The CENTRE for EDUCATION in MATHEMATICS and COMPUTING Le CENTRE d’ÉDUCATION en MATHÉMATIQUES et en INFORMATIQUE cemc.uwaterloo.ca 2017 2017 Results Résultats Fryer Contest Concours Fryer (Grade 9) (9e année – Sec. III) Galois Contest Concours Galois (Grade 10) (10e année – Sec. IV) Hypatia Contest Concours Hypatie (Grade 11) (11e année – Sec. V) ©2017 University of Waterloo Competition Organization Organisation du Concours Centre for Education in Mathematics and Computing Faculty and Staff / Personnel du Centre d’éducation en mathématiques et informatique Ed Anderson Sandy Graham Jeff Anderson Conrad Hewitt Terry Bae Angie Hildebrand Jacquelene Bailey Carrie Knoll Shane Bauman Judith Koeller Carmen Bruni Bev Marshman Ersal Cahit Mike Miniou Serge D’Alessio Brian Moffat Janine Dietrich Dean Murray Jennifer Doucet Jen Nelson Fiona Dunbar J.P. Pretti Mike Eden Kim Schnarr Barry Ferguson Carolyn Sedore Judy Fox Kevin Shonk Steve Furino Ian VanderBurgh Rob Gleeson Troy Vasiga John Galbraith Christine Vender Alain Gamache Heather Vo Robert Garbary Ashley Webster Problems Committee / Comité des problèmes Shawn Godin (Chair / président), Cairine Wilson S.S., Orleans, ON Shane Bauman, University of Waterloo, Waterloo, ON Lino Demasi, Ottawa, ON Rad dePeiza, Toronto, ON Sandy Emms Jones, Forest Heights C.I., Kitchener, ON Rob Gleeson, University of Waterloo, Waterloo, ON Paul Ottaway, Capilano University, North Vancouver, BC Judy Shanks, Pickering H.S., Ajax, ON Tyler Somer, Kitchener, ON Patricia van Donkelaar, Charles P. Allen H.S., Bedford, NS 2 Comments on the Papers Commentaires sur les épreuves Overall Comments The year 2017 marked the fifteenth writings of the Fryer, Galois and Hypatia Contests. The FGH Contests are designed to give students in Grades 9, 10 and 11 an additional mathematical challenge during the school year, and one that requires full written solutions. Being able to do mathematics is an important skill; being able to communicate one’s findings (in mathematics or in any other field) is also very important. We were very pleased this year to see another healthy increase in the number of participants in these Contests. In general, many students writing these Contests are doing a better job each year of writing clear solutions to the problems. That being said, there is always work that can be done to help them improve these skills. Some students reduced their scores because they gave answers rather than solutions or presented untidy or illegible solutions. The averages on the Fryer, Galois and Hypatia Contests this year were 26.1, 23.8 and 22.0, respectively. We will continue to try to keep the averages on these papers in this range. Here are some specific comments on the problems. Fryer Contest 1. Average: 7.9 Parts (a) and (b) were very well done. In part (c), students struggled to earn full marks because their justification was either incomplete or invalid. Statements such as “31 is not divisible by 6 or 9 and therefore you cannot purchase exactly 31 pens" were not correct. This is because there exist integers, such as 15, that are not divisible by 6 or 9 but can be obtained by adding multiples of 6 and 9 together. For example, you could purchase 1 package of red pens and 1 package of blue pens to have 15 pens in total. Students who earned full marks were able to (i) identify that the greatest common divisor of 6 and 9 is 3 and justify why it is important that 31 is not a multiple of 3, or (ii) identify all possible cases, including the zero cases, for the number of red or blue pen packages. 2. Average: 7.3 Students did well on part (a), but seemed to struggle with setting up and solving the more complicated inequalities in parts (b) and (c). One common mistake in parts (b) and (c) was including a value for m and w outside the range of possible values (ex. m = 2; 3; 4 or w = 15; 16; 17; 18; 19; 20; 21). In part (b), a good number of students could set up the inequalities, but were unable to simplify them to a point where the values of m were apparent. In part (c), students could easily identify that w > 15, but found it more 43 challenging to identify w < 2 . In part (c), students were able to earn full marks for identifying w > 15 and using trial and error to identify possible values of w after Fiona’s weekend games. 