the work of Dr. Cynthia Nicol from the University of British Columbia and David Sobel of Antioch University, we have woven together elements of place-
based and culturally-responsive pedagogies with our developing understanding of Reggio-inspired principles and practices. Many of the districts involved in our project are considering ways to incorporate more outdoor learning experiences, and more than just taking math outdoors, we are looking at how place inspires mathematical thinking and connections.
Lauren McLean and Janice Novakowki initiated a series of informal “institutes” on Saturday mornings to be responsive to educators within our network, but also to open up our work to others who might be interested. We held four of these Saturday institutes starting in the spring of 2017, followed by lunch at a local restaurant to continue the conversation for those who were interested.
Spring Institute 2017: Pacific Spirit Park
On April 22, about 30 educators from across BC joined Lauren and Janice for a walk through the forest, noticing the landscape through a mathematical lens. In the group were educators from Vancouver, West Vancouver, Coquitlam, Burnaby, Richmond, Delta, Surrey, Langley, Chilliwack and Campbell River.
We engaged in several different experiences, uncovering and making connections to mathematics. We began with creating
Reggio-Inspired Mathematics: Connecting to Placeby Janice Novakowski on behalf of the educators involved with the BCAMT Reggio-Inspired Mathematics Collaborative Professional Inquiry Project.
● the child is capable, competent and has rights ● the teacher as researcher–provoker, ball tosser ● the child has a hundred or more languages to
express ideas● the environment as third teacher● connectedness to culture, community and
environment● responsive and emergent curriculum; the
uncovering of curriculum
● a pedagogy of listening● socially-constructed learning; collaborative
practice● inquiry-based; projects and investigations● importance of relationships● focus on big ideas and themes● use of loose parts and natural materials● documentation–making learning visible
Reggio-Inspired Principles and Practices
Vector • Spring 2018 35
structures using found materials that could be used for a challenge task. These tasks were linked together to create a course that we could run through, timing each other and trying to improve on our times. Participants mapped the course, and considered the spatial reasoning used in mapping–shape, size, distance, proportion, scale and perspective. In another area of the forest, we looked closely for patterns around us. We used field guides to help identify species, and considered the mathematics embedded in plant and animal identification. Throughout the day, there were many interesting discussions about learning outdoors–how it supports engagement, self-regulation, emotional well-being and opportunities for risky play.
Spring Institute 2017: Pacific Spirit Park
Summer Institute 2017: Jericho Beach Park
On June 10, educators from across the Lower Mainland and Campbell River met at Jericho Beach in Vancouver to think about mathematical experiences inspired by the beach environment. As we walked towards the beach together, we each used a “Can you find…?” bookmark to engage us in using mathematical thinking and language together. For example, some of the prompts were:
Can you find a shell that is smaller than the size as your thumbnail? Can you find a rock that has about half of its surface area covered with barnacles? Can you find ten pebbles that you can hold in your hand all at once? Can you find three different types of seaweed?
As educators engaged in these experiences themselves, they made connections to curricular competencies and content connected to the students they were working with.
We were honoured to be joined by Dr. Bridgette Clarkston of the University of British Columbia who is a seaweed expert. We learned about the three main types of seaweed, their life cycles, and their identifying features. As we moved along the beach, we engaged in different provocations, inspired by the setting and supported with some tools brought along such as measuring tapes, sieves, field guides and sand sculpting tools.
Summer Institute 2017: Jericho Beach Park
Fall Institute 2017: Lynn Canyon Park
On November 18, educators from across the Lower Mainland and Fraser Valley met at Lynn Canyon Park. We walked along Lynn Creek together looking at the trees, plants and leaves through a mathematical frame of size, shape and symmetry. We found lichen samples and discussed growth rates of different species making connections to estimation and creating line graphs. We noticed and wondered about the water flow in the creek, and took some time to sketch out a map of the area, considering size, scale, shape and perspective and making connections to our project’s work around spatial reasoning.
After crossing the suspension bridge, we engaged in creating “land art” inspired by Andy Goldsworthy and James Brunt, thinking about pattern, shape, design, balance and symmetry. As we wove our way back to our starting point, we found several mushrooms along the trail and used field guides and apps to identify them considering identifying features such as size, shape and location.
Vector • Spring 201836
Participants were each provided with a provocation card listing several questions to provoke their thinking about place-based mathematics. Many of these questions are general and can be asked across a variety of contexts. Some examples of the questions are provided above.
