# Problems to Ponder (March edition)

Welcome to the March 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.

Michael Pruner

What is the Shape?
A shape is made with linking cubes. When you look at it from one side, it looks like this:

What might the structure look like?

Source: Small, M. (2012). Good questions: Great ways to differentiate mathematics (2nd ed.). New York, NY: Teachers College Press.

Five Cubes
Using exactly 5 interlocking cubes, make as many shapes as you can so that all five cubes are touching the table. How many different shapes can you make?

Source: Spring 2011 problem set. (2011). Vector, 52(1), 47-49.

Making Ten
Each group of students has a set of playing cards from ace to three (aces have a value of 1). Use these 12 cards to make the numbers from 1-10, using any operations you like.

Sharing Bacon
You are a chef at a summer camp and you are frying 30 identical strips of bacon for this morning’s breakfast. A counselor comes in to inform you that there are only 18 campers coming in for breakfast and they all love bacon. What is the minimum number of cuts necessary? What is the minimum number of pieces?

Extensions: How do you know it is the minimum? What about sharing amongst 17 campers? 16 campers? n campers?

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

Magic Squares
A magic square is a square grid with n rows and n columns, filled with distinct numbers from 1 to n2, such that the sum of the numbers in each row, column, and both long diagonals is the same.

1. Can you come up with a 2×2 magic square?
2. What about a 3×3 magic square?
3. What value does each row, column, and long diagonal need to sum to in a n×n magic square?

Extensions: Investigate magic rectangles and magic triangles.

Source: Urschel, J. (2016, December 14). The Wednesday morning math challenge: Week 14. The Players’ Tribune. Retrieved from http://www.theplayerstribune.com/the-wednesday-morning-math-challenge-week-14/

Box of Marbles
In a box, you have 13 white marbles and 15 black marbles. You also have 28 black marbles outside of the box.

Remove two marbles, randomly, from the box. If they are of different colours, put the white one back in the box. If they are of the same colour, take them out and put a black marble back in the box. Continue this until only one marble remains in the box. What colour is the last marble?

Source: Winter 2014 problem set. (2014). Vector, 55(1), 47-49.

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.