The Same Number in Different Arrangements

Six without counting again

Here are six units. Before counting them one by one, look at their grouping. The small gap divides the line into two groups of three. Three and three make six. The gap helps your eye organize the collection; it does not stand for a missing unit. We will keep this quantity while changing its arrangement.

Now the six units sit at the corners of a hexagon. Trace the boundary and find one unit at each corner. The shape has changed, but there are still six positions occupied by six units. The lines connect those positions. We are counting the units at the corners, not the connecting lines or the region inside.

Here is the same quantity as a rectangle: two rows, with three tiles in each row. Read one row first, then the other. Three plus three gives six. Two times three is a shorter description of those equal groups. The multiplication expression describes the visible arrangement; it does not introduce a different collection.

Return to the line. You can recognize the same two groups of three without having a rectangular boundary around them. Grouping is useful because you can keep one small amount in view and combine equal amounts. The rectangle made the rows especially clear, while the line made the order especially clear. Both showed six.

Different groups can have the same total

Six can also form these triangular rows. The first row has one unit, the second has two, and the third has three. Add the row counts: one plus two plus three gives six. Unlike the rectangle, these rows are unequal. That changes the way we describe the groups, while preserving their combined quantity.

Move closer to the row of three. It is just one part of the triangle. The other rows contain two units and one unit, making three more altogether. So this arrangement also separates six into three and three, though the second group occupies two rows. A useful grouping need not occupy one straight line.

The complete triangle returns. Its three rows contain six units, so the row count and the total count are different. Saying three without naming what you count would be ambiguous here. Three rows, six units: those are two correct descriptions of different features of the same arrangement. Keep the counted object explicit.

Compare the rectangle again. This time two rows contain six units. Changing from three unequal rows to two equal rows changes the grouping, not the total. You can check the invariant quantity by counting every unit, or more efficiently by adding the sizes of the groups. Both methods must agree.

Four units, four vertices

Here are four units in a line, grouped as two and two. We have changed the quantity now: this is four rather than six. A new total is a different operation from rearranging a fixed total. Watch what stays the same during the next few views. Every one of them will contain four counted objects.

The four units form a two-by-two square. There are two rows, each containing two tiles. The whole square contains four unit tiles. Its side length in tile widths is two, while its area in unit tiles is four. Even this small arrangement already separates a length from an area count.

Now four marked vertices form a tetrahedron. Turn around it and locate the vertex that was behind the others. Some positions can be hidden by the view even though they still belong to the object. Here the quantity four counts vertices. It is not a statement about the volume of the solid.

Step back to see the entire tetrahedron. Follow its four corners as the view turns. Rotation changes which corner appears nearest and which edge overlaps another. It cannot add or remove a vertex. A spatial arrangement sometimes needs a change of viewpoint to make its unchanged quantity easy to perceive.

Corners and tiles answer different questions

A cube has eight vertices. Four belong to one square face and four to the opposite face. The connecting edges join corresponding corners. As the cube turns, use those two sets of four to keep track of all eight. A hidden corner still belongs to the cube; visibility and existence are different matters.

Approach one side of the cube. This close view helps you follow the relationships among nearby corners, but it may leave some of the whole object outside the picture. We will return to the overview before counting all eight together. A close view reveals a part; it should not be mistaken for a smaller complete cube.

Here is the whole cube again. Its eight vertices are the counted objects. The faces connect sets of four vertices, but counting the faces would answer a different question. You can describe several features of one solid without expecting their counts to match. In this view, keep following the corners.

Now eight is a rectangle with two rows of four tiles. These are eight unit areas rather than eight vertices, but the quantity of counted objects is the same. Two groups of four appear in both arrangements. The geometry makes the grouping visible in different ways, and naming the counted object keeps the comparison precise.

Notation records a quantity

Twelve tiles fill three rows of four. You can read the total as four plus four plus four, or three times four. Follow one complete row, then the collection of three rows. We will leave all twelve tiles in place while changing the notation used to write their quantity.

The numeral now reads one one zero zero in base two. The tiles have not changed. Binary places stand for groups of eight, four, two and one, so this writing records one eight and one four, with no remaining twos or ones. Those place values describe the twelve already present; this view still groups its tiles in rows of four.

Return to base ten. The same quantity is written one two: one ten and two more. The rectangle still has three rows of four tiles. The displayed arrangement and the written place-value grouping need not use the same group size. Both describe twelve, and you can distinguish what the picture organizes from what the numeral records.

Spread the twelve into a line. The small gaps mark three groups of four, so you can still recover the same grouping as the rectangle. Nothing about the total depends on the outside outline. A line, a rectangle and a numeral each emphasize something different about the same quantity.

Check the rule across new examples

Try ten units in a factor array. The arrangement has two rows of five. Five plus five gives ten, and two times five records those equal groups. Follow one complete row, then the pair of rows. The row length and the row count together determine the total; neither one alone names the whole quantity.

Approach one row of five. Five is one half of the complete collection because the other row has the same length. This part-to-whole description depends on equal row sizes. It would not be enough merely to see two rows if one contained more units than the other. Return to the full arrangement before checking the ten together.

Spread the ten into a line. The total is unchanged even though the rectangle’s boundary is gone. Grouping can help count, but membership in the collection does not depend on the outline of a rectangle. A change of arrangement can move units without creating or discarding any of them.

Return to the array and write its quantity in base three. The numeral is one zero one: one group of nine, no remaining threes and one unit. Nine plus one is ten. These place-value groups differ from the two rows of five on screen. The notation and the spatial grouping give two valid descriptions of the same total.

Now change the quantity to eleven and return to base ten. The line contains one additional unit. Eleven has no whole-number factorization with both factors greater than one, so it cannot make a complete rectangle with two or more equal rows each containing two or more units. That arithmetic statement concerns the exact total, not the length of the line on screen.

Restore ten in its factor array. Removing one unit allows two equal rows of five again. A small change in total can change the available arrangements. Keep the two questions separate: how many objects are present, and how can that exact collection be grouped? The factor array answers the second using the quantity supplied by the first.

Say what changed and what stayed fixed

Come back to six in a rectangle. Try reading it without recounting each tile: two rows of three. Then look at the shape as one complete collection. These two levels of attention, a group and the whole, help you connect addition with multiplication while keeping the actual units in view.

Six becomes a triangle again. Identify the unequal rows and combine them: one, two and three. The arrangement changes, the group sizes change, and the total remains six. You now have two independent descriptions of the same quantity. One uses equal groups; the other uses successive row lengths.

Here is ten in triangular rows. This time the total has changed. The rows contain one, two, three and four units. Compared with the six-unit triangle, the new fourth row contributes four. A change in quantity can have a visible geometric explanation, rather than being merely a different numeral on the panel.

In the next lesson we will follow that growing-row relationship. For now, you can try a small quantity and switch between its available arrangements. Describe the units, the groups and the total separately. Then change the quantity and name what was added. Those distinctions will carry into every sequence we examine.