Welcome to the May edition of Problems to Ponder! This month’s problems have been curated by Michael Pruner, president of the British Columbia Association of Mathematics Teachers (BCAMT). The tasks are released on a weekly basis through the BCAMT listserv, and are also shared via Twitter (@BCAMT) and on the BCAMT website. This post features only a subset of the problems shared by Michael last month – head to the BCAMT website for the full set!
Have an interesting solution? Send it to thevariable@smts.ca for publication in a future issue of The Variable, our monthly periodical.
I am calling these problems ‘competency tasks’ because they seem to fit quite nicely with the curricular competencies in the British Columbia revised curriculum. They are non-content based, so that all students should be able to get started and investigate by drawing pictures, making guesses, or asking questions. When possible, extensions are provided so that you can keep your students in flow during the activity. Although they may not fit under a specific topic for your course, the richness of the mathematics comes out when students explain their thinking or show creativity in their solution strategies.
I think it would be fun and more valuable for everyone if we shared our experiences with the tasks. Take pictures of students’ work and share how the tasks worked with your class through the BCAMT listserv so that others may learn from your experiences.
I hope you and your class have fun with these tasks.
Primary Tasks (Kindergarten-Intermediate)
Sharing Cookies
Charlie, Susan, and Amber get to share six cookies. However, Susan’s mother has told her that she is only allowed to have one cookie. How do you share the cookies?
Source: Winter 2010 problem set. (2010). Vector, 51(1), 40-43. Retrieved from http://www.bcamt.ca/wp-content/uploads/vector/511-Winter-2010.pdf
Number Path
You are on a number path made up of squares of numbers starting at 1 and continuing as far as you wish…
| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
You move some steps forward. Then, you move some steps back. You repeat both moves. You land at 9. How many steps did you take each way?
Source: Small, M. (2017). Mathematical problem solving in all strands [PowerPoint slides]. Retrieved from http://www.onetwoinfinity.ca/wp-content/uploads/2017/03/WinnipegMarch.pdf
How Many?
How many might be in each yellow box? How many in each red box?
Source: Small, M. (2017). Mathematical problem solving in all strands [PowerPoint slides]. Retrieved from http://www.onetwoinfinity.ca/wp-content/uploads/2017/03/WinnipegMarch.pdf
Intermediate and Secondary Tasks (Intermediate-Grade 12)
25 Coins
Twenty-five coins are arranged in a 5 by 5 array. A fly lands on one, and tries to hop on to every coin exactly once, at each stage moving only to the adjacent coin in the same row or column. Is this possible?
Extensions: Can you explain why some starting locations are not possible? What about 3D? What about rectangles?
Source: Mason, J., Burton, L, & Stacey, K. (1985). Thinking mathematically. Essex, England: Prentice Hall.
Split 25
Take the number 25, and break it up into as many pieces as you want. For example,
25 = 10 + 10 + 5
25 = 2 + 23
25 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 9 + 9
What is the biggest product you can make if you multiply those pieces together? Will your strategy work for any number?
Source: Kelly, J., & Yu, X. (n.d.). Play with your math. Retrieved from http://www.playwithyourmath.com/
Consecutive Sums
Some numbers can be expressed as the sum of two or more consecutive positive integers. For example,
9 = 2 + 3 + 4
11 = 5 + 6
18 = 3 + 4 + 5 + 6
Which numbers have this property?
Extensions: In how many ways can a number n be written as the sum of two or more positive integers?
Source: Mason, J., Burton, L, & Stacey, K. (1985). Thinking mathematically. Essex, England: Prentice Hall.
Michael Pruner is the current president of the British Columbia Association of Mathematics Teachers and a full-time mathematics teacher at Windsor Secondary School in North Vancouver. He teaches using the Thinking Classroom model where students work collaboratively on tasks to develop both their mathematical competencies and their understanding of the course content.