Welcome to this month’s edition of Problems to Ponder! Have an interesting solution? Send it to firstname.lastname@example.org for publication in a future issue of The Variable!
Primary Tasks (Kindergarten-Intermediate)
Which One Doesn’t Belong? 
Display the image below 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.
Prime Time 
In this game, students color patterns in a hundred chart, showing multiples of a number they have rolled.
- Two dice
- One hundreds chart
- Two different colored highlighters
- Player 1 and Player 2 each pick a different colored highlighter.
- Player 1 rolls the dice and adds the two numbers. Player 1 then colors in every multiple of that number on the hundred chart. If a player rolls and gets the sum of 2, they color in all of the prime numbers.
- Player 2 rolls the die and colors every multiple of that number on the hundred chart.
- If a number is already colored, the player skips that number and continues coloring any available multiple of their number to 100. If a player rolls and there are no multiples available for their number they lose their turn.
When the hundred chart is completely colored, each player counts the number of squares they have highlighted. The player with the greatest number of colored squares wins the game.
b) Find all of the ways of representing 1/4 by shading in parts of the square above.
c) If the area of the large square is 12 cm2, what is the area of the shaded section in each image below?
d) If the area of the shaded section is 4 cm2 for each image above, what is the area of the large square in each case?
e) Which fractions between 0/16, 1/16, 2/16, …, 16/16 can you represent by shading in one or more of the sections in the first (unshaded) image above?
Intermediate and Secondary Tasks (Intermediate-Grade 12)
Egyptian Fractions 
Ancient Egyptians used unit fractions, such as 1/2 and 1/3, to represent all fractions. For example, they might write the number 2/3 as 1/2 + 1/6. While we often think of 2/3 as 1/3 + 1/3, the ancient Egyptians would not write it this way because they didn’t use the same unit fraction twice.
How many ways can you express 1/6 as the sum of two unique unit fractions? How about 1/8? Can all unit fractions be made in more than one way? Choose different unit fractions of your own to test out your theories.
Fruit Salad 
A fruit salad consists of blueberries, raspberries, grapes, and cherries. The fruit salad has a total of 280 pieces of fruit. There are twice as many raspberries as blueberries, three times as many grapes as cherries, and four times as many cherries as raspberries. How many cherries are there in the fruit salad?
The Prince and the Trolls 
A prince picked a basketful of golden apples in the Enchanted Orchard. On his way home, the prince came to a troll who guarded the orchard. The troll stopped him and demanded payment of one-half of the apples plus 2 more, so the prince gave him the apples and set off again. A little further on, he encountered a second troll. The second troll demanded payment of one-half of the apples the prince now had plus 2 more. The prince paid him, and set off once more. Just before leaving the enchanted orchard, a third troll stopped him and demanded one-half of his remaining apples plus 2 more. The prince paid him and sadly went home. He had only 2 golden apples left. How many apples had he picked?
Extensions: What if the prince had 1 apple left? 3 apples? Design an algorithm to determine the number of apples the prince picked for any number, n, of apples remaining.
 Adapted from Egyptian fractions. (n.d.). Retrieved from the Illustrative Mathematics website at www.illustrativemathematics.org/content-standards/5/NF/A/1/tasks/839 and Keep it simple. (n.d.). Retrieved from the NRICH website at nrich.maths.org/6540
 Fruit salad. (n.d.). Retrieved from the Illustrative Mathematics website at www.illustrativemathematics.org/content-standards/tasks/1032
 Adapted from Kelemanik, G., Lucenta, A., & Janssen Creighton, S. (2016). Routines for reasoning: Fostering the mathematical practices in all students. Portsmouth, NH: Heinemann.