# Problems to Ponder (December 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 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 [which currently connects nearly one thousand educators from across the province, country, and even the world! –Ed.] so that others may learn from your experiences.

Michael Pruner

November 6, 2016

Dragon Fractal
Imagine a long strip of paper folded in the same direction once, twice, and then a third time. When the strip is unfolded, how many creases will be on the paper? In what directions will the creases be pointing? What about n folds?

Extensions: When the paper is unfolded, and the creases are made to equal 90°, what do you notice in the shapes?

November 13, 2016

Cottages
A circular road is 27 km long. On this road are six cottages, owned by 6 friends. The friends visit each other a lot, and they have noticed that every whole number from 1 to 26 (inclusive) is accounted for at least once when they calculate the distances from one cottage to another. Of course, the friends can walk in either direction as required. Your task is to place these cottages at distances that will fulfill these conditions.

Extensions: Can you find more than one solution?

Source: Dudeney, H. E. (1967). 536 puzzles and curious problems. New York, NY: Charles Scribner’s Sons.

November 20, 2016

What is it that makes them surprising?

1. A bat and a ball cost \$1.10 in total. The bat costs \$1.00 more than the ball. How much does the ball cost?
1. If it takes 5 machines 5 minutes to make 5 widgets, how long would it take 100 machines to make 100 widgets?
1. In a lake, there is a patch of lily pads. Every day, the patch doubles in size. If it takes 48 days for the patch to cover the entire lake, how long would it take for the patch to cover half of the lake?

Source: Frederick, S. (2005). Cognitive reflection and decision making. The Journal of Economic Perspectives, 19(4), 25-42.

November 27, 2016

Approaching Midnight

It is 6:00 pm. With a partner, take turns adding one of 15 minutes, 30 minutes, 45 minutes, or 60 minutes to the clock. The first player to reach 12:00 am wins.

Extensions: Is there a winning strategy? What if the first to 12:00 am is the loser?

November 6, 2016

100 Hungry Ants by Elinor J. Pinczes

• Read the story to the students.
• Ask the students to choose one of the following numbers: 12, 24, or 36.
• Ask the students to imagine that this number of ants is going to the picnic.
• Ask how many different ways could the ants arrange themselves into equal rows.
• Have the students draw an array and write an equation for each solution.

November 13, 2016

Up-and-Down Staircase
One block is needed to make an up-and-down staircase, with one step up and one step down:

4 blocks make an up-and-down staircase with 2 steps up and 2 steps down:

How many blocks would be needed to build an up-and-down staircase with 5 steps up and 5 steps down?

Explain how you would work out the number of blocks needed to build a staircase with any number of steps.

November 20, 2016

Creature Legs
At the park, Mike counted (6, 10, 14—choose a quantity appropriate for your students) creatures’ legs.

What creatures could there have been at the park? Which combinations of creatures show the number of legs counted? Show more than one combination.

November 27, 2016

Watermelon Seeds
One hot day, my dad cut a slice of watermelon for me to eat.  I counted 13 (change number to meet the needs of the students—e.g., 23 or 33) black and white seeds in the slice. There were more black than white seeds. How many of each kind of seed might there have been?

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.