In fractured fractions, your students cover a pixelated image with rectangles. You must choose the rectangles carefully so that the fraction covered matches the given fractions. Here we have a yellow submarine with given fractions 1/2 and 1/3.


Does this rectangle help?

Is it half yellow?


Yes! It is half yellow (14 of 28 squares it covers are yellow.)


We are finished if the upper light blue rectangle is one-third yellow.   

The upper rectangle is unfortunately not one-third yellow (only 4 of 21 squares it covers are yellow.)

That’s a failure. 


This time we will try to place three rectangles covering 1/2, 1/3 and 1/4 yellow squares.

Spoiler alert. Try to solve this before going onto the next slide.

This is a solution because the left rectangle is 1/3 yellow; the bottom right is 1/2 yellow; and the top right is 1/4 yellow.

Before you go to the next slide try to find another solution.

This also works.

This will be the first puzzle sheet for the students. They have to figure out the animal before they tackle each puzzle 😉

After students solve the first puzzle sheet they can either design their own puzzles (please send me any beautiful one gord AT or they can try tougher ones like this.

Spolier alert: Try solving this before going to the next slide.

These puzzles have been a big hit with my students probably because they are beautiful to create and solve.

Fractured Fractions

(MathPickle, 2012)

Grade 5 students tackle Fractured Fractions. Excellent engagement of the full spectrum of student ability is typical with beautiful and interesting curricular puzzles. Download the puzzle sheets here.

Standards for Mathematical Practice

MathPickle puzzle and game designs engage a wide spectrum of student abilities while targeting the following Standards for Mathematical Practice:

MP1 Toughen up!

Students develop grit and resiliency in the face of nasty, thorny problems. It is the most sought after skill for our students.

MP2 Think abstractly!

Students take problems and reformat them mathematically. This is helpful because mathematics lets them use powerful operations like addition.

MP3 Work together!

Students discuss their strategies to collaboratively solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP4 Model reality!

Students create a model that mimics the real world. Discoveries made by manipulating the model often hint at something in the real world.

MP5 Use the right tools!

Students should use the right tools: 0-99 wall charts, graph paper, etc.

MP6 Be precise!

Students learn to communicate using precise terminology. MathPickle encourages students not only to use the precise terms of others, but to invent and rigorously define their own terms.

MP7 Be observant!

Students learn to identify patterns. This is one of the things that the human brain does very well. We sometimes even identify patterns that don't really exist 😉

MP8 Be lazy!?!

Students learn to seek for shortcuts. Why would you want to add the numbers one through a hundred if you can find an easier way to do it?


Please use MathPickle in your classrooms. If you have improvements to make, please contact me. I'll give you credit and kudos 😉 For a free poster of MathPickle's ideas on elementary math education go here.

Gordon Hamilton

(MMath, PhD)