Portuguese Man O’ War puzzles are mini-mathematical universes. They are used to teach the scientific method to students learning addition. The next slides are an example of how a Portuguese man o’ war should be dissected in class. Click here to learn more about the philosophy of Mini-Mathematical Universes.
  Students start knowing nothing. Students are asked one by one to try to understand what is going on.
On their turn students may either ask what’s in a circle or take a guess. The first student chooses the highlighted circle and – knowing nothing – chooses not to guess, but just asks what’s in it.  
  I say “2” and reveal the answer. There is no shame in a student asking what’s in a circle. There is no shame in taking a guess and being wrong.  
Another student chooses a circle. Only the circles without empty circles under them are candidates. The student guesses “10.”
  Wrong – the answer is “1.” The student should be made to feel no embarrassment, but you could say that we still know relatively little so that this guess was almost certain to be wrong.
The student picks a circle and says “I don’t know.” You must make sure that the “I don’t know” does not become a wishy-washy meek response, but a strong question.
  “I’ll guess a 4.”
  “Wrong – it is a 1”
  “I guess 2.”
“Correct” If this is a student who needs a success in the class – then emphasize the discovery. If this is a student who is already pretty confident, then move on quickly.
The next student guesses “1,000,000” which is meant by the student to be a joke, but you treat it seriously.
Wrong – the answer is 1.
I’ll fast forward without comments… you can try to figure out what is going on…
After students go through this Portuguese man o’ war they should understand that each positive integer (other than 1,2,4,8,16,32…) is the sum of consecutive integers. Some numbers like 9 may be the sum of three consecutive integers 2+3+4 or two consecutive integers 4+5. Which is correct? 2+3+4 uses more integers than 4+5 and is correct.
Download these Portuguese man o’ war puzzles here.

Download presentation and puzzle-sheets here.

Portuguese Man O’ War Puzzles

(MathPickle, 2012)

These are mini-mathematical universes to teach the scientific method to students learning addition (above) and arithmetic sequences or algebra (above and below). Everyone should start above. Students learning algebra should ask their own questions and try to solve their own questions.

Download presentation and puzzle-sheets here.

The rules governing the creation of these Portuguese men ‘o war are different from those above. The ones above are suitable for grade 2. These ones are suitable for students dabbling in arithmetic sequences.
Here are the answers that your students can uncover circle by circle as done above… Can you figure out what is going on?
  Each number is the sum of the terms in an arithmetic sequence created with positive integers. If two arithmetic sequences are possible – choose the one with the most terms (Example: 80 = 4+10+16+22+28 with five terms is not as good as 80 = 3+5+7+9+11+13+15+17 with eight terms.) If there is still a tie, choose the one containing the biggest term (Example: 9 = 2+3+4 is not as good as 9 = 1+3+5.) Arithmetic sequences of under three terms are not considered. Which numbers do not have such an arithmetic sequence? Algebra will give the answer… 4, 8 and the prime numbers.

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 Know the tools.

Students master the tools at their fingertips - whether it's a pencil or an online app. 

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)