Counting and enumerationGrades 5–6
Draw a figure on squared paper: how many cells does it cover, and how long is its border? In these tasks area is counted in cells and perimeter in cell sides, and the two are easy to mix up: a rectangle can lose a corner cell and keep its perimeter. Counting with nothing missed and nothing counted twice is the whole skill.
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The area of a figure on squared paper is the number of cells it covers. Two half cells make one whole one.
The perimeter is the number of cell sides on the border. A side that two cells of the figure share is inside, and does not count.
Two figures of the same area can have different perimeters: four cells in a row have a perimeter of 10, and four in a square have 8.
Each cell has 4 sides. A side counts towards the perimeter only when no cell of the figure is on its other side.
| The figure | Area, in cells | Perimeter, in sides |
|---|---|---|
| 1 cell | 1 | 4 |
| 2 cells side by side | 2 | 4 + 4 − 2 = 6 |
| 4 cells in a row | 4 | 10 |
| a square of 2 by 2 | 4 | 8 |
Draw the figure on squared paper. For the area, count the cells row by row. For the perimeter, walk round the border and count the cell sides you pass. When a figure is cut, list the pieces in an order that cannot miss one or count one twice. The examples below go from the simplest to an olympiad's.
Example 1 · Grades 5–6
A cross is made of 5 cells: one in the middle and one on each of its four sides. What is the perimeter of the cross, in cell sides?
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+---+Answer 12 cell sides.
Example 2 · Grades 5–6
A strip of 2 by 4 cells is cut along the grid lines into two pieces of 4 cells each, and each piece must hold together. In how many different ways can it be cut?
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Answer 4 ways.
Example 3 · Grades 5–6
A rectangle of 3 by 5 cells is drawn on squared paper, and a line is drawn from one corner to the opposite one. How many cells does the line pass through? A cell counts when the line crosses its inside, not when it only touches a corner.
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Answer 7 cells.
MathTrail explains every wrong answer by the trap it fell into. These are the commonest in this topic, with something you can say to your child.
The same thing is counted twice: the sides the cells share are added to the perimeter, or each cut of the strip is listed once for each of its two pieces.
What to say
“Is any side, or any cut, counted from both sides?”
A case is missed: a cut whose step goes the other way, a piece of border in an inner corner, or the dominoes that stand upright as well as those that lie flat.
What to say
“Is there a way not yet tried, the other way round?”
A condition is lost: a piece that falls apart is counted as a piece, a cell that touches only at a corner is counted as a neighbour, or the line is said to pass through a cell it only touches.
What to say
“Does this piece, or this cell, really fit what the question says?”
Keep squared paper at hand and draw figures of 4 or 5 cells together. Which one has the longest border? Which has the shortest?
Count a perimeter by walking a finger round the border, side by side. Then count the outside sides of each cell, and check that the two agree.
Cut a paper strip into two pieces of the same size in as many ways as you can find. Lay them out so that none is missed or repeated.