Algebra

This is a single variable algebra puzzle in which students try to find squares of different sizes to tile a rectangle. I prefer the two variable algebra variant on the right.

Get students to create their own history puzzles.

Students use linear algebra to build rectangles out of squares. The results are aesthetically pleasing and difficult to reproduce by trial and error.

Lessons should start on graph paper - giving students the challenge of trying to "build a rectangle using squares of all different sizes." Students will find it extremely difficult. After several minutes hold a vote asking if it is an impossible problem.

Download puzzle-sheets here.

Give your students practice working with the addition of square roots and Pythagoras.

Download puzzle-sheets here.

This is a great game to play with your class as they struggle to figure out how to add integers.

PS. Keep the selections between about -5 and +5. Use the word “about” as you do not want students to know for sure that a -6 was not used.

These three exercises go from easy to unsolved. It is this last unsolved problem from the nineteenth century that is absolutely essential in every student’s experience of algebra.

Dots and Boxes xyz

This is an algebraic version of the popular two-player game dots and boxes created by Édouard Lucas in 1889. The rules we will play with here are a good starting point. All students should use these rules for their first games.

This game has now been improved upon. Go to Pillage and Profit.

First, select a board and give an initial value to all variables. Here we start with x=3, y=2 and z=1.

On each turn a player must connect two neighbouring points.

Connections van be made vertically or horizontally.

Once and only once in the game each player may choose not to connect and may instead increase or decrease one of the variables by 10. Players typically hold these special moves till later on in the game.

Moving forward in the game, we see that neither player has yet used their special turn to increase or decrease a variable by 10.

We are reaching a critical point in the game. Whenever a player completes one of the small square - they steal the contents of that square (whether they want to or not.)

In this case, the player on the left definitely wants the lower left box, so will complete that square in the next slide.

It is worth 5 points.

When a square is completed, the player must take another turn. Sometimes they will not want to do this.

In this case the left player will add a connector in to the bottom of the middle square on the next slide.

That is the end of the left players. turn. Let's skip ahead again...

It is Right's turn. On the next slide he will grab the 2x square.

The 2x square is completed...

Its contents are grabbed. Right must take another turn because he has completed a square. He chooses to complete the square around the x.

Now he grabs the contents of that square.

Right again must take another move because he's completed yet another square. This time he prepares to take his special move. He can only do this once throughout the whole game. He's going to choose to increase or decrease one of the variables by 10.

z is now worth minus 9 points!

That is the end of Right's turn. Let's skip ahead a couple of moves...

One last rule - after all the squares are taken you can no longer make your special move. In the present game both Left and Right have already made their special move (you can see that Left has increased the value of y to 12).

Now we'll skip to the end.

All that need to be done now is substitute the values for the variables to see who has won. Here we see that y is the most valuable and z actually gives negative points. Who wins?

Right wins!

After your students have played a few games they should feel free to agree on different rules before the game commences. Instead of the special move being increasing or decreasing by 10 it may be halving and doubling. Instead of the winner having the most points, it may be that the winner has got the fewest points. Let them get creative and agree on the rules before the game starts.

Here are written instructions and a variety of game-boards.

Cartouche Puzzles

(MathPickle, 2012)

These versatile puzzles are great for adults and children alike. They are based on KenKen puzzles, but are superior for the classroom because they offer engaging ways to get at age-appropriate arithmetic skills.

Printable black & white Cartouche puzzles here.

Printable colour Cartouche puzzles here.

PS. These puzzles grow out of KenKen puzzles. In the hands of a master puzzle designer like Thomas Snyder, these are hard fun at its best. However KenKen puzzles lack sufficient flexibility when one is designing for the classroom. For example, there is too much of 5/1 = 5 and 1+2=3 to make them hit curricular targets. Cartouche puzzles solve this.

Use algebra to help find solutions to putting pirates and cutthroats into jail. Sounds righteous, but its darn difficult when each cell has constraints for the fraction of pirates and cutthroats.

Download puzzle-sheets here.