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Ballast Puzzles

The objective of this puzzle is to find those warships that will tip over because of improper ballast. Start by getting them to understand ballast... It is something heavy that you put in the bottom of a boat so it doesn't tip. I sometimes do a demo with a plastic drinking cup. How can you get it to sit up straight in the water? After they understand what it is - ask what would happen if they added too much ballast 😉 That will get everyone engaged.

Draw the five squares on the board, but explain nothing. Ask five students for numbers 1-10.

Printable puzzle-sheets here.

Numbers can be duplicated, but in this case the students chose 10, 2, 3, 4, 1. Now ask the next student which number they want to put in the leftmost square.

PS. They still know none of the rules. Rules introduced too early can bore. Much better to let students guess at what the rules might be for 30 seconds.

Five students in order of seating have put the numbers in the squares. Now you explain that the red dot is multiplication and the green plus is addition and ask them if this warship will sink or float? Is it balanced?

4 x 3   versus  10 + 1 + 2

That seems balanced you might comment. The next student is chosen to answer if he or she thinks it will sink or float. This one will sink because the right side (13) is too heavy. Disaster! Identify the students who failed with a wry grin - of course they failed - they didn't even know the rules!

Notice that you are NOT giving students the option to participate. Hands are not raised and fast students must be controlled not to blurt out the answer thereby robbing slower students of their turn in the sun.

Fast students will be frustrated, but after working with them for a week, they realize that their time will come in the second part of the class when students work alone or in pairs.

The original problem looked hopeless. Let the students discover this. It is MUCH better to start with a Disaster - it is so much more engaging. One student wanted me to replace the 10 with a 9. I did that.

Another student raised her hand. This is okay - it is an insight that she has rather than butting in on another student. She said there was another solution and she was right.

Students do not often start out comfortable enough to share and explore. This class I had been teaching for about 10 periods.

I do not congratulate the girl, rather all the focus is on what a relief it must be to the sailors.  Keep the veneer of story alive!

Now it is time for the puzzle-sheets. In each one there are three warships. Two can be made to float. One will sink no matter what desperate measures you try.

So many students come into math class with some goody-goody mumbo-jumbo about everything being possible if they just try hard enough. I like to break them out of that mind-set with humour. Just like in real life - not everything is possible 😉

Ballast – multiplication puzzles

(MathPickle, 2012)

Once again the valiant 4th grade students of Bradley Meyers (Taft School District, Lockport, IL) have discovered one of my errors. One of these ballast puzzles has two sinkings rather than one. It’s my error, but instead of correcting it we have decided to make it a challenge for others to find. It does not occur in the first several puzzle sheets.


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)