Fall Institute 2017: Lynn Canyon Park
Winter Institute 2018: Cypress Mountain
On February 17, educators from across the Lower Mainland met at Cypress Nordic Park to go snowshoeing and consider what might inspire our mathematical thinking and questions. We began our snowshoeing adventure by considering the questions: What math
do we see? What math lives here? What mathematics is involved in snowshoeing? How does snow provide a context for thinking about mathematics?
It had snowed overnight and was continuing to snow as we snowshoed, giving us the opportunity to look closely at the texture and undulation of the accumulated snow. We considered the conditions that affected how snow formations were created, and Lauren shared her understanding of local indigenous knowledge about how snow accumulations and texture can be used for both locating and tracking. Related questions that were raised by educators included: How do you measure the amount of snow falling? What is the relationship between temperature outside and the size of snowflakes? How do you measure the density and depth of snow on the ground? How does this affect how we move in the snow?
Along the snowshoe trails were several interpretive signs about the region, its animals and trees. Many of the signs provided mathematical information to provoke our thinking, such as how tall some trees grew, leading to the estimation of what percentage of the tree was visible above the snow, as well as information about the territorial range of animals.
As we snowshoed and then had lunch together, we shared different mathematical investigations we could try with our students when it snowed, such as measuring snowfall using graduated cylinders and metre sticks, collecting snow and estimating and measuring how long it takes to melt using
Questions to inspire mathematical thinking when visiting outdoor spaces:
Where does math live here?What do you notice?What do you wonder?What math do you see?How does this place inspire your mathematical thinking?Where do you see (patterns, lines, balance) here?What mathematical story does this place tell?What math to math connections can you find in this place?What connections to shape, size and symmetry do we make when we look closely at plants?What mathematics might we think about as we create a map of this place?What different ways could we measure the creek? rocks? leaves? twigs? shells? cones?How can mathematics help us to understand this place?
Vector • Spring 2018 37
different external temperatures, creating shapes and tracks in the snow, building 3D shapes with snow, creating tracks and trails in the snow and then creating maps of these, and also using field guides to identify trees and animal tracks (which could be included in our maps).
Winter Institute 2018: Cypress Mountain
The Saturday institutes were documented with photographs, observations and notes about the connections and questions that emerged along the way. So far, two e-books have been compiled, which can be accessed by educators unable to join in with our explorations:
Where Does Math Live in the Forest? https://goo.gl/Mbe3y2
Where Does Math Live at the Beach? https://goo.gl/iDUQfJ
For more information, follow #BCAMTreggio or @jnovakowski38 on Twitter, or check out our blog (Reggio-Inspired Mathematics), or “Reggio-Inspired Mathematical Provocations” in the Fall 2014 Elementary Issue of Vector, Vol 2014, issue 2 (http://www.bcamt.ca/communication/vector/past-issues/).
Excerpts in this article were drawn from our project’s Reggio-Inspired Mathematics blog posts found here: http://janicenovkam.typepad.com/reggioinspired_mathematic/
Vector • Spring 2018 38
Book Review
The Solitude of Prime Numbers by Paolo Giordano Translated by Shaun WhitesideReviewed by Shiqi Xu
Alice Della Rocca had very much hated ski school, but wouldn’t have ever dreamed skiing would end up nearly killing her, crippling her permanently, outside and inside. At school, Alice longed to fit in with her peers. School eventually ended, but her anorexia would stay her closest companion.
Mattia Balossino had loved his twin, despite everything. Michela didn't ever emerge from her own bubble, adamantly oblivious of the world around her, and of the consequences this had for her twin. The first and last time that Mattia had let himself be propelled by his desire to be normal, unlike his sister, the price to pay was monstrous, leaving Michela alone in a park, to go to a classmate’s birthday party. They were only eight years old. As he grew older, Mattia did not try to fit in–his only escapes became mathematics and pain–cutting himself.
Alice and Mattia, both prime numbers, divisible only by one and by themselves, each crammed between two even numbers, each belonging nowhere.
The two met in high school: having different needs, and yet alike, each so deeply wrapped up in their own traumas. Alice and Mattia developed a unique relationship, close but never having much conversation or physical contact. They came to understand each other more than anyone else ever would, but something prevented them from completely giving themselves over.
Prime numbers get rarer and rarer as numbers grow larger. More rare are twin primes, a type of prime number, special in that they come in pairs, with one even number in between, setting them apart. As rare as they become, it is commonly believed that there
are an infinite number of pairs of twin primes, lying well beyond the extent of our imagination.
