The Product Game

Players: 2
Ages: 7 and up
Cost: Free! (Printable template and Online version)
Math Ideas: Multiplication, Common Factors
Questions to Ask: 
    Which squares can you color in this turn?
    Why can't you play on this square?
    Where do I want to play? How can you block me?

My first year of teaching, I really had no clue what I was doing. I was terrified to teach my first lesson, since I had no idea what my students did and didn't know. So I grabbed an activity that I had first seen in my Master's program: The Product Game.

Even though this game is playable by any child who knows multiplication, my eighth graders were just as engaged as any third grader who was still practicing their times tables. Four of the boys even created a mini-tournament during class!

The Product Game has been a go-to activity in my classroom ever since that first day. My own kids are still a little too young to play it, but I think the game is great for any kid who knows multiplication, even if their understanding of the multiplication table is still developing.

How to Play

The Product Game is played by two players on a printable game board like the one shown (pdf). The main section of the board is a square grid of all the products that you can get by multiplying the numbers from 1 to 9. The goal of the game is to color in four squares in a row, either vertically, horizontally or diagonally. Along the bottom of the paper is a row of the factors from 1 to 9. 

Quick vocabulary recap: two factors multiply to equal a product. So in the problem 4*5 = 20, the numbers 4 and 5 are the factors and 20 is the product.

Player 1 places a token on any of the factors, but colors in nothing. Then Player 2 places a second token on any of the factors and colors in the product that you get when those two factors are multiplied.

Now it's Player 1's turn. She can move either of the tokens, but not both. So if the first two tokens were placed on 4 and 5, Player 2 got to color in the 20 square. Now Player 1 could, for example, move the 4 token to 7 so that she multiplies 7 by 5 to get 35.

This restriction makes the game particularly fun and strategic. You can't just move anywhere, since your options are constrained by your opponent's previous move. At the same time, you have the ability to constrain your opponent's options! Don't want him to color in 15? Just make sure you never play any 3s or 5s. Although that might be harder than you think...

Here is a quick video of a full Product Game. As you'll see, the game is quick to play, but there's a lot of strategy going on! You can also try a free online version here.

Where's the Math?

The Product Game is all about multiplication, clearly, but there is much more math in the game than the simple recitation of multiplication facts.

Remember the rule where you can only move one token? It can really limit your options! If your opponent just played 4 and 5 to get 20, you can either move the 4 token and play a different multiple of 5, or you can move the 5 token and play a different multiple of 4.

Either way, you have to play on a square that has a common factor with your opponent's move. Common factors are a huge deal in elementary math, and they remain vital in middle school and beyond. 

Remember simplifying fractions? You probably remember the drudgery...

Well, simplifying fractions is a process of dividing out common factors. If your child's teacher asks them to simplify 20/35, they are asking your child to look at 20, look at 35, and try to think of a factor that is in both numbers. In this case, you factor out the 5 to get a simplified fraction of 4/7.

In the product game, you are constantly working in the other direction, looking at two factors and thinking "what other numbers have a factor of 5?" Still, your child will start seeing the connective tissue between different products. How are 28 and 49 connected in The Product Game? Well, they are both options that you can play if you have a token sitting on 7.

I firmly believe that fluency with multiplication is incredibly valuable for students as they move through elementary math and onward. But you don't want every math experience your child has to be one of repetitive fact recall. The Product Game gives your child a reason to remember her math facts, and a way to connect them into a mental framework that will pay off later.

Also, it's just fun. Try it!

Questions to Ask

I think the best questions to ask are those questions that highlight the connections between products. I'd start with a simple "What are some places you can you play on this turn?" Your child, in an effort to find all her options, will likely start trying to find a shortcut for which square she can play on. Guess what? That shortcut is called "practicing her multiplication tables."

You could also reverse the question, asking "Why can't you play on this square?" In order to explain why she can't play on 56, your child might stop and think about which factors multiply to 56, and then she might realize she has a good move the next time a 7 or 8 is played.

Another aspect of the game is defense: keeping your opponent from playing where they want to play. You can ask your child "How do you block me?" and then have a chat about the factors that you want to play, and how they can stymie your efforts.

While writing this newsletter, I came up with a fun little solitaire version of the game. Here's how it works: you get to play as both players, and your goal is to color in the entire board. The only rule is, you can only move one token at a time, and you can't color in a square more than once. Can you figure out a way to color in the whole board?