The Substitution Game features low-floor, high-ceiling cooperative play that works with any age (or with a mixed-age group) — and you can use it while distance learning, too. It’s great for building algebraic thinking.

Excerpted from my new book, *Prealgebra & Geometry: Math Games for Middle School*. Look for it at your favorite online bookstore.

## The Substitution Game

**Math Concepts:** addition, subtraction, multiplication, division, order of operations, integers, fractions, equivalence and substitution.

**Players:** any number (a cooperative game).

**Equipment:** whiteboard and markers (preferred) or pencil and paper to share. Calculator optional.

### Level One

The first player writes a simple equation at the top of the paper, such as “1 + 1 = 2.” Then all players take turns complexifying this equation.

On your turn, copy the equation to the next line, replacing one number with an equivalent expression. For instance, replace the number 2 with:

or 50 ÷ 25

or (1/3) × 6

… or any other calculation that equals two.

Use parentheses or brackets as needed to make your expression perfectly clear. For example, I put parentheses around my fraction above so people can tell I didn’t mean “1/(3×6),” which is definitely not a substitute for two.

If you have colored pencils or markers, circle the number you plan to substitute. Then write the substitution below in the same color. Finally, fill out the rest of the equation using a neutral-colored pencil or marker.

The other players should check to make sure they agree with your math.

After you change part of the equation, it is no longer available for anyone else to use. If you substitute “5 − 3” for the 2, the other players cannot replace your creation with their own version of that number. But they can alter individual numbers within your creation. So the next player may decide to substitute for the 5, writing a new expression in its place.

For young players, an older child or an adult may take dictation. This lets the young one focus on thinking about the numbers without struggling to write an ever-growing equation.

Continue until the paper is full, or until the equation looks satisfyingly complex, or until you run out of time. Save the paper (or copy the final equation from the whiteboard) for playing Level Three.

### Level Two

As students grow comfortable with the basic game, pose additional challenges. Perhaps each substitution must use multiplication, or a fraction, or contain a specific number. If you’re playing with a mixed-ability group, each player may have a different challenge.

For example, if your challenge is to play division, you could replace the number 3 in an existing equation with:

or (70 ÷ 7) − 7

or (1/4) × 300 ÷ 25

… or any other calculation that uses division. Remember to use parentheses or brackets as needed to make your expression perfectly clear.

The other players should check to make sure they agree with your math.

Continue until the paper is full, or until the equation looks satisfyingly complex, or until you run out of time. Save the paper (or copy the final equation from the whiteboard) for playing Level Three.

### Level Three

The first player chooses an old round of the Substitution Game and writes the final equation at the top of the paper or whiteboard.

Players take turns simplifying the equation.

On your turn, copy the equation to the next line, replacing part of it with a simpler expression. The other players should check to make sure they agree with your math.

Continue until you reach the simplest form of the equation — which may not be the same as what the original game started with. For example, the simplest form of the equation “1 + 1 = 2” would be “2 = 2.”

If the final equation is a true statement, then you win. Hooray!

But if anyone made a mistake in either level of play — either complexifying or simplifying the equation — you may end up with a nonsense statement like “2 = 13.” Don’t worry about trying to find the error. All the players still did a lot of mathematical thinking, so count this as a sideways win. Enjoy a good laugh at your silly result.

### Variations

You can also play this as a solitaire game, just for the fun of complexifying equations.

Or do a math composition: Pick a number, write an equal sign, and then write an equivalent expression. Repeat for as long as you can think of new ways to write it.

For example, “6 = 10 – 4 = 36/6 = (100 ÷ 5) – (2 × 7) = …”

How crazy can you make the math?

### History

Homeschooler Sonya Post’s Arithmophobia No More website equips parents to make sense of math, so they can learn to teach their children. About the Substitution Game, she writes:

“The student is learning that there’s nearly an infinite number of ways to write any number. Students are learning the ‘secrets’ to building complex mathematical statements. You know what math a child has mastered when she can generate the math herself.”—Sonya Post

Substitution Game – Forget the Worksheets

CREDITS: “Design work” photo by Kelly Sikkema on Unsplash.

This is such a great idea. I can’t wait to use it with my learners. I especially like how it reinforces the equals sign as meaning is equal to rather than “find the answer”.

Yes, that’s a key idea. Through all the changes, the equation stays nicely balanced and equal (as long as we don’t make any mistakes or typos!)