Arithmetic on a Grid

Grid addition

How many small green squares are on the grid to the left? Click in the first input box to the right, and then type in your answer. Then click the first Next button to see the next example on the grid. Continue doing this for each row in the table. (On a touch screen without a mouse, you can tap instead of clicking.)

In the first row of this table, enter the mathematical sentence that the example on the top grid represents. Then continue to the next row, by clicking its Next button. Enter the new mathematical sentence shown on the top grid. Your equation will be illustrated on the bottom grid as you type. You can switch between input boxes by pressing the tab key. Fill in each row in this way.

So this is how addition is illustrated on the grid: the total number of green squares is calculated by adding together the number in each smaller piece.

Grid subtraction

On the grid to the left, there is a rectangle of 8 squares, which are green except for 2 gray squares. So the number of green squares equals $8-2$.

How many green squares are there? Type in your answer, and continue to the following rows. Click each Next button in turn.

In the first row of this table, enter the mathematical sentence that the example on the top grid represents. Then continue to the next row, clicking its Next button. Your equation will be illustrated on the bottom grid as you type. Fill in each row in this way.

This shows how to subtract a smaller number from a bigger number on the grid. First you draw a green area whose size is the bigger number, and then a gray area inside it whose size is the smaller number. The difference is then the remaining number of green squares.

Grid multiplication

We write multiplication using parentheses. So, when you see “$2(3)$,” think “2 copies of 3,” or “2 times 3.”

Look at the grid to the left. How many green squares are there? Type in your answer, and continue to the following rows. Click each Next button in turn.

In each row of this table, click the Next button and then enter the mathematical sentence that the example on the top grid represents. Your equation will be illustrated on the bottom grid as you type.

So to multiply two numbers on a grid, you think of a rectangle that’s as wide as one number and as tall as the other, and find its area by counting the number of squares inside it.

Grid division

Division works differently on the grid than the other three operations. In each other case, your answer is the total number of green squares. For division, your answer is instead the length of some side of a rectangle.

In this course, we write division of two numbers in fractional form. When you see “$$6/2$$,” think “6 divided by 2.” You should interpret this either as “6 split into 2 groups” or as “the number of 2s in 6.”

Look at the grid to the left. How tall are the green columns? Type in your answer, and continue to the following rows, clicking each Next button in turn. We call this kind of division — where $$6/2$$ means “6 split into 2 groups” — “splitting up.”

In each row of this table, click the Next button and then enter the “splitting up” sentence that the example on the top grid represents. Your equation will be illustrated on the bottom grid as you type. (You can always go back to see a previous example by clicking on its Next Question or Next button.)

So this is how “splitting up” division is illustrated on the grid. To divide one number by another, you draw a green rectangle for the first number which is as wide as the second number. Then their quotient — the answer to the division problem — is the height of the rectangle.

How many green columns are there? Type in your answer, and continue to the following rows. We call this kind of division — where $$6/2$$ means “the number of 2s in 6” — “counting out.”