Math Detours · Grade 5 · Around the corner · 24

The area of your hand

Draw round your hand on squared paper and count. Whole squares, part squares, and how to get a good answer for any shape.

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Tonight's idea

The area of a rectangle is easy: length times width. A hand, a leaf or a puddle has no length and width. So put the shape on squared paper and count squares.

Count in two lots. Whole squares lie completely inside the shape. Part squares are cut by the edge.

a puddle a triangle
Blue squares are whole squares. Orange squares are part squares.

Some part squares are nearly full and some are nearly empty. Taken together they are about half full. So count each part square as a half.

Area ≈ whole squares + half of the part squares. The sign ≈ means "is about".

The puddle has 15 whole squares and 20 part squares. Half of 20 is 10. So its area is about \(15 + 10 = 25\) squares.

Watch one: test the rule on a triangle

Is the rule any good? Try it on a shape whose area we know.

The triangle is half of a rectangle 6 squares long and 4 squares tall. So its area is exactly \(24 \div 2 = 12\) squares.

Now count. There are 8 whole squares and 8 part squares. The rule gives \(8 + 4 = 12\). It agrees.

On a curvy shape, expect the rule to be close, not exact.

Shapes with corners on the grid dots have an exact counting rule. Try it: mathdetours.com/demos/pick.html.

Your turn

A B
Shapes for exercises 1 and 2.
  1. (a) Count the whole squares and the part squares in triangle A. Use the rule to find its area. (b) Triangle A fills half of a box 6 squares wide and 6 squares tall. What is its exact area?
  2. Count the whole squares and the part squares in shape B. About what is its area?
  3. Draw round your hand on paper with 1 cm squares. Keep your fingers together. Count, and find the area of your hand in square centimetres.
  4. Mia uses paper with small squares. Four small squares cover 1 square centimetre. Her hand has 300 whole small squares and 80 part squares. (a) What is its area in small squares? (b) What is that in square centimetres? (c) Why do small squares give a better answer than big ones?

Stretch: trap the answer

The real area is more than the whole squares alone. It is less than the whole squares plus all the part squares. Ben counts 62 whole and 36 part squares. Between which two numbers must his area lie? Where does the rule's answer sit?

Answers

  1. (a) 12 whole squares and 12 part squares, so about \(12 + 6 = 18\) squares. (b) The box holds \(6 \times 6 = 36\) squares, and half of 36 is 18. The rule is exactly right here.
  2. 6 whole squares and 16 part squares, so about \(6 + 8 = 14\) squares. A count that is one or two part squares different is fine. Anything from 13 to 15 is a good answer.
  3. Answers vary. Check the count, then check the sum: whole squares plus half of the part squares. A bigger hand should give a bigger answer, so compare with someone else in the family.
  4. (a) \(300 + 40 = 340\) small squares. (b) \(340 \div 4 = 85\) square centimetres. (c) The guessing happens only in the part squares along the edge. With small squares, the part squares make up less of the shape, so the guess matters less.

Stretch. Between 62 and \(62 + 36 = 98\) squares. The rule gives \(62 + 18 = 80\), exactly half way between the two.

Next year's version counts dots, not squares, and gets exact answers: grade 6, Area by counting dots.

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