Fractured Fraction Puzzles

These puzzles are the best way to engage students with equivalent fractions.

Start with a picture which you must tile with rectangles. In this case we see that one of the rectangles must be half yellow - the other one must be a third yellow.

PS. Here is a printout of this slide show.

Here is a 4x7 rectangle. Does it help?

Yes, half of this rectangle is yellow. We have found one of the rectangles...

Unfortunately, the other rectangle does not look like one third yellow. We failed. In this case the puzzle is impossible. You cannot cover this 7x7 grid with two rectangles - one 1/2 yellow and one 1/3 yellow...

I won't be nasty again 😉

From now on all of my Fractured Fraction puzzles are possible to solve. However, when you are introducing puzzles to students - try to start with a failure. That's much more exciting and gets everybody used to the fact that not everything has a solution.

This yellow submarine is possible. Cover it with three rectangles - 1/2, 1/3 and 1/4 yellow.

Rectangles must cover the whole of the grid and cannot overlap. Spoiler alert... the next slide gives the solution.

Confirm that the three rectangles are 1/2, 1/3 and 1/4 yellow.

There is another solution... If you didn't try to find this one, you still have the opportunity to redeem yourself. The other solution is on the next slide.

Here is the second solution. Confirm that the three rectangles are 1/2, 1/3 and 1/4 yellow.

This is the first puzzle-sheet for your students. You can download all the puzzle-sheets here.

Some students are attracted to beauty - that's why we put time and effort into creating beautiful worksheets.

Other students will enjoy naming the animal as they are solving its puzzle.

The next dozen pages work show the six puzzles on the first puzzle-sheet... followed by the solution(s).

Can you find the other solution for the Piranha?

Here it is. Are not these the most beautiful fraction puzzles? After your students solve the ones on this sheet they may want a tougher sheet or to design their own.

This camel is ornery. Beware!

This one took me quite a while to rediscover... that's okay. It keeps one humble 😉

You would think a rabbit would be kinder than a camel or a piranha...

...but by now you've probably figured out that these cute little animals are nasty cute little animals.

A cute little jelly fish...

Now this one obviously has two solutions. You can flip it around so the horizontal small rectangle becomes vertical on the upper right.

It is a lot of fun making these puzzles. Any of your artistic students should be strongly encouraged to create their own beautiful puzzle.

One last one to go on this first puzzle-sheet.

Many students think this is the toughest puzzle on the first puzzle-sheet, but upon revisiting these - my toughest was the camel and rabbit. I occasionally have a class where I lament my failures and celebrate my successes publicly. I especially do this if students have a realistic chance of beating me.

Sneaky. What made this easy for me to solve is that I realized there were a lot of grey squares. That meant the 1/2 region had to be pretty big.

This is from the second puzzle sheet. It's my favourite Fractured Fraction puzzle I've created. Good puzzles can be easy or difficult. The important thing is that they are fun to solve. I think this one is 😉


I know it looks difficult, but look at those fractions. Are there any that only have one place to go?

Hope you enjoyed that one 😉

If you create any of your own please send them to me: gord$ replacing the $ with something else 😉

The design of this puzzle actually started with something more complex... Each fraction had an associated colour like the puzzle on the left. Try to solve it.


Thanks for sharing Fractured Fractions - one of my favourite puzzle designs over the last years.

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.

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!

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

MP3 Work together!

This is collaborative problem solving in which students discuss their strategies to solve a problem and identify missteps in a failed solution. MathPickle recommends pairing up students for all its puzzles.

MP6 Be precise!

This is where our 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!

One of the things that the human brain does very well is identify pattern. We sometimes do this too well and identify patterns that don't really exist.


Please use MathPickle in your classrooms. If you have improvements to make, please contact us. We'll give you credit 😉

Gordon Hamilton

(MMath, PhD)


Lora Saarnio