Ages 9–10 · 8 little discoveries
Geometry with Piko
Sorting by properties, turning and mirror shapes, perimeter and area with unit squares, cube nets, and finding one shape from clues.
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Sort by sides
Sorts a shape by its number of sides and by whether all its sides are equal.
A polygon is a closed shape with straight sides. This triangle has three sides and three corners.
A square has four equal sides and four right-angle corners. Check the sides first.
Counter-example: a rhombus also has four equal sides, but its corners are not right angles. Equal sides alone do not make a square.
Try it: the rectangle has two pairs of equal opposite sides. Are all four sides equal?
Turn or mirror?
Tells a turned copy of a shape apart from its mirror image.
Sliding moves a shape to another place without changing it. Turning moves its arms around a point. This L shape is our starting point.
A quarter turn changes where the arms point, but the shape stays the same. This L still has a row of three and one square under its left end.
Counter-example: a mirror image is not a turn. This J shape is the mirror image of the L. You cannot turn an L into a J.
Try it: find the foot of the L first, then watch which way it points after each turn.
Fold lines
Identifies the fold lines (symmetry axes) that make both halves match.
A line of symmetry is a fold line. Fold the shape along it and the two halves land exactly on each other. The dashed line is a vertical fold.
This rectangle has two fold lines: one vertical and one horizontal. Each one makes two matching halves.
Counter-example: this P shape has no fold line. Fold it along any line and the halves do not match.
Try it: fold a 2 by 2 square vertically, then horizontally. Both folds match. Check each fold line on its own.
Perimeter: walk the edge
Measures the perimeter of a unit-square shape by counting its outside edges.
Perimeter is the total length around the outside of a shape. Each small square has sides of one unit. Walk around the outside of this 2 by 3 rectangle.
A 2 by 2 square has four outside sides, and each one is 2 units long. The perimeter is 2 + 2 + 2 + 2 = 8 units.
Counter-example: the edge shared by two touching squares is inside, so it is not counted. The 1 by 4 strip has a perimeter of 10 units.
Try it: walk once around the outside of the L and count each outside edge.
Area: count the squares
Measures area by counting the unit squares that fill a shape.
Area is how much space a shape covers. Measure it by counting the unit squares that fill it.
A unit square has sides of one unit. This plus shape holds five unit squares, so its area is 5 square units.
Counter-example: the perimeter measures the outside edge, but the area measures the inside. The 2 by 2 square has an area of 4 and a perimeter of 8.
Try it: count the squares inside the shape, not the lines. Each small square counts once.
Same area, different edge
Explains that equal areas can have different perimeters.
Two shapes can cover the same area and still have different perimeters. The 2 by 3 rectangle covers six unit squares.
Counter-example: the same six squares in one row make a 1 by 6 strip. The area is still 6, but the perimeter grows to 14 units.
A more spread-out shape has more outside edges. This P shape also has an area of 6, but its perimeter is 12 units.
Try it: count the outside edges of the U. It also covers six squares. What is its perimeter?
Cube nets
Tells which flat arrangement of six squares folds into a cube without overlap.
A cube has six square faces. A cube net is a flat arrangement of those six faces that folds into a cube.
Each face is used exactly once. The staircase of three pairs is also a valid net, because every face lands in its own place.
Counter-example: this 2 by 3 block does not fold into a cube. Some faces would land on top of each other.
Try it: this arrangement, two rows of three squares, is also a cube net. Check that every square lands on a different face.
Find the shape from clues
Uses several geometric clues together to identify one shape.
Read each clue one at a time. Each clue removes some shapes from the list.
Clue: four equal sides. Both the square and the rhombus fit this clue, so two shapes are still on the list.
Clue: four right angles. Only rectangles and squares have four right angles. A rhombus does not have to.
Try it: find the triangle with one right angle. The right angle is marked with a small square.