Mattia and Alice were twin primes. As close as was possible, and yet that even number lying between the two, an uncrossable chasm.
A few years later, Mattia got a job offer at a university far, far away. Alice met Fabio, who would fall in love with her. A miserable misunderstanding–or all that it might seem from the surface–flung Alice and Mattia thousands of miles apart.
Enough time passed by, that the two of them could have faded to distant memories in the other’s mind–even though they did not. Even though their lives led utterly different paths. When fate called the two to be brought together again, would Alice and Mattia succumb yet again to the wall between them, the unforgiving even number? How much of their persistent solitude could even love withstand?
Were they really twin primes? Or was solitude their true destiny?
This book is built on a beautiful metaphor–numbers and people. Alice and Mattia are each so lonely and so lost, that a reader would not expect to be able to relate in any significant way, and yet.
The author zeroed in on the negative experiences of the characters, and then each time hurtled forward through time, leaving out how the events unravel. The reader is forced to move forward with only the knowledge that the characters survived–finishing this book definitely left a feeling of unresolvedness.
Vector • Spring 2018 39
I was more hooked by Mattia’s story than Alice’s. He was more withdrawn than Alice, and, I felt, much more scarred. Michela was the one to have drowned, but Mattia suffocated much more, much longer. Could his own father even be to blame, he who had stopped noticing Mattia’s silent existence, with the inevitability in his father’s mind that one day Mattia would finally give in and join his twin?
I can understand Mattia always burying himself in the world of math, where only logic exists, with no emotion in the numbers and symbols laid out across inanimate pages. The author only ever skimmed the surface of the emotional centres of Mattia’s hyper-intelligent brain, I guess because that was Mattia’s choice–to avoid all emotions. I wonder what exactly Mattia found in pain, if it was a way of checking if he was still there and not faded away; to know that he still was able to feel anything; perhaps it was simply an escape, something to bring peace to his mind, by overflowing it–maybe it came from a place not so different compared to an alcoholic.
Mattia was the one to come up with the metaphor, comparing himself and Alice to twin primes.
The list of prime numbers from the first 100 natural numbers are ordered, showing the twin primes in brackets: 2, 3, 5, 7, (11, 13), (17, 19), 23, (29, 31), 37, (41, 43), 47, 53, (59, 61), 67, (71, 73), 79, 83, 89, 97. Of these, 25% are prime numbers; 12% twin primes. Because 3, 5, 7 are a prime triplet, they aren’t included here.
Now if we take the first thousand natural numbers, only 16.8% are primes, and 6.6% are twin primes. In the first million natural numbers, about 7.8% are primes, and only about 1.6% are twin primes.
Natural Numbers Primes
Prime Numbers Twin Primes
10 4 one prime triplet
100 25 12
1 000 168 66
10 000 1 229 406
100 000 9 592 2 444
1 000 000 78 498 16 334
10 000 000 664 579 117 956
100 000 000 5 761 455 880 620
1 000 000 000 50 847 534 6 849 008
As shown above, both prime numbers and twin primes gradually
Twin primes are fascinating, but also extremely complicated. From my research, I learned that it has been proven that there are an infinite number of prime numbers. However, the same has not been done for twin primes. Without too much background knowledge on prime numbers, it’s hard to imagine why. If prime numbers are infinite, why isn’t it obvious that twin primes also are?
According to Mertens’ second theorem, the sum of the infinite series containing the reciprocals of all prime numbers diverges. What? If the denominators get increasingly larger, how does the series diverge? However, Brun’s theorem states that the sum of the series with the reciprocals of twin primes converges. This makes sense, considering that as the denominators get larger and larger, the twin primes eventually become more and more rare.
Going back to the metaphor, I think that every single person in this world has parts of themselves that are prime numbers. For real people, I don’t agree with the metaphor of an entire person as a prime number, even ones like Alice and Mattia.
Looking at prime factorization, the factors of many numbers contain the same prime number multiple times. If everyone had each their own number, from a sequence of non-primes, they would all factor into different prime numbers, different numbers of times. If each prime number was something one has in common with another, some people would have no trouble connecting with many. The small, latter group would meet few others with whom they have common prime factors. Shouldn’t everyone deserve to find–at the very least–one other with common prime factors as themselves, whether they be a family member, a friend, a counterpart . . . ?
For me, The Solitude of Prime Numbers was a lot more than just a cold, depressing read; its chapters held many hidden compartments waiting to be uncovered. Research Bibliography
Kortekaas, Theo. "Prime Numbers." Web Pages of Theo Kortekaas. Web. 30 Nov. 2015. <http://tonjanee.home.xs4all.nl/prime.html>.
