## Making Your Own Product Game

6-8
1

In Part II, students make their own game boards. The task of creating a new game is challenging to most students. They learn a lot by experimenting and by making mistakes about what factors and products to include in a game.

Spend some time reviewing students' strategies for playing the Product Game. Ask students:

We have been playing the Product Game and discussing strategies you can use to win. Look back at the board. Does it contain all the products you can make from the list of factors?

What products would we need to add if we added 10 to the factor list?

Multiplying 10 by the other factors on the list gives the products 10, 20, 30, 40, 50, 60, 70, 80, 90, and 100. Help students to see that 10, 20, 30, and 40 are already on the board. (You might ask why.) Therefore, you would need to add only 50, 60, 70, 80, 90, and 100. You want students to see that every product, including the squares of the factors, must be on the game board to make a good game. Sometimes students have to experience frustration while making their game boards before they realize that every product must be included. This is a good opportunity for students to practice perseverance.
Distribute the Making Your Own Product Game Activity Sheet to each pair of students.

Tell students that they will be making their own game board. After reviewing the worksheet together, offer helpful advice on what students should consider when creating their own board:

Suppose you want to create a product game that takes less time to play or, perhaps, more time to play than the game with the 6 X 6 product grid. You would have to decide what numbers to include in the factor list and what products to include in the product grid.

If, for example, you choose the factors 1, 2, 3, and 4, the products would be 1, 2, 3, 4, 6, 8, 9, 12, and 16. This would create a nice 3 × 3 game board. The rules could be modified so that three in a row would win.

Students need to use enough factors to make their game interesting. For example, the factors 1, 2, and 3 give the products 1, 2, 3, 4, 6, and 9. A 3 × 2 grid would accommodate these six products, but this would not make a very interesting game. Only two markers in a row would be required to win, so the game would end on the second turn of the first player!

Instead of choosing the factors first, students can select the size of the product grid they want, then work backward to find the factors needed to fill the board. Interested students might be challenged to find the factors needed to create a 10 × 10 board (the factors 1 through 16 are needed, and there will be three blank spaces). You might want to help students organize their work as in the table below.

 Factor Products of the factor and numbers less than or equal to the factor Number of products added to the list* Total number of products on the list 1 1 1 1 2 2, 4 2 3 3 3, 6, 9 3 6 4 4, 8, 12, 16 3 9 5 5, 10, 15, 20, 25 5 14 6 6, 12, 18, 24, 30, 36 4 18 7 7, 14, 21, 28, 35, 42, 49 7 25 8 8, 16, 24, 32, 40, 48, 56, 64 5 30 9 9, 18, 27, 36, 45, 54, 63, 72, 81 6 36 10 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 6 42 11 11, 22, 33, 44, 55, 66, 77, 88, 99, 110, 121 11 53 12 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144 6 59 13 13, 26, 39, 52, 65, 78, 91, 104, 117, 130, 143, 156, 169 13 72 14 14, 28, 42, 56, 70, 84, 98, 112, 126, 140, 154, 168, 182, 196 8 80 15 15, 30, 45, 60, 75, 90, 105, 120, 135, 150, 165, 180, 195, 210, 225 9 89 16 16, 32, 48, 64, 80, 96, 112, 128, 144, 160, 176, 192, 208, 224, 240, 256 9 97

Note: *Products in italics have already been used; they are not counted again.

## Explore

If a group is having difficulty, check over their list of products and help them get the products correct.

I notice that the product of 4 and 5 is not on your list. Have you checked to make sure you have all of the products?

Ask questions that will help students focus on the relationship between the list of products and the size of the game board. Some boards will need to have to have blanks or free spaces.

What size game board will hold all of your products? Is this the smallest board you can use?

Circulate while students are making their games, and help keep the groups focused on the task. When students are playing each other's games, remind them that it is very important that they give good feedback.

As you are playing your own or another group's game, if you think it is interesting and should be shared with the rest of the class, let me know.

When students have finished making their boards and trying them, ask them to work on the summary paragraph described in the Activity Sheet.

Alternatively, students may use the Applet once again. Click on the bar called "Customize." Then change the list of factors and the number of squares needed for a winning "run" so it is the same as the game you designed. Click "OK" and the applet will make a game board from your factors. Play the game a few times with your partner. Decide together which game board is better (yours or the applet's) and why.

## Summarize

You can summarize this activity with each group individually. As you interact with a group, observe the problems they are having, and work to help them overcome these problems. Ask the group to explain the steps they went through to create the board. Ask what problems they had and how they solved these problems. Ask how they knew when they had all of the possible products and whether they needed to change the rules to play on their board.

You also could summarize by having groups share their reports with the class. Use the reports to help students focus on characteristics of interesting game boards and the strategies that were used to create them.

This Product Game Investigation was adapted with permission and guidance from:

Prime Time: Factors and Multiples, Connected Mathematics Project, G. Lappan, J. Fey, W. Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications, (1996) pp.17-25.

Assessment Options

1. Use students' game board as a form of assessment.
2. Use informal assessment by circulating the room and taking note of students that understand the connections between factors and products.

Extensions

1. Display and duplicate games the students find interesting so other students can play them on their own.
2. Move on to the next lesson in this unit, Classifying Numbers.

Questions for Students

1. Does your board contain all the products you can make from the list of factors?
2. What products would we need to add if we added [choose a number] to your factor list? What would the dimensions of the board be? How many in a row should we aim for?
3. How did you choose your factors?

Teacher Reflection

• How did students demonstrate their level of engagement?
• How could students be more motivated to create a better game board?
• How much time should be spent working with peers vs. alone?

### Playing the Product Game

6-8
Students learn how to play the Product Game. As they play the game, students develop understanding of factors, multiples, and the relationships between them. Winning strategies are discussed. The Product Game was adapted from Prime Time: Factors and Multiples, part of the Connected Mathematics Project, and was written by G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips (Dale Seymour Publications, 1996, pp.17-25.)

### Classifying Numbers

6-8
Students use Venn diagrams to represent the relationships between the factors or products of two numbers.

### Connections and Extensions

6-8
Students make connections and expand on what they have learned in the first three lessons. Students explain the effects of different moves on the game board. Finally, students "Guess My Number" using various clues.

### Learning Objectives

Students will:

• Learn the connections between factors and products as they create their own game boards.
• Understand that some products are a result of more than one factor pair.

### NCTM Standards and Expectations

• Use factors, multiples, prime factorization, and relatively prime numbers to solve problems.
• Develop and analyze algorithms for computing with fractions, decimals, and integers and develop flue
• Develop and use strategies to estimate the results of rational-number computations and judge the reasonableness of the results.

### Common Core State Standards – Practice

• CCSS.Math.Practice.MP1
Make sense of problems and persevere in solving them.
• CCSS.Math.Practice.MP2
Reason abstractly and quantitatively.