Splitting a rectangle into rows and columns of squares

When you reach for it

Squares in rows and columns are all around a child. Ihapera’s kitchen window is split by wooden bars into square panes. Tomiwa’s hallway floor is laid with square tiles. Esinam’s cupcake box has thin cardboard walls inside it that make a square cell for each cupcake. A sheet of graph paper is the same thing drawn in ink. In each one a rectangle has been cut up, by bars, by the edges of tiles or by walls, into squares that are all the same size and that sit in straight rows and straight columns.

In Arrays: rows and columns of objects a child set out separate things, corn plants or blocks, in rows and columns and counted them by adding the same number again for each row. This lesson runs the other way. It starts with one whole rectangle and asks the child to cut it up. That brings two jobs:

• Drawing the cuts so that every piece is a square and every square is the same size.
• Counting the squares, by rows or by columns, with a sum of equal addends.

Both jobs matter later. The squares a child draws here are the same squares that will measure how much flat space a shape covers, and counting them by rows is the start of the multiplying that comes after. Neither of those ideas is taught here. This lesson is about cutting carefully and counting well.

Give the child real things to work with: paper rectangles to fold, a ruler, square tiles or sticky notes to lay on a rectangle, and graph paper to check against.

The procedure

The lesson runs over three days. Part 1 draws the cuts. Part 2 counts the squares. Part 3 cuts the same rectangle into squares of different sizes, and looks at what happens when the pieces are not all the same.

Part 1: drawing the cuts

A row runs across, from left to right. A column runs up and down. In the picture the rectangle is cut into 3 rows, and each row holds 4 squares. The same squares stand in 4 columns of 3.

A cut across makes rows. A cut up and down makes columns. Each cut is a straight line that runs all the way from one edge of the rectangle to the opposite edge. A cut that stops halfway leaves one big piece where two small ones should be.

One fewer cut than pieces. The edges of the rectangle are already there, so only the lines in between have to be drawn. Two rows need one line across. Three rows need two lines across. Four columns need three lines down. In each direction the number of lines is one less than the number of pieces, just as a log cut into 4 pieces takes 3 cuts.

What makes the pieces same-size squares. Three things must all be true.

• The lines across are evenly spaced, so every row is the same height.
• The lines down are evenly spaced, so every column is the same width.
• The row height and the column width are the same, so each piece is as wide as it is tall. A piece that is wider than it is tall is a rectangle, not a square.

If the first two hold, all the pieces are the same as each other. The third is what makes them squares.

Spacing the cuts by eye. Folding is the surest way to space cuts without a ruler. Fold a paper rectangle in half and press the crease: the two halves lie exactly on top of each other, so the crease is exactly in the middle. Fold the folded paper in half again and open it, and there are three creases cutting the paper into four equal strips. Halving gives 2 pieces, then 4, but never 3 or 5.

For 3 or 5 pieces, draw one square in a corner first, as carefully as you can. Then lay a strip of paper along its side and mark the width on the strip. Step the strip along the edge, making a mark at each step, so every column comes out exactly as wide as the first square. The strip is a home-made ruler with just one length on it.

Spacing the cuts with a ruler. When the sides of the rectangle are a whole number of inches or centimeters, a ruler does the spacing (measuring with a ruler). For squares with 1-inch sides, make a mark at every inch along the top edge, and the same marks along the bottom edge. Then lay the ruler through a top mark and the mark straight below it, and draw the line. Marking both edges matters: two marks fix where a straight line goes, so the line cannot lean. Do the same with marks on the left and right sides for the lines across.

Part 2: counting the squares

Once the cuts are drawn, the squares form an array: equal rows that line up into equal columns. So they are counted the same way.

By rows. Add the number of squares in one row, once for each row. A rectangle cut into 3 rows of 4, as in the first picture, holds

squares. The number of addends is the number of rows, and each addend is the number in a row.

By columns. Add the number of squares in one column, once for each column. The same rectangle has 4 columns of 3:

The two counts always agree. Every square sits in exactly one row and in exactly one column, so each way counts every square once. A child may pick whichever reading is easier to add: counting by fives beats counting by threes, and two rows of the same number make a double.

The top row and the side column decide everything. Every row holds as many squares as the top row, because the lines down run the full height of the rectangle. Every column holds as many as the left column, because the lines across run the full width. So a child who can see the squares along the top edge and down one side knows the whole count, even before the rest of the cuts are drawn (Example 5).

Part 3: big squares, small squares, and pieces that are not the same

One rectangle, many cuts. A paper square 4 inches on each side can be cut into squares with 1-inch sides, or 2-inch sides, or left whole as one 4-inch square (Example 6). The smaller the squares, the more of them there are, and the more rows and columns. Going the other way, each 2-inch square is itself made of 2 rows of 2 one-inch squares.

When the squares fit. Squares fit a rectangle only when each side of the rectangle is a whole number of square-sides long. A side 4 inches long takes 1-inch squares (4 of them), or 2-inch squares (2 + 2 = 4), but not 3-inch squares: one of them leaves 1 inch over, and two would need 3 + 3 = 6 inches. And both sides must fit, the long one and the short one.

