Ages 6–8 · 8 little discoveries
Piko’s Shape Workshop
Eight investigations of sides, corners, triangles, construction, symmetry, covering, plane and solid shapes, and fixed-screen directions.
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Straight sides and corners
Count the straight sides and vertices of a closed simple polygon.
A polygon is closed and has straight sides. Trace this triangle’s boundary once. Three straight sides meet at three corners. Each corner is a meeting point, not the whole side between two points.
On a square, four sides meet at four corners. Start at one corner and return to it. Returning closes the shape; it does not add a fifth corner to the four already there.
A pentagon has five straight sides and five corners. Its colour or position does not change that count. Go around the whole boundary so that no short side is missed or counted twice.
Now follow a new long rectangle. It still has four sides and four corners. Use the numbered corner points to close its boundary. Side lengths may change while the number of corners stays four.
The quadrilateral family
Identify quadrilaterals and distinguish squares using side equality and right corners.
A quadrilateral is a closed shape with four straight sides. This long rectangle has four right corners, like the corner of a sheet of paper. Its opposite sides have equal lengths.
A square has four equal sides and four right corners. It belongs to the rectangle family too. Calling a shape a square does not stop it being a rectangle or a quadrilateral.
This slanted quadrilateral still has four straight sides, but its corners are not right corners. Four sides alone do not make a square. We must check the side lengths and corners together.
Now compare a square with that slanted quadrilateral. Both have four sides. Only the square here has four equal sides and four right corners. Use the stated properties to put each card in its place.
Triangles can turn
Recognise differently oriented and proportioned triangles by three straight sides and three vertices.
A triangle has three straight sides and three corners. This one points upward, but pointing up is not its defining property. Its closed three-sided boundary is what makes it a triangle.
Turn your attention to the triangle pointing right. It still has three sides and three corners. We changed the orientation of a triangular shape, not the number of parts in its boundary.
A wide triangle with a sloping side is also a triangle. Its three sides need not all have equal lengths. A circle has a curved boundary and does not pass this three-straight-side check.
Now organise three triangle cards by the named orientation: upward, rightward, then the wide base below. That order follows this instruction. It does not rank one triangle as more correct than another.
Build a whole from pieces
Compose a square or rectangle from two complementary pieces with a fixed orientation.
This square is split from one corner to the opposite corner. The diagonal makes two triangular halves. Both full halves together reach every edge of the square, with no gap and no overlap.
One piece fills the upper-right part; the other fills the lower-left part. In this task the pieces already have their orientations. Choose a piece, then its named place. Fine dragging is unnecessary.
A smaller triangle cannot replace a full half here: it leaves part of the square empty. Two copies of the upper half would overlap each other and leave the lower half uncovered.
Now a long rectangle is divided into a narrow left piece and a wide right piece. Match both pieces to the new boundary. A shorter right piece leaves a strip empty. Check the whole shape after placing the parts.
Match across a fold line
Complete a simple reflection by matching shape and equal distance across a vertical or horizontal line.
Imagine folding this square along the vertical dashed line. The circle on the left would meet a circle at the same height on the right. Both centres are equally far from the fold line.
The lower-left triangle needs the same triangle at the matching lower-right position. Replacing it with a circle breaks the reflection, even if that circle sits the right distance from the line.
A correct reflection keeps both the shape and its matching position across the line. Here the upper circles match and the lower triangles match. This dashed line is a line of symmetry for the completed pattern.
The fold line can also be horizontal. Now a top-left circle matches a bottom-left circle, and a top-right square matches a bottom-right square. Match across the line, not automatically left to right.
Cover with no gaps
Recognise and construct a bounded rectangular covering with no gaps or overlapping interiors.
Equal square tiles can cover a rectangle. Here each row has three squares and there are two rows. Every part inside the boundary is covered once. The tiles meet edge to edge.
A missing tile leaves a gap. A tile placed over other tiles creates an overlap. Counting tiles alone is not enough to show a correct covering; check where each tile lies inside the boundary.
Two full rows of three tiles contain six tiles altogether. Our pieces keep those units visible: a left column of two squares and a wider right block. Fit each to its region.
A wider new rectangle needs four tiles in each of its two rows. A short right block leaves a gap. Fit the left column and the full wider right block, then check the boundary.
Plane shapes and solid models
Distinguish a plane shape from a drawing representing a three-dimensional solid.
A square is a plane shape. We describe its length and width in a flat plane. A cube is a solid with height, width and depth. A square face is part of a cube, not the whole solid.
Everything displayed on this screen is a flat drawing. This cube picture uses slanting edges and dashed hidden edges to represent depth. We are naming the solid it represents, not claiming the screen drawing has real depth.
A circle is a plane shape. A sphere is a solid, like an ideal ball shape. The curved guide lines in this sphere model suggest its rounded surface. A circle and a sphere are different geometric ideas.
Now match a triangle as a plane shape and this sphere model as a solid representation. A triangle has a flat three-sided boundary. The sphere model represents a solid extending in three dimensions.
Directions on a fixed board
Follow a stated sequence of orthogonal one-cell moves using directions fixed to the screen.
This board has three rows and three columns. Up means toward the top of the screen, and right means toward its right edge. We keep that viewpoint, even if Piko’s face turns.
Starting at the upper left, one step right reaches the top middle. One step down then reaches the centre. Each arrow moves one neighbouring square; diagonal jumps do not follow these arrows.
From the centre, one step left reaches the middle row’s left cell. Left stays screen-left. Starting in the bottom-left cell, up then right reaches the centre by another two-step route.
Now start at the bottom right. Follow left, up, left in that order. You finish at the middle-left cell. Reaching the same final cell by a different arrow order does not follow this particular plan.