Weisstein, Eric W. "Twin Primes." MathWorld--A Wolfram Web Resource. Web. 30 Nov. 2015. <http://mathworld.wolfram.com/TwinPrimes.html>.
get rarer as numbers get higher. The numbers Mattia chose for himself and Alice were: 2,760,889,966,649 and 2,760,889,966,651. Lone amongst the lone–save for each other.
Vector • Spring 2018 40
Book Review
Becoming the Math Teacher You Wish You’d Had by Tracy Johnston ZagerReviewed by Jen Barker
If you were asked to take a moment and think of five words to describe your experiences with mathematics as a student, what might they be? This is an important question, not just for us as teachers, but also for our students. How might your students describe their experiences of your mathematics class? This question is one of many thought-provoking provocations Tracy Zager raises in her powerful book Becoming the Math Teacher You Wish You’d Had.
Zager, a mathematics educator who spent years teaching mathematics, studying mathematics educators, working with pre-service teachers, and working as a mathematics coach, set
out to create a resource that would support teachers in designing mathematics classes where students are engaged in the “doing" of mathematics. Her goal is “to close the gap between math as it is taught and math as it is” (p 7). Rather than putting faith into a a particular program aimed at improving mathematics teaching, Zager places emphasis on the power of teachers. She writes “…programs come and go. The people, though, are at the heart of education” (p 4).
A rich and dense book, Zager explains there are many ways to read her book, and no one right way. The book is organized into ten chapters, with each chapter focussing on a verb that describes key mathematical habits of mind.
Mathematicians: Take Risks Make Mistakes Are Precise Rise to a Challenge Ask Questions Connect Ideas Use Intuition Reason Prove Work Together and Alone
Are you noticing any connections to the BC Curriculum’s curricular and core competencies? BC educators will find Zager’s book profoundly helpful in making sense of the pedagogical shift in our mathematics curriculum, particularly in the shift from context as the main focus, to giving the big ideas and curricular competencies equal weight with content. Through Zager’s easy-
Vector • Spring 2018 41
to-read text, educators will learn practical ways to bring to life the habits of mathematicians. And, although the book draws from experiences of K-8 educators, it is easy to see how the strategies presented could be extended right through to the end of high school.
While each chapter focuses on a different mathematical habit of mind, there are structural similarities. Starting with stories of mathematicians and their habits of mind, Zager always comes back to what is important: getting to understand student thinking about mathematics. She provides a lens into classrooms where habits of mind are at the forefront, with transcripts of classroom discussions that illustrate why these teachers are doing what they’re doing. Throughout each chapter there are practical examples of fostering habits of mind, as well as opportunities to engage in mathematics. There are images of students’ mathematical thinking, talk moves to get at student thinking, formative assessment ideas, guiding questions and suggestions for providing descriptive feedback, and ideas for student self-assessment. Woven into every anecdote and example is research into the thinking of mathematics and mathematics education. Along the way, professional support is offered, through the #MTBoS, websites, blogs, videos and articles. We are not alone in this journey.
Companion website: http://sites.stenhouse.com/becomingmathteacher/
Study Guide: https://www.stenhouse.com/free-resources/study-guides-0
Author website: http://tjzager.com/
In addition to this amazing resource, Zager and the publisher of the book, Stenhouse, have created supporting websites with links to instructional strategies, videos, other books, research, a study guide, and forums where teachers can discuss ideas from the book and questions they might have. The book has become so much more than just a book! It is connecting educators around the world in many ways including through Twitter, where educators are talking about what they are doing in their classrooms using the hashtag #Becomingmath.
This book is an invitation to foster a love of mathematics, a love of teaching mathematics and a love of getting to the heart of students’ thinking about mathematics. In Zager’s words, “It is essential that you take an active role in this journey. That’s the only way to make real and lasting change. You’ll need to examine past experiences, think deeply about new ideas and be willing to explore mathematics with an open mind so you can reconnect to the little kid inside you who once loved playing around with patterns and numbers” (p 8). Consider yourself invited.
Vector • Spring 201842
Problem SetS
Being fluent with decomposing and composing numbers develops a flexibility in understanding and calculating with numbers. It is essential that young mathematicians develop a strong sense of the quantities five, ten and hundred. Decomposing and composing numbers is a critical component of number sense. Students need a strong understanding of parts-whole relationships in order to use mental mathematics strategies to add, subtract, multiply and divide. Experience with composing and decomposing numbers will help to develop computational fluency.
