Exploratorium
Fantastic Four (Tips for Leaders)


Guiding the Game

This game gives people a chance to practice different mathematical operations, such as addition, subtraction, multiplication, and division. It also encourages people to practice writing equations correctly, including making use of parentheses. It can be very competitive—and a lot of fun.

Introducing the Game

We suggest that you introduce this game by showing the group how to play it. Deal out four cards, then a fifth, and ask the group if they can make an equation that uses some or all of the four cards to equal the number on the fifth.
    

Give people a few minutes to work on their own, writing down possibilities that they come up with. Then ask what people’s equations are. Write the equations down for everyone to see.
    

Try to show examples of equations using two numbers (worth 4 points), equations using three numbers (worth 9 points), and equations using all four numbers (worth a whopping 16 points).
Don’t worry if you can’t find an equation that uses all four numbers. Sometimes, it just isn’t possible!

A Word About Equations

Addition, subtraction, multiplication, and division are all acceptable operations to include in equations. Depending on how advanced your players are, you may want to allow other operations, such as square roots, exponentials, factorials, and so on.
   

An equation can yield different answers, depending on the order in which you do the operations. (See Where’s the Math?) That’s why it’s important for players to use parentheses when writing equations. When doing the mathematical operations in an equation, one always does the operations that are in parentheses first. People should use parentheses to make it clear which operations to do first.

Playing the Game

Once players understand the rules, they can play independently in groups of two to five. Alternatively, you can have an entire class or large group play together. The leader deals out the cards and keeps track of time. Once time is up, players calculate their own scores. Ask who has the highest score, then check her or his equations together. Or have players swap papers and check each other’s equations. You might want to remind everyone that a player gets 7 points for finding someone else’s mistake!

Some Helpful Strategies

Are people having trouble finding an equation that works? It may help to focus on finding combinations that lead to 1’s or zeros.

Get Rid of Numbers by Making Zero

Suppose you get stuck trying to use 3, 4, 7, and 7 in an equation that equals 4. You can get rid of unwanted numbers by creating a zero, like this:



Notice how nicely the first term disappears. Dividing anything into zero gives you zero. You may have to remind people that you can divide into zero, but you can’t divide by zero. So you can’t have:

because zero can never be on the bottom of a fraction (in the denominator).

Making a 1

Suppose you’re trying to use 3, 4, 7, and 7 in an equation that equals 7. You could get rid of unwanted numbers by turning them into a "1," like this:



Where's the Math ?

When doing the calculations in an equation, it’s important to do them in the right order. Here are two calculations that look very similar:


(1+2) x 3 =
1 + (2 x 3) =


The numbers and the operations are the same—but the answer you get depends on which operation you do first.

Whenever you make a calculation, you do whatever operation is in parentheses first. So the answer to the first calculation is:


(1+2) x 3 = 3 x 3 = 9


And the answer to the second calculation is:


1 + (2 x 3) = 1 + 6 = 7


Very different answers!


Be sure that people in your group understand this and use parentheses to identify the order
in which they want the operations done.

Using parentheses is just one way to define the order of operations. Mathematicians have a set of rules about the order in which operations are done. Here are those rules:

First, do whatever is inside parentheses.
If there are any exponents , do those next.
Then, do all the multiplication and division, starting at the left-hand side of the equation and working through the operations in the order in which they appear.
Finally, do the addition and subtraction. Once again, start at the left-hand side of the equation and do the operations in the order they appear.
To remember the order of operations, people can memorize the sentence: "Please Excuse My Dear Aunt Sally." Take the first letter of each word in order, and it tells you what to do:

Please P Parentheses
Excuse E Exponents
My Dear M D Multiply and Divide
Aunt Sally A S Add and Subtract

 

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