Why the squares must be the same size. Counting squares is useful because each square is the same as every other. Then “12 squares” says something about the rectangle: it is made of 12 copies of one square. If the pieces are of different sizes, the count only says how many pieces there are, and that number can be anything. Cut a sheet with one line and there are 2 pieces; cut the same sheet with five lines and there are 6. The sheet did not get bigger.

The same holds between two rectangles. Suppose two floors are tiled, one with 12 tiles and one with 15. The floor with 15 tiles is bigger only if the tiles are the same size on both. Small tiles can make a big count on a small floor.

Worked examples

Examples 1 to 3 go with Part 1, Examples 4 and 5 with Part 2, and Examples 6 and 7 with Part 3.

Boundaries

• The pieces must be the same size. A count of pieces of different sizes says only how many pieces there are. It says nothing about the rectangle, and it cannot be compared with another count.
• The pieces must be squares. Evenly spaced lines give pieces that are all alike, but they are squares only when the row height equals the column width. Otherwise the pieces are thinner rectangles, as in C of Example 7.
• Every cut runs from edge to edge. The rows and columns are equal only because each line goes all the way across. A line that stops partway leaves one large piece beside smaller ones.
• The squares must fit both sides. Each side of the rectangle must be a whole number of square-sides long. A 4-inch side takes 1-inch or 2-inch squares, never 3-inch ones.
• Only rectangles split this way. Rows and columns of squares fill a rectangle and nothing else. A triangle or a circle has slanted or curved edges, so squares along its edge are always cut off or stick out.
• The limit of five is a choice. A rectangle can be cut into any number of rows and columns. The examples stay at 5 or fewer so that the sums stay short.

Where this goes next

Cutting a shape into equal pieces carries straight on in halves, thirds and fourths of a shape, where a rectangle or a circle is cut into 2, 3 or 4 equal shares and each share gets a name. The folding and the even spacing from Part 1 are the same moves.

The squares themselves become a measuring tool. Covering a shape with same-size squares and counting them is how flat space is measured (Area as covering and Measuring area with unit squares), and the reason the squares must be the same size is the same reason given in Part 3. Counting the squares of a rectangle by rows leads to the area of a rectangle, and the sum 4 + 4 + 4 gets a shorter name in What multiplication means and Arrays.

Exercises

Part 1: drawing the cuts

1. Tomiwa wants to cut a rectangle into 4 columns. How many lines must he draw from the top edge to the bottom edge?
2. A window has 2 bars running across and 1 bar running up and down. How many rows and how many columns of panes does it have? How many panes are there in all?
3. Ihapera draws a rectangle 5 inches long and 3 inches tall and cuts it into squares with 1-inch sides. Where does she put her marks, and how many squares does she get?
4. A rectangle is 4 inches tall. Esinam wants to cut it into squares with 1-inch sides, in 2 columns. How long must the rectangle be, and how many squares will there be?
5. Esinam folds a paper strip in half, then in half again the same way, then in half a third time, and opens it. How many creases are there, and how many pieces?

Part 2: counting the squares

For Exercises 6 to 8, use the rectangles in the picture. Write a sum for the rows and a sum for the columns, and find how many squares there are.

6. Rectangle A.
7. Rectangle B.
8. Rectangle C.
9. A kitchen floor has square tiles in 4 rows of 5. How many tiles are there? Use the reading that is quicker to add.
10. A cupcake box has square cells in rows and columns. The front row has 2 cells and the left side has 4 cells. How many cupcakes does the box hold?
11. A rectangle cut into same-size squares has 5 rows and 10 squares in all. How many columns does it have?
12. In a rectangle cut into same-size squares, the top row has 5 squares and the left column has 3. How many different squares are in the top row and the left column together? How many squares are there in all?

Part 3: big squares, small squares

13. Ihapera has a paper square 2 inches on each side. How many squares with 1-inch sides can she cut it into?
14. A rectangle is 6 inches long and 4 inches tall. It is cut into squares with 2-inch sides. How many rows, how many columns, and how many squares are there?
15. A paper square is 5 inches on each side. How many squares with 1-inch sides can it be cut into? Can it be cut into squares with 2-inch sides?
16. Tomiwa cuts the rectangle from Exercise 14 into squares with 1-inch sides instead. Will he get more squares or fewer? How many, without counting them one by one?

For Exercises 17 to 19, use the picture. Is the rectangle cut into same-size squares? Say why.

17. Rectangle A.
18. Rectangle B.
19. Rectangle C.
20. Ihapera’s kitchen floor has 12 tiles. Tomiwa’s porch has 15 tiles. Does the porch have to be bigger than the kitchen floor?

Reasoning

21. Why must every cut run all the way from one edge of the rectangle to the other?
22. Esinam wants 3 columns. Can she get them by folding in half? How else can she space the lines?
23. Explain why counting the squares of a rectangle by rows and by columns always gives the same number.
24. A rectangle is 3 inches long and 2 inches tall. In how many ways can it be cut into same-size squares whose sides are a whole number of inches? How many squares does each way give?