3. Average: 7.0 Students who understood the chord property shown in the example did very well on this question. A common error among students was misunderstanding the chord property to involve addition versus multiplication. Very few marks were awarded when this error was made. Students who did well on parts (a) and (b), but poorly on part (c), were able to apply the chord property when there was one point of intersection. The most common error with two points of intersection was multiplying all the line segment lengths on a chord. For example, (3)(n)(6) = (5)(m) or (3)(n)(6) = (8)(n). 4. Average: 4.0 Parts (a), (b), and (c) were done well. There were students who tested smaller powers of two (e.g. 22; 23; 24; 25;:::) 22017 − 2 and then identified a pattern among the remainders in order to conclude the answer . However, full 3 marks were only given to students who could justify the pattern and argue why no more than 2 candies were discarded throughout the entire procedure. 3 Comments on the Papers Commentaires sur les épreuves Galois Contest 1. Average: 8.6 This question was quite well done. 2. Average: 6.3 The first three parts were well done. The majority of mistakes where when students gave the answer in dollars instead of time and vice versa. Many students struggled with part (d). A common mistake was to figure out one of the endpoints of the two correct intervals of the answer and then give one interval as an answer: 5:39 a.m. to 6:59 a.m. Students also struggled to give clear justification for their answers. 3. Average: 7.6 Parts (a) and (b) were well done. Part (c) resulted in many different correct solutions. One common mistake was to misinterpret the areas given as the areas of the circles instead of the areas of the squares. Another common mistake was to assume that the length of OC was the sum of the radii of the two circles. 4. Average: 1.4 Part (a) was reasonably well done. A common mistake was the find the ratio of the shaded area to the total area instead of the ratio of the shaded area to the unshaded area. It was rare to see a correct so- lution to (b) and very rare to see a correct solution to (c). A number of students were able to get to the m − 2 u equation = in part (b) and occasionally with some trial and error managed to find some values of u. m + 2 77 Hypatia Contest 1. Average: 9.1 This question was very well done. In part (a), a common incorrect answer was 92◦ and in part (c) a common incorrect answer was 55◦ by misidentifying the equal angles. In (d), a common error was to not simplify 180◦ − (180◦ − z◦) as z◦. 2. Average: 7.4 In part (a), although some students left out either the integer or the row from the answer box, full marks were given if supporting work included both answers. In parts (c) and (d), some students assumed that only odd integers were in the table. In these parts, students with the correct answer lost marks if they did not explain their work. In part (d), a common incorrect answer was 151, when students neglected to include row 49. 3. Average: 4.3 In part (a), part marks were awarded for listing x-coordinates without y-coordinates, or for listing only one of the two points. In part (c), a nice alternate solution involved a generic equation for a parabola and line, using the fact that x = p and x = q solved both equations. A common error was assuming that the equation for the parabola was y = (x − p)(x − q). In part (d), many students could generate a complicated equation in one variable (either x or y) but were unable to solve. A nice alternate solution involved using several values of k to generate points on the parabola, and then determining the equation of the parabola given its vertex is at the origin. 4. Average: 1.2 This problem was much more challenging, and many students got part (a) only. In part (b), many misin- terpreted an expression like G(6; 3) to represent an actual zigzag number instead of the number of zigzag numbers with certain properties. Many students observed properties such as G(6; 1) = L(6; 6) but without proper justification. Please visit our website at cemc.uwaterloo.ca to download the 2017 Fryer, Galois and Hypatia Contests, plus full solutions. 4 Comments on the Papers Commentaires sur les épreuves Commentaires généraux L’année 2016 est la quinzième édition des concours Fryer, Galois et Hypatie. Les concours de FGH ont été créés pour donner aux étudiants de 9e, 10e et 11e année (sec. III, IV et V) un défi mathématique supplémentaire pendant l’année scolaire qui exige des solutions écrites complètes. Être capable de faire des mathématiques est une aptitude importante. Pouvoir communiquer ses conclusions (en mathématiques ou autres domaines) est aussi très important. Nous sommes ravis cette année de constater une croissance du nombre de participants à ces concours.