Kindergarten–Grade 1
Here are examples of some "open" questions that focus on decomposing and composing numbers (Note: adjust the quantity being used to meet the needs of the students):
1. You have ten counters that are in two piles. How many counters could be in each pile? What do the piles tell you about the number?
For example: The student puts the counters into a pile of six and a pile of four. “There are two more in this pile than the other. I see 4 + 4 + 2 = 10.”
2. A number is made up of two groups of counters. What could the numbers be? How do you know? Think of other ways to make two groups.
For example: The student chooses the number 8 and puts five in one group and three in the other. “I know that 5 and 3 more make 8.”
3. Some counters can be used to show two equal groups with 1 counter left over. How many counters might there be? How do you know? Show different ways.
For example: The student chooses the number 9 and puts four in one group and four in the other with one leftover. “I know that 4 + 4 = 8 and one more makes 9 .”
4. You have 20 counters. You put them into three piles that are not the same. How many counters could be in each pile? How do you know there are still 20 counters without counting by 1s?
Kindergarten–Grade 3Problems Collated by Sandra Ball
Vector • Spring 2018 43
Grade 2–3
Choose a number less than 100. Break it up as many ways as you can. Describe what each way helps you to see about the number. How might you represent the number different ways?
For example: The student chooses the number 54. “When I use five full ten frames and four ones, I can easily see the quantity. When I break the number into 34 and 20, I can count on by 10s to get to 54.”
A number can be modelled with three more ones than tens. What might the number be? Think of many possibilities.
Find a three-digit number you can represent using 15 base ten blocks. The blocks can be all the same or different. Show or describe another way to represent the number using different blocks.
For example: The student uses five of each base ten blocks and builds the quantity 555. The student could build 195 with other blocks.
Vector • Spring 201844
Problem #1
What is the fewest number of squares you can draw inside an 11-by-13 rectangle? Can you prove it is the fewest?
Problem #2
How do you see the figure growing? Where do you see the new squares in each case?
case 1 case 2 case 3
Problem #3: The Collatz Conjecture–an unsolved math problem
• Start with a whole number• If the number is even, divide it by 2 (or halve it)• If the number is odd, multiply it by 3 and add 1• Continue generating numbers until your sequence ends• Choose another number and create the sequence. What do
you think will happen? Problem #4: Number Visual Pennies
Imagine you have pennies stacked in the patterns of the number visuals for 3, 5, 6, 7 and 9.
• Each number visual has stacks of pennies. • The stacks of pennies inside each unique number visual are
equal. • Each number visual can have different stacks of pennies than
the other number visuals. What would be the size of each stack of pennies inside each number visual if the sum of the pennies is $1.00? Is there more than one solution? How many solutions are there?
Problem #5: Circle Fever
• How do you see the pattern growing? Where do you see the new circles being added when you move from case 1 to case 2? Where do you see the new circles being added when you move from case 2 to case 3?
• What would the 10th case look like? • How many circles would be in the 100th case? What would it
look like? • How many circles would be in the 0 case? What would it look
like? • What would the –1 case look like? • Model the pattern with a rule or expression. • Write a rule for the number of circles in the 100th case? References:
All problems are from the Mathematical Mindsets Workshop with Cathy Williams, courtesy of youcubed.org.
The visual in problem #5 is also found on http://www.visualpatterns.org/, pattern #5.
Mathematical Mindsets Workshop Problems
Vector • Spring 2018
Math Links
45
http://teacher.desmos.com
Desmos is more than a calculator! They also have a collection of activities their team (and others) have developed. What makes their activities unique is the ability to see what students are doing and the deliberate designs for interactions between students built into their online activities.
http://fosteringmathpractices.com/
Grace Kelemanik and Amy Lucenta are the authors of this website which contains resources to support the six instructional routines they have developed. These instructional routines are fantastic for helping develop students’ mathematical thinking.
http://www.mathtalks.net/
Are you interested in doing math talks but not sure where to start? This site has many examples of math talks and pattern talks. It also has a teacher section with suggestions on how to implement math talks in your class.
http://mathriddles.williams.edu/
This site contains a collection of mathematical riddles. They are organized by difficulty and by topic.
https://criticalmath.ca/
This website contains a collection of task starters based on the mathematics of a variety of different social issues. For example, in one post the authors describe how the limited supply of McDonald’s Szechuan dipping sauce caused the prices of the sauce to spike on Ebay at $282.
