Students work in pairs to create their own game boards, a worthwhile challenge. They learn by experimenting and by making mistakes about what factors and products to include in a game.
Learning Objectives
Students will:
understand that some products are the result of more than one factor pair
Spend some time reviewing students' strategies for
playing the Product Game.
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.
To create a new game, students might first decide what
factors they want to use and then determine which products are possible. If, for
example, students 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
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. The Product Game applet is below, with one change from the first lesson. Now there is a bar called
"Customize." Click on the "Customize" bar, 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.
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.
Extensions
Display and duplicate games the students find interesting
so other students can play them on their own.
The National Council of Teachers
of Mathematics is a public voice of mathematics education, providing
vision, leadership, and professional development to support teachers
in ensuring mathematics learning of the highest quality for all
students. The views expressed or implied, unless otherwise noted,
should not be interpreted as official positions of the Council.
