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Mathematical Problem Solving for Teens: Past CEMC Contests

Do you want your older children to learn problem solving skills?  While most homeschooling moms do, quality problem solving resources can be difficult to find beyond the elementary level.  After all, very few math curricula spend a huge amount of time on problem solving, and not many helpful books are available.

However, there is an incredible resource, tucked away in the website of the University of Waterloo’s Center for Education in Mathematics and Computing (CEMC).  For the past 50 years the CEMC has worked to promote students’ enjoyment of mathematics and to develop their problem solving skills.  It is Canada’s largest math and computing outreach organization, reaching well over 100,000 students each year with its annual math competitions for grades 7-12.  Since the late 90’s, it has posted both the tests and detailed solutions online.

These CEMC tests and solutions are a treasure trove of math problems at all levels of difficulty.  Using only concepts common to all Canadian math curricula, most of the contest problems test logical thinking and mathematical problem solving rather than content. These tests are well-organized, error-free and have complete and thorough solutions.  We have been using them for 8 years to give our children valuable math practice.  Although our family uses them to prepare for the actual contests, they can also be used as a problem solving unit in math or for math-fun days.

One series of tests, the Gauss (grades 7 and 8), Pascal  (grade 9), Cayley (grade 10), and Fermat (grade 11) tests, each feature 25 multiple choice questions worth 150 points, and are meant to be written in one hour.  Each test is divided into an easy, a moderate, and a difficult section.  All students should be able to do the easy questions but very few students can do all the difficult ones.  In the actual competitions, almost no one achieves a perfect score.

The other series of tests our children have used are the Fryer (grade 9), Galois (grade 10), and Hypatia (grade 11) tests which include both long and short answer segments.  Each 75-minute test is composed of four questions, worth 10 marks each, that have both simple and challenging aspects.

How do we use these great resources?  Very simple.  I just give our children an old test and ask them to see what they can do in the allotted time.  Then we mark it and go over the solutions until we understand them.  The solutions are written up so clearly that my children often do not need my help.  The next day we do another test. Obviously, we do this at least once a year, twice if we’re doing the long answer tests as well as the multiple choice tests.

Usually our children encounter their lowest marks ever when they first try these tests, and this can be discouraging.  It helps them to see that, world-wide, average scores hover around 60%.  After a few weeks of intensive problem solving work, however, my children have typically learned how to think more logically and have greatly improved their scores as well as their confidence and math ability.

These free tests with their clear solutions and explanations are the best problem solving resource I’ve ever come across for teens.  If you want your teens to learn problem solving, do check out the past CEMC Contests.  If your teen is mathematically inclined, it may even be worth your while to organize a homeschool group to take these tests, especially if you live in Canada.

Note:  The CEMC also hosts other math and computing competitions that we have not been involved in, and these contests and solutions are also posted online.

Disclosure: As usual, I am in no way compensated for telling you about this resource.

This post is linked to Trivium Tuesdays.


  1. This sounds like a useful resource. We have something similar in the UK but except for the Primary maths challenge (under 11s) it isn’t easy for home educated children to enter. Their questions are at http://www.mathcomp.leeds.ac.uk/individual-competitions/

  2. Yep, it’s Gauss time again (this Wednesday!). Our homeschool group always gets a group together to write the CEMC competitions. For some reason we’re down to only two Gauss participants this year, but some years we’ve had as many as fourteen grade 7 and 8 students writing.

    The CEMC office is very good about working with homeschoolers. As long as you can find a proctor for the exam (not the parent of a participant) and agree to follow the writing rules, you can register one or more students. Our group contacts them in the fall and they email us a set of homeschooler registration forms.

  3. Annie Kate says:

    Yes, Sarah Elizabeth, your UK contests seem similar. If you want to enter a math competition and the UK one won’t let you, try the CEMC one. They cheerfully allow international entries.

    As Mama Squirrel points out, the CEMC competitions are homeschool friendly, but you must have people besides the parents requesting the exam.

    Miss 12 is writing on Wednesday too, and she seems very well prepared. I was getting more and more excited for her as her practice marks went up and up, but then she had a more normal mark and reality set in again. 🙂

    Parents should not get emotionally involved in their children’s schoolwork; it’s bad all around.

  4. […] as "the ideal bridge from the solving of elementary puzzles to algebra" Kulkarni also . Mathematical Problem Solving for Teens: Past CEMC Contests | Tea … While most homeschooling moms do, quality problem solving resources can be difficult to find beyond […]

  5. Kathleen says:

    Hi Annie Kate, my oldest is writing his first Gauss this May and I am wondering if you have any guidance as to how best to prepare? We did our first test today from the achieves on the CEMC site and it was an underwhelming attempt, but I had braced for this.

    A test a week? Did you go through the test with them or let them mark and correct their own work? Did you ever purchase the hard copy manuals Waterloo has advertised?

    Any advice you are willing to give is always welcome.


    1. Annie Kate says:

      Hi Kathleen,

      We usually allowed a few weeks for practicing these tests, but doing it once a week for now and later spending some time doing it everyday would work just as well. I do recommend dropping other math for the week before the contest and focusing on this to help your son get his mind into that logical thinking groove. There’s something very encouraging for kids when they can feel their minds expanding and are able to do so much better than at first.

      I always marked the tests, but let them go over the correct answers on their own; then I would help them with the hard ones.

      And I had to dish out huge amounts of encouragements, showing them the statistics world wide and encouraging them that getting average on this test was great and anything more was bonus. That helped relieve the stress somewhat, because it is hard for a bright kid to get a low mark, and they have to realize that most bright kids get low marks on this test, especially on the first few tries–and realistically, only bright kids write the test. Sometimes it helped to look up the statistics for that particular test (and that gives a bonus review of basic statistics as well).

      One danger with these tests is that a student might get discouraged about math and give up if he/she is not able to accept a low grade. So the encouragement and comparison to statistics are very, very important.

      We never purchased the manual; the online tests are great as it is.

      I hope that helps somewhat, Kathleen.

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