The Product Game

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SWBAT determine if a number is a multiple of a given number.

Big Idea

Students will practice identifying the multiples of single digit numbers by playing the Product Game.


20 minutes

Today's Number Talk

For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation.pdf. For this Number Talk, I am encouraging students to represent their thinking using an array model.

Task 1: 42/7

For the first task, 42/7, many students experimented with decomposing the 42 into multiples of 7 (such as 35 + 7 or 21 + 21). Here's an example of a student who decomposed the 42 into multiples of 6: 42:7

Task 2: 84/7

For the next task, students immediately responded, "I know! I know! It's going to be 12!" I asked: How do you know? Students explained, "You doubled the dividend which means the quotient will double. Is this always the case? "No... not always... but it's true when you're dividing by the same number over and over." Here, 84:7, a student shows two different ways of dividing 84 by 7 using an array model. 

Task 3: 840/7

During this task, most students drew an array with 840 on the inside. Next, they labeled one side 7 and the other side 120. 

Task 4: 882/7

For the final task, some students were stumped at first. Then, they began to see the connections between the tasks. Here, a student uses two of the prior tasks to solve this task: 882:7.

Throughout every number talk, I continually model student thinking on the board to inspire other students. This also requires students to use math words to explain their thinking instead of relying on a model to represent the math. As students solved each task, I wrote the answers on the board to encourage students to use prior tasks to solve the more complex tasks: Listed Tasks.


Teacher Demonstration

40 minutes

Goal & Introduction

I began today by reminding students of our learning goal: I can determine if a number is a multiple of given number. I explained: Today, you will be playing a game which will require you to identify the multiples of the numbers 1-9. First, let's review the difference between a multiple and factor. 

Multiple vs. Factor Review

I invited students to join me in a circle on the front carpet with their white boards and markers. I modeled how to draw a t-chart on their white boards. I labeled one side Factors and the other side of the t-chart, Multiples. I drew a circle in-between the two columns for the given number: Factors vs Multiples Chart. I continued: Today, we are going to review multiples and factors. I'll give you a number, and I'd like for you list factors of this number on one side of your t-chart and multiples of this number on the other side of your t-chart. Writing the number 3 inside the circle, I said: Your first number is 3. Please list the factors of 3 and the multiples of 3. 

Multiples & Factors for 3

Students went right to work and after a few minutes, most students completed the task: Student White Board for 3. Some students were beginning to run out of room on the multiples side. I asked: Why is it that some of you were running out of room to list multiples but had plenty of room for factors? One student said, "Because there are millions of multiples and only a few factors." Not only was this a great time to encourage Math Practice 3 (Construct Viable Arguments), but it was also the perfect opportunity to sing our Factors & Multiples Song as a class! 

Then, I asked a few students to model their thinking on the front white board for everyone to see: Students Modeling for 3. During this time, I encouraged other students to watch carefully for ideas that they agreed with or respectfully disagreed with. 

Multiples & Factors for 12

We continued the same process, only for the number 12. Again, students completed this task on their white boards: Student White Board for 12 and then students came up to the board to complete the whole-class chart. 

This turned out to be a great way to review factors and multiples. Some misconceptions were cleared up along the way. I heard some students say, "Oh, now I get it!" I noticed one student who wrote 0 under the factors for 12 and another student who skipped listing 4 as a factor. Through turning and talking with a partner or teacher conferencing/questioning, most of these misconceptions were corrected.  

Modeling The Product Game

Before asking students to return to their desks, I modeled The Product Game. This is a fairly simple game that most kids love to play. I projected The Product Game and explained the following Rules I also chose two Student Volunteers to act out the game in front of the class.  

First, I would like you to play Rock-Paper-Scissors. The winner will be Player A. The two volunteers modeled Rock-Paper-Scissors to determine who was Player A.

1. Player A puts a paper clip on a number in the factor list.

  • I asked Player A: Which factor would you like to put the first paper clip on? The student said, "7."

2. Player A does not mark a square on the product grid because only one factor has been marked: it takes two factors to mark a product.

3. Player B puts the other paper clip on any number in the factor list (including the same number marked by Player A) and then covers the product of the two factors on the product grid.

  • I asked Player B: Which factor would you like to put the other paper clip on? The student said, "4." 
  • I asked: What is the product of 7 and 4? "28!" I modeled how to place a red token on 28. 

4. Player A moves either one of the paper clips to another number and then covers the new product. 

  • I asked Player A: You can move one paperclip to a new factor... which paperclip woul you like to move? The student said, "I'd like to move the paperclip on the 4 to the 9." 
  • Okay, now what is the product of 9 and 7? "63!" I modeled how to place a yellow token on 63. 

 5. Each player in turn moves a paper clip and marks a product. If a product is already marked, the player does not get a mark for that turn. The winner is the first player to mark four squares in a row—up and down, across, or diagonally. 

I asked to student volunteers to model a couple more turns before passing out the game boards to the rest of the class. 

As a side note, I used these two-color chips from Really Good Stuff. Since they are red on one side and yellow on the other, I didn't have to take the time to sort colored chips to make sure there was enough for each partner. 

Student Practice

50 minutes


Assigning partners is always easy in my classroom as I already have students strategically placed in desk groups based upon ability levels, communication skills, and behavior. I asked each group to get a Game Board, a cup of colored chips, and two paper clips. 


Getting Started

Students were excited to begin playing and didn't waste any time getting started. Here's an example of Students Playing the Game.

During this time, I conferenced with each pair of students multiple times to ask questions in order to monitor understanding and to challenge students to explain their thinking. By asking students to explain their thinking, I knew they would also be engaging in Math Practice 3: Construct Viable Arguments.

  • Can you tell me what you're thinking?
  • Can you explain this further?
  • What strategies are you using?
  • Why did you choose 5 x 9?


Added Challenge

After students had played a few rounds of this game, I explained: Feel free to make this game more challenging by going for 5 in a row or even 6 in a row! A new excitement filled the air!