*Welcome to the July 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) **

**Which One Doesn’t Belong?
**Display the image and ask the class: “Which one doesn’t belong?”

This task encourages students to use descriptive language in their reasoning and to consider multiple possible answers. See wodb.ca for more details and more images.

*Source: *Bourassa, M. (n.d.). Shape 2 [Digital image]. Retrieved from http://www.wodb.ca/shapes.html

**
Play with 60
**You have 60 items in a bowl. How could you arrange them to make it easier for a friend to count? Can you organize them in different ways?

You want to put the 60 items into bowls so that there is the same number in each bowl; how many different bowls will you need so that there are no items left over?

*Source: *Problem sets. (2017). *Vector, 58*(1), 47. Retrieved from http://www.bcamt.ca/wp-content/uploads/2017/05/581-Spring-2017.pdf

The Tortoise and the Hare** ^{
}**The Tortoise has challenged the Hare to a hopping competition. The challenge is for the Hare to complete 3 equal hops and not land on a red square. Can the Hare succeed in this challenge?

Can the Hare succeed in this challenge?

Does the Hare ever fail? Try each of these:

*A**dapted from *Tortoise and hare—revenge race. (n.d.). Retrieved from http://mathpickle.com/project/tortoise-and-hare-the-revenge-race-skip-counting-pattern/

**Intermediate and Secondary Tasks (Intermediate-Grade 12)**

**The Shoe Sale
**You decide to take advantage of a buy

*2 pair get 1 pair of equal or lesser value for free*sale at the local shoe store. The problem is that you only want to get two pairs of shoes. So, you bring your best friend with you to the store. After much deliberation, you settle on two pairs of shoes – a sporty red pair for $20 and a dressy black pair for $55. Your friend finds a practical cross trainer for $35. When you proceed to the check out desk the cashier tells you that your bill is $90 plus tax (the $20 pair are for free). How much should each of you pay? Justify your decision.

*Source: *Liljedahl, P. (n.d.). Numeracy tasks. Retrieved from http://www.peterliljedahl.com/teachers/numeracy-tasks

**Elections
**There are two parties in an election: Red and Blue. There are only five people voting, and they are numbered 1, 2, 3, 4 and 5. What’s interesting about this election is that the person’s number counts as the number of votes that they cast. When the votes were counted:

- more people had voted for Red;
- Red had the more votes than Blue;
- if any one person had changed their vote, then Blue would have won.

What are all the possible ballot counts for this situation?

*Extensions:* What about 6 people, 7 people, and so on? What if there were three parties?

*Source: *Virtuous democracy. (n.d.). Retrieved from http://www.playwithyourmath.com/

**Truth Tellers
**Truth-tellers always tell the truth and are marked with an ‘X’ on the grid below. Liars always lie and are marked with an ‘O.’ When asked the question, “Are you next to exactly two like yourself?” Everyone responded, “YES!” Where are the truth-tellers and the liars?

*Adapted from *Pinocchio’s playmates. (n.d.). Retrieved from http://mathpickle.com/project/pinoccchios-playmates/

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.