Spring 2018 Math Websites
Links selected and described by David Wees (http://davidwees.com). Previous websites can be viewed here: http://wees.it/mathwebsites
Vector • Spring 2018 46
British Columbia Association of Mathematics
BCAMT Service Award
Selina Millar
Selina Millar recently retired from the Surrey School District where she worked as a classroom teacher and then as a District Numeracy Helping Teacher for a total of 36 years. She loves sharing her passion for learning, particularly in math and continues to be amazed with how students are so capable of making sense of their math learning. Selina looks forward to the next phase of life including new learning that include travelling and exploration abroad.
TEACHERS AWARD WINNERS 2018
Vector • Spring 2018 47
Outstanding Elementary Teacher Award
Kristy Doornbos
Kristy Doornbos has been teaching for almost eight years in the Peace River South School District in northeastern B.C. (Dawson Creek area). She is very dedicated to her students and making sure that each student finds success. She is a life-long learner and is always looking for ways to learn more as a professional and become a better teacher in areas where she feels she may be struggling. Over the last three to four years, Kristy has become involved in both literacy and numeracy professional learning communities, helping to develop best practices in those areas, and helping students become more literate and numerate learners. Kristy is the president of the local Primary Teachers’ Association and has spearheaded a numeracy professional learning community of twenty teachers for the past three years. This group has focused on learning about new curriculum, developing a scope and sequence for teaching primary numeracy, looking at how to assess and knowing when students have achieved numeracy skills, as well as learning on different numeracy topics such as math stations, the link between numeracy and literacy and differentiating mathematics efficiently.
Outstanding Secondary Teacher Award and Ivan L. Johnson Memorial Award
Jennifer Kirkey
Jennifer Kirkey has been a teacher in the Central Okanagan since 2000. She has taught students from grades 7-12. Currently she is teaching at Mount Boucherie Secondary, where she teaches a variety of math courses, from Workplace Math 10 to Calculus 12. She strives to create an air of excitement in math, with her students engaging in inquiry activities, interesting problems, and examples of mathematics in the world around them. She enjoys collaborating with other teachers to share ideas, develop innovative lessons and have fun working with math. She credits all of the amazing teachers she has worked with for making her a better teacher.
Vector • Spring 2018 48
Awards & CriteriaOutstanding 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 Awardhas provided extraordinary service to mathematics education as an active member of the BCAMT for a significant period of time
Ivan L. Johnson Memorial AwardThe 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.
Vector • Spring 2018 49
Selection ProcessAll 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 NominateRequired 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 DriveNorth Vancouver, BC V7H 1N1
Deadline: May 31, 2018
Vector • Spring 2018 50
The BCAMT will be funding several mathematical initiatives throughout BC. These initiatives must meet the goals and objectives of the BCAMT. Please note: this grant is not meant for individual professional development.
BCAMT Goals for 2017-2018:
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 by maintaining current BCAMT membership and recruiting new members from all levels of mathematics education.
The BCAMT values the sharing of ideas, and requests that successful applicants submit a summary of their initiatives with its highlights for publication in a future Vector or newsletter.
All applications must be postmarked no later than November 16, 2017. 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.
Complete this form and be sure to include:
• a rationale for funding request (maximum of one page)• details of your initiative (maximum of one page)• a detailed budget (list expenses, other funding, etc.)
Grant Application Form
Vector • Spring 2018 51
BCAMT MEMBER: [ ] Yes [ ] No Applicant’s Name
School/District
Address
City/Town Postal Code
Work Phone Home Phone
Email Address
Initiative
Dates(s) Location(s)
Please check the target audience:[ ] BCAMT Members[ ] Teachers[ ] Students[ ] Community[ ] Other ___________________________
Please check the type of initiative:[ ] Workshop[ ] Research[ ] Contest[ ] New LSA[ ] Exhibition[ ] Other ___________________________
Applicant Signature
Send the completed application to: Brad Epp Chair, Funding Application Committee BC Association of Mathematics Teachers #51 – 383 Columbia Street West, Kamloops BC V2C 1K5 or email: [email protected]
Deadline: November 16, 2018
We invite you to attend the 2018 meeting of the Canadian Mathematics Education Study Group (CMESG), taking place in Squamish, at Quest University Canada, from June 1st to June 5th.
We are grateful to the BCAMT and the Pacific Institute of Mathematical Sciences (PIMS) for their support.
We hope to see you in Squamish! Conference website: www.cmesg.org