Think Inside the Box
By Michael Naylor

These fun "puzzles" are easy to use with students at any grade level and with any operation. Not only do these help provide practice with facts and operations, but they are also self-checking and lead to some interesting big ideas in algebra.

Addition boxes I
(Grades K-3)
Students are given grids as shown below, with four numbers placed in the upper left boxes. The object is to fill in the missing boxes so that each row and each column shows a correct sum.

After the first two rows and columns are completed, students can then add the numbers in the last row and column and enter it in the final box. The sum should be the same for both the last row and the last column – if it isn't, there's a mistake somewhere!

It's easy to make a sheet with eight or 12 of these. How about asking your students to write their own?

Addition boxes II
(Grades 2-5)
To work on the inverse operation, instead of filling in the four main boxes, fill in the other four boxes as shown in the figure here:

Students must now work backwards through the operations using subtraction or "think-addition."

Subtraction boxes
(Grades 4-6)
Change the operation to subtraction, and you've got a very interesting variation in which students are adding and subtracting with negative numbers. Here's a grid with four random numbers entered. In order to solve this grid, your students will have to solve problems like, "Two minus what equals three?"

Name Number Operations
(Grades 4-8)
Here's an example with a multiplication box. The numbers might get big, so consider letting your students use calculators. They will be practicing not only multiplication, but also division.

MULTADD boxes
(Grades 4-8)
This amazing grid gives practice with multiple operations and can be used to illustrate distribution and other algebraic ideas.

The grid is the same as the addition boxes, except the rows have been extended. Provide four "generator" numbers, two on the top and two on the side. These numbers are multiplied to find entries in the addition boxes, which are then completed as usual.

Generators are given.

Boxes are filled using multiplication.

Rows and columns are summed.

As a final check, fill in the last two spaces with the sums of the generators in that row and column. The product of the numbers in these two spaces is the same as the final sum in the lower right-hand box – in this case, 13 x 10 = 130.

These boxes also illustrate distribution of multiplication over addition. The first columns, for example, show that (6 x 8) + (6 x 2) = 6 x (8 + 2).

Multiplication algorithm
connection (Grades 4-8)
The MULTADD boxes can be used as another way to examine the multiplication algorithm. Suppose you want to multiply 37 x 51. Set up a MULTADD box with the generators (30 + 7) and (50 + 1):

The center four squares show the four partial products that occur when computing using the standard algorithm – and the sum of all four of these numbers is the product of 37 and 51…1887!

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.