Problems to Ponder (January edition)

Welcome to the November 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 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.

Michael Pruner

Intermediate and Secondary Tasks (Grades 5-12)

Milk Crate

A certain milk crate can hold 36 bottles of milk. Can you arrange 14 bottles in the crate so that each row and column has an even number of bottles?

Extensions: What is the smallest array that can fit 14 bottles under this rule? What about 15 bottles?

Source: Mason, J., Burton, L, & Stacey, K. (1985). Thinking mathematically. Essex, England: Prentice Hall.

Mountain Bike Race
In a mountain bike race, there are 25 racers. The track can only fit five racers at any time. Devise a strategy to determine gold, silver, and bronze. How many races are necessary?

Tethered Goat

A goat is tethered by a 6-metre rope to the outside corner of a shed measuring 4 m by 5 m in a grassy field. What area of grass can the goat graze?

Extensions: What if the rope was fastened to the middle of one wall? What if the rope was 20 m long? What if the shed was circular?

Source: Mason, J., Burton, L, & Stacey, K. (1985). Thinking mathematically. Essex, England: Prentice Hall.

Primary Tasks (Grades K-4)

Coloured Dice
Roll 3 different coloured dice. What are all the possible ways to get a total of 5 points?

These are the “double-3 down” dominoes:

Use these dominoes to make the following square, such that each side has eight dots:

Source: Domino square. (n.d.). Retrieved from


Michael Pruner is the current president of the British Columbia Association of Mathematics Teachers (BCAMT) 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.

Leave a Reply

Your email address will not be published. Required fields are marked *